author  nipkow 
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parent 67155  9e5b05d54f9d 
child 67673  c8caefb20564 
permissions  rwrr 
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Theory of homotopic paths (from HOL Light), plus comments and minor refinements
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section \<open>Instantiates the finite Cartesian product of Euclidean spaces as a Euclidean space.\<close> 
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theory Cartesian_Euclidean_Space 
63938  4 
imports Finite_Cartesian_Product Derivative 
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begin 
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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 delta_mult_idempotent: 
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"(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" 
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by simp 
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(*move up?*) 
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lemma sum_UNIV_sum: 
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fixes g :: "'a::finite + 'b::finite \<Rightarrow> _" 
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shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))" 
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apply (subst UNIV_Plus_UNIV [symmetric]) 
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apply (subst sum.Plus) 
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apply simp_all 
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done 
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64267  23 
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] 

26 
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 
49644  41 

67399  42 
definition "( * ) \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))" 
49644  43 
instance .. 
44 

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end 
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instantiation vec :: (one, finite) one 
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begin 
49644  49 

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definition "1 \<equiv> (\<chi> i. 1)" 

51 
instance .. 

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end 
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instantiation vec :: (ord, finite) ord 
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begin 
49644  57 

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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" 
49644  60 
instance .. 
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end 
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text\<open>The ordering on onedimensional vectors is linear.\<close> 
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49197  66 
class cart_one = 
61076  67 
assumes UNIV_one: "card (UNIV :: 'a set) = Suc 0" 
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begin 
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70 
subclass finite 

71 
proof 

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from UNIV_one show "finite (UNIV :: 'a set)" 

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by (auto intro!: card_ge_0_finite) 

74 
qed 

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end 
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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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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" 

85 
proof  

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have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one) 

87 
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) 

90 
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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show "x \<le> y \<or> y \<le> x" by auto 
49197  94 
qed 
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text\<open>Constant Vectors\<close> 
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definition "vec x = (\<chi> i. x)" 
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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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text\<open>Also the scalarvector multiplication.\<close> 
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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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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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64267  115 
hide_fact (open) sum_cong_aux 
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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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simp_tac (put_simpset ss1 ctxt) 
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THEN' (fn i => resolve_tac ctxt @{thms Cartesian_Euclidean_Space.sum_cong_aux} i 
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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  140 
end 
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\<close> "lift trivial vector statements to real arith statements" 
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lemma vec_0[simp]: "vec 0 = 0" by vector 
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lemma vec_1[simp]: "vec 1 = 1" by vector 

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lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector 
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lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto 
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lemma vec_add: "vec(x + y) = vec x + vec y" by vector 
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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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lemma vec_sum: 
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assumes "finite S" 
64267  157 
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 

162 
next 

163 
case insert 

164 
then show ?case by (auto simp add: vec_add) 

165 
qed 

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text\<open>Obvious "componentpushing".\<close> 
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lemma vec_component [simp]: "vec x $ i = x" 
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by vector 
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lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i" 
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by vector 
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lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)" 
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by vector 
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector 
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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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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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instance vec :: (monoid_mult, finite) monoid_mult 
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by standard vector+ 
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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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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult 
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by standard vector 
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instance vec :: (semiring, finite) semiring 
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by standard (vector field_simps)+ 
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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 
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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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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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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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instance vec :: (ring, finite) ring .. 
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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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instance vec :: (ring_1, finite) ring_1 .. 
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instance vec :: (real_algebra, finite) real_algebra 
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by standard (simp_all add: vec_eq_iff) 
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instance vec :: (real_algebra_1, finite) real_algebra_1 .. 
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lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" 
226 
proof (induct n) 

