author | wenzelm |
Sat, 13 Mar 2010 14:44:47 +0100 | |
changeset 35743 | c506c029a082 |
parent 35216 | 7641e8d831d2 |
child 36349 | 39be26d1bc28 |
permissions | -rw-r--r-- |
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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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2 |
Author: Brian Huffman |
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formalization of vector spaces and algebras over the real numbers
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3 |
*) |
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formalization of vector spaces and algebras over the real numbers
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4 |
|
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formalization of vector spaces and algebras over the real numbers
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header {* Vector Spaces and Algebras over the Reals *} |
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formalization of vector spaces and algebras over the real numbers
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6 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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diff
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7 |
theory RealVector |
29197
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adapted HOL source structure to distribution layout
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8 |
imports RealPow |
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formalization of vector spaces and algebras over the real numbers
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9 |
begin |
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formalization of vector spaces and algebras over the real numbers
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parents:
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10 |
|
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formalization of vector spaces and algebras over the real numbers
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11 |
subsection {* Locale for additive functions *} |
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formalization of vector spaces and algebras over the real numbers
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parents:
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12 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
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13 |
locale additive = |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
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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
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parents:
diff
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15 |
assumes add: "f (x + y) = f x + f y" |
27443 | 16 |
begin |
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formalization of vector spaces and algebras over the real numbers
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17 |
|
27443 | 18 |
lemma zero: "f 0 = 0" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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19 |
proof - |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
diff
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|
20 |
have "f 0 = f (0 + 0)" by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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parents:
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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
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|
22 |
finally show "f 0 = 0" by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
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" |
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formalization of vector spaces and algebras over the real numbers
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26 |
proof - |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
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|
27 |
have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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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
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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
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30 |
qed |
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formalization of vector spaces and algebras over the real numbers
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31 |
|
27443 | 32 |
lemma diff: "f (x - y) = f x - f y" |
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formalization of vector spaces and algebras over the real numbers
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33 |
by (simp add: diff_def add minus) |
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formalization of vector spaces and algebras over the real numbers
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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 |
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formalization of vector spaces and algebras over the real numbers
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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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45 |
subsection {* Vector spaces *} |
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46 |
|
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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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47 |
locale vector_space = |
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48 |
fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" |
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declare scaleR distrib rules [algebra_simps]; cleaned up
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parents:
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49 |
assumes scale_right_distrib [algebra_simps]: |
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declare scaleR distrib rules [algebra_simps]; cleaned up
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parents:
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diff
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50 |
"scale a (x + y) = scale a x + scale a y" |
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declare scaleR distrib rules [algebra_simps]; cleaned up
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parents:
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diff
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51 |
and scale_left_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
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parents:
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diff
changeset
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52 |
"scale (a + b) x = scale a x + scale b x" |
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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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53 |
and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" |
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54 |
and scale_one [simp]: "scale 1 x = x" |
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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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55 |
begin |
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
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changeset
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56 |
|
4c55cdec4ce7
simplify definition of vector_space locale (use axclasses instead of inheriting from field and ab_group_add classes)
huffman
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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:
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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:
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diff
changeset
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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:
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diff
changeset
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60 |
|
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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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)
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parents:
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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" |
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
|
65 |
proof - |
29229 | 66 |
interpret s: additive "\<lambda>a. scale a x" |
28823 | 67 |
proof qed (rule scale_left_distrib) |
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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
|
68 |
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
|
69 |
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
|
70 |
show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) |
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
|
71 |
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
|
72 |
|
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
