src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy
author haftmann
Fri Aug 20 17:46:56 2010 +0200 (2010-08-20)
changeset 38621 d6cb7e625d75
parent 37678 0040bafffdef
child 38656 d5d342611edb
permissions -rw-r--r--
more concise characterization of of_nat operation and class semiring_char_0
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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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instantiation prod :: (real_basis, real_basis) real_basis
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begin
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definition "basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
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instance
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proof
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  let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
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  let ?b_a = "basis :: nat \<Rightarrow> 'a"
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  let ?b_b = "basis :: nat \<Rightarrow> 'b"
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  note image_range =
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    image_add_atLeastLessThan[symmetric, of 0 "DIM('a)" "DIM('b)", simplified]
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  have split_range:
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    "{..<DIM('b) + DIM('a)} = {..<DIM('a)} \<union> {DIM('a)..<DIM('b) + DIM('a)}"
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    by auto
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  have *: "?b ` {DIM('a)..<DIM('b) + DIM('a)} = {0} \<times> (?b_b ` {..<DIM('b)})"
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    "?b ` {..<DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0}"
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    unfolding image_range image_image basis_prod_def_raw range_basis
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    by (auto simp: zero_prod_def basis_eq_0_iff)
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  hence b_split:
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    "?b ` {..<DIM('b) + DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0} \<union> {0} \<times> (?b_b ` {..<DIM('b)})" (is "_ = ?prod")
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    by (subst split_range) (simp add: image_Un)
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  have b_0: "?b ` {DIM('b) + DIM('a)..} = {0}" unfolding basis_prod_def_raw
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    by (auto simp: zero_prod_def image_iff basis_eq_0_iff elim!: ballE[of _ _ "DIM('a) + DIM('b)"])
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  have split_UNIV:
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    "UNIV = {..<DIM('b) + DIM('a)} \<union> {DIM('b)+DIM('a)..}"
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    by auto
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  have range_b: "range ?b = ?prod \<union> {0}"
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    by (subst split_UNIV) (simp add: image_Un b_split b_0)
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  have prod: "\<And>f A B. setsum f (A \<times> B) = (\<Sum>a\<in>A. \<Sum>b\<in>B. f (a, b))"
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    by (simp add: setsum_cartesian_product)
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  show "span (range ?b) = UNIV"
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    unfolding span_explicit range_b
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  proof safe
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    fix a::'a and b::'b
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    from in_span_basis[of a] in_span_basis[of b]
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    obtain Sa ua Sb ub where span:
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        "finite Sa" "Sa \<subseteq> basis ` {..<DIM('a)}" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
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        "finite Sb" "Sb \<subseteq> basis ` {..<DIM('b)}" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
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      unfolding span_explicit by auto
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    let ?S = "((Sa - {0}) \<times> {0} \<union> {0} \<times> (Sb - {0}))"
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    have *:
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      "?S \<inter> {v. fst v = 0} \<inter> {v. snd v = 0} = {}"
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      "?S \<inter> - {v. fst v = 0} \<inter> {v. snd v = 0} = (Sa - {0}) \<times> {0}"
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      "?S \<inter> {v. fst v = 0} \<inter> - {v. snd v = 0} = {0} \<times> (Sb - {0})"
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      by (auto simp: zero_prod_def)
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    show "\<exists>S u. finite S \<and> S \<subseteq> ?prod \<union> {0} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (a, b)"
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      apply (rule exI[of _ ?S])
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      apply (rule exI[of _ "\<lambda>(v, w). (if w = 0 then ua v else 0) + (if v = 0 then ub w else 0)"])
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      using span
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      apply (simp add: prod_case_unfold setsum_addf if_distrib cond_application_beta setsum_cases prod *)
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      by (auto simp add: setsum_prod intro!: setsum_mono_zero_cong_left)
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  qed simp
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  show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
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    apply (rule exI[of _ "DIM('b) + DIM('a)"]) unfolding b_0
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  proof (safe intro!: DIM_positive del: notI)
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    show inj_on: "inj_on ?b {..<DIM('b) + DIM('a)}" unfolding split_range
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      using inj_on_iff[OF basis_inj[where 'a='a]] inj_on_iff[OF basis_inj[where 'a='b]]
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      by (auto intro!: inj_onI simp: basis_prod_def basis_eq_0_iff)
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    show "independent (?b ` {..<DIM('b) + DIM('a)})"
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      unfolding independent_eq_inj_on[OF inj_on]
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    proof safe
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      fix i u assume i_upper: "i < DIM('b) + DIM('a)" and
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          "(\<Sum>j\<in>{..<DIM('b) + DIM('a)} - {i}. u (?b j) *\<^sub>R ?b j) = ?b i" (is "?SUM = _")
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      let ?left = "{..<DIM('a)}" and ?right = "{DIM('a)..<DIM('b) + DIM('a)}"
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      show False
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      proof cases
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        assume "i < DIM('a)"
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        hence "(basis i, 0) = ?SUM" unfolding `?SUM = ?b i` unfolding basis_prod_def by auto
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        also have "\<dots> = (\<Sum>j\<in>?left - {i}. u (?b j) *\<^sub>R ?b j) +
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          (\<Sum>j\<in>?right. u (?b j) *\<^sub>R ?b j)"
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          using `i < DIM('a)` by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
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        also have "\<dots> =  (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
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          (\<Sum>j\<in>?right. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
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          unfolding basis_prod_def by auto
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        finally have "basis i = (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R ?b_a j)"
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          by (simp add: setsum_prod)
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        moreover
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        note independent_basis[where 'a='a, unfolded independent_eq_inj_on[OF basis_inj]]
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        note this[rule_format, of i "\<lambda>v. u (v, 0)"]
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        ultimately show False using `i < DIM('a)` by auto
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      next
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        let ?i = "i - DIM('a)"
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        assume not: "\<not> i < DIM('a)" hence "DIM('a) \<le> i" by auto
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        hence "?i < DIM('b)" using `i < DIM('b) + DIM('a)` by auto
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        have inj_on: "inj_on (\<lambda>j. j - DIM('a)) {DIM('a)..<DIM('b) + DIM('a)}"
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          by (auto intro!: inj_onI)
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        with i_upper not have *: "{..<DIM('b)} - {?i} = (\<lambda>j. j-DIM('a))`(?right - {i})"
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          by (auto simp: inj_on_image_set_diff image_minus_const_atLeastLessThan_nat)
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        have "(0, basis ?i) = ?SUM" unfolding `?SUM = ?b i`
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          unfolding basis_prod_def using not `?i < DIM('b)` by auto
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        also have "\<dots> = (\<Sum>j\<in>?left. u (?b j) *\<^sub>R ?b j) +
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          (\<Sum>j\<in>?right - {i}. u (?b j) *\<^sub>R ?b j)"
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          using not by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
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        also have "\<dots> =  (\<Sum>j\<in>?left. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
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          (\<Sum>j\<in>?right - {i}. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
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          unfolding basis_prod_def by auto
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        finally have "basis ?i = (\<Sum>j\<in>{..<DIM('b)} - {?i}. u (0, ?b_b j) *\<^sub>R ?b_b j)"
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          unfolding *
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          by (subst setsum_reindex[OF inj_on[THEN subset_inj_on]])
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             (auto simp: setsum_prod)
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        moreover
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        note independent_basis[where 'a='b, unfolded independent_eq_inj_on[OF basis_inj]]
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        note this[rule_format, of ?i "\<lambda>v. u (0, v)"]
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        ultimately show False using `?i < DIM('b)` by auto
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      qed
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    qed
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  qed
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qed
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end
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lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::real_basis) + DIM('a::real_basis)"
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  by (rule dimension_eq) (auto simp: basis_prod_def zero_prod_def basis_eq_0_iff)
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instance prod :: (euclidean_space, euclidean_space) euclidean_space
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proof (default, safe)
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  let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
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  fix i j assume "i < DIM('a \<times> 'b)" "j < DIM('a \<times> 'b)"
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  thus "?b i \<bullet> ?b j = (if i = j then 1 else 0)"
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    unfolding basis_prod_def by (auto simp: dot_basis)
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qed
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instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
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begin
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definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
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definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
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instance proof qed (auto simp: less_prod_def less_eq_prod_def)
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end
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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 cart :: (times,finite) times
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begin
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  definition vector_mult_def : "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 cart :: (one,finite) one
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begin
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  definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
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  instance ..
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end
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instantiation cart :: (ord,finite) ord
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begin
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  definition vector_le_def:
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    "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
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  definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
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  instance by (intro_classes)
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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 = 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 cart :: (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 * = vector_le_def vector_less_def all Cart_eq
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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 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 = 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 vector_add_def}, @{thm vector_mult_def},
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              @{thm vector_minus_def}, @{thm vector_uminus_def},
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              @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
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              @{thm vector_scaleR_def},
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              @{thm Cart_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 "Cart_eq"}]) 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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   251
 in
hoelzl@37489
   252
  Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
hoelzl@37489
   253
 end
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   254
*} "Lifts trivial vector statements to real arith statements"
hoelzl@37489
   255
hoelzl@37489
   256
lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
hoelzl@37489
   257
lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
hoelzl@37489
   258
hoelzl@37489
   259
lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
hoelzl@37489
   260
hoelzl@37489
   261
lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
hoelzl@37489
   262
hoelzl@37489
   263
lemma vec_add: "vec(x + y) = vec x + vec y"  by (vector vec_def)
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   264
lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
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   265
lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
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   266
lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
hoelzl@37489
   267
hoelzl@37489
   268
lemma vec_setsum: assumes fS: "finite S"
hoelzl@37489
   269
  shows "vec(setsum f S) = setsum (vec o f) S"
hoelzl@37489
   270
  apply (induct rule: finite_induct[OF fS])
hoelzl@37489
   271
  apply (simp)
hoelzl@37489
   272
  apply (auto simp add: vec_add)
hoelzl@37489
   273
  done
hoelzl@37489
   274
hoelzl@37489
   275
text{* Obvious "component-pushing". *}
hoelzl@37489
   276
hoelzl@37489
   277
lemma vec_component [simp]: "vec x $ i = x"
hoelzl@37489
   278
  by (vector vec_def)
hoelzl@37489
   279
hoelzl@37489
   280
lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
hoelzl@37489
   281
  by vector
hoelzl@37489
   282
hoelzl@37489
   283
lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
hoelzl@37489
   284
  by vector
hoelzl@37489
   285
hoelzl@37489
   286
lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
hoelzl@37489
   287
hoelzl@37489
   288
lemmas vector_component =
hoelzl@37489
   289
  vec_component vector_add_component vector_mult_component
hoelzl@37489
   290
  vector_smult_component vector_minus_component vector_uminus_component
hoelzl@37489
   291
  vector_scaleR_component cond_component
hoelzl@37489
   292
hoelzl@37489
   293
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
hoelzl@37489
   294
hoelzl@37489
   295
instance cart :: (semigroup_mult,finite) semigroup_mult
hoelzl@37489
   296
  apply (intro_classes) by (vector mult_assoc)
hoelzl@37489
   297
hoelzl@37489
   298
instance cart :: (monoid_mult,finite) monoid_mult
hoelzl@37489
   299
  apply (intro_classes) by vector+
hoelzl@37489
   300
hoelzl@37489
   301
instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult
hoelzl@37489
   302
  apply (intro_classes) by (vector mult_commute)
hoelzl@37489
   303
hoelzl@37489
   304
instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult
hoelzl@37489
   305
  apply (intro_classes) by (vector mult_idem)
hoelzl@37489
   306
hoelzl@37489
   307
instance cart :: (comm_monoid_mult,finite) comm_monoid_mult
hoelzl@37489
   308
  apply (intro_classes) by vector
hoelzl@37489
   309
hoelzl@37489
   310
instance cart :: (semiring,finite) semiring
hoelzl@37489
   311
  apply (intro_classes) by (vector field_simps)+
hoelzl@37489
   312
hoelzl@37489
   313
instance cart :: (semiring_0,finite) semiring_0
hoelzl@37489
   314
  apply (intro_classes) by (vector field_simps)+
hoelzl@37489
   315
instance cart :: (semiring_1,finite) semiring_1
hoelzl@37489
   316
  apply (intro_classes) by vector
hoelzl@37489
   317
instance cart :: (comm_semiring,finite) comm_semiring
hoelzl@37489
   318
  apply (intro_classes) by (vector field_simps)+
hoelzl@37489
   319
hoelzl@37489
   320
instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes)
hoelzl@37489
   321
instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
hoelzl@37489
   322
instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes)
hoelzl@37489
   323
instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes)
hoelzl@37489
   324
instance cart :: (ring,finite) ring by (intro_classes)
hoelzl@37489
   325
instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes)
hoelzl@37489
   326
instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes)
hoelzl@37489
   327
hoelzl@37489
   328
instance cart :: (ring_1,finite) ring_1 ..
hoelzl@37489
   329
hoelzl@37489
   330
instance cart :: (real_algebra,finite) real_algebra
hoelzl@37489
   331
  apply intro_classes
hoelzl@37489
   332
  apply (simp_all add: vector_scaleR_def field_simps)
hoelzl@37489
   333
  apply vector
hoelzl@37489
   334
  apply vector
hoelzl@37489
   335
  done
hoelzl@37489
   336
hoelzl@37489
   337
instance cart :: (real_algebra_1,finite) real_algebra_1 ..
hoelzl@37489
   338
hoelzl@37489
   339
lemma of_nat_index:
hoelzl@37489
   340
  "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
hoelzl@37489
   341
  apply (induct n)
hoelzl@37489
   342
  apply vector
hoelzl@37489
   343
  apply vector
hoelzl@37489
   344
  done
hoelzl@37489
   345
hoelzl@37489
   346
lemma one_index[simp]:
hoelzl@37489
   347
  "(1 :: 'a::one ^'n)$i = 1" by vector
hoelzl@37489
   348
haftmann@38621
   349
instance cart :: (semiring_char_0, finite) semiring_char_0
haftmann@38621
   350
proof
haftmann@38621
   351
  fix m n :: nat
haftmann@38621
   352
  show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
haftmann@38621
   353
    by (auto intro!: injI simp add: Cart_eq of_nat_index)
hoelzl@37489
   354
qed
hoelzl@37489
   355
haftmann@38621
   356
instance cart :: (comm_ring_1,finite) comm_ring_1 ..
haftmann@38621
   357
instance cart :: (ring_char_0,finite) ring_char_0 ..
hoelzl@37489
   358
hoelzl@37489
   359
instance cart :: (real_vector,finite) real_vector ..