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case 0 

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then show ?case by vector 

229 
next 

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case Suc 

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then show ?case by vector 

232 
qed 

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lemma one_index [simp]: "(1 :: 'a :: one ^ 'n) $ i = 1" 
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by vector 
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lemma neg_one_index [simp]: "( 1 :: 'a :: {one, uminus} ^ 'n) $ i =  1" 
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by vector 
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instance vec :: (semiring_char_0, finite) semiring_char_0 
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proof 
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fix m n :: nat 
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show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)" 
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by (auto intro!: injI simp add: vec_eq_iff of_nat_index) 
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qed 
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instance vec :: (numeral, finite) numeral .. 
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instance vec :: (semiring_numeral, finite) semiring_numeral .. 
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lemma numeral_index [simp]: "numeral w $ i = numeral w" 
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by (induct w) (simp_all only: numeral.simps vector_add_component one_index) 
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lemma neg_numeral_index [simp]: " numeral w $ i =  numeral w" 
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by (simp only: vector_uminus_component numeral_index) 
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instance vec :: (comm_ring_1, finite) comm_ring_1 .. 
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instance vec :: (ring_char_0, finite) ring_char_0 .. 
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lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" 
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by (vector mult.assoc) 
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lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
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by (vector field_simps) 
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lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
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by (vector field_simps) 
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lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector 
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lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector 
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lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x  y) = c *s x  c *s y" 
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by (vector field_simps) 
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lemma vector_smult_rneg: "(c::'a::ring) *s x = (c *s x)" by vector 
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lemma vector_smult_lneg: " (c::'a::ring) *s x = (c *s x)" by vector 
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lemma vector_sneg_minus1: "x = (1::'a::ring_1) *s x" by vector 
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lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector 
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lemma vector_sub_rdistrib: "((a::'a::ring)  b) *s x = a *s x  b *s x" 
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by (vector field_simps) 
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lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
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by (simp add: vec_eq_iff) 
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lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) 
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lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0" 
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by vector 
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lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y" 
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by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) 
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lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0" 
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by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) 
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lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==> a *s x = a *s y ==> (x = y)" 
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by (metis vector_mul_lcancel) 
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lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b" 
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by (metis vector_mul_rcancel) 
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lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x" 
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apply (simp add: norm_vec_def) 
67155  293 
apply (rule member_le_L2_set, simp_all) 
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done 
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lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e" 
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by (metis component_le_norm_cart order_trans) 
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lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e" 
53595  300 
by (metis component_le_norm_cart le_less_trans) 
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64267  302 
lemma norm_le_l1_cart: "norm x <= sum(\<lambda>i. \<bar>x$i\<bar>) UNIV" 
67155  303 
by (simp add: norm_vec_def L2_set_le_sum) 
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lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" 
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unfolding scaleR_vec_def vector_scalar_mult_def by simp 
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lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
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unfolding dist_norm scalar_mult_eq_scaleR 
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unfolding scaleR_right_diff_distrib[symmetric] by simp 
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64267  312 
lemma sum_component [simp]: 
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fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n" 
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shows "(sum f S)$i = sum (\<lambda>x. (f x)$i) S" 
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proof (cases "finite S") 
316 
case True 

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then show ?thesis by induct simp_all 

318 
next 

319 
case False 

320 
then show ?thesis by simp 

321 
qed 

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lemma sum_eq: "sum f S = (\<chi> i. sum (\<lambda>x. (f x)$i ) S)" 
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by (simp add: vec_eq_iff) 
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parents:
diff
changeset

325 

64267  326 
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

327 
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n" 
64267  328 
shows "sum (\<lambda>x. c *s f x) S = c *s sum f S" 
329 
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

330 

64267  331 
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

332 
fixes f:: "'a \<Rightarrow> real ^'n" 
64267  333 
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e" 
334 
shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e" 

335 
using sum_norm_allsubsets_bound[OF assms] 

57865  336 
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

337 

62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

338 
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

339 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

340 
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

341 
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

342 
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

343 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

344 
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

345 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

346 
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

347 
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

348 
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

349 
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

350 
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

351 
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

352 
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

353 
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

354 
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

355 
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

356 
ultimately show ?thesis 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

357 
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

358 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

359 
show ?thesis 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

360 
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

361 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

362 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

363 
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

364 
"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

365 
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

366 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

367 
lemma interior_halfspace_component_le [simp]: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

