src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author haftmann
Fri Nov 01 18:51:14 2013 +0100 (2013-11-01)
changeset 54230 b1d955791529
parent 53600 8fda7ad57466
child 54489 03ff4d1e6784
permissions -rw-r--r--
more simplification rules on unary and binary minus
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header {*Instanciates the finite cartesian product of euclidean spaces as a euclidean space.*}
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theory Cartesian_Euclidean_Space
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imports Finite_Cartesian_Product Integration
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begin
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lemma delta_mult_idempotent:
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  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
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  by (cases "k=a") auto
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lemma setsum_Plus:
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  "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
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    (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
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  unfolding Plus_def
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  by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
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lemma setsum_UNIV_sum:
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  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
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  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
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  apply (subst UNIV_Plus_UNIV [symmetric])
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  apply (rule setsum_Plus [OF finite finite])
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  done
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lemma setsum_mult_product:
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  "setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
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  unfolding 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
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  show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
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  show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
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  proof safe
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    fix j assume "j \<in> {i * B..<i * B + B}"
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    then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
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      by (auto intro!: image_eqI[of _ _ "j - i * B"])
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  qed simp
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qed simp
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subsection{* Basic componentwise operations on vectors. *}
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instantiation vec :: (times, finite) times
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begin
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definition "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
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instance ..
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end
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instantiation vec :: (one, finite) one
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begin
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definition "1 \<equiv> (\<chi> i. 1)"
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instance ..
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end
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instantiation vec :: (ord, finite) ord
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begin
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definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
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definition "x < y \<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 one-dimensional vectors is linear. *}
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class cart_one =
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  assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
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begin
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subclass finite
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proof
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  from UNIV_one show "finite (UNIV :: 'a set)"
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    by (auto intro!: card_ge_0_finite)
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qed
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end
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instantiation vec :: (linorder, cart_one) linorder
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begin
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instance
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proof
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  obtain a :: 'b where all: "\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a"
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  proof -
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    have "card (UNIV :: 'b set) = Suc 0" by (rule UNIV_one)
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    then obtain b :: 'b where "UNIV = {b}" by (auto iff: card_Suc_eq)
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    then have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P b" by auto
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    then show thesis by (auto intro: that)
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  qed
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  note [simp] = less_eq_vec_def less_vec_def all vec_eq_iff field_simps
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  fix x y z :: "'a^'b::cart_one"
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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" by auto
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  { assume "x\<le>y" "y\<le>z" then show "x\<le>z" by auto }
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  { assume "x\<le>y" "y\<le>x" then show "x=y" by auto }
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qed
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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 scalar-vector multiplication. *}
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definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
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  where "c *s x = (\<chi> i. c * (x$i))"
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subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
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method_setup vector = {*
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let
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  val ss1 =
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    simpset_of (put_simpset HOL_basic_ss @{context}
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      addsimps [@{thm setsum_addf} RS sym,
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      @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
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      @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym])
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  val ss2 =
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    simpset_of (@{context} addsimps
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             [@{thm plus_vec_def}, @{thm times_vec_def},
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              @{thm minus_vec_def}, @{thm uminus_vec_def},
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              @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
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              @{thm scaleR_vec_def},
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              @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}])
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  fun vector_arith_tac ctxt ths =
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    simp_tac (put_simpset ss1 ctxt)
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    THEN' (fn i => rtac @{thm setsum_cong2} i
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         ORELSE rtac @{thm setsum_0'} i
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         ORELSE simp_tac (put_simpset HOL_basic_ss ctxt addsimps [@{thm vec_eq_iff}]) i)
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    (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
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    THEN' asm_full_simp_tac (put_simpset ss2 ctxt addsimps ths)
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in
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  Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (vector_arith_tac ctxt ths))
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end
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*} "lift trivial vector statements to real arith statements"
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lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
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lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
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lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
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lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
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lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
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lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
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lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
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lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
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lemma vec_setsum:
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  assumes "finite S"
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  shows "vec(setsum f S) = setsum (vec o f) S"
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  using assms
