author | huffman |
Mon, 01 Jun 2009 16:59:56 -0700 | |
changeset 31388 | e0c05b595d1f |
parent 31339 | b4660351e8e7 |
child 31405 | 1f72869f1a2e |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Product_Vector.thy |
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Author: Brian Huffman |
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*) |
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header {* Cartesian Products as Vector Spaces *} |
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theory Product_Vector |
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imports Inner_Product Product_plus |
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begin |
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subsection {* Product is a real vector space *} |
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instantiation "*" :: (real_vector, real_vector) real_vector |
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begin |
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definition scaleR_prod_def: |
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"scaleR r A = (scaleR r (fst A), scaleR r (snd A))" |
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lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)" |
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unfolding scaleR_prod_def by simp |
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lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)" |
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unfolding scaleR_prod_def by simp |
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lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)" |
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unfolding scaleR_prod_def by simp |
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instance proof |
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fix a b :: real and x y :: "'a \<times> 'b" |
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show "scaleR a (x + y) = scaleR a x + scaleR a y" |
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by (simp add: expand_prod_eq scaleR_right_distrib) |
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show "scaleR (a + b) x = scaleR a x + scaleR b x" |
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by (simp add: expand_prod_eq scaleR_left_distrib) |
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show "scaleR a (scaleR b x) = scaleR (a * b) x" |
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by (simp add: expand_prod_eq) |
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show "scaleR 1 x = x" |
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by (simp add: expand_prod_eq) |
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qed |
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end |
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instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
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subsection {* Product is a metric space *} |
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instantiation |
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"*" :: (metric_space, metric_space) metric_space |
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begin |
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definition dist_prod_def: |
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"dist (x::'a \<times> 'b) y = sqrt ((dist (fst x) (fst y))\<twosuperior> + (dist (snd x) (snd y))\<twosuperior>)" |
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lemma dist_Pair_Pair: "dist (a, b) (c, d) = sqrt ((dist a c)\<twosuperior> + (dist b d)\<twosuperior>)" |
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unfolding dist_prod_def by simp |
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instance proof |
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fix x y :: "'a \<times> 'b" |
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show "dist x y = 0 \<longleftrightarrow> x = y" |
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unfolding dist_prod_def |
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by (simp add: expand_prod_eq) |
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next |
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fix x y z :: "'a \<times> 'b" |
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show "dist x y \<le> dist x z + dist y z" |
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unfolding dist_prod_def |
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apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]) |
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apply (rule real_sqrt_le_mono) |
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apply (rule order_trans [OF add_mono]) |
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apply (rule power_mono [OF dist_triangle2 [of _ _ "fst z"] zero_le_dist]) |
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apply (rule power_mono [OF dist_triangle2 [of _ _ "snd z"] zero_le_dist]) |
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apply (simp only: real_sum_squared_expand) |
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done |
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qed |
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end |
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subsection {* Product is a normed vector space *} |
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instantiation |
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"*" :: (real_normed_vector, real_normed_vector) real_normed_vector |
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begin |
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definition norm_prod_def: |
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"norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)" |
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definition sgn_prod_def: |
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"sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x" |
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lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)" |
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unfolding norm_prod_def by simp |
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instance proof |
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fix r :: real and x y :: "'a \<times> 'b" |
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show "0 \<le> norm x" |
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unfolding norm_prod_def by simp |
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show "norm x = 0 \<longleftrightarrow> x = 0" |
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unfolding norm_prod_def |
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by (simp add: expand_prod_eq) |
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show "norm (x + y) \<le> norm x + norm y" |
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unfolding norm_prod_def |
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apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq]) |
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apply (simp add: add_mono power_mono norm_triangle_ineq) |
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done |
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show "norm (scaleR r x) = \<bar>r\<bar> * norm x" |
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unfolding norm_prod_def |
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apply (simp add: norm_scaleR power_mult_distrib) |
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apply (simp add: right_distrib [symmetric]) |
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apply (simp add: real_sqrt_mult_distrib) |
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done |
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show "sgn x = scaleR (inverse (norm x)) x" |
