src/HOL/Analysis/Determinants.thy
 author wenzelm Mon Mar 25 17:21:26 2019 +0100 (2 months ago) changeset 69981 3dced198b9ec parent 69720 be6634e99e09 child 70136 f03a01a18c6e permissions -rw-r--r--
more strict AFP properties;
```     1 (*  Title:      HOL/Analysis/Determinants.thy
```
```     2     Author:     Amine Chaieb, University of Cambridge; proofs reworked by LCP
```
```     3 *)
```
```     4
```
```     5 section \<open>Traces, Determinant of square matrices and some properties\<close>
```
```     6
```
```     7 theory Determinants
```
```     8 imports
```
```     9   Cartesian_Space
```
```    10   "HOL-Library.Permutations"
```
```    11 begin
```
```    12
```
```    13 subsection  \<open>Trace\<close>
```
```    14
```
```    15 definition%important  trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
```
```    16   where "trace A = sum (\<lambda>i. ((A\$i)\$i)) (UNIV::'n set)"
```
```    17
```
```    18 lemma  trace_0: "trace (mat 0) = 0"
```
```    19   by (simp add: trace_def mat_def)
```
```    20
```
```    21 lemma  trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
```
```    22   by (simp add: trace_def mat_def)
```
```    23
```
```    24 lemma  trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
```
```    25   by (simp add: trace_def sum.distrib)
```
```    26
```
```    27 lemma  trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
```
```    28   by (simp add: trace_def sum_subtractf)
```
```    29
```
```    30 lemma  trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
```
```    31   apply (simp add: trace_def matrix_matrix_mult_def)
```
```    32   apply (subst sum.swap)
```
```    33   apply (simp add: mult.commute)
```
```    34   done
```
```    35
```
```    36 subsubsection%important  \<open>Definition of determinant\<close>
```
```    37
```
```    38 definition%important  det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
```
```    39   "det A =
```
```    40     sum (\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set))
```
```    41       {p. p permutes (UNIV :: 'n set)}"
```
```    42
```
```    43 text \<open>Basic determinant properties\<close>
```
```    44
```
```    45 lemma  det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
```
```    46 proof -
```
```    47   let ?di = "\<lambda>A i j. A\$i\$j"
```
```    48   let ?U = "(UNIV :: 'n set)"
```
```    49   have fU: "finite ?U" by simp
```
```    50   {
```
```    51     fix p
```
```    52     assume p: "p \<in> {p. p permutes ?U}"
```
```    53     from p have pU: "p permutes ?U"
```
```    54       by blast
```
```    55     have sth: "sign (inv p) = sign p"
```
```    56       by (metis sign_inverse fU p mem_Collect_eq permutation_permutes)
```
```    57     from permutes_inj[OF pU]
```
```    58     have pi: "inj_on p ?U"
```
```    59       by (blast intro: subset_inj_on)
```
```    60     from permutes_image[OF pU]
```
```    61     have "prod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U =
```
```    62       prod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)"
```
```    63       by simp
```
```    64     also have "\<dots> = prod ((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) ?U"
```
```    65       unfolding prod.reindex[OF pi] ..