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proof induct
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  case empty
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  then show ?case by simp
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next
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  case insert
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  then show ?case by (auto simp add: vec_add)
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qed
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text{* Obvious "component-pushing". *}
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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 :: (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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  by default (simp_all add: vec_eq_iff)
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instance vec :: (real_algebra_1, finite) real_algebra_1 ..
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lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
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proof (induct n)
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  case 0
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  then show ?case by vector
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next
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  case Suc
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  then show ?case by vector
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qed
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lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1"
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  by vector
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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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lemma vector_sneg_minus1: "-x = (-1::'a::ring_1) *s x" by vector
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lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
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lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
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  by (vector field_simps)
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lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
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  by (simp add: vec_eq_iff)
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lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
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lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
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  by vector
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lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
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lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
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  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
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lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
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  by (metis vector_mul_lcancel)
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lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
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  by (metis vector_mul_rcancel)
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lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
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  apply (simp add: norm_vec_def)
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  apply (rule member_le_setL2, simp_all)
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  done
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lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
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  by (metis component_le_norm_cart order_trans)
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lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
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  by (metis component_le_norm_cart le_less_trans)
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lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
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  by (simp add: norm_vec_def setL2_le_setsum)
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lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
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  unfolding scaleR_vec_def vector_scalar_mult_def by simp
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lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
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   303
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   304
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   305
hoelzl@37489
   306
lemma setsum_component [simp]:
hoelzl@37489
   307
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
hoelzl@37489
   308
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
wenzelm@49644
   309
proof (cases "finite S")
wenzelm@49644
   310
  case True
wenzelm@49644
   311
  then show ?thesis by induct simp_all
wenzelm@49644
   312
next
wenzelm@49644
   313
  case False
wenzelm@49644
   314
  then show ?thesis by simp
wenzelm@49644
   315
qed
hoelzl@37489
   316
hoelzl@37489
   317
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
huffman@44136
   318
  by (simp add: vec_eq_iff)
hoelzl@37489
   319
hoelzl@37489
   320
lemma setsum_cmul:
hoelzl@37489
   321
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
hoelzl@37489
   322
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
huffman@44136
   323
  by (simp add: vec_eq_iff setsum_right_distrib)
hoelzl@37489
   324
hoelzl@37489
   325
lemma setsum_norm_allsubsets_bound_cart:
hoelzl@37489
   326
  fixes f:: "'a \<Rightarrow> real ^'n"
hoelzl@37489
   327
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@37489
   328
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
hoelzl@50526
   329
  using setsum_norm_allsubsets_bound[OF assms]
hoelzl@50526
   330
  by (simp add: DIM_cart Basis_real_def)
hoelzl@37489
   331
huffman@44136
   332
instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
hoelzl@37489
   333
proof
hoelzl@37489
   334
  fix x y::"'a^'b"
hoelzl@50526
   335
  show "(x \<le> y) = (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
hoelzl@50526
   336
    unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: Basis_vec_def inner_axis)
hoelzl@50526
   337
  show"(x < y) = (\<forall>i\<in>Basis. x \<bullet> i < y \<bullet> i)"
hoelzl@50526
   338
    unfolding less_vec_def apply(subst eucl_less) by (simp add: Basis_vec_def inner_axis)
hoelzl@37489
   339
qed
hoelzl@37489
   340
hoelzl@37489
   341
subsection {* Matrix operations *}
hoelzl@37489
   342
hoelzl@37489
   343
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
hoelzl@37489
   344
wenzelm@49644
   345
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
wenzelm@49644
   346
    (infixl "**" 70)
hoelzl@37489
   347
  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   348
wenzelm@49644
   349
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
wenzelm@49644
   350
    (infixl "*v" 70)
hoelzl@37489
   351
  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   352
wenzelm@49644
   353
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
wenzelm@49644
   354
    (infixl "v*" 70)
hoelzl@37489
   355
  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   356
hoelzl@37489
   357
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@37489
   358
definition transpose where 
hoelzl@37489
   359
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   360
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   361
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   362
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   363
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   364
hoelzl@37489
   365
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   366
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
hoelzl@37489
   367
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
hoelzl@37489
   368
hoelzl@37489
   369
lemma matrix_mul_lid:
hoelzl@37489
   370
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   371
  shows "mat 1 ** A = A"
hoelzl@37489
   372
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   373
  apply vector
wenzelm@49644
   374
  apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
wenzelm@49644
   375
    mult_1_left mult_zero_left if_True UNIV_I)
wenzelm@49644
   376
  done
hoelzl@37489
   377
hoelzl@37489
   378
hoelzl@37489
   379
lemma matrix_mul_rid:
hoelzl@37489
   380
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   381
  shows "A ** mat 1 = A"
hoelzl@37489
   382
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   383
  apply vector
wenzelm@49644
   384
  apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
wenzelm@49644
   385
    mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
wenzelm@49644
   386
  done
hoelzl@37489
   387
hoelzl@37489
   388
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
hoelzl@37489
   389
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   390
  apply (subst setsum_commute)
hoelzl@37489
   391
  apply simp
hoelzl@37489
   392
  done
hoelzl@37489
   393
hoelzl@37489
   394
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
wenzelm@49644
   395
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def
wenzelm@49644
   396
    setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   397
  apply (subst setsum_commute)
hoelzl@37489
   398
  apply simp
hoelzl@37489
   399
  done
hoelzl@37489
   400
hoelzl@37489
   401
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   402
  apply (vector matrix_vector_mult_def mat_def)
wenzelm@49644
   403
  apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
wenzelm@49644
   404
  done
hoelzl@37489
   405
wenzelm@49644
   406
lemma matrix_transpose_mul:
wenzelm@49644
   407
    "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
huffman@44136
   408
  by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   409
hoelzl@37489
   410
lemma matrix_eq:
hoelzl@37489
   411
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   412
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   413
  apply auto
huffman@44136
   414
  apply (subst vec_eq_iff)
hoelzl@37489
   415
  apply clarify
hoelzl@50526
   416
  apply (clarsimp simp add: matrix_vector_mult_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
hoelzl@50526
   417
  apply (erule_tac x="axis ia 1" in allE)
hoelzl@37489
   418
  apply (erule_tac x="i" in allE)
