author  wenzelm 
Sat, 07 Apr 2012 16:41:59 +0200  
changeset 47389  e8552cba702d 
parent 47108  2a1953f0d20d 
child 49197  e5224d887e12 
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)" 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 sumr_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 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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thus "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 \<longleftrightarrow> (\<forall>i. x$i < y$i)" 
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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 = 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 from UNIV_one show "finite (UNIV :: 'a set)" 
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by (auto intro!: card_ge_0_finite) qed 
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end 
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instantiation vec :: (linorder,cart_one) linorder begin 
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instance proof 
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guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply by(erule exE)+ 
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hence *:"UNIV = {a}" by auto 
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have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto 
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fix x y z::"'a^'b::cart_one" note * = less_eq_vec_def less_vec_def all vec_eq_iff 
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show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps) 
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{ assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) } 
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{ assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) } 
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qed end 
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text{* Constant Vectors *} 
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definition "vec x = (\<chi> i. x)" 
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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 = HOL_basic_ss 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 = @{simpset} 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 ths = 
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simp_tac ss1 
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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 (HOL_basic_ss 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 (ss2 addsimps ths) 
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in 
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Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths))) 
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end 
42814  110 
*} "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: assumes fS: "finite S" 
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shows "vec(setsum f S) = setsum (vec o f) S" 
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apply (induct rule: finite_induct[OF fS]) 
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apply (simp) 
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apply (auto simp add: vec_add) 
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done 
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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$i)" 
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by vector 
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lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector 
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lemmas vector_component = 
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vec_component vector_add_component vector_mult_component 
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vector_smult_component vector_minus_component vector_uminus_component 
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vector_scaleR_component cond_component 
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subsection {* 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 :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult 
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by default (vector mult_idem) 
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instance vec :: (comm_monoid_mult, finite) comm_monoid_mult 
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by default vector 
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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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apply intro_classes 
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apply (simp_all add: vec_eq_iff) 
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done 
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instance vec :: (real_algebra_1, finite) real_algebra_1 .. 
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lemma of_nat_index: 
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"(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n" 
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apply (induct n) 
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apply vector 
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apply vector 
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done 
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lemma one_index[simp]: 
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"(1 :: 'a::one ^'n)$i = 1" by vector 
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instance vec :: (semiring_char_0, finite) semiring_char_0 
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proof 
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fix m n :: nat 
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show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)" 
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by (auto intro!: injI simp add: vec_eq_iff of_nat_index) 
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qed 
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instance vec :: (numeral, finite) numeral .. 
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instance vec :: (semiring_numeral, finite) semiring_numeral .. 
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lemma numeral_index [simp]: "numeral w $ i = numeral w" 
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by (induct w, simp_all only: numeral.simps vector_add_component one_index) 
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lemma neg_numeral_index [simp]: "neg_numeral w $ i = neg_numeral w" 
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by (simp only: neg_numeral_def vector_uminus_component numeral_index) 
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instance vec :: (comm_ring_1, finite) comm_ring_1 .. 
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instance vec :: (ring_char_0, finite) ring_char_0 .. 
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lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x" 
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by (vector mult_assoc) 
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lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x" 
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by (vector field_simps) 
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lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y" 
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by (vector field_simps) 
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lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector 
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lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector 
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lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x  y) = c *s x  c *s y" 
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by (vector field_simps) 
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lemma vector_smult_rneg: "(c::'a::ring) *s x = (c *s x)" by vector 
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lemma vector_smult_lneg: " (c::'a::ring) *s x = (c *s x)" by vector 
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234 
lemma vector_sneg_minus1: "x = (1::'a::ring_1) *s x" by vector 
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235 
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector 
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236 
lemma vector_sub_rdistrib: "((a::'a::ring)  b) *s x = a *s x  b *s x" 
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237 
by (vector field_simps) 
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changeset

238 

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239 
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)" 
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240 
by (simp add: vec_eq_iff) 
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changeset

241 

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242 
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero) 
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243 
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0" 
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244 
by vector 
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245 
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y" 
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246 
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib) 
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247 
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0" 
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248 
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib) 
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249 
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==> a *s x = a *s y ==> (x = y)" 
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250 
by (metis vector_mul_lcancel) 
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251 
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b" 
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252 
by (metis vector_mul_rcancel) 
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diff
changeset

253 

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254 
lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x" 
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255 
apply (simp add: norm_vec_def) 
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256 
apply (rule member_le_setL2, simp_all) 
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257 
done 
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258 

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259 
lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e" 
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260 
by (metis component_le_norm_cart order_trans) 
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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
changeset

