src/HOL/Multivariate_Analysis/Determinants.thy

simpler proof

(*  Title:      HOL/Multivariate_Analysis/Determinants.thy
    Author:     Amine Chaieb, University of Cambridge
*)

header {* Traces, Determinant of square matrices and some properties *}

theory Determinants
imports
  Cartesian_Euclidean_Space
  "~~/src/HOL/Library/Permutations"
begin

subsection{* First some facts about products*}

lemma setprod_add_split:
  fixes m n :: nat
  assumes mn: "m \<le> n + 1"
  shows "setprod f {m..n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
proof -
  let ?A = "{m..n+p}"
  let ?B = "{m..n}"
  let ?C = "{n+1..n+p}"
  from mn have un: "?B \<union> ?C = ?A"
    by auto
  from mn have dj: "?B \<inter> ?C = {}"
    by auto
  have f: "finite ?B" "finite ?C"
    by simp_all
  from setprod.union_disjoint[OF f dj, of f, unfolded un] show ?thesis .
qed


lemma setprod_offset:
  fixes m n :: nat
  shows "setprod f {m + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
  by (rule setprod.reindex_bij_witness[where i="op + p" and j="\<lambda>i. i - p"]) auto

lemma setprod_singleton: "setprod f {x} = f x"
  by simp

lemma setprod_singleton_nat_seg:
  fixes n :: "'a::order"
  shows "setprod f {n..n} = f n"
  by simp

lemma setprod_numseg:
  "setprod f {m..0} = (if m = 0 then f 0 else 1)"
  "setprod f {m .. Suc n} =
    (if m \<le> Suc n then f (Suc n) * setprod f {m..n} else setprod f {m..n})"
  by (auto simp add: atLeastAtMostSuc_conv)

lemma setprod_le:
  fixes f g :: "'b \<Rightarrow> 'a::linordered_idom"
  assumes fS: "finite S"
    and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> g x"
  shows "setprod f S \<le> setprod g S"
  using fS fg
  apply (induct S)
  apply simp
  apply auto
  apply (rule mult_mono)
  apply (auto intro: setprod_nonneg)
  done

(* FIXME: In Finite_Set there is a useless further assumption *)
lemma setprod_inversef:
  "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: field_inverse_zero)"
  apply (erule finite_induct)
  apply (simp)
  apply simp
  done

lemma setprod_le_1:
  fixes f :: "'b \<Rightarrow> 'a::linordered_idom"
  assumes fS: "finite S"
    and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> 1"
  shows "setprod f S \<le> 1"
  using setprod_le[OF fS f] unfolding setprod.neutral_const .


subsection {* Trace *}

definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
  where "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"

lemma trace_0: "trace (mat 0) = 0"
  by (simp add: trace_def mat_def)

lemma trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
  by (simp add: trace_def mat_def)

lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
  by (simp add: trace_def setsum.distrib)

lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
  by (simp add: trace_def setsum_subtractf)

lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
  apply (simp add: trace_def matrix_matrix_mult_def)
  apply (subst setsum.commute)
  apply (simp add: mult.commute)
  done

text {* Definition of determinant. *}

definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
  "det A =
    setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))
      {p. p permutes (UNIV :: 'n set)}"

text {* A few general lemmas we need below. *}

lemma setprod_permute:
  assumes p: "p permutes S"
  shows "setprod f S = setprod (f \<circ> p) S"
  using assms by (fact setprod.permute)

lemma setproduct_permute_nat_interval:
  fixes m n :: nat
  shows "p permutes {m..n} \<Longrightarrow> setprod f {m..n} = setprod (f \<circ> p) {m..n}"
  by (blast intro!: setprod_permute)

text {* Basic determinant properties. *}

lemma det_transpose: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
proof -
  let ?di = "\<lambda>A i j. A$i$j"
  let ?U = "(UNIV :: 'n set)"
  have fU: "finite ?U" by simp
  {
    fix p
    assume p: "p \<in> {p. p permutes ?U}"
    from p have pU: "p permutes ?U"
      by blast
    have sth: "sign (inv p) = sign p"
      by (metis sign_inverse fU p mem_Collect_eq permutation_permutes)
    from permutes_inj[OF pU]
    have pi: "inj_on p ?U"
      by (blast intro: subset_inj_on)
    from permutes_image[OF pU]
    have "setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U =
      setprod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)"
      by simp
    also have "\<dots> = setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) ?U"
      unfolding setprod.reindex[OF pi] ..
    also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
    proof -
      {
        fix i
        assume i: "i \<in> ?U"
        from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
        have "((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) i = ?di A i (p i)"
          unfolding transpose_def by (simp add: fun_eq_iff)
      }
      then show "setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) ?U =
        setprod (\<lambda>i. ?di A i (p i)) ?U"
        by (auto intro: setprod.cong)
    qed
    finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U) =
      of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)"
      using sth by simp
  }
  then show ?thesis
    unfolding det_def
    apply (subst setsum_permutations_inverse)
    apply (rule setsum.cong)
    apply (rule refl)
    apply blast
    done
qed