hoelzl@50526
   419
  apply (auto simp add: if_distrib cond_application_beta axis_def
wenzelm@49644
   420
    setsum_delta[OF finite] cong del: if_weak_cong)
wenzelm@49644
   421
  done
hoelzl@37489
   422
wenzelm@49644
   423
lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
huffman@44136
   424
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   425
hoelzl@37489
   426
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
huffman@44136
   427
  apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
hoelzl@37489
   428
  apply (subst setsum_commute)
wenzelm@49644
   429
  apply simp
wenzelm@49644
   430
  done
hoelzl@37489
   431
hoelzl@37489
   432
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
   433
  by (vector transpose_def mat_def)
hoelzl@37489
   434
hoelzl@37489
   435
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
   436
  by (vector transpose_def)
hoelzl@37489
   437
hoelzl@37489
   438
lemma row_transpose:
hoelzl@37489
   439
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   440
  shows "row i (transpose A) = column i A"
huffman@44136
   441
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   442
hoelzl@37489
   443
lemma column_transpose:
hoelzl@37489
   444
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
   445
  shows "column i (transpose A) = row i A"
huffman@44136
   446
  by (simp add: row_def column_def transpose_def vec_eq_iff)
hoelzl@37489
   447
hoelzl@37489
   448
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
wenzelm@49644
   449
  by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
hoelzl@37489
   450
wenzelm@49644
   451
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
wenzelm@49644
   452
  by (metis transpose_transpose rows_transpose)
hoelzl@37489
   453
hoelzl@37489
   454
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
hoelzl@37489
   455
hoelzl@37489
   456
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
huffman@44136
   457
  by (simp add: matrix_vector_mult_def inner_vec_def)
hoelzl@37489
   458
wenzelm@49644
   459
lemma matrix_mult_vsum:
wenzelm@49644
   460
  "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
huffman@44136
   461
  by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
hoelzl@37489
   462
hoelzl@37489
   463
lemma vector_componentwise:
hoelzl@50526
   464
  "(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
hoelzl@50526
   465
  by (simp add: axis_def if_distrib setsum_cases vec_eq_iff)
hoelzl@50526
   466
hoelzl@50526
   467
lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
hoelzl@50526
   468
  by (auto simp add: axis_def vec_eq_iff if_distrib setsum_cases cong del: if_weak_cong)
hoelzl@37489
   469
hoelzl@37489
   470
lemma linear_componentwise:
hoelzl@37489
   471
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
   472
  assumes lf: "linear f"
hoelzl@50526
   473
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (axis i 1)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
wenzelm@49644
   474
proof -
hoelzl@37489
   475
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
   476
  let ?N = "(UNIV :: 'n set)"
hoelzl@37489
   477
  have fM: "finite ?M" by simp
hoelzl@50526
   478
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (axis i 1) ) ?M)$j"
hoelzl@50526
   479
    unfolding setsum_component by simp
wenzelm@49644
   480
  then show ?thesis
hoelzl@50526
   481
    unfolding linear_setsum_mul[OF lf fM, symmetric]
hoelzl@50526
   482
    unfolding scalar_mult_eq_scaleR[symmetric]
hoelzl@50526
   483
    unfolding basis_expansion
hoelzl@50526
   484
    by simp
hoelzl@37489
   485
qed
hoelzl@37489
   486
hoelzl@37489
   487
text{* Inverse matrices  (not necessarily square) *}
hoelzl@37489
   488
wenzelm@49644
   489
definition
wenzelm@49644
   490
  "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   491
wenzelm@49644
   492
definition
wenzelm@49644
   493
  "matrix_inv(A:: 'a::semiring_1^'n^'m) =
wenzelm@49644
   494
    (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
   495
hoelzl@37489
   496
text{* Correspondence between matrices and linear operators. *}
hoelzl@37489
   497
wenzelm@49644
   498
definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@50526
   499
  where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
hoelzl@37489
   500
hoelzl@37489
   501
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
huffman@53600
   502
  by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
wenzelm@49644
   503
      field_simps setsum_right_distrib setsum_addf)
hoelzl@37489
   504
wenzelm@49644
   505
lemma matrix_works:
wenzelm@49644
   506
  assumes lf: "linear f"
wenzelm@49644
   507
  shows "matrix f *v x = f (x::real ^ 'n)"
wenzelm@49644
   508
  apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
wenzelm@49644
   509
  apply clarify
wenzelm@49644
   510
  apply (rule linear_componentwise[OF lf, symmetric])
wenzelm@49644
   511
  done
hoelzl@37489
   512
wenzelm@49644
   513
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
wenzelm@49644
   514
  by (simp add: ext matrix_works)
hoelzl@37489
   515
hoelzl@37489
   516
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
   517
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
   518
hoelzl@37489
   519
lemma matrix_compose:
hoelzl@37489
   520
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
wenzelm@49644
   521
    and lg: "linear (g::real^'m \<Rightarrow> real^_)"
hoelzl@37489
   522
  shows "matrix (g o f) = matrix g ** matrix f"
hoelzl@37489
   523
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
wenzelm@49644
   524
  by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
   525
wenzelm@49644
   526
lemma matrix_vector_column:
wenzelm@49644
   527
  "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
huffman@44136
   528
  by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
hoelzl@37489
   529
hoelzl@37489
   530
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
   531
  apply (rule adjoint_unique)
wenzelm@49644
   532
  apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
wenzelm@49644
   533
    setsum_left_distrib setsum_right_distrib)
hoelzl@37489
   534
  apply (subst setsum_commute)
hoelzl@37489
   535
  apply (auto simp add: mult_ac)
hoelzl@37489
   536
  done
hoelzl@37489
   537
hoelzl@37489
   538
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
   539
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
   540
  apply (subst matrix_vector_mul[OF lf])
wenzelm@49644
   541
  unfolding adjoint_matrix matrix_of_matrix_vector_mul
wenzelm@49644
   542
  apply rule
wenzelm@49644
   543
  done
wenzelm@49644
   544
hoelzl@37489
   545
huffman@44360
   546
subsection {* lambda skolemization on cartesian products *}
hoelzl@37489
   547
hoelzl@37489
   548
(* FIXME: rename do choice_cart *)
hoelzl@37489
   549
hoelzl@37489
   550
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
   551
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   552
proof -
hoelzl@37489
   553
  let ?S = "(UNIV :: 'n set)"
wenzelm@49644
   554
  { assume H: "?rhs"
wenzelm@49644
   555
    then have ?lhs by auto }
hoelzl@37489
   556
  moreover
wenzelm@49644
   557
  { assume H: "?lhs"
hoelzl@37489
   558
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
   559
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
wenzelm@49644
   560
    { fix i
hoelzl@37489
   561
      from f have "P i (f i)" by metis
hoelzl@37494
   562
      then have "P i (?x $ i)" by auto
hoelzl@37489
   563
    }
hoelzl@37489
   564
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
   565
    hence ?rhs by metis }
hoelzl@37489
   566
  ultimately show ?thesis by metis
hoelzl@37489
   567
qed
hoelzl@37489
   568
hoelzl@37489
   569
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@50526
   570
  unfolding inner_simps scalar_mult_eq_scaleR by auto
hoelzl@37489
   571
hoelzl@37489
   572
lemma left_invertible_transpose:
hoelzl@37489
   573
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
   574
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   575
hoelzl@37489
   576
lemma right_invertible_transpose:
hoelzl@37489
   577
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
   578
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
   579
hoelzl@37489
   580
lemma matrix_left_invertible_injective:
wenzelm@49644
   581
  "(\<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)"
wenzelm@49644
   582
proof -
wenzelm@49644
   583
  { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
   584
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
   585
    hence "x = y"
wenzelm@49644
   586
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
hoelzl@37489
   587
  moreover
wenzelm@49644
   588
  { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
   589
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
   590
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
hoelzl@37489
   591
    obtain g where g: "linear g" "g o op *v A = id" by blast
hoelzl@37489
   592
    have "matrix g ** A = mat 1"
hoelzl@37489
   593
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   594
      using g(2) by (simp add: fun_eq_iff)
wenzelm@49644
   595
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
hoelzl@37489
   596
  ultimately show ?thesis by blast
hoelzl@37489
   597
qed
hoelzl@37489
   598
hoelzl@37489
   599
lemma matrix_left_invertible_ker:
hoelzl@37489
   600
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
   601
  unfolding matrix_left_invertible_injective
hoelzl@37489
   602
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
   603
  by (simp add: inj_on_def)
hoelzl@37489
   604
hoelzl@37489
   605
lemma matrix_right_invertible_surjective:
wenzelm@49644
   606
  "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
wenzelm@49644
   607
proof -
wenzelm@49644
   608
  { fix B :: "real ^'m^'n"
wenzelm@49644
   609
    assume AB: "A ** B = mat 1"
wenzelm@49644
   610
    { fix x :: "real ^ 'm"
hoelzl@37489
   611
      have "A *v (B *v x) = x"
wenzelm@49644
   612
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hoelzl@37489
   613
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
   614
  moreover
wenzelm@49644
   615
  { assume sf: "surj (op *v A)"
hoelzl@37489
   616
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
hoelzl@37489
   617
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
hoelzl@37489
   618
      by blast
hoelzl@37489
   619
hoelzl@37489
   620
    have "A ** (matrix g) = mat 1"
hoelzl@37489
   621
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
   622
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
huffman@44165
   623
      using g(2) unfolding o_def fun_eq_iff id_def
hoelzl@37489
   624
      .