261 

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262 
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e" 
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263 
by (metis component_le_norm_cart basic_trans_rules(21)) 
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hoelzl
parents:
diff
changeset

264 

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diff
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265 
lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV" 
44136
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changeset

266 
by (simp add: norm_vec_def setL2_le_setsum) 
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changeset

267 

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diff
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268 
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x" 
44136
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diff
changeset

269 
unfolding scaleR_vec_def vector_scalar_mult_def by simp 
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changeset

270 

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271 
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y" 
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272 
unfolding dist_norm scalar_mult_eq_scaleR 
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273 
unfolding scaleR_right_diff_distrib[symmetric] by simp 
44e42d392c6e
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hoelzl
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diff
changeset

274 

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275 
lemma setsum_component [simp]: 
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276 
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n" 
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277 
shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S" 
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278 
by (cases "finite S", induct S set: finite, simp_all) 
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diff
changeset

279 

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280 
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)" 
44136
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diff
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281 
by (simp add: vec_eq_iff) 
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changeset

282 

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283 
lemma setsum_cmul: 
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diff
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284 
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n" 
44e42d392c6e
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parents:
diff
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285 
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S" 
44136
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huffman
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44135
diff
changeset

286 
by (simp add: vec_eq_iff setsum_right_distrib) 
37489
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff
changeset

287 

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288 
(* TODO: use setsum_norm_allsubsets_bound *) 
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289 
lemma setsum_norm_allsubsets_bound_cart: 
44e42d392c6e
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290 
fixes f:: "'a \<Rightarrow> real ^'n" 
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291 
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e" 
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292 
shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e" 
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diff
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293 
proof 
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294 
let ?d = "real CARD('n)" 
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295 
let ?nf = "\<lambda>x. norm (f x)" 
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296 
let ?U = "UNIV :: 'n set" 
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297 
have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U" 
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diff
changeset

298 
by (rule setsum_commute) 
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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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299 
have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_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

300 
have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P" 
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

301 
apply (rule setsum_mono) by (rule norm_le_l1_cart) 
44e42d392c6e
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hoelzl
parents:
diff
changeset

302 
also have "\<dots> \<le> 2 * ?d * e" 
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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

303 
unfolding th0 th1 
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diff
changeset

304 
proof(rule setsum_bounded) 
44e42d392c6e
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hoelzl
parents:
diff
changeset

305 
fix i assume i: "i \<in> ?U" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

306 
let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}" 
44e42d392c6e
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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307 
let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

308 
have thp: "P = ?Pp \<union> ?Pn" by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

309 
have thp0: "?Pp \<inter> ?Pn ={}" by auto 
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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

310 
have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+ 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

311 
have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

312 
using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i] fPs[OF PpP] 
44e42d392c6e
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diff
changeset

313 
by (auto intro: abs_le_D1) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

314 
have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

315 
using component_le_norm_cart[of "setsum (\<lambda>x.  f x) ?Pn" i] fPs[OF PnP] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

316 
by (auto simp add: setsum_negf intro: abs_le_D1) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

317 
have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

318 
apply (subst thp) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

319 
apply (rule setsum_Un_zero) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

320 
using fP thp0 by auto 
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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

321 
also have "\<dots> \<le> 2*e" using Pne Ppe by arith 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

322 
finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

323 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

324 
finally show ?thesis . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

325 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

326 

44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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diff
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327 
lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

328 

44e42d392c6e
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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329 
lemma split_dimensions'[consumes 1]: 
44129  330 
assumes "k < DIM('a::euclidean_space^'b)" 
331 
obtains i j where "i < CARD('b::finite)" and "j < DIM('a::euclidean_space)" and "k = j + i * DIM('a::euclidean_space)" 

37489
44e42d392c6e
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hoelzl
parents:
diff
changeset

332 
using split_times_into_modulo[OF assms[simplified]] . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

333 

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

334 
lemma cart_euclidean_bound[intro]: 
44129  335 
assumes j:"j < DIM('a::euclidean_space)" 
336 
shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::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

337 
using linear_less_than_times[OF pi'_range j, of i] . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

338 

44129  339 
lemma (in euclidean_space) forall_CARD_DIM: 
37489
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diff
changeset

340 
"(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

341 
(is "?l \<longleftrightarrow> ?r") 
44e42d392c6e
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hoelzl
parents:
diff
changeset

342 
proof (safe elim!: split_times_into_modulo) 
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343 
fix i :: 'b and j assume "j < DIM('a)" 
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344 
note linear_less_than_times[OF pi'_range[of i] this] 
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345 
moreover assume "?l" 
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346 
ultimately show "P (j + \<pi>' i * DIM('a))" by auto 
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347 
next 
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348 
fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r" 
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349 
from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)` 
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350 
show "P (j + i * DIM('a))" by simp 
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351 
qed 
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352 

