author | paulson <lp15@cam.ac.uk> |
Tue, 25 Apr 2017 16:39:54 +0100 | |
changeset 65578 | e4997c181cce |
parent 65036 | ab7e11730ad8 |
child 65583 | 8d53b3bebab4 |
permissions | -rw-r--r-- |
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(* Title: HOL/Real_Vector_Spaces.thy |
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Author: Brian Huffman |
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Author: Johannes Hölzl |
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*) |
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section \<open>Vector Spaces and Algebras over the Reals\<close> |
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theory Real_Vector_Spaces |
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imports Real Topological_Spaces |
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begin |
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|
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subsection \<open>Locale for additive functions\<close> |
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|
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locale additive = |
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fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add" |
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assumes add: "f (x + y) = f x + f y" |
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begin |
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|
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lemma zero: "f 0 = 0" |
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proof - |
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have "f 0 = f (0 + 0)" by simp |
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also have "\<dots> = f 0 + f 0" by (rule add) |
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finally show "f 0 = 0" by simp |
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qed |
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formalization of vector spaces and algebras over the real numbers
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|
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lemma minus: "f (- x) = - f x" |
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proof - |
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formalization of vector spaces and algebras over the real numbers
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have "f (- x) + f x = f (- x + x)" by (rule add [symmetric]) |
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also have "\<dots> = - f x + f x" by (simp add: zero) |
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finally show "f (- x) = - f x" by (rule add_right_imp_eq) |
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qed |
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|
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lemma diff: "f (x - y) = f x - f y" |
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using add [of x "- y"] by (simp add: minus) |
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|
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lemma sum: "f (sum g A) = (\<Sum>x\<in>A. f (g x))" |
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by (induct A rule: infinite_finite_induct) (simp_all add: zero add) |
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|
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end |
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|
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|
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subsection \<open>Vector spaces\<close> |
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|
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locale vector_space = |
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fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" |
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assumes scale_right_distrib [algebra_simps]: "scale a (x + y) = scale a x + scale a y" |
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and scale_left_distrib [algebra_simps]: "scale (a + b) x = scale a x + scale b x" |
|
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and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x" |
|
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and scale_one [simp]: "scale 1 x = x" |
|
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begin |
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lemma scale_left_commute: "scale a (scale b x) = scale b (scale a x)" |
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by (simp add: mult.commute) |
|
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|
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lemma scale_zero_left [simp]: "scale 0 x = 0" |
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and scale_minus_left [simp]: "scale (- a) x = - (scale a x)" |
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and scale_left_diff_distrib [algebra_simps]: "scale (a - b) x = scale a x - scale b x" |
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and scale_sum_left: "scale (sum f A) x = (\<Sum>a\<in>A. scale (f a) x)" |
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proof - |
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interpret s: additive "\<lambda>a. scale a x" |
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by standard (rule scale_left_distrib) |
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show "scale 0 x = 0" by (rule s.zero) |
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show "scale (- a) x = - (scale a x)" by (rule s.minus) |
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show "scale (a - b) x = scale a x - scale b x" by (rule s.diff) |
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show "scale (sum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.sum) |
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qed |
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67 |
|
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lemma scale_zero_right [simp]: "scale a 0 = 0" |
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and scale_minus_right [simp]: "scale a (- x) = - (scale a x)" |
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and scale_right_diff_distrib [algebra_simps]: "scale a (x - y) = scale a x - scale a y" |
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and scale_sum_right: "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))" |
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72 |
proof - |
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interpret s: additive "\<lambda>x. scale a x" |
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by standard (rule scale_right_distrib) |
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show "scale a 0 = 0" by (rule s.zero) |
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show "scale a (- x) = - (scale a x)" by (rule s.minus) |
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show "scale a (x - y) = scale a x - scale a y" by (rule s.diff) |
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show "scale a (sum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.sum) |
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79 |
qed |
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80 |
|
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lemma scale_eq_0_iff [simp]: "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0" |
82 |
proof (cases "a = 0") |
|
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case True |
|
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then show ?thesis by simp |
|
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next |
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case False |
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have "x = 0" if "scale a x = 0" |
|
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proof - |
|
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from False that have "scale (inverse a) (scale a x) = 0" by simp |
|
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with False show ?thesis by simp |
|
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qed |
|
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then show ?thesis by force |
|
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93 |
qed |
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94 |
|
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lemma scale_left_imp_eq: |
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assumes nonzero: "a \<noteq> 0" |
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and scale: "scale a x = scale a y" |
|
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shows "x = y" |
|
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proof - |
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from scale have "scale a (x - y) = 0" |
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by (simp add: scale_right_diff_distrib) |
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with nonzero have "x - y = 0" by simp |
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then show "x = y" by (simp only: right_minus_eq) |
|
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104 |
qed |
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105 |
|
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lemma scale_right_imp_eq: |
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assumes nonzero: "x \<noteq> 0" |
108 |
and scale: "scale a x = scale b x" |
|
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shows "a = b" |
|
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110 |
proof - |
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from scale have "scale (a - b) x = 0" |
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by (simp add: scale_left_diff_distrib) |
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with nonzero have "a - b = 0" by simp |
114 |
then show "a = b" by (simp only: right_minus_eq) |
|
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115 |
qed |
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116 |
|
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lemma scale_cancel_left [simp]: "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0" |
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by (auto intro: scale_left_imp_eq) |
|
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119 |
|
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lemma scale_cancel_right [simp]: "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0" |
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by (auto intro: scale_right_imp_eq) |
|
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122 |
|
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123 |
end |
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|
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|
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subsection \<open>Real vector spaces\<close> |
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|
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class scaleR = |
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fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75) |
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begin |
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|
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abbreviation divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70) |
133 |
where "x /\<^sub>R r \<equiv> scaleR (inverse r) x" |
|
24748 | 134 |
|
135 |
end |
|
136 |
||
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class real_vector = scaleR + ab_group_add + |
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assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y" |
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and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x" |
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140 |
and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x" |
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141 |
and scaleR_one: "scaleR 1 x = x" |
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142 |
|
63545 | 143 |
interpretation real_vector: vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector" |
144 |
apply unfold_locales |
|
145 |
apply (rule scaleR_add_right) |
|
146 |
apply (rule scaleR_add_left) |
|
147 |
apply (rule scaleR_scaleR) |
|
148 |
apply (rule scaleR_one) |
|
149 |
done |
|
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150 |
|
60758 | 151 |
text \<open>Recover original theorem names\<close> |
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152 |
|
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153 |
lemmas scaleR_left_commute = real_vector.scale_left_commute |
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154 |
lemmas scaleR_zero_left = real_vector.scale_zero_left |
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lemmas scaleR_minus_left = real_vector.scale_minus_left |
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156 |
lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib |
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lemmas scaleR_sum_left = real_vector.scale_sum_left |
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lemmas scaleR_zero_right = real_vector.scale_zero_right |
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159 |
lemmas scaleR_minus_right = real_vector.scale_minus_right |
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160 |
lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib |
64267 | 161 |
lemmas scaleR_sum_right = real_vector.scale_sum_right |
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162 |
lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff |
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163 |
lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq |
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lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq |
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165 |
lemmas scaleR_cancel_left = real_vector.scale_cancel_left |
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diff
changeset
|
166 |
lemmas scaleR_cancel_right = real_vector.scale_cancel_right |
e93b121074fb
move real_vector class proofs into vector_space and group_hom locales
huffman
parents:
27553
diff
changeset
|
167 |
|
60758 | 168 |
text \<open>Legacy names\<close> |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
169 |
|
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
170 |
lemmas scaleR_left_distrib = scaleR_add_left |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
171 |
lemmas scaleR_right_distrib = scaleR_add_right |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
172 |
lemmas scaleR_left_diff_distrib = scaleR_diff_left |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
173 |
lemmas scaleR_right_diff_distrib = scaleR_diff_right |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
174 |
|
63545 | 175 |
lemma scaleR_minus1_left [simp]: "scaleR (-1) x = - x" |
176 |
for x :: "'a::real_vector" |
|
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
177 |
using scaleR_minus_left [of 1 x] by simp |
62101 | 178 |
|
64788
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
179 |
lemma scaleR_2: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
180 |
fixes x :: "'a::real_vector" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
181 |
shows "scaleR 2 x = x + x" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
182 |
unfolding one_add_one [symmetric] scaleR_left_distrib by simp |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
183 |
|
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
184 |
lemma scaleR_half_double [simp]: |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
185 |
fixes a :: "'a::real_vector" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
186 |
shows "(1 / 2) *\<^sub>R (a + a) = a" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
187 |
proof - |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
188 |
have "\<And>r. r *\<^sub>R (a + a) = (r * 2) *\<^sub>R a" |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
189 |
by (metis scaleR_2 scaleR_scaleR) |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
190 |
then show ?thesis |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
191 |
by simp |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
192 |
qed |
19f3d4af7a7d
New material about path connectedness, etc.
paulson <lp15@cam.ac.uk>
parents:
64272
diff
changeset
|
193 |
|
24588 | 194 |
class real_algebra = real_vector + ring + |
25062 | 195 |
assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)" |
63545 | 196 |
and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
197 |
|
24588 | 198 |
class real_algebra_1 = real_algebra + ring_1 |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
199 |
|
24588 | 200 |
class real_div_algebra = real_algebra_1 + division_ring |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
201 |
|
24588 | 202 |
class real_field = real_div_algebra + field |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
203 |
|
30069 | 204 |
instantiation real :: real_field |
205 |
begin |
|
206 |
||
63545 | 207 |
definition real_scaleR_def [simp]: "scaleR a x = a * x" |
30069 | 208 |
|
63545 | 209 |
instance |
210 |
by standard (simp_all add: algebra_simps) |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
211 |
|
30069 | 212 |
end |
213 |
||
63545 | 214 |
interpretation scaleR_left: additive "(\<lambda>a. scaleR a x :: 'a::real_vector)" |
215 |
by standard (rule scaleR_left_distrib) |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
216 |
|
63545 | 217 |
interpretation scaleR_right: additive "(\<lambda>x. scaleR a x :: 'a::real_vector)" |
218 |
by standard (rule scaleR_right_distrib) |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
219 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
220 |
lemma nonzero_inverse_scaleR_distrib: |
63545 | 221 |
"a \<noteq> 0 \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
222 |
for x :: "'a::real_div_algebra" |
|
223 |
by (rule inverse_unique) simp |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
224 |
|
63545 | 225 |
lemma inverse_scaleR_distrib: "inverse (scaleR a x) = scaleR (inverse a) (inverse x)" |
226 |
for x :: "'a::{real_div_algebra,division_ring}" |
|
227 |
apply (cases "a = 0") |
|
228 |
apply simp |
|
229 |
apply (cases "x = 0") |
|
230 |
apply simp |
|
231 |
apply (erule (1) nonzero_inverse_scaleR_distrib) |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
232 |
done |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
233 |
|
64267 | 234 |
lemma sum_constant_scaleR: "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y" |
63545 | 235 |
for y :: "'a::real_vector" |
63915 | 236 |
by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps) |
63545 | 237 |
|
63927
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
238 |
named_theorems vector_add_divide_simps "to simplify sums of scaled vectors" |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
239 |
|
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
240 |
lemma [vector_add_divide_simps]: |
63545 | 241 |
"v + (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" |
242 |
"a *\<^sub>R v + (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v + b *\<^sub>R w) /\<^sub>R z)" |
|
243 |
"(a / z) *\<^sub>R v + w = (if z = 0 then w else (a *\<^sub>R v + z *\<^sub>R w) /\<^sub>R z)" |
|
244 |
"(a / z) *\<^sub>R v + b *\<^sub>R w = (if z = 0 then b *\<^sub>R w else (a *\<^sub>R v + (b * z) *\<^sub>R w) /\<^sub>R z)" |
|
245 |
"v - (b / z) *\<^sub>R w = (if z = 0 then v else (z *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" |
|
246 |
"a *\<^sub>R v - (b / z) *\<^sub>R w = (if z = 0 then a *\<^sub>R v else ((a * z) *\<^sub>R v - b *\<^sub>R w) /\<^sub>R z)" |
|
247 |
"(a / z) *\<^sub>R v - w = (if z = 0 then -w else (a *\<^sub>R v - z *\<^sub>R w) /\<^sub>R z)" |
|
248 |
"(a / z) *\<^sub>R v - b *\<^sub>R w = (if z = 0 then -b *\<^sub>R w else (a *\<^sub>R v - (b * z) *\<^sub>R w) /\<^sub>R z)" |
|
249 |
for v :: "'a :: real_vector" |
|
250 |
by (simp_all add: divide_inverse_commute scaleR_add_right real_vector.scale_right_diff_distrib) |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63040
diff
changeset
|
251 |
|
63927
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
252 |
|
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
253 |
lemma eq_vector_fraction_iff [vector_add_divide_simps]: |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
254 |
fixes x :: "'a :: real_vector" |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
255 |
shows "(x = (u / v) *\<^sub>R a) \<longleftrightarrow> (if v=0 then x = 0 else v *\<^sub>R x = u *\<^sub>R a)" |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
256 |
by auto (metis (no_types) divide_eq_1_iff divide_inverse_commute scaleR_one scaleR_scaleR) |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
257 |
|
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
258 |
lemma vector_fraction_eq_iff [vector_add_divide_simps]: |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
259 |
fixes x :: "'a :: real_vector" |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
260 |
shows "((u / v) *\<^sub>R a = x) \<longleftrightarrow> (if v=0 then x = 0 else u *\<^sub>R a = v *\<^sub>R x)" |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
261 |
by (metis eq_vector_fraction_iff) |
0efb482beb84
vector_add_divide_simps now a "named theorems" bundle
paulson <lp15@cam.ac.uk>
parents:
63915
diff
changeset
|
262 |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
263 |
lemma real_vector_affinity_eq: |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
264 |
fixes x :: "'a :: real_vector" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
265 |
assumes m0: "m \<noteq> 0" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
266 |
shows "m *\<^sub>R x + c = y \<longleftrightarrow> x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" |
63545 | 267 |
(is "?lhs \<longleftrightarrow> ?rhs") |
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
268 |
proof |
63545 | 269 |
assume ?lhs |
270 |
then have "m *\<^sub>R x = y - c" by (simp add: field_simps) |
|
271 |
then have "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
272 |
then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
273 |
using m0 |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
274 |
by (simp add: real_vector.scale_right_diff_distrib) |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
275 |
next |
63545 | 276 |
assume ?rhs |
277 |
with m0 show "m *\<^sub>R x + c = y" |
|
278 |
by (simp add: real_vector.scale_right_diff_distrib) |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
279 |
qed |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
280 |
|
63545 | 281 |
lemma real_vector_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m *\<^sub>R x + c \<longleftrightarrow> inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x" |
282 |
for x :: "'a::real_vector" |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
283 |
using real_vector_affinity_eq[where m=m and x=x and y=y and c=c] |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
284 |
by metis |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
285 |
|
63545 | 286 |
lemma scaleR_eq_iff [simp]: "b + u *\<^sub>R a = a + u *\<^sub>R b \<longleftrightarrow> a = b \<or> u = 1" |
287 |
for a :: "'a::real_vector" |
|
288 |
proof (cases "u = 1") |
|
289 |
case True |
|
290 |
then show ?thesis by auto |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
291 |
next |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
292 |
case False |
63545 | 293 |
have "a = b" if "b + u *\<^sub>R a = a + u *\<^sub>R b" |
294 |
proof - |
|
295 |
from that have "(u - 1) *\<^sub>R a = (u - 1) *\<^sub>R b" |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
296 |
by (simp add: algebra_simps) |
63545 | 297 |
with False show ?thesis |
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
298 |
by auto |
63545 | 299 |
qed |
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
300 |
then show ?thesis by auto |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
301 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
302 |
|
63545 | 303 |
lemma scaleR_collapse [simp]: "(1 - u) *\<^sub>R a + u *\<^sub>R a = a" |
304 |
for a :: "'a::real_vector" |
|
305 |
by (simp add: algebra_simps) |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
306 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
307 |
|
63545 | 308 |
subsection \<open>Embedding of the Reals into any \<open>real_algebra_1\<close>: \<open>of_real\<close>\<close> |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
309 |
|
63545 | 310 |
definition of_real :: "real \<Rightarrow> 'a::real_algebra_1" |
311 |
where "of_real r = scaleR r 1" |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
312 |
|
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
313 |
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x" |
63545 | 314 |
by (simp add: of_real_def) |
20763 | 315 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
316 |
lemma of_real_0 [simp]: "of_real 0 = 0" |
63545 | 317 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
318 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
319 |
lemma of_real_1 [simp]: "of_real 1 = 1" |
63545 | 320 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
321 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
322 |
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" |
63545 | 323 |
by (simp add: of_real_def scaleR_left_distrib) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
324 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
325 |
lemma of_real_minus [simp]: "of_real (- x) = - of_real x" |
63545 | 326 |
by (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
327 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
328 |
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y" |
63545 | 329 |
by (simp add: of_real_def scaleR_left_diff_distrib) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
330 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
331 |
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" |
63545 | 332 |
by (simp add: of_real_def mult.commute) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
333 |
|
64267 | 334 |
lemma of_real_sum[simp]: "of_real (sum f s) = (\<Sum>x\<in>s. of_real (f x))" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
335 |
by (induct s rule: infinite_finite_induct) auto |
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
336 |
|
64272 | 337 |
lemma of_real_prod[simp]: "of_real (prod f s) = (\<Prod>x\<in>s. of_real (f x))" |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
338 |
by (induct s rule: infinite_finite_induct) auto |
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
339 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
340 |
lemma nonzero_of_real_inverse: |
63545 | 341 |
"x \<noteq> 0 \<Longrightarrow> of_real (inverse x) = inverse (of_real x :: 'a::real_div_algebra)" |
342 |
by (simp add: of_real_def nonzero_inverse_scaleR_distrib) |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
343 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
344 |
lemma of_real_inverse [simp]: |
63545 | 345 |
"of_real (inverse x) = inverse (of_real x :: 'a::{real_div_algebra,division_ring})" |
346 |
by (simp add: of_real_def inverse_scaleR_distrib) |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
347 |
|
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
348 |
lemma nonzero_of_real_divide: |
63545 | 349 |
"y \<noteq> 0 \<Longrightarrow> of_real (x / y) = (of_real x / of_real y :: 'a::real_field)" |
350 |
by (simp add: divide_inverse nonzero_of_real_inverse) |
|
20722 | 351 |
|
352 |
lemma of_real_divide [simp]: |
|
62131
1baed43f453e
nonneg_Reals, nonpos_Reals, Cauchy integral formula, etc.