|
73 |
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
|
74 |
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
|
75 |
and scale_right_diff_distrib [algebra_simps]: |
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
76 |
"scale a (x - y) = scale a x - scale a y" |
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
|
77 |
proof - |
29229 | 78 |
interpret s: additive "\<lambda>x. scale a x" |
28823 | 79 |
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
|
80 |
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
|
81 |
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
|
82 |
show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) |
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 |
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
|
84 |
|
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 |
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
|
86 |
"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
|
87 |
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
|
88 |
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
|
89 |
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
|
90 |
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
|
91 |
{ 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
|
92 |
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
|
93 |
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
|
94 |
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
|
95 |
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
|
96 |
|
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 |
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
|
98 |
"\<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
|
99 |
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
|
100 |
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
|
101 |
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
|
102 |
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
|
103 |
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
|
104 |
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
|
105 |
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
|
106 |
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
|
107 |
|
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 |
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
|
109 |
"\<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
|
110 |
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
|
111 |
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
|
112 |
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
|
113 |
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
|
114 |
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
|
115 |
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
|
116 |
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
|
117 |
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
|
118 |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
119 |
lemma scale_cancel_left [simp]: |
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
|
120 |
"scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 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
|
121 |
by (auto intro: 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
|
122 |
|
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
123 |
lemma scale_cancel_right [simp]: |
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
|
124 |
"scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0" |
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by (auto intro: scale_right_imp_eq) |
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126 |
|
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127 |
end |
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|
128 |
|
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129 |
subsection {* Real vector spaces *} |
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130 |
|
29608 | 131 |
class scaleR = |
25062 | 132 |
fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75) |
24748 | 133 |
begin |
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134 |
|
20763 | 135 |
abbreviation |
25062 | 136 |
divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70) |
24748 | 137 |
where |
25062 | 138 |
"x /\<^sub>R r == scaleR (inverse r) x" |
24748 | 139 |
|
140 |
end |
|
141 |
||
24588 | 142 |
class real_vector = scaleR + ab_group_add + |
25062 | 143 |
assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y" |
144 |
and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x" |
|
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145 |
and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" |
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146 |
and scaleR_one: "scaleR 1 x = x" |
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147 |
|
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interpretation real_vector: |
29229 | 149 |
vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector" |
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150 |
apply unfold_locales |
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151 |
apply (rule scaleR_right_distrib) |
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152 |
apply (rule scaleR_left_distrib) |
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153 |
apply (rule scaleR_scaleR) |
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|
154 |
apply (rule scaleR_one) |
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|
155 |
done |
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|
156 |
|
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|
157 |
text {* Recover original theorem names *} |
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|
158 |
|
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159 |
lemmas scaleR_left_commute = real_vector.scale_left_commute |
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160 |
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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162 |
lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib |
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163 |
lemmas scaleR_zero_right = real_vector.scale_zero_right |
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164 |
lemmas scaleR_minus_right = real_vector.scale_minus_right |
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165 |
lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib |
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166 |
lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff |
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167 |
lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq |
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168 |
lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq |
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|
169 |
lemmas scaleR_cancel_left = real_vector.scale_cancel_left |
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170 |
lemmas scaleR_cancel_right = real_vector.scale_cancel_right |
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|
171 |
|
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|
172 |
lemma scaleR_minus1_left [simp]: |
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|
173 |
fixes x :: "'a::real_vector" |
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|
174 |
shows "scaleR (-1) x = - x" |
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|
175 |
using scaleR_minus_left [of 1 x] by simp |
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|
176 |
|
24588 | 177 |
class real_algebra = real_vector + ring + |
25062 | 178 |
assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" |
179 |
and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" |
|
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180 |
|
24588 | 181 |
class real_algebra_1 = real_algebra + ring_1 |
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182 |
|
24588 | 183 |
class real_div_algebra = real_algebra_1 + division_ring |
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184 |
|
24588 | 185 |
class real_field = real_div_algebra + field |
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|
186 |
|
30069 | 187 |
instantiation real :: real_field |
188 |
begin |
|
189 |
||
190 |
definition |
|
191 |
real_scaleR_def [simp]: "scaleR a x = a * x" |
|
192 |
||
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193 |
instance proof |
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194 |
qed (simp_all add: algebra_simps) |
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195 |
|
30069 | 196 |
end |
197 |
||
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|
198 |
interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)" |
28823 | 199 |
proof qed (rule scaleR_left_distrib) |
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200 |
|
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201 |
interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)" |
28823 | 202 |
proof qed (rule scaleR_right_distrib) |
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203 |
|
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204 |
lemma nonzero_inverse_scaleR_distrib: |
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|
205 |
fixes x :: "'a::real_div_algebra" shows |
