author  hoelzl 
Tue, 18 Mar 2014 15:53:48 +0100  
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parent 56189  c4daa97ac57a 
child 56196  32b7eafc5a52 
permissions  rwrr 
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header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*} 
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theory Cartesian_Euclidean_Space 
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imports Finite_Cartesian_Product Integration 
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begin 
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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 (cases "k=a") auto 

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lemma setsum_Plus: 
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"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow> 
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(\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))" 
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unfolding Plus_def 
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by (subst setsum_Un_disjoint, auto simp add: setsum_reindex) 
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lemma setsum_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 (rule setsum_Plus [OF finite finite]) 
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done 
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lemma setsum_mult_product: 
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"setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))" 
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unfolding setsum_nat_group[of h B A, unfolded atLeast0LessThan, symmetric] 
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proof (rule setsum_cong, simp, rule setsum_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{* Basic componentwise operations on vectors. *} 
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instantiation vec :: (times, finite) times 
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begin 
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definition "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))" 

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instance .. 

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end 
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instantiation vec :: (one, finite) one 
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begin 
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definition "1 \<equiv> (\<chi> i. 1)" 

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instance .. 

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end 
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instantiation vec :: (ord, finite) ord 
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begin 
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definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)" 

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definition "x < (y::'a^'b) \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x" 
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instance .. 
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end 
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text{* The ordering on onedimensional vectors is linear. *} 
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class cart_one = 
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assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0" 

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begin 
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subclass finite 

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proof 

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

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by (auto intro!: card_ge_0_finite) 

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qed 

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end 
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instance vec:: (order, finite) order 
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by default (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" 

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proof  

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

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then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq) 

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then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto 

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then show thesis by (auto intro: that) 

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qed 

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fix x y :: "'a^'b::cart_one" 
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note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps 
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show "x \<le> y \<or> y \<le> x" by auto 
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qed 
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text{* Constant Vectors *} 
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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{* Also the scalarvector multiplication. *} 
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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 {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *} 
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method_setup vector = {* 
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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 setsum_addf} RS sym, 
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@{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib}, 
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@{thm setsum_left_distrib}, @{thm setsum_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 => rtac @{thm setsum_cong2} i 
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ORELSE rtac @{thm setsum_0'} 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  136 
end 
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*} "lift trivial vector statements to real arith statements" 
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lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def) 
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lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def) 
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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 vec_def) 
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lemma vec_sub: "vec(x  y) = vec x  vec y" by (vector vec_def) 
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lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def) 
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lemma vec_neg: "vec( x) =  vec x " by (vector vec_def) 
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lemma vec_setsum: 
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assumes "finite S" 

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shows "vec(setsum f S) = setsum (vec o f) S" 
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using assms 
155 
proof induct 

156 
case empty 

157 
then show ?case by simp 

158 
next 

159 
case insert 

160 
then show ?case by (auto simp add: vec_add) 

161 
qed 

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text{* Obvious "componentpushing". *} 
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lemma vec_component [simp]: "vec x $ i = x" 
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by (vector vec_def) 
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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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173 

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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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49644  181 

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subsection {* Some frequently useful arithmetic lemmas over vectors. *} 
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instance vec :: (semigroup_mult, finite) semigroup_mult 
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by default (vector mult_assoc) 
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instance vec :: (monoid_mult, finite) monoid_mult 
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by default vector+ 
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instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult 
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by default (vector mult_commute) 
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult 
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by default vector 
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195 

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instance vec :: (semiring, finite) semiring 
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by default (vector field_simps)+ 
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instance vec :: (semiring_0, finite) semiring_0 
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by default (vector field_simps)+ 
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instance vec :: (semiring_1, finite) semiring_1 
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by default vector 
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instance vec :: (comm_semiring, finite) comm_semiring 
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by default (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 default (simp_all add: vec_eq_iff) 
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instance vec :: (real_algebra_1, finite) real_algebra_1 .. 
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49644  221 
lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" 
222 
proof (induct n) 

223 
case 0 

224 
then show ?case by vector 

225 
next 

226 
case Suc 

227 
then show ?case by vector 

228 
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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232 