hoelzl@37489
   625
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
   626
  }
hoelzl@37489
   627
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
   628
qed
hoelzl@37489
   629
hoelzl@37489
   630
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
   631
  fixes A :: "real^'n^'m"
wenzelm@49644
   632
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
wenzelm@49644
   633
      (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
wenzelm@49644
   634
    (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@49644
   635
proof -
hoelzl@37489
   636
  let ?U = "UNIV :: 'n set"
wenzelm@49644
   637
  { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
wenzelm@49644
   638
    { fix c i
wenzelm@49644
   639
      assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
hoelzl@37489
   640
      let ?x = "\<chi> i. c i"
hoelzl@37489
   641
      have th0:"A *v ?x = 0"
hoelzl@37489
   642
        using c
huffman@44136
   643
        unfolding matrix_mult_vsum vec_eq_iff
hoelzl@37489
   644
        by auto
hoelzl@37489
   645
      from k[rule_format, OF th0] i
huffman@44136
   646
      have "c i = 0" by (vector vec_eq_iff)}
wenzelm@49644
   647
    hence ?rhs by blast }
hoelzl@37489
   648
  moreover
wenzelm@49644
   649
  { assume H: ?rhs
wenzelm@49644
   650
    { fix x assume x: "A *v x = 0"
hoelzl@37489
   651
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
   652
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
wenzelm@49644
   653
      have "x = 0" by vector }
wenzelm@49644
   654
  }
hoelzl@37489
   655
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
   656
qed
hoelzl@37489
   657
hoelzl@37489
   658
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
   659
  fixes A :: "real^'n^'m"
wenzelm@49644
   660
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   661
    (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
hoelzl@37489
   662
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
   663
    matrix_left_invertible_independent_columns
hoelzl@37489
   664
  by (simp add: column_transpose)
hoelzl@37489
   665
hoelzl@37489
   666
lemma matrix_right_invertible_span_columns:
wenzelm@49644
   667
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
wenzelm@49644
   668
    span (columns A) = UNIV" (is "?lhs = ?rhs")
wenzelm@49644
   669
proof -
hoelzl@37489
   670
  let ?U = "UNIV :: 'm set"
hoelzl@37489
   671
  have fU: "finite ?U" by simp
hoelzl@37489
   672
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
   673
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
wenzelm@49644
   674
    apply (subst eq_commute)
wenzelm@49644
   675
    apply rule
wenzelm@49644
   676
    done
hoelzl@37489
   677
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
wenzelm@49644
   678
  { assume h: ?lhs
wenzelm@49644
   679
    { fix x:: "real ^'n"
wenzelm@49644
   680
      from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
wenzelm@49644
   681
        where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
wenzelm@49644
   682
      have "x \<in> span (columns A)"
wenzelm@49644
   683
        unfolding y[symmetric]
wenzelm@49644
   684
        apply (rule span_setsum[OF fU])
wenzelm@49644
   685
        apply clarify
hoelzl@50526
   686
        unfolding scalar_mult_eq_scaleR
wenzelm@49644
   687
        apply (rule span_mul)
wenzelm@49644
   688
        apply (rule span_superset)
wenzelm@49644
   689
        unfolding columns_def
wenzelm@49644
   690
        apply blast
wenzelm@49644
   691
        done
wenzelm@49644
   692
    }
wenzelm@49644
   693
    then have ?rhs unfolding rhseq by blast }
hoelzl@37489
   694
  moreover
wenzelm@49644
   695
  { assume h:?rhs
hoelzl@37489
   696
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
wenzelm@49644
   697
    { fix y
wenzelm@49644
   698
      have "?P y"
hoelzl@50526
   699
      proof (rule span_induct_alt[of ?P "columns A", folded scalar_mult_eq_scaleR])
hoelzl@37489
   700
        show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
   701
          by (rule exI[where x=0], simp)
hoelzl@37489
   702
      next
wenzelm@49644
   703
        fix c y1 y2
wenzelm@49644
   704
        assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
   705
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
   706
          unfolding columns_def by blast
hoelzl@37489
   707
        from y2 obtain x:: "real ^'m" where
hoelzl@37489
   708
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
   709
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
   710
        show "?P (c*s y1 + y2)"
webertj@49962
   711
        proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib distrib_left cond_application_beta cong del: if_weak_cong)
wenzelm@49644
   712
          fix j
wenzelm@49644
   713
          have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   714
              else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
wenzelm@49644
   715
            using i(1) by (simp add: field_simps)
wenzelm@49644
   716
          have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   717
              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)$j))) ?U"
wenzelm@49644
   718
            apply (rule setsum_cong[OF refl])
wenzelm@49644
   719
            using th apply blast
wenzelm@49644
   720
            done
wenzelm@49644
   721
          also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
   722
            by (simp add: setsum_addf)
wenzelm@49644
   723
          also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
wenzelm@49644
   724
            unfolding setsum_delta[OF fU]
wenzelm@49644
   725
            using i(1) by simp
wenzelm@49644
   726
          finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
wenzelm@49644
   727
            else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
wenzelm@49644
   728
        qed
wenzelm@49644
   729
      next
wenzelm@49644
   730
        show "y \<in> span (columns A)"
wenzelm@49644
   731
          unfolding h by blast
wenzelm@49644
   732
      qed
wenzelm@49644
   733
    }
wenzelm@49644
   734
    then have ?lhs unfolding lhseq ..
wenzelm@49644
   735
  }
hoelzl@37489
   736
  ultimately show ?thesis by blast
hoelzl@37489
   737
qed
hoelzl@37489
   738
hoelzl@37489
   739
lemma matrix_left_invertible_span_rows:
hoelzl@37489
   740
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
   741
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
   742
  unfolding columns_transpose[symmetric]
hoelzl@37489
   743
  unfolding matrix_right_invertible_span_columns
wenzelm@49644
   744
  ..
hoelzl@37489
   745
hoelzl@37489
   746
text {* The same result in terms of square matrices. *}
hoelzl@37489
   747
hoelzl@37489
   748
lemma matrix_left_right_inverse:
hoelzl@37489
   749
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
   750
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
wenzelm@49644
   751
proof -
wenzelm@49644
   752
  { fix A A' :: "real ^'n^'n"
wenzelm@49644
   753
    assume AA': "A ** A' = mat 1"
hoelzl@37489
   754
    have sA: "surj (op *v A)"
hoelzl@37489
   755
      unfolding surj_def
hoelzl@37489
   756
      apply clarify
hoelzl@37489
   757
      apply (rule_tac x="(A' *v y)" in exI)
wenzelm@49644
   758
      apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
wenzelm@49644
   759
      done
hoelzl@37489
   760
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
   761
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
   762
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
   763
    have th: "matrix f' ** A = mat 1"
wenzelm@49644
   764
      by (simp add: matrix_eq matrix_works[OF f'(1)]
wenzelm@49644
   765
          matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
   766
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
wenzelm@49644
   767
    hence "matrix f' = A'"
wenzelm@49644
   768
      by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
   769
    hence "matrix f' ** A = A' ** A" by simp
wenzelm@49644
   770
    hence "A' ** A = mat 1" by (simp add: th)
wenzelm@49644
   771
  }
hoelzl@37489
   772
  then show ?thesis by blast
hoelzl@37489
   773
qed
hoelzl@37489
   774
hoelzl@37489
   775
text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
hoelzl@37489
   776
hoelzl@37489
   777
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
   778
hoelzl@37489
   779
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
   780
wenzelm@49644
   781
lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
huffman@44136
   782
  by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
hoelzl@37489
   783
hoelzl@37489
   784
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
huffman@44136
   785
  by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
hoelzl@37489
   786
wenzelm@49644
   787
lemma dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
   788
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
   789
wenzelm@49644
   790
lemma dot_matrix_product:
wenzelm@49644
   791
  "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
huffman@44136
   792
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
hoelzl@37489
   793
hoelzl@37489
   794
lemma dot_matrix_vector_mul:
hoelzl@37489
   795
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
   796
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
   797
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
wenzelm@49644
   798
  unfolding dot_matrix_product transpose_columnvector[symmetric]
wenzelm@49644
   799
    dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
   800
hoelzl@37489
   801
hoelzl@50526
   802
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in>UNIV}"
hoelzl@50526
   803
  by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
hoelzl@37489
   804
wenzelm@49644
   805
lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@50526
   806
  using Basis_le_infnorm[of "axis i 1" x]
hoelzl@50526
   807
  by (simp add: Basis_vec_def axis_eq_axis inner_axis)
hoelzl@37489
   808
wenzelm@49644
   809
lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
huffman@44647
   810
  unfolding continuous_def by (rule tendsto_vec_nth)
huffman@44213
   811
wenzelm@49644
   812
lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
huffman@44647
   813
  unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
huffman@44213
   814
hoelzl@37489
   815
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
huffman@44233
   816
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
huffman@44213
   817
hoelzl@37489
   818
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
wenzelm@49644
   819
  unfolding bounded_def
wenzelm@49644
   820
  apply clarify
wenzelm@49644
   821
  apply (rule_tac x="x $ i" in exI)
wenzelm@49644
   822
  apply (rule_tac x="e" in exI)
wenzelm@49644
   823
  apply clarify
wenzelm@49644
   824
  apply (rule order_trans [OF dist_vec_nth_le], simp)
wenzelm@49644
   825
  done
hoelzl@37489
   826
hoelzl@37489
   827
lemma compact_lemma_cart:
hoelzl@37489
   828
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@50998
   829
  assumes f: "bounded (range f)"
hoelzl@37489
   830
  shows "\<forall>d.