lemma det_lowerdiagonal:
  fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
  assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
  shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
proof -
  let ?U = "UNIV:: 'n set"
  let ?PU = "{p. p permutes ?U}"
  let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
  have fU: "finite ?U"
    by simp
  from finite_permutations[OF fU] have fPU: "finite ?PU" .
  have id0: "{id} \<subseteq> ?PU"
    by (auto simp add: permutes_id)
  {
    fix p
    assume p: "p \<in> ?PU - {id}"
    from p have pU: "p permutes ?U" and pid: "p \<noteq> id"
      by blast+
    from permutes_natset_le[OF pU] pid obtain i where i: "p i > i"
      by (metis not_le)
    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0"
      by blast
    from setprod_zero[OF fU ex] have "?pp p = 0"
      by simp
  }
  then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
    by blast
  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
    unfolding det_def by (simp add: sign_id)
qed

lemma det_upperdiagonal:
  fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
  assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
  shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
proof -
  let ?U = "UNIV:: 'n set"
  let ?PU = "{p. p permutes ?U}"
  let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
  have fU: "finite ?U"
    by simp
  from finite_permutations[OF fU] have fPU: "finite ?PU" .
  have id0: "{id} \<subseteq> ?PU"
    by (auto simp add: permutes_id)
  {
    fix p
    assume p: "p \<in> ?PU - {id}"
    from p have pU: "p permutes ?U" and pid: "p \<noteq> id"
      by blast+
    from permutes_natset_ge[OF pU] pid obtain i where i: "p i < i"
      by (metis not_le)
    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0"
      by blast
    from setprod_zero[OF fU ex] have "?pp p = 0"
      by simp
  }
  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
    by blast
  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
    unfolding det_def by (simp add: sign_id)
qed

lemma det_diagonal:
  fixes A :: "'a::comm_ring_1^'n^'n"
  assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
  shows "det A = setprod (\<lambda>i. A$i$i) (UNIV::'n set)"
proof -
  let ?U = "UNIV:: 'n set"
  let ?PU = "{p. p permutes ?U}"
  let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
  have fU: "finite ?U" by simp
  from finite_permutations[OF fU] have fPU: "finite ?PU" .
  have id0: "{id} \<subseteq> ?PU"
    by (auto simp add: permutes_id)
  {
    fix p
    assume p: "p \<in> ?PU - {id}"
    then have "p \<noteq> id"
      by simp
    then obtain i where i: "p i \<noteq> i"
      unfolding fun_eq_iff by auto
    from ld [OF i [symmetric]] have ex:"\<exists>i \<in> ?U. A$i$p i = 0"
      by blast
    from setprod_zero [OF fU ex] have "?pp p = 0"
      by simp
  }
  then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
    by blast
  from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
    unfolding det_def by (simp add: sign_id)
qed

lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
proof -
  let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
  let ?U = "UNIV :: 'n set"
  let ?f = "\<lambda>i j. ?A$i$j"
  {
    fix i
    assume i: "i \<in> ?U"
    have "?f i i = 1"
      using i by (vector mat_def)
  }
  then have th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
    by (auto intro: setprod.cong)
  {
    fix i j
    assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
    have "?f i j = 0" using i j ij
      by (vector mat_def)
  }
  then have "det ?A = setprod (\<lambda>i. ?f i i) ?U"
    using det_diagonal by blast
  also have "\<dots> = 1"
    unfolding th setprod.neutral_const ..
  finally show ?thesis .
qed

lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
  by (simp add: det_def setprod_zero)

lemma det_permute_rows:
  fixes A :: "'a::comm_ring_1^'n^'n"
  assumes p: "p permutes (UNIV :: 'n::finite set)"
  shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
  apply (simp add: det_def setsum_right_distrib mult.assoc[symmetric])
  apply (subst sum_permutations_compose_right[OF p])
proof (rule setsum.cong)
  let ?U = "UNIV :: 'n set"
  let ?PU = "{p. p permutes ?U}"
  fix q
  assume qPU: "q \<in> ?PU"
  have fU: "finite ?U"
    by simp
  from qPU have q: "q permutes ?U"
    by blast
  from p q have pp: "permutation p" and qp: "permutation q"
    by (metis fU permutation_permutes)+
  from permutes_inv[OF p] have ip: "inv p permutes ?U" .
  have "setprod (\<lambda>i. A$p i$ (q \<circ> p) i) ?U = setprod ((\<lambda>i. A$p i$(q \<circ> p) i) \<circ> inv p) ?U"
    by (simp only: setprod_permute[OF ip, symmetric])
  also have "\<dots> = setprod (\<lambda>i. A $ (p \<circ> inv p) i $ (q \<circ> (p \<circ> inv p)) i) ?U"
    by (simp only: o_def)
  also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U"
    by (simp only: o_def permutes_inverses[OF p])
  finally have thp: "setprod (\<lambda>i. A$p i$ (q \<circ> p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
    by blast
  show "of_int (sign (q \<circ> p)) * setprod (\<lambda>i. A$ p i$ (q \<circ> p) i) ?U =
    of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
    by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult)
qed rule