hoelzl@37489
   360
hoelzl@37489
   361
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
hoelzl@37489
   362
  by (vector mult_assoc)
hoelzl@37489
   363
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
hoelzl@37489
   364
  by (vector field_simps)
hoelzl@37489
   365
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
hoelzl@37489
   366
  by (vector field_simps)
hoelzl@37489
   367
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
hoelzl@37489
   368
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
hoelzl@37489
   369
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
hoelzl@37489
   370
  by (vector field_simps)
hoelzl@37489
   371
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
hoelzl@37489
   372
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
hoelzl@37489
   373
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
hoelzl@37489
   374
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
hoelzl@37489
   375
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
hoelzl@37489
   376
  by (vector field_simps)
hoelzl@37489
   377
hoelzl@37489
   378
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
hoelzl@37489
   379
  by (simp add: Cart_eq)
hoelzl@37489
   380
hoelzl@37489
   381
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
hoelzl@37489
   382
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
hoelzl@37489
   383
  by vector
hoelzl@37489
   384
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
hoelzl@37489
   385
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
hoelzl@37489
   386
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
hoelzl@37489
   387
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
hoelzl@37489
   388
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
hoelzl@37489
   389
  by (metis vector_mul_lcancel)
hoelzl@37489
   390
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
hoelzl@37489
   391
  by (metis vector_mul_rcancel)
hoelzl@37489
   392
hoelzl@37489
   393
lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
hoelzl@37489
   394
  apply (simp add: norm_vector_def)
hoelzl@37489
   395
  apply (rule member_le_setL2, simp_all)
hoelzl@37489
   396
  done
hoelzl@37489
   397
hoelzl@37489
   398
lemma norm_bound_component_le_cart: "norm x <= e ==> \<bar>x$i\<bar> <= e"
hoelzl@37489
   399
  by (metis component_le_norm_cart order_trans)
hoelzl@37489
   400
hoelzl@37489
   401
lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
hoelzl@37489
   402
  by (metis component_le_norm_cart basic_trans_rules(21))
hoelzl@37489
   403
hoelzl@37489
   404
lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
hoelzl@37489
   405
  by (simp add: norm_vector_def setL2_le_setsum)
hoelzl@37489
   406
hoelzl@37489
   407
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
hoelzl@37489
   408
  unfolding vector_scaleR_def vector_scalar_mult_def by simp
hoelzl@37489
   409
hoelzl@37489
   410
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
hoelzl@37489
   411
  unfolding dist_norm scalar_mult_eq_scaleR
hoelzl@37489
   412
  unfolding scaleR_right_diff_distrib[symmetric] by simp
hoelzl@37489
   413
hoelzl@37489
   414
lemma setsum_component [simp]:
hoelzl@37489
   415
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
hoelzl@37489
   416
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
hoelzl@37489
   417
  by (cases "finite S", induct S set: finite, simp_all)
hoelzl@37489
   418
hoelzl@37489
   419
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
hoelzl@37489
   420
  by (simp add: Cart_eq)
hoelzl@37489
   421
hoelzl@37489
   422
lemma setsum_cmul:
hoelzl@37489
   423
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
hoelzl@37489
   424
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
hoelzl@37489
   425
  by (simp add: Cart_eq setsum_right_distrib)
hoelzl@37489
   426
hoelzl@37489
   427
(* TODO: use setsum_norm_allsubsets_bound *)
hoelzl@37489
   428
lemma setsum_norm_allsubsets_bound_cart:
hoelzl@37489
   429
  fixes f:: "'a \<Rightarrow> real ^'n"
hoelzl@37489
   430
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
hoelzl@37489
   431
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
hoelzl@37489
   432
proof-
hoelzl@37489
   433
  let ?d = "real CARD('n)"
hoelzl@37489
   434
  let ?nf = "\<lambda>x. norm (f x)"
hoelzl@37489
   435
  let ?U = "UNIV :: 'n set"
hoelzl@37489
   436
  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"
hoelzl@37489
   437
    by (rule setsum_commute)
hoelzl@37489
   438
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
hoelzl@37489
   439
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
hoelzl@37489
   440
    apply (rule setsum_mono)    by (rule norm_le_l1_cart)
hoelzl@37489
   441
  also have "\<dots> \<le> 2 * ?d * e"
hoelzl@37489
   442
    unfolding th0 th1
hoelzl@37489
   443
  proof(rule setsum_bounded)
hoelzl@37489
   444
    fix i assume i: "i \<in> ?U"
hoelzl@37489
   445
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
hoelzl@37489
   446
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
hoelzl@37489
   447
    have thp: "P = ?Pp \<union> ?Pn" by auto
hoelzl@37489
   448
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
hoelzl@37489
   449
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
hoelzl@37489
   450
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
hoelzl@37489
   451
      using component_le_norm_cart[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
hoelzl@37489
   452
      by (auto intro: abs_le_D1)
hoelzl@37489
   453
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
hoelzl@37489
   454
      using component_le_norm_cart[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
hoelzl@37489
   455
      by (auto simp add: setsum_negf intro: abs_le_D1)
hoelzl@37489
   456
    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"
hoelzl@37489
   457
      apply (subst thp)
hoelzl@37489
   458
      apply (rule setsum_Un_zero)
hoelzl@37489
   459
      using fP thp0 by auto
hoelzl@37489
   460
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
hoelzl@37489
   461
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
hoelzl@37489
   462
  qed
hoelzl@37489
   463
  finally show ?thesis .
hoelzl@37489
   464
qed
hoelzl@37489
   465
hoelzl@37489
   466
subsection {* A bijection between 'n::finite and {..<CARD('n)} *}
hoelzl@37489
   467
hoelzl@37489
   468
definition cart_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
hoelzl@37489
   469
  "cart_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
hoelzl@37489
   470
hoelzl@37489
   471
abbreviation "\<pi> \<equiv> cart_bij_nat"
hoelzl@37489
   472
definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"
hoelzl@37489
   473
hoelzl@37489
   474
lemma bij_betw_pi:
hoelzl@37489
   475
  "bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
hoelzl@37489
   476
  using ex_bij_betw_nat_finite[of "UNIV::'n set"]
hoelzl@37489
   477
  by (auto simp: cart_bij_nat_def atLeast0LessThan
hoelzl@37489
   478
    intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])
hoelzl@37489
   479
hoelzl@37489
   480
lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
hoelzl@37489
   481
  using bij_betw_inv_into[OF bij_betw_pi] unfolding \<pi>'_def by auto
hoelzl@37489
   482
hoelzl@37489
   483
lemma pi'_inj[intro]: "inj \<pi>'"
hoelzl@37489
   484
  using bij_betw_pi' unfolding bij_betw_def by auto
hoelzl@37489
   485
hoelzl@37489
   486
lemma pi'_range[intro]: "\<And>i::'n. \<pi>' i < CARD('n::finite)"
hoelzl@37489
   487
  using bij_betw_pi' unfolding bij_betw_def by auto
hoelzl@37489
   488
hoelzl@37489
   489
lemma \<pi>\<pi>'[simp]: "\<And>i::'n::finite. \<pi> (\<pi>' i) = i"
hoelzl@37489
   490
  using bij_betw_pi by (auto intro!: f_inv_into_f simp: \<pi>'_def bij_betw_def)
hoelzl@37489
   491
hoelzl@37489
   492
lemma \<pi>'\<pi>[simp]: "\<And>i. i\<in>{..<CARD('n::finite)} \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
hoelzl@37489
   493
  using bij_betw_pi by (auto intro!: inv_into_f_eq simp: \<pi>'_def bij_betw_def)
hoelzl@37489
   494
hoelzl@37489
   495
lemma \<pi>\<pi>'_alt[simp]: "\<And>i. i<CARD('n::finite) \<Longrightarrow> \<pi>' (\<pi> i::'n) = i"
hoelzl@37489
   496
  by auto
hoelzl@37489
   497
hoelzl@37489
   498
lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
hoelzl@37489
   499
  using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)
hoelzl@37489
   500
hoelzl@37489
   501
instantiation cart :: (real_basis,finite) real_basis
hoelzl@37489
   502
begin
hoelzl@37489
   503
hoelzl@37489
   504
definition "(basis i::'a^'b) =
hoelzl@37489
   505
  (if i < (CARD('b) * DIM('a))
hoelzl@37489
   506
  then (\<chi> j::'b. if j = \<pi>(i div DIM('a)) then basis (i mod DIM('a)) else 0)
hoelzl@37489
   507
  else 0)"
hoelzl@37489
   508
hoelzl@37489
   509
lemma basis_eq:
hoelzl@37489
   510
  assumes "i < CARD('b)" and "j < DIM('a)"
hoelzl@37489
   511
  shows "basis (j + i * DIM('a)) = (\<chi> k. if k = \<pi> i then basis j else 0)"
hoelzl@37489
   512
proof -
hoelzl@37489
   513
  have "j + i * DIM('a) <  DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
hoelzl@37489
   514
  also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
hoelzl@37489
   515
  finally show ?thesis
hoelzl@37489
   516
    unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps)
hoelzl@37489
   517
qed
hoelzl@37489
   518
hoelzl@37489
   519
lemma basis_eq_pi':
hoelzl@37489
   520
  assumes "j < DIM('a)"
hoelzl@37489
   521
  shows "basis (j + \<pi>' i * DIM('a)) $ k = (if k = i then basis j else 0)"
hoelzl@37489
   522
  apply (subst basis_eq)
hoelzl@37489
   523
  using pi'_range assms by simp_all
hoelzl@37489
   524
hoelzl@37489
   525
lemma split_times_into_modulo[consumes 1]:
hoelzl@37489
   526
  fixes k :: nat
hoelzl@37489
   527
  assumes "k < A * B"
hoelzl@37489
   528
  obtains i j where "i < A" and "j < B" and "k = j + i * B"
hoelzl@37489
   529
proof
hoelzl@37489
   530
  have "A * B \<noteq> 0"
hoelzl@37489
   531
  proof assume "A * B = 0" with assms show False by simp qed
hoelzl@37489
   532
  hence "0 < B" by auto
hoelzl@37489
   533
  thus "k mod B < B" using `0 < B` by auto
hoelzl@37489
   534
next
hoelzl@37489
   535
  have "k div B * B \<le> k div B * B + k mod B" by (rule le_add1)
hoelzl@37489
   536
  also have "... < A * B" using assms by simp
hoelzl@37489
   537
  finally show "k div B < A" by auto
hoelzl@37489
   538
qed simp
hoelzl@37489
   539
hoelzl@37489
   540
lemma split_CARD_DIM[consumes 1]:
hoelzl@37489
   541
  fixes k :: nat
hoelzl@37489
   542
  assumes k: "k < CARD('b) * DIM('a)"
hoelzl@37489
   543
  obtains i and j::'b where "i < DIM('a)" "k = i + \<pi>' j * DIM('a)"
hoelzl@37489
   544
proof -
hoelzl@37489
   545
  from split_times_into_modulo[OF k] guess i j . note ij = this
hoelzl@37489
   546
  show thesis
hoelzl@37489
   547
  proof
hoelzl@37489
   548
    show "j < DIM('a)" using ij by simp
hoelzl@37489
   549
    show "k = j + \<pi>' (\<pi> i :: 'b) * DIM('a)"
hoelzl@37489
   550
      using ij by simp
hoelzl@37489
   551
  qed
hoelzl@37489
   552
qed
hoelzl@37489
   553
hoelzl@37489
   554
lemma linear_less_than_times:
hoelzl@37489
   555
  fixes i j A B :: nat assumes "i < B" "j < A"
hoelzl@37489
   556
  shows "j + i * A < B * A"
hoelzl@37489
   557
proof -
hoelzl@37489
   558
  have "i * A + j < (Suc i)*A" using `j < A` by simp
hoelzl@37489
   559
  also have "\<dots> \<le> B * A" using `i < B` unfolding mult_le_cancel2 by simp
hoelzl@37489
   560
  finally show ?thesis by simp
hoelzl@37489
   561
qed
hoelzl@37489
   562
hoelzl@37489
   563
instance
hoelzl@37489
   564
proof
hoelzl@37489
   565
  let ?b = "basis :: nat \<Rightarrow> 'a^'b"
hoelzl@37489
   566
  let ?b' = "basis :: nat \<Rightarrow> 'a"
hoelzl@37489
   567
hoelzl@37489
   568
  have setsum_basis:
hoelzl@37489
   569
    "\<And>f. (\<Sum>x\<in>range basis. f (x::'a)) = f 0 + (\<Sum>i<DIM('a). f (basis i))"
hoelzl@37489
   570
    unfolding range_basis apply (subst setsum.insert)
hoelzl@37489
   571
    by (auto simp: basis_eq_0_iff setsum.insert setsum_reindex[OF basis_inj])
hoelzl@37489
   572
hoelzl@37489
   573
  have inj: "inj_on ?b {..<CARD('b)*DIM('a)}"
hoelzl@37489
   574
    by (auto intro!: inj_onI elim!: split_CARD_DIM split: split_if_asm
hoelzl@37489
   575
             simp add: Cart_eq basis_eq_pi' all_conj_distrib basis_neq_0
hoelzl@37489
   576
                       inj_on_iff[OF basis_inj])
hoelzl@37489
   577
  moreover
hoelzl@37489
   578
  hence indep: "independent (?b ` {..<CARD('b) * DIM('a)})"
hoelzl@37664
   579
  proof (rule independent_eq_inj_on[THEN iffD2], safe elim!: split_CARD_DIM del: notI)
hoelzl@37489
   580
    fix j and i :: 'b and u :: "'a^'b \<Rightarrow> real" assume "j < DIM('a)"
hoelzl@37489
   581
    let ?x = "j + \<pi>' i * DIM('a)"
hoelzl@37489
   582
    show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) \<noteq> ?b ?x"
hoelzl@37489
   583
      unfolding Cart_eq not_all
hoelzl@37489
   584
    proof
hoelzl@37489
   585
      have "(\<lambda>j. j + \<pi>' i*DIM('a))`({..<DIM('a)}-{j}) =
hoelzl@37489
   586
        {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)} - {?x}"(is "?S = ?I - _")
hoelzl@37489
   587
      proof safe
hoelzl@37489
   588
        fix y assume "y \<in> ?I"
hoelzl@37489
   589
        moreover def k \<equiv> "y - \<pi>' i*DIM('a)"
hoelzl@37489
   590
        ultimately have "k < DIM('a)" and "y = k + \<pi>' i * DIM('a)" by auto
hoelzl@37489
   591
        moreover assume "y \<notin> ?S"
hoelzl@37489
   592
        ultimately show "y = j + \<pi>' i * DIM('a)" by auto
hoelzl@37489
   593
      qed auto
hoelzl@37489
   594
hoelzl@37489
   595
      have "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i =
hoelzl@37489
   596
          (\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k $ i)" by simp
hoelzl@37489
   597
      also have "\<dots> = (\<Sum>k\<in>?S. u(?b k) *\<^sub>R ?b k $ i)"
hoelzl@37489
   598
        unfolding `?S = ?I - {?x}`
hoelzl@37489
   599
      proof (safe intro!: setsum_mono_zero_cong_right)
hoelzl@37489
   600
        fix y assume "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"
hoelzl@37489
   601
        moreover have "Suc (\<pi>' i) * DIM('a) \<le> CARD('b) * DIM('a)"
hoelzl@37489
   602
          unfolding mult_le_cancel2 using pi'_range[of i] by simp
hoelzl@37489
   603
        ultimately show "y < CARD('b) * DIM('a)" by simp
hoelzl@37489
   604
      next
hoelzl@37489
   605
        fix y assume "y < CARD('b) * DIM('a)"
hoelzl@37489
   606
        with split_CARD_DIM guess l k . note y = this
hoelzl@37489
   607
        moreover assume "u (?b y) *\<^sub>R ?b y $ i \<noteq> 0"
hoelzl@37489
   608
        ultimately show "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"
hoelzl@37489
   609
          by (auto simp: basis_eq_pi' split: split_if_asm)
hoelzl@37489
   610
      qed simp
hoelzl@37489
   611
      also have "\<dots> = (\<Sum>k\<in>{..<DIM('a)} - {j}. u (?b (k + \<pi>' i*DIM('a))) *\<^sub>R (?b' k))"
hoelzl@37489
   612
        by (subst setsum_reindex) (auto simp: basis_eq_pi' intro!: inj_onI)
hoelzl@37489
   613
      also have "\<dots> \<noteq> ?b ?x $ i"
hoelzl@37489
   614
      proof -
hoelzl@37664
   615
        note independent_eq_inj_on[THEN iffD1, OF basis_inj independent_basis, rule_format]
hoelzl@37489
   616
        note this[of j "\<lambda>v. u (\<chi> ka::'b. if ka = i then v else (0\<Colon>'a))"]
hoelzl@37489
   617
        thus ?thesis by (simp add: `j < DIM('a)` basis_eq pi'_range)
hoelzl@37489
   618
      qed
hoelzl@37489
   619
      finally show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i \<noteq> ?b ?x $ i" .