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lemma neg_one_index [simp]: "( 1 :: 'a :: {one, uminus} ^ 'n) $ i =  1" 
49644  234 
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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242 

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instance vec :: (numeral, finite) numeral .. 
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instance vec :: (semiring_numeral, finite) semiring_numeral .. 
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245 

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lemma numeral_index [simp]: "numeral w $ i = numeral w" 
49644  247 
by (induct w) (simp_all only: numeral.simps vector_add_component one_index) 
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248 

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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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251 

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252 
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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254 

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lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" 
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256 
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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271 

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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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274 

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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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283 
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) 
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apply (rule member_le_setL2, simp_all) 
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done 
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291 

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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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294 

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lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e" 
53595  296 
by (metis component_le_norm_cart le_less_trans) 
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lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV" 
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by (simp add: norm_vec_def setL2_le_setsum) 
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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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lemma setsum_component [simp]: 
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fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n" 
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shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S" 
49644  311 
proof (cases "finite S") 
312 
case True 

313 
then show ?thesis by induct simp_all 

314 
next 

315 
case False 

316 
then show ?thesis by simp 

317 
qed 

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lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)" 
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by (simp add: vec_eq_iff) 
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321 

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lemma setsum_cmul: 
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fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n" 
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shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S" 
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by (simp add: vec_eq_iff setsum_right_distrib) 
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326 

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lemma setsum_norm_allsubsets_bound_cart: 
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fixes f:: "'a \<Rightarrow> real ^'n" 
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assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
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shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e" 
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using setsum_norm_allsubsets_bound[OF assms] 
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by (simp add: DIM_cart Basis_real_def) 
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333 

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subsection {* Matrix operations *} 
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text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *} 
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49644  338 
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" 
339 
(infixl "**" 70) 

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where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" 
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49644  342 
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" 
343 
(infixl "*v" 70) 

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where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m" 
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49644  346 
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " 
347 
(infixl "v*" 70) 

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where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n" 
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definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)" 
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definition transpose where 
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"(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))" 
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definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))" 
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definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))" 
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definition "rows(A::'a^'n^'m) = { row i A  i. i \<in> (UNIV :: 'm set)}" 
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definition "columns(A::'a^'n^'m) = { column i A  i. i \<in> (UNIV :: 'n set)}" 
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357 

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lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) 
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lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" 
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by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps) 
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361 

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lemma matrix_mul_lid: 
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fixes A :: "'a::semiring_1 ^ 'm ^ 'n" 
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364 
shows "mat 1 ** A = A" 
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apply (simp add: matrix_matrix_mult_def mat_def) 
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366 
apply vector 
49644  367 
apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite] 
368 
mult_1_left mult_zero_left if_True UNIV_I) 

369 
done 

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370 

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371 

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lemma matrix_mul_rid: 
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fixes A :: "'a::semiring_1 ^ 'm ^ 'n" 
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shows "A ** mat 1 = A" 
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apply (simp add: matrix_matrix_mult_def mat_def) 
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376 
apply vector 
49644  377 
apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite] 
378 
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) 

379 
done 

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380 

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lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" 
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apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) 
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383 
apply (subst setsum_commute) 
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384 
apply simp 
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385 
done 
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386 

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lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" 
49644  388 
apply (vector matrix_matrix_mult_def matrix_vector_mult_def 
389 
setsum_right_distrib setsum_left_distrib mult_assoc) 

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390 
apply (subst setsum_commute) 
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391 
apply simp 
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392 
done 
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393 

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394 
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" 
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395 
apply (vector matrix_vector_mult_def mat_def) 
49644  396 
apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong) 
397 
done 

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398 

49644  399 
lemma matrix_transpose_mul: 
400 
"transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" 

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401 
by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute) 
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402 

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403 
lemma matrix_eq: 
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404 
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm" 
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shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs") 
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406 
apply auto 
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407 
apply (subst vec_eq_iff) 
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408 
apply clarify 
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409 
apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) 
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410 
apply (erule_tac x="axis ia 1" in allE) 
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411 
apply (erule_tac x="i" in allE) 
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412 
apply (auto simp add: if_distrib cond_application_beta axis_def 
49644  413 
setsum_delta[OF finite] cong del: if_weak_cong) 
414 
done 

