author | immler |
Thu, 27 Dec 2018 21:32:36 +0100 | |
changeset 69513 | 42e08052dae8 |
parent 69510 | 0f31dd2e540d |
child 69516 | 09bb8f470959 |
permissions | -rw-r--r-- |
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(* Title: HOL/Analysis/Linear_Algebra.thy |
44133 | 2 |
Author: Amine Chaieb, University of Cambridge |
3 |
*) |
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60420 | 5 |
section \<open>Elementary linear algebra on Euclidean spaces\<close> |
44133 | 6 |
|
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theory Linear_Algebra |
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imports |
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Euclidean_Space |
|
66453
cc19f7ca2ed6
session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
wenzelm
parents:
66447
diff
changeset
|
10 |
"HOL-Library.Infinite_Set" |
44133 | 11 |
begin |
12 |
||
63886
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
13 |
lemma linear_simps: |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
14 |
assumes "bounded_linear f" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
15 |
shows |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
16 |
"f (a + b) = f a + f b" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
17 |
"f (a - b) = f a - f b" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
18 |
"f 0 = 0" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
19 |
"f (- a) = - f a" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
20 |
"f (s *\<^sub>R v) = s *\<^sub>R (f v)" |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
21 |
proof - |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
22 |
interpret f: bounded_linear f by fact |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
23 |
show "f (a + b) = f a + f b" by (rule f.add) |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
24 |
show "f (a - b) = f a - f b" by (rule f.diff) |
685fb01256af
move Henstock-Kurzweil integration after Lebesgue_Measure; replace content by abbreviation measure lborel
hoelzl
parents:
63881
diff
changeset
|
25 |
show "f 0 = 0" by (rule f.zero) |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
26 |
show "f (- a) = - f a" by (rule f.neg) |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
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diff
changeset
|
27 |
show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scale) |
44133 | 28 |
qed |
29 |
||
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36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
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30 |
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}" |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
31 |
using finite finite_image_set by blast |
44133 | 32 |
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53406 | 33 |
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subsection%unimportant \<open>More interesting properties of the norm\<close> |
63050 | 35 |
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notation inner (infix "\<bullet>" 70) |
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37 |
||
67399 | 38 |
text\<open>Equality of vectors in terms of @{term "(\<bullet>)"} products.\<close> |
63050 | 39 |
|
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lemma linear_componentwise: |
|
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fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_inner" |
|
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assumes lf: "linear f" |
|
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shows "(f x) \<bullet> j = (\<Sum>i\<in>Basis. (x\<bullet>i) * (f i\<bullet>j))" (is "?lhs = ?rhs") |
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proof - |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
45 |
interpret linear f by fact |
63050 | 46 |
have "?rhs = (\<Sum>i\<in>Basis. (x\<bullet>i) *\<^sub>R (f i))\<bullet>j" |
64267 | 47 |
by (simp add: inner_sum_left) |
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then show ?thesis |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
49 |
by (simp add: euclidean_representation sum[symmetric] scale[symmetric]) |
63050 | 50 |
qed |
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||
52 |
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" |
|
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(is "?lhs \<longleftrightarrow> ?rhs") |
|
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proof |
|
55 |
assume ?lhs |
|
56 |
then show ?rhs by simp |
|
57 |
next |
|
58 |
assume ?rhs |
|
59 |
then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" |
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by simp |
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61 |
then have "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" |
|
62 |
by (simp add: inner_diff inner_commute) |
|
63 |
then have "(x - y) \<bullet> (x - y) = 0" |
|
64 |
by (simp add: field_simps inner_diff inner_commute) |
|
65 |
then show "x = y" by simp |
|
66 |
qed |
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67 |
||
68 |
lemma norm_triangle_half_r: |
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"norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e" |
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using dist_triangle_half_r unfolding dist_norm[symmetric] by auto |
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||
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lemma norm_triangle_half_l: |
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assumes "norm (x - y) < e / 2" |
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and "norm (x' - y) < e / 2" |
|
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shows "norm (x - x') < e" |
|
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using dist_triangle_half_l[OF assms[unfolded dist_norm[symmetric]]] |
|
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unfolding dist_norm[symmetric] . |
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78 |
||
66420 | 79 |
lemma abs_triangle_half_r: |
80 |
fixes y :: "'a::linordered_field" |
|
81 |
shows "abs (y - x1) < e / 2 \<Longrightarrow> abs (y - x2) < e / 2 \<Longrightarrow> abs (x1 - x2) < e" |
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by linarith |
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83 |
||
84 |
lemma abs_triangle_half_l: |
|
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fixes y :: "'a::linordered_field" |
|
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assumes "abs (x - y) < e / 2" |
|
87 |
and "abs (x' - y) < e / 2" |
|
88 |
shows "abs (x - x') < e" |
|
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using assms by linarith |
|
90 |
||
64267 | 91 |
lemma sum_clauses: |
92 |
shows "sum f {} = 0" |
|
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and "finite S \<Longrightarrow> sum f (insert x S) = (if x \<in> S then sum f S else f x + sum f S)" |
|
63050 | 94 |
by (auto simp add: insert_absorb) |
95 |
||
96 |
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z" |
|
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proof |
|
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assume "\<forall>x. x \<bullet> y = x \<bullet> z" |
|
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then have "\<forall>x. x \<bullet> (y - z) = 0" |
|
100 |
by (simp add: inner_diff) |
|
101 |
then have "(y - z) \<bullet> (y - z) = 0" .. |
|
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then show "y = z" by simp |
|
103 |
qed simp |
|
104 |
||
105 |
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y" |
|
106 |
proof |
|
107 |
assume "\<forall>z. x \<bullet> z = y \<bullet> z" |
|
108 |
then have "\<forall>z. (x - y) \<bullet> z = 0" |
|
109 |
by (simp add: inner_diff) |
|
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then have "(x - y) \<bullet> (x - y) = 0" .. |
|
111 |
then show "x = y" by simp |
|
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qed simp |
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113 |
||
114 |
||
68901 | 115 |
subsection \<open>Orthogonality\<close> |
63050 | 116 |
|
67962 | 117 |
definition%important (in real_inner) "orthogonal x y \<longleftrightarrow> x \<bullet> y = 0" |
118 |
||
63050 | 119 |
context real_inner |
120 |
begin |
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121 |
||
63072 | 122 |
lemma orthogonal_self: "orthogonal x x \<longleftrightarrow> x = 0" |
123 |
by (simp add: orthogonal_def) |
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124 |
||
63050 | 125 |
lemma orthogonal_clauses: |
126 |
"orthogonal a 0" |
|
127 |
"orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)" |
|
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"orthogonal a x \<Longrightarrow> orthogonal a (- x)" |
|
129 |
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)" |
|
130 |
"orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)" |
|
131 |
"orthogonal 0 a" |
|
132 |
"orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a" |
|
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"orthogonal x a \<Longrightarrow> orthogonal (- x) a" |
|
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"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a" |
|
135 |
"orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a" |
|
136 |
unfolding orthogonal_def inner_add inner_diff by auto |
|
137 |
||
138 |
end |
|
139 |
||
140 |
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x" |
|
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by (simp add: orthogonal_def inner_commute) |
|
142 |
||
63114
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Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
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143 |
lemma orthogonal_scaleR [simp]: "c \<noteq> 0 \<Longrightarrow> orthogonal (c *\<^sub>R x) = orthogonal x" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