lemma det_permute_columns:
  fixes A :: "'a::comm_ring_1^'n^'n"
  assumes p: "p permutes (UNIV :: 'n set)"
  shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
proof -
  let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
  let ?At = "transpose A"
  have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A $ p i))"
    unfolding det_permute_rows[OF p, of ?At] det_transpose ..
  moreover
  have "?Ap = transpose (\<chi> i. transpose A $ p i)"
    by (simp add: transpose_def vec_eq_iff)
  ultimately show ?thesis
    by simp
qed

lemma det_identical_rows:
  fixes A :: "'a::linordered_idom^'n^'n"
  assumes ij: "i \<noteq> j"
    and r: "row i A = row j A"
  shows "det A = 0"
proof-
  have tha: "\<And>(a::'a) b. a = b \<Longrightarrow> b = - a \<Longrightarrow> a = 0"
    by simp
  have th1: "of_int (-1) = - 1" by simp
  let ?p = "Fun.swap i j id"
  let ?A = "\<chi> i. A $ ?p i"
  from r have "A = ?A" by (simp add: vec_eq_iff row_def Fun.swap_def)
  then have "det A = det ?A" by simp
  moreover have "det A = - det ?A"
    by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
  ultimately show "det A = 0" by (metis tha)
qed

lemma det_identical_columns:
  fixes A :: "'a::linordered_idom^'n^'n"
  assumes ij: "i \<noteq> j"
    and r: "column i A = column j A"
  shows "det A = 0"
  apply (subst det_transpose[symmetric])
  apply (rule det_identical_rows[OF ij])
  apply (metis row_transpose r)
  done

lemma det_zero_row:
  fixes A :: "'a::{idom, ring_char_0}^'n^'n"
  assumes r: "row i A = 0"
  shows "det A = 0"
  using r
  apply (simp add: row_def det_def vec_eq_iff)
  apply (rule setsum.neutral)
  apply (auto simp: sign_nz)
  done

lemma det_zero_column:
  fixes A :: "'a::{idom,ring_char_0}^'n^'n"
  assumes r: "column i A = 0"
  shows "det A = 0"
  apply (subst det_transpose[symmetric])
  apply (rule det_zero_row [of i])
  apply (metis row_transpose r)
  done

lemma det_row_add:
  fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
  shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
    det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
    det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
  unfolding det_def vec_lambda_beta setsum.distrib[symmetric]
proof (rule setsum.cong)
  let ?U = "UNIV :: 'n set"
  let ?pU = "{p. p permutes ?U}"
  let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
  let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
  let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
  fix p
  assume p: "p \<in> ?pU"
  let ?Uk = "?U - {k}"
  from p have pU: "p permutes ?U"
    by blast
  have kU: "?U = insert k ?Uk"
    by blast
  {
    fix j
    assume j: "j \<in> ?Uk"
    from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
      by simp_all
  }
  then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
    and th2: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?h i $ p i) ?Uk"
    apply -
    apply (rule setprod.cong, simp_all)+
    done
  have th3: "finite ?Uk" "k \<notin> ?Uk"
    by auto
  have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
    unfolding kU[symmetric] ..
  also have "\<dots> = ?f k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk"
    apply (rule setprod.insert)
    apply simp
    apply blast
    done
  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)"
    by (simp add: field_simps)
  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?h i $ p i) ?Uk)"
    by (metis th1 th2)
  also have "\<dots> = setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i $ p i) (insert k ?Uk)"
    unfolding  setprod.insert[OF th3] by simp
  finally have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?g i $ p i) ?U + setprod (\<lambda>i. ?h i $ p i) ?U"
    unfolding kU[symmetric] .
  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U =
    of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i $ p i) ?U"
    by (simp add: field_simps)
qed rule

lemma det_row_mul:
  fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
  shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
    c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
  unfolding det_def vec_lambda_beta setsum_right_distrib
proof (rule setsum.cong)
  let ?U = "UNIV :: 'n set"
  let ?pU = "{p. p permutes ?U}"
  let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
  let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
  fix p
  assume p: "p \<in> ?pU"
  let ?Uk = "?U - {k}"
  from p have pU: "p permutes ?U"
    by blast
  have kU: "?U = insert k ?Uk"
    by blast
  {
    fix j
    assume j: "j \<in> ?Uk"
    from j have "?f j $ p j = ?g j $ p j"
      by simp
  }
  then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
    apply -
    apply (rule setprod.cong)
    apply simp_all
    done
  have th3: "finite ?Uk" "k \<notin> ?Uk"
    by auto
  have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
    unfolding kU[symmetric] ..
  also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
    apply (rule setprod.insert)
    apply simp
    apply blast
    done
  also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk"
    by (simp add: field_simps)
  also have "\<dots> = c* (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk)"
    unfolding th1 by (simp add: ac_simps)
  also have "\<dots> = c* (setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk))"
    unfolding setprod.insert[OF th3] by simp
  finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)"
    unfolding kU[symmetric] .
  then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U =
    c * (of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U)"
    by (simp add: field_simps)
qed rule