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changeset

415 

49644  416 
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x" 
44136
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more uniform naming scheme for finite cartesian product type and related theorems
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changeset

417 
by (simp add: matrix_vector_mult_def inner_vec_def) 
37489
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parents:
diff
changeset

418 

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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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diff
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419 
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
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changeset

420 
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac) 
37489
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changeset

421 
apply (subst setsum_commute) 
49644  422 
apply simp 
423 
done 

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

424 

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

425 
lemma transpose_mat: "transpose (mat n) = mat n" 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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parents:
diff
changeset

426 
by (vector transpose_def mat_def) 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

427 

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

428 
lemma transpose_transpose: "transpose(transpose A) = A" 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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changeset

429 
by (vector transpose_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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parents:
diff
changeset

430 

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

431 
lemma row_transpose: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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diff
changeset

432 
fixes A:: "'a::semiring_1^_^_" 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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parents:
diff
changeset

433 
shows "row i (transpose A) = column i A" 
44136
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more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

434 
by (simp add: row_def column_def transpose_def vec_eq_iff) 
37489
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

435 

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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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changeset

436 
lemma column_transpose: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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437 
fixes A:: "'a::semiring_1^_^_" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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parents:
diff
changeset

438 
shows "column i (transpose A) = row i A" 
44136
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more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

439 
by (simp add: row_def column_def transpose_def vec_eq_iff) 
37489
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
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changeset

440 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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441 
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" 
49644  442 
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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443 

49644  444 
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" 
445 
by (metis transpose_transpose rows_transpose) 

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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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446 

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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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447 
text{* Two sometimes fruitful ways of looking at matrixvector multiplication. *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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changeset

448 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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449 
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)" 
44136
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more uniform naming scheme for finite cartesian product type and related theorems
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parents:
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diff
changeset

450 
by (simp add: matrix_vector_mult_def inner_vec_def) 
37489
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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451 

49644  452 
lemma matrix_mult_vsum: 
453 
"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)" 

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

454 
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

455 

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

456 
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

457 
"(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)" 
899c9c4e4a4c
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diff
changeset

458 
by (simp add: axis_def if_distrib setsum_cases vec_eq_iff) 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
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diff
changeset

459 

899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
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diff
changeset

460 
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
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diff
changeset

461 
by (auto simp add: axis_def vec_eq_iff if_distrib setsum_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.
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parents:
diff
changeset

462 

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

463 
lemma linear_componentwise: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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parents:
diff
changeset

464 
fixes f:: "real ^'m \<Rightarrow> real ^ _" 
44e42d392c6e
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hoelzl
parents:
diff
changeset

465 
assumes lf: "linear f" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
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diff
changeset

466 
shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") 
49644  467 
proof  
37489
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
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changeset

468 
let ?M = "(UNIV :: 'm set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff
changeset

469 
let ?N = "(UNIV :: 'n set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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diff
changeset

470 
have fM: "finite ?M" by simp 
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

471 
have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
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diff
changeset

472 
unfolding setsum_component by simp 
49644  473 
then show ?thesis 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

474 
unfolding linear_setsum_mul[OF lf fM, 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

475 
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:
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diff
changeset

476 
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

477 
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

478 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

479 

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

480 
text{* Inverse matrices (not necessarily square) *} 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

481 

49644  482 
definition 
483 
"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

484 

49644  485 
definition 
486 
"matrix_inv(A:: 'a::semiring_1^'n^'m) = 

487 
(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

488 

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

489 
text{* Correspondence between matrices and linear operators. *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

490 

49644  491 
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

492 
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

493 

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

494 
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

495 
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff 
49644  496 
field_simps setsum_right_distrib setsum_addf) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

497 

49644  498 
lemma matrix_works: 
499 
assumes lf: "linear f" 

500 
shows "matrix f *v x = f (x::real ^ 'n)" 

501 
apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute) 

502 
apply clarify 

503 
apply (rule linear_componentwise[OF lf, symmetric]) 

504 
done 

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

505 

49644  506 
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" 
507 
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