44129  353 
lemma (in euclidean_space) exists_CARD_DIM: 
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354 
"(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))" 
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355 
using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast 
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356 

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357 
lemma forall_CARD: 
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358 
"(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))" 
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359 
using forall_CARD_DIM[where 'a=real, of P] by simp 
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360 

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361 
lemma exists_CARD: 
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362 
"(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))" 
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363 
using exists_CARD_DIM[where 'a=real, of P] by simp 
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364 

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365 
lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD 
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366 

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367 
lemma cart_euclidean_nth[simp]: 
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368 
fixes x :: "('a::euclidean_space, 'b::finite) vec" 
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369 
assumes j:"j < DIM('a)" 
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370 
shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j" 
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371 
unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta 
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372 
by (simp add: setsum_cases) 
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373 

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374 
lemma real_euclidean_nth: 
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375 
fixes x :: "real^'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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376 
shows "x $$ \<pi>' i = (x $ i :: real)" 
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377 
using cart_euclidean_nth[where 'a=real, of 0 x i] by simp 
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378 

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379 
lemmas nth_conv_component = real_euclidean_nth[symmetric] 
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380 

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381 
lemma mult_split_eq: 
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382 
fixes A :: nat assumes "x < A" "y < A" 
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383 
shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j" 
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384 
proof 
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385 
assume *: "x + i * A = y + j * A" 
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386 
{ fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A" 
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387 
hence "x + i * A < Suc i * A" using `x < A` by simp 
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388 
also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp 
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389 
also have "\<dots> \<le> y + j * A" by simp 
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390 
finally have "i = j" using * by simp } 
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391 
note eq = this 
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392 

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393 
have "i = j" 
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394 
proof (cases rule: linorder_cases) 
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395 
assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp 
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396 
next 
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397 
assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp 
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398 
qed simp 
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399 
thus "x = y \<and> i = j" using * by simp 
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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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400 
qed simp 
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diff
changeset

401 

44136
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diff
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402 
instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space 
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403 
proof 
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parents:
diff
changeset

404 
fix x y::"'a^'b" 
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huffman
parents:
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diff
changeset

405 
show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)" 
e63ad7d5158d
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huffman
parents:
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diff
changeset

406 
unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps) 
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diff
changeset

407 
show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)" 
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diff
changeset

408 
unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps) 
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409 
qed 
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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

410 

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411 
subsection{* Basis vectors in coordinate directions. *} 
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412 

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413 
definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)" 
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414 

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415 
lemma basis_component [simp]: "cart_basis k $ i = (if k=i then 1 else 0)" 
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416 
unfolding cart_basis_def by simp 
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diff
changeset

417 

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418 
lemma norm_basis[simp]: 
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419 
shows "norm (cart_basis k :: real ^'n) = 1" 
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420 
apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def 
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421 
apply (vector delta_mult_idempotent) 
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422 
using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto 
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423 

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424 
lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1" 
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425 
by (rule norm_basis) 
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changeset

426 

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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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427 
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c" 
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428 
by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp 
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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
changeset

429 

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430 
lemma vector_choose_dist: assumes e: "0 <= e" 
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431 
shows "\<exists>(y::real^'n). dist x y = e" 
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432 
proof 
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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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433 
from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e" 
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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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434 
by blast 
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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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435 
then have "dist x (x  c) = e" by (simp add: dist_norm) 
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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 
then show ?thesis by blast 
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Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff
changeset

437 
qed 
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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

438 

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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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439 
lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)" 
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440 
by (simp add: inj_on_def vec_eq_iff) 
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441 

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442 
lemma basis_expansion: 
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443 
"setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _") 
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444 
by (auto simp add: vec_eq_iff if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong) 
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445 

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446 
lemma smult_conv_scaleR: "c *s x = scaleR c x" 
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447 
unfolding vector_scalar_mult_def scaleR_vec_def by simp 
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448 

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449 
lemma basis_expansion': 
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450 
"setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x" 
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451 
by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR]) 
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452 

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453 
lemma basis_expansion_unique: 
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changeset

454 
"setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

455 
by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

456 

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

457 
lemma dot_basis: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

458 
shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

459 
by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

460 
cong del: if_weak_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

461 

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

462 
lemma inner_basis: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

463 
fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

464 
shows "inner (cart_basis i) x = inner 1 (x $ i)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

465 
and "inner x (cart_basis i) = inner (x $ i) 1" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

466 
unfolding inner_vec_def cart_basis_def 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