368 
"interior {x. x$k \<le> a} = {x :: (real,'n::finite) vec. x$k < a}" (is "?LE") 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

369 
and interior_halfspace_component_ge [simp]: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

370 
"interior {x. x$k \<ge> a} = {x :: (real,'n::finite) vec. x$k > a}" (is "?GE") 
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 "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

373 
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

374 
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

375 
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

376 
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

377 
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

378 
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

379 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

380 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

381 
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

382 
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

383 
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

384 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

385 
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

386 
by (force simp:) 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

387 
then show ?thesis 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

388 
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

389 
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

390 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

391 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

392 
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

393 
"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

394 
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

395 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

396 
lemma closure_halfspace_component_lt [simp]: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

397 
"closure {x. x$k < a} = {x :: (real,'n::finite) vec. x$k \<le> a}" (is "?LE") 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

398 
and closure_halfspace_component_gt [simp]: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

399 
"closure {x. x$k > a} = {x :: (real,'n::finite) vec. x$k \<ge> a}" (is "?GE") 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

400 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

401 
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

402 
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

403 
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

404 
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

405 
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

406 
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

407 
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

408 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

409 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

410 
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

411 
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

412 
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

413 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

414 
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

415 
by (force simp:) 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

416 
then show ?thesis 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

417 
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

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 frontier_halfspace_le: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

421 
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

422 
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

423 
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

424 
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

425 
next 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

426 
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

427 
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

428 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

429 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

430 
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

431 
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

432 
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

433 
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

434 
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

435 
next 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

436 
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

437 
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

438 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

439 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

440 
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

441 
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

442 
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

443 
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

444 
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

445 
next 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

446 
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

447 
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

448 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

449 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

450 
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

451 
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

452 
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

453 
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

454 
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

455 
next 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

456 
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

457 
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

458 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

459 

5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

460 
lemma interior_standard_hyperplane: 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

461 
"interior {x :: (real,'n::finite) vec. x$k = a} = {}" 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

462 
proof  
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

463 
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

464 
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

465 
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

466 
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

467 
ultimately show ?thesis 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

468 
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

469 
by force 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

470 
qed 
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62127
diff
changeset

471 

60420  472 
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

473 

60420  474 
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

475 

49644  476 
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" 
477 
(infixl "**" 70) 

64267  478 
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

479 

49644  480 
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" 
481 
(infixl "*v" 70) 

64267  482 
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

483 

49644  484 
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " 
485 
(infixl "v*" 70) 

64267  486 
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

487 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

488 
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)" 
63332  489 
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

490 
"(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

491 
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

492 
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

493 
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

494 
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

495 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

496 
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

497 
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" 
64267  498 
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

499 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

500 
lemma matrix_mul_lid: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

501 
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

502 
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

503 
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

504 
apply vector 
64267  505 
apply (auto simp only: if_distrib cond_application_beta sum.delta'[OF finite] 
49644  506 
mult_1_left mult_zero_left if_True UNIV_I) 
507 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

508 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

509 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

510 
lemma matrix_mul_rid: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

511 
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

512 
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

513 
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

514 
apply vector 
64267  515 
apply (auto simp only: if_distrib cond_application_beta sum.delta[OF finite] 
49644  516 
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) 
517 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

518 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

519 
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" 
64267  520 
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

521 
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

522 
apply simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

523 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

524 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

525 
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" 
49644  526 
apply (vector matrix_matrix_mult_def matrix_vector_mult_def 
64267  527 
sum_distrib_left sum_distrib_right mult.assoc) 
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset

528 
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

529 
apply simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

530 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

531 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

532 
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

533 
apply (vector matrix_vector_mult_def mat_def) 
64267  534 
apply (simp add: if_distrib cond_application_beta sum.delta' cong del: if_weak_cong) 
49644  535 
done 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

536 

49644  537 
lemma matrix_transpose_mul: 
538 
"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