hoelzl@37489
   831
        \<exists>l r. subseq r \<and>
hoelzl@37489
   832
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
   833
proof
wenzelm@49644
   834
  fix d :: "'n set"
wenzelm@49644
   835
  have "finite d" by simp
hoelzl@37489
   836
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
hoelzl@37489
   837
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
wenzelm@49644
   838
  proof (induct d)
wenzelm@49644
   839
    case empty
wenzelm@49644
   840
    thus ?case unfolding subseq_def by auto
wenzelm@49644
   841
  next
wenzelm@49644
   842
    case (insert k d)
wenzelm@49644
   843
    obtain l1::"'a^'n" and r1 where r1:"subseq r1"
wenzelm@49644
   844
      and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   845
      using insert(3) by auto
hoelzl@50998
   846
    have s': "bounded ((\<lambda>x. x $ k) ` range f)" using `bounded (range f)`
hoelzl@50998
   847
      by (auto intro!: bounded_component_cart)
hoelzl@50998
   848
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` range f" by simp
hoelzl@50998
   849
    have "bounded (range (\<lambda>i. f (r1 i) $ k))"
hoelzl@50998
   850
      by (metis (lifting) bounded_subset image_subsetI f' s')
hoelzl@50998
   851
    then obtain l2 r2 where r2: "subseq r2"
wenzelm@49644
   852
      and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
hoelzl@50998
   853
      using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) $ k"] by (auto simp: o_def)
wenzelm@49644
   854
    def r \<equiv> "r1 \<circ> r2"
wenzelm@49644
   855
    have r: "subseq r"
hoelzl@37489
   856
      using r1 and r2 unfolding r_def o_def subseq_def by auto
hoelzl@37489
   857
    moreover
hoelzl@37489
   858
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
wenzelm@49644
   859
    { fix e :: real assume "e > 0"
wenzelm@49644
   860
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
wenzelm@49644
   861
        by blast
wenzelm@49644
   862
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
wenzelm@49644
   863
        by (rule tendstoD)
hoelzl@37489
   864
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
   865
        by (rule eventually_subseq)
hoelzl@37489
   866
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
   867
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
hoelzl@37489
   868
    }
hoelzl@37489
   869
    ultimately show ?case by auto
hoelzl@37489
   870
  qed
hoelzl@37489
   871
qed
hoelzl@37489
   872
huffman@44136
   873
instance vec :: (heine_borel, finite) heine_borel
hoelzl@37489
   874
proof
hoelzl@50998
   875
  fix f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@50998
   876
  assume f: "bounded (range f)"
hoelzl@37489
   877
  then obtain l r where r: "subseq r"
wenzelm@49644
   878
      and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@50998
   879
    using compact_lemma_cart [OF f] by blast
hoelzl@37489
   880
  let ?d = "UNIV::'b set"
hoelzl@37489
   881
  { fix e::real assume "e>0"
hoelzl@37489
   882
    hence "0 < e / (real_of_nat (card ?d))"
wenzelm@49644
   883
      using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
   884
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
   885
      by simp
hoelzl@37489
   886
    moreover
wenzelm@49644
   887
    { fix n
wenzelm@49644
   888
      assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
   889
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
huffman@44136
   890
        unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
hoelzl@37489
   891
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
   892
        by (rule setsum_strict_mono) (simp_all add: n)
hoelzl@37489
   893
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
   894
    }
hoelzl@37489
   895
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
hoelzl@37489
   896
      by (rule eventually_elim1)
hoelzl@37489
   897
  }
wenzelm@49644
   898
  hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
hoelzl@37489
   899
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
hoelzl@37489
   900
qed
hoelzl@37489
   901
wenzelm@49644
   902
lemma interval_cart:
wenzelm@49644
   903
  fixes a :: "'a::ord^'n"
wenzelm@49644
   904
  shows "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}"
wenzelm@49644
   905
    and "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
huffman@44136
   906
  by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
   907
wenzelm@49644
   908
lemma mem_interval_cart:
wenzelm@49644
   909
  fixes a :: "'a::ord^'n"
wenzelm@49644
   910
  shows "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
wenzelm@49644
   911
    and "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
wenzelm@49644
   912
  using interval_cart[of a b] by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
hoelzl@37489
   913
wenzelm@49644
   914
lemma interval_eq_empty_cart:
wenzelm@49644
   915
  fixes a :: "real^'n"
wenzelm@49644
   916
  shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
wenzelm@49644
   917
    and "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
wenzelm@49644
   918
proof -
hoelzl@37489
   919
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
hoelzl@37489
   920
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   921
    hence "a$i < b$i" by auto
wenzelm@49644
   922
    hence False using as by auto }
hoelzl@37489
   923
  moreover
hoelzl@37489
   924
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
   925
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   926
    { fix i
hoelzl@37489
   927
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   928
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
   929
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   930
        by auto }
wenzelm@49644
   931
    hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
hoelzl@37489
   932
  ultimately show ?th1 by blast
hoelzl@37489
   933
hoelzl@37489
   934
  { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
hoelzl@37489
   935
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
   936
    hence "a$i \<le> b$i" by auto
wenzelm@49644
   937
    hence False using as by auto }
hoelzl@37489
   938
  moreover
hoelzl@37489
   939
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
   940
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
   941
    { fix i
hoelzl@37489
   942
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
   943
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
   944
        unfolding vector_smult_component and vector_add_component
wenzelm@49644
   945
        by auto }
hoelzl@37489
   946
    hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
   947
  ultimately show ?th2 by blast
hoelzl@37489
   948
qed
hoelzl@37489
   949
wenzelm@49644
   950
lemma interval_ne_empty_cart:
wenzelm@49644
   951
  fixes a :: "real^'n"
wenzelm@49644
   952