lemma det_row_0:
  fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
  shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
  using det_row_mul[of k 0 "\<lambda>i. 1" b]
  apply simp
  apply (simp only: vector_smult_lzero)
  done

lemma det_row_operation:
  fixes A :: "'a::linordered_idom^'n^'n"
  assumes ij: "i \<noteq> j"
  shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
proof -
  let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
  have th: "row i ?Z = row j ?Z" by (vector row_def)
  have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
    by (vector row_def)
  show ?thesis
    unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2
    by simp
qed

lemma det_row_span:
  fixes A :: "real^'n^'n"
  assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
  shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
proof -
  let ?U = "UNIV :: 'n set"
  let ?S = "{row j A |j. j \<noteq> i}"
  let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
  let ?P = "\<lambda>x. ?d (row i A + x) = det A"
  {
    fix k
    have "(if k = i then row i A + 0 else row k A) = row k A"
      by simp
  }
  then have P0: "?P 0"
    apply -
    apply (rule cong[of det, OF refl])
    apply (vector row_def)
    done
  moreover
  {
    fix c z y
    assume zS: "z \<in> ?S" and Py: "?P y"
    from zS obtain j where j: "z = row j A" "i \<noteq> j"
      by blast
    let ?w = "row i A + y"
    have th0: "row i A + (c*s z + y) = ?w + c*s z"
      by vector
    have thz: "?d z = 0"
      apply (rule det_identical_rows[OF j(2)])
      using j
      apply (vector row_def)
      done
    have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)"
      unfolding th0 ..
    then have "?P (c*s z + y)"
      unfolding thz Py det_row_mul[of i] det_row_add[of i]
      by simp
  }
  ultimately show ?thesis
    apply -
    apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR])
    apply blast
    apply (rule x)
    done
qed

text {*
  May as well do this, though it's a bit unsatisfactory since it ignores
  exact duplicates by considering the rows/columns as a set.
*}

lemma det_dependent_rows:
  fixes A:: "real^'n^'n"
  assumes d: "dependent (rows A)"
  shows "det A = 0"
proof -
  let ?U = "UNIV :: 'n set"
  from d obtain i where i: "row i A \<in> span (rows A - {row i A})"
    unfolding dependent_def rows_def by blast
  {
    fix j k
    assume jk: "j \<noteq> k" and c: "row j A = row k A"
    from det_identical_rows[OF jk c] have ?thesis .
  }
  moreover
  {
    assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
    have th0: "- row i A \<in> span {row j A|j. j \<noteq> i}"
      apply (rule span_neg)
      apply (rule set_rev_mp)
      apply (rule i)
      apply (rule span_mono)
      using H i
      apply (auto simp add: rows_def)
      done
    from det_row_span[OF th0]
    have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
      unfolding right_minus vector_smult_lzero ..
    with det_row_mul[of i "0::real" "\<lambda>i. 1"]
    have "det A = 0" by simp
  }
  ultimately show ?thesis by blast
qed

lemma det_dependent_columns:
  assumes d: "dependent (columns (A::real^'n^'n))"
  shows "det A = 0"
  by (metis d det_dependent_rows rows_transpose det_transpose)

text {* Multilinearity and the multiplication formula. *}

lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
  by (rule iffD1[OF vec_lambda_unique]) vector

lemma det_linear_row_setsum:
  assumes fS: "finite S"
  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
    setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
proof (induct rule: finite_induct[OF fS])
  case 1
  then show ?case
    apply simp
    unfolding setsum.empty det_row_0[of k]
    apply rule
    done
next
  case (2 x F)
  then show ?case
    by (simp add: det_row_add cong del: if_weak_cong)
qed

lemma finite_bounded_functions:
  assumes fS: "finite S"
  shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
proof (induct k)
  case 0
  have th: "{f. \<forall>i. f i = i} = {id}"
    by auto
  show ?case
    by (auto simp add: th)
next
  case (Suc k)
  let ?f = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
  let ?S = "?f ` (S \<times> {f. (\<forall>i\<in>{1..k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)})"
  have "?S = {f. (\<forall>i\<in>{1.. Suc k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1.. Suc k} \<longrightarrow> f i = i)}"
    apply (auto simp add: image_iff)
    apply (rule_tac x="x (Suc k)" in bexI)
    apply (rule_tac x = "\<lambda>i. if i = Suc k then i else x i" in exI)
    apply auto
    done
  with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
  show ?case
    by metis
qed


lemma eq_id_iff[simp]: "(\<forall>x. f x = x) \<longleftrightarrow> f = id"
  by auto