paulson
parents:
62102
diff
changeset
|
353 |
"of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)" |
63545 | 354 |
by (simp add: divide_inverse) |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
355 |
|
20722 | 356 |
lemma of_real_power [simp]: |
31017 | 357 |
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n" |
63545 | 358 |
by (induct n) simp_all |
20722 | 359 |
|
63545 | 360 |
lemma of_real_eq_iff [simp]: "of_real x = of_real y \<longleftrightarrow> x = y" |
361 |
by (simp add: of_real_def) |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
362 |
|
63545 | 363 |
lemma inj_of_real: "inj of_real" |
38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
364 |
by (auto intro: injI) |
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37887
diff
changeset
|
365 |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
366 |
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified] |
65578
e4997c181cce
New material from PNT proof, as well as more default [simp] declarations. Also removed duplicate theorems about geometric series
paulson <lp15@cam.ac.uk>
parents:
65036
diff
changeset
|
367 |
lemmas of_real_eq_1_iff [simp] = of_real_eq_iff [of _ 1, simplified] |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
368 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
369 |
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)" |
63545 | 370 |
by (rule ext) (simp add: of_real_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
371 |
|
63545 | 372 |
text \<open>Collapse nested embeddings.\<close> |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
373 |
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" |
63545 | 374 |
by (induct n) auto |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
375 |
|
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
376 |
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" |
63545 | 377 |
by (cases z rule: int_diff_cases) simp |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
378 |
|
60155
91477b3a2d6b
Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents:
60026
diff
changeset
|
379 |
lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w" |
63545 | 380 |
using of_real_of_int_eq [of "numeral w"] by simp |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
381 |
|
60155
91477b3a2d6b
Tidying. Improved simplification for numerals, esp in exponents.
paulson <lp15@cam.ac.uk>
parents:
60026
diff
changeset
|
382 |
lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w" |
63545 | 383 |
using of_real_of_int_eq [of "- numeral w"] by simp |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
384 |
|
63545 | 385 |
text \<open>Every real algebra has characteristic zero.\<close> |
22912 | 386 |
instance real_algebra_1 < ring_char_0 |
387 |
proof |
|
63545 | 388 |
from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" |
389 |
by (rule inj_comp) |
|
390 |
then show "inj (of_nat :: nat \<Rightarrow> 'a)" |
|
391 |
by (simp add: comp_def) |
|
22912 | 392 |
qed |
393 |
||
63967
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
394 |
lemma fraction_scaleR_times [simp]: |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
395 |
fixes a :: "'a::real_algebra_1" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
396 |
shows "(numeral u / numeral v) *\<^sub>R (numeral w * a) = (numeral u * numeral w / numeral v) *\<^sub>R a" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
397 |
by (metis (no_types, lifting) of_real_numeral scaleR_conv_of_real scaleR_scaleR times_divide_eq_left) |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
398 |
|
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
399 |
lemma inverse_scaleR_times [simp]: |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
400 |
fixes a :: "'a::real_algebra_1" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
401 |
shows "(1 / numeral v) *\<^sub>R (numeral w * a) = (numeral w / numeral v) *\<^sub>R a" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
402 |
by (metis divide_inverse_commute inverse_eq_divide of_real_numeral scaleR_conv_of_real scaleR_scaleR) |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
403 |
|
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
404 |
lemma scaleR_times [simp]: |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
405 |
fixes a :: "'a::real_algebra_1" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
406 |
shows "(numeral u) *\<^sub>R (numeral w * a) = (numeral u * numeral w) *\<^sub>R a" |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
407 |
by (simp add: scaleR_conv_of_real) |
2aa42596edc3
new material on paths, etc. Also rationalisation
paulson <lp15@cam.ac.uk>
parents:
63927
diff
changeset
|
408 |
|
27553 | 409 |
instance real_field < field_char_0 .. |
410 |
||
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
411 |
|
60758 | 412 |
subsection \<open>The Set of Real Numbers\<close> |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
413 |
|
61070 | 414 |
definition Reals :: "'a::real_algebra_1 set" ("\<real>") |
415 |
where "\<real> = range of_real" |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
416 |
|
61070 | 417 |
lemma Reals_of_real [simp]: "of_real r \<in> \<real>" |
63545 | 418 |
by (simp add: Reals_def) |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
419 |
|
61070 | 420 |
lemma Reals_of_int [simp]: "of_int z \<in> \<real>" |
63545 | 421 |
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real) |
20718 | 422 |
|
61070 | 423 |
lemma Reals_of_nat [simp]: "of_nat n \<in> \<real>" |
63545 | 424 |
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real) |
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
425 |
|
61070 | 426 |
lemma Reals_numeral [simp]: "numeral w \<in> \<real>" |
63545 | 427 |
by (subst of_real_numeral [symmetric], rule Reals_of_real) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
428 |
|
61070 | 429 |
lemma Reals_0 [simp]: "0 \<in> \<real>" |
63545 | 430 |
apply (unfold Reals_def) |
431 |
apply (rule range_eqI) |
|
432 |
apply (rule of_real_0 [symmetric]) |
|
433 |
done |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
434 |
|
61070 | 435 |
lemma Reals_1 [simp]: "1 \<in> \<real>" |
63545 | 436 |
apply (unfold Reals_def) |
437 |
apply (rule range_eqI) |
|
438 |
apply (rule of_real_1 [symmetric]) |
|
439 |
done |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
440 |
|
63545 | 441 |
lemma Reals_add [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a + b \<in> \<real>" |
442 |
apply (auto simp add: Reals_def) |
|
443 |
apply (rule range_eqI) |
|
444 |
apply (rule of_real_add [symmetric]) |
|
445 |
done |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
446 |
|
61070 | 447 |
lemma Reals_minus [simp]: "a \<in> \<real> \<Longrightarrow> - a \<in> \<real>" |
63545 | 448 |
apply (auto simp add: Reals_def) |
449 |
apply (rule range_eqI) |
|
450 |
apply (rule of_real_minus [symmetric]) |
|
451 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
452 |
|
63545 | 453 |
lemma Reals_diff [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a - b \<in> \<real>" |
454 |
apply (auto simp add: Reals_def) |
|
455 |
apply (rule range_eqI) |
|
456 |
apply (rule of_real_diff [symmetric]) |
|
457 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
458 |
|
63545 | 459 |
lemma Reals_mult [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a * b \<in> \<real>" |
460 |
apply (auto simp add: Reals_def) |
|
461 |
apply (rule range_eqI) |
|
462 |
apply (rule of_real_mult [symmetric]) |
|
463 |
done |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
464 |
|
63545 | 465 |
lemma nonzero_Reals_inverse: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> inverse a \<in> \<real>" |
466 |
for a :: "'a::real_div_algebra" |
|
467 |
apply (auto simp add: Reals_def) |
|
468 |
apply (rule range_eqI) |
|
469 |
apply (erule nonzero_of_real_inverse [symmetric]) |
|
470 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
471 |
|
63545 | 472 |
lemma Reals_inverse: "a \<in> \<real> \<Longrightarrow> inverse a \<in> \<real>" |
473 |
for a :: "'a::{real_div_algebra,division_ring}" |
|
474 |
apply (auto simp add: Reals_def) |
|
475 |
apply (rule range_eqI) |
|
476 |
apply (rule of_real_inverse [symmetric]) |
|
477 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
478 |
|
63545 | 479 |
lemma Reals_inverse_iff [simp]: "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>" |
480 |
for x :: "'a::{real_div_algebra,division_ring}" |
|
481 |
by (metis Reals_inverse inverse_inverse_eq) |
|
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
482 |
|
63545 | 483 |
lemma nonzero_Reals_divide: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a / b \<in> \<real>" |
484 |
for a b :: "'a::real_field" |
|
485 |
apply (auto simp add: Reals_def) |
|
486 |
apply (rule range_eqI) |
|
487 |
apply (erule nonzero_of_real_divide [symmetric]) |
|
488 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
489 |
|
63545 | 490 |
lemma Reals_divide [simp]: "a \<in> \<real> \<Longrightarrow> b \<in> \<real> \<Longrightarrow> a / b \<in> \<real>" |
491 |
for a b :: "'a::{real_field,field}" |
|
492 |
apply (auto simp add: Reals_def) |
|
493 |
apply (rule range_eqI) |
|
494 |
apply (rule of_real_divide [symmetric]) |
|
495 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
496 |
|
63545 | 497 |
lemma Reals_power [simp]: "a \<in> \<real> \<Longrightarrow> a ^ n \<in> \<real>" |
498 |
for a :: "'a::real_algebra_1" |
|
499 |
apply (auto simp add: Reals_def) |
|
500 |
apply (rule range_eqI) |
|
501 |
apply (rule of_real_power [symmetric]) |
|
502 |
done |
|
20722 | 503 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
504 |
lemma Reals_cases [cases set: Reals]: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
505 |
assumes "q \<in> \<real>" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
506 |
obtains (of_real) r where "q = of_real r" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
507 |
unfolding Reals_def |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
508 |
proof - |
60758 | 509 |
from \<open>q \<in> \<real>\<close> have "q \<in> range of_real" unfolding Reals_def . |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
510 |
then obtain r where "q = of_real r" .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
511 |
then show thesis .. |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
512 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
513 |
|
64267 | 514 |
lemma sum_in_Reals [intro,simp]: "(\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>) \<Longrightarrow> sum f s \<in> \<real>" |
63915 | 515 |
proof (induct s rule: infinite_finite_induct) |
516 |
case infinite |
|
64267 | 517 |
then show ?case by (metis Reals_0 sum.infinite) |
63915 | 518 |
qed simp_all |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
519 |
|
64272 | 520 |
lemma prod_in_Reals [intro,simp]: "(\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>) \<Longrightarrow> prod f s \<in> \<real>" |
63915 | 521 |
proof (induct s rule: infinite_finite_induct) |
522 |
case infinite |
|
64272 | 523 |
then show ?case by (metis Reals_1 prod.infinite) |
63915 | 524 |
qed simp_all |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
525 |
|
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
526 |
lemma Reals_induct [case_names of_real, induct set: Reals]: |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
527 |
"q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q" |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
528 |
by (rule Reals_cases) auto |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
529 |
|
63545 | 530 |
|
60758 | 531 |
subsection \<open>Ordered real vector spaces\<close> |
54778 | 532 |
|
533 |
class ordered_real_vector = real_vector + ordered_ab_group_add + |
|
534 |
assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y" |
|
63545 | 535 |
and scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x" |
54778 | 536 |
begin |
537 |
||
63545 | 538 |
lemma scaleR_mono: "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y" |
539 |
apply (erule scaleR_right_mono [THEN order_trans]) |
|
540 |
apply assumption |
|
541 |
apply (erule scaleR_left_mono) |
|
542 |
apply assumption |
|
543 |
done |
|
54778 | 544 |
|
63545 | 545 |
lemma scaleR_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d" |
54778 | 546 |
by (rule scaleR_mono) (auto intro: order.trans) |
547 |
||
54785 | 548 |
lemma pos_le_divideRI: |
549 |
assumes "0 < c" |
|
63545 | 550 |
and "c *\<^sub>R a \<le> b" |
54785 | 551 |
shows "a \<le> b /\<^sub>R c" |
552 |
proof - |
|
553 |
from scaleR_left_mono[OF assms(2)] assms(1) |
|
554 |
have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c" |
|
555 |
by simp |
|
556 |
with assms show ?thesis |
|
557 |
by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) |
|
558 |
qed |
|
559 |
||
560 |
lemma pos_le_divideR_eq: |
|
561 |
assumes "0 < c" |
|
562 |
shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b" |
|
63545 | 563 |
(is "?lhs \<longleftrightarrow> ?rhs") |
564 |
proof |
|
565 |
assume ?lhs |
|
566 |
from scaleR_left_mono[OF this] assms have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)" |
|
54785 | 567 |
by simp |
63545 | 568 |
with assms show ?rhs |
54785 | 569 |
by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide) |
63545 | 570 |
next |
571 |
assume ?rhs |
|
572 |
with assms show ?lhs by (rule pos_le_divideRI) |
|
573 |
qed |
|
54785 | 574 |
|
63545 | 575 |
lemma scaleR_image_atLeastAtMost: "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}" |
54785 | 576 |
apply (auto intro!: scaleR_left_mono) |
577 |
apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI) |
|
63545 | 578 |
apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one) |
54785 | 579 |
done |
580 |
||
54778 | 581 |
end |
582 |
||
60303 | 583 |
lemma neg_le_divideR_eq: |
584 |
fixes a :: "'a :: ordered_real_vector" |
|
585 |
assumes "c < 0" |
|
586 |
shows "a \<le> b /\<^sub>R c \<longleftrightarrow> b \<le> c *\<^sub>R a" |
|
63545 | 587 |