hoelzl@37489
   620
    qed
hoelzl@37489
   621
  qed
hoelzl@37489
   622
  ultimately
hoelzl@37489
   623
  show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
hoelzl@37489
   624
    by (auto intro!: exI[of _ "CARD('b) * DIM('a)"] simp: basis_cart_def)
hoelzl@37489
   625
hoelzl@37489
   626
  from indep have exclude_0: "0 \<notin> ?b ` {..<CARD('b) * DIM('a)}"
hoelzl@37489
   627
    using dependent_0[of "?b ` {..<CARD('b) * DIM('a)}"] by blast
hoelzl@37489
   628
hoelzl@37489
   629
  show "span (range ?b) = UNIV"
hoelzl@37489
   630
  proof -
hoelzl@37489
   631
    { fix x :: "'a^'b"
hoelzl@37489
   632
      let "?if i y" = "(\<chi> k::'b. if k = i then ?b' y else (0\<Colon>'a))"
hoelzl@37489
   633
      have The_if: "\<And>i j. j < DIM('a) \<Longrightarrow> (THE k. (?if i j) $ k \<noteq> 0) = i"
hoelzl@37489
   634
        by (rule the_equality) (simp_all split: split_if_asm add: basis_neq_0)
hoelzl@37489
   635
      { fix x :: 'a
hoelzl@37489
   636
        have "x \<in> span (range basis)"
hoelzl@37664
   637
          using span_basis by (auto simp: range_basis)
hoelzl@37489
   638
        hence "\<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = x"
hoelzl@37489
   639
          by (subst (asm) span_finite) (auto simp: setsum_basis) }
hoelzl@37489
   640
      hence "\<forall>i. \<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = i" by auto
hoelzl@37489
   641
      then obtain u where u: "\<forall>i. (\<Sum>x<DIM('a). u i (?b' x) *\<^sub>R ?b' x) = i"
hoelzl@37489
   642
        by (auto dest: choice)
hoelzl@37489
   643
      have "\<exists>u. \<forall>i. (\<Sum>j<DIM('a). u (?if i j) *\<^sub>R ?b' j) = x $ i"
hoelzl@37489
   644
        apply (rule exI[of _ "\<lambda>v. let i = (THE i. v$i \<noteq> 0) in u (x$i) (v$i)"])
hoelzl@37489
   645
        using The_if u by simp }
hoelzl@37489
   646
    moreover
hoelzl@37489
   647
    have "\<And>i::'b. {..<CARD('b)} \<inter> {x. i = \<pi> x} = {\<pi>' i}"
hoelzl@37489
   648
      using pi'_range[where 'n='b] by auto
hoelzl@37489
   649
    moreover
hoelzl@37489
   650
    have "range ?b = {0} \<union> ?b ` {..<CARD('b) * DIM('a)}"
hoelzl@37489
   651
      by (auto simp: image_def basis_cart_def)
hoelzl@37489
   652
    ultimately
hoelzl@37489
   653
    show ?thesis
hoelzl@37664
   654
      by (auto simp add: Cart_eq setsum_reindex[OF inj] range_basis
hoelzl@37489
   655
          setsum_mult_product basis_eq if_distrib setsum_cases span_finite
hoelzl@37489
   656
          setsum_reindex[OF basis_inj])
hoelzl@37489
   657
  qed
hoelzl@37489
   658
qed
hoelzl@37489
   659
hoelzl@37489
   660
end
hoelzl@37489
   661
hoelzl@37489
   662
lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a::real_basis)"
hoelzl@37489
   663
proof (safe intro!: dimension_eq elim!: split_times_into_modulo del: notI)
hoelzl@37489
   664
  fix i j assume *: "i < CARD('b)" "j < DIM('a)"
hoelzl@37489
   665
  hence A: "(i * DIM('a) + j) div DIM('a) = i"
hoelzl@37489
   666
    by (subst div_add1_eq) simp
hoelzl@37489
   667
  from * have B: "(i * DIM('a) + j) mod DIM('a) = j"
hoelzl@37489
   668
    unfolding mod_mult_self3 by simp
hoelzl@37489
   669
  show "basis (j + i * DIM('a)) \<noteq> (0::'a^'b)" unfolding basis_cart_def
hoelzl@37489
   670
    using * basis_finite[where 'a='a]
hoelzl@37489
   671
      linear_less_than_times[of i "CARD('b)" j "DIM('a)"]
hoelzl@37489
   672
    by (auto simp: A B field_simps Cart_eq basis_eq_0_iff)
hoelzl@37489
   673
qed (auto simp: basis_cart_def)
hoelzl@37489
   674
hoelzl@37489
   675
lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
hoelzl@37489
   676
hoelzl@37489
   677
lemma split_dimensions'[consumes 1]:
hoelzl@37489
   678
  assumes "k < DIM('a::real_basis^'b)"
hoelzl@37489
   679
  obtains i j where "i < CARD('b::finite)" and "j < DIM('a::real_basis)" and "k = j + i * DIM('a::real_basis)"
hoelzl@37489
   680
using split_times_into_modulo[OF assms[simplified]] .
hoelzl@37489
   681
hoelzl@37489
   682
lemma cart_euclidean_bound[intro]:
hoelzl@37489
   683
  assumes j:"j < DIM('a::{real_basis})"
hoelzl@37489
   684
  shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::real_basis)"
hoelzl@37489
   685
  using linear_less_than_times[OF pi'_range j, of i] .
hoelzl@37489
   686
hoelzl@37489
   687
instance cart :: (real_basis_with_inner,finite) real_basis_with_inner ..
hoelzl@37489
   688
hoelzl@37489
   689
lemma (in real_basis) forall_CARD_DIM:
hoelzl@37489
   690
  "(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"
hoelzl@37489
   691
   (is "?l \<longleftrightarrow> ?r")
hoelzl@37489
   692
proof (safe elim!: split_times_into_modulo)
hoelzl@37489
   693
  fix i :: 'b and j assume "j < DIM('a)"
hoelzl@37489
   694
  note linear_less_than_times[OF pi'_range[of i] this]
hoelzl@37489
   695
  moreover assume "?l"
hoelzl@37489
   696
  ultimately show "P (j + \<pi>' i * DIM('a))" by auto
hoelzl@37489
   697
next
hoelzl@37489
   698
  fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r"
hoelzl@37489
   699
  from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`
hoelzl@37489
   700
  show "P (j + i * DIM('a))" by simp
hoelzl@37489
   701
qed
hoelzl@37489
   702
hoelzl@37489
   703
lemma (in real_basis) exists_CARD_DIM:
hoelzl@37489
   704
  "(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"
hoelzl@37489
   705
  using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast
hoelzl@37489
   706
hoelzl@37489
   707
lemma forall_CARD:
hoelzl@37489
   708
  "(\<forall>i<CARD('b). P i) \<longleftrightarrow> (\<forall>i::'b::finite. P (\<pi>' i))"
hoelzl@37489
   709
  using forall_CARD_DIM[where 'a=real, of P] by simp
hoelzl@37489
   710
hoelzl@37489
   711
lemma exists_CARD:
hoelzl@37489
   712
  "(\<exists>i<CARD('b). P i) \<longleftrightarrow> (\<exists>i::'b::finite. P (\<pi>' i))"
hoelzl@37489
   713
  using exists_CARD_DIM[where 'a=real, of P] by simp
hoelzl@37489
   714
hoelzl@37489
   715
lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
hoelzl@37489
   716
hoelzl@37489
   717
lemma cart_euclidean_nth[simp]:
hoelzl@37489
   718
  fixes x :: "('a::real_basis_with_inner, 'b::finite) cart"
hoelzl@37489
   719
  assumes j:"j < DIM('a)"
hoelzl@37489
   720
  shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
hoelzl@37489
   721
  unfolding euclidean_component_def inner_vector_def basis_eq_pi'[OF j] if_distrib cond_application_beta
hoelzl@37489
   722
  by (simp add: setsum_cases)
hoelzl@37489
   723
hoelzl@37489
   724
lemma real_euclidean_nth:
hoelzl@37489
   725
  fixes x :: "real^'n"
hoelzl@37489
   726
  shows "x $$ \<pi>' i = (x $ i :: real)"
hoelzl@37489
   727
  using cart_euclidean_nth[where 'a=real, of 0 x i] by simp
hoelzl@37489
   728
hoelzl@37489
   729
lemmas nth_conv_component = real_euclidean_nth[symmetric]
hoelzl@37489
   730
hoelzl@37489
   731
lemma mult_split_eq:
hoelzl@37489
   732
  fixes A :: nat assumes "x < A" "y < A"
hoelzl@37489
   733
  shows "x + i * A = y + j * A \<longleftrightarrow> x = y \<and> i = j"
hoelzl@37489
   734
proof
hoelzl@37489
   735
  assume *: "x + i * A = y + j * A"
hoelzl@37489
   736
  { fix x y i j assume "i < j" "x < A" and *: "x + i * A = y + j * A"
hoelzl@37489
   737
    hence "x + i * A < Suc i * A" using `x < A` by simp
hoelzl@37489
   738
    also have "\<dots> \<le> j * A" using `i < j` unfolding mult_le_cancel2 by simp
hoelzl@37489
   739
    also have "\<dots> \<le> y + j * A" by simp
hoelzl@37489
   740
    finally have "i = j" using * by simp }
hoelzl@37489
   741
  note eq = this
hoelzl@37489
   742
hoelzl@37489
   743
  have "i = j"
hoelzl@37489
   744
  proof (cases rule: linorder_cases)
hoelzl@37489
   745
    assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp
hoelzl@37489
   746
  next
hoelzl@37489
   747
    assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
hoelzl@37489
   748
  qed simp
hoelzl@37489
   749
  thus "x = y \<and> i = j" using * by simp
hoelzl@37489
   750
qed simp
hoelzl@37489
   751
hoelzl@37489
   752
instance cart :: (euclidean_space,finite) euclidean_space
hoelzl@37489
   753
proof (default, safe elim!: split_dimensions')
hoelzl@37489
   754
  let ?b = "basis :: nat \<Rightarrow> 'a^'b"
hoelzl@37489
   755
  have if_distrib_op: "\<And>f P Q a b c d.