539 
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

540 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

541 
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

542 
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

543 
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

544 
apply auto 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

545 
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

546 
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

547 
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

548 
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

549 
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

550 
apply (auto simp add: if_distrib cond_application_beta axis_def 
64267  551 
sum.delta[OF finite] cong del: if_weak_cong) 
49644  552 
done 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

553 

49644  554 
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

555 
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

556 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

557 
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)" 
64267  558 
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

559 
apply (subst sum.swap) 
49644  560 
apply simp 
561 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

562 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

563 
lemma transpose_mat: "transpose (mat n) = mat n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

564 
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

565 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

566 
lemma transpose_transpose: "transpose(transpose A) = A" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

567 
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

568 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

569 
lemma row_transpose: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

570 
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

571 
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

572 
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

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 column_transpose: 
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:: "'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

576 
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

577 
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

578 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

579 
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" 
49644  580 
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

581 

49644  582 
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" 
583 
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

584 

60420  585 
text\<open>Two sometimes fruitful ways of looking at matrixvector 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

586 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

587 
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

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 

49644  590 
lemma matrix_mult_vsum: 
64267  591 
"(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

592 
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

593 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

594 
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

595 
"(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)" 
64267  596 
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

597 

64267  598 
lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" 
599 
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

600 

63938  601 
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

602 
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

603 
assumes lf: "linear f" 
64267  604 
shows "(f x)$j = sum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") 
49644  605 
proof  
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

606 
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

607 
let ?N = "(UNIV :: 'n set)" 
64267  608 
have "?rhs = (sum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j" 
609 
unfolding sum_component by simp 

49644  610 
then show ?thesis 
64267  611 
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

612 
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

613 
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

614 
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

615 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

616 

60420  617 
text\<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

618 

49644  619 
definition 
620 
"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

621 

49644  622 
definition 
623 
"matrix_inv(A:: 'a::semiring_1^'n^'m) = 

624 
(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

625 

60420  626 
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

627 

49644  628 
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

629 
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

630 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

631 
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

632 
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff 
64267  633 
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

634 

49644  635 
lemma matrix_works: 
636 
assumes lf: "linear f" 

637 
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

638 
apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult.commute) 
63938  639 
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

640 

49644  641 
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" 
642 
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

643 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

644 
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

645 
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

646 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

647 
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

648 
assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)" 
49644  649 
and lg: "linear (g::real^'m \<Rightarrow> real^_)" 
61736  650 
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

651 
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] 
49644  652 
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

653 

49644  654 
lemma matrix_vector_column: 
64267  655 
"(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

656 
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

657 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

658 
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

659 
apply (rule adjoint_unique) 
49644  660 
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def 
64267  661 
sum_distrib_right sum_distrib_left) 
66804
3f9bb52082c4
avoid name clashes on interpretation of abstract locales
haftmann
parents:
66447
diff
changeset

662 
apply (subst sum.swap) 
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset

663 
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

664 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

665 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

666 
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

667 
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

668 
apply (subst matrix_vector_mul[OF lf]) 
49644  669 
unfolding adjoint_matrix matrix_of_matrix_vector_mul 
670 
apply rule 

671 
done 

672 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

673 

60420  674 
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

675 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

676 
(* FIXME: rename do choice_cart *) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

677 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

678 
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow> 
37494  679 
(\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs") 
49644  680 
proof  
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

681 
let ?S = "(UNIV :: 'n set)" 
49644  682 
{ assume H: "?rhs" 
683 
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

684 
moreover 
49644  685 
{ 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

686 
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

687 
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n" 
49644  688 
{ 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

689 
from f have "P i (f i)" by metis 
37494  690 
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

691 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

692 
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

693 
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

694 
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

695 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

696 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

697 
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

698 
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

699 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

700 
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

701 
"(\<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

702 
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

703 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

704 
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

705 
"(\<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

706 
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

707 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

708 
lemma matrix_left_invertible_injective: 
49644  709 
"(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)" 
710 
proof  