508 

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

509 
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

510 
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

511 

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

512 
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

513 
assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)" 
49644  514 
and lg: "linear (g::real^'m \<Rightarrow> real^_)" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

515 
shows "matrix (g o f) = matrix g ** matrix f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

516 
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] 
49644  517 
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

518 

49644  519 
lemma matrix_vector_column: 
520 
"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)" 

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

521 
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

522 

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

523 
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

524 
apply (rule adjoint_unique) 
49644  525 
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def 
526 
setsum_left_distrib setsum_right_distrib) 

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

527 
apply (subst setsum_commute) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

528 
apply (auto simp add: mult_ac) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

530 

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

531 
lemma matrix_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

532 
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

533 
apply (subst matrix_vector_mul[OF lf]) 
49644  534 
unfolding adjoint_matrix matrix_of_matrix_vector_mul 
535 
apply rule 

536 
done 

537 

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

538 

44360  539 
subsection {* lambda skolemization on cartesian products *} 
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 
(* 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

542 

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

543 
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow> 
37494  544 
(\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs") 
49644  545 
proof  
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

546 
let ?S = "(UNIV :: 'n set)" 
49644  547 
{ assume H: "?rhs" 
548 
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

549 
moreover 
49644  550 
{ 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

551 
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

552 
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n" 
49644  553 
{ 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

554 
from f have "P i (f i)" by metis 
37494  555 
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

556 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

557 
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

558 
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

559 
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

560 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

561 

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

562 
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

563 
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

564 

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

565 
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

566 
"(\<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

567 
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

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 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

570 
"(\<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

571 
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

572 

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

573 
lemma matrix_left_invertible_injective: 
49644  574 
"(\<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)" 
575 
proof  

576 
{ 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

577 
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

578 
hence "x = y" 
49644  579 
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

580 
moreover 
49644  581 
{ assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

582 
hence i: "inj (op *v A)" unfolding inj_on_def by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

583 
from linear_injective_left_inverse[OF matrix_vector_mul_linear i] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

584 
obtain g where g: "linear g" "g o op *v A = id" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

585 
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

586 
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  587 
using g(2) by (simp add: fun_eq_iff) 
49644  588 
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

589 
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

590 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

591 

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

592 
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

593 
"(\<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

594 
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

595 
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

596 
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

597 

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

598 
lemma matrix_right_invertible_surjective: 
49644  599 
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)" 
600 
proof  

601 
{ fix B :: "real ^'m^'n" 

602 
assume AB: "A ** B = mat 1" 

603 
{ 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

604 
have "A *v (B *v x) = x" 
49644  605 
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) } 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

606 
hence "surj (op *v A)" unfolding surj_def by metis } 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

607 
moreover 
49644  608 
{ assume sf: "surj (op *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

609 
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

610 
obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

611 
by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

612 

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

613 
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

614 
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

615 
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  616 
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

617 
. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

618 
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

619 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

620 
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

621 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

622 

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

623 
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

624 
fixes A :: "real^'n^'m" 
49644  625 
shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> 
626 
(\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" 

627 
(is "?lhs \<longleftrightarrow> ?rhs") 

628 
proof  

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

629 
let ?U = "UNIV :: 'n set" 
49644  630 
{ assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0" 
631 
{ fix c i 

632 
assume c: "setsum (\<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

633 
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

634 
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

635 
using c 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

636 
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

637 
by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

638 
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

639 
have "c i = 0" by (vector vec_eq_iff)} 
49644  640 
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

641 
moreover 
49644  642 
{ assume H: ?rhs 
643 
{ 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

644 
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

645 
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] 
49644  646 
have "x = 0" by vector } 
647 
} 

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

648 
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

649 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

650 

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

651 
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

652 
fixes A :: "real^'n^'m" 
49644  653 
shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> 
654 
(\<forall>c. setsum (\<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

655 
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

656 
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

657 
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

658 

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

659 
lemma matrix_right_invertible_span_columns: 
49644  660 
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> 
661 
span (columns A) = UNIV" (is "?lhs = ?rhs") 

662 
proof  

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

663 
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

664 
have fU: "finite ?U" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

665 
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

666 
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def 
49644  667 
apply (subst eq_commute) 
668 
apply rule 