using pos_le_divideR_eq [of "-c" a "-b"] assms by simp |
60303 | 588 |
|
63545 | 589 |
lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> a *\<^sub>R x" |
590 |
for x :: "'a::ordered_real_vector" |
|
591 |
using scaleR_left_mono [of 0 x a] by simp |
|
54778 | 592 |
|
63545 | 593 |
lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> x \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0" |
594 |
for x :: "'a::ordered_real_vector" |
|
54778 | 595 |
using scaleR_left_mono [of x 0 a] by simp |
596 |
||
63545 | 597 |
lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> 0" |
598 |
for x :: "'a::ordered_real_vector" |
|
54778 | 599 |
using scaleR_right_mono [of a 0 x] by simp |
600 |
||
63545 | 601 |
lemma split_scaleR_neg_le: "(0 \<le> a \<and> x \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> x) \<Longrightarrow> a *\<^sub>R x \<le> 0" |
602 |
for x :: "'a::ordered_real_vector" |
|
54778 | 603 |
by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg) |
604 |
||
63545 | 605 |
lemma le_add_iff1: "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d" |
606 |
for c d e :: "'a::ordered_real_vector" |
|
54778 | 607 |
by (simp add: algebra_simps) |
608 |
||
63545 | 609 |
lemma le_add_iff2: "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d" |
610 |
for c d e :: "'a::ordered_real_vector" |
|
54778 | 611 |
by (simp add: algebra_simps) |
612 |
||
63545 | 613 |
lemma scaleR_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b" |
614 |
for a b :: "'a::ordered_real_vector" |
|
54778 | 615 |
apply (drule scaleR_left_mono [of _ _ "- c"]) |
63545 | 616 |
apply simp_all |
54778 | 617 |
done |
618 |
||
63545 | 619 |
lemma scaleR_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c" |
620 |
for c :: "'a::ordered_real_vector" |
|
54778 | 621 |
apply (drule scaleR_right_mono [of _ _ "- c"]) |
63545 | 622 |
apply simp_all |
54778 | 623 |
done |
624 |
||
63545 | 625 |
lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b" |
626 |
for b :: "'a::ordered_real_vector" |
|
627 |
using scaleR_right_mono_neg [of a 0 b] by simp |
|
54778 | 628 |
|
63545 | 629 |
lemma split_scaleR_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b" |
630 |
for b :: "'a::ordered_real_vector" |
|
54778 | 631 |
by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos) |
632 |
||
633 |
lemma zero_le_scaleR_iff: |
|
63545 | 634 |
fixes b :: "'a::ordered_real_vector" |
635 |
shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0" |
|
636 |
(is "?lhs = ?rhs") |
|
637 |
proof (cases "a = 0") |
|
638 |
case True |
|
639 |
then show ?thesis by simp |
|
640 |
next |
|
641 |
case False |
|
54778 | 642 |
show ?thesis |
643 |
proof |
|
63545 | 644 |
assume ?lhs |
645 |
from \<open>a \<noteq> 0\<close> consider "a > 0" | "a < 0" by arith |
|
646 |
then show ?rhs |
|
647 |
proof cases |
|
648 |
case 1 |
|
649 |
with \<open>?lhs\<close> have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)" |
|
54778 | 650 |
by (intro scaleR_mono) auto |
63545 | 651 |
with 1 show ?thesis |
54778 | 652 |
by simp |
63545 | 653 |
next |
654 |
case 2 |
|
655 |
with \<open>?lhs\<close> have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)" |
|
54778 | 656 |
by (intro scaleR_mono) auto |
63545 | 657 |
with 2 show ?thesis |
54778 | 658 |
by simp |
63545 | 659 |
qed |
660 |
next |
|
661 |
assume ?rhs |
|
662 |
then show ?lhs |
|
663 |
by (auto simp: not_le \<open>a \<noteq> 0\<close> intro!: split_scaleR_pos_le) |
|
664 |
qed |
|
665 |
qed |
|
54778 | 666 |
|
63545 | 667 |
lemma scaleR_le_0_iff: "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0" |
668 |
for b::"'a::ordered_real_vector" |
|
54778 | 669 |
by (insert zero_le_scaleR_iff [of "-a" b]) force |
670 |
||
63545 | 671 |
lemma scaleR_le_cancel_left: "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)" |
672 |
for b :: "'a::ordered_real_vector" |
|
54778 | 673 |
by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg |
63545 | 674 |
dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"]) |
54778 | 675 |
|
63545 | 676 |
lemma scaleR_le_cancel_left_pos: "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b" |
677 |
for b :: "'a::ordered_real_vector" |
|
54778 | 678 |
by (auto simp: scaleR_le_cancel_left) |
679 |
||
63545 | 680 |
lemma scaleR_le_cancel_left_neg: "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a" |
681 |
for b :: "'a::ordered_real_vector" |
|
54778 | 682 |
by (auto simp: scaleR_le_cancel_left) |
683 |
||
63545 | 684 |
lemma scaleR_left_le_one_le: "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x" |
685 |
for x :: "'a::ordered_real_vector" and a :: real |
|
54778 | 686 |
using scaleR_right_mono[of a 1 x] by simp |
687 |
||
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
688 |
|
60758 | 689 |
subsection \<open>Real normed vector spaces\<close> |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
690 |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
691 |
class dist = |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
692 |
fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
693 |
|
29608 | 694 |
class norm = |
22636 | 695 |
fixes norm :: "'a \<Rightarrow> real" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
696 |
|
24520 | 697 |
class sgn_div_norm = scaleR + norm + sgn + |
25062 | 698 |
assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x" |
24506 | 699 |
|
31289 | 700 |
class dist_norm = dist + norm + minus + |
701 |
assumes dist_norm: "dist x y = norm (x - y)" |
|
702 |
||
62101 | 703 |
class uniformity_dist = dist + uniformity + |
704 |
assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})" |
|
705 |
begin |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
706 |
|
62101 | 707 |
lemma eventually_uniformity_metric: |
708 |
"eventually P uniformity \<longleftrightarrow> (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))" |
|
709 |
unfolding uniformity_dist |
|
710 |
by (subst eventually_INF_base) |
|
711 |
(auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"]) |
|
712 |
||
713 |
end |
|
714 |
||
715 |
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity + |
|
51002
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
716 |
assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0" |
63545 | 717 |
and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y" |
718 |
and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x" |
|
51002
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
719 |
begin |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
720 |
|
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
721 |
lemma norm_ge_zero [simp]: "0 \<le> norm x" |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
722 |
proof - |
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
723 |
have "0 = norm (x + -1 *\<^sub>R x)" |
51002
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
724 |
using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one) |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
725 |
also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq) |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
726 |
finally show ?thesis by simp |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
727 |
qed |
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
728 |
|
496013a6eb38
remove unnecessary assumption from real_normed_vector
hoelzl
parents:
50999
diff
changeset
|
729 |
end |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
730 |
|
24588 | 731 |
class real_normed_algebra = real_algebra + real_normed_vector + |
25062 | 732 |
assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
733 |
|
24588 | 734 |
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra + |
25062 | 735 |
assumes norm_one [simp]: "norm 1 = 1" |
62101 | 736 |
|
63545 | 737 |
lemma (in real_normed_algebra_1) scaleR_power [simp]: "(scaleR x y) ^ n = scaleR (x^n) (y^n)" |
738 |
by (induct n) (simp_all add: scaleR_one scaleR_scaleR mult_ac) |
|
22852 | 739 |
|
24588 | 740 |
class real_normed_div_algebra = real_div_algebra + real_normed_vector + |
25062 | 741 |
assumes norm_mult: "norm (x * y) = norm x * norm y" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
742 |
|
24588 | 743 |
class real_normed_field = real_field + real_normed_div_algebra |
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
744 |
|
22852 | 745 |
instance real_normed_div_algebra < real_normed_algebra_1 |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
746 |
proof |
63545 | 747 |
show "norm (x * y) \<le> norm x * norm y" for x y :: 'a |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
748 |
by (simp add: norm_mult) |
22852 | 749 |
next |
750 |
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)" |
|
751 |
by (rule norm_mult) |
|
63545 | 752 |
then show "norm (1::'a) = 1" by simp |
20554
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
753 |
qed |
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
parents:
20551
diff
changeset
|
754 |
|
22852 | 755 |
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" |
63545 | 756 |
by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
757 |
|
63545 | 758 |
lemma zero_less_norm_iff [simp]: "norm x > 0 \<longleftrightarrow> x \<noteq> 0" |
759 |
for x :: "'a::real_normed_vector" |
|
760 |
by (simp add: order_less_le) |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
761 |
|
63545 | 762 |
lemma norm_not_less_zero [simp]: "\<not> norm x < 0" |
763 |
for x :: "'a::real_normed_vector" |
|
764 |
by (simp add: linorder_not_less) |
|
20828 | 765 |
|
63545 | 766 |
lemma norm_le_zero_iff [simp]: "norm x \<le> 0 \<longleftrightarrow> x = 0" |
767 |
for x :: "'a::real_normed_vector" |
|
768 |
by (simp add: order_le_less) |
|
20828 | 769 |
|
63545 | 770 |
lemma norm_minus_cancel [simp]: "norm (- x) = norm x" |
771 |
for x :: "'a::real_normed_vector" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
772 |
proof - |
21809
4b93e949ac33
remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents:
21404
diff
changeset
|
773 |
have "norm (- x) = norm (scaleR (- 1) x)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
774 |
by (simp only: scaleR_minus_left scaleR_one) |
20533 | 775 |
also have "\<dots> = \<bar>- 1\<bar> * norm x" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
776 |
by (rule norm_scaleR) |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
777 |
finally show ?thesis by simp |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
778 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
779 |
|
63545 | 780 |
lemma norm_minus_commute: "norm (a - b) = norm (b - a)" |
781 |
for a b :: "'a::real_normed_vector" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
782 |
proof - |
22898 | 783 |
have "norm (- (b - a)) = norm (b - a)" |
784 |
by (rule norm_minus_cancel) |
|
63545 | 785 |
then show ?thesis by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
786 |
qed |
63545 | 787 |
|
788 |
lemma dist_add_cancel [simp]: "dist (a + b) (a + c) = dist b c" |
|
789 |
for a :: "'a::real_normed_vector" |
|
790 |
by (simp add: dist_norm) |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63040
diff
changeset
|
791 |
|
63545 | 792 |
lemma dist_add_cancel2 [simp]: "dist (b + a) (c + a) = dist b c" |
793 |
for a :: "'a::real_normed_vector" |
|
794 |
by (simp add: dist_norm) |
|
63114
27afe7af7379
Lots of new material for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
63040
diff
changeset
|
795 |
|
63545 | 796 |
lemma dist_scaleR [simp]: "dist (x *\<^sub>R a) (y *\<^sub>R a) = \<bar>x - y\<bar> * norm a" |
797 |
for a :: "'a::real_normed_vector" |
|
798 |
by (metis dist_norm norm_scaleR scaleR_left.diff) |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
799 |
|
63545 | 800 |
lemma norm_uminus_minus: "norm (- x - y :: 'a :: real_normed_vector) = norm (x + y)" |
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
801 |
by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
802 |
|
63545 | 803 |
lemma norm_triangle_ineq2: "norm a - norm b \<le> norm (a - b)" |
804 |
for a b :: "'a::real_normed_vector" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
805 |
proof - |
20533 | 806 |
have "norm (a - b + b) \<le> norm (a - b) + norm b" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
807 |
by (rule norm_triangle_ineq) |
63545 | 808 |
then show ?thesis by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
809 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
810 |
|
63545 | 811 |
lemma norm_triangle_ineq3: "\<bar>norm a - norm b\<bar> \<le> norm (a - b)" |
812 |
for a b :: "'a::real_normed_vector" |
|
813 |
apply (subst abs_le_iff) |
|
814 |
apply auto |
|
815 |
apply (rule norm_triangle_ineq2) |
|
816 |
apply (subst norm_minus_commute) |
|
817 |
apply (rule norm_triangle_ineq2) |
|
818 |
done |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
819 |
|
63545 | 820 |
lemma norm_triangle_ineq4: "norm (a - b) \<le> norm a + norm b" |
821 |
for a b :: "'a::real_normed_vector" |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
822 |
proof - |
22898 | 823 |
have "norm (a + - b) \<le> norm a + norm (- b)" |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
824 |
by (rule norm_triangle_ineq) |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset
|
825 |
then show ?thesis by simp |
22898 | 826 |
qed |
827 |
||
63545 | 828 |
lemma norm_diff_ineq: "norm a - norm b \<le> norm (a + b)" |
829 |
for a b :: "'a::real_normed_vector" |
|
22898 | 830 |
proof - |
831 |
have "norm a - norm (- b) \<le> norm (a - - b)" |
|
832 |
by (rule norm_triangle_ineq2) |
|
63545 | 833 |
then show ?thesis by simp |
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
834 |
qed |
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
835 |
|
63545 | 836 |
lemma norm_add_leD: "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c" |
837 |
for a b :: "'a::real_normed_vector" |
|
838 |
by (metis add.commute diff_le_eq norm_diff_ineq order.trans) |
|
61762
d50b993b4fb9
Removal of redundant lemmas (diff_less_iff, diff_le_iff) and of the abbreviation Exp. Addition of some new material.