lemma det_linear_rows_setsum_lemma:
  assumes fS: "finite S"
    and fT: "finite T"
  shows "det ((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
    setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
      {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
  using fT
proof (induct T arbitrary: a c set: finite)
  case empty
  have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
    by vector
  from empty.prems show ?case
    unfolding th0 by simp
next
  case (insert z T a c)
  let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
  let ?h = "\<lambda>(y,g) i. if i = z then y else g i"
  let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
  let ?s = "\<lambda> k a c f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n)"
  let ?c = "\<lambda>j i. if i = z then a i j else c i"
  have thif: "\<And>a b c d. (if a \<or> b then c else d) = (if a then c else if b then c else d)"
    by simp
  have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
     (if c then (if a then b else d) else (if a then b else e))"
    by simp
  from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False"
    by auto
  have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
    det (\<chi> i. if i = z then setsum (a i) S else if i \<in> T then setsum (a i) S else c i)"
    unfolding insert_iff thif ..
  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S else if i = z then a i j else c i))"
    unfolding det_linear_row_setsum[OF fS]
    apply (subst thif2)
    using nz
    apply (simp cong del: if_weak_cong cong add: if_cong)
    done
  finally have tha:
    "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
     (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
                                else if i = z then a i j
                                else c i))"
    unfolding insert.hyps unfolding setsum.cartesian_product by blast
  show ?case unfolding tha
    using `z \<notin> T`
    by (intro setsum.reindex_bij_witness[where i="?k" and j="?h"])
       (auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
qed

lemma det_linear_rows_setsum:
  fixes S :: "'n::finite set"
  assumes fS: "finite S"
  shows "det (\<chi> i. setsum (a i) S) =
    setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
proof -
  have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)"
    by vector
  from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
  show ?thesis by simp
qed

lemma matrix_mul_setsum_alt:
  fixes A B :: "'a::comm_ring_1^'n^'n"
  shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
  by (vector matrix_matrix_mult_def setsum_component)

lemma det_rows_mul:
  "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
    setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
proof (simp add: det_def setsum_right_distrib cong add: setprod.cong, rule setsum.cong)
  let ?U = "UNIV :: 'n set"
  let ?PU = "{p. p permutes ?U}"
  fix p
  assume pU: "p \<in> ?PU"
  let ?s = "of_int (sign p)"
  from pU have p: "p permutes ?U"
    by blast
  have "setprod (\<lambda>i. c i * a i $ p i) ?U = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
    unfolding setprod.distrib ..
  then show "?s * (\<Prod>xa\<in>?U. c xa * a xa $ p xa) =
    setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))"
    by (simp add: field_simps)
qed rule

lemma det_mul:
  fixes A B :: "'a::linordered_idom^'n^'n"
  shows "det (A ** B) = det A * det B"
proof -
  let ?U = "UNIV :: 'n set"
  let ?F = "{f. (\<forall>i\<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
  let ?PU = "{p. p permutes ?U}"
  have fU: "finite ?U"
    by simp
  have fF: "finite ?F"
    by (rule finite)
  {
    fix p
    assume p: "p permutes ?U"
    have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
      using p[unfolded permutes_def] by simp
  }
  then have PUF: "?PU \<subseteq> ?F" by blast
  {
    fix f
    assume fPU: "f \<in> ?F - ?PU"
    have fUU: "f ` ?U \<subseteq> ?U"
      using fPU by auto
    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U" "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)"
      unfolding permutes_def by auto

    let ?A = "(\<chi> i. A$i$f i *s B$f i) :: 'a^'n^'n"
    let ?B = "(\<chi> i. B$f i) :: 'a^'n^'n"
    {
      assume fni: "\<not> inj_on f ?U"
      then obtain i j where ij: "f i = f j" "i \<noteq> j"
        unfolding inj_on_def by blast
      from ij
      have rth: "row i ?B = row j ?B"
        by (vector row_def)
      from det_identical_rows[OF ij(2) rth]
      have "det (\<chi> i. A$i$f i *s B$f i) = 0"
        unfolding det_rows_mul by simp
    }
    moreover
    {
      assume fi: "inj_on f ?U"
      from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
        unfolding inj_on_def by metis
      note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
      {
        fix y
        from fs f have "\<exists>x. f x = y"
          by blast
        then obtain x where x: "f x = y"
          by blast
        {
          fix z
          assume z: "f z = y"
          from fith x z have "z = x"
            by metis
        }
        with x have "\<exists>!x. f x = y"
          by blast
      }
      with f(3) have "det (\<chi> i. A$i$f i *s B$f i) = 0"
        by blast
    }
    ultimately have "det (\<chi> i. A$i$f i *s B$f i) = 0"
      by blast
  }
  then have zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A$i$f i *s B$f i) = 0"
    by simp
  {
    fix p
    assume pU: "p \<in> ?PU"
    from pU have p: "p permutes ?U"
      by blast
    let ?s = "\<lambda>p. of_int (sign p)"
    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))"
    have "(setsum (\<lambda>q. ?s q *
        (\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
      (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) * (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
      unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
    proof (rule setsum.cong)
      fix q
      assume qU: "q \<in> ?PU"
      then have q: "q permutes ?U"
        by blast
      from p q have pp: "permutation p" and pq: "permutation q"
        unfolding permutation_permutes by auto
      have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
        "\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
        unfolding mult.assoc[symmetric]
        unfolding of_int_mult[symmetric]
        by (simp_all add: sign_idempotent)
      have ths: "?s q = ?s p * ?s (q \<circ> inv p)"
        using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
        by (simp add:  th00 ac_simps sign_idempotent sign_compose)
      have th001: "setprod (\<lambda>i. B$i$ q (inv p i)) ?U = setprod ((\<lambda>i. B$i$ q (inv p i)) \<circ> p) ?U"
        by (rule setprod_permute[OF p])
      have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U =
        setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
        unfolding th001 setprod.distrib[symmetric] o_def permutes_inverses[OF p]
        apply (rule setprod.cong[OF refl])
        using permutes_in_image[OF q]
        apply vector
        done
      show "?s q * setprod (\<lambda>i. (((\<chi> i. A$i$p i *s B$p i) :: 'a^'n^'n)$i$q i)) ?U =
        ?s p * (setprod (\<lambda>i. A$i$p i) ?U) * (?s (q \<circ> inv p) * setprod (\<lambda>i. B$i$(q \<circ> inv p) i) ?U)"
        using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
        by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
    qed rule
  }
  then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
    unfolding det_def setsum_product
    by (rule setsum.cong [OF refl])
  have "det (A**B) = setsum (\<lambda>f.  det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
    unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU]
    by simp
  also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
    using setsum.mono_neutral_cong_left[OF fF PUF zth, symmetric]
    unfolding det_rows_mul by auto
  finally show ?thesis unfolding th2 .
qed