669 
done 

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

670 
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast 
49644  671 
{ assume h: ?lhs 
672 
{ fix x:: "real ^'n" 

673 
from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m" 

674 
where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast 

675 
have "x \<in> span (columns A)" 

676 
unfolding y[symmetric] 

677 
apply (rule span_setsum[OF fU]) 

678 
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

679 
unfolding scalar_mult_eq_scaleR 
49644  680 
apply (rule span_mul) 
681 
apply (rule span_superset) 

682 
unfolding columns_def 

683 
apply blast 

684 
done 

685 
} 

686 
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

687 
moreover 
49644  688 
{ assume h:?rhs 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

689 
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y" 
49644  690 
{ fix y 
691 
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

692 
proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR]) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

693 
show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

694 
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

695 
next 
49644  696 
fix c y1 y2 
697 
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

698 
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

699 
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

700 
from y2 obtain x:: "real ^'m" where 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

701 
x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

702 
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

703 
show "?P (c*s y1 + y2)" 
49962
a8cc904a6820
Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents:
49644
diff
changeset

704 
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  705 
fix j 
706 
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) 

707 
else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" 

708 
using i(1) by (simp add: field_simps) 

709 
have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 

710 
else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U" 

711 
apply (rule setsum_cong[OF refl]) 

712 
using th apply blast 

713 
done 

714 
also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" 

715 
by (simp add: setsum_addf) 

716 
also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" 

717 
unfolding setsum_delta[OF fU] 

718 
using i(1) by simp 

719 
finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 

720 
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" . 

721 
qed 

722 
next 

723 
show "y \<in> span (columns A)" 

724 
unfolding h by blast 

725 
qed 

726 
} 

727 
then have ?lhs unfolding lhseq .. 

728 
} 

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

729 
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

730 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

731 

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

732 
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

733 
"(\<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

734 
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

735 
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

736 
unfolding matrix_right_invertible_span_columns 
49644  737 
.. 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

738 

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

739 
text {* The same result in terms of square matrices. *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

740 

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

741 
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

742 
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

743 
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1" 
49644  744 
proof  
745 
{ fix A A' :: "real ^'n^'n" 

746 
assume AA': "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

747 
have sA: "surj (op *v A)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

748 
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

749 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

750 
apply (rule_tac x="(A' *v y)" in exI) 
49644  751 
apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid) 
752 
done 

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

753 
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

754 
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

755 
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

756 
have th: "matrix f' ** A = mat 1" 
49644  757 
by (simp add: matrix_eq matrix_works[OF f'(1)] 
758 
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

759 
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp 
49644  760 
hence "matrix f' = A'" 
761 
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

762 
hence "matrix f' ** A = A' ** A" by simp 
49644  763 
hence "A' ** A = mat 1" by (simp add: th) 
764 
} 

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

765 
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

766 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

767 

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

768 
text {* Considering an nelement vector as an nby1 or 1byn matrix. *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

769 

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

770 
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

771 

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

772 
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

773 

49644  774 
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

775 
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

776 

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

777 
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

778 
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

779 

49644  780 
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

781 
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

782 

49644  783 
lemma dot_matrix_product: 
784 
"(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

785 
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

786 

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

787 
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

788 
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

789 
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

790 
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" 
49644  791 
unfolding dot_matrix_product transpose_columnvector[symmetric] 
792 
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

793 

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

794 

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

795 
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) i. i\<in>UNIV}" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

796 
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

797 

49644  798 
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

799 
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

800 
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

801 

49644  802 
lemma continuous_component: "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

803 
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

804 

49644  805 
lemma continuous_on_component: "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

806 
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

807 

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

808 
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}" 
44233  809 
by (simp add: Collect_all_eq closed_INT closed_Collect_le) 
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

810 

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

811 
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)" 
49644  812 
unfolding bounded_def 
813 
apply clarify 

814 
apply (rule_tac x="x $ i" in exI) 

815 
apply (rule_tac x="e" in exI) 

816 
apply clarify 

817 
apply (rule order_trans [OF dist_vec_nth_le], simp) 