hoelzl@37489
   756
    f (if P then a else b) (if Q then c else d) =
hoelzl@37489
   757
      (if P then if Q then f a c else f a d else if Q then f b c else f b d)"
hoelzl@37489
   758
    by simp
hoelzl@37489
   759
hoelzl@37489
   760
  fix i j k l
hoelzl@37489
   761
  assume "i < CARD('b)" "k < CARD('b)" "j < DIM('a)" "l < DIM('a)"
hoelzl@37489
   762
  thus "?b (j + i * DIM('a)) \<bullet> ?b (l + k * DIM('a)) =
hoelzl@37489
   763
    (if j + i * DIM('a) = l + k * DIM('a) then 1 else 0)"
hoelzl@37489
   764
    using inj_on_iff[OF \<pi>_inj_on[where 'n='b], of k i]
hoelzl@37489
   765
    by (auto simp add: basis_eq inner_vector_def if_distrib_op[of inner] setsum_cases basis_orthonormal mult_split_eq)
hoelzl@37489
   766
qed
hoelzl@37489
   767
hoelzl@37489
   768
instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space
hoelzl@37489
   769
proof
hoelzl@37489
   770
  fix x y::"'a^'b"
hoelzl@37489
   771
  show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i \<le> y $$ i)"
hoelzl@37489
   772
    unfolding vector_le_def apply(subst eucl_le) by (simp add: cart_simps)
hoelzl@37489
   773
  show"(x < y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i < y $$ i)"
hoelzl@37489
   774
    unfolding vector_less_def apply(subst eucl_less) by (simp add: cart_simps)
hoelzl@37489
   775
qed
hoelzl@37489
   776
hoelzl@37489
   777
subsection{* Basis vectors in coordinate directions. *}
hoelzl@37489
   778
hoelzl@37489
   779
definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)"
hoelzl@37489
   780
hoelzl@37489
   781
lemma basis_component [simp]: "cart_basis k $ i = (if k=i then 1 else 0)"
hoelzl@37489
   782
  unfolding cart_basis_def by simp
hoelzl@37489
   783
hoelzl@37489
   784
lemma norm_basis[simp]:
hoelzl@37489
   785
  shows "norm (cart_basis k :: real ^'n) = 1"
hoelzl@37489
   786
  apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vector_def
hoelzl@37489
   787
  apply (vector delta_mult_idempotent)
hoelzl@37489
   788
  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto
hoelzl@37489
   789
hoelzl@37489
   790
lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1"
hoelzl@37489
   791
  by (rule norm_basis)
hoelzl@37489
   792
hoelzl@37489
   793
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
hoelzl@37489
   794
  by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp
hoelzl@37489
   795
hoelzl@37489
   796
lemma vector_choose_dist: assumes e: "0 <= e"
hoelzl@37489
   797
  shows "\<exists>(y::real^'n). dist x y = e"
hoelzl@37489
   798
proof-
hoelzl@37489
   799
  from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
hoelzl@37489
   800
    by blast
hoelzl@37489
   801
  then have "dist x (x - c) = e" by (simp add: dist_norm)
hoelzl@37489
   802
  then show ?thesis by blast
hoelzl@37489
   803
qed
hoelzl@37489
   804
hoelzl@37489
   805
lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
hoelzl@37489
   806
  by (simp add: inj_on_def Cart_eq)
hoelzl@37489
   807
hoelzl@37489
   808
lemma basis_expansion:
hoelzl@37489
   809
  "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
hoelzl@37489
   810
  by (auto simp add: Cart_eq if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
hoelzl@37489
   811
hoelzl@37489
   812
lemma smult_conv_scaleR: "c *s x = scaleR c x"
hoelzl@37489
   813
  unfolding vector_scalar_mult_def vector_scaleR_def by simp
hoelzl@37489
   814
hoelzl@37489
   815
lemma basis_expansion':
hoelzl@37489
   816
  "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
hoelzl@37489
   817
  by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])
hoelzl@37489
   818
hoelzl@37489
   819
lemma basis_expansion_unique:
hoelzl@37489
   820
  "setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
hoelzl@37489
   821
  by (simp add: Cart_eq setsum_delta if_distrib cong del: if_weak_cong)
hoelzl@37489
   822
hoelzl@37489
   823
lemma dot_basis:
hoelzl@37489
   824
  shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
hoelzl@37489
   825
  by (auto simp add: inner_vector_def cart_basis_def cond_application_beta if_distrib setsum_delta
hoelzl@37489
   826
           cong del: if_weak_cong)
hoelzl@37489
   827
hoelzl@37489
   828
lemma inner_basis:
hoelzl@37489
   829
  fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
hoelzl@37489
   830
  shows "inner (cart_basis i) x = inner 1 (x $ i)"
hoelzl@37489
   831
    and "inner x (cart_basis i) = inner (x $ i) 1"
hoelzl@37489
   832
  unfolding inner_vector_def cart_basis_def
hoelzl@37489
   833
  by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
hoelzl@37489
   834
hoelzl@37489
   835
lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
hoelzl@37489
   836
  by (auto simp add: Cart_eq)
hoelzl@37489
   837
hoelzl@37489
   838
lemma basis_nonzero:
hoelzl@37489
   839
  shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
hoelzl@37489
   840
  by (simp add: basis_eq_0)
hoelzl@37489
   841
hoelzl@37489
   842
text {* some lemmas to map between Eucl and Cart *}
hoelzl@37489
   843
lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
hoelzl@37489
   844
  unfolding basis_cart_def using pi'_range[where 'n='a]
hoelzl@37489
   845
  by (auto simp: Cart_eq Cart_lambda_beta)
hoelzl@37489
   846
hoelzl@37489
   847
subsection {* Orthogonality on cartesian products *}
hoelzl@37489
   848
hoelzl@37489
   849
lemma orthogonal_basis:
hoelzl@37489
   850
  shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
hoelzl@37489
   851
  by (auto simp add: orthogonal_def inner_vector_def cart_basis_def if_distrib
hoelzl@37489
   852
                     cond_application_beta setsum_delta cong del: if_weak_cong)
hoelzl@37489
   853
hoelzl@37489
   854
lemma orthogonal_basis_basis:
hoelzl@37489
   855
  shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
hoelzl@37489
   856
  unfolding orthogonal_basis[of i] basis_component[of j] by simp
hoelzl@37489
   857
hoelzl@37489
   858
subsection {* Linearity on cartesian products *}
hoelzl@37489
   859
hoelzl@37489
   860
lemma linear_vmul_component:
hoelzl@37489
   861
  assumes lf: "linear f"
hoelzl@37489
   862
  shows "linear (\<lambda>x. f x $ k *\<^sub>R v)"
hoelzl@37489
   863
  using lf
hoelzl@37489
   864
  by (auto simp add: linear_def algebra_simps)
hoelzl@37489
   865
hoelzl@37489
   866
hoelzl@37489
   867
subsection{* Adjoints on cartesian products *}
hoelzl@37489
   868
hoelzl@37489
   869
text {* TODO: The following lemmas about adjoints should hold for any
hoelzl@37489
   870
Hilbert space (i.e. complete inner product space).
hoelzl@37489
   871
(see \url{http://en.wikipedia.org/wiki/Hermitian_adjoint})
hoelzl@37489
   872
*}
hoelzl@37489
   873
hoelzl@37489
   874
lemma adjoint_works_lemma:
hoelzl@37489
   875
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   876
  assumes lf: "linear f"
hoelzl@37489
   877
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
hoelzl@37489
   878
proof-
hoelzl@37489
   879
  let ?N = "UNIV :: 'n set"
hoelzl@37489
   880
  let ?M = "UNIV :: 'm set"
hoelzl@37489
   881
  have fN: "finite ?N" by simp
hoelzl@37489
   882
  have fM: "finite ?M" by simp
hoelzl@37489
   883
  {fix y:: "real ^ 'm"
hoelzl@37489
   884
    let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> y)) :: real ^ 'n"
hoelzl@37489
   885
    {fix x
hoelzl@37489
   886
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"
hoelzl@37489
   887
        by (simp only: basis_expansion')
hoelzl@37489
   888
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"
hoelzl@37489
   889
        unfolding linear_setsum[OF lf fN]
hoelzl@37489
   890
        by (simp add: linear_cmul[OF lf])
hoelzl@37489
   891
      finally have "f x \<bullet> y = x \<bullet> ?w"
hoelzl@37489
   892
        apply (simp only: )
hoelzl@37489
   893
        apply (simp add: inner_vector_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
hoelzl@37489
   894
        done}
hoelzl@37489
   895
  }
hoelzl@37489
   896
  then show ?thesis unfolding adjoint_def
hoelzl@37489
   897
    some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
hoelzl@37489
   898
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
hoelzl@37489
   899
    by metis
hoelzl@37489
   900
qed
hoelzl@37489
   901
hoelzl@37489
   902
lemma adjoint_works:
hoelzl@37489
   903
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   904
  assumes lf: "linear f"
hoelzl@37489
   905
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@37489
   906
  using adjoint_works_lemma[OF lf] by metis
hoelzl@37489
   907
hoelzl@37489
   908
lemma adjoint_linear:
hoelzl@37489
   909
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   910
  assumes lf: "linear f"
hoelzl@37489
   911
  shows "linear (adjoint f)"
hoelzl@37489
   912
  unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe
hoelzl@37489
   913
  unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto
hoelzl@37489
   914
hoelzl@37489
   915
lemma adjoint_clauses:
hoelzl@37489
   916
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   917
  assumes lf: "linear f"
hoelzl@37489
   918
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
hoelzl@37489
   919
  and "adjoint f y \<bullet> x = y \<bullet> f x"
hoelzl@37489
   920
  by (simp_all add: adjoint_works[OF lf] inner_commute)
hoelzl@37489
   921
hoelzl@37489
   922
lemma adjoint_adjoint:
hoelzl@37489
   923
  fixes f:: "real ^'n \<Rightarrow> real ^'m"
hoelzl@37489
   924
  assumes lf: "linear f"
hoelzl@37489
   925
  shows "adjoint (adjoint f) = f"
hoelzl@37489
   926
  by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
hoelzl@37489
   927
hoelzl@37489
   928
hoelzl@37489
   929
subsection {* Matrix operations *}
hoelzl@37489
   930
hoelzl@37489
   931
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
hoelzl@37489
   932
hoelzl@37489
   933
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
hoelzl@37489
   934
  where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
hoelzl@37489
   935
hoelzl@37489
   936
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
hoelzl@37489
   937
  where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
hoelzl@37489
   938
hoelzl@37489
   939
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
hoelzl@37489
   940
  where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
hoelzl@37489
   941
hoelzl@37489
   942
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
hoelzl@37489
   943
definition transpose where 
hoelzl@37489
   944
  "(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
hoelzl@37489
   945
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
hoelzl@37489
   946
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
hoelzl@37489
   947
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
hoelzl@37489
   948
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
hoelzl@37489
   949
hoelzl@37489
   950
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
hoelzl@37489
   951
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
hoelzl@37489
   952
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
hoelzl@37489
   953
hoelzl@37489
   954
lemma matrix_mul_lid:
hoelzl@37489
   955
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   956
  shows "mat 1 ** A = A"
hoelzl@37489
   957
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   958
  apply vector
hoelzl@37489
   959
  by (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
hoelzl@37489
   960
hoelzl@37489
   961
hoelzl@37489
   962
lemma matrix_mul_rid:
hoelzl@37489
   963
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
hoelzl@37489
   964
  shows "A ** mat 1 = A"
hoelzl@37489
   965
  apply (simp add: matrix_matrix_mult_def mat_def)
hoelzl@37489
   966
  apply vector
hoelzl@37489
   967
  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)
hoelzl@37489
   968
hoelzl@37489
   969
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
hoelzl@37489
   970
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   971
  apply (subst setsum_commute)
hoelzl@37489
   972
  apply simp
hoelzl@37489
   973
  done
hoelzl@37489
   974
hoelzl@37489
   975
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
hoelzl@37489
   976
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
hoelzl@37489
   977
  apply (subst setsum_commute)
hoelzl@37489
   978
  apply simp
hoelzl@37489
   979
  done
hoelzl@37489
   980
hoelzl@37489
   981
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
hoelzl@37489
   982
  apply (vector matrix_vector_mult_def mat_def)
hoelzl@37489
   983
  by (simp add: if_distrib cond_application_beta
hoelzl@37489
   984
    setsum_delta' cong del: if_weak_cong)
hoelzl@37489
   985
hoelzl@37489
   986
lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
hoelzl@37489
   987
  by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)
hoelzl@37489
   988
hoelzl@37489
   989
lemma matrix_eq:
hoelzl@37489
   990
  fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
hoelzl@37489
   991
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
   992
  apply auto
hoelzl@37489
   993
  apply (subst Cart_eq)
hoelzl@37489
   994
  apply clarify
hoelzl@37489
   995
  apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta Cart_eq cong del: if_weak_cong)
hoelzl@37489
   996
  apply (erule_tac x="cart_basis ia" in allE)
hoelzl@37489
   997
  apply (erule_tac x="i" in allE)
hoelzl@37489
   998
  by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
hoelzl@37489
   999
hoelzl@37489
  1000
lemma matrix_vector_mul_component:
hoelzl@37489
  1001
  shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
hoelzl@37489
  1002
  by (simp add: matrix_vector_mult_def inner_vector_def)
hoelzl@37489
  1003
hoelzl@37489
  1004
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
hoelzl@37489
  1005
  apply (simp add: inner_vector_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
hoelzl@37489
  1006
  apply (subst setsum_commute)
hoelzl@37489
  1007
  by simp
hoelzl@37489
  1008
hoelzl@37489
  1009
lemma transpose_mat: "transpose (mat n) = mat n"
hoelzl@37489
  1010
  by (vector transpose_def mat_def)
hoelzl@37489
  1011
hoelzl@37489
  1012
lemma transpose_transpose: "transpose(transpose A) = A"
hoelzl@37489
  1013
  by (vector transpose_def)
hoelzl@37489
  1014
hoelzl@37489
  1015
lemma row_transpose:
hoelzl@37489
  1016
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
  1017
  shows "row i (transpose A) = column i A"
hoelzl@37489
  1018
  by (simp add: row_def column_def transpose_def Cart_eq)
hoelzl@37489
  1019
hoelzl@37489
  1020
lemma column_transpose:
hoelzl@37489
  1021
  fixes A:: "'a::semiring_1^_^_"
hoelzl@37489
  1022
  shows "column i (transpose A) = row i A"
hoelzl@37489
  1023
  by (simp add: row_def column_def transpose_def Cart_eq)
hoelzl@37489
  1024
hoelzl@37489
  1025
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
hoelzl@37489
  1026
by (auto simp add: rows_def columns_def row_transpose intro: set_ext)
hoelzl@37489
  1027
hoelzl@37489
  1028
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose)
hoelzl@37489
  1029
hoelzl@37489
  1030
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
hoelzl@37489
  1031
hoelzl@37489
  1032
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
hoelzl@37489
  1033
  by (simp add: matrix_vector_mult_def inner_vector_def)
hoelzl@37489
  1034
hoelzl@37489
  1035
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)"
hoelzl@37489
  1036
  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
hoelzl@37489
  1037
hoelzl@37489
  1038
lemma vector_componentwise:
hoelzl@37489
  1039
  "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
hoelzl@37489
  1040
  apply (subst basis_expansion[symmetric])
hoelzl@37489
  1041
  by (vector Cart_eq setsum_component)
hoelzl@37489
  1042
hoelzl@37489
  1043
lemma linear_componentwise:
hoelzl@37489
  1044
  fixes f:: "real ^'m \<Rightarrow> real ^ _"
hoelzl@37489
  1045
  assumes lf: "linear f"
hoelzl@37489
  1046
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
hoelzl@37489
  1047
proof-
hoelzl@37489
  1048
  let ?M = "(UNIV :: 'm set)"
hoelzl@37489
  1049
  let ?N = "(UNIV :: 'n set)"
hoelzl@37489
  1050
  have fM: "finite ?M" by simp
hoelzl@37489
  1051
  have "?rhs = (setsum (\<lambda>i.(x$i) *\<^sub>R f (cart_basis i) ) ?M)$j"
hoelzl@37489
  1052
    unfolding vector_smult_component[symmetric] smult_conv_scaleR
hoelzl@37489
  1053
    unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]
hoelzl@37489
  1054
    ..