144 |
by (rule ext) (simp add: orthogonal_def) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
145 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
146 |
lemma pairwise_ortho_scaleR: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
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147 |
"pairwise (\<lambda>i j. orthogonal (f i) (g j)) B |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
148 |
\<Longrightarrow> pairwise (\<lambda>i j. orthogonal (a i *\<^sub>R f i) (a j *\<^sub>R g j)) B" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
149 |
by (auto simp: pairwise_def orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
150 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
151 |
lemma orthogonal_rvsum: |
64267 | 152 |
"\<lbrakk>finite s; \<And>y. y \<in> s \<Longrightarrow> orthogonal x (f y)\<rbrakk> \<Longrightarrow> orthogonal x (sum f s)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
153 |
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
154 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
155 |
lemma orthogonal_lvsum: |
64267 | 156 |
"\<lbrakk>finite s; \<And>x. x \<in> s \<Longrightarrow> orthogonal (f x) y\<rbrakk> \<Longrightarrow> orthogonal (sum f s) y" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
157 |
by (induction s rule: finite_induct) (auto simp: orthogonal_clauses) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
158 |
|
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
159 |
lemma norm_add_Pythagorean: |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
160 |
assumes "orthogonal a b" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
161 |
shows "norm(a + b) ^ 2 = norm a ^ 2 + norm b ^ 2" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
162 |
proof - |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
163 |
from assms have "(a - (0 - b)) \<bullet> (a - (0 - b)) = a \<bullet> a - (0 - b \<bullet> b)" |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
164 |
by (simp add: algebra_simps orthogonal_def inner_commute) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
165 |
then show ?thesis |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
166 |
by (simp add: power2_norm_eq_inner) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
167 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
168 |
|
64267 | 169 |
lemma norm_sum_Pythagorean: |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
170 |
assumes "finite I" "pairwise (\<lambda>i j. orthogonal (f i) (f j)) I" |
64267 | 171 |
shows "(norm (sum f I))\<^sup>2 = (\<Sum>i\<in>I. (norm (f i))\<^sup>2)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
172 |
using assms |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
173 |
proof (induction I rule: finite_induct) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
174 |
case empty then show ?case by simp |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
175 |
next |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
176 |
case (insert x I) |
64267 | 177 |
then have "orthogonal (f x) (sum f I)" |
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
178 |
by (metis pairwise_insert orthogonal_rvsum) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
179 |
with insert show ?case |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
180 |
by (simp add: pairwise_insert norm_add_Pythagorean) |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
181 |
qed |
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63075
diff
changeset
|
182 |
|
63050 | 183 |
|
68901 | 184 |
subsection \<open>Bilinear functions\<close> |
63050 | 185 |
|
67962 | 186 |
definition%important "bilinear f \<longleftrightarrow> (\<forall>x. linear (\<lambda>y. f x y)) \<and> (\<forall>y. linear (\<lambda>x. f x y))" |
63050 | 187 |
|
188 |
lemma bilinear_ladd: "bilinear h \<Longrightarrow> h (x + y) z = h x z + h y z" |
|
189 |
by (simp add: bilinear_def linear_iff) |
|
190 |
||
191 |
lemma bilinear_radd: "bilinear h \<Longrightarrow> h x (y + z) = h x y + h x z" |
|
192 |
by (simp add: bilinear_def linear_iff) |
|
193 |
||
194 |
lemma bilinear_lmul: "bilinear h \<Longrightarrow> h (c *\<^sub>R x) y = c *\<^sub>R h x y" |
|
195 |
by (simp add: bilinear_def linear_iff) |
|
196 |
||
197 |
lemma bilinear_rmul: "bilinear h \<Longrightarrow> h x (c *\<^sub>R y) = c *\<^sub>R h x y" |
|
198 |
by (simp add: bilinear_def linear_iff) |
|
199 |
||
200 |
lemma bilinear_lneg: "bilinear h \<Longrightarrow> h (- x) y = - h x y" |
|
201 |
by (drule bilinear_lmul [of _ "- 1"]) simp |
|
202 |
||
203 |
lemma bilinear_rneg: "bilinear h \<Longrightarrow> h x (- y) = - h x y" |
|
204 |
by (drule bilinear_rmul [of _ _ "- 1"]) simp |
|
205 |
||
206 |
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0" |
|
207 |
using add_left_imp_eq[of x y 0] by auto |
|
208 |
||
209 |
lemma bilinear_lzero: |
|
210 |
assumes "bilinear h" |
|
211 |
shows "h 0 x = 0" |
|
212 |
using bilinear_ladd [OF assms, of 0 0 x] by (simp add: eq_add_iff field_simps) |
|
213 |
||
214 |
lemma bilinear_rzero: |
|
215 |
assumes "bilinear h" |
|
216 |
shows "h x 0 = 0" |
|
217 |
using bilinear_radd [OF assms, of x 0 0 ] by (simp add: eq_add_iff field_simps) |
|
218 |
||
219 |
lemma bilinear_lsub: "bilinear h \<Longrightarrow> h (x - y) z = h x z - h y z" |
|
220 |
using bilinear_ladd [of h x "- y"] by (simp add: bilinear_lneg) |
|
221 |
||
222 |
lemma bilinear_rsub: "bilinear h \<Longrightarrow> h z (x - y) = h z x - h z y" |
|
223 |
using bilinear_radd [of h _ x "- y"] by (simp add: bilinear_rneg) |
|
224 |
||
64267 | 225 |
lemma bilinear_sum: |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
226 |
assumes "bilinear h" |
64267 | 227 |
shows "h (sum f S) (sum g T) = sum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) " |
63050 | 228 |
proof - |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
229 |
interpret l: linear "\<lambda>x. h x y" for y using assms by (simp add: bilinear_def) |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
230 |
interpret r: linear "\<lambda>y. h x y" for x using assms by (simp add: bilinear_def) |
64267 | 231 |
have "h (sum f S) (sum g T) = sum (\<lambda>x. h (f x) (sum g T)) S" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
232 |
by (simp add: l.sum) |
64267 | 233 |
also have "\<dots> = sum (\<lambda>x. sum (\<lambda>y. h (f x) (g y)) T) S" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
234 |
by (rule sum.cong) (simp_all add: r.sum) |
63050 | 235 |
finally show ?thesis |
64267 | 236 |
unfolding sum.cartesian_product . |
63050 | 237 |
qed |
238 |
||
239 |
||
68901 | 240 |
subsection \<open>Adjoints\<close> |
63050 | 241 |
|
67962 | 242 |
definition%important "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)" |
63050 | 243 |
|
244 |
lemma adjoint_unique: |
|
245 |
assumes "\<forall>x y. inner (f x) y = inner x (g y)" |
|
246 |
shows "adjoint f = g" |
|
247 |
unfolding adjoint_def |
|
248 |
proof (rule some_equality) |
|
249 |
show "\<forall>x y. inner (f x) y = inner x (g y)" |
|
250 |
by (rule assms) |
|
251 |
next |
|
252 |
fix h |
|
253 |
assume "\<forall>x y. inner (f x) y = inner x (h y)" |
|
254 |
then have "\<forall>x y. inner x (g y) = inner x (h y)" |
|
255 |
using assms by simp |
|
256 |
then have "\<forall>x y. inner x (g y - h y) = 0" |
|
257 |
by (simp add: inner_diff_right) |
|
258 |
then have "\<forall>y. inner (g y - h y) (g y - h y) = 0" |
|
259 |
by simp |
|
260 |
then have "\<forall>y. h y = g y" |
|
261 |
by simp |
|
262 |
then show "h = g" by (simp add: ext) |
|
263 |
qed |
|
264 |
||
265 |
text \<open>TODO: The following lemmas about adjoints should hold for any |
|
63680 | 266 |
Hilbert space (i.e. complete inner product space). |
68224 | 267 |
(see \<^url>\<open>https://en.wikipedia.org/wiki/Hermitian_adjoint\<close>) |
63050 | 268 |
\<close> |
269 |
||
270 |
lemma adjoint_works: |
|
271 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
272 |
assumes lf: "linear f" |
|
273 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
|
274 |
proof - |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
275 |
interpret linear f by fact |
63050 | 276 |
have "\<forall>y. \<exists>w. \<forall>x. f x \<bullet> y = x \<bullet> w" |
277 |
proof (intro allI exI) |
|
278 |
fix y :: "'m" and x |
|
279 |
let ?w = "(\<Sum>i\<in>Basis. (f i \<bullet> y) *\<^sub>R i) :: 'n" |
|
280 |
have "f x \<bullet> y = f (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i) \<bullet> y" |
|
281 |
by (simp add: euclidean_representation) |
|
282 |
also have "\<dots> = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R f i) \<bullet> y" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
283 |
by (simp add: sum scale) |
63050 | 284 |
finally show "f x \<bullet> y = x \<bullet> ?w" |
64267 | 285 |
by (simp add: inner_sum_left inner_sum_right mult.commute) |
63050 | 286 |
qed |
287 |
then show ?thesis |
|
288 |
unfolding adjoint_def choice_iff |
|
289 |
by (intro someI2_ex[where Q="\<lambda>f'. x \<bullet> f' y = f x \<bullet> y"]) auto |
|
290 |
qed |
|
291 |
||
292 |
lemma adjoint_clauses: |
|
293 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
294 |
assumes lf: "linear f" |
|
295 |
shows "x \<bullet> adjoint f y = f x \<bullet> y" |
|
296 |
and "adjoint f y \<bullet> x = y \<bullet> f x" |
|
297 |
by (simp_all add: adjoint_works[OF lf] inner_commute) |
|
298 |
||
299 |
lemma adjoint_linear: |
|
300 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
301 |
assumes lf: "linear f" |
|
302 |
shows "linear (adjoint f)" |
|
303 |
by (simp add: lf linear_iff euclidean_eq_iff[where 'a='n] euclidean_eq_iff[where 'a='m] |
|
304 |
adjoint_clauses[OF lf] inner_distrib) |
|
305 |
||
306 |
lemma adjoint_adjoint: |
|
307 |
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space" |
|
308 |
assumes lf: "linear f" |
|
309 |
shows "adjoint (adjoint f) = f" |
|
310 |
