src/HOL/Library/Determinants.thy
author nipkow
Wed Apr 01 22:29:10 2009 +0200 (2009-04-01)
changeset 30843 3419ca741dbf
parent 30837 3d4832d9f7e4
child 31280 8ef7ba78bf26
permissions -rw-r--r--
cleaned up setprod_zero-related lemmas
     1 (* Title:      Determinants
     2    Author:     Amine Chaieb, University of Cambridge
     3 *)
     4 
     5 header {* Traces, Determinant of square matrices and some properties *}
     6 
     7 theory Determinants
     8 imports Euclidean_Space Permutations
     9 begin
    10 
    11 subsection{* First some facts about products*}
    12 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)"
    13 apply clarsimp
    14 by(subgoal_tac "insert a A = A", auto)
    15 
    16 lemma setprod_add_split:
    17   assumes mn: "(m::nat) <= n + 1"
    18   shows "setprod f {m.. n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
    19 proof-
    20   let ?A = "{m .. n+p}"
    21   let ?B = "{m .. n}"
    22   let ?C = "{n+1..n+p}"
    23   from mn have un: "?B \<union> ?C = ?A" by auto
    24   from mn have dj: "?B \<inter> ?C = {}" by auto
    25   have f: "finite ?B" "finite ?C" by simp_all
    26   from setprod_Un_disjoint[OF f dj, of f, unfolded un] show ?thesis .
    27 qed
    28 
    29 
    30 lemma setprod_offset: "setprod f {(m::nat) + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
    31 apply (rule setprod_reindex_cong[where f="op + p"])
    32 apply (auto simp add: image_iff Bex_def inj_on_def)
    33 apply arith
    34 apply (rule ext)
    35 apply (simp add: add_commute)
    36 done
    37 
    38 lemma setprod_singleton: "setprod f {x} = f x" by simp
    39 
    40 lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
    41 
    42 lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
    43   "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
    44                              else setprod f {m..n})"
    45   by (auto simp add: atLeastAtMostSuc_conv)
    46 
    47 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)"
    48   shows "setprod f S \<le> setprod g S"
    49 using fS fg
    50 apply(induct S)
    51 apply simp
    52 apply auto
    53 apply (rule mult_mono)
    54 apply (auto intro: setprod_nonneg)
    55 done
    56 
    57   (* FIXME: In Finite_Set there is a useless further assumption *)
    58 lemma setprod_inversef: "finite A ==> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: {division_by_zero, field})"
    59   apply (erule finite_induct)
    60   apply (simp)
    61   apply simp
    62   done
    63 
    64 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)"
    65   shows "setprod f S \<le> 1"
    66 using setprod_le[OF fS f] unfolding setprod_1 .
    67 
    68 subsection{* Trace *}
    69 
    70 definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
    71   "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
    72 
    73 lemma trace_0: "trace(mat 0) = 0"
    74   by (simp add: trace_def mat_def)
    75 
    76 lemma trace_I: "trace(mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
    77   by (simp add: trace_def mat_def)
    78 
    79 lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
    80   by (simp add: trace_def setsum_addf)
    81 
    82 lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
    83   by (simp add: trace_def setsum_subtractf)
    84 
    85 lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'n) ** B) = trace (B**A)"
    86   apply (simp add: trace_def matrix_matrix_mult_def)
    87   apply (subst setsum_commute)
    88   by (simp add: mult_commute)
    89 
    90 (* ------------------------------------------------------------------------- *)
    91 (* Definition of determinant.                                                *)
    92 (* ------------------------------------------------------------------------- *)
    93 
    94 definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
    95   "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)}"
    96 
    97 (* ------------------------------------------------------------------------- *)
    98 (* A few general lemmas we need below.                                       *)
    99 (* ------------------------------------------------------------------------- *)
   100 
   101 lemma setprod_permute:
   102   assumes p: "p permutes S"
   103   shows "setprod f S = setprod (f o p) S"
   104 proof-
   105   {assume "\<not> finite S" hence ?thesis by simp}
   106   moreover
   107   {assume fS: "finite S"
   108     then have ?thesis
   109       apply (simp add: setprod_def cong del:strong_setprod_cong)
   110       apply (rule ab_semigroup_mult.fold_image_permute)
   111       apply (auto simp add: p)
   112       apply unfold_locales
   113       done}
   114   ultimately show ?thesis by blast
   115 qed
   116 
   117 lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
   118   by (blast intro!: setprod_permute)
   119 
   120 (* ------------------------------------------------------------------------- *)
   121 (* Basic determinant properties.                                             *)
   122 (* ------------------------------------------------------------------------- *)
   123 
   124 lemma det_transp: "det (transp A) = det (A::'a::comm_ring_1 ^'n^'n::finite)"
   125 proof-
   126   let ?di = "\<lambda>A i j. A$i$j"
   127   let ?U = "(UNIV :: 'n set)"
   128   have fU: "finite ?U" by simp
   129   {fix p assume p: "p \<in> {p. p permutes ?U}"
   130     from p have pU: "p permutes ?U" by blast
   131     have sth: "sign (inv p) = sign p"
   132       by (metis sign_inverse fU p mem_def Collect_def permutation_permutes)
   133     from permutes_inj[OF pU]
   134     have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
   135     from permutes_image[OF pU]
   136     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
   137     also have "\<dots> = setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U"
   138       unfolding setprod_reindex[OF pi] ..