text {* Relation to invertibility. *}

lemma invertible_left_inverse:
  fixes A :: "real^'n^'n"
  shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). B** A = mat 1)"
  by (metis invertible_def matrix_left_right_inverse)

lemma invertible_righ_inverse:
  fixes A :: "real^'n^'n"
  shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). A** B = mat 1)"
  by (metis invertible_def matrix_left_right_inverse)

lemma invertible_det_nz:
  fixes A::"real ^'n^'n"
  shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
proof -
  {
    assume "invertible A"
    then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
      unfolding invertible_righ_inverse by blast
    then have "det (A ** B) = det (mat 1 :: real ^'n^'n)"
      by simp
    then have "det A \<noteq> 0"
      by (simp add: det_mul det_I) algebra
  }
  moreover
  {
    assume H: "\<not> invertible A"
    let ?U = "UNIV :: 'n set"
    have fU: "finite ?U"
      by simp
    from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
      and iU: "i \<in> ?U"
      and ci: "c i \<noteq> 0"
      unfolding invertible_righ_inverse
      unfolding matrix_right_invertible_independent_rows
      by blast
    have *: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
      apply (drule_tac f="op + (- a)" in cong[OF refl])
      apply (simp only: ab_left_minus add.assoc[symmetric])
      apply simp
      done
    from c ci
    have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
      unfolding setsum.remove[OF fU iU] setsum_cmul
      apply -
      apply (rule vector_mul_lcancel_imp[OF ci])
      apply (auto simp add: field_simps)
      unfolding *
      apply rule
      done
    have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
      unfolding thr0
      apply (rule span_setsum)
      apply simp
      apply (rule ballI)
      apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
      apply (rule span_superset)
      apply auto
      done
    let ?B = "(\<chi> k. if k = i then 0 else row k A) :: real ^'n^'n"
    have thrb: "row i ?B = 0" using iU by (vector row_def)
    have "det A = 0"
      unfolding det_row_span[OF thr, symmetric] right_minus
      unfolding det_zero_row[OF thrb] ..
  }
  ultimately show ?thesis
    by blast
qed

text {* Cramer's rule. *}

lemma cramer_lemma_transpose:
  fixes A:: "real^'n^'n"
    and x :: "real^'n"
  shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
                             else row i A)::real^'n^'n) = x$k * det A"
  (is "?lhs = ?rhs")
proof -
  let ?U = "UNIV :: 'n set"
  let ?Uk = "?U - {k}"
  have U: "?U = insert k ?Uk"
    by blast
  have fUk: "finite ?Uk"
    by simp
  have kUk: "k \<notin> ?Uk"
    by simp
  have th00: "\<And>k s. x$k *s row k A + s = (x$k - 1) *s row k A + row k A + s"
    by (vector field_simps)
  have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f"
    by auto
  have "(\<chi> i. row i A) = A" by (vector row_def)
  then have thd1: "det (\<chi> i. row i A) = det A"
    by simp
  have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
    apply (rule det_row_span)
    apply (rule span_setsum)
    apply (rule ballI)
    apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
    apply (rule span_superset)
    apply auto
    done
  show "?lhs = x$k * det A"
    apply (subst U)
    unfolding setsum.insert[OF fUk kUk]
    apply (subst th00)
    unfolding add.assoc
    apply (subst det_row_add)
    unfolding thd0
    unfolding det_row_mul
    unfolding th001[of k "\<lambda>i. row i A"]
    unfolding thd1
    apply (simp add: field_simps)
    done
qed