818 
done 

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

819 

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

820 
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

821 
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n" 
50998  822 
assumes f: "bounded (range f)" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

823 
shows "\<forall>d. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

824 
\<exists>l r. subseq r \<and> 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

825 
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

826 
proof 
49644  827 
fix d :: "'n set" 
828 
have "finite 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

829 
thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and> 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

830 
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)" 
49644  831 
proof (induct d) 
832 
case empty 

833 
thus ?case unfolding subseq_def by auto 

834 
next 

835 
case (insert k d) 

836 
obtain l1::"'a^'n" and r1 where r1:"subseq r1" 

837 
and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" 

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

838 
using insert(3) by auto 
50998  839 
have s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)` 
840 
by (auto intro!: bounded_component_cart) 

841 
have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp 

842 
have "bounded (range (\<lambda>i. f (r1 i) $ k))" 

843 
by (metis (lifting) bounded_subset image_subsetI f' s') 

844 
then obtain l2 r2 where r2: "subseq r2" 

49644  845 
and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) > l2) sequentially" 
50998  846 
using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def) 
49644  847 
def r \<equiv> "r1 \<circ> r2" 
848 
have r: "subseq r" 

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

849 
using r1 and r2 unfolding r_def o_def subseq_def by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

850 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

851 
def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n" 
49644  852 
{ fix e :: real assume "e > 0" 
853 
from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" 

854 
by blast 

855 
from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" 

856 
by (rule tendstoD) 

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

857 
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

858 
by (rule eventually_subseq) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

859 
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

860 
using N1' N2 by (rule eventually_elim2, simp add: l_def r_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

861 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

862 
ultimately show ?case by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

863 
qed 
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 

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

866 
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

867 
proof 
50998  868 
fix f :: "nat \<Rightarrow> 'a ^ 'b" 
869 
assume f: "bounded (range f)" 

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

870 
then obtain l r where r: "subseq r" 
49644  871 
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially" 
50998  872 
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

873 
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

874 
{ 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

875 
hence "0 < e / (real_of_nat (card ?d))" 
49644  876 
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

877 
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

878 
by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

879 
moreover 
49644  880 
{ fix n 
881 
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

882 
have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

883 
unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

884 
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

885 
by (rule setsum_strict_mono) (simp_all add: n) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

886 
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

887 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

888 
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

889 
by (rule eventually_elim1) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

890 
} 
49644  891 
hence "((f \<circ> r) > l) sequentially" unfolding o_def tendsto_iff 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

892 
with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) > l) sequentially" by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

893 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

894 

49644  895 
lemma interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

896 
fixes a :: "real^'n" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

897 
shows "box a b = {x::real^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" 
56188  898 
and "cbox a b = {x::real^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}" 
899 
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

900 

49644  901 
lemma mem_interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

902 
fixes a :: "real^'n" 
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

903 
shows "x \<in> box a b \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)" 
56188  904 
and "x \<in> cbox a b \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)" 
49644  905 
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

906 

49644  907 
lemma interval_eq_empty_cart: 
908 
fixes a :: "real^'n" 

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

909 
shows "(box a b = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) 
56188  910 
and "(cbox a b = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2) 
49644  911 
proof  
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

912 
{ 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

913 
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

914 
hence "a$i < b$i" by auto 
49644  915 
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

916 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

917 
{ 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

918 
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

919 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

920 
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

921 
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

922 
unfolding vector_smult_component and vector_add_component 
49644  923 
by auto } 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

924 
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

925 
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

926 

56188  927 
{ 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

928 
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

929 
hence "a$i \<le> b$i" by auto 
49644  930 
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

931 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

932 
{ 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

933 
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

934 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

935 
have "a$i \<le> b$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

936 
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

937 
unfolding vector_smult_component and vector_add_component 
49644  938 
by auto } 
56188  939 
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

940 
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

941 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

942 

49644  943 
lemma interval_ne_empty_cart: 
944 
fixes a :: "real^'n" 

56188  945 
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

946 
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

947 
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

948 
(* 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

949 

49644  950 
lemma subset_interval_imp_cart: 
951 
fixes a :: "real^'n" 

56188  952 
shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> cbox c d \<subseteq> cbox a b" 
953 
and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> cbox c d \<subseteq> box a b" 