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
839 |
|
63545 | 840 |
lemma norm_diff_triangle_ineq: "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)" |
841 |
for a b c d :: "'a::real_normed_vector" |
|
20551 | 842 |
proof - |
843 |
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset
|
844 |
by (simp add: algebra_simps) |
20551 | 845 |
also have "\<dots> \<le> norm (a - c) + norm (b - d)" |
846 |
by (rule norm_triangle_ineq) |
|
847 |
finally show ?thesis . |
|
848 |
qed |
|
849 |
||
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
850 |
lemma norm_diff_triangle_le: |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
851 |
fixes x y z :: "'a::real_normed_vector" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
852 |
assumes "norm (x - y) \<le> e1" "norm (y - z) \<le> e2" |
63545 | 853 |
shows "norm (x - z) \<le> e1 + e2" |
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
854 |
using norm_diff_triangle_ineq [of x y y z] assms by simp |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
855 |
|
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
856 |
lemma norm_diff_triangle_less: |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
857 |
fixes x y z :: "'a::real_normed_vector" |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
858 |
assumes "norm (x - y) < e1" "norm (y - z) < e2" |
63545 | 859 |
shows "norm (x - z) < e1 + e2" |
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
860 |
using norm_diff_triangle_ineq [of x y y z] assms by simp |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
861 |
|
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
862 |
lemma norm_triangle_mono: |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
863 |
fixes a b :: "'a::real_normed_vector" |
63545 | 864 |
shows "norm a \<le> r \<Longrightarrow> norm b \<le> s \<Longrightarrow> norm (a + b) \<le> r + s" |
865 |
by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans) |
|
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
866 |
|
64267 | 867 |
lemma norm_sum: |
56194 | 868 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
64267 | 869 |
shows "norm (sum f A) \<le> (\<Sum>i\<in>A. norm (f i))" |
56194 | 870 |
by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono) |
871 |
||
64267 | 872 |
lemma sum_norm_le: |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56194
diff
changeset
|
873 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56194
diff
changeset
|
874 |
assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x" |
64267 | 875 |
shows "norm (sum f S) \<le> sum g S" |
876 |
by (rule order_trans [OF norm_sum sum_mono]) (simp add: fg) |
|
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56194
diff
changeset
|
877 |
|
63545 | 878 |
lemma abs_norm_cancel [simp]: "\<bar>norm a\<bar> = norm a" |
879 |
for a :: "'a::real_normed_vector" |
|
880 |
by (rule abs_of_nonneg [OF norm_ge_zero]) |
|
22857 | 881 |
|
63545 | 882 |
lemma norm_add_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x + y) < r + s" |
883 |
for x y :: "'a::real_normed_vector" |
|
884 |
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono]) |
|
22880 | 885 |
|
63545 | 886 |
lemma norm_mult_less: "norm x < r \<Longrightarrow> norm y < s \<Longrightarrow> norm (x * y) < r * s" |
887 |
for x y :: "'a::real_normed_algebra" |
|
888 |
by (rule order_le_less_trans [OF norm_mult_ineq]) (simp add: mult_strict_mono') |
|
22880 | 889 |
|
63545 | 890 |
lemma norm_of_real [simp]: "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>" |
891 |
by (simp add: of_real_def) |
|
20560 | 892 |
|
63545 | 893 |
lemma norm_numeral [simp]: "norm (numeral w::'a::real_normed_algebra_1) = numeral w" |
894 |
by (subst of_real_numeral [symmetric], subst norm_of_real, simp) |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46868
diff
changeset
|
895 |
|
63545 | 896 |
lemma norm_neg_numeral [simp]: "norm (- numeral w::'a::real_normed_algebra_1) = numeral w" |
897 |
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp) |
|
22876
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
898 |
|
63545 | 899 |
lemma norm_of_real_add1 [simp]: "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = \<bar>x + 1\<bar>" |
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
900 |
by (metis norm_of_real of_real_1 of_real_add) |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
901 |
|
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
902 |
lemma norm_of_real_addn [simp]: |
63545 | 903 |
"norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = \<bar>x + numeral b\<bar>" |
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
904 |
by (metis norm_of_real of_real_add of_real_numeral) |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
905 |
|
63545 | 906 |
lemma norm_of_int [simp]: "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>" |
907 |
by (subst of_real_of_int_eq [symmetric], rule norm_of_real) |
|
22876
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
908 |
|
63545 | 909 |
lemma norm_of_nat [simp]: "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n" |
910 |
apply (subst of_real_of_nat_eq [symmetric]) |
|
911 |
apply (subst norm_of_real, simp) |
|
912 |
done |
|
22876
2b4c831ceca7
add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents:
22857
diff
changeset
|
913 |
|
63545 | 914 |
lemma nonzero_norm_inverse: "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)" |
915 |
for a :: "'a::real_normed_div_algebra" |
|
916 |
apply (rule inverse_unique [symmetric]) |
|
917 |
apply (simp add: norm_mult [symmetric]) |
|
918 |
done |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
919 |
|
63545 | 920 |
lemma norm_inverse: "norm (inverse a) = inverse (norm a)" |
921 |
for a :: "'a::{real_normed_div_algebra,division_ring}" |
|
922 |
apply (cases "a = 0") |
|
923 |
apply simp |
|
924 |
apply (erule nonzero_norm_inverse) |
|
925 |
done |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
926 |
|
63545 | 927 |
lemma nonzero_norm_divide: "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b" |
928 |
for a b :: "'a::real_normed_field" |
|
929 |
by (simp add: divide_inverse norm_mult nonzero_norm_inverse) |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
930 |
|
63545 | 931 |
lemma norm_divide: "norm (a / b) = norm a / norm b" |
932 |
for a b :: "'a::{real_normed_field,field}" |
|
933 |
by (simp add: divide_inverse norm_mult norm_inverse) |
|
20584
60b1d52a455d
added classes real_div_algebra and real_field; added lemmas
huffman
parents:
20560
diff
changeset
|
934 |
|
63545 | 935 |
lemma norm_power_ineq: "norm (x ^ n) \<le> norm x ^ n" |
936 |
for x :: "'a::real_normed_algebra_1" |
|
22852 | 937 |
proof (induct n) |
63545 | 938 |
case 0 |
939 |
show "norm (x ^ 0) \<le> norm x ^ 0" by simp |
|
22852 | 940 |
next |
941 |
case (Suc n) |
|
942 |
have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)" |
|
943 |
by (rule norm_mult_ineq) |
|
944 |
also from Suc have "\<dots> \<le> norm x * norm x ^ n" |
|
945 |
using norm_ge_zero by (rule mult_left_mono) |
|
946 |
finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n" |
|
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents:
30242
diff
changeset
|
947 |
by simp |
22852 | 948 |
qed |
949 |
||
63545 | 950 |
lemma norm_power: "norm (x ^ n) = norm x ^ n" |
951 |
for x :: "'a::real_normed_div_algebra" |
|
952 |
by (induct n) (simp_all add: norm_mult) |
|
20684 | 953 |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
954 |
lemma power_eq_imp_eq_norm: |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
955 |
fixes w :: "'a::real_normed_div_algebra" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
956 |
assumes eq: "w ^ n = z ^ n" and "n > 0" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
957 |
shows "norm w = norm z" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
958 |
proof - |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
959 |
have "norm w ^ n = norm z ^ n" |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
960 |
by (metis (no_types) eq norm_power) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
961 |
then show ?thesis |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
962 |
using assms by (force intro: power_eq_imp_eq_base) |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
963 |
qed |
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
964 |
|
63545 | 965 |
lemma norm_mult_numeral1 [simp]: "norm (numeral w * a) = numeral w * norm a" |
966 |
for a b :: "'a::{real_normed_field,field}" |
|
967 |
by (simp add: norm_mult) |
|
60762 | 968 |
|
63545 | 969 |
lemma norm_mult_numeral2 [simp]: "norm (a * numeral w) = norm a * numeral w" |
970 |
for a b :: "'a::{real_normed_field,field}" |
|
971 |
by (simp add: norm_mult) |
|
60762 | 972 |
|
63545 | 973 |
lemma norm_divide_numeral [simp]: "norm (a / numeral w) = norm a / numeral w" |
974 |
for a b :: "'a::{real_normed_field,field}" |
|
975 |
by (simp add: norm_divide) |
|
60762 | 976 |
|
977 |
lemma norm_of_real_diff [simp]: |
|
63545 | 978 |
"norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \<le> \<bar>b - a\<bar>" |
60762 | 979 |
by (metis norm_of_real of_real_diff order_refl) |
980 |
||
63545 | 981 |
text \<open>Despite a superficial resemblance, \<open>norm_eq_1\<close> is not relevant.\<close> |
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
982 |
lemma square_norm_one: |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
983 |
fixes x :: "'a::real_normed_div_algebra" |
63545 | 984 |
assumes "x\<^sup>2 = 1" |
985 |
shows "norm x = 1" |
|
59613
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
986 |
by (metis assms norm_minus_cancel norm_one power2_eq_1_iff) |
7103019278f0
The function frac. Various lemmas about limits, series, the exp function, etc.
paulson <lp15@cam.ac.uk>
parents:
59587
diff
changeset
|
987 |
|
63545 | 988 |
lemma norm_less_p1: "norm x < norm (of_real (norm x) + 1 :: 'a)" |
989 |
for x :: "'a::real_normed_algebra_1" |
|
59658
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
990 |
proof - |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
991 |
have "norm x < norm (of_real (norm x + 1) :: 'a)" |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
992 |
by (simp add: of_real_def) |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
993 |
then show ?thesis |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
994 |
by simp |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
995 |
qed |
0cc388370041
sin, cos generalised from type real to any "'a::{real_normed_field,banach}", including complex
paulson <lp15@cam.ac.uk>
parents:
59613
diff
changeset
|
996 |
|
64272 | 997 |
lemma prod_norm: "prod (\<lambda>x. norm (f x)) A = norm (prod f A)" |
63545 | 998 |
for f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
999 |
by (induct A rule: infinite_finite_induct) (auto simp: norm_mult) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1000 |
|
64272 | 1001 |
lemma norm_prod_le: |
1002 |
"norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1,comm_monoid_mult}))" |
|
63545 | 1003 |
proof (induct A rule: infinite_finite_induct) |
1004 |
case empty |
|
1005 |
then show ?case by simp |
|
1006 |
next |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1007 |
case (insert a A) |
64272 | 1008 |
then have "norm (prod f (insert a A)) \<le> norm (f a) * norm (prod f A)" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1009 |
by (simp add: norm_mult_ineq) |
64272 | 1010 |
also have "norm (prod f A) \<le> (\<Prod>a\<in>A. norm (f a))" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1011 |
by (rule insert) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1012 |
finally show ?case |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1013 |
by (simp add: insert mult_left_mono) |
63545 | 1014 |
next |
1015 |
case infinite |
|
1016 |
then show ?case by simp |
|
1017 |
qed |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1018 |
|
64272 | 1019 |
lemma norm_prod_diff: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1020 |
fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1021 |
shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow> |
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1022 |
norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1023 |
proof (induction I rule: infinite_finite_induct) |
63545 | 1024 |
case empty |
1025 |
then show ?case by simp |
|
1026 |
next |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1027 |
case (insert i I) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1028 |
note insert.hyps[simp] |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1029 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1030 |
have "norm ((\<Prod>i\<in>insert i I. z i) - (\<Prod>i\<in>insert i I. w i)) = |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1031 |
norm ((\<Prod>i\<in>I. z i) * (z i - w i) + ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * w i)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1032 |
(is "_ = norm (?t1 + ?t2)") |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1033 |
by (auto simp add: field_simps) |
63545 | 1034 |
also have "\<dots> \<le> norm ?t1 + norm ?t2" |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1035 |
by (rule norm_triangle_ineq) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1036 |
also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1037 |
by (rule norm_mult_ineq) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1038 |
also have "\<dots> \<le> (\<Prod>i\<in>I. norm (z i)) * norm(z i - w i)" |
64272 | 1039 |
by (rule mult_right_mono) (auto intro: norm_prod_le) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1040 |
also have "(\<Prod>i\<in>I. norm (z i)) \<le> (\<Prod>i\<in>I. 1)" |
64272 | 1041 |
by (intro prod_mono) (auto intro!: insert) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1042 |
also have "norm ?t2 \<le> norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * norm (w i)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1043 |
by (rule norm_mult_ineq) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1044 |
also have "norm (w i) \<le> 1" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1045 |
by (auto intro: insert) |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1046 |
also have "norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1047 |
using insert by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1048 |
finally show ?case |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1049 |
by (auto simp add: ac_simps mult_right_mono mult_left_mono) |
63545 | 1050 |
next |
1051 |
case infinite |
|
1052 |
then show ?case by simp |
|
1053 |
qed |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1054 |
|
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1055 |
lemma norm_power_diff: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1056 |
fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1057 |
assumes "norm z \<le> 1" "norm w \<le> 1" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1058 |
shows "norm (z^m - w^m) \<le> m * norm (z - w)" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1059 |
proof - |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1060 |
have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))" |
64272 | 1061 |
by (simp add: prod_constant) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1062 |
also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))" |
64272 | 1063 |
by (intro norm_prod_diff) (auto simp add: assms) |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1064 |
also have "\<dots> = m * norm (z - w)" |
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:
61531
diff
changeset
|
1065 |
by simp |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1066 |
finally show ?thesis . |
55719
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
1067 |
qed |
cdddd073bff8
Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents:
54890
diff
changeset
|
1068 |
|
63545 | 1069 |
|
60758 | 1070 |
subsection \<open>Metric spaces\<close> |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1071 |
|
62101 | 1072 |
class metric_space = uniformity_dist + open_uniformity + |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1073 |
assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y" |
63545 | 1074 |
and dist_triangle2: "dist x y \<le> dist x z + dist y z" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1075 |
begin |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1076 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1077 |
lemma dist_self [simp]: "dist x x = 0" |
63545 | 1078 |
by simp |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1079 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1080 |
lemma zero_le_dist [simp]: "0 \<le> dist x y" |
63545 | 1081 |
using dist_triangle2 [of x x y] by simp |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1082 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1083 |
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y" |
63545 | 1084 |
by (simp add: less_le) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1085 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1086 |
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0" |
63545 | 1087 |
by (simp add: not_less) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1088 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1089 |
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y" |
63545 | 1090 |
by (simp add: le_less) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1091 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1092 |
lemma dist_commute: "dist x y = dist y x" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1093 |
proof (rule order_antisym) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1094 |
show "dist x y \<le> dist y x" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1095 |
using dist_triangle2 [of x y x] by simp |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1096 |
show "dist y x \<le> dist x y" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1097 |
using dist_triangle2 [of y x y] by simp |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1098 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1099 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1100 |
lemma dist_commute_lessI: "dist y x < e \<Longrightarrow> dist x y < e" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1101 |
by (simp add: dist_commute) |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1102 |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1103 |
lemma dist_triangle: "dist x z \<le> dist x y + dist y z" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1104 |
using dist_triangle2 [of x z y] by (simp add: dist_commute) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1105 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1106 |
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y" |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1107 |
using dist_triangle2 [of x y a] by (simp add: dist_commute) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1108 |
|
63545 | 1109 |
lemma dist_pos_lt: "x \<noteq> y \<Longrightarrow> 0 < dist x y" |
1110 |
by (simp add: zero_less_dist_iff) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1111 |
|
63545 | 1112 |
lemma dist_nz: "x \<noteq> y \<longleftrightarrow> 0 < dist x y" |
1113 |
by (simp add: zero_less_dist_iff) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1114 |
|
62087
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
62049
diff
changeset
|
1115 |
declare dist_nz [symmetric, simp] |
44841d07ef1d
revisions to limits and derivatives, plus new lemmas
paulson
parents:
62049
diff
changeset
|
1116 |
|
63545 | 1117 |
lemma dist_triangle_le: "dist x z + dist y z \<le> e \<Longrightarrow> dist x y \<le> e" |
1118 |
by (rule order_trans [OF dist_triangle2]) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1119 |
|
63545 | 1120 |
lemma dist_triangle_lt: "dist x z + dist y z < e \<Longrightarrow> dist x y < e" |
1121 |
by (rule le_less_trans [OF dist_triangle2]) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1122 |
|
63545 | 1123 |
lemma dist_triangle_less_add: "dist x1 y < e1 \<Longrightarrow> dist x2 y < e2 \<Longrightarrow> dist x1 x2 < e1 + e2" |
1124 |
by (rule dist_triangle_lt [where z=y]) simp |
|
62948
7700f467892b
lots of new theorems for multivariate analysis
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
1125 |
|
63545 | 1126 |
lemma dist_triangle_half_l: "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e" |
1127 |
by (rule dist_triangle_lt [where z=y]) simp |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1128 |
|
63545 | 1129 |
lemma dist_triangle_half_r: "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e" |
1130 |
by (rule dist_triangle_half_l) (simp_all add: dist_commute) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1131 |
|
65036
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1132 |
lemma dist_triangle_third: |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1133 |
assumes "dist x1 x2 < e/3" "dist x2 x3 < e/3" "dist x3 x4 < e/3" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1134 |
shows "dist x1 x4 < e" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1135 |
proof - |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1136 |
have "dist x1 x3 < e/3 + e/3" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1137 |
by (metis assms(1) assms(2) dist_commute dist_triangle_less_add) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1138 |
then have "dist x1 x4 < (e/3 + e/3) + e/3" |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1139 |
by (metis assms(3) dist_commute dist_triangle_less_add) |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1140 |
then show ?thesis |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1141 |