hoelzl@37489
  1055
  then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
hoelzl@37489
  1056
qed
hoelzl@37489
  1057
hoelzl@37489
  1058
text{* Inverse matrices  (not necessarily square) *}
hoelzl@37489
  1059
hoelzl@37489
  1060
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
  1061
hoelzl@37489
  1062
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
hoelzl@37489
  1063
        (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
hoelzl@37489
  1064
hoelzl@37489
  1065
text{* Correspondence between matrices and linear operators. *}
hoelzl@37489
  1066
hoelzl@37489
  1067
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
hoelzl@37489
  1068
where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
hoelzl@37489
  1069
hoelzl@37489
  1070
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
hoelzl@37489
  1071
  by (simp add: linear_def matrix_vector_mult_def Cart_eq field_simps setsum_right_distrib setsum_addf)
hoelzl@37489
  1072
hoelzl@37489
  1073
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)"
hoelzl@37489
  1074
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
hoelzl@37489
  1075
apply clarify
hoelzl@37489
  1076
apply (rule linear_componentwise[OF lf, symmetric])
hoelzl@37489
  1077
done
hoelzl@37489
  1078
hoelzl@37489
  1079
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" by (simp add: ext matrix_works)
hoelzl@37489
  1080
hoelzl@37489
  1081
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
hoelzl@37489
  1082
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
hoelzl@37489
  1083
hoelzl@37489
  1084
lemma matrix_compose:
hoelzl@37489
  1085
  assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
hoelzl@37489
  1086
  and lg: "linear (g::real^'m \<Rightarrow> real^_)"
hoelzl@37489
  1087
  shows "matrix (g o f) = matrix g ** matrix f"
hoelzl@37489
  1088
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
hoelzl@37489
  1089
  by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
hoelzl@37489
  1090
hoelzl@37489
  1091
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
hoelzl@37489
  1092
  by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)
hoelzl@37489
  1093
hoelzl@37489
  1094
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
hoelzl@37489
  1095
  apply (rule adjoint_unique)
hoelzl@37489
  1096
  apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
hoelzl@37489
  1097
  apply (subst setsum_commute)
hoelzl@37489
  1098
  apply (auto simp add: mult_ac)
hoelzl@37489
  1099
  done
hoelzl@37489
  1100
hoelzl@37489
  1101
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
hoelzl@37489
  1102
  shows "matrix(adjoint f) = transpose(matrix f)"
hoelzl@37489
  1103
  apply (subst matrix_vector_mul[OF lf])
hoelzl@37489
  1104
  unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
hoelzl@37489
  1105
hoelzl@37494
  1106
section {* lambda skolemization on cartesian products *}
hoelzl@37489
  1107
hoelzl@37489
  1108
(* FIXME: rename do choice_cart *)
hoelzl@37489
  1109
hoelzl@37489
  1110
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
hoelzl@37494
  1111
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
  1112
proof-
hoelzl@37489
  1113
  let ?S = "(UNIV :: 'n set)"
hoelzl@37489
  1114
  {assume H: "?rhs"
hoelzl@37489
  1115
    then have ?lhs by auto}
hoelzl@37489
  1116
  moreover
hoelzl@37489
  1117
  {assume H: "?lhs"
hoelzl@37489
  1118
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
hoelzl@37489
  1119
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
hoelzl@37489
  1120
    {fix i
hoelzl@37489
  1121
      from f have "P i (f i)" by metis
hoelzl@37494
  1122
      then have "P i (?x $ i)" by auto
hoelzl@37489
  1123
    }
hoelzl@37489
  1124
    hence "\<forall>i. P i (?x$i)" by metis
hoelzl@37489
  1125
    hence ?rhs by metis }
hoelzl@37489
  1126
  ultimately show ?thesis by metis
hoelzl@37489
  1127
qed
hoelzl@37489
  1128
hoelzl@37489
  1129
subsection {* Standard bases are a spanning set, and obviously finite. *}
hoelzl@37489
  1130
hoelzl@37489
  1131
lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
hoelzl@37489
  1132
apply (rule set_ext)
hoelzl@37489
  1133
apply auto
hoelzl@37489
  1134
apply (subst basis_expansion'[symmetric])
hoelzl@37489
  1135
apply (rule span_setsum)
hoelzl@37489
  1136
apply simp
hoelzl@37489
  1137
apply auto
hoelzl@37489
  1138
apply (rule span_mul)
hoelzl@37489
  1139
apply (rule span_superset)
hoelzl@37489
  1140
apply (auto simp add: Collect_def mem_def)
hoelzl@37489
  1141
done
hoelzl@37489
  1142
hoelzl@37489
  1143
lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
hoelzl@37489
  1144
proof-
hoelzl@37489
  1145
  have eq: "?S = cart_basis ` UNIV" by blast
hoelzl@37489
  1146
  show ?thesis unfolding eq by auto
hoelzl@37489
  1147
qed
hoelzl@37489
  1148
hoelzl@37489
  1149
lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
hoelzl@37489
  1150
proof-
hoelzl@37489
  1151
  have eq: "?S = cart_basis ` UNIV" by blast
hoelzl@37489
  1152
  show ?thesis unfolding eq using card_image[OF basis_inj] by simp
hoelzl@37489
  1153
qed
hoelzl@37489
  1154
hoelzl@37489
  1155
hoelzl@37489
  1156
lemma independent_stdbasis_lemma:
hoelzl@37489
  1157
  assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"
hoelzl@37489
  1158
  and iS: "i \<notin> S"
hoelzl@37489
  1159
  shows "(x$i) = 0"
hoelzl@37489
  1160
proof-
hoelzl@37489
  1161
  let ?U = "UNIV :: 'n set"
hoelzl@37489
  1162
  let ?B = "cart_basis ` S"
hoelzl@37489
  1163
  let ?P = "\<lambda>(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
hoelzl@37489
  1164
 {fix x::"real^_" assume xS: "x\<in> ?B"
hoelzl@37489
  1165
   from xS have "?P x" by auto}
hoelzl@37489
  1166
 moreover
hoelzl@37489
  1167
 have "subspace ?P"
hoelzl@37489
  1168
   by (auto simp add: subspace_def Collect_def mem_def)
hoelzl@37489
  1169
 ultimately show ?thesis
hoelzl@37489
  1170
   using x span_induct[of ?B ?P x] iS by blast
hoelzl@37489
  1171
qed
hoelzl@37489
  1172
hoelzl@37489
  1173
lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
hoelzl@37489
  1174
proof-
hoelzl@37489
  1175
  let ?I = "UNIV :: 'n set"
hoelzl@37489
  1176
  let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"
hoelzl@37489
  1177
  let ?B = "?b ` ?I"
hoelzl@37489
  1178
  have eq: "{?b i|i. i \<in> ?I} = ?B"
hoelzl@37489
  1179
    by auto
hoelzl@37489
  1180
  {assume d: "dependent ?B"
hoelzl@37489
  1181
    then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
hoelzl@37489
  1182
      unfolding dependent_def by auto
hoelzl@37489
  1183
    have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
hoelzl@37489
  1184
    have eq2: "?B - {?b k} = ?b ` (?I - {k})"
hoelzl@37489
  1185
      unfolding eq1
hoelzl@37489
  1186
      apply (rule inj_on_image_set_diff[symmetric])
hoelzl@37489
  1187
      apply (rule basis_inj) using k(1) by auto
hoelzl@37489
  1188
    from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
hoelzl@37489
  1189
    from independent_stdbasis_lemma[OF th0, of k, simplified]
hoelzl@37489
  1190
    have False by simp}
hoelzl@37489
  1191
  then show ?thesis unfolding eq dependent_def ..
hoelzl@37489
  1192
qed
hoelzl@37489
  1193
hoelzl@37489
  1194
lemma vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
hoelzl@37489
  1195
  unfolding inner_simps smult_conv_scaleR by auto
hoelzl@37489
  1196
hoelzl@37489
  1197
lemma linear_eq_stdbasis_cart:
hoelzl@37489
  1198
  assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g"
hoelzl@37489
  1199
  and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
hoelzl@37489
  1200
  shows "f = g"
hoelzl@37489
  1201
proof-
hoelzl@37489
  1202
  let ?U = "UNIV :: 'm set"
hoelzl@37489
  1203
  let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"
hoelzl@37489
  1204
  {fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
hoelzl@37489
  1205
    from equalityD2[OF span_stdbasis]
hoelzl@37489
  1206
    have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast
hoelzl@37489
  1207
    from linear_eq[OF lf lg IU] fg x
hoelzl@37489
  1208
    have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
hoelzl@37489
  1209
  then show ?thesis by (auto intro: ext)
hoelzl@37489
  1210
qed
hoelzl@37489
  1211
hoelzl@37489
  1212
lemma bilinear_eq_stdbasis_cart:
hoelzl@37489
  1213
  assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"
hoelzl@37489
  1214
  and bg: "bilinear g"
hoelzl@37489
  1215
  and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
hoelzl@37489
  1216
  shows "f = g"
hoelzl@37489
  1217
proof-
hoelzl@37489
  1218
  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
hoelzl@37489
  1219
  from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
hoelzl@37489
  1220
qed
hoelzl@37489
  1221
hoelzl@37489
  1222
lemma left_invertible_transpose:
hoelzl@37489
  1223
  "(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
hoelzl@37489
  1224
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
  1225
hoelzl@37489
  1226
lemma right_invertible_transpose:
hoelzl@37489
  1227
  "(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
hoelzl@37489
  1228
  by (metis matrix_transpose_mul transpose_mat transpose_transpose)
hoelzl@37489
  1229
hoelzl@37489
  1230
lemma matrix_left_invertible_injective:
hoelzl@37489
  1231
"(\<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)"
hoelzl@37489
  1232
proof-
hoelzl@37489
  1233
  {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
hoelzl@37489
  1234
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hoelzl@37489
  1235
    hence "x = y"
hoelzl@37489
  1236
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
hoelzl@37489
  1237
  moreover
hoelzl@37489
  1238
  {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hoelzl@37489
  1239
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
hoelzl@37489
  1240
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
hoelzl@37489
  1241
    obtain g where g: "linear g" "g o op *v A = id" by blast
hoelzl@37489
  1242
    have "matrix g ** A = mat 1"
hoelzl@37489
  1243
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
hoelzl@37489
  1244
      using g(2) by (simp add: o_def id_def stupid_ext)
hoelzl@37489
  1245
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
hoelzl@37489
  1246
  ultimately show ?thesis by blast
hoelzl@37489
  1247
qed
hoelzl@37489
  1248
hoelzl@37489
  1249
lemma matrix_left_invertible_ker:
hoelzl@37489
  1250
  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
hoelzl@37489
  1251
  unfolding matrix_left_invertible_injective
hoelzl@37489
  1252
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
hoelzl@37489
  1253
  by (simp add: inj_on_def)
hoelzl@37489
  1254
hoelzl@37489
  1255
lemma matrix_right_invertible_surjective:
hoelzl@37489
  1256
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
hoelzl@37489
  1257
proof-
hoelzl@37489
  1258
  {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
hoelzl@37489
  1259
    {fix x :: "real ^ 'm"
hoelzl@37489
  1260
      have "A *v (B *v x) = x"
hoelzl@37489
  1261
        by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
hoelzl@37489
  1262
    hence "surj (op *v A)" unfolding surj_def by metis }
hoelzl@37489
  1263
  moreover
hoelzl@37489
  1264
  {assume sf: "surj (op *v A)"
hoelzl@37489
  1265
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
hoelzl@37489
  1266
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
hoelzl@37489
  1267
      by blast
hoelzl@37489
  1268
hoelzl@37489
  1269
    have "A ** (matrix g) = mat 1"
hoelzl@37489
  1270
      unfolding matrix_eq  matrix_vector_mul_lid
hoelzl@37489
  1271
        matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
hoelzl@37489
  1272
      using g(2) unfolding o_def stupid_ext[symmetric] id_def
hoelzl@37489
  1273
      .
hoelzl@37489
  1274
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
hoelzl@37489
  1275
  }
hoelzl@37489
  1276
  ultimately show ?thesis unfolding surj_def by blast
hoelzl@37489
  1277
qed
hoelzl@37489
  1278
hoelzl@37489
  1279
lemma matrix_left_invertible_independent_columns:
hoelzl@37489
  1280
  fixes A :: "real^'n^'m"
hoelzl@37489
  1281
  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))"
hoelzl@37489
  1282
   (is "?lhs \<longleftrightarrow> ?rhs")
hoelzl@37489
  1283
proof-
hoelzl@37489
  1284
  let ?U = "UNIV :: 'n set"
hoelzl@37489
  1285
  {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
hoelzl@37489
  1286
    {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
hoelzl@37489
  1287
      and i: "i \<in> ?U"
hoelzl@37489
  1288
      let ?x = "\<chi> i. c i"
hoelzl@37489
  1289
      have th0:"A *v ?x = 0"
hoelzl@37489
  1290
        using c
hoelzl@37489
  1291
        unfolding matrix_mult_vsum Cart_eq
hoelzl@37489
  1292
        by auto
hoelzl@37489
  1293
      from k[rule_format, OF th0] i
hoelzl@37489
  1294
      have "c i = 0" by (vector Cart_eq)}
hoelzl@37489
  1295
    hence ?rhs by blast}
hoelzl@37489
  1296
  moreover
hoelzl@37489
  1297
  {assume H: ?rhs
hoelzl@37489
  1298
    {fix x assume x: "A *v x = 0"
hoelzl@37489
  1299
      let ?c = "\<lambda>i. ((x$i ):: real)"
hoelzl@37489
  1300
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
hoelzl@37489
  1301
      have "x = 0" by vector}}
hoelzl@37489
  1302
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
hoelzl@37489
  1303
qed
hoelzl@37489
  1304
hoelzl@37489
  1305
lemma matrix_right_invertible_independent_rows:
hoelzl@37489
  1306
  fixes A :: "real^'n^'m"
hoelzl@37489
  1307
  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))"
hoelzl@37489
  1308
  unfolding left_invertible_transpose[symmetric]
hoelzl@37489
  1309
    matrix_left_invertible_independent_columns
hoelzl@37489
  1310
  by (simp add: column_transpose)
hoelzl@37489
  1311
hoelzl@37489
  1312
lemma matrix_right_invertible_span_columns:
hoelzl@37489
  1313
  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
hoelzl@37489
  1314
proof-
hoelzl@37489
  1315
  let ?U = "UNIV :: 'm set"
hoelzl@37489
  1316
  have fU: "finite ?U" by simp
hoelzl@37489
  1317
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
hoelzl@37489
  1318
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
hoelzl@37489
  1319
    apply (subst eq_commute) ..