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) |
|
311 |
||
312 |
||
67962 | 313 |
subsection%unimportant \<open>Interlude: Some properties of real sets\<close> |
63050 | 314 |
|
315 |
lemma seq_mono_lemma: |
|
316 |
assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" |
|
317 |
and "\<forall>n \<ge> m. e n \<le> e m" |
|
318 |
shows "\<forall>n \<ge> m. d n < e m" |
|
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
319 |
using assms by force |
63050 | 320 |
|
321 |
lemma infinite_enumerate: |
|
322 |
assumes fS: "infinite S" |
|
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
66420
diff
changeset
|
323 |
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)" |
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
66420
diff
changeset
|
324 |
unfolding strict_mono_def |
63050 | 325 |
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto |
326 |
||
327 |
lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)" |
|
328 |
apply auto |
|
329 |
apply (rule_tac x="d/2" in exI) |
|
330 |
apply auto |
|
331 |
done |
|
332 |
||
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67399
diff
changeset
|
333 |
lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close> |
63050 | 334 |
"(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)" |
335 |
apply auto |
|
336 |
apply (rule_tac x="d/2" in exI, auto) |
|
337 |
done |
|
338 |
||
339 |
lemma triangle_lemma: |
|
340 |
fixes x y z :: real |
|
341 |
assumes x: "0 \<le> x" |
|
342 |
and y: "0 \<le> y" |
|
343 |
and z: "0 \<le> z" |
|
344 |
and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2" |
|
345 |
shows "x \<le> y + z" |
|
346 |
proof - |
|
347 |
have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2" |
|
348 |
using z y by simp |
|
349 |
with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2" |
|
350 |
by (simp add: power2_eq_square field_simps) |
|
351 |
from y z have yz: "y + z \<ge> 0" |
|
352 |
by arith |
|
353 |
from power2_le_imp_le[OF th yz] show ?thesis . |
|
354 |
qed |
|
355 |
||
356 |
||
357 |
||
358 |
subsection \<open>Archimedean properties and useful consequences\<close> |
|
359 |
||
360 |
text\<open>Bernoulli's inequality\<close> |
|
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68224
diff
changeset
|
361 |
proposition Bernoulli_inequality: |
63050 | 362 |
fixes x :: real |
363 |
assumes "-1 \<le> x" |
|
364 |
shows "1 + n * x \<le> (1 + x) ^ n" |
|
68607
67bb59e49834
make theorem, corollary, and proposition %important for HOL-Analysis manual
immler
parents:
68224
diff
changeset
|
365 |
proof (induct n) |
63050 | 366 |
case 0 |
367 |
then show ?case by simp |
|
368 |
next |
|
369 |
case (Suc n) |
|
370 |
have "1 + Suc n * x \<le> 1 + (Suc n)*x + n * x^2" |
|
371 |
by (simp add: algebra_simps) |
|
372 |
also have "... = (1 + x) * (1 + n*x)" |
|
373 |
by (auto simp: power2_eq_square algebra_simps of_nat_Suc) |
|
374 |
also have "... \<le> (1 + x) ^ Suc n" |
|
375 |
using Suc.hyps assms mult_left_mono by fastforce |
|
376 |
finally show ?case . |
|
377 |
qed |
|
378 |
||
379 |
corollary Bernoulli_inequality_even: |
|
380 |
fixes x :: real |
|
381 |
assumes "even n" |
|
382 |
shows "1 + n * x \<le> (1 + x) ^ n" |
|
383 |
proof (cases "-1 \<le> x \<or> n=0") |
|
384 |
case True |
|
385 |
then show ?thesis |
|
386 |
by (auto simp: Bernoulli_inequality) |
|
387 |
next |
|
388 |
case False |
|
389 |
then have "real n \<ge> 1" |
|
390 |
by simp |
|
391 |
with False have "n * x \<le> -1" |
|
392 |
by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one) |
|
393 |
then have "1 + n * x \<le> 0" |
|
394 |
by auto |
|
395 |
also have "... \<le> (1 + x) ^ n" |
|
396 |
using assms |
|
397 |
using zero_le_even_power by blast |
|
398 |
finally show ?thesis . |
|
399 |
qed |
|
400 |
||
401 |
corollary real_arch_pow: |
|
402 |
fixes x :: real |
|
403 |
assumes x: "1 < x" |
|
404 |
shows "\<exists>n. y < x^n" |
|
405 |
proof - |
|
406 |
from x have x0: "x - 1 > 0" |
|
407 |
by arith |
|
408 |
from reals_Archimedean3[OF x0, rule_format, of y] |
|
409 |
obtain n :: nat where n: "y < real n * (x - 1)" by metis |
|
410 |
from x0 have x00: "x- 1 \<ge> -1" by arith |
|
411 |
from Bernoulli_inequality[OF x00, of n] n |
|
412 |
have "y < x^n" by auto |
|
413 |
then show ?thesis by metis |
|
414 |
qed |
|
415 |
||
416 |
corollary real_arch_pow_inv: |
|
417 |
fixes x y :: real |
|
418 |
assumes y: "y > 0" |
|
419 |
and x1: "x < 1" |
|
420 |
shows "\<exists>n. x^n < y" |
|
421 |
proof (cases "x > 0") |
|
422 |
case True |
|
423 |
with x1 have ix: "1 < 1/x" by (simp add: field_simps) |
|
424 |
from real_arch_pow[OF ix, of "1/y"] |
|
425 |
obtain n where n: "1/y < (1/x)^n" by blast |
|
426 |
then show ?thesis using y \<open>x > 0\<close> |
|
427 |
by (auto simp add: field_simps) |
|
428 |
next |
|
429 |
case False |
|
430 |
with y x1 show ?thesis |
|
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
431 |
by (metis less_le_trans not_less power_one_right) |
63050 | 432 |
qed |
433 |
||
434 |
lemma forall_pos_mono: |
|
435 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
|
436 |
(\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)" |
|
437 |
by (metis real_arch_inverse) |
|
438 |
||
439 |
lemma forall_pos_mono_1: |
|
440 |
"(\<And>d e::real. d < e \<Longrightarrow> P d \<Longrightarrow> P e) \<Longrightarrow> |
|
441 |
(\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e" |
|
442 |
apply (rule forall_pos_mono) |
|
443 |
apply auto |
|
444 |
apply (metis Suc_pred of_nat_Suc) |
|
445 |
done |
|
446 |
||
447 |
||
67962 | 448 |
subsection%unimportant \<open>Euclidean Spaces as Typeclass\<close> |
44133 | 449 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
450 |
lemma independent_Basis: "independent Basis" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
451 |
by (rule independent_Basis) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
452 |
|
53939 | 453 |
lemma span_Basis [simp]: "span Basis = UNIV" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
454 |
by (rule span_Basis) |
44133 | 455 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
456 |
lemma in_span_Basis: "x \<in> span Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
457 |
unfolding span_Basis .. |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
458 |
|
53406 | 459 |
|
67962 | 460 |
subsection%unimportant \<open>Linearity and Bilinearity continued\<close> |
44133 | 461 |
|
462 |
lemma linear_bounded: |
|
56444 | 463 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
44133 | 464 |
assumes lf: "linear f" |
465 |
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
53939 | 466 |
proof |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
467 |
interpret linear f by fact |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
468 |
let ?B = "\<Sum>b\<in>Basis. norm (f b)" |
53939 | 469 |
show "\<forall>x. norm (f x) \<le> ?B * norm x" |
470 |
proof |
|
53406 | 471 |
fix x :: 'a |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
472 |
let ?g = "\<lambda>b. (x \<bullet> b) *\<^sub>R f b" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
473 |
have "norm (f x) = norm (f (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b))" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
474 |
unfolding euclidean_representation .. |
64267 | 475 |
also have "\<dots> = norm (sum ?g Basis)" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
476 |
by (simp add: sum scale) |
64267 | 477 |
finally have th0: "norm (f x) = norm (sum ?g Basis)" . |
64773
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
478 |
have th: "norm (?g i) \<le> norm (f i) * norm x" if "i \<in> Basis" for i |
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
479 |
proof - |
223b2ebdda79
Many new theorems, and more tidying
paulson <lp15@cam.ac.uk>
parents:
64267
diff
changeset
|
480 |
from Basis_le_norm[OF that, of x] |
53939 | 481 |
show "norm (?g i) \<le> norm (f i) * norm x" |
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
482 |
unfolding norm_scaleR by (metis mult.commute mult_left_mono norm_ge_zero) |
53939 | 483 |
qed |
64267 | 484 |
from sum_norm_le[of _ ?g, OF th] |
53939 | 485 |
show "norm (f x) \<le> ?B * norm x" |
64267 | 486 |
unfolding th0 sum_distrib_right by metis |
53939 | 487 |
qed |
44133 | 488 |
qed |
489 |
||
490 |
lemma linear_conv_bounded_linear: |
|
491 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
|
492 |
shows "linear f \<longleftrightarrow> bounded_linear f" |
|
493 |
proof |
|
494 |
assume "linear f" |
|
53939 | 495 |
then interpret f: linear f . |
44133 | 496 |
show "bounded_linear f" |
497 |
proof |
|
498 |
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x" |
|
60420 | 499 |
using \<open>linear f\<close> by (rule linear_bounded) |
49522 | 500 |
then show "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
501 |
by (simp add: mult.commute) |
44133 | 502 |
qed |
503 |
next |
|
504 |
assume "bounded_linear f" |
|
505 |
then interpret f: bounded_linear f . |
|
53939 | 506 |
show "linear f" .. |
507 |
qed |
|
508 |
||
61518
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61306
diff
changeset
|
509 |
lemmas linear_linear = linear_conv_bounded_linear[symmetric] |
ff12606337e9
new lemmas about topology, etc., for Cauchy integral formula
paulson
parents:
61306
diff
changeset
|
510 |
|
53939 | 511 |
lemma linear_bounded_pos: |
56444 | 512 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
53939 | 513 |
assumes lf: "linear f" |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
514 |
obtains B where "B > 0" "\<And>x. norm (f x) \<le> B * norm x" |
53939 | 515 |
proof - |
516 |
have "\<exists>B > 0. \<forall>x. norm (f x) \<le> norm x * B" |
|
517 |
using lf unfolding linear_conv_bounded_linear |
|
518 |
by (rule bounded_linear.pos_bounded) |
|
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
519 |
with that show ?thesis |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
520 |
by (auto simp: mult.commute) |