467 
by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

468 

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

469 
lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

470 
by (auto simp add: vec_eq_iff) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

471 

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

472 
lemma basis_nonzero: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

473 
shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

474 
by (simp add: basis_eq_0) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

475 

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

476 
text {* some lemmas to map between Eucl and Cart *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

477 
lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

478 
unfolding basis_vec_def using pi'_range[where 'n='a] 
44166
d12d89a66742
modify euclidean_space class to include basis set
huffman
parents:
44165
diff
changeset

479 
by (auto simp: vec_eq_iff axis_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

480 

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

481 
subsection {* Orthogonality on cartesian products *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

482 

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

483 
lemma orthogonal_basis: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

484 
shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

485 
by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

486 
cond_application_beta setsum_delta cong del: if_weak_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

487 

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

488 
lemma orthogonal_basis_basis: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

489 
shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

490 
unfolding orthogonal_basis[of i] basis_component[of j] by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

491 

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

492 
subsection {* Linearity on cartesian products *} 
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 linear_vmul_component: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

495 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

496 
shows "linear (\<lambda>x. f x $ k *\<^sub>R v)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

497 
using lf 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

498 
by (auto simp add: linear_def algebra_simps) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

499 

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

500 

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

501 
subsection{* Adjoints on cartesian products *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

502 

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

503 
text {* TODO: The following lemmas about adjoints should hold for any 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

504 
Hilbert space (i.e. complete inner product space). 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

505 
(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint}) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

506 
*} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

507 

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

508 
lemma adjoint_works_lemma: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

509 
fixes 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

510 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

511 
shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

512 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

513 
let ?N = "UNIV :: 'n set" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

514 
let ?M = "UNIV :: 'm set" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

515 
have fN: "finite ?N" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

516 
have fM: "finite ?M" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

517 
{fix y:: "real ^ 'm" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

518 
let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> 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

519 
{fix x 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

520 
have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

521 
by (simp only: basis_expansion') 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

522 
also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

523 
unfolding linear_setsum[OF lf fN] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

524 
by (simp add: linear_cmul[OF lf]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

525 
finally have "f x \<bullet> y = x \<bullet> ?w" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

526 
apply (simp only: ) 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

527 
apply (simp add: inner_vec_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

528 
done} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

530 
then show ?thesis unfolding adjoint_def 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

531 
some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

532 
using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

533 
by metis 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

534 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

535 

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

536 
lemma adjoint_works: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

537 
fixes 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

538 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

539 
shows "x \<bullet> adjoint f y = f x \<bullet> y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

540 
using adjoint_works_lemma[OF lf] by metis 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

541 

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

542 
lemma adjoint_linear: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

543 
fixes 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

544 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

545 
shows "linear (adjoint f)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

546 
unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

547 
unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

548 

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

549 
lemma adjoint_clauses: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

550 
fixes 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

551 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

552 
shows "x \<bullet> adjoint f y = f x \<bullet> y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

553 
and "adjoint f y \<bullet> x = y \<bullet> f x" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

554 
by (simp_all add: adjoint_works[OF lf] inner_commute) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

555 

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

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

557 
fixes 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

558 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

559 
shows "adjoint (adjoint f) = f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

560 
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf]) 
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 

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

563 
subsection {* Matrix operations *} 
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 
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

566 

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

567 
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" (infixl "**" 70) 
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

568 
where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

569 

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

570 
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" (infixl "*v" 70) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

571 
where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m" 
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 
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " (infixl "v*" 70) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

574 
where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

575 

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

576 
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

577 
definition transpose where 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

578 
"(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

579 
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

580 
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

581 
definition "rows(A::'a^'n^'m) = { row i A  i. i \<in> (UNIV :: 'm set)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

582 
definition "columns(A::'a^'n^'m) = { column i A  i. i \<in> (UNIV :: 'n set)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

583 

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

584 
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

585 
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

586 
by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

587 

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

588 
lemma matrix_mul_lid: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

589 
fixes A :: "'a::semiring_1 ^ 'm ^ 'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

590 
shows "mat 1 ** A = A" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

591 
apply (simp add: matrix_matrix_mult_def mat_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

592 
apply vector 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

593 
by (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite] mult_1_left mult_zero_left if_True UNIV_I) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

594 

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

595 

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

596 
lemma matrix_mul_rid: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

597 
fixes A :: "'a::semiring_1 ^ 'm ^ 'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

598 
shows "A ** mat 1 = A" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

599 
apply (simp add: matrix_matrix_mult_def mat_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

600 
apply vector 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

601 
by (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite] mult_1_right mult_zero_right if_True UNIV_I cong: if_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