711 
{ fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

712 
from xy have "B*v (A *v x) = B *v (A*v y)" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

713 
hence "x = y" 
49644  714 
unfolding matrix_vector_mul_assoc B matrix_vector_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

715 
moreover 
49644  716 
{ assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y" 
67399  717 
hence i: "inj (( *v) A)" unfolding inj_on_def 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

718 
from linear_injective_left_inverse[OF matrix_vector_mul_linear i] 
67399  719 
obtain g where g: "linear g" "g \<circ> ( *v) A = id" 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

720 
have "matrix g ** 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

721 
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  722 
using g(2) by (simp add: fun_eq_iff) 
49644  723 
then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" 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

724 
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

725 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

726 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

727 
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

728 
"(\<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

729 
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

730 
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

731 
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

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_right_invertible_surjective: 
49644  734 
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)" 
735 
proof  

736 
{ fix B :: "real ^'m^'n" 

737 
assume AB: "A ** B = mat 1" 

738 
{ 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

739 
have "A *v (B *v x) = x" 
49644  740 
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } 
67399  741 
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

742 
moreover 
67399  743 
{ 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

744 
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] 
67399  745 
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

746 
by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

747 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

748 
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

749 
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

750 
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  751 
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

752 
. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

753 
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

754 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

755 
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

756 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

757 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

758 
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

759 
fixes A :: "real^'n^'m" 
49644  760 
shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> 
64267  761 
(\<forall>c. sum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" 
49644  762 
(is "?lhs \<longleftrightarrow> ?rhs") 
763 
proof  

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

764 
let ?U = "UNIV :: 'n set" 
49644  765 
{ assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0" 
766 
{ fix c i 

64267  767 
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

768 
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

769 
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

770 
using c 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

771 
unfolding matrix_mult_vsum 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

772 
by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

773 
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

774 
have "c i = 0" by (vector vec_eq_iff)} 
49644  775 
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

776 
moreover 
49644  777 
{ assume H: ?rhs 
778 
{ 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

779 
let ?c = "\<lambda>i. ((x$i ):: real)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

780 
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] 
49644  781 
have "x = 0" by vector } 
782 
} 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

783 
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

784 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

785 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

786 
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

787 
fixes A :: "real^'n^'m" 
49644  788 
shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> 
64267  789 
(\<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

790 
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

791 
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

792 
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

793 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

794 
lemma matrix_right_invertible_span_columns: 
49644  795 
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> 
796 
span (columns A) = UNIV" (is "?lhs = ?rhs") 

797 
proof  

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

798 
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

799 
have fU: "finite ?U" by simp 
64267  800 
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y)" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

801 
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def 
49644  802 
apply (subst eq_commute) 
803 
apply rule 

804 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

805 
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast 
49644  806 
{ assume h: ?lhs 
807 
{ fix x:: "real ^'n" 

808 
from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" 

64267  809 
where y: "sum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast 
49644  810 
have "x \<in> span (columns A)" 
811 
unfolding y[symmetric] 

64267  812 
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

813 
unfolding scalar_mult_eq_scaleR 
49644  814 
apply (rule span_mul) 
815 
apply (rule span_superset) 

816 
unfolding columns_def 

817 
apply blast 

818 
done 

819 
} 

820 
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

821 
moreover 
49644  822 
{ assume h:?rhs 
64267  823 
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). sum (\<lambda>i. (x$i) *s column i A) ?U = y" 
49644  824 
{ fix y 
825 
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

826 
proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) 
64267  827 
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

828 
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

829 
next 
49644  830 
fix c y1 y2 
831 
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

832 
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

833 
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

834 
from y2 obtain x:: "real ^'m" where 
64267  835 
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

836 
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

837 
show "?P (c*s y1 + y2)" 
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49644
diff
changeset

838 
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  839 
fix j 
840 
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) 

841 
else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" 

842 
using i(1) by (simp add: field_simps) 

64267  843 
have "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 
844 
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" 

845 
apply (rule sum.cong[OF refl]) 

49644  846 
using th apply blast 
847 
done 

64267  848 
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" 
849 
by (simp add: sum.distrib) 

850 
also have "\<dots> = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" 

851 
unfolding sum.delta[OF fU] 

49644  852 
using i(1) by simp 
64267  853 
finally show "sum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 
854 
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + sum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" . 