44133 | 521 |
qed |
522 |
||
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
523 |
lemma linear_invertible_bounded_below_pos: |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
524 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
525 |
assumes "linear f" "linear g" "g \<circ> f = id" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
526 |
obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
527 |
proof - |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
528 |
obtain B where "B > 0" and B: "\<And>x. norm (g x) \<le> B * norm x" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
529 |
using linear_bounded_pos [OF \<open>linear g\<close>] by blast |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
530 |
show thesis |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
531 |
proof |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
532 |
show "0 < 1/B" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
533 |
by (simp add: \<open>B > 0\<close>) |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
534 |
show "1/B * norm x \<le> norm (f x)" for x |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
535 |
proof - |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
536 |
have "1/B * norm x = 1/B * norm (g (f x))" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
537 |
using assms by (simp add: pointfree_idE) |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
538 |
also have "\<dots> \<le> norm (f x)" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
539 |
using B [of "f x"] by (simp add: \<open>B > 0\<close> mult.commute pos_divide_le_eq) |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
540 |
finally show ?thesis . |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
541 |
qed |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
542 |
qed |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
543 |
qed |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
544 |
|
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
545 |
lemma linear_inj_bounded_below_pos: |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
546 |
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
547 |
assumes "linear f" "inj f" |
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
548 |
obtains B where "B > 0" "\<And>x. B * norm x \<le> norm(f x)" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
549 |
using linear_injective_left_inverse [OF assms] |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
550 |
linear_invertible_bounded_below_pos assms by blast |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
551 |
|
49522 | 552 |
lemma bounded_linearI': |
56444 | 553 |
fixes f ::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
53406 | 554 |
assumes "\<And>x y. f (x + y) = f x + f y" |
555 |
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" |
|
49522 | 556 |
shows "bounded_linear f" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
557 |
using assms linearI linear_conv_bounded_linear by blast |
44133 | 558 |
|
559 |
lemma bilinear_bounded: |
|
56444 | 560 |
fixes h :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector" |
44133 | 561 |
assumes bh: "bilinear h" |
562 |
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
563 |
proof (clarify intro!: exI[of _ "\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)"]) |
53406 | 564 |
fix x :: 'm |
565 |
fix y :: 'n |
|
64267 | 566 |
have "norm (h x y) = norm (h (sum (\<lambda>i. (x \<bullet> i) *\<^sub>R i) Basis) (sum (\<lambda>i. (y \<bullet> i) *\<^sub>R i) Basis))" |
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
567 |
by (simp add: euclidean_representation) |
64267 | 568 |
also have "\<dots> = norm (sum (\<lambda> (i,j). h ((x \<bullet> i) *\<^sub>R i) ((y \<bullet> j) *\<^sub>R j)) (Basis \<times> Basis))" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
569 |
unfolding bilinear_sum[OF bh] .. |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
570 |
finally have th: "norm (h x y) = \<dots>" . |
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
571 |
have "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
572 |
\<Longrightarrow> \<bar>x \<bullet> i\<bar> * (\<bar>y \<bullet> j\<bar> * norm (h i j)) \<le> norm x * (norm y * norm (h i j))" |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
573 |
by (auto simp add: zero_le_mult_iff Basis_le_norm mult_mono) |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
574 |
then show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y" |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
575 |
unfolding sum_distrib_right th sum.cartesian_product |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
576 |
by (clarsimp simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] |
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
577 |
field_simps simp del: scaleR_scaleR intro!: sum_norm_le) |
44133 | 578 |
qed |
579 |
||
580 |
lemma bilinear_conv_bounded_bilinear: |
|
581 |
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
|
582 |
shows "bilinear h \<longleftrightarrow> bounded_bilinear h" |
|
583 |
proof |
|
584 |
assume "bilinear h" |
|
585 |
show "bounded_bilinear h" |
|
586 |
proof |
|
53406 | 587 |
fix x y z |
588 |
show "h (x + y) z = h x z + h y z" |
|
60420 | 589 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp |
44133 | 590 |
next |
53406 | 591 |
fix x y z |
592 |
show "h x (y + z) = h x y + h x z" |
|
60420 | 593 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff by simp |
44133 | 594 |
next |
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
595 |
show "h (scaleR r x) y = scaleR r (h x y)" "h x (scaleR r y) = scaleR r (h x y)" for r x y |
60420 | 596 |
using \<open>bilinear h\<close> unfolding bilinear_def linear_iff |
68069
36209dfb981e
tidying up and using real induction methods
paulson <lp15@cam.ac.uk>
parents:
68062
diff
changeset
|
597 |
by simp_all |
44133 | 598 |
next |
599 |
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
60420 | 600 |
using \<open>bilinear h\<close> by (rule bilinear_bounded) |
49522 | 601 |
then show "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
602 |
by (simp add: ac_simps) |
44133 | 603 |
qed |
604 |
next |
|
605 |
assume "bounded_bilinear h" |
|
606 |
then interpret h: bounded_bilinear h . |
|
607 |
show "bilinear h" |
|
608 |
unfolding bilinear_def linear_conv_bounded_linear |
|
49522 | 609 |
using h.bounded_linear_left h.bounded_linear_right by simp |
44133 | 610 |
qed |
611 |
||
53939 | 612 |
lemma bilinear_bounded_pos: |
56444 | 613 |
fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector" |
53939 | 614 |
assumes bh: "bilinear h" |
615 |
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y" |
|
616 |
proof - |
|
617 |
have "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> norm x * norm y * B" |
|
618 |
using bh [unfolded bilinear_conv_bounded_bilinear] |
|
619 |
by (rule bounded_bilinear.pos_bounded) |
|
620 |
then show ?thesis |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
621 |
by (simp only: ac_simps) |
53939 | 622 |
qed |
623 |
||
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
624 |
lemma bounded_linear_imp_has_derivative: "bounded_linear f \<Longrightarrow> (f has_derivative f) net" |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
625 |
by (auto simp add: has_derivative_def linear_diff linear_linear linear_def |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
626 |
dest: bounded_linear.linear) |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
627 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
628 |
lemma linear_imp_has_derivative: |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
629 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
630 |
shows "linear f \<Longrightarrow> (f has_derivative f) net" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
631 |
by (simp add: bounded_linear_imp_has_derivative linear_conv_bounded_linear) |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
632 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
633 |
lemma bounded_linear_imp_differentiable: "bounded_linear f \<Longrightarrow> f differentiable net" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
634 |
using bounded_linear_imp_has_derivative differentiable_def by blast |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
635 |
|
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
636 |
lemma linear_imp_differentiable: |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
637 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector" |
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
638 |
shows "linear f \<Longrightarrow> f differentiable net" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
639 |
by (metis linear_imp_has_derivative differentiable_def) |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63170
diff
changeset
|
640 |
|
49522 | 641 |
|
68901 | 642 |
subsection%unimportant \<open>We continue\<close> |
44133 | 643 |
|
644 |
lemma independent_bound: |
|
53716 | 645 |
fixes S :: "'a::euclidean_space set" |
646 |
shows "independent S \<Longrightarrow> finite S \<and> card S \<le> DIM('a)" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
647 |
by (metis dim_subset_UNIV finiteI_independent dim_span_eq_card_independent) |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
648 |
|
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
649 |
lemmas independent_imp_finite = finiteI_independent |
44133 | 650 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61520
diff
changeset
|
651 |
corollary |
60303 | 652 |
fixes S :: "'a::euclidean_space set" |
653 |
assumes "independent S" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
654 |
shows independent_card_le:"card S \<le> DIM('a)" |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
655 |
using assms independent_bound by auto |
63075
60a42a4166af
lemmas about dimension, hyperplanes, span, etc.