hoelzl@37489
  1320
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
hoelzl@37489
  1321
  {assume h: ?lhs
hoelzl@37489
  1322
    {fix x:: "real ^'n"
hoelzl@37489
  1323
        from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
hoelzl@37489
  1324
          where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
hoelzl@37489
  1325
        have "x \<in> span (columns A)"
hoelzl@37489
  1326
          unfolding y[symmetric]
hoelzl@37489
  1327
          apply (rule span_setsum[OF fU])
hoelzl@37489
  1328
          apply clarify
hoelzl@37489
  1329
          unfolding smult_conv_scaleR
hoelzl@37489
  1330
          apply (rule span_mul)
hoelzl@37489
  1331
          apply (rule span_superset)
hoelzl@37489
  1332
          unfolding columns_def
hoelzl@37489
  1333
          by blast}
hoelzl@37489
  1334
    then have ?rhs unfolding rhseq by blast}
hoelzl@37489
  1335
  moreover
hoelzl@37489
  1336
  {assume h:?rhs
hoelzl@37489
  1337
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
hoelzl@37489
  1338
    {fix y have "?P y"
hoelzl@37489
  1339
      proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
hoelzl@37489
  1340
        show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
hoelzl@37489
  1341
          by (rule exI[where x=0], simp)
hoelzl@37489
  1342
      next
hoelzl@37489
  1343
        fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
hoelzl@37489
  1344
        from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
hoelzl@37489
  1345
          unfolding columns_def by blast
hoelzl@37489
  1346
        from y2 obtain x:: "real ^'m" where
hoelzl@37489
  1347
          x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
hoelzl@37489
  1348
        let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
hoelzl@37489
  1349
        show "?P (c*s y1 + y2)"
hoelzl@37489
  1350
          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)
hoelzl@37489
  1351
            fix j
hoelzl@37489
  1352
            have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
hoelzl@37489
  1353
           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)
hoelzl@37489
  1354
              by (simp add: field_simps)
hoelzl@37489
  1355
            have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
hoelzl@37489
  1356
           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"
hoelzl@37489
  1357
              apply (rule setsum_cong[OF refl])
hoelzl@37489
  1358
              using th by blast
hoelzl@37489
  1359
            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"
hoelzl@37489
  1360
              by (simp add: setsum_addf)
hoelzl@37489
  1361
            also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
hoelzl@37489
  1362
              unfolding setsum_delta[OF fU]
hoelzl@37489
  1363
              using i(1) by simp
hoelzl@37489
  1364
            finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
hoelzl@37489
  1365
           else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
hoelzl@37489
  1366
          qed
hoelzl@37489
  1367
        next
hoelzl@37489
  1368
          show "y \<in> span (columns A)" unfolding h by blast
hoelzl@37489
  1369
        qed}
hoelzl@37489
  1370
    then have ?lhs unfolding lhseq ..}
hoelzl@37489
  1371
  ultimately show ?thesis by blast
hoelzl@37489
  1372
qed
hoelzl@37489
  1373
hoelzl@37489
  1374
lemma matrix_left_invertible_span_rows:
hoelzl@37489
  1375
  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
hoelzl@37489
  1376
  unfolding right_invertible_transpose[symmetric]
hoelzl@37489
  1377
  unfolding columns_transpose[symmetric]
hoelzl@37489
  1378
  unfolding matrix_right_invertible_span_columns
hoelzl@37489
  1379
 ..
hoelzl@37489
  1380
hoelzl@37489
  1381
text {* The same result in terms of square matrices. *}
hoelzl@37489
  1382
hoelzl@37489
  1383
lemma matrix_left_right_inverse:
hoelzl@37489
  1384
  fixes A A' :: "real ^'n^'n"
hoelzl@37489
  1385
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
hoelzl@37489
  1386
proof-
hoelzl@37489
  1387
  {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
hoelzl@37489
  1388
    have sA: "surj (op *v A)"
hoelzl@37489
  1389
      unfolding surj_def
hoelzl@37489
  1390
      apply clarify
hoelzl@37489
  1391
      apply (rule_tac x="(A' *v y)" in exI)
hoelzl@37489
  1392
      by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
hoelzl@37489
  1393
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
hoelzl@37489
  1394
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
hoelzl@37489
  1395
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
hoelzl@37489
  1396
    have th: "matrix f' ** A = mat 1"
hoelzl@37489
  1397
      by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hoelzl@37489
  1398
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
hoelzl@37489
  1399
    hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hoelzl@37489
  1400
    hence "matrix f' ** A = A' ** A" by simp
hoelzl@37489
  1401
    hence "A' ** A = mat 1" by (simp add: th)}
hoelzl@37489
  1402
  then show ?thesis by blast
hoelzl@37489
  1403
qed
hoelzl@37489
  1404
hoelzl@37489
  1405
text {* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *}
hoelzl@37489
  1406
hoelzl@37489
  1407
definition "rowvector v = (\<chi> i j. (v$j))"
hoelzl@37489
  1408
hoelzl@37489
  1409
definition "columnvector v = (\<chi> i j. (v$i))"
hoelzl@37489
  1410
hoelzl@37489
  1411
lemma transpose_columnvector:
hoelzl@37489
  1412
 "transpose(columnvector v) = rowvector v"
hoelzl@37489
  1413
  by (simp add: transpose_def rowvector_def columnvector_def Cart_eq)
hoelzl@37489
  1414
hoelzl@37489
  1415
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
hoelzl@37489
  1416
  by (simp add: transpose_def columnvector_def rowvector_def Cart_eq)
hoelzl@37489
  1417
hoelzl@37489
  1418
lemma dot_rowvector_columnvector:
hoelzl@37489
  1419
  "columnvector (A *v v) = A ** columnvector v"
hoelzl@37489
  1420
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
hoelzl@37489
  1421
hoelzl@37489
  1422
lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
hoelzl@37489
  1423
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def)
hoelzl@37489
  1424
hoelzl@37489
  1425
lemma dot_matrix_vector_mul:
hoelzl@37489
  1426
  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
hoelzl@37489
  1427
  shows "(A *v x) \<bullet> (B *v y) =
hoelzl@37489
  1428
      (((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
hoelzl@37489
  1429
unfolding dot_matrix_product transpose_columnvector[symmetric]
hoelzl@37489
  1430
  dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
hoelzl@37489
  1431
hoelzl@37489
  1432
hoelzl@37489
  1433
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
hoelzl@37489
  1434
  unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe
hoelzl@37489
  1435
  apply(rule_tac x="\<pi> i" in exI) defer
hoelzl@37489
  1436
  apply(rule_tac x="\<pi>' i" in exI) unfolding nth_conv_component using pi'_range by auto
hoelzl@37489
  1437
hoelzl@37489
  1438
lemma infnorm_set_image_cart:
hoelzl@37489
  1439
  "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
hoelzl@37489
  1440
  (\<lambda>i. abs(x$i)) ` (UNIV)" by blast
hoelzl@37489
  1441
hoelzl@37489
  1442
lemma infnorm_set_lemma_cart:
hoelzl@37489
  1443
  shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
hoelzl@37489
  1444
  and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
hoelzl@37489
  1445
  unfolding  infnorm_set_image_cart
hoelzl@37489
  1446
  by (auto intro: finite_imageI)
hoelzl@37489
  1447
hoelzl@37489
  1448
lemma component_le_infnorm_cart:
hoelzl@37489
  1449
  shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
hoelzl@37489
  1450
  unfolding nth_conv_component
hoelzl@37489
  1451
  using component_le_infnorm[of x] .
hoelzl@37489
  1452
hoelzl@37489
  1453
lemma dist_nth_le_cart: "dist (x $ i) (y $ i) \<le> dist x y"
hoelzl@37489
  1454
  unfolding dist_vector_def
hoelzl@37489
  1455
  by (rule member_le_setL2) simp_all
hoelzl@37489
  1456
hoelzl@37489
  1457
instance cart :: (perfect_space, finite) perfect_space
hoelzl@37489
  1458
proof
hoelzl@37489
  1459
  fix x :: "'a ^ 'b"
hoelzl@37489
  1460
  {
hoelzl@37489
  1461
    fix e :: real assume "0 < e"
hoelzl@37489
  1462
    def a \<equiv> "x $ undefined"
hoelzl@37489
  1463
    have "a islimpt UNIV" by (rule islimpt_UNIV)
hoelzl@37489
  1464
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
hoelzl@37489
  1465
      unfolding islimpt_approachable by auto
hoelzl@37489
  1466
    def y \<equiv> "Cart_lambda ((Cart_nth x)(undefined := b))"
hoelzl@37489
  1467
    from `b \<noteq> a` have "y \<noteq> x"
hoelzl@37489
  1468
      unfolding a_def y_def by (simp add: Cart_eq)
hoelzl@37489
  1469
    from `dist b a < e` have "dist y x < e"
hoelzl@37489
  1470
      unfolding dist_vector_def a_def y_def
hoelzl@37489
  1471
      apply simp
hoelzl@37489
  1472
      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
hoelzl@37489
  1473
      apply (subst setsum_diff1' [where a=undefined], simp, simp, simp)
hoelzl@37489
  1474
      done
hoelzl@37489
  1475
    from `y \<noteq> x` and `dist y x < e`
hoelzl@37489
  1476
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
hoelzl@37489
  1477
  }
hoelzl@37489
  1478
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
hoelzl@37489
  1479
qed
hoelzl@37489
  1480
hoelzl@37489
  1481
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
hoelzl@37489
  1482
proof-
hoelzl@37489
  1483
  let ?U = "UNIV :: 'n set"
hoelzl@37489
  1484
  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
hoelzl@37489
  1485
  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
hoelzl@37489
  1486
    and xi: "x$i < 0"
hoelzl@37489
  1487
    from xi have th0: "-x$i > 0" by arith
hoelzl@37489
  1488
    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
hoelzl@37489
  1489
      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
hoelzl@37489
  1490
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
hoelzl@37489
  1491
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
hoelzl@37489
  1492
        apply (simp only: vector_component)
hoelzl@37489
  1493
        by (rule th') auto
hoelzl@37489
  1494
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm_cart[of "x'-x" i]
hoelzl@37489
  1495
        apply (simp add: dist_norm) by norm
hoelzl@37489
  1496
      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
hoelzl@37489
  1497
  then show ?thesis unfolding closed_limpt islimpt_approachable
hoelzl@37489
  1498
    unfolding not_le[symmetric] by blast
hoelzl@37489
  1499
qed
hoelzl@37489
  1500
lemma Lim_component_cart:
hoelzl@37489
  1501
  fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n"
hoelzl@37489
  1502
  shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
hoelzl@37489
  1503
  unfolding tendsto_iff
hoelzl@37489
  1504
  apply (clarify)
hoelzl@37489
  1505
  apply (drule spec, drule (1) mp)
hoelzl@37489
  1506
  apply (erule eventually_elim1)
hoelzl@37489
  1507
  apply (erule le_less_trans [OF dist_nth_le_cart])
hoelzl@37489
  1508
  done
hoelzl@37489
  1509
hoelzl@37489
  1510
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
hoelzl@37489
  1511
unfolding bounded_def
hoelzl@37489
  1512
apply clarify
hoelzl@37489
  1513
apply (rule_tac x="x $ i" in exI)
hoelzl@37489
  1514
apply (rule_tac x="e" in exI)
hoelzl@37489
  1515
apply clarify
hoelzl@37489
  1516
apply (rule order_trans [OF dist_nth_le_cart], simp)
hoelzl@37489
  1517
done
hoelzl@37489
  1518
hoelzl@37489
  1519
lemma compact_lemma_cart:
hoelzl@37489
  1520
  fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
hoelzl@37489
  1521
  assumes "bounded s" and "\<forall>n. f n \<in> s"
hoelzl@37489
  1522
  shows "\<forall>d.