49644  855 
qed 
856 
next 

857 
show "y \<in> span (columns A)" 

858 
unfolding h by blast 

859 
qed 

860 
} 

861 
then have ?lhs unfolding lhseq .. 

862 
} 

37489
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 blast 
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 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

866 
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

867 
"(\<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

868 
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

869 
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

870 
unfolding matrix_right_invertible_span_columns 
49644  871 
.. 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

872 

60420  873 
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

874 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

875 
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

876 
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

877 
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1" 
49644  878 
proof  
879 
{ fix A A' :: "real ^'n^'n" 

880 
assume AA': "A ** A' = mat 1" 

67399  881 
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

882 
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

883 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

884 
apply (rule_tac x="(A' *v y)" in exI) 
49644  885 
apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) 
886 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

887 
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

888 
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

889 
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

890 
have th: "matrix f' ** A = mat 1" 
49644  891 
by (simp add: matrix_eq matrix_works[OF f'(1)] 
892 
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

893 
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp 
49644  894 
hence "matrix f' = A'" 
895 
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

896 
hence "matrix f' ** A = A' ** A" by simp 
49644  897 
hence "A' ** A = mat 1" by (simp add: th) 
898 
} 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

899 
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

900 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

901 

60420  902 
text \<open>Considering an nelement vector as an nby1 or 1byn 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

903 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

904 
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

905 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

906 
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

907 

49644  908 
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

909 
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

910 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

911 
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

912 
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

913 

49644  914 
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

915 
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

916 

49644  917 
lemma dot_matrix_product: 
918 
"(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

919 
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

920 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

921 
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

922 
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

923 
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

924 
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" 
49644  925 
unfolding dot_matrix_product transpose_columnvector[symmetric] 
926 
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

927 

61945  928 
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

929 
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

930 

49644  931 
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

932 
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

933 
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

934 

63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset

935 
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

936 
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

937 

63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset

938 
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

939 
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

940 

63334
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset

941 
lemma continuous_on_vec_lambda[continuous_intros]: 
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset

942 
"(\<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

943 
unfolding continuous_on_def by (auto intro: tendsto_vec_lambda) 
bd37a72a940a
Multivariate_Analysis: add continuous_on_vec_lambda
hoelzl
parents:
63332
diff
changeset

944 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

945 
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}" 
63332  946 
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

947 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

948 
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)" 
49644  949 
unfolding bounded_def 
950 
apply clarify 

951 
apply (rule_tac x="x $ i" in exI) 

952 
apply (rule_tac x="e" in exI) 

953 
apply clarify 

954 
apply (rule order_trans [OF dist_vec_nth_le], simp) 

955 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

956 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

957 
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

958 
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n" 
50998  959 
assumes f: "bounded (range f)" 
66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
64267
diff
changeset

960 
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

961 
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)" 
62127  962 
(is "?th d") 
963 
proof  

964 
have "\<forall>d' \<subseteq> d. ?th d'" 

965 
by (rule compact_lemma_general[where unproj=vec_lambda]) 

966 
(auto intro!: f bounded_component_cart simp: vec_lambda_eta) 

967 
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

968 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

969 

44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

970 
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

971 
proof 
50998  972 
fix f :: "nat \<Rightarrow> 'a ^ 'b" 
973 
assume f: "bounded (range f)" 

66447
a1f5c5c26fa6
Replaced subseq with strict_mono
eberlm <eberlm@in.tum.de>
parents:
64267
diff
changeset

974 
then obtain l r where r: "strict_mono r" 
49644  975 
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially" 
50998  976 
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

977 
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

978 
{ 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

979 
hence "0 < e / (real_of_nat (card ?d))" 
49644  980 
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