paulson <lp15@cam.ac.uk>
parents:
63072
diff
changeset
|
656 |
|
49663 | 657 |
lemma dependent_biggerset: |
56444 | 658 |
fixes S :: "'a::euclidean_space set" |
659 |
shows "(finite S \<Longrightarrow> card S > DIM('a)) \<Longrightarrow> dependent S" |
|
44133 | 660 |
by (metis independent_bound not_less) |
661 |
||
60420 | 662 |
text \<open>Picking an orthogonal replacement for a spanning set.\<close> |
44133 | 663 |
|
53406 | 664 |
lemma vector_sub_project_orthogonal: |
665 |
fixes b x :: "'a::euclidean_space" |
|
666 |
shows "b \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0" |
|
44133 | 667 |
unfolding inner_simps by auto |
668 |
||
44528 | 669 |
lemma pairwise_orthogonal_insert: |
670 |
assumes "pairwise orthogonal S" |
|
49522 | 671 |
and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" |
44528 | 672 |
shows "pairwise orthogonal (insert x S)" |
673 |
using assms unfolding pairwise_def |
|
674 |
by (auto simp add: orthogonal_commute) |
|
675 |
||
44133 | 676 |
lemma basis_orthogonal: |
53406 | 677 |
fixes B :: "'a::real_inner set" |
44133 | 678 |
assumes fB: "finite B" |
679 |
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C" |
|
680 |
(is " \<exists>C. ?P B C") |
|
49522 | 681 |
using fB |
682 |
proof (induct rule: finite_induct) |
|
683 |
case empty |
|
53406 | 684 |
then show ?case |
685 |
apply (rule exI[where x="{}"]) |
|
686 |
apply (auto simp add: pairwise_def) |
|
687 |
done |
|
44133 | 688 |
next |
49522 | 689 |
case (insert a B) |
60420 | 690 |
note fB = \<open>finite B\<close> and aB = \<open>a \<notin> B\<close> |
691 |
from \<open>\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C\<close> |
|
44133 | 692 |
obtain C where C: "finite C" "card C \<le> card B" |
693 |
"span C = span B" "pairwise orthogonal C" by blast |
|
64267 | 694 |
let ?a = "a - sum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x) C" |
44133 | 695 |
let ?C = "insert ?a C" |
53406 | 696 |
from C(1) have fC: "finite ?C" |
697 |
by simp |
|
49522 | 698 |
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" |
699 |
by (simp add: card_insert_if) |
|
53406 | 700 |
{ |
701 |
fix x k |
|
49522 | 702 |
have th0: "\<And>(a::'a) b c. a - (b - c) = c + (a - b)" |
703 |
by (simp add: field_simps) |
|
44133 | 704 |
have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C" |
705 |
apply (simp only: scaleR_right_diff_distrib th0) |
|
706 |
apply (rule span_add_eq) |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
707 |
apply (rule span_scale) |
64267 | 708 |
apply (rule span_sum) |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
709 |
apply (rule span_scale) |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
710 |
apply (rule span_base) |
49522 | 711 |
apply assumption |
53406 | 712 |
done |
713 |
} |
|
44133 | 714 |
then have SC: "span ?C = span (insert a B)" |
715 |
unfolding set_eq_iff span_breakdown_eq C(3)[symmetric] by auto |
|
53406 | 716 |
{ |
717 |
fix y |
|
718 |
assume yC: "y \<in> C" |
|
719 |
then have Cy: "C = insert y (C - {y})" |
|
720 |
by blast |
|
721 |
have fth: "finite (C - {y})" |
|
722 |
using C by simp |
|
44528 | 723 |
have "orthogonal ?a y" |
724 |
unfolding orthogonal_def |
|
64267 | 725 |
unfolding inner_diff inner_sum_left right_minus_eq |
726 |
unfolding sum.remove [OF \<open>finite C\<close> \<open>y \<in> C\<close>] |
|
44528 | 727 |
apply (clarsimp simp add: inner_commute[of y a]) |
64267 | 728 |
apply (rule sum.neutral) |
44528 | 729 |
apply clarsimp |
730 |
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
60420 | 731 |
using \<open>y \<in> C\<close> by auto |
53406 | 732 |
} |
60420 | 733 |
with \<open>pairwise orthogonal C\<close> have CPO: "pairwise orthogonal ?C" |
44528 | 734 |
by (rule pairwise_orthogonal_insert) |
53406 | 735 |
from fC cC SC CPO have "?P (insert a B) ?C" |
736 |
by blast |
|
44133 | 737 |
then show ?case by blast |
738 |
qed |
|
739 |
||
740 |
lemma orthogonal_basis_exists: |
|
741 |
fixes V :: "('a::euclidean_space) set" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
742 |
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
743 |
(card B = dim V) \<and> pairwise orthogonal B" |
49663 | 744 |
proof - |
49522 | 745 |
from basis_exists[of V] obtain B where |
53406 | 746 |
B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" |
68073
fad29d2a17a5
merged; resolved conflicts manually (esp. lemmas that have been moved from Linear_Algebra and Cartesian_Euclidean_Space)
immler
diff
changeset
|
747 |
by force |
53406 | 748 |
from B have fB: "finite B" "card B = dim V" |
749 |
using independent_bound by auto |
|
44133 | 750 |
from basis_orthogonal[OF fB(1)] obtain C where |
53406 | 751 |
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" |
752 |
by blast |
|
753 |
from C B have CSV: "C \<subseteq> span V" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
754 |
by (metis span_superset span_mono subset_trans) |
53406 | 755 |
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" |
756 |
by (simp add: span_span) |
|
44133 | 757 |
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB |
53406 | 758 |
have iC: "independent C" |
44133 | 759 |
by (simp add: dim_span) |
53406 | 760 |
from C fB have "card C \<le> dim V" |
761 |
by simp |
|
762 |
moreover have "dim V \<le> card C" |
|
763 |
using span_card_ge_dim[OF CSV SVC C(1)] |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
764 |
by simp |
53406 | 765 |
ultimately have CdV: "card C = dim V" |
766 |
using C(1) by simp |
|
767 |
from C B CSV CdV iC show ?thesis |
|
768 |
by auto |
|
44133 | 769 |
qed |
770 |
||
60420 | 771 |
text \<open>Low-dimensional subset is in a hyperplane (weak orthogonal complement).\<close> |
44133 | 772 |
|
49522 | 773 |
lemma span_not_univ_orthogonal: |
53406 | 774 |
fixes S :: "'a::euclidean_space set" |
44133 | 775 |
assumes sU: "span S \<noteq> UNIV" |
56444 | 776 |
shows "\<exists>a::'a. a \<noteq> 0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)" |
49522 | 777 |
proof - |
53406 | 778 |
from sU obtain a where a: "a \<notin> span S" |
779 |
by blast |
|
44133 | 780 |
from orthogonal_basis_exists obtain B where |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
781 |
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
782 |
"card B = dim S" "pairwise orthogonal B" |
44133 | 783 |
by blast |
53406 | 784 |
from B have fB: "finite B" "card B = dim S" |
785 |
using independent_bound by auto |
|
44133 | 786 |
from span_mono[OF B(2)] span_mono[OF B(3)] |
53406 | 787 |
have sSB: "span S = span B" |
788 |
by (simp add: span_span) |
|
64267 | 789 |
let ?a = "a - sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B" |
790 |
have "sum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R b) B \<in> span S" |
|
44133 | 791 |
unfolding sSB |
64267 | 792 |
apply (rule span_sum) |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
793 |
apply (rule span_scale) |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
794 |
apply (rule span_base) |
49522 | 795 |
apply assumption |
796 |
done |
|
53406 | 797 |
with a have a0:"?a \<noteq> 0" |
798 |
by auto |
|
68058 | 799 |
have "?a \<bullet> x = 0" if "x\<in>span B" for x |
800 |
proof (rule span_induct [OF that]) |
|
49522 | 801 |
show "subspace {x. ?a \<bullet> x = 0}" |
802 |
by (auto simp add: subspace_def inner_add) |
|
803 |
next |
|
53406 | 804 |
{ |
805 |
fix x |
|
806 |
assume x: "x \<in> B" |
|
807 |
from x have B': "B = insert x (B - {x})" |
|
808 |
by blast |
|
809 |
have fth: "finite (B - {x})" |
|
810 |
using fB by simp |
|
44133 | 811 |
have "?a \<bullet> x = 0" |
53406 | 812 |