   139     also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
   140     proof-
   141       {fix i assume i: "i \<in> ?U"
   142 	from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
   143 	have "((\<lambda>i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)"
   144 	  unfolding transp_def by (simp add: expand_fun_eq)}
   145       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)
   146     qed
   147     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
   148       by simp}
   149   then show ?thesis unfolding det_def apply (subst setsum_permutations_inverse)
   150   apply (rule setsum_cong2) by blast
   151 qed
   152 
   153 lemma det_lowerdiagonal:
   154   fixes A :: "'a::comm_ring_1^'n^'n::{finite,wellorder}"
   155   assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
   156   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
   157 proof-
   158   let ?U = "UNIV:: 'n set"
   159   let ?PU = "{p. p permutes ?U}"
   160   let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
   161   have fU: "finite ?U" by simp
   162   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   163   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
   164   {fix p assume p: "p \<in> ?PU -{id}"
   165     from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
   166     from permutes_natset_le[OF pU] pid obtain i where
   167       i: "p i > i" by (metis not_le)
   168     from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
   169     from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
   170   then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
   171   from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
   172     unfolding det_def by (simp add: sign_id)
   173 qed
   174 
   175 lemma det_upperdiagonal:
   176   fixes A :: "'a::comm_ring_1^'n^'n::{finite,wellorder}"
   177   assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
   178   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
   179 proof-
   180   let ?U = "UNIV:: 'n set"
   181   let ?PU = "{p. p permutes ?U}"
   182   let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
   183   have fU: "finite ?U" by simp
   184   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   185   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
   186   {fix p assume p: "p \<in> ?PU -{id}"
   187     from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
   188     from permutes_natset_ge[OF pU] pid obtain i where
   189       i: "p i < i" by (metis not_le)
   190     from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
   191     from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
   192   then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
   193   from   setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
   194     unfolding det_def by (simp add: sign_id)
   195 qed
   196 
   197 lemma det_diagonal:
   198   fixes A :: "'a::comm_ring_1^'n^'n::finite"
   199   assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
   200   shows "det A = setprod (\<lambda>i. A$i$i) (UNIV::'n set)"
   201 proof-
   202   let ?U = "UNIV:: 'n set"
   203   let ?PU = "{p. p permutes ?U}"
   204   let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
   205   have fU: "finite ?U" by simp
   206   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   207   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
   208   {fix p assume p: "p \<in> ?PU - {id}"
   209     then have "p \<noteq> id" by simp
   210     then obtain i where i: "p i \<noteq> i" unfolding expand_fun_eq by auto
   211     from ld [OF i [symmetric]] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
   212     from setprod_zero [OF fU ex] have "?pp p = 0" by simp}
   213   then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"  by blast
   214   from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
   215     unfolding det_def by (simp add: sign_id)
   216 qed
   217 
   218 lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n::finite) = 1"
   219 proof-
   220   let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
   221   let ?U = "UNIV :: 'n set"
   222   let ?f = "\<lambda>i j. ?A$i$j"
   223   {fix i assume i: "i \<in> ?U"
   224     have "?f i i = 1" using i by (vector mat_def)}
   225   hence th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
   226     by (auto intro: setprod_cong)
   227   {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
   228     have "?f i j = 0" using i j ij by (vector mat_def) }
   229   then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_diagonal
   230     by blast
   231   also have "\<dots> = 1" unfolding th setprod_1 ..
   232   finally show ?thesis .
   233 qed
   234 
   235 lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n::finite) = 0"
   236   by (simp add: det_def setprod_zero)
   237 
   238 lemma det_permute_rows:
   239   fixes A :: "'a::comm_ring_1^'n^'n::finite"
   240   assumes p: "p permutes (UNIV :: 'n::finite set)"
   241   shows "det(\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
   242   apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric])
   243   apply (subst sum_permutations_compose_right[OF p])
   244 proof(rule setsum_cong2)
   245   let ?U = "UNIV :: 'n set"
   246   let ?PU = "{p. p permutes ?U}"
   247   fix q assume qPU: "q \<in> ?PU"
   248   have fU: "finite ?U" by simp
   249   from qPU have q: "q permutes ?U" by blast
   250   from p q have pp: "permutation p" and qp: "permutation q"
   251     by (metis fU permutation_permutes)+
   252   from permutes_inv[OF p] have ip: "inv p permutes ?U" .
   253     have "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
   254       by (simp only: setprod_permute[OF ip, symmetric])
   255     also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
   256       by (simp only: o_def)
   257     also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
   258     finally   have thp: "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
   259       by blast
   260   show "of_int (sign (q o p)) * setprod (\<lambda>i. A$ p i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
   261     by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
   262 qed
   263 
   264 lemma det_permute_columns:
   265   fixes A :: "'a::comm_ring_1^'n^'n::finite"
   266   assumes p: "p permutes (UNIV :: 'n set)"
   267   shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
   268 proof-
   269   let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
   270   let ?At = "transp A"
   271   have "of_int (sign p) * det A = det (transp (\<chi> i. transp A $ p i))"
   272     unfolding det_permute_rows[OF p, of ?At] det_transp ..