by simp |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1142 |
qed |
ab7e11730ad8
Some new lemmas. Existing lemmas modified to use uniform_limit rather than its expansion
paulson <lp15@cam.ac.uk>
parents:
64788
diff
changeset
|
1143 |
|
62101 | 1144 |
subclass uniform_space |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1145 |
proof |
63545 | 1146 |
fix E x |
1147 |
assume "eventually E uniformity" |
|
62101 | 1148 |
then obtain e where E: "0 < e" "\<And>x y. dist x y < e \<Longrightarrow> E (x, y)" |
63545 | 1149 |
by (auto simp: eventually_uniformity_metric) |
62101 | 1150 |
then show "E (x, x)" "\<forall>\<^sub>F (x, y) in uniformity. E (y, x)" |
63545 | 1151 |
by (auto simp: eventually_uniformity_metric dist_commute) |
62101 | 1152 |
show "\<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))" |
63545 | 1153 |
using E dist_triangle_half_l[where e=e] |
1154 |
unfolding eventually_uniformity_metric |
|
62101 | 1155 |
by (intro exI[of _ "\<lambda>(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI) |
63545 | 1156 |
(auto simp: dist_commute) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1157 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1158 |
|
62101 | 1159 |
lemma open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)" |
63545 | 1160 |
by (simp add: dist_commute open_uniformity eventually_uniformity_metric) |
62101 | 1161 |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1162 |
lemma open_ball: "open {y. dist x y < d}" |
63545 | 1163 |
unfolding open_dist |
1164 |
proof (intro ballI) |
|
1165 |
fix y |
|
1166 |
assume *: "y \<in> {y. dist x y < d}" |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1167 |
then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1168 |
by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1169 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1170 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1171 |
subclass first_countable_topology |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1172 |
proof |
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1173 |
fix x |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1174 |
show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1175 |
proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"]) |
63545 | 1176 |
fix S |
1177 |
assume "open S" "x \<in> S" |
|
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
52381
diff
changeset
|
1178 |
then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1179 |
by (auto simp: open_dist subset_eq dist_commute) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1180 |
moreover |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
52381
diff
changeset
|
1181 |
from e obtain i where "inverse (Suc i) < e" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1182 |
by (auto dest!: reals_Archimedean) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1183 |
then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1184 |
by auto |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1185 |
ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1186 |
by blast |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1187 |
qed (auto intro: open_ball) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1188 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1189 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1190 |
end |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1191 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1192 |
instance metric_space \<subseteq> t2_space |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1193 |
proof |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1194 |
fix x y :: "'a::metric_space" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1195 |
assume xy: "x \<noteq> y" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1196 |
let ?U = "{y'. dist x y' < dist x y / 2}" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1197 |
let ?V = "{x'. dist y x' < dist x y / 2}" |
63545 | 1198 |
have *: "d x z \<le> d x y + d y z \<Longrightarrow> d y z = d z y \<Longrightarrow> \<not> (d x y * 2 < d x z \<and> d z y * 2 < d x z)" |
1199 |
for d :: "'a \<Rightarrow> 'a \<Rightarrow> real" and x y z :: 'a |
|
1200 |
by arith |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1201 |
have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}" |
63545 | 1202 |
using dist_pos_lt[OF xy] *[of dist, OF dist_triangle dist_commute] |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1203 |
using open_ball[of _ "dist x y / 2"] by auto |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1204 |
then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1205 |
by blast |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1206 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1207 |
|
60758 | 1208 |
text \<open>Every normed vector space is a metric space.\<close> |
31289 | 1209 |
instance real_normed_vector < metric_space |
1210 |
proof |
|
63545 | 1211 |
fix x y z :: 'a |
1212 |
show "dist x y = 0 \<longleftrightarrow> x = y" |
|
1213 |
by (simp add: dist_norm) |
|
1214 |
show "dist x y \<le> dist x z + dist y z" |
|
1215 |
using norm_triangle_ineq4 [of "x - z" "y - z"] by (simp add: dist_norm) |
|
31289 | 1216 |
qed |
31285
0a3f9ee4117c
generalize dist function to class real_normed_vector
huffman
parents:
31017
diff
changeset
|
1217 |
|
63545 | 1218 |
|
60758 | 1219 |
subsection \<open>Class instances for real numbers\<close> |
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1220 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1221 |
instantiation real :: real_normed_field |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1222 |
begin |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1223 |
|
63545 | 1224 |
definition dist_real_def: "dist x y = \<bar>x - y\<bar>" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1225 |
|
62101 | 1226 |
definition uniformity_real_def [code del]: |
1227 |
"(uniformity :: (real \<times> real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})" |
|
1228 |
||
52381
63eec9cea2c7
pragmatic executability for instance real :: open
haftmann
parents:
51775
diff
changeset
|
1229 |
definition open_real_def [code del]: |
62101 | 1230 |
"open (U :: real set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1231 |
|
63545 | 1232 |
definition real_norm_def [simp]: "norm r = \<bar>r\<bar>" |
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1233 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1234 |
instance |
63545 | 1235 |
apply intro_classes |
1236 |
apply (unfold real_norm_def real_scaleR_def) |
|
1237 |
apply (rule dist_real_def) |
|
1238 |
apply (simp add: sgn_real_def) |
|
1239 |
apply (rule uniformity_real_def) |
|
1240 |
apply (rule open_real_def) |
|
1241 |
apply (rule abs_eq_0) |
|
1242 |
apply (rule abs_triangle_ineq) |
|
1243 |
apply (rule abs_mult) |
|
1244 |
apply (rule abs_mult) |
|
1245 |
done |
|
31564
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1246 |
|
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1247 |
end |
d2abf6f6f619
subsection for real instances; new lemmas for open sets of reals
huffman
parents:
31494
diff
changeset
|
1248 |
|
62102
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1249 |
declare uniformity_Abort[where 'a=real, code] |
877463945ce9
fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents:
62101
diff
changeset
|
1250 |
|
63545 | 1251 |
lemma dist_of_real [simp]: "dist (of_real x :: 'a) (of_real y) = dist x y" |
1252 |
for a :: "'a::real_normed_div_algebra" |
|
1253 |
by (metis dist_norm norm_of_real of_real_diff real_norm_def) |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
1254 |
|
54890
cb892d835803
fundamental treatment of undefined vs. universally partial replaces code_abort
haftmann
parents:
54863
diff
changeset
|
1255 |
declare [[code abort: "open :: real set \<Rightarrow> bool"]] |
52381
63eec9cea2c7
pragmatic executability for instance real :: open
haftmann
parents:
51775
diff
changeset
|
1256 |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1257 |
instance real :: linorder_topology |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1258 |
proof |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1259 |
show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1260 |
proof (rule ext, safe) |
63545 | 1261 |
fix S :: "real set" |
1262 |
assume "open S" |
|
53381 | 1263 |
then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)" |
62101 | 1264 |
unfolding open_dist bchoice_iff .. |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1265 |
then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1266 |
by (fastforce simp: dist_real_def) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1267 |
show "generate_topology (range lessThan \<union> range greaterThan) S" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1268 |
apply (subst *) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1269 |
apply (intro generate_topology_Union generate_topology.Int) |
63545 | 1270 |
apply (auto intro: generate_topology.Basis) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1271 |
done |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1272 |
next |
63545 | 1273 |
fix S :: "real set" |
1274 |
assume "generate_topology (range lessThan \<union> range greaterThan) S" |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1275 |
moreover have "\<And>a::real. open {..<a}" |
62101 | 1276 |
unfolding open_dist dist_real_def |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1277 |
proof clarify |
63545 | 1278 |
fix x a :: real |
1279 |
assume "x < a" |
|
1280 |
then have "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto |
|
1281 |
then show "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" .. |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1282 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1283 |
moreover have "\<And>a::real. open {a <..}" |
62101 | 1284 |
unfolding open_dist dist_real_def |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1285 |
proof clarify |
63545 | 1286 |
fix x a :: real |
1287 |
assume "a < x" |
|
1288 |
then have "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto |
|
1289 |
then show "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" .. |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1290 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1291 |
ultimately show "open S" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1292 |
by induct auto |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1293 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1294 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1295 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
1296 |
instance real :: linear_continuum_topology .. |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
1297 |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1298 |
lemmas open_real_greaterThan = open_greaterThan[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1299 |
lemmas open_real_lessThan = open_lessThan[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1300 |
lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1301 |
lemmas closed_real_atMost = closed_atMost[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1302 |
lemmas closed_real_atLeast = closed_atLeast[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1303 |
lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1304 |
|
63545 | 1305 |
|
60758 | 1306 |
subsection \<open>Extra type constraints\<close> |
31446 | 1307 |
|
61799 | 1308 |
text \<open>Only allow @{term "open"} in class \<open>topological_space\<close>.\<close> |
60758 | 1309 |
setup \<open>Sign.add_const_constraint |
1310 |
(@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close> |
|
31492
5400beeddb55
replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents:
31490
diff
changeset
|
1311 |
|
62101 | 1312 |
text \<open>Only allow @{term "uniformity"} in class \<open>uniform_space\<close>.\<close> |
1313 |
setup \<open>Sign.add_const_constraint |
|
1314 |
(@{const_name "uniformity"}, SOME @{typ "('a::uniformity \<times> 'a) filter"})\<close> |
|
1315 |
||
61799 | 1316 |
text \<open>Only allow @{term dist} in class \<open>metric_space\<close>.\<close> |
60758 | 1317 |
setup \<open>Sign.add_const_constraint |
1318 |
(@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"})\<close> |
|
31446 | 1319 |
|
61799 | 1320 |
text \<open>Only allow @{term norm} in class \<open>real_normed_vector\<close>.\<close> |
60758 | 1321 |
setup \<open>Sign.add_const_constraint |
1322 |
(@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"})\<close> |
|
31446 | 1323 |
|
63545 | 1324 |
|
60758 | 1325 |
subsection \<open>Sign function\<close> |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1326 |
|
63545 | 1327 |
lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)" |
1328 |
for x :: "'a::real_normed_vector" |
|
1329 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1330 |
|
63545 | 1331 |
lemma sgn_zero [simp]: "sgn (0::'a::real_normed_vector) = 0" |
1332 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1333 |
|
63545 | 1334 |
lemma sgn_zero_iff: "sgn x = 0 \<longleftrightarrow> x = 0" |
1335 |
for x :: "'a::real_normed_vector" |
|
1336 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1337 |
|
63545 | 1338 |
lemma sgn_minus: "sgn (- x) = - sgn x" |
1339 |
for x :: "'a::real_normed_vector" |
|
1340 |
by (simp add: sgn_div_norm) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1341 |
|
63545 | 1342 |
lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)" |
1343 |
for x :: "'a::real_normed_vector" |
|
1344 |
by (simp add: sgn_div_norm ac_simps) |
|
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
1345 |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1346 |
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1" |
63545 | 1347 |
by (simp add: sgn_div_norm) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1348 |
|
63545 | 1349 |
lemma sgn_of_real: "sgn (of_real r :: 'a::real_normed_algebra_1) = of_real (sgn r)" |
1350 |
unfolding of_real_def by (simp only: sgn_scaleR sgn_one) |
|
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1351 |
|
63545 | 1352 |
lemma sgn_mult: "sgn (x * y) = sgn x * sgn y" |
1353 |
for x y :: "'a::real_normed_div_algebra" |
|
1354 |
by (simp add: sgn_div_norm norm_mult mult.commute) |
|
22973
64d300e16370
add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents:
22972
diff
changeset
|
1355 |
|
64240 | 1356 |
hide_fact (open) sgn_mult |
1357 |
||
63545 | 1358 |
lemma real_sgn_eq: "sgn x = x / \<bar>x\<bar>" |
1359 |
for x :: real |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1360 |
by (simp add: sgn_div_norm divide_inverse) |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1361 |
|
63545 | 1362 |
lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> x" |
1363 |
for x :: real |
|
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
1364 |
by (cases "0::real" x rule: linorder_cases) simp_all |
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1365 |
|
63545 | 1366 |
lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> x \<le> 0" |
1367 |
for x :: real |
|
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56479
diff
changeset
|
1368 |
by (cases "0::real" x rule: linorder_cases) simp_all |
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1369 |
|
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51472
diff
changeset
|
1370 |
lemma norm_conv_dist: "norm x = dist x 0" |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51472
diff
changeset
|
1371 |
unfolding dist_norm by simp |
22972
3e96b98d37c6
generalized sgn function to work on any real normed vector space
huffman
parents:
22942
diff
changeset
|
1372 |
|
62379
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
1373 |
declare norm_conv_dist [symmetric, simp] |
340738057c8c
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
paulson <lp15@cam.ac.uk>
parents:
62368
diff
changeset
|
1374 |
|
63545 | 1375 |
lemma dist_0_norm [simp]: "dist 0 x = norm x" |
1376 |
for x :: "'a::real_normed_vector" |
|
1377 |
by (simp add: dist_norm) |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62379
diff
changeset
|
1378 |
|
60307
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
1379 |
lemma dist_diff [simp]: "dist a (a - b) = norm b" "dist (a - b) a = norm b" |
75e1aa7a450e
Convex hulls: theorems about interior, etc. And a few simple lemmas.
paulson <lp15@cam.ac.uk>
parents:
60303
diff
changeset
|
1380 |
by (simp_all add: dist_norm) |
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:
61531
diff
changeset
|
1381 |
|
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1382 |
lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \<bar>m - n\<bar>" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1383 |
proof - |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1384 |
have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1385 |
by simp |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1386 |
also have "\<dots> = of_int \<bar>m - n\<bar>" by (subst dist_diff, subst norm_of_int) simp |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1387 |
finally show ?thesis . |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1388 |
qed |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1389 |
|
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:
61531
diff
changeset
|
1390 |
lemma dist_of_nat: |
61524
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1391 |
"dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \<bar>int m - int n\<bar>" |
f2e51e704a96
added many small lemmas about setsum/setprod/powr/...
eberlm
parents:
61169
diff
changeset
|
1392 |
by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int) |
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:
61531
diff
changeset
|
1393 |
|
63545 | 1394 |
|
60758 | 1395 |
subsection \<open>Bounded Linear and Bilinear Operators\<close> |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1396 |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1397 |
locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" + |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1398 |
assumes scaleR: "f (scaleR r x) = scaleR r (f x)" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1399 |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
1400 |
lemma linear_imp_scaleR: |
63545 | 1401 |
assumes "linear D" |
1402 |
obtains d where "D = (\<lambda>x. x *\<^sub>R d)" |
|
60800
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
1403 |
by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def) |
7d04351c795a
New material for Cauchy's integral theorem
paulson <lp15@cam.ac.uk>
parents:
60762
diff
changeset
|
1404 |
|
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1405 |
corollary real_linearD: |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1406 |
fixes f :: "real \<Rightarrow> real" |
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1407 |
assumes "linear f" obtains c where "f = op* c" |
63545 | 1408 |
by (rule linear_imp_scaleR [OF assms]) (force simp: scaleR_conv_of_real) |
62533
bc25f3916a99
new material to Blochj's theorem, as well as supporting lemmas
paulson <lp15@cam.ac.uk>
parents:
62397
diff
changeset
|
1409 |
|
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1410 |
lemma linearI: |
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1411 |
assumes "\<And>x y. f (x + y) = f x + f y" |
63545 | 1412 |
and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1413 |
shows "linear f" |
61169 | 1414 |
by standard (rule assms)+ |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1415 |
|
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53381
diff
changeset
|
1416 |
locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" + |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1417 |
assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K" |
27443 | 1418 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1419 |
|
63545 | 1420 |
lemma pos_bounded: "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1421 |
proof - |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1422 |
obtain K where K: "\<And>x. norm (f x) \<le> norm x * K" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1423 |
using bounded by blast |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1424 |
show ?thesis |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1425 |
proof (intro exI impI conjI allI) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1426 |
show "0 < max 1 K" |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54785
diff
changeset
|
1427 |
by (rule order_less_le_trans [OF zero_less_one max.cobounded1]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1428 |
next |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1429 |
fix x |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1430 |
have "norm (f x) \<le> norm x * K" using K . |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1431 |
also have "\<dots> \<le> norm x * max 1 K" |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54785
diff
changeset
|
1432 |
by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1433 |
finally show "norm (f x) \<le> norm x * max 1 K" . |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1434 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1435 |
qed |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1436 |
|
63545 | 1437 |
lemma nonneg_bounded: "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K" |
1438 |
using pos_bounded by (auto intro: order_less_imp_le) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1439 |
|
63545 | 1440 |
lemma linear: "linear f" |
63469
b6900858dcb9
lots of new theorems about differentiable_on, retracts, ANRs, etc.
