author | paulson <lp15@cam.ac.uk> |
Sun, 15 Apr 2018 21:41:40 +0100 | |
changeset 67986 | b65c4a6a015e |
parent 67982 | 7643b005b29a |
child 68038 | 20b713cff87a |
child 68072 | 493b818e8e10 |
permissions | -rw-r--r-- |
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section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space\<close> |
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|
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theory Cartesian_Euclidean_Space |
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imports Finite_Cartesian_Product Derivative |
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begin |
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|
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lemma norm_le_componentwise: |
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"(\<And>b. b \<in> Basis \<Longrightarrow> abs(x \<bullet> b) \<le> abs(y \<bullet> b)) \<Longrightarrow> norm x \<le> norm y" |
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by (auto simp: norm_le euclidean_inner [of x x] euclidean_inner [of y y] abs_le_square_iff power2_eq_square intro!: sum_mono) |
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|
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lemma norm_le_componentwise_cart: |
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fixes x :: "real^'n" |
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shows "(\<And>i. abs(x$i) \<le> abs(y$i)) \<Longrightarrow> norm x \<le> norm y" |
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unfolding cart_eq_inner_axis |
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by (rule norm_le_componentwise) (metis axis_index) |
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|
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numerous theorems about affine hulls, hyperplanes, etc.
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lemma subspace_special_hyperplane: "subspace {x. x $ k = 0}" |
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by (simp add: subspace_def) |
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lemma sum_mult_product: |
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"sum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))" |
|
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unfolding sum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric] |
|
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proof (rule sum.cong, simp, rule sum.reindex_cong) |
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fix i |
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show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI) |
|
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show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}" |
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proof safe |
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fix j assume "j \<in> {i * B..<i * B + B}" |
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then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}" |
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by (auto intro!: image_eqI[of _ _ "j - i * B"]) |
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qed simp |
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qed simp |
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subsection\<open>Basic componentwise operations on vectors\<close> |
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instantiation vec :: (times, finite) times |
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begin |
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|
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definition "( * ) \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))" |
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instance .. |
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||
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end |
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|
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instantiation vec :: (one, finite) one |
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begin |
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|
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definition "1 \<equiv> (\<chi> i. 1)" |
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instance .. |
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||
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end |
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instantiation vec :: (ord, finite) ord |
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begin |
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|
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definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)" |
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definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" |
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instance .. |
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||
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end |
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text\<open>The ordering on one-dimensional vectors is linear.\<close> |
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class cart_one = |
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assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0" |
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begin |
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|
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subclass finite |
|
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proof |
|
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from UNIV_one show "finite (UNIV :: 'a set)" |
|
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by (auto intro!: card_ge_0_finite) |
|
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qed |
|
72 |
||
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end |
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|
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instance vec:: (order, finite) order |
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by standard (auto simp: less_eq_vec_def less_vec_def vec_eq_iff |
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intro: order.trans order.antisym order.strict_implies_order) |
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|
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instance vec :: (linorder, cart_one) linorder |
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proof |
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obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" |
|
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proof - |
|
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have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one) |
|
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then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq) |
|
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then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto |
|
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then show thesis by (auto intro: that) |
|
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qed |
|
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fix x y :: "'a^'b::cart_one" |
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note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps |
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ordered_euclidean_space compatible with more standard pointwise ordering on products; conditionally complete lattice with product order
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show "x \<le> y \<or> y \<le> x" by auto |
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qed |
92 |
||
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text\<open>Constant Vectors\<close> |
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|
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definition "vec x = (\<chi> i. x)" |
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|
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lemma interval_cbox_cart: "{a::real^'n..b} = cbox a b" |
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by (auto simp add: less_eq_vec_def mem_box Basis_vec_def inner_axis) |
|
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||
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text\<open>Also the scalar-vector multiplication.\<close> |
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|
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definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70) |
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where "c *s x = (\<chi> i. c * (x$i))" |
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104 |
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subsection \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close> |
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lemma sum_cong_aux: |
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"(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A" |
|
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by (auto intro: sum.cong) |
|
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|
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hide_fact (open) sum_cong_aux |
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|
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method_setup vector = \<open> |
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let |
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val ss1 = |
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simpset_of (put_simpset HOL_basic_ss @{context} |
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addsimps [@{thm sum.distrib} RS sym, |
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@{thm sum_subtractf} RS sym, @{thm sum_distrib_left}, |
|
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@{thm sum_distrib_right}, @{thm sum_negf} RS sym]) |
|
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val ss2 = |
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simpset_of (@{context} addsimps |
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[@{thm plus_vec_def}, @{thm times_vec_def}, |
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@{thm minus_vec_def}, @{thm uminus_vec_def}, |
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@{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def}, |
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@{thm scaleR_vec_def}, |
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@{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]) |
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fun vector_arith_tac ctxt ths = |
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129 |
simp_tac (put_simpset ss1 ctxt) |
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THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i |
131 |
ORELSE resolve_tac ctxt @{thms sum.neutral} i |
|
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ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i) |
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(* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *) |
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THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths) |
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in |
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Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths)) |
49644 | 137 |
end |
60420 | 138 |
\<close> "lift trivial vector statements to real arith statements" |
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139 |
|
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lemma vec_0[simp]: "vec 0 = 0" by vector |
141 |
lemma vec_1[simp]: "vec 1 = 1" by vector |
|
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|
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143 |
lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector |
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|
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lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto |
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146 |
|
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lemma vec_add: "vec(x + y) = vec x + vec y" by vector |
148 |
lemma vec_sub: "vec(x - y) = vec x - vec y" by vector |
|
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lemma vec_cmul: "vec(c * x) = c *s vec x " by vector |
|
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lemma vec_neg: "vec(- x) = - vec x " by vector |
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|
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lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x" |
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by vector |
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|
64267 | 155 |
lemma vec_sum: |
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assumes "finite S" |
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shows "vec(sum f S) = sum (vec \<circ> f) S" |
49644 | 158 |
using assms |
159 |
proof induct |
|
160 |
case empty |
|
161 |
then show ?case by simp |
|
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next |
|
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case insert |
|
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then show ?case by (auto simp add: vec_add) |
|
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qed |
|
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|
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text\<open>Obvious "component-pushing".\<close> |
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|
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lemma vec_component [simp]: "vec x $ i = x" |
57865 | 170 |
by vector |
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|
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lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i" |
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by vector |
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|
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lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)" |
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by vector |
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|
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector |
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|
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lemmas vector_component = |
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vec_component vector_add_component vector_mult_component |
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vector_smult_component vector_minus_component vector_uminus_component |
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vector_scaleR_component cond_component |
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|
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|
67968 | 186 |
subsection \<open>Some frequently useful arithmetic lemmas over vectors\<close> |
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instance vec :: (semigroup_mult, finite) semigroup_mult |
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by standard (vector mult.assoc) |
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|
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instance vec :: (monoid_mult, finite) monoid_mult |
61169 | 192 |
by standard vector+ |
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193 |
|
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instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult |
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by standard (vector mult.commute) |
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196 |
|
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult |
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by standard vector |
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199 |
|
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instance vec :: (semiring, finite) semiring |
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by standard (vector field_simps)+ |
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202 |
|
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instance vec :: (semiring_0, finite) semiring_0 |
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by standard (vector field_simps)+ |
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instance vec :: (semiring_1, finite) semiring_1 |
61169 | 206 |
by standard vector |
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instance vec :: (comm_semiring, finite) comm_semiring |
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by standard (vector field_simps)+ |
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209 |
|
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210 |
instance vec :: (comm_semiring_0, finite) comm_semiring_0 .. |
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instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add .. |
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212 |
instance vec :: (semiring_0_cancel, finite) semiring_0_cancel .. |
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instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel .. |
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214 |
instance vec :: (ring, finite) ring .. |
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215 |
instance vec :: (semiring_1_cancel, finite) semiring_1_cancel .. |
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instance vec :: (comm_semiring_1, finite) comm_semiring_1 .. |
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217 |
|
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instance vec :: (ring_1, finite) ring_1 .. |
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219 |
|
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instance vec :: (real_algebra, finite) real_algebra |
61169 | 221 |
by standard (simp_all add: vec_eq_iff) |
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222 |
|
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223 |
instance vec :: (real_algebra_1, finite) real_algebra_1 .. |
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|
49644 | 225 |
lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" |
226 |
proof (induct n) |
|
227 |
case 0 |
|
228 |
then show ?case by vector |
|
229 |
next |
|
230 |
case Suc |
|
231 |
then show ?case by vector |
|
232 |
qed |
|
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233 |
|
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234 |
lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1" |
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235 |
by vector |
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236 |
|
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237 |
lemma neg_one_index [simp]: "(- 1 :: 'a :: {one, uminus} ^ 'n) $ i = - 1" |
49644 | 238 |
by vector |
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239 |
|
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240 |
instance vec :: (semiring_char_0, finite) semiring_char_0 |
38621
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241 |
proof |
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242 |
fix m n :: nat |
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243 |
show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)" |
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244 |
by (auto intro!: injI simp add: vec_eq_iff of_nat_index) |
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245 |
qed |
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246 |
|
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247 |
instance vec :: (numeral, finite) numeral .. |
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248 |
instance vec :: (semiring_numeral, finite) semiring_numeral .. |
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249 |
|
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250 |
lemma numeral_index [simp]: "numeral w $ i = numeral w" |
49644 | 251 |
by (induct w) (simp_all only: numeral.simps vector_add_component one_index) |
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252 |
|
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253 |
lemma neg_numeral_index [simp]: "- numeral w $ i = - numeral w" |
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254 |
by (simp only: vector_uminus_component numeral_index) |
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255 |
|
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256 |
instance vec :: (comm_ring_1, finite) comm_ring_1 .. |
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257 |
instance vec :: (ring_char_0, finite) ring_char_0 .. |
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258 |
|
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259 |
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" |
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260 |
by (vector mult.assoc) |
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261 |
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" |
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262 |
by (vector field_simps) |
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263 |
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" |
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264 |
by (vector field_simps) |
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265 |
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector |
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266 |
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector |
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267 |
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y" |
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268 |
by (vector field_simps) |
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269 |
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector |
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270 |
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector |
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271 |
lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector |
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272 |
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector |
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273 |
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x" |
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274 |
by (vector field_simps) |
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275 |
|
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276 |
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" |
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277 |
by (simp add: vec_eq_iff) |
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278 |
|
67979
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parents:
67971
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|
279 |
lemma linear_vec [simp]: "linear vec" |
53323937ee25
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paulson <lp15@cam.ac.uk>
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280 |
by (simp add: linearI vec_add vec_eq_iff) |
53323937ee25
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paulson <lp15@cam.ac.uk>
parents:
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|
281 |
|
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|
282 |
lemma differentiable_vec: |
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parents:
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|
283 |
fixes S :: "'a::euclidean_space set" |
53323937ee25
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parents:
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|
284 |
shows "vec differentiable_on S" |
53323937ee25
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|
285 |
by (simp add: linear_linear bounded_linear_imp_differentiable_on) |
53323937ee25
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parents:
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286 |
|
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287 |
lemma continuous_vec [continuous_intros]: |
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|
288 |
fixes x :: "'a::euclidean_space" |
53323937ee25
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parents:
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|
289 |
shows "isCont vec x" |
53323937ee25
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changeset
|
290 |
apply (clarsimp simp add: continuous_def LIM_def dist_vec_def L2_set_def) |
53323937ee25
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parents:
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changeset
|
291 |
apply (rule_tac x="r / sqrt (real CARD('b))" in exI) |
53323937ee25
new material about vec, real^1, etc.