   273   moreover
   274   have "?Ap = transp (\<chi> i. transp A $ p i)"
   275     by (simp add: transp_def Cart_eq)
   276   ultimately show ?thesis by simp
   277 qed
   278 
   279 lemma det_identical_rows:
   280   fixes A :: "'a::ordered_idom^'n^'n::finite"
   281   assumes ij: "i \<noteq> j"
   282   and r: "row i A = row j A"
   283   shows	"det A = 0"
   284 proof-
   285   have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
   286     by simp
   287   have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min)
   288   let ?p = "Fun.swap i j id"
   289   let ?A = "\<chi> i. A $ ?p i"
   290   from r have "A = ?A" by (simp add: Cart_eq row_def swap_def)
   291   hence "det A = det ?A" by simp
   292   moreover have "det A = - det ?A"
   293     by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
   294   ultimately show "det A = 0" by (metis tha)
   295 qed
   296 
   297 lemma det_identical_columns:
   298   fixes A :: "'a::ordered_idom^'n^'n::finite"
   299   assumes ij: "i \<noteq> j"
   300   and r: "column i A = column j A"
   301   shows	"det A = 0"
   302 apply (subst det_transp[symmetric])
   303 apply (rule det_identical_rows[OF ij])
   304 by (metis row_transp r)
   305 
   306 lemma det_zero_row:
   307   fixes A :: "'a::{idom, ring_char_0}^'n^'n::finite"
   308   assumes r: "row i A = 0"
   309   shows "det A = 0"
   310 using r
   311 apply (simp add: row_def det_def Cart_eq)
   312 apply (rule setsum_0')
   313 apply (auto simp: sign_nz)
   314 done
   315 
   316 lemma det_zero_column:
   317   fixes A :: "'a::{idom,ring_char_0}^'n^'n::finite"
   318   assumes r: "column i A = 0"
   319   shows "det A = 0"
   320   apply (subst det_transp[symmetric])
   321   apply (rule det_zero_row [of i])
   322   by (metis row_transp r)
   323 
   324 lemma det_row_add:
   325   fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
   326   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
   327              det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
   328              det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
   329 unfolding det_def Cart_lambda_beta setsum_addf[symmetric]
   330 proof (rule setsum_cong2)
   331   let ?U = "UNIV :: 'n set"
   332   let ?pU = "{p. p permutes ?U}"
   333   let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   334   let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   335   let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   336   fix p assume p: "p \<in> ?pU"
   337   let ?Uk = "?U - {k}"
   338   from p have pU: "p permutes ?U" by blast
   339   have kU: "?U = insert k ?Uk" by blast
   340   {fix j assume j: "j \<in> ?Uk"
   341     from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
   342       by simp_all}
   343   then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
   344     and th2: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?h i $ p i) ?Uk"
   345     apply -
   346     apply (rule setprod_cong, simp_all)+
   347     done
   348   have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
   349   have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
   350     unfolding kU[symmetric] ..
   351   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
   352     apply (rule setprod_insert)
   353     apply simp
   354     by blast
   355   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: ring_simps)
   356   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)
   357   also have "\<dots> = setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i $ p i) (insert k ?Uk)"
   358     unfolding  setprod_insert[OF th3] by simp
   359   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] .
   360   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"
   361     by (simp add: ring_simps)
   362 qed
   363 
   364 lemma det_row_mul:
   365   fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
   366   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
   367              c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
   368 
   369 unfolding det_def Cart_lambda_beta setsum_right_distrib
   370 proof (rule setsum_cong2)
   371   let ?U = "UNIV :: 'n set"
   372   let ?pU = "{p. p permutes ?U}"
   373   let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   374   let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   375   fix p assume p: "p \<in> ?pU"
   376   let ?Uk = "?U - {k}"
   377   from p have pU: "p permutes ?U" by blast
   378   have kU: "?U = insert k ?Uk" by blast
   379   {fix j assume j: "j \<in> ?Uk"
   380     from j have "?f j $ p j = ?g j $ p j" by simp}
   381   then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
   382     apply -
   383     apply (rule setprod_cong, simp_all)
   384     done
   385   have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
   386   have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
   387     unfolding kU[symmetric] ..
   388   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
   389     apply (rule setprod_insert)
   390     apply simp
   391     by blast
   392   also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk" by (simp add: ring_simps)
   393   also have "\<dots> = c* (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk)"
   394     unfolding th1 by (simp add: mult_ac)
   395   also have "\<dots> = c* (setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk))"
   396     unfolding  setprod_insert[OF th3] by simp
   397   finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)" unfolding kU[symmetric] .
   398   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)"
   399     by (simp add: ring_simps)
   400 qed
   401 
   402 lemma det_row_0:
   403   fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
   404   shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
   405 using det_row_mul[of k 0 "\<lambda>i. 1" b]
   406 apply (simp)
   407   unfolding vector_smult_lzero .