981 
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

982 
by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

983 
moreover 
49644  984 
{ fix n 
985 
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

986 
have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))" 
67155  987 
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

988 
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))" 
64267  989 
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

990 
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

991 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

992 
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially" 
61810  993 
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

994 
} 
61973  995 
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

996 
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

997 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

998 

49644  999 
lemma interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1000 
fixes a :: "real^'n" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1001 
shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" 
56188  1002 
and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}" 
1003 
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

1004 

49644  1005 
lemma mem_interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1006 
fixes a :: "real^'n" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1007 
shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)" 
56188  1008 
and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)" 
49644  1009 
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

1010 

49644  1011 
lemma interval_eq_empty_cart: 
1012 
fixes a :: "real^'n" 

54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1013 
shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) 
56188  1014 
and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2) 
49644  1015 
proof  
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1016 
{ fix i x assume as:"b$i \<le> a$i" and x:"x\<in>box a b" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1017 
hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1018 
hence "a$i < b$i" by auto 
49644  1019 
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

1020 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1021 
{ 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

1022 
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

1023 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1024 
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

1025 
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

1026 
unfolding vector_smult_component and vector_add_component 
49644  1027 
by auto } 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1028 
hence "box a b \<noteq> {}" using mem_interval_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

1029 
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

1030 

56188  1031 
{ fix i x assume as:"b$i < a$i" and x:"x\<in>cbox a b" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1032 
hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1033 
hence "a$i \<le> b$i" by auto 
49644  1034 
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

1035 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1036 
{ 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

1037 
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

1038 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1039 
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

1040 
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

1041 
unfolding vector_smult_component and vector_add_component 
49644  1042 
by auto } 
56188  1043 
hence "cbox a b \<noteq> {}" using mem_interval_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

1044 
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

1045 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1046 

49644  1047 
lemma interval_ne_empty_cart: 
1048 
fixes a :: "real^'n" 

56188  1049 
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

1050 
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

1051 
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

1052 
(* 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

1053 

49644  1054 
lemma subset_interval_imp_cart: 
1055 
fixes a :: "real^'n" 

56188  1056 
shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b" 
1057 
and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b" 

1058 
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

1059 
and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> box c d \<subseteq> box a b" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1060 
unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1061 
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

1062 

49644  1063 
lemma interval_sing: 
1064 
fixes a :: "'a::linorder^'n" 

1065 
shows "{a .. a} = {a} \<and> {a<..<a} = {}" 

1066 
apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) 

1067 
done 

37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1068 

49644  1069 
lemma subset_interval_cart: 
1070 
fixes a :: "real^'n" 

56188  1071 
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) 
1072 
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) 

1073 
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

1074 
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  1075 
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

1076 

49644  1077 
lemma disjoint_interval_cart: 
1078 
fixes a::"real^'n" 

56188  1079 
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) 
1080 
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) 

1081 
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

1082 
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

1083 
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

1084 

49644  1085 
lemma inter_interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

1086 
fixes a :: "real^'n" 
56188  1087 
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

1088 
unfolding Int_interval 
56188  1089 
by (auto simp: mem_box less_eq_vec_def) 
1090 
(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

1091 

49644  1092 
lemma closed_interval_left_cart: 
1093 
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

1094 
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}" 
63332  1095 
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

1096 

49644  1097 
lemma closed_interval_right_cart: 
1098 
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

1099 
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}" 
63332  1100 
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

1101 

49644  1102 
lemma is_interval_cart: 
1103 
"is_interval (s::(real^'n) set) \<longleftrightarrow> 

1104 
(\<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

1105 
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

1106 

49644  1107 
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}" 
63332  1108 
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

1109 

49644  1110 
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}" 
63332  1111 
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

1112 

49644  1113 
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}" 
63332  1114 
by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component) 
49644  1115 

1116 
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}" 

63332  1117 
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

1118 

49644  1119 
lemma Lim_component_le_cart: 