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parents:
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|
292 |
by (simp add: mult.commute pos_less_divide_eq real_sqrt_mult) |
53323937ee25
new material about vec, real^1, etc.
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parents:
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293 |
|
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
294 |
lemma box_vec_eq_empty [simp]: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
295 |
shows "cbox (vec a) (vec b) = {} \<longleftrightarrow> cbox a b = {}" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
296 |
"box (vec a) (vec b) = {} \<longleftrightarrow> box a b = {}" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
297 |
by (auto simp: Basis_vec_def mem_box box_eq_empty inner_axis) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
298 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
299 |
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) |
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
300 |
|
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
301 |
lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1" |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
302 |
by (simp add: inner_axis' norm_eq_1) |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
303 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
304 |
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
305 |
by vector |
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
306 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
307 |
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
308 |
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) |
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
309 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
310 |
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
311 |
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) |
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
312 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
313 |
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==> a *s x = a *s y ==> (x = y)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
314 |
by (metis vector_mul_lcancel) |
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
315 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
316 |
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
317 |
by (metis vector_mul_rcancel) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
318 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
319 |
lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
320 |
apply (simp add: norm_vec_def) |
67155 | 321 |
apply (rule member_le_L2_set, simp_all) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
322 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
323 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
324 |
lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
325 |
by (metis component_le_norm_cart order_trans) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
326 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
327 |
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e" |
53595 | 328 |
by (metis component_le_norm_cart le_less_trans) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
329 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
330 |
lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV" |
67155 | 331 |
by (simp add: norm_vec_def L2_set_le_sum) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
332 |
|
67969
83c8cafdebe8
Syntax for the special cases Min(A`I) and Max (A`I)
paulson <lp15@cam.ac.uk>
parents:
67962
diff
changeset
|
333 |
lemma scalar_mult_eq_scaleR [simp]: "c *s x = c *\<^sub>R x" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
334 |
unfolding scaleR_vec_def vector_scalar_mult_def by simp |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
335 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
336 |
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
337 |
unfolding dist_norm scalar_mult_eq_scaleR |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
338 |
unfolding scaleR_right_diff_distrib[symmetric] by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
339 |
|
64267 | 340 |
lemma sum_component [simp]: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
341 |
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n" |
64267 | 342 |
shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S" |
49644 | 343 |
proof (cases "finite S") |
344 |
case True |
|
345 |
then show ?thesis by induct simp_all |
|
346 |
next |
|
347 |
case False |
|
348 |
then show ?thesis by simp |
|
349 |
qed |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
350 |
|
64267 | 351 |
lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
352 |
by (simp add: vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
353 |
|
64267 | 354 |
lemma sum_cmul: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
355 |
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n" |
64267 | 356 |
shows "sum (\<lambda>x. c *s f x) S = c *s sum f S" |
357 |
by (simp add: vec_eq_iff sum_distrib_left) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
358 |
|
64267 | 359 |
lemma sum_norm_allsubsets_bound_cart: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
360 |
fixes f:: "'a \<Rightarrow> real ^'n" |
64267 | 361 |
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e" |
362 |
shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e" |
|
363 |
using sum_norm_allsubsets_bound[OF assms] |
|
57865 | 364 |
by simp |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
365 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
366 |
subsection\<open>Closures and interiors of halfspaces\<close> |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
367 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
368 |
lemma interior_halfspace_le [simp]: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
369 |
assumes "a \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
370 |
shows "interior {x. a \<bullet> x \<le> b} = {x. a \<bullet> x < b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
371 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
372 |
have *: "a \<bullet> x < b" if x: "x \<in> S" and S: "S \<subseteq> {x. a \<bullet> x \<le> b}" and "open S" for S x |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
373 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
374 |
obtain e where "e>0" and e: "cball x e \<subseteq> S" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
375 |
using \<open>open S\<close> open_contains_cball x by blast |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
376 |
then have "x + (e / norm a) *\<^sub>R a \<in> cball x e" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
377 |
by (simp add: dist_norm) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
378 |
then have "x + (e / norm a) *\<^sub>R a \<in> S" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
379 |
using e by blast |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
380 |
then have "x + (e / norm a) *\<^sub>R a \<in> {x. a \<bullet> x \<le> b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
381 |
using S by blast |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
382 |
moreover have "e * (a \<bullet> a) / norm a > 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
383 |
by (simp add: \<open>0 < e\<close> assms) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
384 |
ultimately show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
385 |
by (simp add: algebra_simps) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
386 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
387 |
show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
388 |
by (rule interior_unique) (auto simp: open_halfspace_lt *) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
389 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
390 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
391 |
lemma interior_halfspace_ge [simp]: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
392 |
"a \<noteq> 0 \<Longrightarrow> interior {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x > b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
393 |
using interior_halfspace_le [of "-a" "-b"] by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
394 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
395 |
lemma interior_halfspace_component_le [simp]: |
67731 | 396 |
"interior {x. x$k \<le> a} = {x :: (real^'n). x$k < a}" (is "?LE") |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
397 |
and interior_halfspace_component_ge [simp]: |
67731 | 398 |
"interior {x. x$k \<ge> a} = {x :: (real^'n). x$k > a}" (is "?GE") |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
399 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
400 |
have "axis k (1::real) \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
401 |
by (simp add: axis_def vec_eq_iff) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
402 |
moreover have "axis k (1::real) \<bullet> x = x$k" for x |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
403 |
by (simp add: cart_eq_inner_axis inner_commute) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
404 |
ultimately show ?LE ?GE |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
405 |
using interior_halfspace_le [of "axis k (1::real)" a] |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
406 |
interior_halfspace_ge [of "axis k (1::real)" a] by auto |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
407 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
408 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
409 |
lemma closure_halfspace_lt [simp]: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
410 |
assumes "a \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
411 |
shows "closure {x. a \<bullet> x < b} = {x. a \<bullet> x \<le> b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
412 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
413 |
have [simp]: "-{x. a \<bullet> x < b} = {x. a \<bullet> x \<ge> b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
414 |
by (force simp:) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
415 |
then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
416 |
using interior_halfspace_ge [of a b] assms |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
417 |
by (force simp: closure_interior) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
418 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
419 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
420 |
lemma closure_halfspace_gt [simp]: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
421 |
"a \<noteq> 0 \<Longrightarrow> closure {x. a \<bullet> x > b} = {x. a \<bullet> x \<ge> b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
422 |
using closure_halfspace_lt [of "-a" "-b"] by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
423 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
424 |
lemma closure_halfspace_component_lt [simp]: |
67731 | 425 |
"closure {x. x$k < a} = {x :: (real^'n). x$k \<le> a}" (is "?LE") |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
426 |
and closure_halfspace_component_gt [simp]: |
67731 | 427 |
"closure {x. x$k > a} = {x :: (real^'n). x$k \<ge> a}" (is "?GE") |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
428 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
429 |
have "axis k (1::real) \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
430 |
by (simp add: axis_def vec_eq_iff) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
431 |
moreover have "axis k (1::real) \<bullet> x = x$k" for x |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
432 |
by (simp add: cart_eq_inner_axis inner_commute) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
433 |
ultimately show ?LE ?GE |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
434 |
using closure_halfspace_lt [of "axis k (1::real)" a] |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
435 |
closure_halfspace_gt [of "axis k (1::real)" a] by auto |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
436 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
437 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
438 |
lemma interior_hyperplane [simp]: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
439 |
assumes "a \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
440 |
shows "interior {x. a \<bullet> x = b} = {}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
441 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
442 |
have [simp]: "{x. a \<bullet> x = b} = {x. a \<bullet> x \<le> b} \<inter> {x. a \<bullet> x \<ge> b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
443 |
by (force simp:) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
444 |
then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
445 |
by (auto simp: assms) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
446 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
447 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
448 |
lemma frontier_halfspace_le: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
449 |
assumes "a \<noteq> 0 \<or> b \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
450 |
shows "frontier {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
451 |
proof (cases "a = 0") |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
452 |
case True with assms show ?thesis by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
453 |
next |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
454 |
case False then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
455 |
by (force simp: frontier_def closed_halfspace_le) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
456 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
457 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
458 |
lemma frontier_halfspace_ge: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
459 |
assumes "a \<noteq> 0 \<or> b \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
460 |
shows "frontier {x. a \<bullet> x \<ge> b} = {x. a \<bullet> x = b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
461 |
proof (cases "a = 0") |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
462 |
case True with assms show ?thesis by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
463 |
next |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
464 |
case False then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
465 |
by (force simp: frontier_def closed_halfspace_ge) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
466 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
467 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
468 |
lemma frontier_halfspace_lt: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
469 |
assumes "a \<noteq> 0 \<or> b \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
470 |
shows "frontier {x. a \<bullet> x < b} = {x. a \<bullet> x = b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
471 |
proof (cases "a = 0") |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
472 |
case True with assms show ?thesis by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
473 |
next |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
474 |
case False then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
475 |
by (force simp: frontier_def interior_open open_halfspace_lt) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
476 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
477 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
478 |
lemma frontier_halfspace_gt: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
479 |
assumes "a \<noteq> 0 \<or> b \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
480 |
shows "frontier {x. a \<bullet> x > b} = {x. a \<bullet> x = b}" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
481 |
proof (cases "a = 0") |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
482 |
case True with assms show ?thesis by simp |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
483 |
next |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
484 |
case False then show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
485 |
by (force simp: frontier_def interior_open open_halfspace_gt) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
486 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
487 |
|
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
488 |
lemma interior_standard_hyperplane: |
67731 | 489 |
"interior {x :: (real^'n). x$k = a} = {}" |
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
490 |
proof - |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
491 |
have "axis k (1::real) \<noteq> 0" |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
492 |
by (simp add: axis_def vec_eq_iff) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
493 |
moreover have "axis k (1::real) \<bullet> x = x$k" for x |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
494 |
by (simp add: cart_eq_inner_axis inner_commute) |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