   408 
   409 lemma det_row_operation:
   410   fixes A :: "'a::ordered_idom^'n^'n::finite"
   411   assumes ij: "i \<noteq> j"
   412   shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
   413 proof-
   414   let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
   415   have th: "row i ?Z = row j ?Z" by (vector row_def)
   416   have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
   417     by (vector row_def)
   418   show ?thesis
   419     unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2
   420     by simp
   421 qed
   422 
   423 lemma det_row_span:
   424   fixes A :: "'a:: ordered_idom^'n^'n::finite"
   425   assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
   426   shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
   427 proof-
   428   let ?U = "UNIV :: 'n set"
   429   let ?S = "{row j A |j. j \<noteq> i}"
   430   let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
   431   let ?P = "\<lambda>x. ?d (row i A + x) = det A"
   432   {fix k
   433 
   434     have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
   435   then have P0: "?P 0"
   436     apply -
   437     apply (rule cong[of det, OF refl])
   438     by (vector row_def)
   439   moreover
   440   {fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
   441     from zS obtain j where j: "z = row j A" "i \<noteq> j" by blast
   442     let ?w = "row i A + y"
   443     have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector
   444     have thz: "?d z = 0"
   445       apply (rule det_identical_rows[OF j(2)])
   446       using j by (vector row_def)
   447     have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
   448     then have "?P (c*s z + y)" unfolding thz Py det_row_mul[of i] det_row_add[of i]
   449       by simp }
   450 
   451   ultimately show ?thesis
   452     apply -
   453     apply (rule span_induct_alt[of ?P ?S, OF P0])
   454     apply blast
   455     apply (rule x)
   456     done
   457 qed
   458 
   459 (* ------------------------------------------------------------------------- *)
   460 (* May as well do this, though it's a bit unsatisfactory since it ignores    *)
   461 (* exact duplicates by considering the rows/columns as a set.                *)
   462 (* ------------------------------------------------------------------------- *)
   463 
   464 lemma det_dependent_rows:
   465   fixes A:: "'a::ordered_idom^'n^'n::finite"
   466   assumes d: "dependent (rows A)"
   467   shows "det A = 0"
   468 proof-
   469   let ?U = "UNIV :: 'n set"
   470   from d obtain i where i: "row i A \<in> span (rows A - {row i A})"
   471     unfolding dependent_def rows_def by blast
   472   {fix j k assume jk: "j \<noteq> k"
   473     and c: "row j A = row k A"
   474     from det_identical_rows[OF jk c] have ?thesis .}
   475   moreover
   476   {assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
   477     have th0: "- row i A \<in> span {row j A|j. j \<noteq> i}"
   478       apply (rule span_neg)
   479       apply (rule set_rev_mp)
   480       apply (rule i)
   481       apply (rule span_mono)
   482       using H i by (auto simp add: rows_def)
   483     from det_row_span[OF th0]
   484     have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
   485       unfolding right_minus vector_smult_lzero ..
   486     with det_row_mul[of i "0::'a" "\<lambda>i. 1"]
   487     have "det A = 0" by simp}
   488   ultimately show ?thesis by blast
   489 qed
   490 
   491 lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n::finite))" shows "det A = 0"
   492 by (metis d det_dependent_rows rows_transp det_transp)
   493 
   494 (* ------------------------------------------------------------------------- *)
   495 (* Multilinearity and the multiplication formula.                            *)
   496 (* ------------------------------------------------------------------------- *)
   497 
   498 lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
   499   apply (rule iffD1[OF Cart_lambda_unique]) by vector
   500 
   501 lemma det_linear_row_setsum:
   502   assumes fS: "finite S"
   503   shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n::finite) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
   504 proof(induct rule: finite_induct[OF fS])
   505   case 1 thus ?case apply simp  unfolding setsum_empty det_row_0[of k] ..
   506 next
   507   case (2 x F)
   508   then  show ?case by (simp add: det_row_add cong del: if_weak_cong)
   509 qed
   510 
   511 lemma finite_bounded_functions:
   512   assumes fS: "finite S"
   513   shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
   514 proof(induct k)
   515   case 0
   516   have th: "{f. \<forall>i. f i = i} = {id}" by (auto intro: ext)
   517   show ?case by (auto simp add: th)
   518 next
   519   case (Suc k)
   520   let ?f = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
   521   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)})"
   522   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)}"
   523     apply (auto simp add: image_iff)
   524     apply (rule_tac x="x (Suc k)" in bexI)
   525     apply (rule_tac x = "\<lambda>i. if i = Suc k then i else x i" in exI)
   526     apply (auto intro: ext)
   527     done
   528   with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
   529   show ?case by metis
   530 qed
   531 
   532 
   533 lemma eq_id_iff[simp]: "(\<forall>x. f x = x) = (f = id)" by (auto intro: ext)
   534 
   535 lemma det_linear_rows_setsum_lemma:
   536   assumes fS: "finite S" and fT: "finite T"
   537   shows "det((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n::finite) =
   538              setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
   539                  {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
   540 using fT
   541 proof(induct T arbitrary: a c set: finite)
   542   case empty
   543   have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)" by vector
   544   from "empty.prems"  show ?case unfolding th0 by simp
   545 next
   546   case (insert z T a c)
   547   let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
   548   let ?h = "\<lambda>(y,g) i. if i = z then y else g i"
   549   let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
   550   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)"
   551   let ?c = "\<lambda>i. if i = z then a i j else c i"
   552   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
   553   have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
   554      (if c then (if a then b else d) else (if a then b else e))" by simp
   555   from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False" by auto
   556   have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
   557         det (\<chi> i. if i = z then setsum (a i) S
   558                  else if i \<in> T then setsum (a i) S else c i)"
   559     unfolding insert_iff thif ..