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|
206 |
"\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
20763 | 207 |
by (rule inverse_unique, simp) |
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|
208 |
|
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|
209 |
lemma inverse_scaleR_distrib: |
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|
210 |
fixes x :: "'a::{real_div_algebra,division_by_zero}" |
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|
211 |
shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
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|
212 |
apply (case_tac "a = 0", simp) |
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|
213 |
apply (case_tac "x = 0", simp) |
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|
214 |
apply (erule (1) nonzero_inverse_scaleR_distrib) |
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|
215 |
done |
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|
216 |
|
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|
217 |
|
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|
218 |
subsection {* Embedding of the Reals into any @{text real_algebra_1}: |
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|
219 |
@{term of_real} *} |
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|
220 |
|
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|
221 |
definition |
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|
222 |
of_real :: "real \<Rightarrow> 'a::real_algebra_1" where |
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223 |
"of_real r = scaleR r 1" |
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|
224 |
|
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|
225 |
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" |
20763 | 226 |
by (simp add: of_real_def) |
227 |
||
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|
228 |
lemma of_real_0 [simp]: "of_real 0 = 0" |
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|
229 |
by (simp add: of_real_def) |
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|
230 |
|
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|
231 |
lemma of_real_1 [simp]: "of_real 1 = 1" |
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|
232 |
by (simp add: of_real_def) |
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|
233 |
|
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|
234 |
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" |
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|
235 |
by (simp add: of_real_def scaleR_left_distrib) |
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|
236 |
|
c433e78d4203
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|
237 |
lemma of_real_minus [simp]: "of_real (- x) = - of_real x" |
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|
238 |
by (simp add: of_real_def) |
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|
239 |
|
c433e78d4203
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|
240 |
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" |
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|
241 |
by (simp add: of_real_def scaleR_left_diff_distrib) |
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|
242 |
|
c433e78d4203
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|
243 |
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" |
20763 | 244 |
by (simp add: of_real_def mult_commute) |
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|
245 |
|
20584
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|
246 |
lemma nonzero_of_real_inverse: |
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|
247 |
"x \<noteq> 0 \<Longrightarrow> of_real (inverse x) = |
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|
248 |
inverse (of_real x :: 'a::real_div_algebra)" |
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|
249 |
by (simp add: of_real_def nonzero_inverse_scaleR_distrib) |
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|
250 |
|
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|
251 |
lemma of_real_inverse [simp]: |
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|
252 |
"of_real (inverse x) = |
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|
253 |
inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})" |
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|
254 |
by (simp add: of_real_def inverse_scaleR_distrib) |
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|
255 |
|
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|
256 |
lemma nonzero_of_real_divide: |
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|
257 |
"y \<noteq> 0 \<Longrightarrow> of_real (x / y) = |
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|
258 |
(of_real x / of_real y :: 'a::real_field)" |
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|
259 |
by (simp add: divide_inverse nonzero_of_real_inverse) |
20722 | 260 |
|
261 |
lemma of_real_divide [simp]: |
|
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262 |
"of_real (x / y) = |
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|
263 |
(of_real x / of_real y :: 'a::{real_field,division_by_zero})" |
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264 |
by (simp add: divide_inverse) |
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|
265 |
|
20722 | 266 |
lemma of_real_power [simp]: |
31017 | 267 |
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" |
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|
268 |
by (induct n) simp_all |
20722 | 269 |
|
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diff
changeset
|
270 |
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" |
35216 | 271 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
272 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
273 |
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
|
274 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
275 |
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
|
276 |
proof |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
277 |
fix r |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
278 |
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
|
279 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
280 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
281 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
282 |
text{*Collapse nested embeddings*} |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
283 |
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" |
20772 | 284 |
by (induct n) auto |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
285 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
286 |
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
|
287 |
by (cases z rule: int_diff_cases, simp) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
288 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
289 |
lemma of_real_number_of_eq: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
290 |
"of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
291 |
by (simp add: number_of_eq) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
292 |
|
22912 | 293 |
text{*Every real algebra has characteristic zero*} |
294 |
instance real_algebra_1 < ring_char_0 |
|
295 |
proof |
|
23282
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23127
diff
changeset
|
296 |
fix m n :: nat |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23127
diff
changeset
|
297 |
have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)" |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23127
diff
changeset
|
298 |
by (simp only: of_real_eq_iff of_nat_eq_iff) |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23127
diff
changeset
|
299 |
thus "(of_nat m = (of_nat n::'a)) = (m = n)" |
dfc459989d24
add axclass semiring_char_0 for types where of_nat is injective
huffman
parents:
23127
diff
changeset
|
300 |
by (simp only: of_real_of_nat_eq) |
22912 | 301 |
qed |
302 |
||
27553 | 303 |
instance real_field < field_char_0 .. |
304 |
||
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
305 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
306 |
subsection {* The Set of Real Numbers *} |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
307 |
|
20772 | 308 |
definition |
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset
|
309 |
Reals :: "'a::real_algebra_1 set" where |
30070
76cee7c62782
declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents:
30069
diff
changeset
|
310 |
[code del]: "Reals = range of_real" |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
311 |
|
21210 | 312 |
notation (xsymbols) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
313 |
Reals ("\<real>") |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
314 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
315 |
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
|
316 |
by (simp add: Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
317 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
318 |
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
|
319 |
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) |
20718 | 320 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
321 |
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
|
322 |
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