602 

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

603 
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

604 
apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

605 
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

606 
apply simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

607 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

608 

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

609 
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

610 
apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

611 
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

612 
apply simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

613 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

614 

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

615 
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

616 
apply (vector matrix_vector_mult_def mat_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

618 
setsum_delta' cong del: if_weak_cong) 
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 
lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

621 
by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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_eq: 
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 B :: "'a::semiring_1 ^ 'n ^ 'm" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

625 
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

626 
apply auto 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

627 
apply (subst vec_eq_iff) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

628 
apply clarify 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

629 
apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

630 
apply (erule_tac x="cart_basis ia" in allE) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

631 
apply (erule_tac x="i" in allE) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

632 
by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

633 

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

634 
lemma matrix_vector_mul_component: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

635 
shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

636 
by (simp add: matrix_vector_mult_def inner_vec_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

637 

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

638 
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
huffman
parents:
44135
diff
changeset

639 
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

640 
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

641 
by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

642 

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

643 
lemma transpose_mat: "transpose (mat n) = mat n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

644 
by (vector transpose_def mat_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

645 

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

646 
lemma transpose_transpose: "transpose(transpose A) = A" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

647 
by (vector transpose_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

648 

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

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

650 
fixes A:: "'a::semiring_1^_^_" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

651 
shows "row i (transpose A) = column i A" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

652 
by (simp add: row_def column_def transpose_def vec_eq_iff) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

653 

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

654 
lemma column_transpose: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

655 
fixes A:: "'a::semiring_1^_^_" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

656 
shows "column i (transpose A) = row i A" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

657 
by (simp add: row_def column_def transpose_def vec_eq_iff) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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 rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

660 
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

661 

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

662 
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

663 

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

664 
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.
hoelzl
parents:
diff
changeset

665 

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

666 
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

667 
by (simp add: matrix_vector_mult_def inner_vec_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

668 

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

669 
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

670 
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

671 

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

672 
lemma vector_componentwise: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

673 
"(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

674 
apply (subst basis_expansion[symmetric]) 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

675 
by (vector vec_eq_iff setsum_component) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

676 

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

677 
lemma linear_componentwise: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

678 
fixes f:: "real ^'m \<Rightarrow> real ^ _" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

679 
assumes lf: "linear f" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

680 
shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

681 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

682 
let ?M = "(UNIV :: 'm set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

683 
let ?N = "(UNIV :: 'n set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

684 
have fM: "finite ?M" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

685 
have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

686 
unfolding vector_smult_component[symmetric] smult_conv_scaleR 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

687 
unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

688 
.. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

689 
then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' .. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

690 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

691 

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

692 
text{* Inverse matrices (not necessarily square) *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

693 

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

694 
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** 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

695 

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

696 
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) = 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

697 
(SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** 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

698 

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

699 
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

700 

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

701 
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

702 
where "matrix f = (\<chi> i j. (f(cart_basis j))$i)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

703 

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

704 
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

705 
by (simp add: linear_def matrix_vector_mult_def vec_eq_iff 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

706 

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

707 
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

708 
apply (simp add: matrix_def matrix_vector_mult_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

709 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

710 
apply (rule linear_componentwise[OF lf, symmetric]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

711 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

712 

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

713 
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" by (simp add: ext matrix_works) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

714 

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

715 
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

716 
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

717 

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

718 
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

719 
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

720 
and lg: "linear (g::real^'m \<Rightarrow> real^_)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

721 
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

722 
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

723 
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

724 

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

725 
lemma matrix_vector_column:"(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

726 
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

727 

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

728 
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

729 
apply (rule adjoint_unique) 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

730 
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def 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

731 
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

732 
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

733 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

734 

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

735 
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

736 
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

737 
apply (subst matrix_vector_mul[OF lf]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

738 
unfolding adjoint_matrix matrix_of_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

739 

44360  740 
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

741 

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

742 
(* 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

743 

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

744 
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow> 
37494  745 
(\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs") 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

747 
let ?S = "(UNIV :: 'n set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

748 
{assume H: "?rhs" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

749 
then have ?lhs by auto} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

750 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

751 
{assume H: "?lhs" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

752 
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

753 
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

755 
from f have "P i (f i)" by metis 
37494  756 
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

757 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

758 
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

759 
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

760 
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

761 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

762 

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

763 
subsection {* Standard bases are a spanning set, and obviously finite. *} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

764 

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

765 
lemma span_stdbasis:"span {cart_basis i :: real^'n  i. i \<in> (UNIV :: 'n set)} = UNIV" 
39302
d7728f65b353
renamed lemmas: ext_iff > fun_eq_iff, set_ext_iff > set_eq_iff, set_ext > set_eqI
nipkow
parents:
39198
diff
changeset