  shows "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
wenzelm@49644
   953
    and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
   954
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
   955
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   956
wenzelm@49644
   957
lemma subset_interval_imp_cart:
wenzelm@49644
   958
  fixes a :: "real^'n"
wenzelm@49644
   959
  shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
wenzelm@49644
   960
    and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"
wenzelm@49644
   961
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"
wenzelm@49644
   962
    and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
hoelzl@37489
   963
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
   964
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
   965
wenzelm@49644
   966
lemma interval_sing:
wenzelm@49644
   967
  fixes a :: "'a::linorder^'n"
wenzelm@49644
   968
  shows "{a .. a} = {a} \<and> {a<..<a} = {}"
wenzelm@49644
   969
  apply (auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
wenzelm@49644
   970
  apply (simp add: order_eq_iff)
wenzelm@49644
   971
  apply (auto simp add: not_less less_imp_le)
wenzelm@49644
   972
  done
hoelzl@37489
   973
wenzelm@49644
   974
lemma interval_open_subset_closed_cart:
wenzelm@49644
   975
  fixes a :: "'a::preorder^'n"
wenzelm@49644
   976
  shows "{a<..<b} \<subseteq> {a .. b}"
wenzelm@49644
   977
proof (simp add: subset_eq, rule)
hoelzl@37489
   978
  fix x
wenzelm@49644
   979
  assume x: "x \<in>{a<..<b}"
hoelzl@37489
   980
  { fix i
hoelzl@37489
   981
    have "a $ i \<le> x $ i"
hoelzl@37489
   982
      using x order_less_imp_le[of "a$i" "x$i"]
huffman@44136
   983
      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
   984
  }
hoelzl@37489
   985
  moreover
hoelzl@37489
   986
  { fix i
hoelzl@37489
   987
    have "x $ i \<le> b $ i"
hoelzl@37489
   988
      using x order_less_imp_le[of "x$i" "b$i"]
huffman@44136
   989
      by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
   990
  }
hoelzl@37489
   991
  ultimately
hoelzl@37489
   992
  show "a \<le> x \<and> x \<le> b"
huffman@44136
   993
    by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
hoelzl@37489
   994
qed
hoelzl@37489
   995
wenzelm@49644
   996
lemma subset_interval_cart:
wenzelm@49644
   997
  fixes a :: "real^'n"
wenzelm@49644
   998
  shows "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
wenzelm@49644
   999
    and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
wenzelm@49644
  1000
    and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
wenzelm@49644
  1001
    and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
hoelzl@50526
  1002
  using subset_interval[of c d a b] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1003
wenzelm@49644
  1004
lemma disjoint_interval_cart:
wenzelm@49644
  1005
  fixes a::"real^'n"
wenzelm@49644
  1006
  shows "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
wenzelm@49644
  1007
    and "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
wenzelm@49644
  1008
    and "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
wenzelm@49644
  1009
    and "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
hoelzl@50526
  1010
  using disjoint_interval[of a b c d] by (simp_all add: Basis_vec_def inner_axis)
hoelzl@37489
  1011
wenzelm@49644
  1012
lemma inter_interval_cart:
wenzelm@49644
  1013
  fixes a :: "'a::linorder^'n"
wenzelm@49644
  1014
  shows "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
nipkow@39302
  1015
  unfolding set_eq_iff and Int_iff and mem_interval_cart
hoelzl@37489
  1016
  by auto
hoelzl@37489
  1017
wenzelm@49644
  1018
lemma closed_interval_left_cart:
wenzelm@49644
  1019
  fixes b :: "real^'n"
hoelzl@37489
  1020
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
huffman@44233
  1021
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
  1022
wenzelm@49644
  1023
lemma closed_interval_right_cart:
wenzelm@49644
  1024
  fixes a::"real^'n"
hoelzl@37489
  1025
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
huffman@44233
  1026
  by (simp add: Collect_all_eq closed_INT closed_Collect_le)
hoelzl@37489
  1027
wenzelm@49644
  1028
lemma is_interval_cart:
wenzelm@49644
  1029
  "is_interval (s::(real^'n) set) \<longleftrightarrow>
wenzelm@49644
  1030
    (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
hoelzl@50526
  1031
  by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
hoelzl@37489
  1032
wenzelm@49644
  1033
lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
huffman@44233
  1034
  by (simp add: closed_Collect_le)
hoelzl@37489
  1035
wenzelm@49644
  1036
lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
huffman@44233
  1037
  by (simp add: closed_Collect_le)
hoelzl@37489
  1038
wenzelm@49644
  1039
lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
wenzelm@49644
  1040
  by (simp add: open_Collect_less)
wenzelm@49644
  1041
wenzelm@49644
  1042
lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i  > a}"
huffman@44233
  1043
  by (simp add: open_Collect_less)
hoelzl@37489
  1044
wenzelm@49644
  1045
lemma Lim_component_le_cart:
wenzelm@49644
  1046
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@50526
  1047
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f x $i \<le> b) net"
hoelzl@37489
  1048
  shows "l$i \<le> b"
hoelzl@50526
  1049
  by (rule tendsto_le[OF assms(2) tendsto_const tendsto_vec_nth, OF assms(1, 3)])
hoelzl@37489
  1050
wenzelm@49644
  1051
lemma Lim_component_ge_cart:
wenzelm@49644
  1052
  fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1053
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1054
  shows "b \<le> l$i"
hoelzl@50526
  1055
  by (rule tendsto_le[OF assms(2) tendsto_vec_nth tendsto_const, OF assms(1, 3)])
hoelzl@37489
  1056
wenzelm@49644
  1057
lemma Lim_component_eq_cart:
wenzelm@49644
  1058
  fixes f :: "'a \<Rightarrow> real^'n"
wenzelm@49644
  1059
  assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1060
  shows "l$i = b"
wenzelm@49644
  1061
  using ev[unfolded order_eq_iff eventually_conj_iff] and
wenzelm@49644
  1062
    Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1063
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1064
wenzelm@49644
  1065
lemma connected_ivt_component_cart:
wenzelm@49644
  1066
  fixes x :: "real^'n"
wenzelm@49644
  1067
  shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
hoelzl@50526
  1068
  using connected_ivt_hyperplane[of s x y "axis k 1" a]
hoelzl@50526
  1069
  by (auto simp add: inner_axis inner_commute)
hoelzl@37489
  1070
wenzelm@49644
  1071
lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1072
  unfolding subspace_def by auto
hoelzl@37489
  1073
hoelzl@37489
  1074
lemma closed_substandard_cart:
huffman@44213
  1075
  "closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1076