954 
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

955 
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

956 
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

957 
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

958 

49644  959 
lemma interval_sing: 
960 
fixes a :: "'a::linorder^'n" 

961 
shows "{a .. a} = {a} \<and> {a<..<a} = {}" 

962 
apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff) 

963 
done 

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

964 

49644  965 
lemma subset_interval_cart: 
966 
fixes a :: "real^'n" 

56188  967 
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) 
968 
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) 

969 
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

970 
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  971 
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

972 

49644  973 
lemma disjoint_interval_cart: 
974 
fixes a::"real^'n" 

56188  975 
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) 
976 
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) 

977 
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

978 
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

979 
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

980 

49644  981 
lemma inter_interval_cart: 
54775
2d3df8633dad
prefer box over greaterThanLessThan on euclidean_space
immler
parents:
54489
diff
changeset

982 
fixes a :: "real^'n" 
56188  983 
shows "cbox a b \<inter> cbox c d = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}" 
984 
unfolding inter_interval 

985 
by (auto simp: mem_box less_eq_vec_def) 

986 
(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

987 

49644  988 
lemma closed_interval_left_cart: 
989 
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

990 
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}" 
44233  991 
by (simp add: Collect_all_eq closed_INT closed_Collect_le) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

992 

49644  993 
lemma closed_interval_right_cart: 
994 
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

995 
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}" 
44233  996 
by (simp add: Collect_all_eq closed_INT closed_Collect_le) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

997 

49644  998 
lemma is_interval_cart: 
999 
"is_interval (s::(real^'n) set) \<longleftrightarrow> 

1000 
(\<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

1001 
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

1002 

49644  1003 
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}" 
44233  1004 
by (simp add: closed_Collect_le) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1005 

49644  1006 
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}" 
44233  1007 
by (simp add: closed_Collect_le) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1008 

49644  1009 
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}" 
1010 
by (simp add: open_Collect_less) 

1011 

1012 
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}" 

44233  1013 
by (simp add: open_Collect_less) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1014 

49644  1015 
lemma Lim_component_le_cart: 
1016 
fixes f :: "'a \<Rightarrow> 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

1017 
assumes "(f > l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f x $i \<le> b) net" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1018 
shows "l$i \<le> b" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1019 
by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)]) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1020 

49644  1021 
lemma Lim_component_ge_cart: 
1022 
fixes f :: "'a \<Rightarrow> 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

1023 
assumes "(f > l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1024 
shows "b \<le> l$i" 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1025 
by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)]) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1026 

49644  1027 
lemma Lim_component_eq_cart: 
1028 
fixes f :: "'a \<Rightarrow> real^'n" 

1029 
assumes net: "(f > l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net" 

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

1030 
shows "l$i = b" 
49644  1031 
using ev[unfolded order_eq_iff eventually_conj_iff] and 
1032 
Lim_component_ge_cart[OF net, of b i] and 

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

1033 
Lim_component_le_cart[OF net, of i b] by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1034 

49644  1035 
lemma connected_ivt_component_cart: 
1036 
fixes x :: "real^'n" 

1037 
shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)" 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1038 
using connected_ivt_hyperplane[of s x y "axis k 1" a] 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1039 
by (auto simp add: inner_axis inner_commute) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1040 

49644  1041 
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1042 
unfolding subspace_def by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1043 

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

1044 
lemma closed_substandard_cart: 
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

1045 
"closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}" 
49644  1046 
proof  
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

1047 
{ fix i::'n 
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

1048 
have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}" 
49644  1049 
by (cases "P i") (simp_all add: closed_Collect_eq) } 
44213
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

1050 
thus ?thesis 
6fb54701a11b
add lemmas open_Collect_less, closed_Collect_le, closed_Collect_eq;
huffman
parents:
44211
diff
changeset

1051 
unfolding Collect_all_eq by (simp add: closed_INT) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1052 
qed 
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 dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" 
1055 
(is "dim ?A = _") 