495 |
ultimately show ?thesis |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
496 |
using interior_hyperplane [of "axis k (1::real)" a] |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
497 |
by force |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
498 |
qed |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset
|
499 |
|
60420 | 500 |
subsection \<open>Matrix operations\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
501 |
|
60420 | 502 |
text\<open>Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"}\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
503 |
|
67962 | 504 |
definition map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where |
505 |
"map_matrix f x = (\<chi> i j. f (x $ i $ j))" |
|
506 |
||
507 |
lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)" |
|
508 |
by (simp add: map_matrix_def) |
|
509 |
||
49644 | 510 |
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" |
511 |
(infixl "**" 70) |
|
64267 | 512 |
where "m ** m' == (\<chi> i j. sum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
513 |
|
49644 | 514 |
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" |
515 |
(infixl "*v" 70) |
|
64267 | 516 |
where "m *v x \<equiv> (\<chi> i. sum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
517 |
|
49644 | 518 |
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " |
519 |
(infixl "v*" 70) |
|
64267 | 520 |
where "v v* m == (\<chi> j. sum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
521 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
522 |
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)" |
63332 | 523 |
definition transpose where |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
524 |
"(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
525 |
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
526 |
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
527 |
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
528 |
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
529 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
530 |
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
531 |
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" |
64267 | 532 |
by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
533 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
534 |
lemma matrix_mul_lid [simp]: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
535 |
fixes A :: "'a::semiring_1 ^ 'm ^ 'n" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
536 |
shows "mat 1 ** A = A" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
537 |
apply (simp add: matrix_matrix_mult_def mat_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
538 |
apply vector |
64267 | 539 |
apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite] |
49644 | 540 |
mult_1_left mult_zero_left if_True UNIV_I) |
541 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
542 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
543 |
lemma matrix_mul_rid [simp]: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
544 |
fixes A :: "'a::semiring_1 ^ 'm ^ 'n" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
545 |
shows "A ** mat 1 = A" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
546 |
apply (simp add: matrix_matrix_mult_def mat_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
547 |
apply vector |
64267 | 548 |
apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite] |
49644 | 549 |
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) |
550 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
551 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
552 |
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" |
64267 | 553 |
apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc) |
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset
|
554 |
apply (subst sum.swap) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
555 |
apply simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
556 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
557 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
558 |
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" |
49644 | 559 |
apply (vector matrix_matrix_mult_def matrix_vector_mult_def |
64267 | 560 |
sum_distrib_left sum_distrib_right mult.assoc) |
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset
|
561 |
apply (subst sum.swap) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
562 |
apply simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
563 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
564 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
565 |
lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
566 |
apply (vector matrix_vector_mult_def mat_def) |
64267 | 567 |
apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong) |
49644 | 568 |
done |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
569 |
|
49644 | 570 |
lemma matrix_transpose_mul: |
571 |
"transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
572 |
by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult.commute) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
573 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
574 |
lemma matrix_eq: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
575 |
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
576 |
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs") |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
577 |
apply auto |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
578 |
apply (subst vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
579 |
apply clarify |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
580 |
apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
581 |
apply (erule_tac x="axis ia 1" in allE) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
582 |
apply (erule_tac x="i" in allE) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
583 |
apply (auto simp add: if_distrib cond_application_beta axis_def |
64267 | 584 |
sum.delta[OF finite] cong del: if_weak_cong) |
49644 | 585 |
done |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
586 |
|
49644 | 587 |
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
588 |
by (simp add: matrix_vector_mult_def inner_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
589 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
590 |
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)" |
64267 | 591 |
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps) |
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset
|
592 |
apply (subst sum.swap) |
49644 | 593 |
apply simp |
594 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
595 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
596 |
lemma transpose_mat [simp]: "transpose (mat n) = mat n" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
597 |
by (vector transpose_def mat_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
598 |
|
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
599 |
lemma transpose_transpose [simp]: "transpose(transpose A) = A" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
600 |
by (vector transpose_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
601 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
602 |
lemma row_transpose [simp]: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
603 |
fixes A:: "'a::semiring_1^_^_" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
604 |
shows "row i (transpose A) = column i A" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
605 |
by (simp add: row_def column_def transpose_def vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
606 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
607 |
lemma column_transpose [simp]: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
608 |
fixes A:: "'a::semiring_1^_^_" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
609 |
shows "column i (transpose A) = row i A" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
610 |
by (simp add: row_def column_def transpose_def vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
611 |
|
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
612 |
lemma rows_transpose [simp]: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" |
49644 | 613 |
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
614 |
|
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
615 |
lemma columns_transpose [simp]: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" |
49644 | 616 |
by (metis transpose_transpose rows_transpose) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
617 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
618 |
lemma matrix_mult_transpose_dot_column: |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
619 |
fixes A :: "real^'n^'n" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
620 |
shows "transpose A ** A = (\<chi> i j. (column i A) \<bullet> (column j A))" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
621 |
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
622 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
623 |
lemma matrix_mult_transpose_dot_row: |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
624 |
fixes A :: "real^'n^'n" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
625 |
shows "A ** transpose A = (\<chi> i j. (row i A) \<bullet> (row j A))" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
626 |
by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
627 |
|
60420 | 628 |
text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
629 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
630 |
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
631 |
by (simp add: matrix_vector_mult_def inner_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
632 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
633 |
lemma matrix_mult_sum: |
64267 | 634 |
"(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
635 |
by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
636 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
637 |
lemma vector_componentwise: |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
638 |
"(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)" |
64267 | 639 |
by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
640 |
|
64267 | 641 |
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" |
642 |
by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
643 |
|
63938 | 644 |
lemma linear_componentwise_expansion: |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
645 |
fixes f:: "real ^'m \<Rightarrow> real ^ _" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
646 |
assumes lf: "linear f" |
64267 | 647 |
shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") |
49644 | 648 |
proof - |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
649 |
let ?M = "(UNIV :: 'm set)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
650 |
let ?N = "(UNIV :: 'n set)" |
64267 | 651 |
have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j" |
652 |
unfolding sum_component by simp |
|
49644 | 653 |
then show ?thesis |
64267 | 654 |
unfolding linear_sum_mul[OF lf, symmetric] |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
655 |
unfolding scalar_mult_eq_scaleR[symmetric] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
656 |
unfolding basis_expansion |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
657 |
by simp |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
658 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
659 |
|
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
660 |
subsection\<open>Inverse matrices (not necessarily square)\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
661 |
|
49644 | 662 |
definition |
663 |
"invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
664 |
|
49644 | 665 |
definition |
666 |
"matrix_inv(A:: 'a::semiring_1^'n^'m) = |
|
667 |
(SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
668 |
|
60420 | 669 |
text\<open>Correspondence between matrices and linear operators.\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
670 |
|
49644 | 671 |
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
672 |
where "matrix f = (\<chi> i j. (f(axis j 1))$i)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
673 |
|
67986
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
674 |
lemma matrix_id_mat_1: "matrix id = mat 1" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
675 |
by (simp add: mat_def matrix_def axis_def) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
676 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
677 |
lemma matrix_scaleR: "(matrix (( *\<^sub>R) r)) = mat r" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
678 |
by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
679 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
680 |
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))" |
53600
8fda7ad57466
make 'linear' into a sublocale of 'bounded_linear';
huffman
parents:
53595
diff
changeset
|
681 |
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff |
64267 | 682 |
field_simps sum_distrib_left sum.distrib) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
683 |
|
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
684 |
lemma |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
685 |
fixes A :: "real^'n^'m" |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
686 |
shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont (( *v) A) z" |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
687 |
and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S (( *v) A)" |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
688 |
by (simp_all add: linear_linear linear_continuous_at linear_continuous_on matrix_vector_mul_linear) |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
689 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
690 |
lemma matrix_vector_mult_add_distrib [algebra_simps]: |
67728 | 691 |
"A *v (x + y) = A *v x + A *v y" |
692 |
by (vector matrix_vector_mult_def sum.distrib distrib_left) |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
693 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
694 |
lemma matrix_vector_mult_diff_distrib [algebra_simps]: |
67728 | 695 |
fixes A :: "'a::ring_1^'n^'m" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
696 |
shows "A *v (x - y) = A *v x - A *v y" |
67728 | 697 |
by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
698 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
699 |
lemma matrix_vector_mult_scaleR[algebra_simps]: |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
700 |
fixes A :: "real^'n^'m" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
701 |
shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
702 |
using linear_iff matrix_vector_mul_linear by blast |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
703 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
704 |
lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
705 |
by (simp add: matrix_vector_mult_def vec_eq_iff) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
706 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
707 |
lemma matrix_vector_mult_0 [simp]: "0 *v w = 0" |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
708 |
by (simp add: matrix_vector_mult_def vec_eq_iff) |
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
709 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
710 |
lemma matrix_vector_mult_add_rdistrib [algebra_simps]: |
67728 | 711 |
"(A + B) *v x = (A *v x) + (B *v x)" |
712 |
by (vector matrix_vector_mult_def sum.distrib distrib_right) |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
713 |
|
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
714 |
lemma matrix_vector_mult_diff_rdistrib [algebra_simps]: |
67728 | 715 |
fixes A :: "'a :: ring_1^'n^'m" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
716 |
shows "(A - B) *v x = (A *v x) - (B *v x)" |
67728 | 717 |
by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
718 |
|
49644 | 719 |
lemma matrix_works: |
720 |
assumes lf: "linear f" |
|
721 |
shows "matrix f *v x = f (x::real ^ 'n)" |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
722 |
apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute) |
63938 | 723 |
by (simp add: linear_componentwise_expansion lf) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
724 |
|
49644 | 725 |
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" |
726 |
by (simp add: ext matrix_works) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
727 |
|
67683
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
728 |
declare matrix_vector_mul [symmetric, simp] |
817944aeac3f
Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67673
diff
changeset
|
729 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
730 |
lemma matrix_of_matrix_vector_mul [simp]: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
731 |
by (simp add: matrix_eq matrix_vector_mul_linear matrix_works) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
732 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
733 |