|
323 |
|
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
324 |
lemma Reals_number_of [simp]: |
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
325 |
"(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals" |
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
326 |
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real) |
20718 | 327 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
328 |
lemma Reals_0 [simp]: "0 \<in> Reals" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
329 |
apply (unfold Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
330 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
331 |
apply (rule of_real_0 [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
332 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
333 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
334 |
lemma Reals_1 [simp]: "1 \<in> Reals" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
335 |
apply (unfold Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
336 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
337 |
apply (rule of_real_1 [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
338 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
339 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
340 |
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
|
341 |
apply (auto simp add: Reals_def) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
342 |
apply (rule range_eqI) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
343 |
apply (rule of_real_add [symmetric]) |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
344 |
done |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
345 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
346 |
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
|
347 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
348 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
349 |
apply (rule of_real_minus [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
350 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
351 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
352 |
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
|
353 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
354 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
355 |
apply (rule of_real_diff [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
356 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
357 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
358 |
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
|
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_mult [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 nonzero_Reals_inverse: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
365 |
fixes a :: "'a::real_div_algebra" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
366 |
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
|
367 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
368 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
369 |
apply (erule nonzero_of_real_inverse [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
370 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
371 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
372 |
lemma Reals_inverse [simp]: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
373 |
fixes a :: "'a::{real_div_algebra,division_by_zero}" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
374 |
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
|
375 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
376 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
377 |
apply (rule of_real_inverse [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
378 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
379 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
380 |
lemma nonzero_Reals_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
381 |
fixes a b :: "'a::real_field" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
382 |
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
|
383 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
384 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
385 |
apply (erule nonzero_of_real_divide [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
386 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
387 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
388 |
lemma Reals_divide [simp]: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
389 |
fixes a b :: "'a::{real_field,division_by_zero}" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
390 |
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
|
391 |
apply (auto simp add: Reals_def) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
392 |
apply (rule range_eqI) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
393 |
apply (rule of_real_divide [symmetric]) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
394 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
395 |
|
20722 | 396 |
lemma Reals_power [simp]: |
31017 | 397 |
fixes a :: "'a::{real_algebra_1}" |
20722 | 398 |
shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals" |
399 |
apply (auto simp add: Reals_def) |
|
400 |
apply (rule range_eqI) |
|
401 |
apply (rule of_real_power [symmetric]) |
|
402 |
done |
|
403 |
||
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
404 |
lemma Reals_cases [cases set: Reals]: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
405 |
assumes "q \<in> \<real>" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
406 |
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
|
407 |
unfolding Reals_def |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
408 |
proof - |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
409 |
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
|
410 |
then obtain r where "q = of_real r" .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
411 |
then show thesis .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
412 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
413 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
414 |
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
|
415 |
"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
|
416 |
by (rule Reals_cases) auto |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
417 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
418 |
|
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
419 |
subsection {* Topological spaces *} |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
420 |
|
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
421 |
class "open" = |
31494 | 422 |
fixes "open" :: "'a set \<Rightarrow> bool" |
31490
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
423 |
|
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
424 |
class topological_space = "open" + |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
425 |
assumes open_UNIV [simp, intro]: "open UNIV" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
426 |
assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
427 |
assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)" |
31490
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
428 |
begin |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
429 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
430 |
definition |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
431 |
closed :: "'a set \<Rightarrow> bool" where |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
432 |
"closed S \<longleftrightarrow> open (- S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
433 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
434 |
lemma open_empty [intro, simp]: "open {}" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
435 |
using open_Union [of "{}"] by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
436 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
437 |
lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
438 |
using open_Union [of "{S, T}"] by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
439 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
440 |
lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
441 |
unfolding UN_eq by (rule open_Union) auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
442 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
443 |
lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
444 |
by (induct set: finite) auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
445 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
446 |
lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
447 |
unfolding Inter_def by (rule open_INT) |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
448 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
449 |
lemma closed_empty [intro, simp]: "closed {}" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
450 |
unfolding closed_def by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
451 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
452 |
lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
453 |
unfolding closed_def by auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
454 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
455 |
lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