paulson <lp15@cam.ac.uk>
parents:
63128
diff
changeset
|
1441 |
by (fact local.linear_axioms) |
56369
2704ca85be98
moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents:
56194
diff
changeset
|
1442 |
|
27443 | 1443 |
end |
1444 |
||
44127 | 1445 |
lemma bounded_linear_intro: |
1446 |
assumes "\<And>x y. f (x + y) = f x + f y" |
|
63545 | 1447 |
and "\<And>r x. f (scaleR r x) = scaleR r (f x)" |
1448 |
and "\<And>x. norm (f x) \<le> norm x * K" |
|
44127 | 1449 |
shows "bounded_linear f" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1450 |
by standard (blast intro: assms)+ |
44127 | 1451 |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1452 |
locale bounded_bilinear = |
63545 | 1453 |
fixes prod :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector \<Rightarrow> 'c::real_normed_vector" |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1454 |
(infixl "**" 70) |
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1455 |
assumes add_left: "prod (a + a') b = prod a b + prod a' b" |
63545 | 1456 |
and add_right: "prod a (b + b') = prod a b + prod a b'" |
1457 |
and scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)" |
|
1458 |
and scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)" |
|
1459 |
and bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" |
|
27443 | 1460 |
begin |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1461 |
|
63545 | 1462 |
lemma pos_bounded: "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
1463 |
apply (insert bounded) |
|
1464 |
apply (erule exE) |
|
1465 |
apply (rule_tac x="max 1 K" in exI) |
|
1466 |
apply safe |
|
1467 |
apply (rule order_less_le_trans [OF zero_less_one max.cobounded1]) |
|
1468 |
apply (drule spec) |
|
1469 |
apply (drule spec) |
|
1470 |
apply (erule order_trans) |
|
1471 |
apply (rule mult_left_mono [OF max.cobounded2]) |
|
1472 |
apply (intro mult_nonneg_nonneg norm_ge_zero) |
|
1473 |
done |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1474 |
|
63545 | 1475 |
lemma nonneg_bounded: "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K" |
1476 |
using pos_bounded by (auto intro: order_less_imp_le) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1477 |
|
27443 | 1478 |
lemma additive_right: "additive (\<lambda>b. prod a b)" |
63545 | 1479 |
by (rule additive.intro, rule add_right) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1480 |
|
27443 | 1481 |
lemma additive_left: "additive (\<lambda>a. prod a b)" |
63545 | 1482 |
by (rule additive.intro, rule add_left) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1483 |
|
27443 | 1484 |
lemma zero_left: "prod 0 b = 0" |
63545 | 1485 |
by (rule additive.zero [OF additive_left]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1486 |
|
27443 | 1487 |
lemma zero_right: "prod a 0 = 0" |
63545 | 1488 |
by (rule additive.zero [OF additive_right]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1489 |
|
27443 | 1490 |
lemma minus_left: "prod (- a) b = - prod a b" |
63545 | 1491 |
by (rule additive.minus [OF additive_left]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1492 |
|
27443 | 1493 |
lemma minus_right: "prod a (- b) = - prod a b" |
63545 | 1494 |
by (rule additive.minus [OF additive_right]) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1495 |
|
63545 | 1496 |
lemma diff_left: "prod (a - a') b = prod a b - prod a' b" |
1497 |
by (rule additive.diff [OF additive_left]) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1498 |
|
63545 | 1499 |
lemma diff_right: "prod a (b - b') = prod a b - prod a b'" |
1500 |
by (rule additive.diff [OF additive_right]) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1501 |
|
64267 | 1502 |
lemma sum_left: "prod (sum g S) x = sum ((\<lambda>i. prod (g i) x)) S" |
1503 |
by (rule additive.sum [OF additive_left]) |
|
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1504 |
|
64267 | 1505 |
lemma sum_right: "prod x (sum g S) = sum ((\<lambda>i. (prod x (g i)))) S" |
1506 |
by (rule additive.sum [OF additive_right]) |
|
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1507 |
|
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1508 |
|
63545 | 1509 |
lemma bounded_linear_left: "bounded_linear (\<lambda>a. a ** b)" |
1510 |
apply (insert bounded) |
|
1511 |
apply safe |
|
1512 |
apply (rule_tac K="norm b * K" in bounded_linear_intro) |
|
1513 |
apply (rule add_left) |
|
1514 |
apply (rule scaleR_left) |
|
1515 |
apply (simp add: ac_simps) |
|
1516 |
done |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1517 |
|
63545 | 1518 |
lemma bounded_linear_right: "bounded_linear (\<lambda>b. a ** b)" |
1519 |
apply (insert bounded) |
|
1520 |
apply safe |
|
1521 |
apply (rule_tac K="norm a * K" in bounded_linear_intro) |
|
1522 |
apply (rule add_right) |
|
1523 |
apply (rule scaleR_right) |
|
1524 |
apply (simp add: ac_simps) |
|
1525 |
done |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1526 |
|
63545 | 1527 |
lemma prod_diff_prod: "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)" |
1528 |
by (simp add: diff_left diff_right) |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1529 |
|
61916 | 1530 |
lemma flip: "bounded_bilinear (\<lambda>x y. y ** x)" |
1531 |
apply standard |
|
63545 | 1532 |
apply (rule add_right) |
1533 |
apply (rule add_left) |
|
1534 |
apply (rule scaleR_right) |
|
1535 |
apply (rule scaleR_left) |
|
61916 | 1536 |
apply (subst mult.commute) |
63545 | 1537 |
apply (insert bounded) |
61916 | 1538 |
apply blast |
1539 |
done |
|
1540 |
||
1541 |
lemma comp1: |
|
1542 |
assumes "bounded_linear g" |
|
1543 |
shows "bounded_bilinear (\<lambda>x. op ** (g x))" |
|
1544 |
proof unfold_locales |
|
1545 |
interpret g: bounded_linear g by fact |
|
1546 |
show "\<And>a a' b. g (a + a') ** b = g a ** b + g a' ** b" |
|
1547 |
"\<And>a b b'. g a ** (b + b') = g a ** b + g a ** b'" |
|
1548 |
"\<And>r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)" |
|
1549 |
"\<And>a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)" |
|
1550 |
by (auto simp: g.add add_left add_right g.scaleR scaleR_left scaleR_right) |
|
63545 | 1551 |
from g.nonneg_bounded nonneg_bounded obtain K L |
1552 |
where nn: "0 \<le> K" "0 \<le> L" |
|
1553 |
and K: "\<And>x. norm (g x) \<le> norm x * K" |
|
1554 |
and L: "\<And>a b. norm (a ** b) \<le> norm a * norm b * L" |
|
61916 | 1555 |
by auto |
1556 |
have "norm (g a ** b) \<le> norm a * K * norm b * L" for a b |
|
1557 |
by (auto intro!: order_trans[OF K] order_trans[OF L] mult_mono simp: nn) |
|
1558 |
then show "\<exists>K. \<forall>a b. norm (g a ** b) \<le> norm a * norm b * K" |
|
1559 |
by (auto intro!: exI[where x="K * L"] simp: ac_simps) |
|
1560 |
qed |
|
1561 |
||
63545 | 1562 |
lemma comp: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_bilinear (\<lambda>x y. f x ** g y)" |
61916 | 1563 |
by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]]) |
1564 |
||
27443 | 1565 |
end |
1566 |
||
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1567 |
lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)" |
61169 | 1568 |
by standard (auto intro!: exI[of _ 1]) |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1569 |
|
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1570 |
lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)" |
61169 | 1571 |
by standard (auto intro!: exI[of _ 1]) |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1572 |
|
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1573 |
lemma bounded_linear_add: |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1574 |
assumes "bounded_linear f" |
63545 | 1575 |
and "bounded_linear g" |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1576 |
shows "bounded_linear (\<lambda>x. f x + g x)" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1577 |
proof - |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1578 |
interpret f: bounded_linear f by fact |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1579 |
interpret g: bounded_linear g by fact |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1580 |
show ?thesis |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1581 |
proof |
63545 | 1582 |
from f.bounded obtain Kf where Kf: "norm (f x) \<le> norm x * Kf" for x |
1583 |
by blast |
|
1584 |
from g.bounded obtain Kg where Kg: "norm (g x) \<le> norm x * Kg" for x |
|
1585 |
by blast |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1586 |
show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1587 |
using add_mono[OF Kf Kg] |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1588 |
by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1589 |
qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1590 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1591 |
|
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1592 |
lemma bounded_linear_minus: |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1593 |
assumes "bounded_linear f" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1594 |
shows "bounded_linear (\<lambda>x. - f x)" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1595 |
proof - |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1596 |
interpret f: bounded_linear f by fact |
63545 | 1597 |
show ?thesis |
1598 |
apply unfold_locales |
|
1599 |
apply (simp add: f.add) |
|
1600 |
apply (simp add: f.scaleR) |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1601 |
apply (simp add: f.bounded) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1602 |
done |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1603 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1604 |
|
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1605 |
lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. f x - g x)" |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1606 |
using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1607 |
by (auto simp add: algebra_simps) |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1608 |
|
64267 | 1609 |
lemma bounded_linear_sum: |
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1610 |
fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" |
63915 | 1611 |
shows "(\<And>i. i \<in> I \<Longrightarrow> bounded_linear (f i)) \<Longrightarrow> bounded_linear (\<lambda>x. \<Sum>i\<in>I. f i x)" |
1612 |
by (induct I rule: infinite_finite_induct) (auto intro!: bounded_linear_add) |
|
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1613 |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1614 |
lemma bounded_linear_compose: |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1615 |
assumes "bounded_linear f" |
63545 | 1616 |
and "bounded_linear g" |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1617 |
shows "bounded_linear (\<lambda>x. f (g x))" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1618 |
proof - |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1619 |
interpret f: bounded_linear f by fact |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1620 |
interpret g: bounded_linear g by fact |
63545 | 1621 |
show ?thesis |
1622 |
proof unfold_locales |
|
1623 |
show "f (g (x + y)) = f (g x) + f (g y)" for x y |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1624 |
by (simp only: f.add g.add) |
63545 | 1625 |
show "f (g (scaleR r x)) = scaleR r (f (g x))" for r x |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1626 |
by (simp only: f.scaleR g.scaleR) |
63545 | 1627 |
from f.pos_bounded obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" |
1628 |
by blast |
|
1629 |
from g.pos_bounded obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" |
|
1630 |
by blast |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1631 |
show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1632 |
proof (intro exI allI) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1633 |
fix x |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1634 |
have "norm (f (g x)) \<le> norm (g x) * Kf" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1635 |
using f . |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1636 |
also have "\<dots> \<le> (norm x * Kg) * Kf" |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1637 |
using g Kf [THEN order_less_imp_le] by (rule mult_right_mono) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1638 |
also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57448
diff
changeset
|
1639 |
by (rule mult.assoc) |
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1640 |
finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" . |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1641 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1642 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1643 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1644 |
|
63545 | 1645 |
lemma bounded_bilinear_mult: "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)" |
1646 |
apply (rule bounded_bilinear.intro) |
|
1647 |
apply (rule distrib_right) |
|
1648 |
apply (rule distrib_left) |
|
1649 |
apply (rule mult_scaleR_left) |
|
1650 |
apply (rule mult_scaleR_right) |
|
1651 |
apply (rule_tac x="1" in exI) |
|
1652 |
apply (simp add: norm_mult_ineq) |
|
1653 |
done |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1654 |
|
63545 | 1655 |
lemma bounded_linear_mult_left: "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)" |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1656 |
using bounded_bilinear_mult |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1657 |
by (rule bounded_bilinear.bounded_linear_left) |
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1658 |
|
63545 | 1659 |
lemma bounded_linear_mult_right: "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)" |
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1660 |
using bounded_bilinear_mult |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1661 |
by (rule bounded_bilinear.bounded_linear_right) |
23127 | 1662 |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1663 |
lemmas bounded_linear_mult_const = |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1664 |
bounded_linear_mult_left [THEN bounded_linear_compose] |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1665 |
|
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1666 |
lemmas bounded_linear_const_mult = |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1667 |
bounded_linear_mult_right [THEN bounded_linear_compose] |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1668 |
|
63545 | 1669 |
lemma bounded_linear_divide: "bounded_linear (\<lambda>x. x / y)" |
1670 |
for y :: "'a::real_normed_field" |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1671 |
unfolding divide_inverse by (rule bounded_linear_mult_left) |
23120 | 1672 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1673 |
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR" |
63545 | 1674 |
apply (rule bounded_bilinear.intro) |
1675 |
apply (rule scaleR_left_distrib) |
|
1676 |
apply (rule scaleR_right_distrib) |
|
1677 |
apply simp |
|
1678 |
apply (rule scaleR_left_commute) |
|
1679 |
apply (rule_tac x="1" in exI) |
|
1680 |
apply simp |
|
1681 |
done |
|
22442
15d9ed9b5051
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents:
21809
diff
changeset
|
1682 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1683 |
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1684 |
using bounded_bilinear_scaleR |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1685 |
by (rule bounded_bilinear.bounded_linear_left) |
23127 | 1686 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1687 |
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1688 |
using bounded_bilinear_scaleR |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1689 |
by (rule bounded_bilinear.bounded_linear_right) |
23127 | 1690 |
|
61915
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1691 |
lemmas bounded_linear_scaleR_const = |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1692 |
bounded_linear_scaleR_left[THEN bounded_linear_compose] |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1693 |
|
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1694 |
lemmas bounded_linear_const_scaleR = |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1695 |
bounded_linear_scaleR_right[THEN bounded_linear_compose] |
e9812a95d108
theory for type of bounded linear functions; differentiation under the integral sign
immler
parents:
61799
diff
changeset
|
1696 |
|
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1697 |
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)" |
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
44127
diff
changeset
|
1698 |
unfolding of_real_def by (rule bounded_linear_scaleR_left) |
22625 | 1699 |
|
63545 | 1700 |
lemma real_bounded_linear: "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))" |
1701 |
for f :: "real \<Rightarrow> real" |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1702 |
proof - |
63545 | 1703 |
{ |
1704 |
fix x |
|
1705 |
assume "bounded_linear f" |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1706 |
then interpret bounded_linear f . |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1707 |
from scaleR[of x 1] have "f x = x * f 1" |
63545 | 1708 |
by simp |
1709 |
} |
|
51642
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1710 |
then show ?thesis |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1711 |
by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left) |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1712 |
qed |
400ec5ae7f8f
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents:
51641
diff
changeset
|
1713 |
|
63545 | 1714 |
lemma bij_linear_imp_inv_linear: "linear f \<Longrightarrow> bij f \<Longrightarrow> linear (inv f)" |
1715 |
by (auto simp: linear_def linear_axioms_def additive_def bij_is_surj bij_is_inj surj_f_inv_f |
|
1716 |
intro!: Hilbert_Choice.inv_f_eq) |
|
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:
61531
diff
changeset
|
1717 |
|
44571 | 1718 |
instance real_normed_algebra_1 \<subseteq> perfect_space |
1719 |
proof |
|
63545 | 1720 |
show "\<not> open {x}" for x :: 'a |
1721 |
apply (simp only: open_dist dist_norm) |
|
1722 |
apply clarsimp |
|
1723 |
apply (rule_tac x = "x + of_real (e/2)" in exI) |
|
1724 |
apply simp |
|
1725 |
done |
|
44571 | 1726 |
qed |
1727 |
||
63545 | 1728 |
|
60758 | 1729 |
subsection \<open>Filters and Limits on Metric Space\<close> |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1730 |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1731 |
lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1732 |
unfolding nhds_def |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1733 |
proof (safe intro!: INF_eq) |
63545 | 1734 |
fix S |
1735 |