lemma matrix_compose: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
734 |
assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)" |
49644 | 735 |
and lg: "linear (g::real^'m \<Rightarrow> real^_)" |
61736 | 736 |
shows "matrix (g \<circ> f) = matrix g ** matrix f" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
737 |
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] |
49644 | 738 |
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
739 |
|
49644 | 740 |
lemma matrix_vector_column: |
64267 | 741 |
"(A::'a::comm_semiring_1^'n^_) *v x = sum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)" |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
742 |
by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult.commute) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
743 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
744 |
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
745 |
apply (rule adjoint_unique) |
49644 | 746 |
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def |
64267 | 747 |
sum_distrib_right sum_distrib_left) |
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset
|
748 |
apply (subst sum.swap) |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
749 |
apply (auto simp add: ac_simps) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
750 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
751 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
752 |
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
753 |
shows "matrix(adjoint f) = transpose(matrix f)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
754 |
apply (subst matrix_vector_mul[OF lf]) |
49644 | 755 |
unfolding adjoint_matrix matrix_of_matrix_vector_mul |
756 |
apply rule |
|
757 |
done |
|
758 |
||
67981
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
759 |
lemma inj_matrix_vector_mult: |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
760 |
fixes A::"'a::field^'n^'m" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
761 |
assumes "invertible A" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
762 |
shows "inj (( *v) A)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
763 |
by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
764 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
765 |
|
67968 | 766 |
subsection\<open>Some bounds on components etc. relative to operator norm\<close> |
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
767 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
768 |
lemma norm_column_le_onorm: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
769 |
fixes A :: "real^'n^'m" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
770 |
shows "norm(column i A) \<le> onorm(( *v) A)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
771 |
proof - |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
772 |
have bl: "bounded_linear (( *v) A)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
773 |
by (simp add: linear_linear matrix_vector_mul_linear) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
774 |
have "norm (\<chi> j. A $ j $ i) \<le> norm (A *v axis i 1)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
775 |
by (simp add: matrix_mult_dot cart_eq_inner_axis) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
776 |
also have "\<dots> \<le> onorm (( *v) A)" |
67982
7643b005b29a
various new results on measures, integrals, etc., and some simplified proofs
paulson <lp15@cam.ac.uk>
parents:
67981
diff
changeset
|
777 |
using onorm [OF bl, of "axis i 1"] by auto |
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
778 |
finally have "norm (\<chi> j. A $ j $ i) \<le> onorm (( *v) A)" . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
779 |
then show ?thesis |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
780 |
unfolding column_def . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
781 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
782 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
783 |
lemma matrix_component_le_onorm: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
784 |
fixes A :: "real^'n^'m" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
785 |
shows "\<bar>A $ i $ j\<bar> \<le> onorm(( *v) A)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
786 |
proof - |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
787 |
have "\<bar>A $ i $ j\<bar> \<le> norm (\<chi> n. (A $ n $ j))" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
788 |
by (metis (full_types, lifting) component_le_norm_cart vec_lambda_beta) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
789 |
also have "\<dots> \<le> onorm (( *v) A)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
790 |
by (metis (no_types) column_def norm_column_le_onorm) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
791 |
finally show ?thesis . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
792 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
793 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
794 |
lemma component_le_onorm: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
795 |
fixes f :: "real^'m \<Rightarrow> real^'n" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
796 |
shows "linear f \<Longrightarrow> \<bar>matrix f $ i $ j\<bar> \<le> onorm f" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
797 |
by (metis matrix_component_le_onorm matrix_vector_mul) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
798 |
|
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
799 |
lemma onorm_le_matrix_component_sum: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
800 |
fixes A :: "real^'n^'m" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
801 |
shows "onorm(( *v) A) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
802 |
proof (rule onorm_le) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
803 |
fix x |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
804 |
have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
805 |
by (rule norm_le_l1_cart) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
806 |
also have "\<dots> \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
807 |
proof (rule sum_mono) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
808 |
fix i |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
809 |
have "\<bar>(A *v x) $ i\<bar> \<le> \<bar>\<Sum>j\<in>UNIV. A $ i $ j * x $ j\<bar>" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
810 |
by (simp add: matrix_vector_mult_def) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
811 |
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j * x $ j\<bar>)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
812 |
by (rule sum_abs) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
813 |
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
814 |
by (rule sum_mono) (simp add: abs_mult component_le_norm_cart mult_left_mono) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
815 |
finally show "\<bar>(A *v x) $ i\<bar> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar> * norm x)" . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
816 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
817 |
finally show "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
818 |
by (simp add: sum_distrib_right) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
819 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
820 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
821 |
lemma onorm_le_matrix_component: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
822 |
fixes A :: "real^'n^'m" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
823 |
assumes "\<And>i j. abs(A$i$j) \<le> B" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
824 |
shows "onorm(( *v) A) \<le> real (CARD('m)) * real (CARD('n)) * B" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
825 |
proof (rule onorm_le) |
67731 | 826 |
fix x :: "real^'n::_" |
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
827 |
have "norm (A *v x) \<le> (\<Sum>i\<in>UNIV. \<bar>(A *v x) $ i\<bar>)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
828 |
by (rule norm_le_l1_cart) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
829 |
also have "\<dots> \<le> (\<Sum>i::'m \<in>UNIV. real (CARD('n)) * B * norm x)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
830 |
proof (rule sum_mono) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
831 |
fix i |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
832 |
have "\<bar>(A *v x) $ i\<bar> \<le> norm(A $ i) * norm x" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
833 |
by (simp add: matrix_mult_dot Cauchy_Schwarz_ineq2) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
834 |
also have "\<dots> \<le> (\<Sum>j\<in>UNIV. \<bar>A $ i $ j\<bar>) * norm x" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
835 |
by (simp add: mult_right_mono norm_le_l1_cart) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
836 |
also have "\<dots> \<le> real (CARD('n)) * B * norm x" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
837 |
by (simp add: assms sum_bounded_above mult_right_mono) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
838 |
finally show "\<bar>(A *v x) $ i\<bar> \<le> real (CARD('n)) * B * norm x" . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
839 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
840 |
also have "\<dots> \<le> CARD('m) * real (CARD('n)) * B * norm x" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
841 |
by simp |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
842 |
finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
843 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
844 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
845 |
subsection \<open>lambda skolemization on cartesian products\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
846 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
847 |
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow> |
37494 | 848 |
(\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs") |
49644 | 849 |
proof - |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
850 |
let ?S = "(UNIV :: 'n set)" |
49644 | 851 |
{ assume H: "?rhs" |
852 |
then have ?lhs by auto } |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
853 |
moreover |
49644 | 854 |
{ assume H: "?lhs" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
855 |
then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
856 |
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n" |
49644 | 857 |
{ fix i |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
858 |
from f have "P i (f i)" by metis |
37494 | 859 |
then have "P i (?x $ i)" by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
860 |
} |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
861 |
hence "\<forall>i. P i (?x$i)" by metis |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
862 |
hence ?rhs by metis } |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
863 |
ultimately show ?thesis by metis |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
864 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
865 |
|
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
866 |
lemma rational_approximation: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
867 |
assumes "e > 0" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
868 |
obtains r::real where "r \<in> \<rat>" "\<bar>r - x\<bar> < e" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
869 |
using Rats_dense_in_real [of "x - e/2" "x + e/2"] assms by auto |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
870 |
|
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
871 |
lemma matrix_rational_approximation: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
872 |
fixes A :: "real^'n^'m" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
873 |
assumes "e > 0" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
874 |
obtains B where "\<And>i j. B$i$j \<in> \<rat>" "onorm(\<lambda>x. (A - B) *v x) < e" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
875 |
proof - |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
876 |
have "\<forall>i j. \<exists>q \<in> \<rat>. \<bar>q - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
877 |
using assms by (force intro: rational_approximation [of "e / (2 * CARD('m) * CARD('n))"]) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
878 |
then obtain B where B: "\<And>i j. B$i$j \<in> \<rat>" and Bclo: "\<And>i j. \<bar>B$i$j - A $ i $ j\<bar> < e / (2 * CARD('m) * CARD('n))" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
879 |
by (auto simp: lambda_skolem Bex_def) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
880 |
show ?thesis |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
881 |
proof |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
882 |
have "onorm (( *v) (A - B)) \<le> real CARD('m) * real CARD('n) * |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
883 |
(e / (2 * real CARD('m) * real CARD('n)))" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
884 |
apply (rule onorm_le_matrix_component) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
885 |
using Bclo by (simp add: abs_minus_commute less_imp_le) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
886 |
also have "\<dots> < e" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
887 |
using \<open>0 < e\<close> by (simp add: divide_simps) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
888 |
finally show "onorm (( *v) (A - B)) < e" . |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
889 |
qed (use B in auto) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
890 |
qed |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
891 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
892 |
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
893 |
unfolding inner_simps scalar_mult_eq_scaleR by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
894 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
895 |
lemma left_invertible_transpose: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
896 |
"(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
897 |
by (metis matrix_transpose_mul transpose_mat transpose_transpose) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
898 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
899 |
lemma right_invertible_transpose: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
900 |
"(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
901 |
by (metis matrix_transpose_mul transpose_mat transpose_transpose) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
902 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
903 |
lemma matrix_left_invertible_injective: |
67986
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
904 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
905 |
shows "(\<exists>B. B ** A = mat 1) \<longleftrightarrow> inj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
906 |
proof safe |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
907 |
fix B |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
908 |
assume B: "B ** A = mat 1" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
909 |
show "inj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
910 |
unfolding inj_on_def |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
911 |
by (metis B matrix_vector_mul_assoc matrix_vector_mul_lid) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
912 |
next |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
913 |
assume "inj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
914 |
with linear_injective_left_inverse[OF matrix_vector_mul_linear] |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
915 |
obtain g where "linear g" and g: "g \<circ> ( *v) A = id" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
916 |
by blast |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
917 |
have "matrix g ** A = mat 1" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
918 |
by (metis \<open>linear g\<close> g matrix_compose matrix_id_mat_1 matrix_of_matrix_vector_mul matrix_vector_mul_linear) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
919 |
then show "\<exists>B. B ** A = mat 1" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
920 |
by metis |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
921 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
922 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
923 |
lemma matrix_left_invertible_ker: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
924 |
"(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
925 |
unfolding matrix_left_invertible_injective |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
926 |
using linear_injective_0[OF matrix_vector_mul_linear, of A] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
927 |
by (simp add: inj_on_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
928 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
929 |
lemma matrix_right_invertible_surjective: |
49644 | 930 |
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)" |
931 |
proof - |
|
932 |
{ fix B :: "real ^'m^'n" |