lemma cramer_lemma:
  fixes A :: "real^'n^'n"
  shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: real^'n^'n) = x$k * det A"
proof -
  let ?U = "UNIV :: 'n set"
  have *: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
    by (auto simp add: row_transpose intro: setsum.cong)
  show ?thesis
    unfolding matrix_mult_vsum
    unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
    unfolding *[of "\<lambda>i. x$i"]
    apply (subst det_transpose[symmetric])
    apply (rule cong[OF refl[of det]])
    apply (vector transpose_def column_def row_def)
    done
qed

lemma cramer:
  fixes A ::"real^'n^'n"
  assumes d0: "det A \<noteq> 0"
  shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
proof -
  from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
    unfolding invertible_det_nz[symmetric] invertible_def
    by blast
  have "(A ** B) *v b = b"
    by (simp add: B matrix_vector_mul_lid)
  then have "A *v (B *v b) = b"
    by (simp add: matrix_vector_mul_assoc)
  then have xe: "\<exists>x. A *v x = b"
    by blast
  {
    fix x
    assume x: "A *v x = b"
    have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
      unfolding x[symmetric]
      using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)
  }
  with xe show ?thesis
    by auto
qed

text {* Orthogonality of a transformation and matrix. *}

definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"

lemma orthogonal_transformation:
  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
  unfolding orthogonal_transformation_def
  apply auto
  apply (erule_tac x=v in allE)+
  apply (simp add: norm_eq_sqrt_inner)
  apply (simp add: dot_norm  linear_add[symmetric])
  done

definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
  transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"

lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
  by (metis matrix_left_right_inverse orthogonal_matrix_def)

lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
  by (simp add: orthogonal_matrix_def transpose_mat matrix_mul_lid)

lemma orthogonal_matrix_mul:
  fixes A :: "real ^'n^'n"
  assumes oA : "orthogonal_matrix A"
    and oB: "orthogonal_matrix B"
  shows "orthogonal_matrix(A ** B)"
  using oA oB
  unfolding orthogonal_matrix matrix_transpose_mul
  apply (subst matrix_mul_assoc)
  apply (subst matrix_mul_assoc[symmetric])
  apply (simp add: matrix_mul_rid)
  done

lemma orthogonal_transformation_matrix:
  fixes f:: "real^'n \<Rightarrow> real^'n"
  shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
  (is "?lhs \<longleftrightarrow> ?rhs")
proof -
  let ?mf = "matrix f"
  let ?ot = "orthogonal_transformation f"
  let ?U = "UNIV :: 'n set"
  have fU: "finite ?U" by simp
  let ?m1 = "mat 1 :: real ^'n^'n"
  {
    assume ot: ?ot
    from ot have lf: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
      unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
    {
      fix i j
      let ?A = "transpose ?mf ** ?mf"
      have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
        "\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
        by simp_all
      from fd[rule_format, of "axis i 1" "axis j 1", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
      have "?A$i$j = ?m1 $ i $ j"
        by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
            th0 setsum.delta[OF fU] mat_def axis_def)
    }
    then have "orthogonal_matrix ?mf"
      unfolding orthogonal_matrix
      by vector
    with lf have ?rhs
      by blast
  }
  moreover
  {
    assume lf: "linear f" and om: "orthogonal_matrix ?mf"
    from lf om have ?lhs
      unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
      unfolding matrix_works[OF lf, symmetric]
      apply (subst dot_matrix_vector_mul)
      apply (simp add: dot_matrix_product matrix_mul_lid)
      done
  }
  ultimately show ?thesis
    by blast
qed

lemma det_orthogonal_matrix:
  fixes Q:: "'a::linordered_idom^'n^'n"
  assumes oQ: "orthogonal_matrix Q"
  shows "det Q = 1 \<or> det Q = - 1"
proof -
  have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
  proof -
    fix x:: 'a
    have th0: "x * x - 1 = (x - 1) * (x + 1)"
      by (simp add: field_simps)
    have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
      apply (subst eq_iff_diff_eq_0)
      apply simp
      done
    have "x * x = 1 \<longleftrightarrow> x * x - 1 = 0"
      by simp
    also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1"
      unfolding th0 th1 by simp
    finally show "?ths x" ..
  qed
  from oQ have "Q ** transpose Q = mat 1"
    by (metis orthogonal_matrix_def)
  then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
    by simp
  then have "det Q * det Q = 1"
    by (simp add: det_mul det_I det_transpose)
  then show ?thesis unfolding th .
qed

text {* Linearity of scaling, and hence isometry, that preserves origin. *}

lemma scaling_linear:
  fixes f :: "real ^'n \<Rightarrow> real ^'n"
  assumes f0: "f 0 = 0"
    and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
  shows "linear f"
proof -
  {
    fix v w
    {
      fix x
      note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right]
    }
    note th0 = this
    have "f v \<bullet> f w = c\<^sup>2 * (v \<bullet> w)"
      unfolding dot_norm_neg dist_norm[symmetric]
      unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
  note fc = this
  show ?thesis
    unfolding linear_iff vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR
    by (simp add: inner_add fc field_simps)
qed

lemma isometry_linear:
  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
  by (rule scaling_linear[where c=1]) simp_all