1056 
proof  

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1057 
let ?a = "\<lambda>x. axis x 1 :: real^'n" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1058 
have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1059 
by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis) 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1060 
have "?a ` d \<subseteq> Basis" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1061 
by (auto simp: Basis_vec_def) 
49644  1062 
thus ?thesis 
50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1063 
using dim_substandard[of "?a ` d"] card_image[of ?a d] 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1064 
by (auto simp: axis_eq_axis inj_on_def *) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1065 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1066 

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

1067 
lemma affinity_inverses: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1068 
assumes m0: "m \<noteq> (0::'a::field)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1069 
shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + ((inverse(m) *s c))) = id" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1070 
"(\<lambda>x. inverse(m) *s x + ((inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1071 
using m0 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset

1072 
apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff) 
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset

1073 
apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric]) 
49644  1074 
done 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1075 

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

1076 
lemma vector_affinity_eq: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1077 
assumes m0: "(m::'a::field) \<noteq> 0" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1078 
shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + (inverse m *s c)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1079 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1080 
assume h: "m *s x + c = y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1081 
hence "m *s x = y  c" by (simp add: field_simps) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1082 
hence "inverse m *s (m *s x) = inverse m *s (y  c)" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1083 
then show "x = inverse m *s y +  (inverse m *s c)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1084 
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1085 
next 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1086 
assume h: "x = inverse m *s y +  (inverse m *s c)" 
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53600
diff
changeset

1087 
show "m *s x + c = y" unfolding h 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1088 
using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1089 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1090 

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

1091 
lemma vector_eq_affinity: 
49644  1092 
"(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + (inverse(m) *s c) = x)" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1093 
using vector_affinity_eq[where m=m and x=x and y=y and c=c] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1094 
by metis 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1095 

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

1096 
lemma vector_cart: 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1097 
fixes f :: "real^'n \<Rightarrow> real" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1098 
shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1099 
unfolding euclidean_eq_iff[where 'a="real^'n"] 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1100 
by simp (simp add: Basis_vec_def inner_axis) 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1101 

899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1102 
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1103 
by (rule vector_cart) 
49644  1104 

44360  1105 
subsection "Convex Euclidean Space" 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1106 

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

1107 
lemma Cart_1:"(1::real^'n) = \<Sum>Basis" 
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1108 
using const_vector_cart[of 1] by (simp add: one_vec_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1109 

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

1110 
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1111 
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1112 

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

1113 
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1114 

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

1115 
lemma convex_box_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1116 
assumes "\<And>i. convex {x. P i x}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1117 
shows "convex {x. \<forall>i. P i (x$i)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1118 
using assms unfolding convex_def by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1119 

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

1120 
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1121 
by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1122 

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

1123 
lemma unit_interval_convex_hull_cart: 
56188  1124 
"cbox (0::real^'n) 1 = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" 
1125 
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"] box_real[symmetric] 

50526
899c9c4e4a4c
Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents:
49962
diff
changeset

1126 
by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1127 

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

1128 
lemma cube_convex_hull_cart: 
49644  1129 
assumes "0 < d" 
1130 
obtains s::"(real^'n) set" 

56188  1131 
where "finite s" "cbox (x  (\<chi> i. d)) (x + (\<chi> i. d)) = convex hull s" 
49644  1132 
proof  
55522  1133 
from assms obtain s where "finite s" 
56188  1134 
and "cbox (x  setsum (op *\<^sub>R d) Basis) (x + setsum (op *\<^sub>R d) Basis) = convex hull s" 
55522  1135 
by (rule cube_convex_hull) 
1136 
with that[of s] show thesis 

1137 
by (simp add: const_vector_cart) 

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

1138 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1139 

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

1140 

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

1141 
subsection "Derivative" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1142 

49644  1143 
lemma differentiable_at_imp_differentiable_on: 
1144 
"(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s" 

51641
cd05e9fcc63d
remove the withinfilter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51489
diff
changeset

1145 
by (metis differentiable_at_withinI differentiable_on_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1146 

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

1147 
definition "jacobian f net = matrix(frechet_derivative f net)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1148 

49644  1149 
lemma jacobian_works: 
1150 
"(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> 

1151 
(f has_derivative (\<lambda>h. (jacobian f net) *v h)) net" 

1152 
apply rule 

1153 
unfolding jacobian_def 