|
933 |
assume AB: "A ** B = mat 1" |
|
934 |
{ fix x :: "real ^ 'm" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
935 |
have "A *v (B *v x) = x" |
49644 | 936 |
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } |
67399 | 937 |
hence "surj (( *v) A)" unfolding surj_def by metis } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
938 |
moreover |
67399 | 939 |
{ assume sf: "surj (( *v) A)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
940 |
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] |
67399 | 941 |
obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "( *v) A \<circ> g = id" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
942 |
by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
943 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
944 |
have "A ** (matrix g) = mat 1" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
945 |
unfolding matrix_eq matrix_vector_mul_lid |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
946 |
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] |
44165 | 947 |
using g(2) unfolding o_def fun_eq_iff id_def |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
948 |
. |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
949 |
hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
950 |
} |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
951 |
ultimately show ?thesis unfolding surj_def by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
952 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
953 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
954 |
lemma matrix_left_invertible_independent_columns: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
955 |
fixes A :: "real^'n^'m" |
49644 | 956 |
shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> |
64267 | 957 |
(\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" |
49644 | 958 |
(is "?lhs \<longleftrightarrow> ?rhs") |
959 |
proof - |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
960 |
let ?U = "UNIV :: 'n set" |
49644 | 961 |
{ assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0" |
962 |
{ fix c i |
|
64267 | 963 |
assume c: "sum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
964 |
let ?x = "\<chi> i. c i" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
965 |
have th0:"A *v ?x = 0" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
966 |
using c |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
967 |
unfolding matrix_mult_sum vec_eq_iff |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
968 |
by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
969 |
from k[rule_format, OF th0] i |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
970 |
have "c i = 0" by (vector vec_eq_iff)} |
49644 | 971 |
hence ?rhs by blast } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
972 |
moreover |
49644 | 973 |
{ assume H: ?rhs |
974 |
{ fix x assume x: "A *v x = 0" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
975 |
let ?c = "\<lambda>i. ((x$i ):: real)" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
976 |
from H[rule_format, of ?c, unfolded matrix_mult_sum[symmetric], OF x] |
49644 | 977 |
have "x = 0" by vector } |
978 |
} |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
979 |
ultimately show ?thesis unfolding matrix_left_invertible_ker by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
980 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
981 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
982 |
lemma matrix_right_invertible_independent_rows: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
983 |
fixes A :: "real^'n^'m" |
49644 | 984 |
shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> |
64267 | 985 |
(\<forall>c. sum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
986 |
unfolding left_invertible_transpose[symmetric] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
987 |
matrix_left_invertible_independent_columns |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
988 |
by (simp add: column_transpose) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
989 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
990 |
lemma matrix_right_invertible_span_columns: |
49644 | 991 |
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> |
992 |
span (columns A) = UNIV" (is "?lhs = ?rhs") |
|
993 |
proof - |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
994 |
let ?U = "UNIV :: 'm set" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
995 |
have fU: "finite ?U" by simp |
64267 | 996 |
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
997 |
unfolding matrix_right_invertible_surjective matrix_mult_sum surj_def |
49644 | 998 |
apply (subst eq_commute) |
999 |
apply rule |
|
1000 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1001 |
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast |
49644 | 1002 |
{ assume h: ?lhs |
1003 |
{ fix x:: "real ^'n" |
|
1004 |
from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" |
|
64267 | 1005 |
where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast |
49644 | 1006 |
have "x \<in> span (columns A)" |
1007 |
unfolding y[symmetric] |
|
64267 | 1008 |
apply (rule span_sum) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1009 |
unfolding scalar_mult_eq_scaleR |
49644 | 1010 |
apply (rule span_mul) |
1011 |
apply (rule span_superset) |
|
1012 |
unfolding columns_def |
|
1013 |
apply blast |
|
1014 |
done |
|
1015 |
} |
|
1016 |
then have ?rhs unfolding rhseq by blast } |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1017 |
moreover |
49644 | 1018 |
{ assume h:?rhs |
64267 | 1019 |
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y" |
49644 | 1020 |
{ fix y |
1021 |
have "?P y" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1022 |
proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) |
64267 | 1023 |
show "\<exists>x::real ^ 'm. sum (\<lambda>i. (x$i) *s column i A) ?U = 0" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1024 |
by (rule exI[where x=0], simp) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1025 |
next |
49644 | 1026 |
fix c y1 y2 |
1027 |
assume y1: "y1 \<in> columns A" and y2: "?P y2" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1028 |
from y1 obtain i where i: "i \<in> ?U" "y1 = column i A" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1029 |
unfolding columns_def by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1030 |
from y2 obtain x:: "real ^'m" where |
64267 | 1031 |
x: "sum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1032 |
let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1033 |
show "?P (c*s y1 + y2)" |
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49644
diff
changeset
|
1034 |
proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong) |
49644 | 1035 |
fix j |
1036 |
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) |
|
1037 |
else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" |
|
1038 |
using i(1) by (simp add: field_simps) |
|
64267 | 1039 |
have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) |
1040 |
else (x$xa) * ((column xa A$j))) ?U = sum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U" |
|
1041 |
apply (rule sum.cong[OF refl]) |
|
49644 | 1042 |
using th apply blast |
1043 |
done |
|
64267 | 1044 |
also have "\<dots> = sum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" |
1045 |
by (simp add: sum.distrib) |
|
1046 |
also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" |
|
1047 |
unfolding sum.delta[OF fU] |
|
49644 | 1048 |
using i(1) by simp |
64267 | 1049 |
finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) |
1050 |
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" . |
|
49644 | 1051 |
qed |
1052 |
next |
|
1053 |
show "y \<in> span (columns A)" |
|
1054 |
unfolding h by blast |
|
1055 |
qed |
|
1056 |
} |
|
1057 |
then have ?lhs unfolding lhseq .. |
|
1058 |
} |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1059 |
ultimately show ?thesis by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1060 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1061 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1062 |
lemma matrix_left_invertible_span_rows: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1063 |
"(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1064 |
unfolding right_invertible_transpose[symmetric] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1065 |
unfolding columns_transpose[symmetric] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1066 |
unfolding matrix_right_invertible_span_columns |
49644 | 1067 |
.. |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1068 |
|
60420 | 1069 |
text \<open>The same result in terms of square matrices.\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1070 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1071 |
lemma matrix_left_right_inverse: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1072 |
fixes A A' :: "real ^'n^'n" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1073 |
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1" |
49644 | 1074 |
proof - |
1075 |
{ fix A A' :: "real ^'n^'n" |
|
1076 |
assume AA': "A ** A' = mat 1" |
|
67399 | 1077 |
have sA: "surj (( *v) A)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1078 |
unfolding surj_def |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1079 |
apply clarify |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1080 |
apply (rule_tac x="(A' *v y)" in exI) |
49644 | 1081 |
apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) |
1082 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1083 |
from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1084 |
obtain f' :: "real ^'n \<Rightarrow> real ^'n" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1085 |
where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1086 |
have th: "matrix f' ** A = mat 1" |
49644 | 1087 |
by (simp add: matrix_eq matrix_works[OF f'(1)] |
1088 |
matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1089 |
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp |
49644 | 1090 |
hence "matrix f' = A'" |
1091 |
by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1092 |
hence "matrix f' ** A = A' ** A" by simp |
49644 | 1093 |
hence "A' ** A = mat 1" by (simp add: th) |
1094 |
} |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1095 |
then show ?thesis by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1096 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1097 |
|
60420 | 1098 |
text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1099 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1100 |
definition "rowvector v = (\<chi> i j. (v$j))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1101 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1102 |
definition "columnvector v = (\<chi> i j. (v$i))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1103 |
|
49644 | 1104 |
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1105 |
by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1106 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1107 |
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1108 |
by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1109 |
|
49644 | 1110 |
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1111 |
by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1112 |
|
49644 | 1113 |
lemma dot_matrix_product: |
1114 |
"(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1" |
|
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1115 |
by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1116 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1117 |
lemma dot_matrix_vector_mul: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1118 |
fixes A B :: "real ^'n ^'n" and x y :: "real ^'n" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1119 |
shows "(A *v x) \<bullet> (B *v y) = |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1120 |
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" |
49644 | 1121 |
unfolding dot_matrix_product transpose_columnvector[symmetric] |
1122 |
dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1123 |
|
61945 | 1124 |
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1125 |
by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1126 |
|
49644 | 1127 |
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1128 |
using Basis_le_infnorm[of "axis i 1" x] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1129 |
by (simp add: Basis_vec_def axis_eq_axis inner_axis) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1130 |
|
63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1131 |
lemma continuous_component[continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)" |
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset
|
1132 |
unfolding continuous_def by (rule tendsto_vec_nth) |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1133 |
|
63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1134 |
lemma continuous_on_component[continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)" |
44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset
|
1135 |
unfolding continuous_on_def by (fast intro: tendsto_vec_nth) |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1136 |
|
63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1137 |
lemma continuous_on_vec_lambda[continuous_intros]: |
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1138 |
"(\<And>i. continuous_on S (f i)) \<Longrightarrow> continuous_on S (\<lambda>x. \<chi> i. f i x)" |
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1139 |
unfolding continuous_on_def by (auto intro: tendsto_vec_lambda) |
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1140 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1141 |
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}" |
63332 | 1142 |
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1143 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1144 |
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)" |
49644 | 1145 |
unfolding bounded_def |
1146 |
apply clarify |
|
1147 |
apply (rule_tac x="x $ i" in exI) |
|
1148 |
apply (rule_tac x="e" in exI) |
|
1149 |
apply clarify |
|
1150 |
apply (rule order_trans [OF dist_vec_nth_le], simp) |
|
1151 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1152 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1153 |
lemma compact_lemma_cart: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1154 |
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n" |
50998 | 1155 |
assumes f: "bounded (range f)" |
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
64267
diff
changeset
|
1156 |
shows "\<exists>l r. strict_mono r \<and> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1157 |
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)" |
62127 | 1158 |
(is "?th d") |
1159 |
proof - |
|
1160 |
have "\<forall>d' \<subseteq> d. ?th d'" |
|
1161 |
by (rule compact_lemma_general[where unproj=vec_lambda]) |
|
1162 |
(auto intro!: f bounded_component_cart simp: vec_lambda_eta) |
|
1163 |
then show "?th d" by simp |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1164 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1165 |
|
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1166 |
instance vec :: (heine_borel, finite) heine_borel |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1167 |
proof |
50998 | 1168 |
fix f :: "nat \<Rightarrow> 'a ^ 'b" |
1169 |
assume f: "bounded (range f)" |
|
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
64267
diff
changeset
|
1170 |
then obtain l r where r: "strict_mono r" |
49644 | 1171 |
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially" |
50998 | 1172 |
using compact_lemma_cart [OF f] by blast |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1173 |
let ?d = "UNIV::'b set" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1174 |
{ fix e::real assume "e>0" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1175 |
hence "0 < e / (real_of_nat (card ?d))" |
49644 | 1176 |
using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1177 |
with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1178 |
by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1179 |
moreover |
49644 | 1180 |
{ fix n |
1181 |
assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1182 |
have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))" |
67155 | 1183 |
unfolding dist_vec_def using zero_le_dist by (rule L2_set_le_sum) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1184 |
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))" |
64267 | 1185 |
by (rule sum_strict_mono) (simp_all add: n) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1186 |
finally have "dist (f (r n)) l < e" by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1187 |
} |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1188 |
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially" |
61810 | 1189 |
by (rule eventually_mono) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1190 |
} |
61973 | 1191 |
hence "((f \<circ> r) \<longlongrightarrow> l) sequentially" unfolding o_def tendsto_iff by simp |