   560   also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S
   561                     else if i = z then a i j else c i))"
   562     unfolding det_linear_row_setsum[OF fS]
   563     apply (subst thif2)
   564     using nz by (simp cong del: if_weak_cong cong add: if_cong)
   565   finally have tha:
   566     "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
   567      (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
   568                                 else if i = z then a i j
   569                                 else c i))"
   570     unfolding  insert.hyps unfolding setsum_cartesian_product by blast
   571   show ?case unfolding tha
   572     apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
   573       blast intro: finite_cartesian_product fS finite,
   574       blast intro: finite_cartesian_product fS finite)
   575     using `z \<notin> T`
   576     apply (auto intro: ext)
   577     apply (rule cong[OF refl[of det]])
   578     by vector
   579 qed
   580 
   581 lemma det_linear_rows_setsum:
   582   assumes fS: "finite (S::'n::finite set)"
   583   shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n::finite)) {f. \<forall>i. f i \<in> S}"
   584 proof-
   585   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
   586 
   587   from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp
   588 qed
   589 
   590 lemma matrix_mul_setsum_alt:
   591   fixes A B :: "'a::comm_ring_1^'n^'n::finite"
   592   shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
   593   by (vector matrix_matrix_mult_def setsum_component)
   594 
   595 lemma det_rows_mul:
   596   "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n::finite) =
   597   setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
   598 proof (simp add: det_def setsum_right_distrib cong add: setprod_cong, rule setsum_cong2)
   599   let ?U = "UNIV :: 'n set"
   600   let ?PU = "{p. p permutes ?U}"
   601   fix p assume pU: "p \<in> ?PU"
   602   let ?s = "of_int (sign p)"
   603   from pU have p: "p permutes ?U" by blast
   604   have "setprod (\<lambda>i. c i * a i $ p i) ?U = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
   605     unfolding setprod_timesf ..
   606   then show "?s * (\<Prod>xa\<in>?U. c xa * a xa $ p xa) =
   607         setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))" by (simp add: ring_simps)
   608 qed
   609 
   610 lemma det_mul:
   611   fixes A B :: "'a::ordered_idom^'n^'n::finite"
   612   shows "det (A ** B) = det A * det B"
   613 proof-
   614   let ?U = "UNIV :: 'n set"
   615   let ?F = "{f. (\<forall>i\<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
   616   let ?PU = "{p. p permutes ?U}"
   617   have fU: "finite ?U" by simp
   618   have fF: "finite ?F" by (rule finite)
   619   {fix p assume p: "p permutes ?U"
   620 
   621     have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
   622       using p[unfolded permutes_def] by simp}
   623   then have PUF: "?PU \<subseteq> ?F"  by blast
   624   {fix f assume fPU: "f \<in> ?F - ?PU"
   625     have fUU: "f ` ?U \<subseteq> ?U" using fPU by auto
   626     from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U"
   627       "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
   628       by auto
   629 
   630     let ?A = "(\<chi> i. A$i$f i *s B$f i) :: 'a^'n^'n"
   631     let ?B = "(\<chi> i. B$f i) :: 'a^'n^'n"
   632     {assume fni: "\<not> inj_on f ?U"
   633       then obtain i j where ij: "f i = f j" "i \<noteq> j"
   634 	unfolding inj_on_def by blast
   635       from ij
   636       have rth: "row i ?B = row j ?B" by (vector row_def)
   637       from det_identical_rows[OF ij(2) rth]
   638       have "det (\<chi> i. A$i$f i *s B$f i) = 0"
   639 	unfolding det_rows_mul by simp}
   640     moreover
   641     {assume fi: "inj_on f ?U"
   642       from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
   643 	unfolding inj_on_def by metis
   644       note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
   645 
   646       {fix y
   647 	from fs f have "\<exists>x. f x = y" by blast
   648 	then obtain x where x: "f x = y" by blast
   649 	{fix z assume z: "f z = y" from fith x z have "z = x" by metis}
   650 	with x have "\<exists>!x. f x = y" by blast}
   651       with f(3) have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
   652     ultimately have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
   653   hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A$i$f i *s B$f i) = 0" by simp
   654   {fix p assume pU: "p \<in> ?PU"
   655     from pU have p: "p permutes ?U" by blast
   656     let ?s = "\<lambda>p. of_int (sign p)"
   657     let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) *
   658                (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))"
   659     have "(setsum (\<lambda>q. ?s q *
   660             (\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
   661         (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) *
   662                (?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
   663       unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
   664     proof(rule setsum_cong2)
   665       fix q assume qU: "q \<in> ?PU"
   666       hence q: "q permutes ?U" by blast
   667       from p q have pp: "permutation p" and pq: "permutation q"
   668 	unfolding permutation_permutes by auto
   669       have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
   670 	"\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
   671 	unfolding mult_assoc[symmetric]	unfolding of_int_mult[symmetric]
   672 	by (simp_all add: sign_idempotent)
   673       have ths: "?s q = ?s p * ?s (q o inv p)"
   674 	using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
   675 	by (simp add:  th00 mult_ac sign_idempotent sign_compose)
   676       have th001: "setprod (\<lambda>i. B$i$ q (inv p i)) ?U = setprod ((\<lambda>i. B$i$ q (inv p i)) o p) ?U"
   677 	by (rule setprod_permute[OF p])
   678       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"
   679 	unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
   680 	apply (rule setprod_cong[OF refl])
   681 	using permutes_in_image[OF q] by vector
   682       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$(q o inv p) i) ?U)"
   683 	using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
   684 	by (simp add: sign_nz th00 ring_simps sign_idempotent sign_compose)
   685     qed
   686   }
   687   then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
   688     unfolding det_def setsum_product
   689     by (rule setsum_cong2)
   690   have "det (A**B) = setsum (\<lambda>f.  det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
   691     unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] by simp
   692   also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
   693     using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric]
   694     unfolding det_rows_mul by auto
   695   finally show ?thesis unfolding th2 .
   696 qed
   697 
   698 (* ------------------------------------------------------------------------- *)
   699 (* Relation to invertibility.                                                *)
   700 (* ------------------------------------------------------------------------- *)
   701 
   702 lemma invertible_left_inverse:
   703   fixes A :: "real^'n^'n::finite"
   704   shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). B** A = mat 1)"
   705   by (metis invertible_def matrix_left_right_inverse)
   706 
   707 lemma invertible_righ_inverse:
   708   fixes A :: "real^'n^'n::finite"
   709   shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). A** B = mat 1)"
   710   by (metis invertible_def matrix_left_right_inverse)
   711 
   712 lemma invertible_det_nz:
   713   fixes A::"real ^'n^'n::finite"
   714   shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
   715 proof-
   716   {assume "invertible A"
   717     then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
   718       unfolding invertible_righ_inverse by blast
   719     hence "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp
   720     hence "det A \<noteq> 0"
   721       apply (simp add: det_mul det_I) by algebra }
   722   moreover
   723   {assume H: "\<not> invertible A"
   724     let ?U = "UNIV :: 'n set"
   725     have fU: "finite ?U" by simp
   726     from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
   727       and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
   728       unfolding invertible_righ_inverse
   729       unfolding matrix_right_invertible_independent_rows by blast
   730     have stupid: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
   731       apply (drule_tac f="op + (- a)" in cong[OF refl])
   732       apply (simp only: ab_left_minus add_assoc[symmetric])
   733       apply simp
   734       done
   735     from c ci
   736     have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s c j *s row j A) (?U - {i})"
   737       unfolding setsum_diff1'[OF fU iU] setsum_cmul
   738       apply -
   739       apply (rule vector_mul_lcancel_imp[OF ci])
   740       apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps)
   741       unfolding stupid ..
   742     have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
   743       unfolding thr0
   744       apply (rule span_setsum)
   745       apply simp
   746       apply (rule ballI)
   747       apply (rule span_mul)+
   748       apply (rule span_superset)
   749       apply auto
   750       done
   751     let ?B = "(\<chi> k. if k = i then 0 else row k A) :: real ^'n^'n"
   752     have thrb: "row i ?B = 0" using iU by (vector row_def)
   753     have "det A = 0"
   754       unfolding det_row_span[OF thr, symmetric] right_minus
   755       unfolding  det_zero_row[OF thrb]  ..}
   756   ultimately show ?thesis by blast
   757 qed
   758 
   759 (* ------------------------------------------------------------------------- *)
   760 (* Cramer's rule.                                                            *)
   761 (* ------------------------------------------------------------------------- *)
   762 
   763 lemma cramer_lemma_transp:
   764   fixes A:: "'a::ordered_idom^'n^'n::finite" and x :: "'a ^'n::finite"
   765   shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
   766                            else row i A)::'a^'n^'n) = x$k * det A"
   767   (is "?lhs = ?rhs")
   768 proof-
   769   let ?U = "UNIV :: 'n set"
   770   let ?Uk = "?U - {k}"
   771   have U: "?U = insert k ?Uk" by blast
   772   have fUk: "finite ?Uk" by simp
   773   have kUk: "k \<notin> ?Uk" by simp
   774   have th00: "\<And>k s. x$k *s row k A + s = (x$k - 1) *s row k A + row k A + s"
   775     by (vector ring_simps)
   776   have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by (auto intro: ext)
   777   have "(\<chi> i. row i A) = A" by (vector row_def)
   778   then have thd1: "det (\<chi> i. row i A) = det A"  by simp
   779   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"
   780     apply (rule det_row_span)
   781     apply (rule span_setsum[OF fUk])
   782     apply (rule ballI)
   783     apply (rule span_mul)
   784     apply (rule span_superset)
   785     apply auto
   786     done
   787   show "?lhs = x$k * det A"
   788     apply (subst U)
   789     unfolding setsum_insert[OF fUk kUk]
   790     apply (subst th00)
   791     unfolding add_assoc
   792     apply (subst det_row_add)
   793     unfolding thd0
   794     unfolding det_row_mul
   795     unfolding th001[of k "\<lambda>i. row i A"]
   796     unfolding thd1  by (simp add: ring_simps)
   797 qed
   798 
   799 lemma cramer_lemma:
   800   fixes A :: "'a::ordered_idom ^'n^'n::finite"
   801   shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a^'n^'n) = x$k * det A"
   802 proof-
   803   let ?U = "UNIV :: 'n set"
   804   have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transp A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
   805     by (auto simp add: row_transp intro: setsum_cong2)
   806   show ?thesis  unfolding matrix_mult_vsum
   807   unfolding cramer_lemma_transp[of k x "transp A", unfolded det_transp, symmetric]
   808   unfolding stupid[of "\<lambda>i. x$i"]
   809   apply (subst det_transp[symmetric])
   810   apply (rule cong[OF refl[of det]]) by (vector transp_def column_def row_def)
   811 qed
   812 
   813 lemma cramer:
   814   fixes A ::"real^'n^'n::finite"
   815   assumes d0: "det A \<noteq> 0"
   816   shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
   817 proof-
   818   from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
   819     unfolding invertible_det_nz[symmetric] invertible_def by blast
   820   have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
   821   hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
   822   then have xe: "\<exists>x. A*v x = b" by blast
   823   {fix x assume x: "A *v x = b"
   824   have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
   825     unfolding x[symmetric]
   826     using d0 by (simp add: Cart_eq cramer_lemma field_simps)}
   827   with xe show ?thesis by auto
   828 qed
   829 
   830 (* ------------------------------------------------------------------------- *)
   831 (* Orthogonality of a transformation and matrix.                             *)
   832 (* ------------------------------------------------------------------------- *)
   833 
   834 definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
   835 
   836 lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
   837   unfolding orthogonal_transformation_def
   838   apply auto
   839   apply (erule_tac x=v in allE)+
   840   apply (simp add: real_vector_norm_def)
   841   by (simp add: dot_norm  linear_add[symmetric])
   842 
   843 definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transp Q ** Q = mat 1 \<and> Q ** transp Q = mat 1"
   844 
   845 lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n::finite)  \<longleftrightarrow> transp Q ** Q = mat 1"
   846   by (metis matrix_left_right_inverse orthogonal_matrix_def)
   847 
   848 lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n::finite)"
   849   by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid)
   850 
   851 lemma orthogonal_matrix_mul:
   852   fixes A :: "real ^'n^'n::finite"
   853   assumes oA : "orthogonal_matrix A"
   854   and oB: "orthogonal_matrix B"
   855   shows "orthogonal_matrix(A ** B)"
   856   using oA oB
   857   unfolding orthogonal_matrix matrix_transp_mul
   858   apply (subst matrix_mul_assoc)
   859   apply (subst matrix_mul_assoc[symmetric])
   860   by (simp add: matrix_mul_rid)
   861 
   862 lemma orthogonal_transformation_matrix:
   863   fixes f:: "real^'n \<Rightarrow> real^'n::finite"
   864   shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
   865   (is "?lhs \<longleftrightarrow> ?rhs")
   866 proof-
   867   let ?mf = "matrix f"
   868   let ?ot = "orthogonal_transformation f"
   869   let ?U = "UNIV :: 'n set"
   870   have fU: "finite ?U" by simp
   871   let ?m1 = "mat 1 :: real ^'n^'n"
   872   {assume ot: ?ot
   873     from ot have lf: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
   874       unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
   875     {fix i j
   876       let ?A = "transp ?mf ** ?mf"
   877       have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
   878 	"\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
   879 	by simp_all
   880       from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
   881       have "?A$i$j = ?m1 $ i $ j"
   882 	by (simp add: dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def)}
   883     hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
   884     with lf have ?rhs by blast}
   885   moreover
   886   {assume lf: "linear f" and om: "orthogonal_matrix ?mf"
   887     from lf om have ?lhs
   888       unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
   889       unfolding matrix_works[OF lf, symmetric]
   890       apply (subst dot_matrix_vector_mul)
   891       by (simp add: dot_matrix_product matrix_mul_lid)}
   892   ultimately show ?thesis by blast
   893 qed
   894 
   895 lemma det_orthogonal_matrix:
   896   fixes Q:: "'a::ordered_idom^'n^'n::finite"
   897   assumes oQ: "orthogonal_matrix Q"
   898   shows "det Q = 1 \<or> det Q = - 1"
   899 proof-
   900 
   901   have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
   902   proof-
   903     fix x:: 'a
   904     have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: ring_simps)
   905     have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
   906       apply (subst eq_iff_diff_eq_0) by simp
   907     have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
   908     also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
   909     finally show "?ths x" ..
   910   qed
   911   from oQ have "Q ** transp Q = mat 1" by (metis orthogonal_matrix_def)
   912   hence "det (Q ** transp Q) = det (mat 1:: 'a^'n^'n)" by simp
   913   hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transp)
   914   then show ?thesis unfolding th .
   915 qed
   916 
   917 (* ------------------------------------------------------------------------- *)
   918 (* Linearity of scaling, and hence isometry, that preserves origin.          *)
   919 (* ------------------------------------------------------------------------- *)
   920 lemma scaling_linear:
   921   fixes f :: "real ^'n \<Rightarrow> real ^'n::finite"
   922   assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
   923   shows "linear f"
   924 proof-
   925   {fix v w
   926     {fix x note fd[rule_format, of x 0, unfolded dist_def f0 diff_0_right] }
   927     note th0 = this
   928     have "f v \<bullet> f w = c^2 * (v \<bullet> w)"
   929       unfolding dot_norm_neg dist_def[symmetric]
   930       unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
   931   note fc = this
   932   show ?thesis unfolding linear_def vector_eq
   933     by (simp add: dot_lmult dot_ladd dot_rmult dot_radd fc ring_simps)
   934 qed
   935 
   936 lemma isometry_linear:
   937   "f (0:: real^'n) = (0:: real^'n::finite) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
   938         \<Longrightarrow> linear f"
   939 by (rule scaling_linear[where c=1]) simp_all
   940 
   941 (* ------------------------------------------------------------------------- *)
   942 (* Hence another formulation of orthogonal transformation.                   *)
   943 (* ------------------------------------------------------------------------- *)
   944 
   945 lemma orthogonal_transformation_isometry:
   946   "orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n::finite) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
   947   unfolding orthogonal_transformation
   948   apply (rule iffI)
   949   apply clarify
   950   apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_def)
   951   apply (rule conjI)
   952   apply (rule isometry_linear)
   953   apply simp
   954   apply simp
   955   apply clarify
   956   apply (erule_tac x=v in allE)
   957   apply (erule_tac x=0 in allE)
   958   by (simp add: dist_def)
   959 
   960 (* ------------------------------------------------------------------------- *)
   961 (* Can extend an isometry from unit sphere.                                  *)
   962 (* ------------------------------------------------------------------------- *)
   963 
   964 lemma isometry_sphere_extend:
   965   fixes f:: "real ^'n \<Rightarrow> real ^'n::finite"
   966   assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
   967   and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
   968   shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
   969 proof-
   970   {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
   971     assume H: "x = norm x *s x0" "y = norm y *s y0"
   972     "x' = norm x *s x0'" "y' = norm y *s y0'"
   973     "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
   974     "norm(x0' - y0') = norm(x0 - y0)"
   975 
   976     have "norm(x' - y') = norm(x - y)"
   977       apply (subst H(1))
   978       apply (subst H(2))
   979       apply (subst H(3))
   980       apply (subst H(4))
   981       using H(5-9)
   982       apply (simp add: norm_eq norm_eq_1)
   983       apply (simp add: dot_lsub dot_rsub dot_lmult dot_rmult)
   984       apply (simp add: ring_simps)
   985       by (simp only: right_distrib[symmetric])}
   986   note th0 = this
   987   let ?g = "\<lambda>x. if x = 0 then 0 else norm x *s f (inverse (norm x) *s x)"
   988   {fix x:: "real ^'n" assume nx: "norm x = 1"
   989     have "?g x = f x" using nx by auto}
   990   hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
   991   have g0: "?g 0 = 0" by simp
   992   {fix x y :: "real ^'n"
   993     {assume "x = 0" "y = 0"
   994       then have "dist (?g x) (?g y) = dist x y" by simp }
   995     moreover
   996     {assume "x = 0" "y \<noteq> 0"
   997       then have "dist (?g x) (?g y) = dist x y"
   998 	apply (simp add: dist_def norm_mul)
   999 	apply (rule f1[rule_format])
  1000 	by(simp add: norm_mul field_simps)}
  1001     moreover
  1002     {assume "x \<noteq> 0" "y = 0"
  1003       then have "dist (?g x) (?g y) = dist x y"
  1004 	apply (simp add: dist_def norm_mul)
  1005 	apply (rule f1[rule_format])
  1006 	by(simp add: norm_mul field_simps)}
  1007     moreover
  1008     {assume z: "x \<noteq> 0" "y \<noteq> 0"
  1009       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)"
  1010 	"norm y *s f (inverse (norm y) *s y) = norm y *s f (inverse (norm y) *s y)"
  1011 	"norm (inverse (norm x) *s x) = 1"
  1012 	"norm (f (inverse (norm x) *s x)) = 1"
  1013 	"norm (inverse (norm y) *s y) = 1"
  1014 	"norm (f (inverse (norm y) *s y)) = 1"
  1015 	"norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
  1016 	norm (inverse (norm x) *s x - inverse (norm y) *s y)"
  1017 	using z
  1018 	by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
  1019       from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
  1020 	by (simp add: dist_def)}
  1021     ultimately have "dist (?g x) (?g y) = dist x y" by blast}
  1022   note thd = this
  1023     show ?thesis
  1024     apply (rule exI[where x= ?g])
  1025     unfolding orthogonal_transformation_isometry
  1026       using  g0 thfg thd by metis
  1027 qed
  1028 
  1029 (* ------------------------------------------------------------------------- *)
  1030 (* Rotation, reflection, rotoinversion.                                      *)
  1031 (* ------------------------------------------------------------------------- *)
  1032 
  1033 definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
  1034 definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
  1035 
  1036 lemma orthogonal_rotation_or_rotoinversion:
  1037   fixes Q :: "'a::ordered_idom^'n^'n::finite"
  1038   shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
  1039   by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
  1040 (* ------------------------------------------------------------------------- *)
  1041 (* Explicit formulas for low dimensions.                                     *)
  1042 (* ------------------------------------------------------------------------- *)
  1043 
  1044 lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
  1045 
  1046 lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
  1047   by (simp add: nat_number setprod_numseg mult_commute)
  1048 lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
  1049   by (simp add: nat_number setprod_numseg mult_commute)
  1050 
  1051 lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
  1052   by (simp add: det_def permutes_sing sign_id UNIV_1)
  1053 
  1054 lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
  1055 proof-
  1056   have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
  1057   show ?thesis
  1058   unfolding det_def UNIV_2
  1059   unfolding setsum_over_permutations_insert[OF f12]
  1060   unfolding permutes_sing
  1061   apply (simp add: sign_swap_id sign_id swap_id_eq)
  1062   by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
  1063 qed
  1064 
  1065 lemma det_3: "det (A::'a::comm_ring_1^3^3) =
  1066   A$1$1 * A$2$2 * A$3$3 +
  1067   A$1$2 * A$2$3 * A$3$1 +
  1068   A$1$3 * A$2$1 * A$3$2 -
  1069   A$1$1 * A$2$3 * A$3$2 -
  1070   A$1$2 * A$2$1 * A$3$3 -
  1071   A$1$3 * A$2$2 * A$3$1"
  1072 proof-
  1073   have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}" by auto
  1074   have f23: "finite {3::3}" "2 \<notin> {3::3}" by auto
  1075 
  1076   show ?thesis
  1077   unfolding det_def UNIV_3
  1078   unfolding setsum_over_permutations_insert[OF f123]
  1079   unfolding setsum_over_permutations_insert[OF f23]
  1080 
  1081   unfolding permutes_sing
  1082   apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
  1083   apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
  1084   by (simp add: ring_simps)
  1085 qed
  1086 
  1087 end