456 |
unfolding closed_def Inter_def by auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
457 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
458 |
lemma closed_UNIV [intro, simp]: "closed UNIV" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
459 |
unfolding closed_def by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
460 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
461 |
lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
462 |
unfolding closed_def by auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
463 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
464 |
lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
465 |
unfolding closed_def by auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
466 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
467 |
lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
468 |
by (induct set: finite) auto |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
469 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
470 |
lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
471 |
unfolding Union_def by (rule closed_UN) |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
472 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
473 |
lemma open_closed: "open S \<longleftrightarrow> closed (- S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
474 |
unfolding closed_def by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
475 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
476 |
lemma closed_open: "closed S \<longleftrightarrow> open (- S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
477 |
unfolding closed_def by simp |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
478 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
479 |
lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
480 |
unfolding closed_open Diff_eq by (rule open_Int) |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
481 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
482 |
lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
483 |
unfolding open_closed Diff_eq by (rule closed_Int) |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
484 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
485 |
lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
486 |
unfolding closed_open . |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
487 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
488 |
lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)" |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
489 |
unfolding open_closed . |
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
490 |
|
c350f3ad6b0d
move definitions of open, closed to RealVector.thy
huffman
parents:
31446
diff
changeset
|
491 |
end |
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
492 |
|
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
493 |
|
31289 | 494 |
subsection {* Metric spaces *} |
495 |
||
496 |
class dist = |
|
497 |
fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real" |
|
498 |
||
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
499 |
class open_dist = "open" + dist + |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
500 |
assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" |
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
501 |
|
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
502 |
class metric_space = open_dist + |
31289 | 503 |
assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y" |
504 |
assumes dist_triangle2: "dist x y \<le> dist x z + dist y z" |
|
505 |
begin |
|
506 |
||
507 |
lemma dist_self [simp]: "dist x x = 0" |
|
508 |
by simp |
|
509 |
||
510 |
lemma zero_le_dist [simp]: "0 \<le> dist x y" |
|
511 |
using dist_triangle2 [of x x y] by simp |
|
512 |
||
513 |
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y" |
|
514 |
by (simp add: less_le) |
|
515 |
||
516 |
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0" |
|
517 |
by (simp add: not_less) |
|
518 |
||
519 |
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y" |
|
520 |
by (simp add: le_less) |
|
521 |
||
522 |
lemma dist_commute: "dist x y = dist y x" |
|
523 |
proof (rule order_antisym) |
|
524 |
show "dist x y \<le> dist y x" |
|
525 |
using dist_triangle2 [of x y x] by simp |
|
526 |
show "dist y x \<le> dist x y" |
|
527 |
using dist_triangle2 [of y x y] by simp |
|
528 |
qed |
|
529 |
||
530 |
lemma dist_triangle: "dist x z \<le> dist x y + dist y z" |
|
531 |
using dist_triangle2 [of x z y] by (simp add: dist_commute) |
|
532 |
||
31565 | 533 |
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y" |
534 |
using dist_triangle2 [of x y a] by (simp add: dist_commute) |
|
535 |
||
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
536 |
subclass topological_space |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
537 |
proof |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
538 |
have "\<exists>e::real. 0 < e" |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
539 |
by (fast intro: zero_less_one) |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
540 |
then show "open UNIV" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
541 |
unfolding open_dist by simp |
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
542 |
next |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
543 |
fix S T assume "open S" "open T" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
544 |
then show "open (S \<inter> T)" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
545 |
unfolding open_dist |
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
546 |
apply clarify |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
547 |
apply (drule (1) bspec)+ |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
548 |
apply (clarify, rename_tac r s) |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
549 |
apply (rule_tac x="min r s" in exI, simp) |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
550 |
done |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
551 |
next |
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
552 |
fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)" |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
553 |
unfolding open_dist by fast |
31413
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
554 |
qed |
729d90a531e4
introduce class topological_space as a superclass of metric_space
huffman
parents:
31289
diff
changeset
|
555 |
|
31289 | 556 |
end |
557 |
||
558 |
||
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
559 |
subsection {* Real normed vector spaces *} |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
560 |
|
29608 | 561 |
class norm = |
22636 | 562 |
fixes norm :: "'a \<Rightarrow> real" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
563 |
|
24520 | 564 |
class sgn_div_norm = scaleR + norm + sgn + |
25062 | 565 |
assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" |
24506 | 566 |
|
31289 | 567 |
class dist_norm = dist + norm + minus + |
568 |
assumes dist_norm: "dist x y = norm (x - y)" |
|
569 |
||
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
570 |
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist + |
24588 | 571 |
assumes norm_ge_zero [simp]: "0 \<le> norm x" |
25062 | 572 |
and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0" |
573 |
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
|
574 |
and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
575 |
|
24588 | 576 |
class real_normed_algebra = real_algebra + real_normed_vector + |
25062 | 577 |
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
|
578 |
|
24588 | 579 |
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + |
25062 | 580 |
assumes norm_one [simp]: "norm 1 = 1" |
22852 | 581 |
|
24588 | 582 |
class real_normed_div_algebra = real_div_algebra + real_normed_vector + |
25062 | 583 |
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
|
584 |
|
24588 | 585 |
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
|
586 |
|
22852 | 587 |
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
|
588 |
proof |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
589 |
fix x y :: 'a |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
590 |
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
|
591 |
by (simp add: norm_mult) |
22852 | 592 |
next |
593 |
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" |
|
594 |
by (rule norm_mult) |
|
595 |
thus "norm (1::'a) = 1" by simp |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
596 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
597 |
|
22852 | 598 |
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
|
599 |
by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
600 |
|
22852 | 601 |
lemma zero_less_norm_iff [simp]: |
602 |
fixes x :: "'a::real_normed_vector" |
|
603 |
shows "(0 < norm x) = (x \<noteq> 0)" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
604 |
by (simp add: order_less_le) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
605 |
|
22852 | 606 |
lemma norm_not_less_zero [simp]: |
607 |
fixes x :: "'a::real_normed_vector" |
|
608 |
shows "\<not> norm x < 0" |
|
20828 | 609 |
by (simp add: linorder_not_less) |
610 |
||
22852 | 611 |
lemma norm_le_zero_iff [simp]: |
612 |