apply (subst B') |
813 |
using fB fth |
|
64267 | 814 |
unfolding sum_clauses(2)[OF fth] |
44133 | 815 |
apply simp unfolding inner_simps |
64267 | 816 |
apply (clarsimp simp add: inner_add inner_sum_left) |
817 |
apply (rule sum.neutral, rule ballI) |
|
63170 | 818 |
apply (simp only: inner_commute) |
49711 | 819 |
apply (auto simp add: x field_simps |
820 |
intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format]) |
|
53406 | 821 |
done |
822 |
} |
|
68058 | 823 |
then show "?a \<bullet> x = 0" if "x \<in> B" for x |
824 |
using that by blast |
|
825 |
qed |
|
53406 | 826 |
with a0 show ?thesis |
827 |
unfolding sSB by (auto intro: exI[where x="?a"]) |
|
44133 | 828 |
qed |
829 |
||
830 |
lemma span_not_univ_subset_hyperplane: |
|
53406 | 831 |
fixes S :: "'a::euclidean_space set" |
832 |
assumes SU: "span S \<noteq> UNIV" |
|
44133 | 833 |
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
834 |
using span_not_univ_orthogonal[OF SU] by auto |
|
835 |
||
49663 | 836 |
lemma lowdim_subset_hyperplane: |
53406 | 837 |
fixes S :: "'a::euclidean_space set" |
44133 | 838 |
assumes d: "dim S < DIM('a)" |
56444 | 839 |
shows "\<exists>a::'a. a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}" |
49522 | 840 |
proof - |
53406 | 841 |
{ |
842 |
assume "span S = UNIV" |
|
843 |
then have "dim (span S) = dim (UNIV :: ('a) set)" |
|
844 |
by simp |
|
845 |
then have "dim S = DIM('a)" |
|
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
846 |
by (metis Euclidean_Space.dim_UNIV dim_span) |
53406 | 847 |
with d have False by arith |
848 |
} |
|
849 |
then have th: "span S \<noteq> UNIV" |
|
850 |
by blast |
|
44133 | 851 |
from span_not_univ_subset_hyperplane[OF th] show ?thesis . |
852 |
qed |
|
853 |
||
854 |
lemma linear_eq_stdbasis: |
|
56444 | 855 |
fixes f :: "'a::euclidean_space \<Rightarrow> _" |
856 |
assumes lf: "linear f" |
|
49663 | 857 |
and lg: "linear g" |
68058 | 858 |
and fg: "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b" |
44133 | 859 |
shows "f = g" |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
860 |
using linear_eq_on_span[OF lf lg, of Basis] fg |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
861 |
by auto |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
862 |
|
44133 | 863 |
|
60420 | 864 |
text \<open>Similar results for bilinear functions.\<close> |
44133 | 865 |
|
866 |
lemma bilinear_eq: |
|
867 |
assumes bf: "bilinear f" |
|
49522 | 868 |
and bg: "bilinear g" |
53406 | 869 |
and SB: "S \<subseteq> span B" |
870 |
and TC: "T \<subseteq> span C" |
|
68058 | 871 |
and "x\<in>S" "y\<in>T" |
872 |
and fg: "\<And>x y. \<lbrakk>x \<in> B; y\<in> C\<rbrakk> \<Longrightarrow> f x y = g x y" |
|
873 |
shows "f x y = g x y" |
|
49663 | 874 |
proof - |
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44166
diff
changeset
|
875 |
let ?P = "{x. \<forall>y\<in> span C. f x y = g x y}" |
44133 | 876 |
from bf bg have sp: "subspace ?P" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
877 |
unfolding bilinear_def linear_iff subspace_def bf bg |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
878 |
by (auto simp add: span_zero bilinear_lzero[OF bf] bilinear_lzero[OF bg] |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
879 |
span_add Ball_def |
49663 | 880 |
intro: bilinear_ladd[OF bf]) |
68058 | 881 |
have sfg: "\<And>x. x \<in> B \<Longrightarrow> subspace {a. f x a = g x a}" |
44133 | 882 |
apply (auto simp add: subspace_def) |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53596
diff
changeset
|
883 |
using bf bg unfolding bilinear_def linear_iff |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
884 |
apply (auto simp add: span_zero bilinear_rzero[OF bf] bilinear_rzero[OF bg] |
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
885 |
span_add Ball_def |
49663 | 886 |
intro: bilinear_ladd[OF bf]) |
49522 | 887 |
done |
68058 | 888 |
have "\<forall>y\<in> span C. f x y = g x y" if "x \<in> span B" for x |
889 |
apply (rule span_induct [OF that sp]) |
|
68062 | 890 |
using fg sfg span_induct by blast |
53406 | 891 |
then show ?thesis |
68058 | 892 |
using SB TC assms by auto |
44133 | 893 |
qed |
894 |
||
49522 | 895 |
lemma bilinear_eq_stdbasis: |
53406 | 896 |
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _" |
44133 | 897 |
assumes bf: "bilinear f" |
49522 | 898 |
and bg: "bilinear g" |
68058 | 899 |
and fg: "\<And>i j. i \<in> Basis \<Longrightarrow> j \<in> Basis \<Longrightarrow> f i j = g i j" |
44133 | 900 |
shows "f = g" |
68074 | 901 |
using bilinear_eq[OF bf bg equalityD2[OF span_Basis] equalityD2[OF span_Basis]] fg by blast |
49522 | 902 |
|
60420 | 903 |
subsection \<open>Infinity norm\<close> |
44133 | 904 |
|
67962 | 905 |
definition%important "infnorm (x::'a::euclidean_space) = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}" |
44133 | 906 |
|
907 |
lemma infnorm_set_image: |
|
53716 | 908 |
fixes x :: "'a::euclidean_space" |
56444 | 909 |
shows "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} = (\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
910 |
by blast |
44133 | 911 |
|
53716 | 912 |
lemma infnorm_Max: |
913 |
fixes x :: "'a::euclidean_space" |
|
56444 | 914 |
shows "infnorm x = Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis)" |
62343
24106dc44def
prefer abbreviations for compound operators INFIMUM and SUPREMUM
haftmann
parents:
61973
diff
changeset
|
915 |
by (simp add: infnorm_def infnorm_set_image cSup_eq_Max) |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
916 |
|
44133 | 917 |
lemma infnorm_set_lemma: |
53716 | 918 |
fixes x :: "'a::euclidean_space" |
56444 | 919 |
shows "finite {\<bar>x \<bullet> i\<bar> |i. i \<in> Basis}" |
920 |
and "{\<bar>x \<bullet> i\<bar> |i. i \<in> Basis} \<noteq> {}" |
|
44133 | 921 |
unfolding infnorm_set_image |
922 |
by auto |
|
923 |
||
53406 | 924 |
lemma infnorm_pos_le: |
925 |
fixes x :: "'a::euclidean_space" |
|
926 |
shows "0 \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
927 |
by (simp add: infnorm_Max Max_ge_iff ex_in_conv) |
44133 | 928 |
|
53406 | 929 |
lemma infnorm_triangle: |
930 |
fixes x :: "'a::euclidean_space" |
|
931 |
shows "infnorm (x + y) \<le> infnorm x + infnorm y" |
|
49522 | 932 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
933 |
have *: "\<And>a b c d :: real. \<bar>a\<bar> \<le> c \<Longrightarrow> \<bar>b\<bar> \<le> d \<Longrightarrow> \<bar>a + b\<bar> \<le> c + d" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
934 |
by simp |
44133 | 935 |
show ?thesis |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
936 |
by (auto simp: infnorm_Max inner_add_left intro!: *) |
44133 | 937 |
qed |
938 |
||
53406 | 939 |
lemma infnorm_eq_0: |
940 |
fixes x :: "'a::euclidean_space" |
|
941 |
shows "infnorm x = 0 \<longleftrightarrow> x = 0" |
|
49522 | 942 |
proof - |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
943 |
have "infnorm x \<le> 0 \<longleftrightarrow> x = 0" |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
944 |
unfolding infnorm_Max by (simp add: euclidean_all_zero_iff) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
945 |
then show ?thesis |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
946 |
using infnorm_pos_le[of x] by simp |
44133 | 947 |
qed |
948 |
||
949 |
lemma infnorm_0: "infnorm 0 = 0" |
|
950 |
by (simp add: infnorm_eq_0) |
|
951 |
||
952 |
lemma infnorm_neg: "infnorm (- x) = infnorm x" |
|
68062 | 953 |
unfolding infnorm_def by simp |
44133 | 954 |
|
955 |
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)" |
|
68062 | 956 |
by (metis infnorm_neg minus_diff_eq) |