proof -
huffman@44213
  1077
  { fix i::'n
huffman@44213
  1078
    have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
wenzelm@49644
  1079
      by (cases "P i") (simp_all add: closed_Collect_eq) }
huffman@44213
  1080
  thus ?thesis
huffman@44213
  1081
    unfolding Collect_all_eq by (simp add: closed_INT)
hoelzl@37489
  1082
qed
hoelzl@37489
  1083
wenzelm@49644
  1084
lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
wenzelm@49644
  1085
  (is "dim ?A = _")
wenzelm@49644
  1086
proof -
hoelzl@50526
  1087
  let ?a = "\<lambda>x. axis x 1 :: real^'n"
hoelzl@50526
  1088
  have *: "{x. \<forall>i\<in>Basis. i \<notin> ?a ` d \<longrightarrow> x \<bullet> i = 0} = ?A"
hoelzl@50526
  1089
    by (auto simp: image_iff Basis_vec_def axis_eq_axis inner_axis)
hoelzl@50526
  1090
  have "?a ` d \<subseteq> Basis"
hoelzl@50526
  1091
    by (auto simp: Basis_vec_def)
wenzelm@49644
  1092
  thus ?thesis
hoelzl@50526
  1093
    using dim_substandard[of "?a ` d"] card_image[of ?a d]
hoelzl@50526
  1094
    by (auto simp: axis_eq_axis inj_on_def *)
hoelzl@37489
  1095
qed
hoelzl@37489
  1096
hoelzl@37489
  1097
lemma affinity_inverses:
hoelzl@37489
  1098
  assumes m0: "m \<noteq> (0::'a::field)"
hoelzl@37489
  1099
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
hoelzl@37489
  1100
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1101
  using m0
haftmann@54230
  1102
  apply (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
haftmann@54230
  1103
  apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1 [symmetric])
wenzelm@49644
  1104
  done
hoelzl@37489
  1105
hoelzl@37489
  1106
lemma vector_affinity_eq:
hoelzl@37489
  1107
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1108
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1109
proof
hoelzl@37489
  1110
  assume h: "m *s x + c = y"
hoelzl@37489
  1111
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1112
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1113
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1114
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1115
next
hoelzl@37489
  1116
  assume h: "x = inverse m *s y + - (inverse m *s c)"
haftmann@54230
  1117
  show "m *s x + c = y" unfolding h
hoelzl@37489
  1118
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1119
qed
hoelzl@37489
  1120
hoelzl@37489
  1121
lemma vector_eq_affinity:
wenzelm@49644
  1122
    "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1123
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1124
  by metis
hoelzl@37489
  1125
hoelzl@50526
  1126
lemma vector_cart:
hoelzl@50526
  1127
  fixes f :: "real^'n \<Rightarrow> real"
hoelzl@50526
  1128
  shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
hoelzl@50526
  1129
  unfolding euclidean_eq_iff[where 'a="real^'n"]
hoelzl@50526
  1130
  by simp (simp add: Basis_vec_def inner_axis)
hoelzl@50526
  1131
  
hoelzl@50526
  1132
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
hoelzl@50526
  1133
  by (rule vector_cart)
wenzelm@49644
  1134
huffman@44360
  1135
subsection "Convex Euclidean Space"
hoelzl@37489
  1136
hoelzl@50526
  1137
lemma Cart_1:"(1::real^'n) = \<Sum>Basis"
hoelzl@50526
  1138
  using const_vector_cart[of 1] by (simp add: one_vec_def)
hoelzl@37489
  1139
hoelzl@37489
  1140
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1141
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1142
hoelzl@50526
  1143
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta vector_uminus_component
hoelzl@37489
  1144
hoelzl@37489
  1145
lemma convex_box_cart:
hoelzl@37489
  1146
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1147
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1148
  using assms unfolding convex_def by auto
hoelzl@37489
  1149
hoelzl@37489
  1150
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@37489
  1151
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
hoelzl@37489
  1152
hoelzl@37489
  1153
lemma unit_interval_convex_hull_cart:
hoelzl@37489
  1154
  "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
hoelzl@37489
  1155
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
hoelzl@50526
  1156
  by (rule arg_cong[where f="\<lambda>x. convex hull x"]) (simp add: Basis_vec_def inner_axis)
hoelzl@37489
  1157
hoelzl@37489
  1158
lemma cube_convex_hull_cart:
wenzelm@49644
  1159
  assumes "0 < d"
wenzelm@49644
  1160
  obtains s::"(real^'n) set"
wenzelm@49644
  1161
    where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
wenzelm@49644
  1162
proof -
hoelzl@50526
  1163
  from cube_convex_hull [OF assms, of x] guess s .
hoelzl@50526
  1164
  with that[of s] show thesis by (simp add: const_vector_cart)
hoelzl@37489
  1165
qed
hoelzl@37489
  1166
hoelzl@37489
  1167
hoelzl@37489
  1168
subsection "Derivative"
hoelzl@37489
  1169
wenzelm@49644
  1170
lemma differentiable_at_imp_differentiable_on:
wenzelm@49644
  1171
  "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
hoelzl@51641
  1172
  by (metis differentiable_at_withinI differentiable_on_def)
hoelzl@37489
  1173
hoelzl@37489
  1174
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1175
wenzelm@49644
  1176
lemma jacobian_works:
wenzelm@49644
  1177
  "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
wenzelm@49644
  1178
    (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
wenzelm@49644
  1179
  apply rule
wenzelm@49644
  1180
  unfolding jacobian_def
wenzelm@49644
  1181
  apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
wenzelm@49644
  1182
  apply (rule differentiableI)
wenzelm@49644
  1183
  apply assumption
wenzelm@49644
  1184
  unfolding frechet_derivative_works
wenzelm@49644
  1185
  apply assumption
wenzelm@49644
  1186
  done
hoelzl@37489
  1187
hoelzl@37489
  1188
wenzelm@49644
  1189
subsection {* Component of the differential must be zero if it exists at a local
wenzelm@49644
  1190
  maximum or minimum for that corresponding component. *}
hoelzl@37489
  1191
hoelzl@50526
  1192
lemma differential_zero_maxmin_cart:
wenzelm@49644
  1193
  fixes f::"real^'a \<Rightarrow> real^'b"
wenzelm@49644
  1194
  assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
hoelzl@50526
  1195
    "f differentiable (at x)"
hoelzl@50526
  1196
  shows "jacobian f (at x) $ k = 0"
hoelzl@50526
  1197
  using differential_zero_maxmin_component[of "axis k 1" e x f] assms
hoelzl@50526
  1198
    vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
hoelzl@50526
  1199
  by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
wenzelm@49644
  1200
hoelzl@37494
  1201
subsection {* Lemmas for working on @{typ "real^1"} *}
hoelzl@37489
  1202
hoelzl@37489
  1203
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
wenzelm@49644
  1204
  by (metis (full_types) num1_eq_iff)
hoelzl@37489
  1205
hoelzl@37489
  1206
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