fixes x :: "'a::real_normed_vector" |
|
613 |
shows "(norm x \<le> 0) = (x = 0)" |
|
20828 | 614 |
by (simp add: order_le_less) |
615 |
||
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
616 |
lemma norm_minus_cancel [simp]: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
617 |
fixes x :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
618 |
shows "norm (- x) = norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
619 |
proof - |
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
620 |
have "norm (- x) = norm (scaleR (- 1) x)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
621 |
by (simp only: scaleR_minus_left scaleR_one) |
20533 | 622 |
also have "\<dots> = \<bar>- 1\<bar> * norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
623 |
by (rule norm_scaleR) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
624 |
finally show ?thesis by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
625 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
626 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
627 |
lemma norm_minus_commute: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
628 |
fixes a b :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
629 |
shows "norm (a - b) = norm (b - a)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
630 |
proof - |
22898 | 631 |
have "norm (- (b - a)) = norm (b - a)" |
632 |
by (rule norm_minus_cancel) |
|
633 |
thus ?thesis by simp |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
634 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
635 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
636 |
lemma norm_triangle_ineq2: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
637 |
fixes a b :: "'a::real_normed_vector" |
20533 | 638 |
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
|
639 |
proof - |
20533 | 640 |
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
|
641 |
by (rule norm_triangle_ineq) |
22898 | 642 |
thus ?thesis by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
643 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
644 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
645 |
lemma norm_triangle_ineq3: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
646 |
fixes a b :: "'a::real_normed_vector" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
647 |
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
|
648 |
apply (subst abs_le_iff) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
649 |
apply auto |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
650 |
apply (rule norm_triangle_ineq2) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
651 |
apply (subst norm_minus_commute) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
652 |
apply (rule norm_triangle_ineq2) |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
653 |
done |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
654 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
655 |
lemma norm_triangle_ineq4: |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
656 |
fixes a b :: "'a::real_normed_vector" |
20533 | 657 |
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
|
658 |
proof - |
22898 | 659 |
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
|
660 |
by (rule norm_triangle_ineq) |
22898 | 661 |
thus ?thesis |
662 |
by (simp only: diff_minus norm_minus_cancel) |
|
663 |
qed |
|
664 |
||
665 |
lemma norm_diff_ineq: |
|
666 |
fixes a b :: "'a::real_normed_vector" |
|
667 |
shows "norm a - norm b \<le> norm (a + b)" |
|
668 |
proof - |
|
669 |
have "norm a - norm (- b) \<le> norm (a - - b)" |
|
670 |
by (rule norm_triangle_ineq2) |
|
671 |
thus ?thesis by simp |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
672 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
673 |
|
20551 | 674 |
lemma norm_diff_triangle_ineq: |
675 |
fixes a b c d :: "'a::real_normed_vector" |
|
676 |
shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)" |
|
677 |
proof - |
|
678 |
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" |
|
679 |
by (simp add: diff_minus add_ac) |
|
680 |
also have "\<dots> \<le> norm (a - c) + norm (b - d)" |
|
681 |
by (rule norm_triangle_ineq) |
|
682 |
finally show ?thesis . |
|
683 |
qed |
|
684 |
||
22857 | 685 |
lemma abs_norm_cancel [simp]: |
686 |
fixes a :: "'a::real_normed_vector" |
|
687 |
shows "\<bar>norm a\<bar> = norm a" |
|
688 |
by (rule abs_of_nonneg [OF norm_ge_zero]) |
|
689 |
||
22880 | 690 |
lemma norm_add_less: |
691 |
fixes x y :: "'a::real_normed_vector" |
|
692 |
shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s" |
|
693 |
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) |
|
694 |
||
695 |
lemma norm_mult_less: |
|
696 |
fixes x y :: "'a::real_normed_algebra" |
|
697 |
shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s" |
|
698 |
apply (rule order_le_less_trans [OF norm_mult_ineq]) |
|
699 |
apply (simp add: mult_strict_mono') |
|
700 |
done |
|
701 |
||
22857 | 702 |
lemma norm_of_real [simp]: |
703 |
"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
|
704 |
unfolding of_real_def by simp |
20560 | 705 |
|
22876
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
706 |
lemma norm_number_of [simp]: |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
707 |
"norm (number_of w::'a::{number_ring,real_normed_algebra_1}) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
708 |
= \<bar>number_of w\<bar>" |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
709 |
by (subst of_real_number_of_eq [symmetric], rule norm_of_real) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
710 |
|
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
711 |
lemma norm_of_int [simp]: |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
712 |
"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
|
713 |
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
|
714 |
|
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
715 |
lemma norm_of_nat [simp]: |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
716 |
"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
|
717 |
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
|
718 |
apply (subst norm_of_real, simp) |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
719 |
done |
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
720 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
721 |
lemma nonzero_norm_inverse: |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
722 |
fixes a :: "'a::real_normed_div_algebra" |
20533 | 723 |
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
|
724 |
apply (rule inverse_unique [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
725 |
apply (simp add: norm_mult [symmetric]) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
726 |
done |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
727 |
|
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
728 |
lemma norm_inverse: |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
729 |
fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" |
20533 | 730 |
shows "norm (inverse a) = inverse (norm a)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
731 |
apply (case_tac "a = 0", simp) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
732 |
apply (erule nonzero_norm_inverse) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
733 |
done |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
734 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
735 |
lemma nonzero_norm_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
736 |
fixes a b :: "'a::real_normed_field" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
737 |
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
|
738 |
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
|
739 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
740 |
lemma norm_divide: |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
741 |
fixes a b :: "'a::{real_normed_field,division_by_zero}" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
742 |
shows "norm (a / b) = norm a / norm b" |
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
743 |
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
|
744 |
|
22852 | 745 |
lemma norm_power_ineq: |
31017 | 746 |
fixes x :: "'a::{real_normed_algebra_1}" |
22852 | 747 |
shows "norm (x ^ n) \<le> norm x ^ n" |
748 |
proof (induct n) |
|
749 |
case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp |
|
750 |
next |
|
751 |
case (Suc n) |
|
752 |
have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)" |
|
753 |
by (rule norm_mult_ineq) |
|
754 |
also from Suc have "\<dots> \<le> norm x * norm x ^ n" |
|
755 |
using norm_ge_zero by (rule mult_left_mono) |
|
756 |
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
|
757 |
by simp |
22852 | 758 |
qed |
759 |
||
20684 | 760 |
lemma norm_power: |
31017 | 761 |
fixes x :: "'a::{real_normed_div_algebra}" |
20684 | 762 |
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
|
763 |
by (induct n) (simp_all add: norm_mult) |
20684 | 764 |
|
31289 | 765 |
text {* Every normed vector space is a metric space. *} |
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
766 |
|
31289 | 767 |
instance real_normed_vector < metric_space |
768 |
proof |
|
769 |
fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y" |
|
770 |