hoelzl@37489
  1523
        \<exists>l r. subseq r \<and>
hoelzl@37489
  1524
        (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
  1525
proof
hoelzl@37489
  1526
  fix d::"'n set" have "finite d" by simp
hoelzl@37489
  1527
  thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
hoelzl@37489
  1528
      (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
hoelzl@37489
  1529
  proof(induct d) case empty thus ?case unfolding subseq_def by auto
hoelzl@37489
  1530
  next case (insert k d)
hoelzl@37489
  1531
    have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component_cart)
hoelzl@37489
  1532
    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"
hoelzl@37489
  1533
      using insert(3) by auto
hoelzl@37489
  1534
    have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
hoelzl@37489
  1535
    obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
hoelzl@37489
  1536
      using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
hoelzl@37489
  1537
    def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
hoelzl@37489
  1538
      using r1 and r2 unfolding r_def o_def subseq_def by auto
hoelzl@37489
  1539
    moreover
hoelzl@37489
  1540
    def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
hoelzl@37489
  1541
    { fix e::real assume "e>0"
hoelzl@37489
  1542
      from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
hoelzl@37489
  1543
      from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
hoelzl@37489
  1544
      from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
hoelzl@37489
  1545
        by (rule eventually_subseq)
hoelzl@37489
  1546
      have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
  1547
        using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
hoelzl@37489
  1548
    }
hoelzl@37489
  1549
    ultimately show ?case by auto
hoelzl@37489
  1550
  qed
hoelzl@37489
  1551
qed
hoelzl@37489
  1552
hoelzl@37489
  1553
instance cart :: (heine_borel, finite) heine_borel
hoelzl@37489
  1554
proof
hoelzl@37489
  1555
  fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
hoelzl@37489
  1556
  assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
hoelzl@37489
  1557
  then obtain l r where r: "subseq r"
hoelzl@37489
  1558
    and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
hoelzl@37489
  1559
    using compact_lemma_cart [OF s f] by blast
hoelzl@37489
  1560
  let ?d = "UNIV::'b set"
hoelzl@37489
  1561
  { fix e::real assume "e>0"
hoelzl@37489
  1562
    hence "0 < e / (real_of_nat (card ?d))"
hoelzl@37489
  1563
      using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
hoelzl@37489
  1564
    with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
hoelzl@37489
  1565
      by simp
hoelzl@37489
  1566
    moreover
hoelzl@37489
  1567
    { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
hoelzl@37489
  1568
      have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
hoelzl@37489
  1569
        unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
hoelzl@37489
  1570
      also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
hoelzl@37489
  1571
        by (rule setsum_strict_mono) (simp_all add: n)
hoelzl@37489
  1572
      finally have "dist (f (r n)) l < e" by simp
hoelzl@37489
  1573
    }
hoelzl@37489
  1574
    ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
hoelzl@37489
  1575
      by (rule eventually_elim1)
hoelzl@37489
  1576
  }
hoelzl@37489
  1577
  hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
hoelzl@37489
  1578
  with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
hoelzl@37489
  1579
qed
hoelzl@37489
  1580
hoelzl@37489
  1581
lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
hoelzl@37489
  1582
unfolding continuous_at by (intro tendsto_intros)
hoelzl@37489
  1583
hoelzl@37489
  1584
lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
hoelzl@37489
  1585
unfolding continuous_on_def by (intro ballI tendsto_intros)
hoelzl@37489
  1586
hoelzl@37489
  1587
lemma interval_cart: fixes a :: "'a::ord^'n" shows
hoelzl@37489
  1588
  "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
hoelzl@37489
  1589
  "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
hoelzl@37489
  1590
  by (auto simp add: expand_set_eq vector_less_def vector_le_def)
hoelzl@37489
  1591
hoelzl@37489
  1592
lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows
hoelzl@37489
  1593
  "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
hoelzl@37489
  1594
  "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
hoelzl@37489
  1595
  using interval_cart[of a b] by(auto simp add: expand_set_eq vector_less_def vector_le_def)
hoelzl@37489
  1596
hoelzl@37489
  1597
lemma interval_eq_empty_cart: fixes a :: "real^'n" shows
hoelzl@37489
  1598
 "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
hoelzl@37489
  1599
 "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
hoelzl@37489
  1600
proof-
hoelzl@37489
  1601
  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
hoelzl@37489
  1602
    hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1603
    hence "a$i < b$i" by auto
hoelzl@37489
  1604
    hence False using as by auto  }
hoelzl@37489
  1605
  moreover
hoelzl@37489
  1606
  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
hoelzl@37489
  1607
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1608
    { fix i
hoelzl@37489
  1609
      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1610
      hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
hoelzl@37489
  1611
        unfolding vector_smult_component and vector_add_component
hoelzl@37489
  1612
        by auto  }
hoelzl@37489
  1613
    hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto  }
hoelzl@37489
  1614
  ultimately show ?th1 by blast
hoelzl@37489
  1615
hoelzl@37489
  1616
  { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
hoelzl@37489
  1617
    hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hoelzl@37489
  1618
    hence "a$i \<le> b$i" by auto
hoelzl@37489
  1619
    hence False using as by auto  }
hoelzl@37489
  1620
  moreover
hoelzl@37489
  1621
  { assume as:"\<forall>i. \<not> (b$i < a$i)"
hoelzl@37489
  1622
    let ?x = "(1/2) *\<^sub>R (a + b)"
hoelzl@37489
  1623
    { fix i
hoelzl@37489
  1624
      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hoelzl@37489
  1625
      hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
hoelzl@37489
  1626
        unfolding vector_smult_component and vector_add_component
hoelzl@37489
  1627
        by auto  }
hoelzl@37489
  1628
    hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto  }
hoelzl@37489
  1629
  ultimately show ?th2 by blast
hoelzl@37489
  1630
qed
hoelzl@37489
  1631
hoelzl@37489
  1632
lemma interval_ne_empty_cart: fixes a :: "real^'n" shows
hoelzl@37489
  1633
  "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
hoelzl@37489
  1634
  "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
hoelzl@37489
  1635
  unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
hoelzl@37489
  1636
    (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1637
hoelzl@37489
  1638
lemma subset_interval_imp_cart: fixes a :: "real^'n" shows
hoelzl@37489
  1639
 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
hoelzl@37489
  1640
 "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
hoelzl@37489
  1641
 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
hoelzl@37489
  1642
 "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
hoelzl@37489
  1643
  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
hoelzl@37489
  1644
  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
hoelzl@37489
  1645
hoelzl@37489
  1646
lemma interval_sing: fixes a :: "'a::linorder^'n" shows
hoelzl@37489
  1647
 "{a .. a} = {a} \<and> {a<..<a} = {}"
hoelzl@37489
  1648
apply(auto simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
hoelzl@37489
  1649
apply (simp add: order_eq_iff)
hoelzl@37489
  1650
apply (auto simp add: not_less less_imp_le)
hoelzl@37489
  1651
done
hoelzl@37489
  1652
hoelzl@37489
  1653
lemma interval_open_subset_closed_cart:  fixes a :: "'a::preorder^'n" shows
hoelzl@37489
  1654
 "{a<..<b} \<subseteq> {a .. b}"
hoelzl@37489
  1655
proof(simp add: subset_eq, rule)
hoelzl@37489
  1656
  fix x
hoelzl@37489
  1657
  assume x:"x \<in>{a<..<b}"
hoelzl@37489
  1658
  { fix i
hoelzl@37489
  1659
    have "a $ i \<le> x $ i"
hoelzl@37489
  1660
      using x order_less_imp_le[of "a$i" "x$i"]
hoelzl@37489
  1661
      by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
hoelzl@37489
  1662
  }
hoelzl@37489
  1663
  moreover
hoelzl@37489
  1664
  { fix i
hoelzl@37489
  1665
    have "x $ i \<le> b $ i"
hoelzl@37489
  1666
      using x order_less_imp_le[of "x$i" "b$i"]
hoelzl@37489
  1667
      by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
hoelzl@37489
  1668
  }
hoelzl@37489
  1669
  ultimately
hoelzl@37489
  1670
  show "a \<le> x \<and> x \<le> b"
hoelzl@37489
  1671
    by(simp add: expand_set_eq vector_less_def vector_le_def Cart_eq)
hoelzl@37489
  1672
qed
hoelzl@37489
  1673
hoelzl@37489
  1674
lemma subset_interval_cart: fixes a :: "real^'n" shows
hoelzl@37489
  1675
 "{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) and
hoelzl@37489
  1676
 "{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) and
hoelzl@37489
  1677
 "{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) and
hoelzl@37489
  1678
 "{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@37489
  1679
  using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)
hoelzl@37489
  1680
hoelzl@37489
  1681
lemma disjoint_interval_cart: fixes a::"real^'n" shows
hoelzl@37489
  1682
  "{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) and
hoelzl@37489
  1683
  "{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) and
hoelzl@37489
  1684
  "{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) and
hoelzl@37489
  1685
  "{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@37489
  1686
  using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)
hoelzl@37489
  1687
hoelzl@37489
  1688
lemma inter_interval_cart: fixes a :: "'a::linorder^'n" shows
hoelzl@37489
  1689
 "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
hoelzl@37489
  1690
  unfolding expand_set_eq and Int_iff and mem_interval_cart
hoelzl@37489
  1691
  by auto
hoelzl@37489
  1692
hoelzl@37489
  1693
lemma closed_interval_left_cart: fixes b::"real^'n"
hoelzl@37489
  1694
  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
hoelzl@37489
  1695
proof-
hoelzl@37489
  1696
  { fix i
hoelzl@37489
  1697
    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
hoelzl@37489
  1698
    { assume "x$i > b$i"
hoelzl@37489
  1699
      then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto
hoelzl@37489
  1700
      hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
hoelzl@37489
  1701
    hence "x$i \<le> b$i" by(rule ccontr)auto  }
hoelzl@37489
  1702
  thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
hoelzl@37489
  1703
qed
hoelzl@37489
  1704
hoelzl@37489
  1705
lemma closed_interval_right_cart: fixes a::"real^'n"
hoelzl@37489
  1706
  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
hoelzl@37489
  1707
proof-
hoelzl@37489
  1708
  { fix i
hoelzl@37489
  1709
    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
hoelzl@37489
  1710
    { assume "a$i > x$i"
hoelzl@37489
  1711
      then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto
hoelzl@37489
  1712
      hence False using component_le_norm_cart[of "y - x" i] unfolding dist_norm and vector_minus_component by auto   }
hoelzl@37489
  1713
    hence "a$i \<le> x$i" by(rule ccontr)auto  }
hoelzl@37489
  1714
  thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
hoelzl@37489
  1715
qed
hoelzl@37489
  1716
hoelzl@37489
  1717
lemma is_interval_cart:"is_interval (s::(real^'n) set) \<longleftrightarrow>
hoelzl@37489
  1718
  (\<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@37489
  1719
  unfolding is_interval_def Ball_def by (simp add: cart_simps real_euclidean_nth)
hoelzl@37489
  1720
hoelzl@37489
  1721
lemma closed_halfspace_component_le_cart:
hoelzl@37489
  1722
  shows "closed {x::real^'n. x$i \<le> a}"
hoelzl@37489
  1723
  using closed_halfspace_le[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
hoelzl@37489
  1724
hoelzl@37489
  1725
lemma closed_halfspace_component_ge_cart:
hoelzl@37489
  1726
  shows "closed {x::real^'n. x$i \<ge> a}"
hoelzl@37489
  1727
  using closed_halfspace_ge[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
hoelzl@37489
  1728
hoelzl@37489
  1729
lemma open_halfspace_component_lt_cart:
hoelzl@37489
  1730
  shows "open {x::real^'n. x$i < a}"
hoelzl@37489
  1731
  using open_halfspace_lt[of "(cart_basis i)::real^'n" a] unfolding inner_basis[OF assms] by auto
hoelzl@37489
  1732
hoelzl@37489
  1733
lemma open_halfspace_component_gt_cart:
hoelzl@37489
  1734
  shows "open {x::real^'n. x$i  > a}"
hoelzl@37489
  1735
  using open_halfspace_gt[of a "(cart_basis i)::real^'n"] unfolding inner_basis[OF assms] by auto
hoelzl@37489
  1736
hoelzl@37489
  1737
lemma Lim_component_le_cart: fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1738
  assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
hoelzl@37489
  1739
  shows "l$i \<le> b"
hoelzl@37489
  1740
proof-
hoelzl@37489
  1741
  { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
hoelzl@37489
  1742
  show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l] unfolding *
hoelzl@37489
  1743
    using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto
hoelzl@37489
  1744
qed
hoelzl@37489
  1745
hoelzl@37489
  1746
lemma Lim_component_ge_cart: fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1747
  assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
hoelzl@37489
  1748
  shows "b \<le> l$i"
hoelzl@37489
  1749
proof-
hoelzl@37489
  1750
  { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
hoelzl@37489
  1751
  show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l] unfolding *
hoelzl@37489
  1752
    using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto
hoelzl@37489
  1753
qed
hoelzl@37489
  1754
hoelzl@37489
  1755
lemma Lim_component_eq_cart: fixes f :: "'a \<Rightarrow> real^'n"
hoelzl@37489
  1756
  assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
hoelzl@37489
  1757
  shows "l$i = b"
hoelzl@37489
  1758
  using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge_cart[OF net, of b i] and
hoelzl@37489
  1759
    Lim_component_le_cart[OF net, of i b] by auto
hoelzl@37489
  1760
hoelzl@37489
  1761
lemma connected_ivt_component_cart: fixes x::"real^'n" shows
hoelzl@37489
  1762
 "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@37489
  1763
  using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] by (auto simp add: inner_basis)
hoelzl@37489
  1764
hoelzl@37489
  1765
lemma subspace_substandard_cart:
hoelzl@37489
  1766
 "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
hoelzl@37489
  1767
  unfolding subspace_def by auto
hoelzl@37489
  1768
hoelzl@37489
  1769
lemma closed_substandard_cart:
hoelzl@37489
  1770
 "closed {x::real^'n. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
hoelzl@37489
  1771
proof-
hoelzl@37489
  1772
  let ?D = "{i. P i}"
hoelzl@37489
  1773
  let ?Bs = "{{x::real^'n. inner (cart_basis i) x = 0}| i. i \<in> ?D}"
hoelzl@37489
  1774
  { fix x
hoelzl@37489
  1775
    { assume "x\<in>?A"
hoelzl@37489
  1776
      hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
hoelzl@37489
  1777
      hence "x\<in> \<Inter> ?Bs" by(auto simp add: inner_basis x) }
hoelzl@37489
  1778
    moreover
hoelzl@37489
  1779
    { assume x:"x\<in>\<Inter>?Bs"
hoelzl@37489
  1780
      { fix i assume i:"i \<in> ?D"
hoelzl@37489
  1781
        then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. inner (cart_basis i) x = 0}" by auto
hoelzl@37489
  1782
        hence "x $ i = 0" unfolding B using x unfolding inner_basis by auto  }
hoelzl@37489
  1783
      hence "x\<in>?A" by auto }
hoelzl@37489
  1784
    ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" .. }
hoelzl@37489
  1785
  hence "?A = \<Inter> ?Bs" by auto
hoelzl@37489
  1786
  thus ?thesis by(auto simp add: closed_Inter closed_hyperplane)
hoelzl@37489
  1787
qed
hoelzl@37489
  1788
hoelzl@37489
  1789
lemma dim_substandard_cart:
hoelzl@37489
  1790
  shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
hoelzl@37489
  1791
proof- have *:"{x. \<forall>i<DIM((real, 'n) cart). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} = 
hoelzl@37489
  1792
    {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe
hoelzl@37489
  1793
    apply(erule_tac x="\<pi>' i" in allE) defer
hoelzl@37489
  1794
    apply(erule_tac x="\<pi> i" in allE)
hoelzl@37489
  1795
    unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])
hoelzl@37489
  1796
  have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) cart)}" using pi'_range[where 'n='n] by auto
hoelzl@37489
  1797
  thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"] 
hoelzl@37489
  1798
    unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto
hoelzl@37489
  1799
qed
hoelzl@37489
  1800
hoelzl@37489
  1801
lemma affinity_inverses:
hoelzl@37489
  1802
  assumes m0: "m \<noteq> (0::'a::field)"
hoelzl@37489
  1803
  shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
hoelzl@37489
  1804
  "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
hoelzl@37489
  1805
  using m0
hoelzl@37489
  1806
apply (auto simp add: expand_fun_eq vector_add_ldistrib)
hoelzl@37489
  1807
by (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
hoelzl@37489
  1808
hoelzl@37489
  1809
lemma vector_affinity_eq:
hoelzl@37489
  1810
  assumes m0: "(m::'a::field) \<noteq> 0"
hoelzl@37489
  1811
  shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
hoelzl@37489
  1812
proof
hoelzl@37489
  1813
  assume h: "m *s x + c = y"
hoelzl@37489
  1814
  hence "m *s x = y - c" by (simp add: field_simps)
hoelzl@37489
  1815
  hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
hoelzl@37489
  1816
  then show "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1817
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1818
next
hoelzl@37489
  1819
  assume h: "x = inverse m *s y + - (inverse m *s c)"
hoelzl@37489
  1820
  show "m *s x + c = y" unfolding h diff_minus[symmetric]
hoelzl@37489
  1821
    using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
hoelzl@37489
  1822
qed
hoelzl@37489
  1823
hoelzl@37489
  1824
lemma vector_eq_affinity:
hoelzl@37489
  1825
 "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
hoelzl@37489
  1826
  using vector_affinity_eq[where m=m and x=x and y=y and c=c]
hoelzl@37489
  1827
  by metis
hoelzl@37489
  1828
hoelzl@37489
  1829
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"
hoelzl@37489
  1830
  apply(subst euclidean_eq)
hoelzl@37489
  1831
proof safe case goal1
hoelzl@37489
  1832
  hence *:"(basis i::real^'n) = cart_basis (\<pi> i)"
hoelzl@37489
  1833
    unfolding basis_real_n[THEN sym] by auto
hoelzl@37489
  1834
  have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *
hoelzl@37489
  1835
    unfolding dot_basis by auto
hoelzl@37489
  1836
  thus ?case using goal1 by auto
hoelzl@37489
  1837
qed
hoelzl@37489
  1838
hoelzl@37489
  1839
section "Convex Euclidean Space"
hoelzl@37489
  1840
hoelzl@37489
  1841
lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"
hoelzl@37489
  1842
  apply(subst euclidean_eq)
hoelzl@37489
  1843
proof safe case goal1 thus ?case using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
hoelzl@37489
  1844
qed
hoelzl@37489
  1845
hoelzl@37489
  1846
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
hoelzl@37489
  1847
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
hoelzl@37489
  1848
hoelzl@37489
  1849
lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component
hoelzl@37489
  1850
hoelzl@37489
  1851
lemma convex_box_cart:
hoelzl@37489
  1852
  assumes "\<And>i. convex {x. P i x}"
hoelzl@37489
  1853
  shows "convex {x. \<forall>i. P i (x$i)}"
hoelzl@37489
  1854
  using assms unfolding convex_def by auto
hoelzl@37489
  1855
hoelzl@37489
  1856
lemma convex_positive_orthant_cart: "convex {x::real^'n. (\<forall>i. 0 \<le> x$i)}"
hoelzl@37489
  1857
  by (rule convex_box_cart) (simp add: atLeast_def[symmetric] convex_real_interval)
hoelzl@37489
  1858
hoelzl@37489
  1859
lemma unit_interval_convex_hull_cart:
hoelzl@37489
  1860
  "{0::real^'n .. 1} = convex hull {x. \<forall>i. (x$i = 0) \<or> (x$i = 1)}" (is "?int = convex hull ?points")
hoelzl@37489
  1861
  unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
hoelzl@37489
  1862
  apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_ext) unfolding mem_Collect_eq
hoelzl@37489
  1863
  apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer
hoelzl@37489
  1864
  apply(erule_tac x="\<pi> i" in allE) by auto
hoelzl@37489
  1865
hoelzl@37489
  1866
lemma cube_convex_hull_cart:
hoelzl@37489
  1867
  assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s" 
hoelzl@37489
  1868
proof- from cube_convex_hull[OF assms, where 'a="real^'n" and x=x] guess s . note s=this
hoelzl@37489
  1869
  show thesis apply(rule that[OF s(1)]) unfolding s(2)[THEN sym] const_vector_cart ..