wenzelm@49644
  1207
  by auto (metis (full_types) num1_eq_iff)
hoelzl@37489
  1208
hoelzl@37489
  1209
lemma exhaust_2:
wenzelm@49644
  1210
  fixes x :: 2
wenzelm@49644
  1211
  shows "x = 1 \<or> x = 2"
hoelzl@37489
  1212
proof (induct x)
hoelzl@37489
  1213
  case (of_int z)
hoelzl@37489
  1214
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1215
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1216
  then show ?case by auto
hoelzl@37489
  1217
qed
hoelzl@37489
  1218
hoelzl@37489
  1219
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1220
  by (metis exhaust_2)
hoelzl@37489
  1221
hoelzl@37489
  1222
lemma exhaust_3:
wenzelm@49644
  1223
  fixes x :: 3
wenzelm@49644
  1224
  shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1225
proof (induct x)
hoelzl@37489
  1226
  case (of_int z)
hoelzl@37489
  1227
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1228
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1229
  then show ?case by auto
hoelzl@37489
  1230
qed
hoelzl@37489
  1231
hoelzl@37489
  1232
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1233
  by (metis exhaust_3)
hoelzl@37489
  1234
hoelzl@37489
  1235
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1236
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1237
hoelzl@37489
  1238
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1239
  using exhaust_2 by auto
hoelzl@37489
  1240
hoelzl@37489
  1241
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  1242
  using exhaust_3 by auto
hoelzl@37489
  1243
hoelzl@37489
  1244
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
hoelzl@37489
  1245
  unfolding UNIV_1 by simp
hoelzl@37489
  1246
hoelzl@37489
  1247
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  1248
  unfolding UNIV_2 by simp
hoelzl@37489
  1249
hoelzl@37489
  1250
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
hoelzl@37489
  1251
  unfolding UNIV_3 by (simp add: add_ac)
hoelzl@37489
  1252
wenzelm@49644
  1253
instantiation num1 :: cart_one
wenzelm@49644
  1254
begin
wenzelm@49644
  1255
wenzelm@49644
  1256
instance
wenzelm@49644
  1257
proof
hoelzl@37489
  1258
  show "CARD(1) = Suc 0" by auto
wenzelm@49644
  1259
qed
wenzelm@49644
  1260
wenzelm@49644
  1261
end
hoelzl@37489
  1262
hoelzl@37489
  1263
subsection{* The collapse of the general concepts to dimension one. *}
hoelzl@37489
  1264
hoelzl@37489
  1265
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
huffman@44136
  1266
  by (simp add: vec_eq_iff)
hoelzl@37489
  1267
hoelzl@37489
  1268
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  1269
  apply auto
hoelzl@37489
  1270
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  1271
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  1272
  done
hoelzl@37489
  1273
hoelzl@37489
  1274
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
huffman@44136
  1275
  by (simp add: norm_vec_def)
hoelzl@37489
  1276
hoelzl@37489
  1277
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
hoelzl@37489
  1278
  by (simp add: norm_vector_1)
hoelzl@37489
  1279
hoelzl@37489
  1280
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
hoelzl@37489
  1281
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  1282
wenzelm@49644
  1283
hoelzl@37489
  1284
subsection{* Explicit vector construction from lists. *}
hoelzl@37489
  1285
hoelzl@43995
  1286
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
hoelzl@37489
  1287
hoelzl@37489
  1288
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  1289
  unfolding vector_def by simp
hoelzl@37489
  1290
hoelzl@37489
  1291
lemma vector_2:
hoelzl@37489
  1292
 "(vector[x,y]) $1 = x"
hoelzl@37489
  1293
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
hoelzl@37489
  1294
  unfolding vector_def by simp_all
hoelzl@37489
  1295
hoelzl@37489
  1296
lemma vector_3:
hoelzl@37489
  1297
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
hoelzl@37489
  1298
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
hoelzl@37489
  1299
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
hoelzl@37489
  1300
  unfolding vector_def by simp_all
hoelzl@37489
  1301
hoelzl@37489
  1302
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
hoelzl@37489
  1303
  apply auto
hoelzl@37489
  1304
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1305
  apply (subgoal_tac "vector [v$1] = v")
hoelzl@37489
  1306
  apply simp
hoelzl@37489
  1307
  apply (vector vector_def)
hoelzl@37489
  1308
  apply simp
hoelzl@37489
  1309
  done
hoelzl@37489
  1310
hoelzl@37489
  1311
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
hoelzl@37489
  1312
  apply auto
hoelzl@37489
  1313
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1314
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1315
  apply (subgoal_tac "vector [v$1, v$2] = v")
hoelzl@37489
  1316
  apply simp
hoelzl@37489
  1317
  apply (vector vector_def)
hoelzl@37489
  1318
  apply (simp add: forall_2)
hoelzl@37489
  1319
  done
hoelzl@37489
  1320
hoelzl@37489
  1321
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
hoelzl@37489
  1322
  apply auto
hoelzl@37489
  1323
  apply (erule_tac x="v$1" in allE)
hoelzl@37489
  1324
  apply (erule_tac x="v$2" in allE)
hoelzl@37489
  1325
  apply (erule_tac x="v$3" in allE)
hoelzl@37489
  1326
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
hoelzl@37489
  1327
  apply simp
hoelzl@37489
  1328
  apply (vector vector_def)
hoelzl@37489
  1329
  apply (simp add: forall_3)
hoelzl@37489
  1330
  done
hoelzl@37489
  1331
hoelzl@37489
  1332
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
wenzelm@49644
  1333
  apply (rule bounded_linearI[where K=1])
hoelzl@37489
  1334
  using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
hoelzl@37489
  1335
wenzelm@49644
  1336
lemma integral_component_eq_cart[simp]:
wenzelm@49644
  1337
  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> real^'m"
wenzelm@49644
  1338
  assumes "f integrable_on s"
wenzelm@49644
  1339
  shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
hoelzl@37489
  1340
  using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
hoelzl@37489
  1341
hoelzl@37489
  1342
lemma interval_split_cart:
hoelzl@37489
  1343
  "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
hoelzl@37489
  1344
  "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
wenzelm@49644
  1345
  apply (rule_tac[!] set_eqI)
wenzelm@49644
  1346
  unfolding Int_iff mem_interval_cart mem_Collect_eq
wenzelm@49644
  1347
  unfolding vec_lambda_beta
wenzelm@49644
  1348
  by auto
hoelzl@37489
  1349
hoelzl@50526
  1350
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i" 
hoelzl@50526
  1351
  shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
hoelzl@50526
  1352
  using assms by (intro interval_bij_bij) (auto simp: Basis_vec_def inner_axis)
hoelzl@37489
  1353
hoelzl@37489
  1354
end