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by (rule sgn_prod_def) |
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show "dist x y = norm (x - y)" |
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unfolding dist_prod_def norm_prod_def |
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by (simp add: dist_norm) |
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qed |
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end |
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subsection {* Product is an inner product space *} |
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instantiation "*" :: (real_inner, real_inner) real_inner |
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begin |
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definition inner_prod_def: |
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"inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)" |
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lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d" |
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unfolding inner_prod_def by simp |
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126 |
|
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huffman
parents:
diff
changeset
|
127 |
instance proof |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
128 |
fix r :: real |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
129 |
fix x y z :: "'a::real_inner * 'b::real_inner" |
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huffman
parents:
diff
changeset
|
130 |
show "inner x y = inner y x" |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
131 |
unfolding inner_prod_def |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
132 |
by (simp add: inner_commute) |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
133 |
show "inner (x + y) z = inner x z + inner y z" |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
134 |
unfolding inner_prod_def |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
135 |
by (simp add: inner_left_distrib) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
136 |
show "inner (scaleR r x) y = r * inner x y" |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
137 |
unfolding inner_prod_def |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
138 |
by (simp add: inner_scaleR_left right_distrib) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
139 |
show "0 \<le> inner x x" |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
140 |
unfolding inner_prod_def |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
141 |
by (intro add_nonneg_nonneg inner_ge_zero) |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
142 |
show "inner x x = 0 \<longleftrightarrow> x = 0" |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
143 |
unfolding inner_prod_def expand_prod_eq |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
144 |
by (simp add: add_nonneg_eq_0_iff) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
145 |
show "norm x = sqrt (inner x x)" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
146 |
unfolding norm_prod_def inner_prod_def |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
147 |
by (simp add: power2_norm_eq_inner) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
148 |
qed |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
149 |
|
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
150 |
end |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
151 |
|
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
152 |
subsection {* Pair operations are linear and continuous *} |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
153 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30019
diff
changeset
|
154 |
interpretation fst: bounded_linear fst |
30019
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
155 |
apply (unfold_locales) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
156 |
apply (rule fst_add) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
157 |
apply (rule fst_scaleR) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
158 |
apply (rule_tac x="1" in exI, simp add: norm_Pair) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
159 |
done |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
160 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30019
diff
changeset
|
161 |
interpretation snd: bounded_linear snd |
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
162 |
apply (unfold_locales) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
163 |
apply (rule snd_add) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
164 |
apply (rule snd_scaleR) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
165 |
apply (rule_tac x="1" in exI, simp add: norm_Pair) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
166 |
done |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
167 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
168 |
text {* TODO: move to NthRoot *} |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
169 |
lemma sqrt_add_le_add_sqrt: |
a2f19e0a28b2
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huffman
parents:
diff
changeset
|
170 |
assumes x: "0 \<le> x" and y: "0 \<le> y" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
171 |
shows "sqrt (x + y) \<le> sqrt x + sqrt y" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
172 |
apply (rule power2_le_imp_le) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
173 |
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
174 |
apply (simp add: mult_nonneg_nonneg x y) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
175 |
apply (simp add: add_nonneg_nonneg x y) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
176 |
done |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
177 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
178 |
lemma bounded_linear_Pair: |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
179 |
assumes f: "bounded_linear f" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
180 |
assumes g: "bounded_linear g" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
181 |
shows "bounded_linear (\<lambda>x. (f x, g x))" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
182 |
proof |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
183 |
interpret f: bounded_linear f by fact |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
184 |
interpret g: bounded_linear g by fact |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
185 |
fix x y and r :: real |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
186 |
show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
187 |
by (simp add: f.add g.add) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
188 |
show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
189 |
by (simp add: f.scaleR g.scaleR) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
190 |
obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
191 |
using f.pos_bounded by fast |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
192 |
obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
193 |
using g.pos_bounded by fast |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
194 |
have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
195 |
apply (rule allI) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
196 |
apply (simp add: norm_Pair) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
197 |
apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
198 |
apply (simp add: right_distrib) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
199 |
apply (rule add_mono [OF norm_f norm_g]) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
200 |
done |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
201 |
then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" .. |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
202 |
qed |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