957 |
||
958 |
lemma absdiff_infnorm: "\<bar>infnorm x - infnorm y\<bar> \<le> infnorm (x - y)" |
|
49522 | 959 |
proof - |
68062 | 960 |
have *: "\<And>(nx::real) n ny. nx \<le> n + ny \<Longrightarrow> ny \<le> n + nx \<Longrightarrow> \<bar>nx - ny\<bar> \<le> n" |
44133 | 961 |
by arith |
68062 | 962 |
show ?thesis |
963 |
proof (rule *) |
|
964 |
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"] |
|
965 |
show "infnorm x \<le> infnorm (x - y) + infnorm y" "infnorm y \<le> infnorm (x - y) + infnorm x" |
|
966 |
by (simp_all add: field_simps infnorm_neg) |
|
967 |
qed |
|
44133 | 968 |
qed |
969 |
||
53406 | 970 |
lemma real_abs_infnorm: "\<bar>infnorm x\<bar> = infnorm x" |
44133 | 971 |
using infnorm_pos_le[of x] by arith |
972 |
||
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
973 |
lemma Basis_le_infnorm: |
53406 | 974 |
fixes x :: "'a::euclidean_space" |
975 |
shows "b \<in> Basis \<Longrightarrow> \<bar>x \<bullet> b\<bar> \<le> infnorm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
976 |
by (simp add: infnorm_Max) |
44133 | 977 |
|
56444 | 978 |
lemma infnorm_mul: "infnorm (a *\<^sub>R x) = \<bar>a\<bar> * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
979 |
unfolding infnorm_Max |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
980 |
proof (safe intro!: Max_eqI) |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
981 |
let ?B = "(\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis" |
68062 | 982 |
{ fix b :: 'a |
53406 | 983 |
assume "b \<in> Basis" |
984 |
then show "\<bar>a *\<^sub>R x \<bullet> b\<bar> \<le> \<bar>a\<bar> * Max ?B" |
|
985 |
by (simp add: abs_mult mult_left_mono) |
|
986 |
next |
|
987 |
from Max_in[of ?B] obtain b where "b \<in> Basis" "Max ?B = \<bar>x \<bullet> b\<bar>" |
|
988 |
by (auto simp del: Max_in) |
|
989 |
then show "\<bar>a\<bar> * Max ((\<lambda>i. \<bar>x \<bullet> i\<bar>) ` Basis) \<in> (\<lambda>i. \<bar>a *\<^sub>R x \<bullet> i\<bar>) ` Basis" |
|
990 |
by (intro image_eqI[where x=b]) (auto simp: abs_mult) |
|
991 |
} |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
992 |
qed simp |
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
993 |
|
53406 | 994 |
lemma infnorm_mul_lemma: "infnorm (a *\<^sub>R x) \<le> \<bar>a\<bar> * infnorm x" |
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
995 |
unfolding infnorm_mul .. |
44133 | 996 |
|
997 |
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0" |
|
998 |
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith |
|
999 |
||
60420 | 1000 |
text \<open>Prove that it differs only up to a bound from Euclidean norm.\<close> |
44133 | 1001 |
|
1002 |
lemma infnorm_le_norm: "infnorm x \<le> norm x" |
|
51475
ebf9d4fd00ba
introduct the conditional_complete_lattice type class; generalize theorems about real Sup and Inf to it
hoelzl
parents:
50526
diff
changeset
|
1003 |
by (simp add: Basis_le_norm infnorm_Max) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
50105
diff
changeset
|
1004 |
|
53716 | 1005 |
lemma norm_le_infnorm: |
1006 |
fixes x :: "'a::euclidean_space" |
|
1007 |
shows "norm x \<le> sqrt DIM('a) * infnorm x" |
|
68062 | 1008 |
unfolding norm_eq_sqrt_inner id_def |
1009 |
proof (rule real_le_lsqrt[OF inner_ge_zero]) |
|
1010 |
show "sqrt DIM('a) * infnorm x \<ge> 0" |
|
44133 | 1011 |
by (simp add: zero_le_mult_iff infnorm_pos_le) |
68062 | 1012 |
have "x \<bullet> x \<le> (\<Sum>b\<in>Basis. x \<bullet> b * (x \<bullet> b))" |
1013 |
by (metis euclidean_inner order_refl) |
|
1014 |
also have "... \<le> DIM('a) * \<bar>infnorm x\<bar>\<^sup>2" |
|
1015 |
by (rule sum_bounded_above) (metis Basis_le_infnorm abs_le_square_iff power2_eq_square real_abs_infnorm) |
|
1016 |
also have "... \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" |
|
1017 |
by (simp add: power_mult_distrib) |
|
1018 |
finally show "x \<bullet> x \<le> (sqrt DIM('a) * infnorm x)\<^sup>2" . |
|
44133 | 1019 |
qed |
1020 |
||
44646 | 1021 |
lemma tendsto_infnorm [tendsto_intros]: |
61973 | 1022 |
assumes "(f \<longlongrightarrow> a) F" |
1023 |
shows "((\<lambda>x. infnorm (f x)) \<longlongrightarrow> infnorm a) F" |
|
44646 | 1024 |
proof (rule tendsto_compose [OF LIM_I assms]) |
53406 | 1025 |
fix r :: real |
1026 |
assume "r > 0" |
|
49522 | 1027 |
then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (infnorm x - infnorm a) < r" |
68062 | 1028 |
by (metis real_norm_def le_less_trans absdiff_infnorm infnorm_le_norm) |
44646 | 1029 |
qed |
1030 |
||
60420 | 1031 |
text \<open>Equality in Cauchy-Schwarz and triangle inequalities.\<close> |
44133 | 1032 |
|
53406 | 1033 |
lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
1034 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
68062 | 1035 |
proof (cases "x=0") |
1036 |
case True |
|
1037 |
then show ?thesis |
|
1038 |
by auto |
|
1039 |
next |
|
1040 |
case False |
|
1041 |
from inner_eq_zero_iff[of "norm y *\<^sub>R x - norm x *\<^sub>R y"] |
|
1042 |
have "?rhs \<longleftrightarrow> |
|
49522 | 1043 |
(norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - |
1044 |
norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) = 0)" |
|
68062 | 1045 |
using False unfolding inner_simps |
1046 |
by (auto simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) |
|
1047 |
also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" |
|
1048 |
using False by (simp add: field_simps inner_commute) |
|
1049 |
also have "\<dots> \<longleftrightarrow> ?lhs" |
|
1050 |
using False by auto |
|
1051 |
finally show ?thesis by metis |
|
44133 | 1052 |
qed |
1053 |
||
1054 |
lemma norm_cauchy_schwarz_abs_eq: |
|
56444 | 1055 |
"\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> |
53716 | 1056 |
norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm x *\<^sub>R y = - norm y *\<^sub>R x" |
53406 | 1057 |
(is "?lhs \<longleftrightarrow> ?rhs") |
49522 | 1058 |
proof - |
56444 | 1059 |
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> \<bar>x\<bar> = a \<longleftrightarrow> x = a \<or> x = - a" |
53406 | 1060 |
by arith |
44133 | 1061 |
have "?rhs \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)" |
1062 |
by simp |
|
68062 | 1063 |
also have "\<dots> \<longleftrightarrow> (x \<bullet> y = norm x * norm y \<or> (- x) \<bullet> y = norm x * norm y)" |
44133 | 1064 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
1065 |
unfolding norm_minus_cancel norm_scaleR .. |
|
1066 |
also have "\<dots> \<longleftrightarrow> ?lhs" |
|
53406 | 1067 |
unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps |
1068 |
by auto |
|
44133 | 1069 |
finally show ?thesis .. |
1070 |
qed |
|
1071 |
||
1072 |
lemma norm_triangle_eq: |
|
1073 |
fixes x y :: "'a::real_inner" |
|
53406 | 1074 |
shows "norm (x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
68062 | 1075 |
proof (cases "x = 0 \<or> y = 0") |
1076 |
case True |
|
1077 |
then show ?thesis |
|
1078 |
by force |
|
1079 |
next |
|
1080 |
case False |
|
1081 |
then have n: "norm x > 0" "norm y > 0" |
|
1082 |
by auto |
|
1083 |
have "norm (x + y) = norm x + norm y \<longleftrightarrow> (norm (x + y))\<^sup>2 = (norm x + norm y)\<^sup>2" |
|
1084 |
by simp |
|
1085 |
also have "\<dots> \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R x" |
|
1086 |
unfolding norm_cauchy_schwarz_eq[symmetric] |
|
1087 |
unfolding power2_norm_eq_inner inner_simps |
|
1088 |
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps) |
|
1089 |
finally show ?thesis . |
|
44133 | 1090 |
qed |
1091 |
||
49522 | 1092 |
|
60420 | 1093 |
subsection \<open>Collinearity\<close> |
44133 | 1094 |