766 
apply (rule set_eqI) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

767 
apply auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

768 
apply (subst basis_expansion'[symmetric]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

769 
apply (rule span_setsum) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

770 
apply simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

771 
apply auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

772 
apply (rule span_mul) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

773 
apply (rule span_superset) 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

774 
apply auto 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

775 
done 
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 finite_stdbasis: "finite {cart_basis i ::real^'n i. i\<in> (UNIV:: 'n set)}" (is "finite ?S") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

778 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

779 
have eq: "?S = cart_basis ` UNIV" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

780 
show ?thesis unfolding eq by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

781 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

782 

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

783 
lemma card_stdbasis: "card {cart_basis i ::real^'n i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

785 
have eq: "?S = cart_basis ` UNIV" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

786 
show ?thesis unfolding eq using card_image[OF basis_inj] by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

787 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

788 

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

789 

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

790 
lemma independent_stdbasis_lemma: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

791 
assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

792 
and iS: "i \<notin> S" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

793 
shows "(x$i) = 0" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

794 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

795 
let ?U = "UNIV :: 'n set" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

796 
let ?B = "cart_basis ` S" 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

797 
let ?P = "{(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<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

798 
{fix x::"real^_" assume xS: "x\<in> ?B" 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

799 
from xS have "x \<in> ?P" 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

800 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

801 
have "subspace ?P" 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

802 
by (auto simp add: subspace_def) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

803 
ultimately show ?thesis 
44521  804 
using x span_induct[of x ?B ?P] iS 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

805 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

806 

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

807 
lemma independent_stdbasis: "independent {cart_basis i ::real^'n i. i\<in> (UNIV :: 'n set)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

808 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

809 
let ?I = "UNIV :: 'n set" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

810 
let ?b = "cart_basis :: _ \<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

811 
let ?B = "?b ` ?I" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

812 
have eq: "{?b ii. i \<in> ?I} = ?B" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

813 
by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

814 
{assume d: "dependent ?B" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

815 
then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B  {?b k})" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

816 
unfolding dependent_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

817 
have eq1: "?B  {?b k} = ?B  ?b ` {k}" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

818 
have eq2: "?B  {?b k} = ?b ` (?I  {k})" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

819 
unfolding eq1 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

820 
apply (rule inj_on_image_set_diff[symmetric]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

821 
apply (rule basis_inj) using k(1) by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

822 
from k(2) have th0: "?b k \<in> span (?b ` (?I  {k}))" unfolding eq2 . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

823 
from independent_stdbasis_lemma[OF th0, of k, simplified] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

824 
have False by simp} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

825 
then show ?thesis unfolding eq dependent_def .. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

826 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

827 

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

828 
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x  ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

829 
unfolding inner_simps smult_conv_scaleR by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

830 

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

831 
lemma linear_eq_stdbasis_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

832 
assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

833 
and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

834 
shows "f = g" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

835 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

836 
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

837 
let ?I = "{cart_basis i:: real^'mi. i \<in> ?U}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

838 
{fix x assume x: "x \<in> (UNIV :: (real^'m) set)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

839 
from equalityD2[OF span_stdbasis] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

840 
have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

841 
from linear_eq[OF lf lg IU] fg x 
44170
510ac30f44c0
make Multivariate_Analysis work with separate set type
huffman
parents:
44167
diff
changeset

842 
have "f x = g x" unfolding Ball_def mem_Collect_eq by metis} 
44457
d366fa5551ef
declare euclidean_simps [simp] at the point they are proved;
huffman
parents:
44452
diff
changeset

843 
then show ?thesis 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

844 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

845 

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

846 
lemma bilinear_eq_stdbasis_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

847 
assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

848 
and bg: "bilinear g" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

849 
and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

851 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

852 
from fg have th: "\<forall>x \<in> {cart_basis i i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in> {cart_basis j j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast 
44457
d366fa5551ef
declare euclidean_simps [simp] at the point they are proved;
huffman
parents:
44452
diff
changeset

853 
from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis 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

854 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

855 

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

856 
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

857 
"(\<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

858 
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

859 

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

860 
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

861 
"(\<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

862 
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

863 

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

864 
lemma 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

865 
"(\<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)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

866 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

867 
{fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

868 
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

869 
hence "x = y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

870 
unfolding matrix_vector_mul_assoc B 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

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

872 
{assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

873 
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

874 
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

875 
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

876 
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

877 
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  878 
using g(2) by (simp add: fun_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

879 
then have "\<exists>B. (B::real ^'m^'n) ** A = 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