hoelzl@37489
  1870
qed
hoelzl@37489
  1871
hoelzl@37489
  1872
lemma std_simplex_cart:
hoelzl@37489
  1873
  "(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
hoelzl@37489
  1874
  (insert 0 { basis i | i. i<DIM((real,'n) cart)})"
hoelzl@37489
  1875
  apply(rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
hoelzl@37489
  1876
  unfolding basis_real_n[THEN sym] apply safe
hoelzl@37489
  1877
  apply(rule_tac x="\<pi>' i" in exI) defer
hoelzl@37489
  1878
  apply(rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n] by auto
hoelzl@37489
  1879
hoelzl@37489
  1880
subsection "Brouwer Fixpoint"
hoelzl@37489
  1881
hoelzl@37489
  1882
lemma kuhn_labelling_lemma_cart:
hoelzl@37489
  1883
  assumes "(\<forall>x::real^_. P x \<longrightarrow> P (f x))"  "\<forall>x. P x \<longrightarrow> (\<forall>i. Q i \<longrightarrow> 0 \<le> x$i \<and> x$i \<le> 1)"
hoelzl@37489
  1884
  shows "\<exists>l. (\<forall>x i. l x i \<le> (1::nat)) \<and>
hoelzl@37489
  1885
             (\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
hoelzl@37489
  1886
             (\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
hoelzl@37489
  1887
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
hoelzl@37489
  1888
             (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-
hoelzl@37489
  1889
  have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
hoelzl@37489
  1890
  have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto
hoelzl@37489
  1891
  show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
hoelzl@37489
  1892
    let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
hoelzl@37489
  1893
        (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
hoelzl@37489
  1894
    { assume "P x" "Q xa" hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
hoelzl@37489
  1895
        apply(drule_tac assms(1)[rule_format]) by auto }
hoelzl@37489
  1896
    hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed 
hoelzl@37489
  1897
hoelzl@37489
  1898
lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
hoelzl@37489
  1899
    (\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
hoelzl@37489
  1900
  unfolding interval_bij_def apply(rule ext)+ apply safe
hoelzl@37489
  1901
  unfolding Cart_eq Cart_lambda_beta unfolding nth_conv_component
hoelzl@37489
  1902
  apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto
hoelzl@37489
  1903
hoelzl@37489
  1904
lemma interval_bij_affine_cart:
hoelzl@37489
  1905
 "interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
hoelzl@37489
  1906
            (\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
hoelzl@37489
  1907
  apply rule unfolding Cart_eq interval_bij_cart vector_component_simps
hoelzl@37489
  1908
  by(auto simp add: field_simps add_divide_distrib[THEN sym]) 
hoelzl@37489
  1909
hoelzl@37489
  1910
subsection "Derivative"
hoelzl@37489
  1911
hoelzl@37489
  1912
lemma has_derivative_vmul_component_cart: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
hoelzl@37489
  1913
  assumes "(c has_derivative c') net"
hoelzl@37489
  1914
  shows "((\<lambda>x. c(x)$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$k *\<^sub>R v)) net" 
hoelzl@37489
  1915
  using has_derivative_vmul_component[OF assms] 
hoelzl@37489
  1916
  unfolding nth_conv_component .
hoelzl@37489
  1917
hoelzl@37489
  1918
lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
hoelzl@37489
  1919
  unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
hoelzl@37489
  1920
hoelzl@37489
  1921
definition "jacobian f net = matrix(frechet_derivative f net)"
hoelzl@37489
  1922
hoelzl@37489
  1923
lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
hoelzl@37489
  1924
  apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
hoelzl@37489
  1925
  apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
hoelzl@37489
  1926
hoelzl@37489
  1927
subsection {* Component of the differential must be zero if it exists at a local        *)
hoelzl@37489
  1928
(* maximum or minimum for that corresponding component. *}
hoelzl@37489
  1929
hoelzl@37489
  1930
lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"
hoelzl@37489
  1931
  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@37489
  1932
  "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
hoelzl@37489
  1933
(* FIXME: reuse proof of generic differential_zero_maxmin_component*)
hoelzl@37489
  1934
hoelzl@37489
  1935
proof(rule ccontr)
hoelzl@37489
  1936
  def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
hoelzl@37489
  1937
  then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
hoelzl@37489
  1938
  hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
hoelzl@37489
  1939
  note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
hoelzl@37489
  1940
  guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
hoelzl@37489
  1941
  guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
hoelzl@37489
  1942
  { fix c assume "abs c \<le> d" 
hoelzl@37489
  1943
    hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" using norm_basis[of j] d by auto
hoelzl@37489
  1944
    have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))" 
hoelzl@37489
  1945
      by(rule component_le_norm_cart)
hoelzl@37489
  1946
    also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
hoelzl@37489
  1947
    finally have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
hoelzl@37489
  1948
    hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
hoelzl@37489
  1949
      unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric] 
hoelzl@37489
  1950
      unfolding inner_simps dot_basis smult_conv_scaleR by simp  } note * = this
hoelzl@37489
  1951
  have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"
hoelzl@37489
  1952
    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
hoelzl@37489
  1953
  hence **:"((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
hoelzl@37489
  1954
         ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)" using assms(2) by auto
hoelzl@37489
  1955
  have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
hoelzl@37489
  1956
  show False apply(rule ***[OF **, where dx="d * D $ k $ j" and d="\<bar>D $ k $ j\<bar> / 2 * \<bar>d\<bar>"]) 
hoelzl@37489
  1957
    using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
hoelzl@37489
  1958
    unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
hoelzl@37489
  1959
qed
hoelzl@37489
  1960
hoelzl@37494
  1961
subsection {* Lemmas for working on @{typ "real^1"} *}
hoelzl@37489
  1962
hoelzl@37489
  1963
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
hoelzl@37489
  1964
  by (metis num1_eq_iff)
hoelzl@37489
  1965
hoelzl@37489
  1966
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
hoelzl@37489
  1967
  by auto (metis num1_eq_iff)
hoelzl@37489
  1968
hoelzl@37489
  1969
lemma exhaust_2:
hoelzl@37489
  1970
  fixes x :: 2 shows "x = 1 \<or> x = 2"
hoelzl@37489
  1971
proof (induct x)
hoelzl@37489
  1972
  case (of_int z)
hoelzl@37489
  1973
  then have "0 <= z" and "z < 2" by simp_all
hoelzl@37489
  1974
  then have "z = 0 | z = 1" by arith
hoelzl@37489
  1975
  then show ?case by auto
hoelzl@37489
  1976
qed
hoelzl@37489
  1977
hoelzl@37489
  1978
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
hoelzl@37489
  1979
  by (metis exhaust_2)
hoelzl@37489
  1980
hoelzl@37489
  1981
lemma exhaust_3:
hoelzl@37489
  1982
  fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
hoelzl@37489
  1983
proof (induct x)
hoelzl@37489
  1984
  case (of_int z)
hoelzl@37489
  1985
  then have "0 <= z" and "z < 3" by simp_all
hoelzl@37489
  1986
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
hoelzl@37489
  1987
  then show ?case by auto
hoelzl@37489
  1988
qed
hoelzl@37489
  1989
hoelzl@37489
  1990
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
hoelzl@37489
  1991
  by (metis exhaust_3)
hoelzl@37489
  1992
hoelzl@37489
  1993
lemma UNIV_1 [simp]: "UNIV = {1::1}"
hoelzl@37489
  1994
  by (auto simp add: num1_eq_iff)
hoelzl@37489
  1995
hoelzl@37489
  1996
lemma UNIV_2: "UNIV = {1::2, 2::2}"
hoelzl@37489
  1997
  using exhaust_2 by auto
hoelzl@37489
  1998
hoelzl@37489
  1999
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
hoelzl@37489
  2000
  using exhaust_3 by auto
hoelzl@37489
  2001
hoelzl@37489
  2002
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
hoelzl@37489
  2003
  unfolding UNIV_1 by simp
hoelzl@37489
  2004
hoelzl@37489
  2005
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
hoelzl@37489
  2006
  unfolding UNIV_2 by simp
hoelzl@37489
  2007
hoelzl@37489
  2008
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
hoelzl@37489
  2009
  unfolding UNIV_3 by (simp add: add_ac)
hoelzl@37489
  2010
hoelzl@37489
  2011
instantiation num1 :: cart_one begin
hoelzl@37489
  2012
instance proof
hoelzl@37489
  2013
  show "CARD(1) = Suc 0" by auto
hoelzl@37489
  2014
qed end
hoelzl@37489
  2015
hoelzl@37489
  2016
(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
hoelzl@37489
  2017
hoelzl@37489
  2018
abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"
hoelzl@37489
  2019
hoelzl@37489
  2020
abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a"
hoelzl@37489
  2021
  where "dest_vec1 x \<equiv> (x$1)"
hoelzl@37489
  2022
hoelzl@37489
  2023
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
hoelzl@37489
  2024
  by (simp_all add:  Cart_eq)
hoelzl@37489
  2025
hoelzl@37489
  2026
lemma vec1_component[simp]: "(vec1 x)$1 = x"
hoelzl@37489
  2027
  by (simp_all add:  Cart_eq)
hoelzl@37489
  2028
hoelzl@37489
  2029
declare vec1_dest_vec1(1) [simp]
hoelzl@37489
  2030
hoelzl@37489
  2031
lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))"
hoelzl@37489
  2032
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  2033
hoelzl@37489
  2034
lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))"
hoelzl@37489
  2035
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  2036
hoelzl@37489
  2037
lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y"
hoelzl@37489
  2038
  by (metis vec1_dest_vec1(2))
hoelzl@37489
  2039
hoelzl@37489
  2040
lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
hoelzl@37489
  2041
  by (metis vec1_dest_vec1(1))
hoelzl@37489
  2042
hoelzl@37489
  2043
subsection{* The collapse of the general concepts to dimension one. *}
hoelzl@37489
  2044
hoelzl@37489
  2045
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
hoelzl@37489
  2046
  by (simp add: Cart_eq)
hoelzl@37489
  2047
hoelzl@37489
  2048
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
hoelzl@37489
  2049
  apply auto
hoelzl@37489
  2050
  apply (erule_tac x= "x$1" in allE)
hoelzl@37489
  2051
  apply (simp only: vector_one[symmetric])
hoelzl@37489
  2052
  done
hoelzl@37489
  2053
hoelzl@37489
  2054
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
hoelzl@37489
  2055
  by (simp add: norm_vector_def)
hoelzl@37489
  2056
hoelzl@37489
  2057
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
hoelzl@37489
  2058
  by (simp add: norm_vector_1)
hoelzl@37489
  2059
hoelzl@37489
  2060
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
hoelzl@37489
  2061
  by (auto simp add: norm_real dist_norm)
hoelzl@37489
  2062
hoelzl@37489
  2063
subsection{* Explicit vector construction from lists. *}
hoelzl@37489
  2064
hoelzl@37489
  2065
primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
hoelzl@37489
  2066
where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
hoelzl@37489
  2067
hoelzl@37489
  2068
lemma from_nat [simp]: "from_nat = of_nat"
hoelzl@37489
  2069
by (rule ext, induct_tac x, simp_all)
hoelzl@37489
  2070
hoelzl@37489
  2071
primrec
hoelzl@37489
  2072
  list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
hoelzl@37489
  2073
where
hoelzl@37489
  2074
  "list_fun n [] = (\<lambda>x. 0)"
hoelzl@37489
  2075
| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
hoelzl@37489
  2076
hoelzl@37489
  2077
definition "vector l = (\<chi> i. list_fun 1 l i)"
hoelzl@37489
  2078
(*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
hoelzl@37489
  2079
hoelzl@37489
  2080
lemma vector_1: "(vector[x]) $1 = x"
hoelzl@37489
  2081
  unfolding vector_def by simp
hoelzl@37489
  2082
hoelzl@37489
  2083
lemma vector_2: