author chaieb Mon Feb 09 17:21:19 2009 +0000 (2009-02-09) changeset 29846 57dccccc37b3 parent 29845 5ef75225c9c2 child 29847 af32126ee729
Traces, Determinant of square matrices and some properties
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Library/Determinants.thy	Mon Feb 09 17:21:19 2009 +0000
1.3 @@ -0,0 +1,1151 @@
1.4 +(* Title:      Determinants
1.5 +   ID:         \$Id:
1.6 +   Author:     Amine Chaieb, University of Cambridge
1.7 +*)
1.8 +
1.9 +header {* Traces, Determinant of square matrices and some properties *}
1.10 +
1.11 +theory Determinants
1.12 +  imports Euclidean_Space Permutations
1.13 +begin
1.14 +
1.15 +subsection{* First some facts about products*}
1.16 +lemma setprod_insert_eq: "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
1.17 +apply clarsimp
1.18 +by(subgoal_tac "insert a A = A", auto)
1.19 +
1.21 +  assumes mn: "(m::nat) <= n + 1"
1.22 +  shows "setprod f {m.. n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
1.23 +proof-
1.24 +  let ?A = "{m .. n+p}"
1.25 +  let ?B = "{m .. n}"
1.26 +  let ?C = "{n+1..n+p}"
1.27 +  from mn have un: "?B \<union> ?C = ?A" by auto
1.28 +  from mn have dj: "?B \<inter> ?C = {}" by auto
1.29 +  have f: "finite ?B" "finite ?C" by simp_all
1.30 +  from setprod_Un_disjoint[OF f dj, of f, unfolded un] show ?thesis .
1.31 +qed
1.32 +
1.33 +
1.34 +lemma setprod_offset: "setprod f {(m::nat) + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
1.35 +apply (rule setprod_reindex_cong[where f="op + p"])
1.36 +apply (auto simp add: image_iff Bex_def inj_on_def)
1.37 +apply arith
1.38 +apply (rule ext)
1.40 +done
1.41 +
1.42 +lemma setprod_singleton: "setprod f {x} = f x" by simp
1.43 +
1.44 +lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
1.45 +
1.46 +lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
1.47 +  "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
1.48 +                             else setprod f {m..n})"
1.49 +  by (auto simp add: atLeastAtMostSuc_conv)
1.50 +
1.51 +lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::ordered_idom)"
1.52 +  shows "setprod f S \<le> setprod g S"
1.53 +using fS fg
1.54 +apply(induct S)
1.55 +apply simp
1.56 +apply auto
1.57 +apply (rule mult_mono)
1.58 +apply (auto intro: setprod_nonneg)
1.59 +done
1.60 +
1.61 +  (* FIXME: In Finite_Set there is a useless further assumption *)
1.62 +lemma setprod_inversef: "finite A ==> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: {division_by_zero, field})"
1.63 +  apply (erule finite_induct)
1.64 +  apply (simp)
1.65 +  apply simp
1.66 +  done
1.67 +
1.68 +lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::ordered_idom)"
1.69 +  shows "setprod f S \<le> 1"
1.70 +using setprod_le[OF fS f] unfolding setprod_1 .
1.71 +
1.72 +subsection{* Trace *}
1.73 +
1.74 +definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
1.75 +  "trace A = setsum (\<lambda>i. ((A\$i)\$i)) {1..dimindex(UNIV::'n set)}"
1.76 +
1.77 +lemma trace_0: "trace(mat 0) = 0"
1.78 +  by (simp add: trace_def mat_def Cart_lambda_beta setsum_0)
1.79 +
1.80 +lemma trace_I: "trace(mat 1 :: 'a::semiring_1^'n^'n) = of_nat(dimindex(UNIV::'n set))"
1.81 +  by (simp add: trace_def mat_def Cart_lambda_beta)
1.82 +
1.83 +lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
1.85 +
1.86 +lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
1.87 +  by (simp add: trace_def setsum_subtractf Cart_lambda_beta vector_component)
1.88 +
1.89 +lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'n) ** B) = trace (B**A)"
1.90 +  apply (simp add: trace_def matrix_matrix_mult_def Cart_lambda_beta)
1.91 +  apply (subst setsum_commute)
1.92 +  by (simp add: mult_commute)
1.93 +
1.94 +(* ------------------------------------------------------------------------- *)
1.95 +(* Definition of determinant.                                                *)
1.96 +(* ------------------------------------------------------------------------- *)
1.97 +
1.98 +definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
1.99 +  "det A = setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A\$i\$p i) {1 .. dimindex(UNIV :: 'n set)}) {p. p permutes {1 .. dimindex(UNIV :: 'n set)}}"
1.100 +
1.101 +(* ------------------------------------------------------------------------- *)
1.102 +(* A few general lemmas we need below.                                       *)
1.103 +(* ------------------------------------------------------------------------- *)
1.104 +
1.105 +lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}"
1.106 +  and i: "i \<in> {1..dimindex(UNIV::'n set)}"
1.107 +  shows "Cart_nth (Cart_lambda g ::'a^'n) (p i) = g(p i)"
1.108 +  using permutes_in_image[OF p] i
1.109 +  by (simp add:  Cart_lambda_beta permutes_in_image[OF p])
1.110 +
1.111 +lemma setprod_permute:
1.112 +  assumes p: "p permutes S"
1.113 +  shows "setprod f S = setprod (f o p) S"
1.114 +proof-
1.115 +  {assume "\<not> finite S" hence ?thesis by simp}
1.116 +  moreover
1.117 +  {assume fS: "finite S"
1.118 +    then have ?thesis
1.119 +      apply (simp add: setprod_def)
1.120 +      apply (rule ab_semigroup_mult.fold_image_permute)
1.121 +      apply (auto simp add: p)
1.122 +      apply unfold_locales
1.123 +      done}
1.124 +  ultimately show ?thesis by blast
1.125 +qed
1.126 +
1.127 +lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
1.128 +  by (auto intro: setprod_permute)
1.129 +
1.130 +(* ------------------------------------------------------------------------- *)
1.131 +(* Basic determinant properties.                                             *)
1.132 +(* ------------------------------------------------------------------------- *)
1.133 +
1.134 +lemma det_transp: "det (transp A) = det (A::'a::comm_ring_1 ^'n^'n)"
1.135 +proof-
1.136 +  let ?di = "\<lambda>A i j. A\$i\$j"
1.137 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.138 +  have fU: "finite ?U" by blast
1.139 +  {fix p assume p: "p \<in> {p. p permutes ?U}"
1.140 +    from p have pU: "p permutes ?U" by blast
1.141 +    have sth: "sign (inv p) = sign p"
1.142 +      by (metis sign_inverse fU p mem_def Collect_def permutation_permutes)
1.143 +    from permutes_inj[OF pU]
1.144 +    have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
1.145 +    from permutes_image[OF pU]
1.146 +    have "setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U = setprod (\<lambda>i. ?di (transp A) i (inv p i)) (p ` ?U)" by simp
1.147 +    also have "\<dots> = setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U"
1.148 +      unfolding setprod_reindex[OF pi] ..
1.149 +    also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
1.150 +    proof-
1.151 +      {fix i assume i: "i \<in> ?U"
1.152 +	from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
1.153 +	have "((\<lambda>i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)"
1.154 +	  unfolding transp_def by (simp add: Cart_lambda_beta expand_fun_eq)}
1.155 +      then show "setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
1.156 +    qed
1.157 +    finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
1.158 +      by simp}
1.159 +  then show ?thesis unfolding det_def apply (subst setsum_permutations_inverse)
1.160 +  apply (rule setsum_cong2) by blast
1.161 +qed
1.162 +
1.163 +lemma det_lowerdiagonal:
1.164 +  fixes A :: "'a::comm_ring_1^'n^'n"
1.165 +  assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i < j \<Longrightarrow> A\$i\$j = 0"
1.166 +  shows "det A = setprod (\<lambda>i. A\$i\$i) {1..dimindex(UNIV:: 'n set)}"
1.167 +proof-
1.168 +  let ?U = "{1..dimindex(UNIV:: 'n set)}"
1.169 +  let ?PU = "{p. p permutes ?U}"
1.170 +  let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A\$i\$p i) {1 .. dimindex(UNIV :: 'n set)}"
1.171 +  have fU: "finite ?U" by blast
1.172 +  from finite_permutations[OF fU] have fPU: "finite ?PU" .
1.173 +  have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
1.174 +  {fix p assume p: "p \<in> ?PU -{id}"
1.175 +    from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
1.176 +    from permutes_natset_le[OF pU] pid obtain i where
1.177 +      i: "i \<in> ?U" "p i > i" by (metis not_le)
1.178 +    from permutes_in_image[OF pU] i(1) have piU: "p i \<in> ?U" by blast
1.179 +    from ld[OF i(1) piU i(2)] i(1) have ex:"\<exists>i \<in> ?U. A\$i\$p i = 0" by blast
1.180 +    from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
1.181 +  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
1.182 +  from setsum_superset[OF fPU id0 p0] show ?thesis
1.183 +    unfolding det_def by (simp add: sign_id)
1.184 +qed
1.185 +
1.186 +lemma det_upperdiagonal:
1.187 +  fixes A :: "'a::comm_ring_1^'n^'n"
1.188 +  assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i > j \<Longrightarrow> A\$i\$j = 0"
1.189 +  shows "det A = setprod (\<lambda>i. A\$i\$i) {1..dimindex(UNIV:: 'n set)}"
1.190 +proof-
1.191 +  let ?U = "{1..dimindex(UNIV:: 'n set)}"
1.192 +  let ?PU = "{p. p permutes ?U}"
1.193 +  let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A\$i\$p i) {1 .. dimindex(UNIV :: 'n set)})"
1.194 +  have fU: "finite ?U" by blast
1.195 +  from finite_permutations[OF fU] have fPU: "finite ?PU" .
1.196 +  have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
1.197 +  {fix p assume p: "p \<in> ?PU -{id}"
1.198 +    from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
1.199 +    from permutes_natset_ge[OF pU] pid obtain i where
1.200 +      i: "i \<in> ?U" "p i < i" by (metis not_le)
1.201 +    from permutes_in_image[OF pU] i(1) have piU: "p i \<in> ?U" by blast
1.202 +    from ld[OF i(1) piU i(2)] i(1) have ex:"\<exists>i \<in> ?U. A\$i\$p i = 0" by blast
1.203 +    from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
1.204 +  then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
1.205 +  from setsum_superset[OF fPU id0 p0] show ?thesis
1.206 +    unfolding det_def by (simp add: sign_id)
1.207 +qed
1.208 +
1.209 +lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
1.210 +proof-
1.211 +  let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
1.212 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.213 +  let ?f = "\<lambda>i j. ?A\$i\$j"
1.214 +  {fix i assume i: "i \<in> ?U"
1.215 +    have "?f i i = 1" using i by (vector mat_def)}
1.216 +  hence th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
1.217 +    by (auto intro: setprod_cong)
1.218 +  {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i < j"
1.219 +    have "?f i j = 0" using i j ij by (vector mat_def) }
1.220 +  then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_lowerdiagonal
1.221 +    by blast
1.222 +  also have "\<dots> = 1" unfolding th setprod_1 ..
1.223 +  finally show ?thesis .
1.224 +qed
1.225 +
1.226 +lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
1.227 +proof-
1.228 +  let ?A = "mat 0 :: 'a::comm_ring_1^'n^'n"
1.229 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.230 +  let ?f = "\<lambda>i j. ?A\$i\$j"
1.231 +  have th:"setprod (\<lambda>i. ?f i i) ?U = 0"
1.232 +    apply (rule setprod_zero)
1.233 +    apply simp
1.234 +    apply (rule bexI[where x=1])
1.235 +    using dimindex_ge_1[of "UNIV :: 'n set"]
1.236 +    by (simp_all add: mat_def Cart_lambda_beta)
1.237 +  {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i < j"
1.238 +    have "?f i j = 0" using i j ij by (vector mat_def) }
1.239 +  then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_lowerdiagonal
1.240 +    by blast
1.241 +  also have "\<dots> = 0" unfolding th  ..
1.242 +  finally show ?thesis .
1.243 +qed
1.244 +
1.245 +lemma det_permute_rows:
1.246 +  fixes A :: "'a::comm_ring_1^'n^'n"
1.247 +  assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}"
1.248 +  shows "det(\<chi> i. A\$p i :: 'a^'n^'n) = of_int (sign p) * det A"
1.249 +  apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric] del: One_nat_def)
1.250 +  apply (subst sum_permutations_compose_right[OF p])
1.251 +proof(rule setsum_cong2)
1.252 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.253 +  let ?PU = "{p. p permutes ?U}"
1.254 +  let ?Ap = "(\<chi> i. A\$p i :: 'a^'n^'n)"
1.255 +  fix q assume qPU: "q \<in> ?PU"
1.256 +  have fU: "finite ?U" by blast
1.257 +  from qPU have q: "q permutes ?U" by blast
1.258 +  from p q have pp: "permutation p" and qp: "permutation q"
1.259 +    by (metis fU permutation_permutes)+
1.260 +  from permutes_inv[OF p] have ip: "inv p permutes ?U" .
1.261 +    {fix i assume i: "i \<in> ?U"
1.262 +      from Cart_lambda_beta[rule_format, OF i, of "\<lambda>i. A\$ p i"]
1.263 +      have "?Ap\$i\$ (q o p) i = A \$ p i \$ (q o p) i " by simp}
1.264 +    hence "setprod (\<lambda>i. ?Ap\$i\$ (q o p) i) ?U = setprod (\<lambda>i. A\$p i\$(q o p) i) ?U"
1.265 +      by (auto intro: setprod_cong)
1.266 +    also have "\<dots> = setprod ((\<lambda>i. A\$p i\$(q o p) i) o inv p) ?U"
1.267 +      by (simp only: setprod_permute[OF ip, symmetric])
1.268 +    also have "\<dots> = setprod (\<lambda>i. A \$ (p o inv p) i \$ (q o (p o inv p)) i) ?U"
1.269 +      by (simp only: o_def)
1.270 +    also have "\<dots> = setprod (\<lambda>i. A\$i\$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
1.271 +    finally   have thp: "setprod (\<lambda>i. ?Ap\$i\$ (q o p) i) ?U = setprod (\<lambda>i. A\$i\$q i) ?U"
1.272 +      by blast
1.273 +  show "of_int (sign (q o p)) * setprod (\<lambda>i. ?Ap\$i\$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A\$i\$q i) ?U"
1.274 +    by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
1.275 +qed
1.276 +
1.277 +lemma det_permute_columns:
1.278 +  fixes A :: "'a::comm_ring_1^'n^'n"
1.279 +  assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}"
1.280 +  shows "det(\<chi> i j. A\$i\$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
1.281 +proof-
1.282 +  let ?Ap = "\<chi> i j. A\$i\$ p j :: 'a^'n^'n"
1.283 +  let ?At = "transp A"
1.284 +  have "of_int (sign p) * det A = det (transp (\<chi> i. transp A \$ p i))"
1.285 +    unfolding det_permute_rows[OF p, of ?At] det_transp ..
1.286 +  moreover
1.287 +  have "?Ap = transp (\<chi> i. transp A \$ p i)"
1.288 +    by (simp add: transp_def Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF p])
1.289 +  ultimately show ?thesis by simp
1.290 +qed
1.291 +
1.292 +lemma det_identical_rows:
1.293 +  fixes A :: "'a::ordered_idom^'n^'n"
1.294 +  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.295 +  and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.296 +  and ij: "i \<noteq> j"
1.297 +  and r: "row i A = row j A"
1.298 +  shows	"det A = 0"
1.299 +proof-
1.300 +  have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
1.301 +    by simp
1.302 +  have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min)
1.303 +  let ?p = "Fun.swap i j id"
1.304 +  let ?A = "\<chi> i. A \$ ?p i"
1.305 +  from r have "A = ?A" by (simp add: Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF permutes_swap_id[OF i j]] row_def swap_def)
1.306 +  hence "det A = det ?A" by simp
1.307 +  moreover have "det A = - det ?A"
1.308 +    by (simp add: det_permute_rows[OF permutes_swap_id[OF i j]] sign_swap_id ij th1)
1.309 +  ultimately show "det A = 0" by (metis tha)
1.310 +qed
1.311 +
1.312 +lemma det_identical_columns:
1.313 +  fixes A :: "'a::ordered_idom^'n^'n"
1.314 +  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.315 +  and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.316 +  and ij: "i \<noteq> j"
1.317 +  and r: "column i A = column j A"
1.318 +  shows	"det A = 0"
1.319 +apply (subst det_transp[symmetric])
1.320 +apply (rule det_identical_rows[OF i j ij])
1.321 +by (metis row_transp i j r)
1.322 +
1.323 +lemma det_zero_row:
1.324 +  fixes A :: "'a::{idom, ring_char_0}^'n^'n"
1.325 +  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.326 +  and r: "row i A = 0"
1.327 +  shows "det A = 0"
1.328 +using i r
1.329 +apply (simp add: row_def det_def Cart_lambda_beta Cart_eq vector_component del: One_nat_def)
1.330 +apply (rule setsum_0')
1.331 +apply (clarsimp simp add: sign_nz simp del: One_nat_def)
1.332 +apply (rule setprod_zero)
1.333 +apply simp
1.334 +apply (rule bexI[where x=i])
1.335 +apply (erule_tac x="a i" in ballE)
1.336 +apply (subgoal_tac "(0\<Colon>'a ^ 'n) \$ a i = 0")
1.337 +apply simp
1.338 +apply (rule zero_index)
1.339 +apply (drule permutes_in_image[of _ _ i])
1.340 +apply simp
1.341 +apply (drule permutes_in_image[of _ _ i])
1.342 +apply simp
1.343 +apply simp
1.344 +done
1.345 +
1.346 +lemma det_zero_column:
1.347 +  fixes A :: "'a::{idom,ring_char_0}^'n^'n"
1.348 +  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
1.349 +  and r: "column i A = 0"
1.350 +  shows "det A = 0"
1.351 +  apply (subst det_transp[symmetric])
1.352 +  apply (rule det_zero_row[OF i])
1.353 +  by (metis row_transp r i)
1.354 +
1.355 +lemma setsum_lambda_beta[simp]: "setsum (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_add}^'n) \$ i ) {1 .. dimindex (UNIV :: 'n set)} = setsum g {1 .. dimindex (UNIV :: 'n set)}"
1.356 +  by (simp add: Cart_lambda_beta)
1.357 +
1.358 +lemma setprod_lambda_beta[simp]: "setprod (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_mult}^'n) \$ i ) {1 .. dimindex (UNIV :: 'n set)} = setprod g {1 .. dimindex (UNIV :: 'n set)}"
1.359 +  apply (rule setprod_cong)
1.360 +  apply simp
1.361 +  apply (simp add: Cart_lambda_beta')
1.362 +  done
1.363 +
1.364 +lemma setprod_lambda_beta2[simp]: "setprod (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_mult}^'n^'n) \$ i\$ f i ) {1 .. dimindex (UNIV :: 'n set)} = setprod (\<lambda>i. g i \$ f i) {1 .. dimindex (UNIV :: 'n set)}"
1.365 +proof(rule setprod_cong[OF refl])
1.366 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.367 +  fix i assume i: "i \<in> ?U"
1.368 +  from Cart_lambda_beta'[OF i, of g] have
1.369 +    "((\<chi> i. g i) :: 'a^'n^'n) \$ i = g i" .
1.370 +  hence "((\<chi> i. g i) :: 'a^'n^'n) \$ i \$ f i = g i \$ f i" by simp
1.371 +  then
1.372 +  show "((\<chi> i. g i):: 'a^'n^'n) \$ i \$ f i = g i \$ f i"   .
1.373 +qed
1.374 +
1.376 +  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1.377 +  shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
1.378 +             det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
1.379 +             det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
1.381 +proof (rule setsum_cong2)
1.382 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.383 +  let ?pU = "{p. p permutes ?U}"
1.384 +  let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
1.385 +  let ?g = "(\<lambda> i. if i = k then a i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
1.386 +  let ?h = "(\<lambda> i. if i = k then b i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
1.387 +  fix p assume p: "p \<in> ?pU"
1.388 +  let ?Uk = "?U - {k}"
1.389 +  from p have pU: "p permutes ?U" by blast
1.390 +  from k have pkU: "p k \<in> ?U" by (simp only: permutes_in_image[OF pU])
1.391 +  note pin[simp] = permutes_in_image[OF pU]
1.392 +  have kU: "?U = insert k ?Uk" using k by blast
1.393 +  {fix j assume j: "j \<in> ?Uk"
1.394 +    from j have "?f j \$ p j = ?g j \$ p j" and "?f j \$ p j= ?h j \$ p j"
1.395 +      by simp_all}
1.396 +  then have th1: "setprod (\<lambda>i. ?f i \$ p i) ?Uk = setprod (\<lambda>i. ?g i \$ p i) ?Uk"
1.397 +    and th2: "setprod (\<lambda>i. ?f i \$ p i) ?Uk = setprod (\<lambda>i. ?h i \$ p i) ?Uk"
1.398 +    apply -
1.399 +    apply (rule setprod_cong, simp_all)+
1.400 +    done
1.401 +  have th3: "finite ?Uk" "k \<notin> ?Uk" using k by auto
1.402 +  have "setprod (\<lambda>i. ?f i \$ p i) ?U = setprod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
1.403 +    unfolding kU[symmetric] ..
1.404 +  also have "\<dots> = ?f k \$ p k  * setprod (\<lambda>i. ?f i \$ p i) ?Uk"
1.405 +    apply (rule setprod_insert)
1.406 +    apply simp
1.407 +    using k by blast
1.408 +  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)" using pkU by (simp add: ring_simps vector_component)
1.409 +  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)
1.410 +  also have "\<dots> = setprod (\<lambda>i. ?g i \$ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i \$ p i) (insert k ?Uk)"
1.411 +    unfolding  setprod_insert[OF th3] by simp
1.412 +  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] .
1.413 +  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"
1.414 +    by (simp add: ring_simps)
1.415 +qed
1.416 +
1.417 +lemma det_row_mul:
1.418 +  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1.419 +  shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
1.420 +             c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
1.421 +
1.422 +unfolding det_def setprod_lambda_beta2 setsum_right_distrib
1.423 +proof (rule setsum_cong2)
1.424 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.425 +  let ?pU = "{p. p permutes ?U}"
1.426 +  let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
1.427 +  let ?g = "(\<lambda> i. if i = k then a i else b i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
1.428 +  fix p assume p: "p \<in> ?pU"
1.429 +  let ?Uk = "?U - {k}"
1.430 +  from p have pU: "p permutes ?U" by blast
1.431 +  from k have pkU: "p k \<in> ?U" by (simp only: permutes_in_image[OF pU])
1.432 +  note pin[simp] = permutes_in_image[OF pU]
1.433 +  have kU: "?U = insert k ?Uk" using k by blast
1.434 +  {fix j assume j: "j \<in> ?Uk"
1.435 +    from j have "?f j \$ p j = ?g j \$ p j" by simp}
1.436 +  then have th1: "setprod (\<lambda>i. ?f i \$ p i) ?Uk = setprod (\<lambda>i. ?g i \$ p i) ?Uk"
1.437 +    apply -
1.438 +    apply (rule setprod_cong, simp_all)
1.439 +    done
1.440 +  have th3: "finite ?Uk" "k \<notin> ?Uk" using k by auto
1.441 +  have "setprod (\<lambda>i. ?f i \$ p i) ?U = setprod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
1.442 +    unfolding kU[symmetric] ..
1.443 +  also have "\<dots> = ?f k \$ p k  * setprod (\<lambda>i. ?f i \$ p i) ?Uk"
1.444 +    apply (rule setprod_insert)
1.445 +    apply simp
1.446 +    using k by blast
1.447 +  also have "\<dots> = (c*s a k) \$ p k * setprod (\<lambda>i. ?f i \$ p i) ?Uk" using pkU by (simp add: ring_simps vector_component)
1.448 +  also have "\<dots> = c* (a k \$ p k * setprod (\<lambda>i. ?g i \$ p i) ?Uk)"
1.449 +    unfolding th1 using pkU by (simp add: vector_component mult_ac)
1.450 +  also have "\<dots> = c* (setprod (\<lambda>i. ?g i \$ p i) (insert k ?Uk))"
1.451 +    unfolding  setprod_insert[OF th3] by simp
1.452 +  finally have "setprod (\<lambda>i. ?f i \$ p i) ?U = c* (setprod (\<lambda>i. ?g i \$ p i) ?U)" unfolding kU[symmetric] .
1.453 +  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)"
1.454 +    by (simp add: ring_simps)
1.455 +qed
1.456 +
1.457 +lemma det_row_0:
1.458 +  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1.459 +  shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
1.460 +using det_row_mul[OF k, of 0 "\<lambda>i. 1" b]
1.461 +apply (simp)
1.462 +  unfolding vector_smult_lzero .
1.463 +
1.464 +lemma det_row_operation:
1.465 +  fixes A :: "'a::ordered_idom^'n^'n"
1.466 +  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
1.467 +  and j: "j \<in> {1 .. dimindex(UNIV :: 'n set)}"
1.468 +  and ij: "i \<noteq> j"
1.469 +  shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
1.470 +proof-
1.471 +  let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
1.472 +  have th: "row i ?Z = row j ?Z" using i j by (vector row_def)
1.473 +  have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
1.474 +    using i j by (vector row_def)
1.475 +  show ?thesis
1.476 +    unfolding det_row_add [OF i] det_row_mul[OF i] det_identical_rows[OF i j ij th] th2
1.477 +    by simp
1.478 +qed
1.479 +
1.480 +lemma det_row_span:
1.481 +  fixes A :: "'a:: ordered_idom^'n^'n"
1.482 +  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
1.483 +  and x: "x \<in> span {row j A |j. j\<in> {1 .. dimindex(UNIV :: 'n set)} \<and> j\<noteq> i}"
1.484 +  shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
1.485 +proof-
1.486 +  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
1.487 +  let ?S = "{row j A |j. j\<in> ?U \<and> j\<noteq> i}"
1.488 +  let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
1.489 +  let ?P = "\<lambda>x. ?d (row i A + x) = det A"
1.490 +  {fix k
1.491 +
1.492 +    have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
1.493 +  then have P0: "?P 0"
1.494 +    apply -
1.495 +    apply (rule cong[of det, OF refl])
1.496 +    using i by (vector row_def)
1.497 +  moreover
1.498 +  {fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
1.499 +    from zS obtain j where j: "z = row j A" "j \<in> ?U" "i \<noteq> j" by blast
1.500 +    let ?w = "row i A + y"
1.501 +    have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector
1.502 +    have thz: "?d z = 0"
1.503 +      apply (rule det_identical_rows[OF i j(2,3)])
1.504 +      using i j by (vector row_def)
1.505 +    have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
1.506 +    then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i]
1.507 +      by simp }
1.508 +
1.509 +  ultimately show ?thesis
1.510 +    apply -
1.511 +    apply (rule span_induct_alt[of ?P ?S, OF P0])
1.512 +    apply blast
1.513 +    apply (rule x)
1.514 +    done
1.515 +qed
1.516 +
1.517 +(* ------------------------------------------------------------------------- *)
1.518 +(* May as well do this, though it's a bit unsatisfactory since it ignores    *)
1.519 +(* exact duplicates by considering the rows/columns as a set.                *)
1.520 +(* ------------------------------------------------------------------------- *)
1.521 +
1.522 +lemma det_dependent_rows:
1.523 +  fixes A:: "'a::ordered_idom^'n^'n"
1.524 +  assumes d: "dependent (rows A)"
1.525 +  shows "det A = 0"
1.526 +proof-
1.527 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.528 +  from d obtain i where i: "i \<in> ?U" "row i A \<in> span (rows A - {row i A})"
1.529 +    unfolding dependent_def rows_def by blast
1.530 +  {fix j k assume j: "j \<in>?U" and k: "k \<in> ?U" and jk: "j \<noteq> k"
1.531 +    and c: "row j A = row k A"
1.532 +    from det_identical_rows[OF j k jk c] have ?thesis .}
1.533 +  moreover
1.534 +  {assume H: "\<And> i j. i\<in> ?U \<Longrightarrow> j \<in> ?U \<Longrightarrow> i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
1.535 +    have th0: "- row i A \<in> span {row j A|j. j \<in> ?U \<and> j \<noteq> i}"
1.536 +      apply (rule span_neg)
1.537 +      apply (rule set_rev_mp)
1.538 +      apply (rule i(2))
1.539 +      apply (rule span_mono)
1.540 +      using H i by (auto simp add: rows_def)
1.541 +    from det_row_span[OF i(1) th0]
1.542 +    have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
1.543 +      unfolding right_minus vector_smult_lzero ..
1.544 +    with det_row_mul[OF i(1), of "0::'a" "\<lambda>i. 1"]
1.545 +    have "det A = 0" by simp}
1.546 +  ultimately show ?thesis by blast
1.547 +qed
1.548 +
1.549 +lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n))" shows "det A = 0"
1.550 +by (metis d det_dependent_rows rows_transp det_transp)
1.551 +
1.552 +(* ------------------------------------------------------------------------- *)
1.553 +(* Multilinearity and the multiplication formula.                            *)
1.554 +(* ------------------------------------------------------------------------- *)
1.555 +
1.556 +lemma Cart_lambda_cong: "(\<And>x. x \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
1.557 +  apply (rule iffD1[OF Cart_lambda_unique]) by vector
1.558 +
1.559 +lemma det_linear_row_setsum:
1.560 +  assumes fS: "finite S" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
1.561 +  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"
1.562 +  using k
1.563 +proof(induct rule: finite_induct[OF fS])
1.564 +  case 1 thus ?case apply simp  unfolding setsum_empty det_row_0[OF k] ..
1.565 +next
1.566 +  case (2 x F)
1.567 +  then  show ?case by (simp add: det_row_add cong del: if_weak_cong)
1.568 +qed
1.569 +
1.570 +lemma finite_bounded_functions:
1.571 +  assumes fS: "finite S"
1.572 +  shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
1.573 +proof(induct k)
1.574 +  case 0
1.575 +  have th: "{f. \<forall>i. f i = i} = {id}" by (auto intro: ext)
1.576 +  show ?case by (auto simp add: th)
1.577 +next
1.578 +  case (Suc k)
1.579 +  let ?f = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
1.580 +  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)})"
1.581 +  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)}"
1.582 +    apply (auto simp add: image_iff)
1.583 +    apply (rule_tac x="x (Suc k)" in bexI)
1.584 +    apply (rule_tac x = "\<lambda>i. if i = Suc k then i else x i" in exI)
1.585 +    apply (auto intro: ext)
1.586 +    done
1.587 +  with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
1.588 +  show ?case by metis
1.589 +qed
1.590 +
1.591 +
1.592 +lemma eq_id_iff[simp]: "(\<forall>x. f x = x) = (f = id)" by (auto intro: ext)
1.593 +
1.594 +lemma det_linear_rows_setsum_lemma:
1.595 +  assumes fS: "finite S" and k: "k \<le> dimindex (UNIV :: 'n set)"
1.596 +  shows "det((\<chi> i. if i <= k then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
1.597 +             setsum (\<lambda>f. det((\<chi> i. if i <= k then a i (f i) else c i)::'a^'n^'n))
1.598 +                 {f. (\<forall>i \<in> {1 .. k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)}"
1.599 +using k
1.600 +proof(induct k arbitrary: a c)
1.601 +  case 0
1.602 +  have th0: "\<And>x y. (\<chi> i. if i <= 0 then x i else y i) = (\<chi> i. y i)" by vector
1.603 +  from "0.prems"  show ?case unfolding th0 by simp
1.604 +next
1.605 +  case (Suc k a c)
1.606 +  let ?F = "\<lambda>k. {f. (\<forall>i \<in> {1 .. k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)}"
1.607 +  let ?h = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
1.608 +  let ?k = "\<lambda>h. (h(Suc k),(\<lambda>i. if i = Suc k then i else h i))"
1.609 +  let ?s = "\<lambda> k a c f. det((\<chi> i. if i <= k then a i (f i) else c i)::'a^'n^'n)"
1.610 +  let ?c = "\<lambda>i. if i = Suc k then a i j else c i"
1.611 +  from Suc.prems have Sk: "Suc k \<in> {1 .. dimindex (UNIV :: 'n set)}" by simp
1.612 +  from Suc.prems have k': "k \<le> dimindex (UNIV :: 'n set)" by arith
1.613 +  have thif: "\<And>a b c d. (if b \<or> a then c else d) = (if a then c else if b then c else d)" by simp
1.614 +  have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
1.615 +     (if c then (if a then b else d) else (if a then b else e))" by simp
1.616 +  have "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
1.617 +        det (\<chi> i. if i = Suc k then setsum (a i) S
1.618 +                 else if i \<le> k then setsum (a i) S else c i)"
1.619 +    unfolding le_Suc_eq thif  ..
1.620 +  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<le> k then setsum (a i) S
1.621 +                    else if i = Suc k then a i j else c i))"
1.622 +    unfolding det_linear_row_setsum[OF fS Sk]
1.623 +    apply (subst thif2)
1.624 +    by (simp cong del: if_weak_cong cong add: if_cong)
1.625 +  finally have tha:
1.626 +    "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
1.627 +     (\<Sum>(j, f)\<in>S \<times> ?F k. det (\<chi> i. if i \<le> k then a i (f i)
1.628 +                                else if i = Suc k then a i j
1.629 +                                else c i))"
1.630 +    unfolding  Suc.hyps[OF k'] unfolding setsum_cartesian_product by blast
1.631 +  show ?case unfolding tha
1.632 +    apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
1.633 +      blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS],
1.634 +      blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS], auto intro: ext)
1.635 +    apply (rule cong[OF refl[of det]])
1.636 +    by vector
1.637 +qed
1.638 +
1.639 +lemma det_linear_rows_setsum:
1.640 +  assumes fS: "finite S"
1.641 +  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 \<in> {1 .. dimindex (UNIV :: 'n set)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1.. dimindex (UNIV :: 'n set)} \<longrightarrow> f i = i)}"
1.642 +proof-
1.643 +  have th0: "\<And>x y. ((\<chi> i. if i <= dimindex(UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
1.644 +
1.645 +  from det_linear_rows_setsum_lemma[OF fS, of "dimindex (UNIV :: 'n set)" a, unfolded th0, OF order_refl] show ?thesis by blast
1.646 +qed
1.647 +
1.648 +lemma matrix_mul_setsum_alt:
1.649 +  fixes A B :: "'a::comm_ring_1^'n^'n"
1.650 +  shows "A ** B = (\<chi> i. setsum (\<lambda>k. A\$i\$k *s B \$ k) {1 .. dimindex (UNIV :: 'n set)})"
1.651 +  by (vector matrix_matrix_mult_def setsum_component)
1.652 +
1.653 +lemma det_rows_mul:
1.654 +  "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
1.655 +  setprod (\<lambda>i. c i) {1..dimindex(UNIV:: 'n set)} * det((\<chi> i. a i)::'a^'n^'n)"
1.656 +proof (simp add: det_def Cart_lambda_beta' setsum_right_distrib vector_component cong add: setprod_cong del: One_nat_def, rule setsum_cong2)
1.657 +  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
1.658 +  let ?PU = "{p. p permutes ?U}"
1.659 +  fix p assume pU: "p \<in> ?PU"
1.660 +  let ?s = "of_int (sign p)"
1.661 +  from pU have p: "p permutes ?U" by blast
1.662 +  have "setprod (\<lambda>i. (c i *s a i) \$ p i) ?U = setprod (\<lambda>i. c i * a i \$ p i) ?U"
1.663 +    apply (rule setprod_cong, blast)
1.664 +    by (auto simp only: permutes_in_image[OF p] intro: vector_smult_component)
1.665 +  also have "\<dots> = setprod c ?U * setprod (\<lambda>i. a i \$ p i) ?U"
1.666 +    unfolding setprod_timesf ..
1.667 +  finally show "?s * (\<Prod>xa\<in>?U. (c xa *s a xa) \$ p xa) =
1.668 +        setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa \$ p xa))" by (simp add: ring_simps)
1.669 +qed
1.670 +
1.671 +lemma det_mul:
1.672 +  fixes A B :: "'a::ordered_idom^'n^'n"
1.673 +  shows "det (A ** B) = det A * det B"
1.674 +proof-
1.675 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.676 +  let ?F = "{f. (\<forall>i\<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
1.677 +  let ?PU = "{p. p permutes ?U}"
1.678 +  have fU: "finite ?U" by simp
1.679 +  have fF: "finite ?F"  using finite_bounded_functions[OF fU] .
1.680 +  {fix p assume p: "p permutes ?U"
1.681 +
1.682 +    have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
1.683 +      using p[unfolded permutes_def] by simp}
1.684 +  then have PUF: "?PU \<subseteq> ?F"  by blast
1.685 +  {fix f assume fPU: "f \<in> ?F - ?PU"
1.686 +    have fUU: "f ` ?U \<subseteq> ?U" using fPU by auto
1.687 +    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U"
1.688 +      "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
1.689 +      by auto
1.690 +
1.691 +    let ?A = "(\<chi> i. A\$i\$f i *s B\$f i) :: 'a^'n^'n"
1.692 +    let ?B = "(\<chi> i. B\$f i) :: 'a^'n^'n"
1.693 +    {assume fni: "\<not> inj_on f ?U"
1.694 +      then obtain i j where ij: "i \<in> ?U" "j \<in> ?U" "f i = f j" "i \<noteq> j"
1.695 +	unfolding inj_on_def by blast
1.696 +      from ij
1.697 +      have rth: "row i ?B = row j ?B" by (vector row_def)
1.698 +      from det_identical_rows[OF ij(1,2,4) rth]
1.699 +      have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
1.700 +	unfolding det_rows_mul by simp}
1.701 +    moreover
1.702 +    {assume fi: "inj_on f ?U"
1.703 +      from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
1.704 +	unfolding inj_on_def
1.705 +	apply (case_tac "i \<in> ?U")
1.706 +	apply (case_tac "j \<in> ?U") by metis+
1.707 +      note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
1.708 +
1.709 +      {fix y
1.710 +	from fs f have "\<exists>x. f x = y" by (cases "y \<in> ?U") blast+
1.711 +	then obtain x where x: "f x = y" by blast
1.712 +	{fix z assume z: "f z = y" from fith x z have "z = x" by metis}
1.713 +	with x have "\<exists>!x. f x = y" by blast}
1.714 +      with f(3) have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast}
1.715 +    ultimately have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast}
1.716 +  hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by simp
1.717 +  {fix p assume pU: "p \<in> ?PU"
1.718 +    from pU have p: "p permutes ?U" by blast
1.719 +    let ?s = "\<lambda>p. of_int (sign p)"
1.720 +    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) *
1.721 +               (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))"
1.722 +    have "(setsum (\<lambda>q. ?s q *
1.723 +            (\<Prod>i\<in> ?U. (\<chi> i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
1.724 +        (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) *
1.725 +               (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))) ?PU)"
1.726 +      unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
1.727 +    proof(rule setsum_cong2)
1.728 +      fix q assume qU: "q \<in> ?PU"
1.729 +      hence q: "q permutes ?U" by blast
1.730 +      from p q have pp: "permutation p" and pq: "permutation q"
1.731 +	unfolding permutation_permutes by auto
1.732 +      have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
1.733 +	"\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
1.734 +	unfolding mult_assoc[symmetric]	unfolding of_int_mult[symmetric]
1.735 +	by (simp_all add: sign_idempotent)
1.736 +      have ths: "?s q = ?s p * ?s (q o inv p)"
1.737 +	using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
1.738 +	by (simp add:  th00 mult_ac sign_idempotent sign_compose)
1.739 +      have th001: "setprod (\<lambda>i. B\$i\$ q (inv p i)) ?U = setprod ((\<lambda>i. B\$i\$ q (inv p i)) o p) ?U"
1.740 +	by (rule setprod_permute[OF p])
1.741 +      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"
1.742 +	unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
1.743 +	apply (rule setprod_cong[OF refl])
1.744 +	using permutes_in_image[OF q] by vector
1.745 +      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 o inv p) * setprod (\<lambda>i. B\$i\$(q o inv p) i) ?U)"
1.746 +	using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
1.747 +	by (simp add: sign_nz th00 ring_simps sign_idempotent sign_compose)
1.748 +    qed
1.749 +  }
1.750 +  then have th2: "setsum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU = det A * det B"
1.751 +    unfolding det_def setsum_product
1.752 +    by (rule setsum_cong2)
1.753 +  have "det (A**B) = setsum (\<lambda>f.  det (\<chi> i. A \$ i \$ f i *s B \$ f i)) ?F"
1.754 +    unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] ..
1.755 +  also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU"
1.756 +    unfolding setsum_superset[OF fF PUF zth, symmetric]
1.757 +    unfolding det_rows_mul ..
1.758 +  finally show ?thesis unfolding th2 .
1.759 +qed
1.760 +
1.761 +(* ------------------------------------------------------------------------- *)
1.762 +(* Relation to invertibility.                                                *)
1.763 +(* ------------------------------------------------------------------------- *)
1.764 +
1.765 +lemma invertible_left_inverse:
1.766 +  fixes A :: "real^'n^'n"
1.767 +  shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). B** A = mat 1)"
1.768 +  by (metis invertible_def matrix_left_right_inverse)
1.769 +
1.770 +lemma invertible_righ_inverse:
1.771 +  fixes A :: "real^'n^'n"
1.772 +  shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). A** B = mat 1)"
1.773 +  by (metis invertible_def matrix_left_right_inverse)
1.774 +
1.775 +lemma invertible_det_nz:
1.776 +  fixes A::"real ^'n^'n"
1.777 +  shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
1.778 +proof-
1.779 +  {assume "invertible A"
1.780 +    then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
1.781 +      unfolding invertible_righ_inverse by blast
1.782 +    hence "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp
1.783 +    hence "det A \<noteq> 0"
1.784 +      apply (simp add: det_mul det_I) by algebra }
1.785 +  moreover
1.786 +  {assume H: "\<not> invertible A"
1.787 +    let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
1.788 +    have fU: "finite ?U" by simp
1.789 +    from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
1.790 +      and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
1.791 +      unfolding invertible_righ_inverse
1.792 +      unfolding matrix_right_invertible_independent_rows by blast
1.793 +    have stupid: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
1.794 +      apply (drule_tac f="op + (- a)" in cong[OF refl])
1.795 +      apply (simp only: ab_left_minus add_assoc[symmetric])
1.796 +      apply simp
1.797 +      done
1.798 +    from c ci
1.799 +    have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s c j *s row j A) (?U - {i})"
1.800 +      unfolding setsum_diff1'[OF fU iU] setsum_cmul
1.801 +      apply (simp add: field_simps)
1.802 +      apply (rule vector_mul_lcancel_imp[OF ci])
1.803 +      apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps)
1.804 +      unfolding stupid ..
1.805 +    have thr: "- row i A \<in> span {row j A| j. j\<in> ?U \<and> j \<noteq> i}"
1.806 +      unfolding thr0
1.807 +      apply (rule span_setsum)
1.808 +      apply simp
1.809 +      apply (rule ballI)
1.810 +      apply (rule span_mul)+
1.811 +      apply (rule span_superset)
1.812 +      apply auto
1.813 +      done
1.814 +    let ?B = "(\<chi> k. if k = i then 0 else row k A) :: real ^'n^'n"
1.815 +    have thrb: "row i ?B = 0" using iU by (vector row_def)
1.816 +    have "det A = 0"
1.817 +      unfolding det_row_span[OF iU thr, symmetric] right_minus
1.818 +      unfolding  det_zero_row[OF iU thrb]  ..}
1.819 +  ultimately show ?thesis by blast
1.820 +qed
1.821 +
1.822 +(* ------------------------------------------------------------------------- *)
1.823 +(* Cramer's rule.                                                            *)
1.824 +(* ------------------------------------------------------------------------- *)
1.825 +
1.826 +lemma cramer_lemma_transp:
1.827 +  fixes A:: "'a::ordered_idom^'n^'n" and x :: "'a ^'n"
1.828 +  assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
1.829 +  shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x\$i *s row i A) {1 .. dimindex(UNIV::'n set)}
1.830 +                           else row i A)::'a^'n^'n) = x\$k * det A"
1.831 +  (is "?lhs = ?rhs")
1.832 +proof-
1.833 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.834 +  let ?Uk = "?U - {k}"
1.835 +  have U: "?U = insert k ?Uk" using k by blast
1.836 +  have fUk: "finite ?Uk" by simp
1.837 +  have kUk: "k \<notin> ?Uk" by simp
1.838 +  have th00: "\<And>k s. x\$k *s row k A + s = (x\$k - 1) *s row k A + row k A + s"
1.839 +    by (vector ring_simps)
1.840 +  have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by (auto intro: ext)
1.841 +  have "(\<chi> i. row i A) = A" by (vector row_def)
1.842 +  then have thd1: "det (\<chi> i. row i A) = det A"  by simp
1.843 +  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"
1.844 +    apply (rule det_row_span[OF k])
1.845 +    apply (rule span_setsum[OF fUk])
1.846 +    apply (rule ballI)
1.847 +    apply (rule span_mul)
1.848 +    apply (rule span_superset)
1.849 +    apply auto
1.850 +    done
1.851 +  show "?lhs = x\$k * det A"
1.852 +    apply (subst U)
1.853 +    unfolding setsum_insert[OF fUk kUk]
1.854 +    apply (subst th00)
1.856 +    apply (subst det_row_add[OF k])
1.857 +    unfolding thd0
1.858 +    unfolding det_row_mul[OF k]
1.859 +    unfolding th001[of k "\<lambda>i. row i A"]
1.860 +    unfolding thd1  by (simp add: ring_simps)
1.861 +qed
1.862 +
1.863 +lemma cramer_lemma:
1.864 +  fixes A :: "'a::ordered_idom ^'n^'n"
1.865 +  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}" (is " _ \<in> ?U")
1.866 +  shows "det((\<chi> i j. if j = k then (A *v x)\$i else A\$i\$j):: 'a^'n^'n) = x\$k * det A"
1.867 +proof-
1.868 +  have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transp A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
1.869 +    by (auto simp add: row_transp intro: setsum_cong2)
1.870 +  show ?thesis
1.871 +  unfolding matrix_mult_vsum
1.872 +  unfolding cramer_lemma_transp[OF k, of x "transp A", unfolded det_transp, symmetric]
1.873 +  unfolding stupid[of "\<lambda>i. x\$i"]
1.874 +  apply (subst det_transp[symmetric])
1.875 +  apply (rule cong[OF refl[of det]]) by (vector transp_def column_def row_def)
1.876 +qed
1.877 +
1.878 +lemma cramer:
1.879 +  fixes A ::"real^'n^'n"
1.880 +  assumes d0: "det A \<noteq> 0"
1.881 +  shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j :: real^'n^'n) / det A)"
1.882 +proof-
1.883 +  from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
1.884 +    unfolding invertible_det_nz[symmetric] invertible_def by blast
1.885 +  have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
1.886 +  hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
1.887 +  then have xe: "\<exists>x. A*v x = b" by blast
1.888 +  {fix x assume x: "A *v x = b"
1.889 +  have "x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j :: real^'n^'n) / det A)"
1.890 +    unfolding x[symmetric]
1.891 +    using d0 by (simp add: Cart_eq Cart_lambda_beta' cramer_lemma field_simps)}
1.892 +  with xe show ?thesis by auto
1.893 +qed
1.894 +
1.895 +(* ------------------------------------------------------------------------- *)
1.896 +(* Orthogonality of a transformation and matrix.                             *)
1.897 +(* ------------------------------------------------------------------------- *)
1.898 +
1.899 +definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
1.900 +
1.901 +lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^'n). norm (f v) = norm v)"
1.902 +  unfolding orthogonal_transformation_def
1.903 +  apply auto
1.904 +  apply (erule_tac x=v in allE)+
1.905 +  apply (simp add: real_vector_norm_def)
1.907 +
1.908 +definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transp Q ** Q = mat 1 \<and> Q ** transp Q = mat 1"
1.909 +
1.910 +lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n)  \<longleftrightarrow> transp Q ** Q = mat 1"
1.911 +  by (metis matrix_left_right_inverse orthogonal_matrix_def)
1.912 +
1.913 +lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1)"
1.914 +  by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid)
1.915 +
1.916 +lemma orthogonal_matrix_mul:
1.917 +  fixes A :: "real ^'n^'n"
1.918 +  assumes oA : "orthogonal_matrix A"
1.919 +  and oB: "orthogonal_matrix B"
1.920 +  shows "orthogonal_matrix(A ** B)"
1.921 +  using oA oB
1.922 +  unfolding orthogonal_matrix matrix_transp_mul
1.923 +  apply (subst matrix_mul_assoc)
1.924 +  apply (subst matrix_mul_assoc[symmetric])
1.925 +  by (simp add: matrix_mul_rid)
1.926 +
1.927 +lemma orthogonal_transformation_matrix:
1.928 +  fixes f:: "real^'n \<Rightarrow> real^'n"
1.929 +  shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
1.930 +  (is "?lhs \<longleftrightarrow> ?rhs")
1.931 +proof-
1.932 +  let ?mf = "matrix f"
1.933 +  let ?ot = "orthogonal_transformation f"
1.934 +  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
1.935 +  have fU: "finite ?U" by simp
1.936 +  let ?m1 = "mat 1 :: real ^'n^'n"
1.937 +  {assume ot: ?ot
1.938 +    from ot have lf: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
1.939 +      unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
1.940 +    {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U"
1.941 +      let ?A = "transp ?mf ** ?mf"
1.942 +      have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
1.943 +	"\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
1.944 +	by simp_all
1.945 +      from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul] i j
1.946 +      have "?A\$i\$j = ?m1 \$ i \$ j"
1.947 +	by (simp add: Cart_lambda_beta' dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def del: One_nat_def)}
1.948 +    hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
1.949 +    with lf have ?rhs by blast}
1.950 +  moreover
1.951 +  {assume lf: "linear f" and om: "orthogonal_matrix ?mf"
1.952 +    from lf om have ?lhs
1.953 +      unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
1.954 +      unfolding matrix_works[OF lf, symmetric]
1.955 +      apply (subst dot_matrix_vector_mul)
1.956 +      by (simp add: dot_matrix_product matrix_mul_lid del: One_nat_def)}
1.957 +  ultimately show ?thesis by blast
1.958 +qed
1.959 +
1.960 +lemma det_orthogonal_matrix:
1.961 +  fixes Q:: "'a::ordered_idom^'n^'n"
1.962 +  assumes oQ: "orthogonal_matrix Q"
1.963 +  shows "det Q = 1 \<or> det Q = - 1"
1.964 +proof-
1.965 +
1.966 +  have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
1.967 +  proof-
1.968 +    fix x:: 'a
1.969 +    have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: ring_simps)
1.970 +    have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
1.971 +      apply (subst eq_iff_diff_eq_0) by simp
1.972 +    have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
1.973 +    also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
1.974 +    finally show "?ths x" ..
1.975 +  qed
1.976 +  from oQ have "Q ** transp Q = mat 1" by (metis orthogonal_matrix_def)
1.977 +  hence "det (Q ** transp Q) = det (mat 1:: 'a^'n^'n)" by simp
1.978 +  hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transp)
1.979 +  then show ?thesis unfolding th .