assume "open S" "x \<in> S" |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1736 |
then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1737 |
by (auto simp: open_dist subset_eq) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1738 |
then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1739 |
by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1740 |
qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1741 |
|
63545 | 1742 |
lemma (in metric_space) tendsto_iff: "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1743 |
unfolding nhds_metric filterlim_INF filterlim_principal by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1744 |
|
63545 | 1745 |
lemma (in metric_space) tendstoI [intro?]: |
1746 |
"(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F" |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1747 |
by (auto simp: tendsto_iff) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1748 |
|
61973 | 1749 |
lemma (in metric_space) tendstoD: "(f \<longlongrightarrow> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1750 |
by (auto simp: tendsto_iff) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1751 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1752 |
lemma (in metric_space) eventually_nhds_metric: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1753 |
"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1754 |
unfolding nhds_metric |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1755 |
by (subst eventually_INF_base) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
1756 |
(auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b]) |
51531
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changeset
|
1757 |
|
63545 | 1758 |
lemma eventually_at: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)" |
1759 |
for a :: "'a :: metric_space" |
|
1760 |
by (auto simp: eventually_at_filter eventually_nhds_metric) |
|
51531
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changeset
|
1761 |
|
63545 | 1762 |
lemma eventually_at_le: "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)" |
1763 |
for a :: "'a::metric_space" |
|
1764 |
apply (simp only: eventually_at_filter eventually_nhds_metric) |
|
51641
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hoelzl
parents:
51531
diff
changeset
|
1765 |
apply auto |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51531
diff
changeset
|
1766 |
apply (rule_tac x="d / 2" in exI) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51531
diff
changeset
|
1767 |
apply auto |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51531
diff
changeset
|
1768 |
done |
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parents:
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changeset
|
1769 |
|
61531
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Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1770 |
lemma eventually_at_left_real: "a > (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {b<..<a}) (at_left a)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1771 |
by (subst eventually_at, rule exI[of _ "a - b"]) (force simp: dist_real_def) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1772 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1773 |
lemma eventually_at_right_real: "a < (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {a<..<b}) (at_right a)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1774 |
by (subst eventually_at, rule exI[of _ "b - a"]) (force simp: dist_real_def) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1775 |
|
51531
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hoelzl
parents:
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diff
changeset
|
1776 |
lemma metric_tendsto_imp_tendsto: |
63545 | 1777 |
fixes a :: "'a :: metric_space" |
1778 |
and b :: "'b :: metric_space" |
|
61973 | 1779 |
assumes f: "(f \<longlongrightarrow> a) F" |
63545 | 1780 |
and le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F" |
61973 | 1781 |
shows "(g \<longlongrightarrow> b) F" |
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parents:
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diff
changeset
|
1782 |
proof (rule tendstoI) |
63545 | 1783 |
fix e :: real |
1784 |
assume "0 < e" |
|
51531
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changeset
|
1785 |
with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD) |
f415febf4234
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parents:
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changeset
|
1786 |
with le show "eventually (\<lambda>x. dist (g x) b < e) F" |
f415febf4234
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parents:
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diff
changeset
|
1787 |
using le_less_trans by (rule eventually_elim2) |
f415febf4234
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parents:
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changeset
|
1788 |
qed |
f415febf4234
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parents:
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diff
changeset
|
1789 |
|
f415febf4234
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parents:
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changeset
|
1790 |
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top" |
63545 | 1791 |
apply (simp only: filterlim_at_top) |
51531
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parents:
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diff
changeset
|
1792 |
apply (intro allI) |
61942 | 1793 |
apply (rule_tac c="nat \<lceil>Z + 1\<rceil>" in eventually_sequentiallyI) |
1794 |
apply linarith |
|
1795 |
done |
|
1796 |
||
63556 | 1797 |
lemma filterlim_nat_sequentially: "filterlim nat sequentially at_top" |
1798 |
unfolding filterlim_at_top |
|
1799 |
apply (rule allI) |
|
1800 |
subgoal for Z by (auto intro!: eventually_at_top_linorderI[where c="int Z"]) |
|
1801 |
done |
|
1802 |
||
1803 |
lemma filterlim_floor_sequentially: "filterlim floor at_top at_top" |
|
1804 |
unfolding filterlim_at_top |
|
1805 |
apply (rule allI) |
|
1806 |
subgoal for Z by (auto simp: le_floor_iff intro!: eventually_at_top_linorderI[where c="of_int Z"]) |
|
1807 |
done |
|
1808 |
||
1809 |
lemma filterlim_sequentially_iff_filterlim_real: |
|
1810 |
"filterlim f sequentially F \<longleftrightarrow> filterlim (\<lambda>x. real (f x)) at_top F" |
|
1811 |
apply (rule iffI) |
|
1812 |
subgoal using filterlim_compose filterlim_real_sequentially by blast |
|
1813 |
subgoal premises prems |
|
1814 |
proof - |
|
1815 |
have "filterlim (\<lambda>x. nat (floor (real (f x)))) sequentially F" |
|
1816 |
by (intro filterlim_compose[OF filterlim_nat_sequentially] |
|
1817 |
filterlim_compose[OF filterlim_floor_sequentially] prems) |
|
1818 |
then show ?thesis by simp |
|
1819 |
qed |
|
1820 |
done |
|
1821 |
||
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changeset
|
1822 |
|
60758 | 1823 |
subsubsection \<open>Limits of Sequences\<close> |
51531
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changeset
|
1824 |
|
63545 | 1825 |
lemma lim_sequentially: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)" |
1826 |
for L :: "'a::metric_space" |
|
51531
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changeset
|
1827 |
unfolding tendsto_iff eventually_sequentially .. |
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changeset
|
1828 |
|
60026
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1829 |
lemmas LIMSEQ_def = lim_sequentially (*legacy binding*) |
41d81b4a0a21
Restored LIMSEQ_def as legacy binding. [The other changes are whitespace only.]
paulson <lp15@cam.ac.uk>
parents:
60017
diff
changeset
|
1830 |
|
63545 | 1831 |
lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> L \<longleftrightarrow> (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)" |
1832 |
for L :: "'a::metric_space" |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
1833 |
unfolding lim_sequentially by (metis Suc_leD zero_less_Suc) |
51531
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changeset
|
1834 |
|
63545 | 1835 |
lemma metric_LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> L" |
1836 |
for L :: "'a::metric_space" |
|
1837 |
by (simp add: lim_sequentially) |
|
51531
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parents:
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changeset
|
1838 |
|
63545 | 1839 |
lemma metric_LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r" |
1840 |
for L :: "'a::metric_space" |
|
1841 |
by (simp add: lim_sequentially) |
|
51531
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parents:
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changeset
|
1842 |
|
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parents:
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changeset
|
1843 |
|
60758 | 1844 |
subsubsection \<open>Limits of Functions\<close> |
51531
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parents:
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changeset
|
1845 |
|
63545 | 1846 |
lemma LIM_def: "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)" |
1847 |
for a :: "'a::metric_space" and L :: "'b::metric_space" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51531
diff
changeset
|
1848 |
unfolding tendsto_iff eventually_at by simp |
51531
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hoelzl
parents:
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diff
changeset
|
1849 |
|
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parents:
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diff
changeset
|
1850 |
lemma metric_LIM_I: |
63545 | 1851 |
"(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L" |
1852 |
for a :: "'a::metric_space" and L :: "'b::metric_space" |
|
1853 |
by (simp add: LIM_def) |
|
51531
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parents:
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diff
changeset
|
1854 |
|
63545 | 1855 |
lemma metric_LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r" |
1856 |
for a :: "'a::metric_space" and L :: "'b::metric_space" |
|
1857 |
by (simp add: LIM_def) |
|
51531
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parents:
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diff
changeset
|
1858 |
|
f415febf4234
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parents:
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diff
changeset
|
1859 |
lemma metric_LIM_imp_LIM: |
63545 | 1860 |
fixes l :: "'a::metric_space" |
1861 |
and m :: "'b::metric_space" |
|
1862 |
assumes f: "f \<midarrow>a\<rightarrow> l" |
|
1863 |
and le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l" |
|
1864 |
shows "g \<midarrow>a\<rightarrow> m" |
|
51531
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hoelzl
parents:
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diff
changeset
|
1865 |
by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le) |
f415febf4234
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hoelzl
parents:
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changeset
|
1866 |
|
f415febf4234
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parents:
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changeset
|
1867 |
lemma metric_LIM_equal2: |
63545 | 1868 |
fixes a :: "'a::metric_space" |
1869 |
assumes "0 < R" |
|
1870 |
and "\<And>x. x \<noteq> a \<Longrightarrow> dist x a < R \<Longrightarrow> f x = g x" |
|
1871 |
shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l" |
|
1872 |
apply (rule topological_tendstoI) |
|
1873 |
apply (drule (2) topological_tendstoD) |
|
1874 |
apply (simp add: eventually_at) |
|
1875 |
apply safe |
|
1876 |
apply (rule_tac x="min d R" in exI) |
|
1877 |
apply safe |
|
1878 |
apply (simp add: assms(1)) |
|
1879 |
apply (simp add: assms(2)) |
|
1880 |
done |
|
51531
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parents:
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diff
changeset
|
1881 |
|
f415febf4234
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parents:
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diff
changeset
|
1882 |
lemma metric_LIM_compose2: |
63545 | 1883 |
fixes a :: "'a::metric_space" |
1884 |
assumes f: "f \<midarrow>a\<rightarrow> b" |
|
1885 |
and g: "g \<midarrow>b\<rightarrow> c" |
|
1886 |
and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b" |
|
61976 | 1887 |
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c" |
63545 | 1888 |
using inj by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at) |
51531
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diff
changeset
|
1889 |
|
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parents:
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changeset
|
1890 |
lemma metric_isCont_LIM_compose2: |
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parents:
51524
diff
changeset
|
1891 |
fixes f :: "'a :: metric_space \<Rightarrow> _" |
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
1892 |
assumes f [unfolded isCont_def]: "isCont f a" |
63545 | 1893 |
and g: "g \<midarrow>f a\<rightarrow> l" |
1894 |
and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a" |
|
61976 | 1895 |
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l" |
63545 | 1896 |
by (rule metric_LIM_compose2 [OF f g inj]) |
1897 |
||
51531
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changeset
|
1898 |
|
60758 | 1899 |
subsection \<open>Complete metric spaces\<close> |
51531
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changeset
|
1900 |
|
60758 | 1901 |
subsection \<open>Cauchy sequences\<close> |
51531
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changeset
|
1902 |
|
62101 | 1903 |
lemma (in metric_space) Cauchy_def: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)" |
1904 |
proof - |
|
63545 | 1905 |
have *: "eventually P (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) \<longleftrightarrow> |
62101 | 1906 |
(\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. P (X m, X n))" for P |
63545 | 1907 |
apply (subst eventually_INF_base) |
1908 |
subgoal by simp |
|
1909 |
subgoal for a b |
|
62101 | 1910 |
by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq) |
63545 | 1911 |
subgoal by (auto simp: eventually_principal, blast) |
1912 |
done |
|
62101 | 1913 |
have "Cauchy X \<longleftrightarrow> (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) \<le> uniformity" |
1914 |
unfolding Cauchy_uniform_iff le_filter_def * .. |
|
1915 |
also have "\<dots> = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)" |
|
1916 |
unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal) |
|
1917 |
finally show ?thesis . |
|
1918 |
qed |
|
51531
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changeset
|
1919 |
|
63545 | 1920 |
lemma (in metric_space) Cauchy_altdef: "Cauchy f \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)" |
1921 |
(is "?lhs \<longleftrightarrow> ?rhs") |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1922 |
proof |
63545 | 1923 |
assume ?rhs |
1924 |
show ?lhs |
|
1925 |
unfolding Cauchy_def |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1926 |
proof (intro allI impI) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1927 |
fix e :: real assume e: "e > 0" |
63545 | 1928 |
with \<open>?rhs\<close> obtain M where M: "m \<ge> M \<Longrightarrow> n > m \<Longrightarrow> dist (f m) (f n) < e" for m n |
1929 |
by blast |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1930 |
have "dist (f m) (f n) < e" if "m \<ge> M" "n \<ge> M" for m n |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1931 |
using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute) |
63545 | 1932 |
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m) (f n) < e" |
1933 |
by blast |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1934 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1935 |
next |
63545 | 1936 |
assume ?lhs |
1937 |
show ?rhs |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1938 |
proof (intro allI impI) |
63545 | 1939 |
fix e :: real |
1940 |
assume e: "e > 0" |
|
61799 | 1941 |
with \<open>Cauchy f\<close> obtain M where "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> dist (f m) (f n) < e" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1942 |
unfolding Cauchy_def by blast |
63545 | 1943 |
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e" |
1944 |
by (intro exI[of _ M]) force |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1945 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1946 |
qed |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1947 |
|
62101 | 1948 |
lemma (in metric_space) metric_CauchyI: |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1949 |
"(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1950 |
by (simp add: Cauchy_def) |
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
1951 |
|
63545 | 1952 |
lemma (in metric_space) CauchyI': |
1953 |
"(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X" |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1954 |
unfolding Cauchy_altdef by blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61524
diff
changeset
|
1955 |
|
62101 | 1956 |
lemma (in metric_space) metric_CauchyD: |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1957 |
"Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1958 |
by (simp add: Cauchy_def) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1959 |
|
62101 | 1960 |
lemma (in metric_space) metric_Cauchy_iff2: |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1961 |
"Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))" |
63545 | 1962 |
apply (simp add: Cauchy_def) |
1963 |
apply auto |
|
1964 |
apply (drule reals_Archimedean) |
|
1965 |
apply safe |
|
1966 |
apply (drule_tac x = n in spec) |
|
1967 |
apply auto |
|
1968 |
apply (rule_tac x = M in exI) |
|
1969 |
apply auto |
|
1970 |
apply (drule_tac x = m in spec) |
|
1971 |
apply simp |
|
1972 |
apply (drule_tac x = na in spec) |
|
1973 |
apply auto |
|
1974 |
done |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1975 |
|
63545 | 1976 |
lemma Cauchy_iff2: "Cauchy X \<longleftrightarrow> (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse (real (Suc j))))" |
1977 |
by (simp only: metric_Cauchy_iff2 dist_real_def) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1978 |
|
62101 | 1979 |
lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially" |
1980 |
proof (subst lim_sequentially, intro allI impI exI) |
|
63545 | 1981 |
fix e :: real |
1982 |
assume e: "e > 0" |
|
1983 |
fix n :: nat |
|
1984 |
assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>" |
|
62101 | 1985 |
have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith |
1986 |
also note n |