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
64267
diff
changeset
|
1192 |
with r show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1193 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1194 |
|
49644 | 1195 |
lemma interval_cart: |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1196 |
fixes a :: "real^'n" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1197 |
shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" |
56188 | 1198 |
and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}" |
1199 |
by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def mem_box Basis_vec_def inner_axis) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1200 |
|
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1201 |
lemma mem_box_cart: |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1202 |
fixes a :: "real^'n" |
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1203 |
shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)" |
56188 | 1204 |
and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)" |
49644 | 1205 |
using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1206 |
|
49644 | 1207 |
lemma interval_eq_empty_cart: |
1208 |
fixes a :: "real^'n" |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1209 |
shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) |
56188 | 1210 |
and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2) |
49644 | 1211 |
proof - |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1212 |
{ fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1213 |
hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_box_cart by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1214 |
hence "a$i < b$i" by auto |
49644 | 1215 |
hence False using as by auto } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1216 |
moreover |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1217 |
{ assume as:"\<forall>i. \<not> (b$i \<le> a$i)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1218 |
let ?x = "(1/2) *\<^sub>R (a + b)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1219 |
{ fix i |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1220 |
have "a$i < b$i" using as[THEN spec[where x=i]] by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1221 |
hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1222 |
unfolding vector_smult_component and vector_add_component |
49644 | 1223 |
by auto } |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1224 |
hence "box a b \<noteq> {}" using mem_box_cart(1)[of "?x" a b] by auto } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1225 |
ultimately show ?th1 by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1226 |
|
56188 | 1227 |
{ fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1228 |
hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_box_cart by auto |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1229 |
hence "a$i \<le> b$i" by auto |
49644 | 1230 |
hence False using as by auto } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1231 |
moreover |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1232 |
{ assume as:"\<forall>i. \<not> (b$i < a$i)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1233 |
let ?x = "(1/2) *\<^sub>R (a + b)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1234 |
{ fix i |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1235 |
have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1236 |
hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1237 |
unfolding vector_smult_component and vector_add_component |
49644 | 1238 |
by auto } |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1239 |
hence "cbox a b \<noteq> {}" using mem_box_cart(2)[of "?x" a b] by auto } |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1240 |
ultimately show ?th2 by blast |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1241 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1242 |
|
49644 | 1243 |
lemma interval_ne_empty_cart: |
1244 |
fixes a :: "real^'n" |
|
56188 | 1245 |
shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1246 |
and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1247 |
unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1248 |
(* BH: Why doesn't just "auto" work here? *) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1249 |
|
49644 | 1250 |
lemma subset_interval_imp_cart: |
1251 |
fixes a :: "real^'n" |
|
56188 | 1252 |
shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b" |
1253 |
and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b" |
|
1254 |
and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> cbox a b" |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1255 |
and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b" |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1256 |
unfolding subset_eq[unfolded Ball_def] unfolding mem_box_cart |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1257 |
by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1258 |
|
49644 | 1259 |
lemma interval_sing: |
1260 |
fixes a :: "'a::linorder^'n" |
|
1261 |
shows "{a .. a} = {a} \<and> {a<..<a} = {}" |
|
1262 |
apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) |
|
1263 |
done |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1264 |
|
49644 | 1265 |
lemma subset_interval_cart: |
1266 |
fixes a :: "real^'n" |
|
56188 | 1267 |
shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) |
1268 |
and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) |
|
1269 |
and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1270 |
and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4) |
56188 | 1271 |
using subset_box[of c d a b] by (simp_all add: Basis_vec_def inner_axis) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1272 |
|
49644 | 1273 |
lemma disjoint_interval_cart: |
1274 |
fixes a::"real^'n" |
|
56188 | 1275 |
shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) |
1276 |
and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) |
|
1277 |
and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) |
|
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1278 |
and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4) |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1279 |
using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1280 |
|
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1281 |
lemma Int_interval_cart: |
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset
|
1282 |
fixes a :: "real^'n" |
56188 | 1283 |
shows "cbox a b \<inter> cbox c d = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}" |
63945
444eafb6e864
a few new theorems and a renaming
paulson <lp15@cam.ac.uk>
parents:
63938
diff
changeset
|
1284 |
unfolding Int_interval |
56188 | 1285 |
by (auto simp: mem_box less_eq_vec_def) |
1286 |
(auto simp: Basis_vec_def inner_axis) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1287 |
|
49644 | 1288 |
lemma closed_interval_left_cart: |
1289 |
fixes b :: "real^'n" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1290 |
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}" |
63332 | 1291 |
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1292 |
|
49644 | 1293 |
lemma closed_interval_right_cart: |
1294 |
fixes a::"real^'n" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1295 |
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}" |
63332 | 1296 |
by (simp add: Collect_all_eq closed_INT closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1297 |
|
49644 | 1298 |
lemma is_interval_cart: |
1299 |
"is_interval (s::(real^'n) set) \<longleftrightarrow> |
|
1300 |
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1301 |
by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1302 |
|
49644 | 1303 |
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}" |
63332 | 1304 |
by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1305 |
|
49644 | 1306 |
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}" |
63332 | 1307 |
by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1308 |
|
49644 | 1309 |
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}" |
63332 | 1310 |
by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) |
49644 | 1311 |
|
1312 |
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}" |
|
63332 | 1313 |
by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1314 |
|
49644 | 1315 |
lemma Lim_component_le_cart: |
1316 |
fixes f :: "'a \<Rightarrow> real^'n" |
|
61973 | 1317 |
assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f x $i \<le> b) net" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1318 |
shows "l$i \<le> b" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1319 |
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)]) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1320 |
|
49644 | 1321 |
lemma Lim_component_ge_cart: |
1322 |
fixes f :: "'a \<Rightarrow> real^'n" |
|
61973 | 1323 |
assumes "(f \<longlongrightarrow> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1324 |
shows "b \<le> l$i" |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1325 |
by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)]) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1326 |
|
49644 | 1327 |
lemma Lim_component_eq_cart: |
1328 |
fixes f :: "'a \<Rightarrow> real^'n" |
|
61973 | 1329 |
assumes net: "(f \<longlongrightarrow> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1330 |
shows "l$i = b" |
49644 | 1331 |
using ev[unfolded order_eq_iff eventually_conj_iff] and |
1332 |
Lim_component_ge_cart[OF net, of b i] and |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1333 |
Lim_component_le_cart[OF net, of i b] by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1334 |
|
49644 | 1335 |
lemma connected_ivt_component_cart: |
1336 |
fixes x :: "real^'n" |
|
1337 |
shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1338 |
using connected_ivt_hyperplane[of s x y "axis k 1" a] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1339 |
by (auto simp add: inner_axis inner_commute) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1340 |
|
49644 | 1341 |
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1342 |
unfolding subspace_def by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1343 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1344 |
lemma closed_substandard_cart: |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1345 |
"closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}" |
49644 | 1346 |
proof - |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1347 |
{ fix i::'n |
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1348 |
have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}" |
63332 | 1349 |
by (cases "P i") (simp_all add: closed_Collect_eq continuous_on_const continuous_on_id continuous_on_component) } |
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1350 |
thus ?thesis |
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset
|
1351 |
unfolding Collect_all_eq by (simp add: closed_INT) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1352 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1353 |
|
49644 | 1354 |
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" |
1355 |
(is "dim ?A = _") |
|
1356 |
proof - |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1357 |
let ?a = "\<lambda>x. axis x 1 :: real^'n" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1358 |
have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1359 |
by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis) |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1360 |
have "?a ` d \<subseteq> Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1361 |
by (auto simp: Basis_vec_def) |
49644 | 1362 |
thus ?thesis |
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1363 |
using dim_substandard[of "?a ` d"] card_image[of ?a d] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1364 |
by (auto simp: axis_eq_axis inj_on_def *) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1365 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1366 |
|
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1367 |
lemma dim_subset_UNIV_cart: |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1368 |
fixes S :: "(real^'n) set" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1369 |
shows "dim S \<le> CARD('n)" |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1370 |
by (metis dim_subset_UNIV DIM_cart DIM_real mult.right_neutral) |
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1371 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1372 |
lemma affinity_inverses: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1373 |
assumes m0: "m \<noteq> (0::'a::field)" |
61736 | 1374 |
shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id" |
1375 |
"(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1376 |
using m0 |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset
|
1377 |
apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset
|
1378 |
apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric]) |
49644 | 1379 |
done |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1380 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1381 |
lemma vector_affinity_eq: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1382 |
assumes m0: "(m::'a::field) \<noteq> 0" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1383 |
shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1384 |
proof |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1385 |
assume h: "m *s x + c = y" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1386 |
hence "m *s x = y - c" by (simp add: field_simps) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1387 |
hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1388 |
then show "x = inverse m *s y + - (inverse m *s c)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1389 |
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1390 |
next |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1391 |
assume h: "x = inverse m *s y + - (inverse m *s c)" |
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset
|
1392 |
show "m *s x + c = y" unfolding h |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1393 |
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1394 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1395 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1396 |
lemma vector_eq_affinity: |
49644 | 1397 |
"(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1398 |
using vector_affinity_eq[where m=m and x=x and y=y and c=c] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1399 |
by metis |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1400 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1401 |
lemma vector_cart: |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1402 |
fixes f :: "real^'n \<Rightarrow> real" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1403 |
shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1404 |
unfolding euclidean_eq_iff[where 'a="real^'n"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1405 |
by simp (simp add: Basis_vec_def inner_axis) |
63332 | 1406 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1407 |
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1408 |
by (rule vector_cart) |
49644 | 1409 |
|
44360 | 1410 |
subsection "Convex Euclidean Space" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1411 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1412 |
lemma Cart_1:"(1::real^'n) = \<Sum>Basis" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1413 |
using const_vector_cart[of 1] by (simp add: one_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1414 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1415 |
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1416 |
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1417 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1418 |
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1419 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1420 |
lemma convex_box_cart: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1421 |
assumes "\<And>i. convex {x. P i x}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1422 |
shows "convex {x. \<forall>i. P i (x$i)}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1423 |
using assms unfolding convex_def by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1424 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1425 |
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}" |
63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset
|
1426 |
by (rule convex_box_cart) (simp add: atLeast_def[symmetric]) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1427 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1428 |
lemma unit_interval_convex_hull_cart: |
56188 | 1429 |
"cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" |
1430 |
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric] |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1431 |
by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1432 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1433 |