880 
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

881 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

882 

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

883 
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

884 
"(\<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

885 
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

886 
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

887 
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

888 

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

889 
lemma matrix_right_invertible_surjective: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

890 
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. 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

891 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

892 
{fix B :: "real ^'m^'n" assume AB: "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

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

894 
have "A *v (B *v x) = x" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

895 
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

896 
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

897 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

898 
{assume sf: "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

899 
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

900 
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

901 
by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

902 

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

903 
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

904 
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

905 
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)] 
44165  906 
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

907 
. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

908 
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

909 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

910 
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

911 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

912 

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

913 
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

914 
fixes A :: "real^'n^'m" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

915 
shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

916 
(is "?lhs \<longleftrightarrow> ?rhs") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

917 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

918 
let ?U = "UNIV :: 'n set" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

919 
{assume k: "\<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

920 
{fix c i assume c: "setsum (\<lambda>i. c 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

921 
and i: "i \<in> ?U" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

922 
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

923 
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

924 
using c 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

925 
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

926 
by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

927 
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

928 
have "c i = 0" by (vector 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

929 
hence ?rhs by blast} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

930 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

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

932 
{fix x assume x: "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

933 
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

934 
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

935 
have "x = 0" by vector}} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

936 
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

937 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

938 

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

939 
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

940 
fixes A :: "real^'n^'m" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

941 
shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

942 
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

943 
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

944 
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

945 

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

946 
lemma matrix_right_invertible_span_columns: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

947 
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs") 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

948 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

949 
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

950 
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

951 
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

952 
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

953 
apply (subst eq_commute) .. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

954 
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

955 
{assume h: ?lhs 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

956 
{fix x:: "real ^'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

957 
from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

958 
where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = 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

959 
have "x \<in> span (columns A)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

960 
unfolding y[symmetric] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

961 
apply (rule span_setsum[OF fU]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

962 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

963 
unfolding smult_conv_scaleR 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

964 
apply (rule span_mul) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

965 
apply (rule span_superset) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

966 
unfolding columns_def 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

967 
by blast} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

968 
then have ?rhs unfolding rhseq by blast} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

969 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

970 
{assume h:?rhs 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

971 
let ?P = "\<lambda>(y::real ^'n). \<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

972 
{fix y have "?P y" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

973 
proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

974 
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

975 
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

976 
next 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

977 
fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

978 
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

979 
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

980 
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

981 
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

982 
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

983 
show "?P (c*s y1 + y2)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

984 
proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

985 
fix j 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

986 
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

987 
else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

988 
by (simp add: field_simps) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

989 
have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

990 
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" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

991 
apply (rule setsum_cong[OF refl]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

992 
using th by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

993 
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" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

994 
by (simp add: setsum_addf) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

995 
also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

996 
unfolding setsum_delta[OF fU] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

997 
using i(1) by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

998 
finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

999 
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1000 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1001 
next 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1002 
show "y \<in> span (columns A)" unfolding h by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1003 
qed} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1004 
then have ?lhs unfolding lhseq ..} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1005 
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

1006 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1007 

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

1008 
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

1009 
"(\<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

1010 
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

1011 
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

1012 
unfolding matrix_right_invertible_span_columns 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1013 
.. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1014 

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

1015 
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

1016 

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

1017 
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

1018 
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

1019 
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** 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

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

1021 
{fix A A' :: "real ^'n^'n" assume AA': "A ** 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

1022 
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

1023 
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

1024 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1025 
apply (rule_tac x="(A' *v y)" in exI) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1026 
by (simp add: matrix_vector_mul_assoc AA' 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

1027 
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

1028 
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

1029 
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

1030 
have th: "matrix f' ** 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

1031 
by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format]) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1032 
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1033 
hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1034 
hence "matrix f' ** A = A' ** A" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1035 
hence "A' ** A = mat 1" by (simp add: th)} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1036 
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

1037 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1038 

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

1039 
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

1040 

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

1041 
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

1042 

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

1043 
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

1044 

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

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

1046 
"transpose(columnvector v) = rowvector v" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

1047 
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

1048 

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

1049 
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

1050 
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

1051 

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

1052 
lemma dot_rowvector_columnvector: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1053 
"columnvector (A *v v) = A ** columnvector v" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1054 
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

1055 

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

1056 
lemma dot_matrix_product: "(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

1057 
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

1058 

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

1059 
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

1060 
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

1061 
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

1062 
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1063 
unfolding dot_matrix_product transpose_columnvector[symmetric] 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1064 
dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc .. 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1065 

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 infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) i. i\<in> (UNIV :: 'n set)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1068 
unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1069 
apply(rule_tac x="\<pi> i" in exI) defer 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1070 
apply(rule_tac x="\<pi>' i" in exI) unfolding nth_conv_component using pi'_range by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1071 