|
63545 | 1987 |
finally show "dist (1 / of_nat n :: 'a) 0 < e" |
1988 |
using e by (simp add: divide_simps mult.commute norm_divide) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1989 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
1990 |
|
62101 | 1991 |
lemma (in metric_space) complete_def: |
1992 |
shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))" |
|
1993 |
unfolding complete_uniform |
|
1994 |
proof safe |
|
63545 | 1995 |
fix f :: "nat \<Rightarrow> 'a" |
1996 |
assume f: "\<forall>n. f n \<in> S" "Cauchy f" |
|
62101 | 1997 |
and *: "\<forall>F\<le>principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x)" |
1998 |
then show "\<exists>l\<in>S. f \<longlonglongrightarrow> l" |
|
1999 |
unfolding filterlim_def using f |
|
2000 |
by (intro *[rule_format]) |
|
2001 |
(auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform) |
|
2002 |
next |
|
63545 | 2003 |
fix F :: "'a filter" |
2004 |
assume "F \<le> principal S" "F \<noteq> bot" "cauchy_filter F" |
|
62101 | 2005 |
assume seq: "\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l)" |
2006 |
||
63545 | 2007 |
from \<open>F \<le> principal S\<close> \<open>cauchy_filter F\<close> |
2008 |
have FF_le: "F \<times>\<^sub>F F \<le> uniformity_on S" |
|
62101 | 2009 |
by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono) |
2010 |
||
2011 |
let ?P = "\<lambda>P e. eventually P F \<and> (\<forall>x. P x \<longrightarrow> x \<in> S) \<and> (\<forall>x y. P x \<longrightarrow> P y \<longrightarrow> dist x y < e)" |
|
63545 | 2012 |
have P: "\<exists>P. ?P P \<epsilon>" if "0 < \<epsilon>" for \<epsilon> :: real |
2013 |
proof - |
|
2014 |
from that have "eventually (\<lambda>(x, y). x \<in> S \<and> y \<in> S \<and> dist x y < \<epsilon>) (uniformity_on S)" |
|
2015 |
by (auto simp: eventually_inf_principal eventually_uniformity_metric) |
|
2016 |
from filter_leD[OF FF_le this] show ?thesis |
|
2017 |
by (auto simp: eventually_prod_same) |
|
2018 |
qed |
|
62101 | 2019 |
|
2020 |
have "\<exists>P. \<forall>n. ?P (P n) (1 / Suc n) \<and> P (Suc n) \<le> P n" |
|
2021 |
proof (rule dependent_nat_choice) |
|
2022 |
show "\<exists>P. ?P P (1 / Suc 0)" |
|
2023 |
using P[of 1] by auto |
|
2024 |
next |
|
2025 |
fix P n assume "?P P (1/Suc n)" |
|
2026 |
moreover obtain Q where "?P Q (1 / Suc (Suc n))" |
|
2027 |
using P[of "1/Suc (Suc n)"] by auto |
|
2028 |
ultimately show "\<exists>Q. ?P Q (1 / Suc (Suc n)) \<and> Q \<le> P" |
|
2029 |
by (intro exI[of _ "\<lambda>x. P x \<and> Q x"]) (auto simp: eventually_conj_iff) |
|
2030 |
qed |
|
63545 | 2031 |
then obtain P where P: "eventually (P n) F" "P n x \<Longrightarrow> x \<in> S" |
2032 |
"P n x \<Longrightarrow> P n y \<Longrightarrow> dist x y < 1 / Suc n" "P (Suc n) \<le> P n" |
|
2033 |
for n x y |
|
62101 | 2034 |
by metis |
2035 |
have "antimono P" |
|
2036 |
using P(4) unfolding decseq_Suc_iff le_fun_def by blast |
|
2037 |
||
63545 | 2038 |
obtain X where X: "P n (X n)" for n |
62101 | 2039 |
using P(1)[THEN eventually_happens'[OF \<open>F \<noteq> bot\<close>]] by metis |
2040 |
have "Cauchy X" |
|
2041 |
unfolding metric_Cauchy_iff2 inverse_eq_divide |
|
2042 |
proof (intro exI allI impI) |
|
63545 | 2043 |
fix j m n :: nat |
2044 |
assume "j \<le> m" "j \<le> n" |
|
62101 | 2045 |
with \<open>antimono P\<close> X have "P j (X m)" "P j (X n)" |
2046 |
by (auto simp: antimono_def) |
|
2047 |
then show "dist (X m) (X n) < 1 / Suc j" |
|
2048 |
by (rule P) |
|
2049 |
qed |
|
2050 |
moreover have "\<forall>n. X n \<in> S" |
|
2051 |
using P(2) X by auto |
|
2052 |
ultimately obtain x where "X \<longlonglongrightarrow> x" "x \<in> S" |
|
2053 |
using seq by blast |
|
2054 |
||
2055 |
show "\<exists>x\<in>S. F \<le> nhds x" |
|
2056 |
proof (rule bexI) |
|
63545 | 2057 |
have "eventually (\<lambda>y. dist y x < e) F" if "0 < e" for e :: real |
2058 |
proof - |
|
2059 |
from that have "(\<lambda>n. 1 / Suc n :: real) \<longlonglongrightarrow> 0 \<and> 0 < e / 2" |
|
62101 | 2060 |
by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n) |
2061 |
then have "\<forall>\<^sub>F n in sequentially. dist (X n) x < e / 2 \<and> 1 / Suc n < e / 2" |
|
63545 | 2062 |
using \<open>X \<longlonglongrightarrow> x\<close> |
2063 |
unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff |
|
2064 |
by blast |
|
62101 | 2065 |
then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2" |
2066 |
by (auto simp: eventually_sequentially dist_commute) |
|
63545 | 2067 |
show ?thesis |
62101 | 2068 |
using \<open>eventually (P n) F\<close> |
2069 |
proof eventually_elim |
|
63545 | 2070 |
case (elim y) |
62101 | 2071 |
then have "dist y (X n) < 1 / Suc n" |
2072 |
by (intro X P) |
|
2073 |
also have "\<dots> < e / 2" by fact |
|
2074 |
finally show "dist y x < e" |
|
2075 |
by (rule dist_triangle_half_l) fact |
|
63545 | 2076 |
qed |
2077 |
qed |
|
62101 | 2078 |
then show "F \<le> nhds x" |
2079 |
unfolding nhds_metric le_INF_iff le_principal by auto |
|
2080 |
qed fact |
|
2081 |
qed |
|
2082 |
||
2083 |
lemma (in metric_space) totally_bounded_metric: |
|
2084 |
"totally_bounded S \<longleftrightarrow> (\<forall>e>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. {y. dist x y < e}))" |
|
63545 | 2085 |
apply (simp only: totally_bounded_def eventually_uniformity_metric imp_ex) |
62101 | 2086 |
apply (subst all_comm) |
2087 |
apply (intro arg_cong[where f=All] ext) |
|
2088 |
apply safe |
|
2089 |
subgoal for e |
|
2090 |
apply (erule allE[of _ "\<lambda>(x, y). dist x y < e"]) |
|
2091 |
apply auto |
|
2092 |
done |
|
2093 |
subgoal for e P k |
|
2094 |
apply (intro exI[of _ k]) |
|
2095 |
apply (force simp: subset_eq) |
|
2096 |
done |
|
2097 |
done |
|
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2098 |
|
63545 | 2099 |
|
60758 | 2100 |
subsubsection \<open>Cauchy Sequences are Convergent\<close> |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2101 |
|
62101 | 2102 |
(* TODO: update to uniform_space *) |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2103 |
class complete_space = metric_space + |
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2104 |
assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X" |
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2105 |
|
63545 | 2106 |
lemma Cauchy_convergent_iff: "Cauchy X \<longleftrightarrow> convergent X" |
2107 |
for X :: "nat \<Rightarrow> 'a::complete_space" |
|
2108 |
by (blast intro: Cauchy_convergent convergent_Cauchy) |
|
2109 |
||
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2110 |
|
60758 | 2111 |
subsection \<open>The set of real numbers is a complete metric space\<close> |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2112 |
|
60758 | 2113 |
text \<open> |
63545 | 2114 |
Proof that Cauchy sequences converge based on the one from |
63680 | 2115 |
\<^url>\<open>http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html\<close> |
60758 | 2116 |
\<close> |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2117 |
|
60758 | 2118 |
text \<open> |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2119 |
If sequence @{term "X"} is Cauchy, then its limit is the lub of |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2120 |
@{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"} |
60758 | 2121 |
\<close> |
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2122 |
lemma increasing_LIMSEQ: |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2123 |
fixes f :: "nat \<Rightarrow> real" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2124 |
assumes inc: "\<And>n. f n \<le> f (Suc n)" |
63545 | 2125 |
and bdd: "\<And>n. f n \<le> l" |
2126 |
and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e" |
|
61969 | 2127 |
shows "f \<longlonglongrightarrow> l" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2128 |
proof (rule increasing_tendsto) |
63545 | 2129 |
fix x |
2130 |
assume "x < l" |
|
51531
f415febf4234
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hoelzl
parents:
51524
diff
changeset
|
2131 |
with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2132 |
by auto |
60758 | 2133 |
from en[OF \<open>0 < e\<close>] obtain n where "l - e \<le> f n" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2134 |
by (auto simp: field_simps) |
63545 | 2135 |
with \<open>e < l - x\<close> \<open>0 < e\<close> have "x < f n" |
2136 |
by simp |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2137 |
with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2138 |
by (auto simp: eventually_sequentially incseq_def intro: less_le_trans) |
63545 | 2139 |
qed (use bdd in auto) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2140 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2141 |
lemma real_Cauchy_convergent: |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2142 |
fixes X :: "nat \<Rightarrow> real" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2143 |
assumes X: "Cauchy X" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2144 |
shows "convergent X" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2145 |
proof - |
63040 | 2146 |
define S :: "real set" where "S = {x. \<exists>N. \<forall>n\<ge>N. x < X n}" |
63545 | 2147 |
then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" |
2148 |
by auto |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2149 |
|
63545 | 2150 |
have bound_isUb: "y \<le> x" if N: "\<forall>n\<ge>N. X n < x" and "y \<in> S" for N and x y :: real |
2151 |
proof - |
|
2152 |
from that have "\<exists>M. \<forall>n\<ge>M. y < X n" |
|
2153 |
by (simp add: S_def) |
|
2154 |
then obtain M where "\<forall>n\<ge>M. y < X n" .. |
|
2155 |
then have "y < X (max M N)" by simp |
|
2156 |
also have "\<dots> < x" using N by simp |
|
2157 |
finally show ?thesis by (rule order_less_imp_le) |
|
2158 |
qed |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2159 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2160 |
obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2161 |
using X[THEN metric_CauchyD, OF zero_less_one] by auto |
63545 | 2162 |
then have N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp |
54263
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
2163 |
have [simp]: "S \<noteq> {}" |
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
2164 |
proof (intro exI ex_in_conv[THEN iffD1]) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2165 |
from N have "\<forall>n\<ge>N. X N - 1 < X n" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2166 |
by (simp add: abs_diff_less_iff dist_real_def) |
63545 | 2167 |
then show "X N - 1 \<in> S" by (rule mem_S) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2168 |
qed |
54263
c4159fe6fa46
move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents:
54230
diff
changeset
|
2169 |
have [simp]: "bdd_above S" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2170 |
proof |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2171 |
from N have "\<forall>n\<ge>N. X n < X N + 1" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2172 |
by (simp add: abs_diff_less_iff dist_real_def) |
63545 | 2173 |
then show "\<And>s. s \<in> S \<Longrightarrow> s \<le> X N + 1" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2174 |
by (rule bound_isUb) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2175 |
qed |
61969 | 2176 |
have "X \<longlonglongrightarrow> Sup S" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2177 |
proof (rule metric_LIMSEQ_I) |
63545 | 2178 |
fix r :: real |
2179 |
assume "0 < r" |
|
2180 |
then have r: "0 < r/2" by simp |
|
2181 |
obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2" |
|
2182 |
using metric_CauchyD [OF X r] by auto |
|
2183 |
then have "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp |
|
2184 |
then have N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2" |
|
2185 |
by (simp only: dist_real_def abs_diff_less_iff) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2186 |
|
63545 | 2187 |
from N have "\<forall>n\<ge>N. X N - r/2 < X n" by blast |
2188 |
then have "X N - r/2 \<in> S" by (rule mem_S) |
|
2189 |
then have 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper) |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2190 |
|
63545 | 2191 |
from N have "\<forall>n\<ge>N. X n < X N + r/2" by blast |
2192 |
from bound_isUb[OF this] |
|
2193 |
have 2: "Sup S \<le> X N + r/2" |
|
2194 |
by (intro cSup_least) simp_all |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2195 |
|
63545 | 2196 |
show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r" |
2197 |
proof (intro exI allI impI) |
|
2198 |
fix n |
|
2199 |
assume n: "N \<le> n" |
|
2200 |
from N n have "X n < X N + r/2" and "X N - r/2 < X n" |
|
2201 |
by simp_all |
|
2202 |
then show "dist (X n) (Sup S) < r" using 1 2 |
|
2203 |
by (simp add: abs_diff_less_iff dist_real_def) |
|
2204 |
qed |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2205 |
qed |
63545 | 2206 |
then show ?thesis by (auto simp: convergent_def) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2207 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2208 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2209 |
instance real :: complete_space |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2210 |
by intro_classes (rule real_Cauchy_convergent) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2211 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2212 |
class banach = real_normed_vector + complete_space |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2213 |
|
61169 | 2214 |
instance real :: banach .. |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2215 |
|
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2216 |
lemma tendsto_at_topI_sequentially: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2217 |
fixes f :: "real \<Rightarrow> 'b::first_countable_topology" |
61969 | 2218 |
assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y" |
61973 | 2219 |
shows "(f \<longlongrightarrow> y) at_top" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2220 |
proof - |
63545 | 2221 |
obtain A where A: "decseq A" "open (A n)" "y \<in> A n" "nhds y = (INF n. principal (A n))" for n |
2222 |
by (rule nhds_countable[of y]) (rule that) |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2223 |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2224 |
have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2225 |
proof (rule ccontr) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2226 |
assume "\<not> (\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2227 |
then obtain m where "\<And>k. \<exists>x\<ge>k. f x \<notin> A m" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2228 |
by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2229 |
then have "\<exists>X. \<forall>n. (f (X n) \<notin> A m) \<and> max n (X n) + 1 \<le> X (Suc n)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2230 |
by (intro dependent_nat_choice) (auto simp del: max.bounded_iff) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2231 |
then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2232 |
by auto |
63545 | 2233 |
have "1 \<le> n \<Longrightarrow> real n \<le> X n" for n |
2234 |
using X[of "n - 1"] by auto |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2235 |
then have "filterlim X at_top sequentially" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2236 |
by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially] |
63545 | 2237 |
simp: eventually_sequentially) |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2238 |
from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2239 |
by auto |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2240 |
qed |
63545 | 2241 |
then obtain k where "k m \<le> x \<Longrightarrow> f x \<in> A m" for m x |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2242 |
by metis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57418
diff
changeset
|
2243 |
then show ?thesis |
63545 | 2244 |
unfolding at_top_def A by (intro filterlim_base[where i=k]) auto |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2245 |
qed |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2246 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
2247 |
lemma tendsto_at_topI_sequentially_real: |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2248 |
fixes f :: "real \<Rightarrow> real" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2249 |
assumes mono: "mono f" |
63545 | 2250 |
and limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y" |
61973 | 2251 |
shows "(f \<longlongrightarrow> y) at_top" |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2252 |
proof (rule tendstoI) |
63545 | 2253 |
fix e :: real |
2254 |
assume "0 < e" |
|
2255 |
with limseq obtain N :: nat where N: "N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e" for n |
|
60017
b785d6d06430
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
paulson <lp15@cam.ac.uk>
parents:
59867
diff
changeset
|
2256 |
by (auto simp: lim_sequentially dist_real_def) |
63545 | 2257 |
have le: "f x \<le> y" for x :: real |
2258 |
proof - |
|
53381 | 2259 |
obtain n where "x \<le> real_of_nat n" |
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62533
diff
changeset
|
2260 |
using real_arch_simple[of x] .. |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2261 |
note monoD[OF mono this] |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2262 |
also have "f (real_of_nat n) \<le> y" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
2263 |
by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono]) |
63545 | 2264 |
finally show ?thesis . |
2265 |
qed |
|
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2266 |
have "eventually (\<lambda>x. real N \<le> x) at_top" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2267 |
by (rule eventually_ge_at_top) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2268 |
then show "eventually (\<lambda>x. dist (f x) y < e) at_top" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2269 |
proof eventually_elim |
63545 | 2270 |
case (elim x) |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2271 |
with N[of N] le have "y - f (real N) < e" by auto |
63545 | 2272 |
moreover note monoD[OF mono elim] |
51531
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2273 |
ultimately show "dist (f x) y < e" |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2274 |
using le[of x] by (auto simp: dist_real_def field_simps) |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2275 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2276 |
qed |
f415febf4234
remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents:
51524
diff
changeset
|
2277 |
|
20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff
changeset
|
2278 |
end |