lemma cube_convex_hull_cart: |
49644 | 1434 |
assumes "0 < d" |
1435 |
obtains s::"(real^'n) set" |
|
56188 | 1436 |
where "finite s" "cbox (x - (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s" |
49644 | 1437 |
proof - |
55522 | 1438 |
from assms obtain s where "finite s" |
67399 | 1439 |
and "cbox (x - sum (( *\<^sub>R) d) Basis) (x + sum (( *\<^sub>R) d) Basis) = convex hull s" |
55522 | 1440 |
by (rule cube_convex_hull) |
1441 |
with that[of s] show thesis |
|
1442 |
by (simp add: const_vector_cart) |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1443 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1444 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1445 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1446 |
subsection "Derivative" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1447 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1448 |
definition "jacobian f net = matrix(frechet_derivative f net)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1449 |
|
49644 | 1450 |
lemma jacobian_works: |
1451 |
"(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> |
|
67986
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1452 |
(f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" (is "?lhs = ?rhs") |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1453 |
proof |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1454 |
assume ?lhs then show ?rhs |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1455 |
by (simp add: frechet_derivative_works has_derivative_linear jacobian_def) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1456 |
next |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1457 |
assume ?rhs then show ?lhs |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1458 |
by (rule differentiableI) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1459 |
qed |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1460 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1461 |
|
60420 | 1462 |
subsection \<open>Component of the differential must be zero if it exists at a local |
67968 | 1463 |
maximum or minimum for that corresponding component\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1464 |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1465 |
lemma differential_zero_maxmin_cart: |
49644 | 1466 |
fixes f::"real^'a \<Rightarrow> real^'b" |
1467 |
assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))" |
|
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1468 |
"f differentiable (at x)" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1469 |
shows "jacobian f (at x) $ k = 0" |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1470 |
using differential_zero_maxmin_component[of "axis k 1" e x f] assms |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1471 |
vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"] |
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset
|
1472 |
by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def) |
49644 | 1473 |
|
60420 | 1474 |
subsection \<open>Lemmas for working on @{typ "real^1"}\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1475 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1476 |
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1" |
49644 | 1477 |
by (metis (full_types) num1_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1478 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1479 |
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1" |
49644 | 1480 |
by auto (metis (full_types) num1_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1481 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1482 |
lemma exhaust_2: |
49644 | 1483 |
fixes x :: 2 |
1484 |
shows "x = 1 \<or> x = 2" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1485 |
proof (induct x) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1486 |
case (of_int z) |
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1487 |
then have "0 \<le> z" and "z < 2" by simp_all |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1488 |
then have "z = 0 | z = 1" by arith |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1489 |
then show ?case by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1490 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1491 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1492 |
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1493 |
by (metis exhaust_2) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1494 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1495 |
lemma exhaust_3: |
49644 | 1496 |
fixes x :: 3 |
1497 |
shows "x = 1 \<or> x = 2 \<or> x = 3" |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1498 |
proof (induct x) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1499 |
case (of_int z) |
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1500 |
then have "0 \<le> z" and "z < 3" by simp_all |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1501 |
then have "z = 0 \<or> z = 1 \<or> z = 2" by arith |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1502 |
then show ?case by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1503 |
qed |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1504 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1505 |
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1506 |
by (metis exhaust_3) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1507 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1508 |
lemma UNIV_1 [simp]: "UNIV = {1::1}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1509 |
by (auto simp add: num1_eq_iff) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1510 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1511 |
lemma UNIV_2: "UNIV = {1::2, 2::2}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1512 |
using exhaust_2 by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1513 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1514 |
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1515 |
using exhaust_3 by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1516 |
|
64267 | 1517 |
lemma sum_1: "sum f (UNIV::1 set) = f 1" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1518 |
unfolding UNIV_1 by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1519 |
|
64267 | 1520 |
lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1521 |
unfolding UNIV_2 by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1522 |
|
64267 | 1523 |
lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1524 |
unfolding UNIV_3 by (simp add: ac_simps) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1525 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1526 |
lemma num1_eqI: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1527 |
fixes a::num1 shows "a = b" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1528 |
by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1529 |
|
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1530 |
lemma num1_eq1 [simp]: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1531 |
fixes a::num1 shows "a = 1" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1532 |
by (rule num1_eqI) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1533 |
|
49644 | 1534 |
instantiation num1 :: cart_one |
1535 |
begin |
|
1536 |
||
1537 |
instance |
|
1538 |
proof |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1539 |
show "CARD(1) = Suc 0" by auto |
49644 | 1540 |
qed |
1541 |
||
1542 |
end |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1543 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1544 |
instantiation num1 :: linorder begin |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1545 |
definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1546 |
definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1547 |
instance |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1548 |
by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1549 |
end |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1550 |
|
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1551 |
instance num1 :: wellorder |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1552 |
by intro_classes (auto simp: less_eq_num1_def less_num1_def) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1553 |
|
67968 | 1554 |
subsection\<open>The collapse of the general concepts to dimension one\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1555 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1556 |
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1557 |
by (simp add: vec_eq_iff) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1558 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1559 |
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1560 |
apply auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1561 |
apply (erule_tac x= "x$1" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1562 |
apply (simp only: vector_one[symmetric]) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1563 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1564 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1565 |
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)" |
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset
|
1566 |
by (simp add: norm_vec_def) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1567 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1568 |
lemma dist_vector_1: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1569 |
fixes x :: "'a::real_normed_vector^1" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1570 |
shows "dist x y = dist (x$1) (y$1)" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1571 |
by (simp add: dist_norm norm_vector_1) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1572 |
|
61945 | 1573 |
lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1574 |
by (simp add: norm_vector_1) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1575 |
|
61945 | 1576 |
lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1577 |
by (auto simp add: norm_real dist_norm) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1578 |
|
67986
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1579 |
subsection\<open> Rank of a matrix\<close> |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1580 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1581 |
text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close> |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1582 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1583 |
lemma matrix_vector_mult_in_columnspace: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1584 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1585 |
shows "(A *v x) \<in> span(columns A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1586 |
apply (simp add: matrix_vector_column columns_def transpose_def column_def) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1587 |
apply (intro span_sum span_mul) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1588 |
apply (force intro: span_superset) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1589 |
done |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1590 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1591 |
lemma orthogonal_nullspace_rowspace: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1592 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1593 |
assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1594 |
shows "orthogonal x y" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1595 |
proof (rule span_induct [OF y]) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1596 |
show "subspace {a. orthogonal x a}" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1597 |
by (simp add: subspace_orthogonal_to_vector) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1598 |
next |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1599 |
fix v |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1600 |
assume "v \<in> rows A" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1601 |
then obtain i where "v = row i A" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1602 |
by (auto simp: rows_def) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1603 |
with 0 show "orthogonal x v" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1604 |
unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1605 |
by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1606 |
qed |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1607 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1608 |
lemma nullspace_inter_rowspace: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1609 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1610 |
shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1611 |
using orthogonal_nullspace_rowspace orthogonal_self by auto |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1612 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1613 |
lemma matrix_vector_mul_injective_on_rowspace: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1614 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1615 |
shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1616 |
using nullspace_inter_rowspace [of A "x-y"] |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1617 |
by (metis eq_iff_diff_eq_0 matrix_vector_mult_diff_distrib span_diff) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1618 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1619 |
definition rank :: "real^'n^'m=>nat" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1620 |
where "rank A \<equiv> dim(columns A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1621 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1622 |
lemma dim_rows_le_dim_columns: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1623 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1624 |
shows "dim(rows A) \<le> dim(columns A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1625 |
proof - |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1626 |
have "dim (span (rows A)) \<le> dim (span (columns A))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1627 |
proof - |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1628 |
obtain B where "independent B" "span(rows A) \<subseteq> span B" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1629 |
and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1630 |
using basis_exists [of "span(rows A)"] by blast |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1631 |
with span_subspace have eq: "span B = span(rows A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1632 |
by auto |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1633 |
then have inj: "inj_on (( *v) A) (span B)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1634 |
using inj_on_def matrix_vector_mul_injective_on_rowspace by blast |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1635 |
then have ind: "independent (( *v) A ` B)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1636 |
by (rule independent_inj_on_image [OF \<open>independent B\<close> matrix_vector_mul_linear]) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1637 |
then have "finite (( *v) A ` B) \<and> card (( *v) A ` B) \<le> dim (( *v) A ` B)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1638 |
by (rule independent_bound_general) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1639 |
then show ?thesis |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1640 |
by (metis (no_types, lifting) B ind inj eq card_image image_subset_iff independent_card_le_dim inj_on_subset matrix_vector_mult_in_columnspace) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1641 |
qed |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1642 |
then show ?thesis |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1643 |
by simp |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1644 |
qed |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1645 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1646 |
lemma rank_row: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1647 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1648 |
shows "rank A = dim(rows A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1649 |
unfolding rank_def |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1650 |
by (metis dim_rows_le_dim_columns columns_transpose dual_order.antisym rows_transpose) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1651 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1652 |
lemma rank_transpose: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1653 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1654 |
shows "rank(transpose A) = rank A" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1655 |
by (metis rank_def rank_row rows_transpose) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1656 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1657 |
lemma matrix_vector_mult_basis: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1658 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1659 |
shows "A *v (axis k 1) = column k A" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1660 |