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

1072 
lemma infnorm_set_image_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1073 
"{abs(x$i) i. i\<in> (UNIV :: _ set)} = 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1074 
(\<lambda>i. abs(x$i)) ` (UNIV)" by blast 
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 infnorm_set_lemma_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1077 
shows "finite {abs((x::'a::abs ^'n)$i) i. i\<in> (UNIV :: 'n set)}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1078 
and "{abs(x$i) i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1079 
unfolding infnorm_set_image_cart 
40786
0a54cfc9add3
gave more standard finite set rules simp and intro attribute
nipkow
parents:
39302
diff
changeset

1080 
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

1081 

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

1082 
lemma component_le_infnorm_cart: 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1083 
shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1084 
unfolding nth_conv_component 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1085 
using component_le_infnorm[of x] . 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1086 

44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1087 
lemma continuous_component: 
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1088 
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)" 
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1089 
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

1090 

44647
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1091 
lemma continuous_on_component: 
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1092 
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)" 
e4de7750cdeb
modernize lemmas about 'continuous' and 'continuous_on';
huffman
parents:
44571
diff
changeset

1093 
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

1094 

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

1095 
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}" 
44233  1096 
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

1097 

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

1098 
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1099 
unfolding bounded_def 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1100 
apply clarify 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1101 
apply (rule_tac x="x $ i" in exI) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1102 
apply (rule_tac x="e" in exI) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1103 
apply clarify 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

1104 
apply (rule order_trans [OF dist_vec_nth_le], simp) 
37489
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1105 
done 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1106 

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

1107 
lemma 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

1108 
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1109 
assumes "bounded s" and "\<forall>n. f n \<in> s" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1110 
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

1111 
\<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

1112 
(\<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

1113 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1114 
fix d::"'n set" have "finite d" by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1115 
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

1116 
(\<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

1117 
proof(induct d) case empty thus ?case unfolding 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

1118 
next case (insert k d) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1119 
have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component_cart) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1120 
obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 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

1121 
using insert(3) by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1122 
have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1123 
obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) > l2) sequentially" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1124 
using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_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

1125 
def r \<equiv> "r1 \<circ> r2" have r:"subseq r" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1126 
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

1127 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1128 
def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1129 
{ 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

1130 
from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1131 
from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1132 
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

1133 
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

1134 
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

1135 
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

1136 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1137 
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

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

1140 

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

1141 
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

1142 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1143 
fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1144 
assume s: "bounded s" and f: "\<forall>n. f n \<in> s" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1145 
then obtain l r where r: "subseq r" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1146 
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. 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

1147 
using compact_lemma_cart [OF s f] by blast 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1148 
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

1149 
{ 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

1150 
hence "0 < 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

1151 
using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1152 
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

1153 
by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1154 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1155 
{ fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < 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

1156 
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

1157 
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

1158 
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

1159 
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

1160 
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

1161 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1162 
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

1163 
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

1164 
} 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1165 
hence *:"((f \<circ> r) > l) sequentially" unfolding o_def tendsto_iff by simp 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1166 
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

1167 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1168 

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

1169 
lemma interval_cart: fixes a :: "'a::ord^'n" shows 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1170 
"{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1171 
"{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

1172 
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

1173 

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

1174 
lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1175 
"x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1176 
"x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)" 
44136
e63ad7d5158d
more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents:
44135
diff
changeset

1177 
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

1178 

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

1179 
lemma interval_eq_empty_cart: fixes a :: "real^'n" shows 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1180 
"({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1181 
"({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2) 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1182 
proof 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1183 
{ fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1184 
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

1185 
hence "a$i < b$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

1186 
hence False using as by auto } 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1187 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1188 
{ 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

1189 
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

1190 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1191 
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

1192 
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

1193 
unfolding vector_smult_component and vector_add_component 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1194 
by auto } 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1195 
hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a 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

1196 
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

1197 

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

1198 
{ fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1199 
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

1200 
hence "a$i \<le> b$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

1201 
hence False using as by auto } 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1202 
moreover 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1203 
{ 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

1204 
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

1205 
{ fix i 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1206 
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

1207 
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

1208 
unfolding vector_smult_component and vector_add_component 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1209 
by auto } 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1210 
hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a 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

1211 
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

1212 
qed 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1213 

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

1214 
lemma interval_ne_empty_cart: fixes a :: "real^'n" shows 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1215 
"{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1216 
"{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)" 
44e42d392c6e
Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents:
diff
changeset

1217 
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

1218 
(* BH: Why doesn't just "auto" work here? *) 