1.980 +qed
1.981 +
1.982 +(* ------------------------------------------------------------------------- *)
1.983 +(* Linearity of scaling, and hence isometry, that preserves origin.          *)
1.984 +(* ------------------------------------------------------------------------- *)
1.985 +lemma scaling_linear:
1.986 +  fixes f :: "real ^'n \<Rightarrow> real ^'n"
1.987 +  assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
1.988 +  shows "linear f"
1.989 +proof-
1.990 +  {fix v w
1.991 +    {fix x note fd[rule_format, of x 0, unfolded dist_def f0 diff_0_right] }
1.992 +    note th0 = this
1.993 +    have "f v \<bullet> f w = c^2 * (v \<bullet> w)"
1.994 +      unfolding dot_norm_neg dist_def[symmetric]
1.995 +      unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
1.996 +  note fc = this
1.997 +  show ?thesis unfolding linear_def vector_eq
1.999 +qed
1.1000 +
1.1001 +lemma isometry_linear:
1.1002 +  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
1.1003 +        \<Longrightarrow> linear f"
1.1004 +by (rule scaling_linear[where c=1]) simp_all
1.1005 +
1.1006 +(* ------------------------------------------------------------------------- *)
1.1007 +(* Hence another formulation of orthogonal transformation.                   *)
1.1008 +(* ------------------------------------------------------------------------- *)
1.1009 +
1.1010 +lemma orthogonal_transformation_isometry:
1.1011 +  "orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
1.1012 +  unfolding orthogonal_transformation
1.1013 +  apply (rule iffI)
1.1014 +  apply clarify
1.1015 +  apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_def)
1.1016 +  apply (rule conjI)
1.1017 +  apply (rule isometry_linear)
1.1018 +  apply simp
1.1019 +  apply simp
1.1020 +  apply clarify
1.1021 +  apply (erule_tac x=v in allE)
1.1022 +  apply (erule_tac x=0 in allE)
1.1023 +  by (simp add: dist_def)
1.1024 +
1.1025 +(* ------------------------------------------------------------------------- *)
1.1026 +(* Can extend an isometry from unit sphere.                                  *)
1.1027 +(* ------------------------------------------------------------------------- *)
1.1028 +
1.1029 +lemma isometry_sphere_extend:
1.1030 +  fixes f:: "real ^'n \<Rightarrow> real ^'n"
1.1031 +  assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
1.1032 +  and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
1.1033 +  shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
1.1034 +proof-
1.1035 +  {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
1.1036 +    assume H: "x = norm x *s x0" "y = norm y *s y0"
1.1037 +    "x' = norm x *s x0'" "y' = norm y *s y0'"
1.1038 +    "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
1.1039 +    "norm(x0' - y0') = norm(x0 - y0)"
1.1040 +
1.1041 +    have "norm(x' - y') = norm(x - y)"
1.1042 +      apply (subst H(1))
1.1043 +      apply (subst H(2))
1.1044 +      apply (subst H(3))
1.1045 +      apply (subst H(4))
1.1046 +      using H(5-9)
1.1047 +      apply (simp add: norm_eq norm_eq_1)
1.1048 +      apply (simp add: dot_lsub dot_rsub dot_lmult dot_rmult)
1.1049 +      apply (simp add: ring_simps)
1.1050 +      by (simp only: right_distrib[symmetric])}
1.1051 +  note th0 = this
1.1052 +  let ?g = "\<lambda>x. if x = 0 then 0 else norm x *s f (inverse (norm x) *s x)"
1.1053 +  {fix x:: "real ^'n" assume nx: "norm x = 1"
1.1054 +    have "?g x = f x" using nx by (simp add: norm_eq_0[symmetric])}
1.1055 +  hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
1.1056 +  have g0: "?g 0 = 0" by simp
1.1057 +  {fix x y :: "real ^'n"
1.1058 +    {assume "x = 0" "y = 0"
1.1059 +      then have "dist (?g x) (?g y) = dist x y" by simp }
1.1060 +    moreover
1.1061 +    {assume "x = 0" "y \<noteq> 0"
1.1062 +      then have "dist (?g x) (?g y) = dist x y"
1.1063 +	apply (simp add: dist_def norm_neg norm_mul norm_eq_0)
1.1064 +	apply (rule f1[rule_format])
1.1065 +	by(simp add: norm_mul norm_eq_0 field_simps)}
1.1066 +    moreover
1.1067 +    {assume "x \<noteq> 0" "y = 0"
1.1068 +      then have "dist (?g x) (?g y) = dist x y"
1.1069 +	apply (simp add: dist_def norm_neg norm_mul norm_eq_0)
1.1070 +	apply (rule f1[rule_format])
1.1071 +	by(simp add: norm_mul norm_eq_0 field_simps)}
1.1072 +    moreover
1.1073 +    {assume z: "x \<noteq> 0" "y \<noteq> 0"
1.1074 +      have th00: "x = norm x *s inverse (norm x) *s x" "y = norm y *s inverse (norm y) *s y" "norm x *s f (inverse (norm x) *s x) = norm x *s f (inverse (norm x) *s x)"
1.1075 +	"norm y *s f (inverse (norm y) *s y) = norm y *s f (inverse (norm y) *s y)"
1.1076 +	"norm (inverse (norm x) *s x) = 1"
1.1077 +	"norm (f (inverse (norm x) *s x)) = 1"
1.1078 +	"norm (inverse (norm y) *s y) = 1"
1.1079 +	"norm (f (inverse (norm y) *s y)) = 1"
1.1080 +	"norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
1.1081 +	norm (inverse (norm x) *s x - inverse (norm y) *s y)"
1.1082 +	using z
1.1083 +	by (auto simp add: norm_eq_0 vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
1.1084 +      from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
1.1085 +	by (simp add: dist_def)}
1.1086 +    ultimately have "dist (?g x) (?g y) = dist x y" by blast}
1.1087 +  note thd = this
1.1088 +    show ?thesis
1.1089 +    apply (rule exI[where x= ?g])
1.1090 +    unfolding orthogonal_transformation_isometry
1.1091 +      using  g0 thfg thd by metis
1.1092 +qed
1.1093 +
1.1094 +(* ------------------------------------------------------------------------- *)
1.1095 +(* Rotation, reflection, rotoinversion.                                      *)
1.1096 +(* ------------------------------------------------------------------------- *)
1.1097 +
1.1098 +definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
1.1099 +definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
1.1100 +
1.1101 +lemma orthogonal_rotation_or_rotoinversion:
1.1102 +  fixes Q :: "'a::ordered_idom^'n^'n"
1.1103 +  shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
1.1104 +  by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
1.1105 +(* ------------------------------------------------------------------------- *)
1.1106 +(* Explicit formulas for low dimensions.                                     *)
1.1107 +(* ------------------------------------------------------------------------- *)
1.1108 +
1.1109 +lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
1.1110 +
1.1111 +lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
1.1112 +  by (simp add: nat_number setprod_numseg mult_commute)
1.1113 +lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
1.1114 +  by (simp add: nat_number setprod_numseg mult_commute)
1.1115 +
1.1116 +lemma det_1: "det (A::'a::comm_ring_1^1^1) = A\$1\$1"
1.1117 +  by (simp add: det_def dimindex_def permutes_sing sign_id del: One_nat_def)
1.1118 +
1.1119 +lemma det_2: "det (A::'a::comm_ring_1^2^2) = A\$1\$1 * A\$2\$2 - A\$1\$2 * A\$2\$1"
1.1120 +proof-
1.1121 +  have f12: "finite {2::nat}" "1 \<notin> {2::nat}" by auto
1.1122 +  have th12: "{1 .. 2} = insert (1::nat) {2}" by auto
1.1123 +  show ?thesis
1.1124 +  apply (simp add: det_def dimindex_def th12 del: One_nat_def)
1.1125 +  unfolding setsum_over_permutations_insert[OF f12]
1.1126 +  unfolding permutes_sing
1.1127 +  apply (simp add: sign_swap_id sign_id swap_id_eq del: One_nat_def)
1.1128 +  by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
1.1129 +qed
1.1130 +
1.1131 +lemma det_3: "det (A::'a::comm_ring_1^3^3) =
1.1132 +  A\$1\$1 * A\$2\$2 * A\$3\$3 +
1.1133 +  A\$1\$2 * A\$2\$3 * A\$3\$1 +
1.1134 +  A\$1\$3 * A\$2\$1 * A\$3\$2 -
1.1135 +  A\$1\$1 * A\$2\$3 * A\$3\$2 -
1.1136 +  A\$1\$2 * A\$2\$1 * A\$3\$3 -
1.1137 +  A\$1\$3 * A\$2\$2 * A\$3\$1"
1.1138 +proof-
1.1139 +  have f123: "finite {(2::nat), 3}" "1 \<notin> {(2::nat), 3}" by auto
1.1140 +  have f23: "finite {(3::nat)}" "2 \<notin> {(3::nat)}" by auto
1.1141 +  have th12: "{1 .. 3} = insert (1::nat) (insert 2 {3})" by auto
1.1142 +
1.1143 +  show ?thesis
1.1144 +  apply (simp add: det_def dimindex_def th12 del: One_nat_def)
1.1145 +  unfolding setsum_over_permutations_insert[OF f123]
1.1146 +  unfolding setsum_over_permutations_insert[OF f23]
1.1147 +
1.1148 +  unfolding permutes_sing
1.1149 +  apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq del: One_nat_def)
1.1150 +  apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31) One_nat_def)
1.1151 +  by (simp add: ring_simps)
1.1152 +qed
1.1153 +
1.1154 +end
1.1155 \ No newline at end of file
```