203 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
204 |
text {* |
31388 | 205 |
TODO: The "tendsto" notion generalizes LIM and LIMSEQ. |
206 |
But the Cauchy proof still requires a lot of duplication. |
|
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
207 |
Is there a good way to factor out the common parts? |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
208 |
*} |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
209 |
|
31388 | 210 |
lemma tendsto_Pair: |
211 |
assumes "tendsto X a net" and "tendsto Y b net" |
|
212 |
shows "tendsto (\<lambda>i. (X i, Y i)) (a, b) net" |
|
213 |
proof (rule tendstoI) |
|
214 |
fix r :: real assume "0 < r" |
|
215 |
then have "0 < r / sqrt 2" (is "0 < ?s") |
|
216 |
by (simp add: divide_pos_pos) |
|
217 |
have "eventually (\<lambda>i. dist (X i) a < ?s) net" |
|
218 |
using `tendsto X a net` `0 < ?s` by (rule tendstoD) |
|
219 |
moreover |
|
220 |
have "eventually (\<lambda>i. dist (Y i) b < ?s) net" |
|
221 |
using `tendsto Y b net` `0 < ?s` by (rule tendstoD) |
|
222 |
ultimately |
|
223 |
show "eventually (\<lambda>i. dist (X i, Y i) (a, b) < r) net" |
|
224 |
by (rule eventually_elim2) |
|
225 |
(simp add: real_sqrt_sum_squares_less dist_Pair_Pair) |
|
226 |
qed |
|
227 |
||
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
228 |
lemma LIMSEQ_Pair: |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
229 |
assumes "X ----> a" and "Y ----> b" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
230 |
shows "(\<lambda>n. (X n, Y n)) ----> (a, b)" |
31388 | 231 |
using assms unfolding LIMSEQ_conv_tendsto |
232 |
by (rule tendsto_Pair) |
|
233 |
||
234 |
lemma LIM_Pair: |
|
235 |
assumes "f -- x --> a" and "g -- x --> b" |
|
236 |
shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)" |
|
237 |
using assms unfolding LIM_conv_tendsto |
|
238 |
by (rule tendsto_Pair) |
|
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
239 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
240 |
lemma Cauchy_Pair: |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
241 |
assumes "Cauchy X" and "Cauchy Y" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
242 |
shows "Cauchy (\<lambda>n. (X n, Y n))" |
31339
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
243 |
proof (rule metric_CauchyI) |
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
244 |
fix r :: real assume "0 < r" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
245 |
then have "0 < r / sqrt 2" (is "0 < ?s") |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
246 |
by (simp add: divide_pos_pos) |
31339
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
247 |
obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < ?s" |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
248 |
using metric_CauchyD [OF `Cauchy X` `0 < ?s`] .. |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
249 |
obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (Y m) (Y n) < ?s" |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
250 |
using metric_CauchyD [OF `Cauchy Y` `0 < ?s`] .. |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
251 |
have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. dist (X m, Y m) (X n, Y n) < r" |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
252 |
using M N by (simp add: real_sqrt_sum_squares_less dist_Pair_Pair) |
b4660351e8e7
instance * :: (metric_space, metric_space) metric_space; generalize lemmas to class metric_space
huffman
parents:
31290
diff
changeset
|
253 |
then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. dist (X m, Y m) (X n, Y n) < r" .. |
30019
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
254 |
qed |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
255 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
256 |
lemma isCont_Pair [simp]: |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
257 |
"\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
258 |
unfolding isCont_def by (rule LIM_Pair) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
259 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
260 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
261 |
subsection {* Product is a complete vector space *} |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
262 |
|
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
263 |
instance "*" :: (banach, banach) banach |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
264 |
proof |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
265 |
fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
266 |
have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
267 |
using fst.Cauchy [OF `Cauchy X`] |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
268 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
269 |
have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))" |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
270 |
using snd.Cauchy [OF `Cauchy X`] |
a2f19e0a28b2
add theory of products as real vector spaces to Library
huffman
parents:
diff
changeset
|
271 |
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff) |
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272 |
have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))" |
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273 |
using LIMSEQ_Pair [OF 1 2] by simp |
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274 |
then show "convergent X" |
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by (rule convergentI) |
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qed |
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277 |
|
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278 |
subsection {* Frechet derivatives involving pairs *} |
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279 |
|
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280 |
lemma FDERIV_Pair: |
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assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'" |
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282 |
shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))" |
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283 |
apply (rule FDERIV_I) |
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284 |
apply (rule bounded_linear_Pair) |
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285 |
apply (rule FDERIV_bounded_linear [OF f]) |
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286 |
apply (rule FDERIV_bounded_linear [OF g]) |
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287 |
apply (simp add: norm_Pair) |
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288 |
apply (rule real_LIM_sandwich_zero) |
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289 |
apply (rule LIM_add_zero) |
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290 |
apply (rule FDERIV_D [OF f]) |
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291 |
apply (rule FDERIV_D [OF g]) |
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292 |
apply (rename_tac h) |
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293 |
apply (simp add: divide_nonneg_pos) |
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294 |
apply (rename_tac h) |
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295 |
apply (subst add_divide_distrib [symmetric]) |
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296 |
apply (rule divide_right_mono [OF _ norm_ge_zero]) |
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297 |
apply (rule order_trans [OF sqrt_add_le_add_sqrt]) |
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298 |
apply simp |
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299 |
apply simp |
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300 |
apply simp |
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301 |
done |
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302 |
|
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303 |
end |