by (simp add: cart_eq_inner_axis column_def matrix_mult_dot) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1661 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1662 |
lemma columns_image_basis: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1663 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1664 |
shows "columns A = ( *v) A ` (range (\<lambda>i. axis i 1))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1665 |
by (force simp: columns_def matrix_vector_mult_basis [symmetric]) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1666 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1667 |
lemma rank_dim_range: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1668 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1669 |
shows "rank A = dim(range (\<lambda>x. A *v x))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1670 |
unfolding rank_def |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1671 |
proof (rule span_eq_dim) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1672 |
show "span (columns A) = span (range (( *v) A))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1673 |
apply (auto simp: columns_image_basis span_linear_image matrix_vector_mul_linear) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1674 |
by (metis columns_image_basis matrix_vector_mul_linear matrix_vector_mult_in_columnspace span_linear_image) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1675 |
qed |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1676 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1677 |
lemma rank_bound: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1678 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1679 |
shows "rank A \<le> min CARD('m) (CARD('n))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1680 |
by (metis (mono_tags, hide_lams) min.bounded_iff DIM_cart DIM_real dim_subset_UNIV mult.right_neutral rank_def rank_transpose) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1681 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1682 |
lemma full_rank_injective: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1683 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1684 |
shows "rank A = CARD('n) \<longleftrightarrow> inj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1685 |
by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows rank_row dim_eq_full [symmetric]) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1686 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1687 |
lemma full_rank_surjective: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1688 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1689 |
shows "rank A = CARD('m) \<longleftrightarrow> surj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1690 |
by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric] |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1691 |
matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1692 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1693 |
lemma rank_I: "rank(mat 1::real^'n^'n) = CARD('n)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1694 |
by (simp add: full_rank_injective inj_on_def) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1695 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1696 |
lemma less_rank_noninjective: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1697 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1698 |
shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj (( *v) A)" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1699 |
using less_le rank_bound by (auto simp: full_rank_injective [symmetric]) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1700 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1701 |
lemma matrix_nonfull_linear_equations_eq: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1702 |
fixes A :: "real^'n^'m" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1703 |
shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> ~(rank A = CARD('n))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1704 |
by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1705 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1706 |
lemma rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank 0 = 0" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1707 |
by (auto simp: rank_dim_range matrix_eq) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1708 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1709 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1710 |
lemma rank_mul_le_right: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1711 |
fixes A :: "real^'n^'m" and B :: "real^'p^'n" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1712 |
shows "rank(A ** B) \<le> rank B" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1713 |
proof - |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1714 |
have "rank(A ** B) \<le> dim (( *v) A ` range (( *v) B))" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1715 |
by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1716 |
also have "\<dots> \<le> rank B" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1717 |
by (simp add: rank_dim_range matrix_vector_mul_linear dim_image_le) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1718 |
finally show ?thesis . |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1719 |
qed |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1720 |
|
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1721 |
lemma rank_mul_le_left: |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1722 |
fixes A :: "real^'n^'m" and B :: "real^'p^'n" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1723 |
shows "rank(A ** B) \<le> rank A" |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1724 |
by (metis matrix_transpose_mul rank_mul_le_right rank_transpose) |
b65c4a6a015e
quite a few more results about negligibility, etc., and a bit of tidying up
paulson <lp15@cam.ac.uk>
parents:
67982
diff
changeset
|
1725 |
|
67981
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1726 |
subsection\<open>Routine results connecting the types @{typ "real^1"} and @{typ real}\<close> |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1727 |
|
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1728 |
lemma vector_one_nth [simp]: |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1729 |
fixes x :: "'a^1" shows "vec (x $ 1) = x" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1730 |
by (metis vec_def vector_one) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1731 |
|
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1732 |
lemma vec_cbox_1_eq [simp]: |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1733 |
shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1734 |
by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1735 |
|
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1736 |
lemma vec_nth_cbox_1_eq [simp]: |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1737 |
fixes u v :: "'a::euclidean_space^1" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1738 |
shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u$1) (v$1)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1739 |
by (auto simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box image_iff Bex_def inner_axis) (metis vec_component) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1740 |
|
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1741 |
lemma vec_nth_1_iff_cbox [simp]: |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1742 |
fixes a b :: "'a::euclidean_space" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1743 |
shows "(\<lambda>x::'a^1. x $ 1) ` S = cbox a b \<longleftrightarrow> S = cbox (vec a) (vec b)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1744 |
(is "?lhs = ?rhs") |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1745 |
proof |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1746 |
assume L: ?lhs show ?rhs |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1747 |
proof (intro equalityI subsetI) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1748 |
fix x |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1749 |
assume "x \<in> S" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1750 |
then have "x $ 1 \<in> (\<lambda>v. v $ (1::1)) ` cbox (vec a) (vec b)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1751 |
using L by auto |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1752 |
then show "x \<in> cbox (vec a) (vec b)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1753 |
by (metis (no_types, lifting) imageE vector_one_nth) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1754 |
next |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1755 |
fix x :: "'a^1" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1756 |
assume "x \<in> cbox (vec a) (vec b)" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1757 |
then show "x \<in> S" |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1758 |
by (metis (no_types, lifting) L imageE imageI vec_component vec_nth_cbox_1_eq vector_one_nth) |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1759 |
qed |
349c639e593c
more new theorems on real^1, matrices, etc.
paulson <lp15@cam.ac.uk>
parents:
67979
diff
changeset
|
1760 |
qed simp |
49644 | 1761 |
|
67979
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1762 |
lemma tendsto_at_within_vector_1: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1763 |
fixes S :: "'a :: metric_space set" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1764 |
assumes "(f \<longlongrightarrow> fx) (at x within S)" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1765 |
shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1766 |
proof (rule topological_tendstoI) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1767 |
fix T :: "('a^1) set" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1768 |
assume "open T" "vec fx \<in> T" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1769 |
have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1770 |
using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1771 |
then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1772 |
unfolding eventually_at dist_norm [symmetric] |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1773 |
by (rule ex_forward) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1774 |
(use \<open>open T\<close> in |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1775 |
\<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1776 |
qed |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1777 |
|
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1778 |
lemma has_derivative_vector_1: |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1779 |
assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1780 |
shows "((\<lambda>x. vec (g (x $ 1))) has_derivative ( *\<^sub>R) (g' a)) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1781 |
(at ((vec a)::real^1) within vec ` S)" |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1782 |
using der_g |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1783 |
apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1784 |
apply (drule tendsto_at_within_vector_1, vector) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1785 |
apply (auto simp: algebra_simps eventually_at tendsto_def) |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1786 |
done |
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1787 |
|
53323937ee25
new material about vec, real^1, etc.
paulson <lp15@cam.ac.uk>
parents:
67971
diff
changeset
|
1788 |
|
67968 | 1789 |
subsection\<open>Explicit vector construction from lists\<close> |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1790 |
|
43995
c479836d9048
simplified definition of vector (also removed Cartesian_Euclidean_Space.from_nat which collides with Countable.from_nat)
hoelzl
parents:
42814
diff
changeset
|
1791 |
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)" |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1792 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1793 |
lemma vector_1: "(vector[x]) $1 = x" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1794 |
unfolding vector_def by simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1795 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1796 |
lemma vector_2: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1797 |
"(vector[x,y]) $1 = x" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1798 |
"(vector[x,y] :: 'a^2)$2 = (y::'a::zero)" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1799 |
unfolding vector_def by simp_all |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1800 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1801 |
lemma vector_3: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1802 |
"(vector [x,y,z] ::('a::zero)^3)$1 = x" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1803 |
"(vector [x,y,z] ::('a::zero)^3)$2 = y" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1804 |
"(vector [x,y,z] ::('a::zero)^3)$3 = z" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1805 |
unfolding vector_def by simp_all |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1806 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1807 |
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))" |
67719
bffb7482faaa
new material on matrices, etc., and consolidating duplicate results about of_nat
paulson <lp15@cam.ac.uk>
parents:
67686
diff
changeset
|
1808 |
by (metis vector_1 vector_one) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1809 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1810 |
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1811 |
apply auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1812 |
apply (erule_tac x="v$1" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1813 |
apply (erule_tac x="v$2" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1814 |
apply (subgoal_tac "vector [v$1, v$2] = v") |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1815 |
apply simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1816 |
apply (vector vector_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1817 |
apply (simp add: forall_2) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1818 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1819 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1820 |
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))" |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1821 |
apply auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1822 |
apply (erule_tac x="v$1" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1823 |
apply (erule_tac x="v$2" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1824 |
apply (erule_tac x="v$3" in allE) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1825 |
apply (subgoal_tac "vector [v$1, v$2, v$3] = v") |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1826 |
apply simp |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1827 |
apply (vector vector_def) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1828 |
apply (simp add: forall_3) |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1829 |
done |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1830 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1831 |
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)" |
49644 | 1832 |
apply (rule bounded_linearI[where K=1]) |
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1833 |
using component_le_norm_cart[of _ k] unfolding real_norm_def by auto |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1834 |
|
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1835 |
lemma interval_split_cart: |
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1836 |
"{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}" |
56188 | 1837 |
"cbox a b \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}" |
49644 | 1838 |
apply (rule_tac[!] set_eqI) |
67673
c8caefb20564
lots of new material, ultimately related to measure theory
paulson <lp15@cam.ac.uk>
parents:
67399
diff
changeset
|
1839 |
unfolding Int_iff mem_box_cart mem_Collect_eq interval_cbox_cart |
49644 | 1840 |
unfolding vec_lambda_beta |
1841 |
by auto |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1842 |
|
67685
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67673
diff
changeset
|
1843 |
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] = |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67673
diff
changeset
|
1844 |
bounded_linear.uniform_limit[OF blinfun.bounded_linear_right] |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67673
diff
changeset
|
1845 |
bounded_linear.uniform_limit[OF bounded_linear_vec_nth] |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67673
diff
changeset
|
1846 |
bounded_linear.uniform_limit[OF bounded_linear_component_cart] |
bdff8bf0a75b
moved theorems from AFP/Affine_Arithmetic and AFP/Ordinary_Differential_Equations
immler
parents:
67673
diff
changeset
|
1847 |
|
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset
|
1848 |
end |