unfolding dist_norm by simp |
|
771 |
next |
|
772 |
fix x y z :: 'a show "dist x y \<le> dist x z + dist y z" |
|
773 |
unfolding dist_norm |
|
774 |
using norm_triangle_ineq4 [of "x - z" "y - z"] by simp |
|
775 |
qed |
|
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
776 |
|
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
777 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
778 |
subsection {* Class instances for real numbers *} |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
779 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
780 |
instantiation real :: real_normed_field |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
781 |
begin |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
782 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
783 |
definition real_norm_def [simp]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
784 |
"norm r = \<bar>r\<bar>" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
785 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
786 |
definition dist_real_def: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
787 |
"dist x y = \<bar>x - y\<bar>" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
788 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
789 |
definition open_real_def [code del]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
790 |
"open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
791 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
792 |
instance |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
793 |
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
|
794 |
apply (rule dist_real_def) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
795 |
apply (rule open_real_def) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
796 |
apply (simp add: real_sgn_def) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
797 |
apply (rule abs_ge_zero) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
798 |
apply (rule abs_eq_0) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
799 |
apply (rule abs_triangle_ineq) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
800 |
apply (rule abs_mult) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
801 |
apply (rule abs_mult) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
802 |
done |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
803 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
804 |
end |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
805 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
806 |
lemma open_real_lessThan [simp]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
807 |
fixes a :: real shows "open {..<a}" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
808 |
unfolding open_real_def dist_real_def |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
809 |
proof (clarify) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
810 |
fix x assume "x < a" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
811 |
hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
812 |
thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" .. |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
813 |
qed |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
814 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
815 |
lemma open_real_greaterThan [simp]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
816 |
fixes a :: real shows "open {a<..}" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
817 |
unfolding open_real_def dist_real_def |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
818 |
proof (clarify) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
819 |
fix x assume "a < x" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
820 |
hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
821 |
thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" .. |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
822 |
qed |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
823 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
824 |
lemma open_real_greaterThanLessThan [simp]: |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
825 |
fixes a b :: real shows "open {a<..<b}" |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
826 |
proof - |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
827 |
have "{a<..<b} = {a<..} \<inter> {..<b}" by auto |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
828 |
thus "open {a<..<b}" by (simp add: open_Int) |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
829 |
qed |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
830 |
|
31567 | 831 |
lemma closed_real_atMost [simp]: |
832 |
fixes a :: real shows "closed {..a}" |
|
833 |
unfolding closed_open by simp |
|
834 |
||
835 |
lemma closed_real_atLeast [simp]: |
|
836 |
fixes a :: real shows "closed {a..}" |
|
837 |
unfolding closed_open by simp |
|
838 |
||
839 |
lemma closed_real_atLeastAtMost [simp]: |
|
840 |
fixes a b :: real shows "closed {a..b}" |
|
841 |
proof - |
|
842 |
have "{a..b} = {a..} \<inter> {..b}" by auto |
|
843 |
thus "closed {a..b}" by (simp add: closed_Int) |
|
844 |
qed |
|
845 |
||
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
846 |
|
31446 | 847 |
subsection {* Extra type constraints *} |
848 |
||
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
849 |
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
|
850 |
|
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
851 |
setup {* Sign.add_const_constraint |
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
852 |
(@{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
|
853 |
|
31446 | 854 |
text {* Only allow @{term dist} in class @{text metric_space}. *} |
855 |
||
856 |
setup {* Sign.add_const_constraint |
|
857 |
(@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *} |
|
858 |
||
859 |
text {* Only allow @{term norm} in class @{text real_normed_vector}. *} |
|
860 |
||
861 |
setup {* Sign.add_const_constraint |
|
862 |
(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *} |
|
863 |
||
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
864 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
865 |
subsection {* Sign function *} |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
866 |
|
24506 | 867 |
lemma norm_sgn: |
868 |
"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
|
869 |
by (simp add: sgn_div_norm) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
870 |
|
24506 | 871 |
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0" |
872 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
873 |
|
24506 | 874 |
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)" |
875 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
876 |
|
24506 | 877 |
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)" |
878 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
879 |
|
24506 | 880 |
lemma sgn_scaleR: |
881 |
"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
|
882 |
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
|
883 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
884 |
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" |
24506 | 885 |
by (simp add: sgn_div_norm) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
886 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
887 |
lemma sgn_of_real: |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
888 |
"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
|
889 |
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
|
890 |
|
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
891 |
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
|
892 |
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
|
893 |
shows "sgn (x * y) = sgn x * sgn y" |
24506 | 894 |
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
|
895 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
896 |
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>" |
24506 | 897 |
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
|
898 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
899 |
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
|
900 |
unfolding real_sgn_eq by simp |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
901 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
902 |
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
|
903 |
unfolding real_sgn_eq by simp |
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
904 |
|
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
905 |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
906 |
subsection {* Bounded Linear and Bilinear Operators *} |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
907 |
|
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
908 |
locale bounded_linear = additive + |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
909 |
constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
910 |
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
|
911 |
assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
27443 | 912 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
913 |
|
27443 | 914 |
lemma pos_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
915 |
"\<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
|
916 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