|
67962 | 1095 |
definition%important collinear :: "'a::real_vector set \<Rightarrow> bool" |
49522 | 1096 |
where "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)" |
44133 | 1097 |
|
66287
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1098 |
lemma collinear_alt: |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1099 |
"collinear S \<longleftrightarrow> (\<exists>u v. \<forall>x \<in> S. \<exists>c. x = u + c *\<^sub>R v)" (is "?lhs = ?rhs") |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1100 |
proof |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1101 |
assume ?lhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1102 |
then show ?rhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1103 |
unfolding collinear_def by (metis Groups.add_ac(2) diff_add_cancel) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1104 |
next |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1105 |
assume ?rhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1106 |
then obtain u v where *: "\<And>x. x \<in> S \<Longrightarrow> \<exists>c. x = u + c *\<^sub>R v" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1107 |
by (auto simp: ) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1108 |
have "\<exists>c. x - y = c *\<^sub>R v" if "x \<in> S" "y \<in> S" for x y |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1109 |
by (metis *[OF \<open>x \<in> S\<close>] *[OF \<open>y \<in> S\<close>] scaleR_left.diff add_diff_cancel_left) |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1110 |
then show ?lhs |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1111 |
using collinear_def by blast |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1112 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1113 |
|
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1114 |
lemma collinear: |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1115 |
fixes S :: "'a::{perfect_space,real_vector} set" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1116 |
shows "collinear S \<longleftrightarrow> (\<exists>u. u \<noteq> 0 \<and> (\<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u))" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1117 |
proof - |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1118 |
have "\<exists>v. v \<noteq> 0 \<and> (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v)" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1119 |
if "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R u" "u=0" for u |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1120 |
proof - |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1121 |
have "\<forall>x\<in>S. \<forall>y\<in>S. x = y" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1122 |
using that by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1123 |
moreover |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1124 |
obtain v::'a where "v \<noteq> 0" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1125 |
using UNIV_not_singleton [of 0] by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1126 |
ultimately have "\<forall>x\<in>S. \<forall>y\<in>S. \<exists>c. x - y = c *\<^sub>R v" |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1127 |
by auto |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1128 |
then show ?thesis |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1129 |
using \<open>v \<noteq> 0\<close> by blast |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1130 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1131 |
then show ?thesis |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1132 |
apply (clarsimp simp: collinear_def) |
68072
493b818e8e10
added Johannes' generalizations Modules.thy and Vector_Spaces.thy; adapted HOL and HOL-Analysis accordingly
immler
parents:
67982
diff
changeset
|
1133 |
by (metis scaleR_zero_right vector_fraction_eq_iff) |
66287
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1134 |
qed |
005a30862ed0
new material: Colinearity, convex sets, polytopes
paulson <lp15@cam.ac.uk>
parents:
65680
diff
changeset
|
1135 |
|
63881
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
1136 |
lemma collinear_subset: "\<lbrakk>collinear T; S \<subseteq> T\<rbrakk> \<Longrightarrow> collinear S" |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
1137 |
by (meson collinear_def subsetCE) |
b746b19197bd
lots of new results about topology, affine dimension etc
paulson <lp15@cam.ac.uk>
parents:
63680
diff
changeset
|
1138 |
|
60762 | 1139 |
lemma collinear_empty [iff]: "collinear {}" |
53406 | 1140 |
by (simp add: collinear_def) |
44133 | 1141 |
|
60762 | 1142 |
lemma collinear_sing [iff]: "collinear {x}" |
44133 | 1143 |
by (simp add: collinear_def) |
1144 |
||
60762 | 1145 |
lemma collinear_2 [iff]: "collinear {x, y}" |
44133 | 1146 |
apply (simp add: collinear_def) |
1147 |
apply (rule exI[where x="x - y"]) |
|
68062 | 1148 |
by (metis minus_diff_eq scaleR_left.minus scaleR_one) |
44133 | 1149 |
|
56444 | 1150 |
lemma collinear_lemma: "collinear {0, x, y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R x)" |
53406 | 1151 |
(is "?lhs \<longleftrightarrow> ?rhs") |
68062 | 1152 |
proof (cases "x = 0 \<or> y = 0") |
1153 |
case True |
|
1154 |
then show ?thesis |
|
1155 |
by (auto simp: insert_commute) |
|
1156 |
next |
|
1157 |
case False |
|
1158 |
show ?thesis |
|
1159 |
proof |
|
1160 |
assume h: "?lhs" |
|
1161 |
then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *\<^sub>R u" |
|
1162 |
unfolding collinear_def by blast |
|
1163 |
from u[rule_format, of x 0] u[rule_format, of y 0] |
|
1164 |
obtain cx and cy where |
|
1165 |
cx: "x = cx *\<^sub>R u" and cy: "y = cy *\<^sub>R u" |
|
1166 |
by auto |
|
1167 |
from cx cy False have cx0: "cx \<noteq> 0" and cy0: "cy \<noteq> 0" by auto |
|
1168 |
let ?d = "cy / cx" |
|
1169 |
from cx cy cx0 have "y = ?d *\<^sub>R x" |
|
1170 |
by simp |
|
1171 |
then show ?rhs using False by blast |
|
1172 |
next |
|
1173 |
assume h: "?rhs" |
|
1174 |
then obtain c where c: "y = c *\<^sub>R x" |
|
1175 |
using False by blast |
|
1176 |
show ?lhs |
|
1177 |
unfolding collinear_def c |
|
1178 |
apply (rule exI[where x=x]) |
|
1179 |
apply auto |
|
1180 |
apply (rule exI[where x="- 1"], simp) |
|
1181 |
apply (rule exI[where x= "-c"], simp) |
|
44133 | 1182 |
apply (rule exI[where x=1], simp) |
68062 | 1183 |
apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib) |
1184 |
apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib) |
|
1185 |
done |
|
1186 |
qed |
|
44133 | 1187 |
qed |
1188 |
||
56444 | 1189 |
lemma norm_cauchy_schwarz_equal: "\<bar>x \<bullet> y\<bar> = norm x * norm y \<longleftrightarrow> collinear {0, x, y}" |
68062 | 1190 |
proof (cases "x=0") |
1191 |
case True |
|
1192 |
then show ?thesis |
|
1193 |
by (auto simp: insert_commute) |
|
1194 |
next |
|
1195 |
case False |
|
1196 |
then have nnz: "norm x \<noteq> 0" |
|
1197 |
by auto |
|
1198 |
show ?thesis |
|
1199 |
proof |
|
1200 |
assume "\<bar>x \<bullet> y\<bar> = norm x * norm y" |
|
1201 |
then show "collinear {0, x, y}" |
|
1202 |
unfolding norm_cauchy_schwarz_abs_eq collinear_lemma |
|
1203 |
by (meson eq_vector_fraction_iff nnz) |
|
1204 |
next |
|
1205 |
assume "collinear {0, x, y}" |
|
1206 |
with False show "\<bar>x \<bullet> y\<bar> = norm x * norm y" |
|
1207 |
unfolding norm_cauchy_schwarz_abs_eq collinear_lemma by (auto simp: abs_if) |
|
1208 |
qed |
|
1209 |
qed |
|
49522 | 1210 |
|
54776
db890d9fc5c2
ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
immler
parents:
54703
diff
changeset
|
1211 |
end |