text {* Hence another formulation of orthogonal transformation. *}

lemma orthogonal_transformation_isometry:
  "orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
  unfolding orthogonal_transformation
  apply (rule iffI)
  apply clarify
  apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_norm)
  apply (rule conjI)
  apply (rule isometry_linear)
  apply simp
  apply simp
  apply clarify
  apply (erule_tac x=v in allE)
  apply (erule_tac x=0 in allE)
  apply (simp add: dist_norm)
  done

text {* Can extend an isometry from unit sphere. *}

lemma isometry_sphere_extend:
  fixes f:: "real ^'n \<Rightarrow> real ^'n"
  assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
    and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
  shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
proof -
  {
    fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
    assume H:
      "x = norm x *\<^sub>R x0"
      "y = norm y *\<^sub>R y0"
      "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'"
      "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
      "norm(x0' - y0') = norm(x0 - y0)"
    then have *: "x0 \<bullet> y0 = x0' \<bullet> y0' + y0' \<bullet> x0' - y0 \<bullet> x0 "
      by (simp add: norm_eq norm_eq_1 inner_add inner_diff)
    have "norm(x' - y') = norm(x - y)"
      apply (subst H(1))
      apply (subst H(2))
      apply (subst H(3))
      apply (subst H(4))
      using H(5-9)
      apply (simp add: norm_eq norm_eq_1)
      apply (simp add: inner_diff scalar_mult_eq_scaleR)
      unfolding *
      apply (simp add: field_simps)
      done
  }
  note th0 = this
  let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
  {
    fix x:: "real ^'n"
    assume nx: "norm x = 1"
    have "?g x = f x"
      using nx by auto
  }
  then have thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x"
    by blast
  have g0: "?g 0 = 0"
    by simp
  {
    fix x y :: "real ^'n"
    {
      assume "x = 0" "y = 0"
      then have "dist (?g x) (?g y) = dist x y"
        by simp
    }
    moreover
    {
      assume "x = 0" "y \<noteq> 0"
      then have "dist (?g x) (?g y) = dist x y"
        apply (simp add: dist_norm)
        apply (rule f1[rule_format])
        apply (simp add: field_simps)
        done
    }
    moreover
    {
      assume "x \<noteq> 0" "y = 0"
      then have "dist (?g x) (?g y) = dist x y"
        apply (simp add: dist_norm)
        apply (rule f1[rule_format])
        apply (simp add: field_simps)
        done
    }
    moreover
    {
      assume z: "x \<noteq> 0" "y \<noteq> 0"
      have th00:
        "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)"
        "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)"
        "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
        "norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y) = norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y)"
        "norm (inverse (norm x) *\<^sub>R x) = 1"
        "norm (f (inverse (norm x) *\<^sub>R x)) = 1"
        "norm (inverse (norm y) *\<^sub>R y) = 1"
        "norm (f (inverse (norm y) *\<^sub>R y)) = 1"
        "norm (f (inverse (norm x) *\<^sub>R x) - f (inverse (norm y) *\<^sub>R y)) =
          norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)"
        using z
        by (auto simp add: field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
      from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
        by (simp add: dist_norm)
    }
    ultimately have "dist (?g x) (?g y) = dist x y"
      by blast
  }
  note thd = this
    show ?thesis
    apply (rule exI[where x= ?g])
    unfolding orthogonal_transformation_isometry
    using g0 thfg thd
    apply metis
    done
qed

text {* Rotation, reflection, rotoinversion. *}

definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"

lemma orthogonal_rotation_or_rotoinversion:
  fixes Q :: "'a::linordered_idom^'n^'n"
  shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
  by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)

text {* Explicit formulas for low dimensions. *}

lemma setprod_neutral_const: "setprod f {(1::nat)..1} = f 1"
  by (fact setprod_singleton_nat_seg)

lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
  by (simp add: eval_nat_numeral setprod_numseg mult.commute)

lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
  by (simp add: eval_nat_numeral setprod_numseg mult.commute)

lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
  by (simp add: det_def sign_id)

lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
proof -
  have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
  show ?thesis
    unfolding det_def UNIV_2
    unfolding setsum_over_permutations_insert[OF f12]
    unfolding permutes_sing
    by (simp add: sign_swap_id sign_id swap_id_eq)
qed

lemma det_3:
  "det (A::'a::comm_ring_1^3^3) =
    A$1$1 * A$2$2 * A$3$3 +
    A$1$2 * A$2$3 * A$3$1 +
    A$1$3 * A$2$1 * A$3$2 -
    A$1$1 * A$2$3 * A$3$2 -
    A$1$2 * A$2$1 * A$3$3 -
    A$1$3 * A$2$2 * A$3$1"
proof -
  have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}"
    by auto
  have f23: "finite {3::3}" "2 \<notin> {3::3}"
    by auto

  show ?thesis
    unfolding det_def UNIV_3
    unfolding setsum_over_permutations_insert[OF f123]
    unfolding setsum_over_permutations_insert[OF f23]
    unfolding permutes_sing
    by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
qed

end