917 |
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
|
918 |
using bounded by fast |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
919 |
show ?thesis |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
920 |
proof (intro exI impI conjI allI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
921 |
show "0 < max 1 K" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
922 |
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
|
923 |
next |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
924 |
fix x |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
925 |
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
|
926 |
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
|
927 |
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
|
928 |
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
|
929 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
930 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
931 |
|
27443 | 932 |
lemma nonneg_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
933 |
"\<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
|
934 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
935 |
from pos_bounded |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
936 |
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
|
937 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
938 |
|
27443 | 939 |
end |
940 |
||
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
941 |
locale bounded_bilinear = |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
942 |
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
|
943 |
\<Rightarrow> 'c::real_normed_vector" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
944 |
(infixl "**" 70) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
945 |
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
|
946 |
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
|
947 |
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
|
948 |
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
|
949 |
assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" |
27443 | 950 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
951 |
|
27443 | 952 |
lemma pos_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
953 |
"\<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
|
954 |
apply (cut_tac bounded, erule exE) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
955 |
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
|
956 |
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
|
957 |
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
|
958 |
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
|
959 |
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
|
960 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
961 |
|
27443 | 962 |
lemma nonneg_bounded: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
963 |
"\<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
|
964 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
965 |
from pos_bounded |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
966 |
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
|
967 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
968 |
|
27443 | 969 |
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
|
970 |
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
|
971 |
|
27443 | 972 |
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
|
973 |
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
|
974 |
|
27443 | 975 |
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
|
976 |
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
|
977 |
|
27443 | 978 |
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
|
979 |
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
|
980 |
|
27443 | 981 |
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
|
982 |
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
|
983 |
|
27443 | 984 |
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
|
985 |
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
|
986 |
|
27443 | 987 |
lemma diff_left: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
988 |
"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
|
989 |
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
|
990 |
|
27443 | 991 |
lemma diff_right: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
992 |
"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
|
993 |
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
|
994 |
|
27443 | 995 |
lemma bounded_linear_left: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
996 |
"bounded_linear (\<lambda>a. a ** b)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
997 |
apply (unfold_locales) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
998 |
apply (rule add_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
999 |
apply (rule scaleR_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1000 |
apply (cut_tac bounded, safe) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1001 |
apply (rule_tac x="norm b * K" in exI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1002 |
apply (simp add: mult_ac) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1003 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1004 |
|
27443 | 1005 |
lemma bounded_linear_right: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1006 |
"bounded_linear (\<lambda>b. a ** b)" |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1007 |
apply (unfold_locales) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1008 |
apply (rule add_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1009 |
apply (rule scaleR_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1010 |
apply (cut_tac bounded, safe) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1011 |
apply (rule_tac x="norm a * K" in exI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1012 |
apply (simp add: mult_ac) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1013 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1014 |
|
27443 | 1015 |
lemma prod_diff_prod: |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1016 |
"(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
|
1017 |
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
|
1018 |
|
27443 | 1019 |
end |
1020 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1021 |
interpretation mult: |
29229 | 1022 |
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
|
1023 |
apply (rule bounded_bilinear.intro) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1024 |
apply (rule left_distrib) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1025 |
apply (rule right_distrib) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1026 |
apply (rule mult_scaleR_left) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1027 |
apply (rule mult_scaleR_right) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1028 |
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
|
1029 |
apply (simp add: norm_mult_ineq) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1030 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1031 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1032 |
interpretation mult_left: |
29229 | 1033 |
bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)" |
23127 | 1034 |
by (rule mult.bounded_linear_left) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1035 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1036 |
interpretation mult_right: |
29229 | 1037 |
bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)" |
23127 | 1038 |
by (rule mult.bounded_linear_right) |
1039 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1040 |
interpretation divide: |
29229 | 1041 |
bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)" |
23127 | 1042 |
unfolding divide_inverse by (rule mult.bounded_linear_left) |
23120 | 1043 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1044 |
interpretation scaleR: bounded_bilinear "scaleR" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1045 |
apply (rule bounded_bilinear.intro) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1046 |
apply (rule scaleR_left_distrib) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1047 |
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
|
1048 |
apply simp |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1049 |
apply (rule scaleR_left_commute) |
31586
d4707b99e631
declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents:
31567
diff
changeset
|
1050 |
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
|
1051 |
done |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1052 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1053 |
interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x" |
23127 | 1054 |
by (rule scaleR.bounded_linear_left) |
1055 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1056 |
interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x" |
23127 | 1057 |
by (rule scaleR.bounded_linear_right) |
1058 |
||
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30273
diff
changeset
|
1059 |
interpretation of_real: bounded_linear "\<lambda>r. of_real r" |
23127 | 1060 |
unfolding of_real_def by (rule scaleR.bounded_linear_left) |
22625 | 1061 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
1062 |
end |