```
```    66     also have "\<dots> = prod (\<lambda>i. ?di A i (p i)) ?U"
```
```    67     proof -
```
```    68       have "((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) i = ?di A i (p i)" if "i \<in> ?U" for i
```
```    69         using that permutes_inv_o[OF pU] permutes_in_image[OF pU]
```
```    70         unfolding transpose_def by (simp add: fun_eq_iff)
```
```    71       then show "prod ((\<lambda>i. ?di (transpose A) i (inv p i)) \<circ> p) ?U = prod (\<lambda>i. ?di A i (p i)) ?U"
```
```    72         by (auto intro: prod.cong)
```
```    73     qed
```
```    74     finally have "of_int (sign (inv p)) * (prod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U) =
```
```    75       of_int (sign p) * (prod (\<lambda>i. ?di A i (p i)) ?U)"
```
```    76       using sth by simp
```
```    77   }
```
```    78   then show ?thesis
```
```    79     unfolding det_def
```
```    80     by (subst sum_permutations_inverse) (blast intro: sum.cong)
```
```    81 qed
```
```    82
```
```    83 lemma  det_lowerdiagonal:
```
```    84   fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
```
```    85   assumes ld: "\<And>i j. i < j \<Longrightarrow> A\$i\$j = 0"
```
```    86   shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV:: 'n set)"
```
```    87 proof -
```
```    88   let ?U = "UNIV:: 'n set"
```
```    89   let ?PU = "{p. p permutes ?U}"
```
```    90   let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set)"
```
```    91   have fU: "finite ?U"
```
```    92     by simp
```
```    93   have id0: "{id} \<subseteq> ?PU"
```
```    94     by (auto simp: permutes_id)
```
```    95   have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
```
```    96   proof
```
```    97     fix p
```
```    98     assume "p \<in> ?PU - {id}"
```
```    99     then obtain i where i: "p i > i"
```
```   100       by clarify (meson leI permutes_natset_le)
```
```   101     from ld[OF i] have "\<exists>i \<in> ?U. A\$i\$p i = 0"
```
```   102       by blast
```
```   103     with prod_zero[OF fU] show "?pp p = 0"
```
```   104       by force
```
```   105   qed
```
```   106   from sum.mono_neutral_cong_left[OF finite_permutations[OF fU] id0 p0] show ?thesis
```
```   107     unfolding det_def by (simp add: sign_id)
```
```   108 qed
```
```   109
```
```   110 lemma  det_upperdiagonal:
```
```   111   fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
```
```   112   assumes ld: "\<And>i j. i > j \<Longrightarrow> A\$i\$j = 0"
```
```   113   shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV:: 'n set)"
```
```   114 proof -
```
```   115   let ?U = "UNIV:: 'n set"
```
```   116   let ?PU = "{p. p permutes ?U}"
```
```   117   let ?pp = "(\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set))"
```
```   118   have fU: "finite ?U"
```
```   119     by simp
```
```   120   have id0: "{id} \<subseteq> ?PU"
```
```   121     by (auto simp: permutes_id)
```
```   122   have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
```
```   123   proof
```
```   124     fix p
```
```   125     assume p: "p \<in> ?PU - {id}"
```
```   126     then obtain i where i: "p i < i"
```
```   127       by clarify (meson leI permutes_natset_ge)
```
```   128     from ld[OF i] have "\<exists>i \<in> ?U. A\$i\$p i = 0"
```
```   129       by blast
```
```   130     with prod_zero[OF fU]  show "?pp p = 0"
```
```   131       by force
```
```   132   qed
```
```   133   from sum.mono_neutral_cong_left[OF finite_permutations[OF fU] id0 p0] show ?thesis
```
```   134     unfolding det_def by (simp add: sign_id)
```
```   135 qed
```
```   136
```
```   137 proposition  det_diagonal:
```
```   138   fixes A :: "'a::comm_ring_1^'n^'n"
```
```   139   assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0"
```
```   140   shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV::'n set)"
```
```   141 proof -
```
```   142   let ?U = "UNIV:: 'n set"
```
```   143   let ?PU = "{p. p permutes ?U}"
```
```   144   let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set)"
```
```   145   have fU: "finite ?U" by simp
```
```   146   from finite_permutations[OF fU] have fPU: "finite ?PU" .
```
```   147   have id0: "{id} \<subseteq> ?PU"
```
```   148     by (auto simp: permutes_id)
```
```   149   have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
```
```   150   proof
```
```   151     fix p
```
```   152     assume p: "p \<in> ?PU - {id}"
```
```   153     then obtain i where i: "p i \<noteq> i"
```
```   154       by fastforce
```
```   155     with ld have "\<exists>i \<in> ?U. A\$i\$p i = 0"
```
```   156       by (metis UNIV_I)
```
```   157     with prod_zero [OF fU] show "?pp p = 0"
```
```   158       by force
```
```   159   qed
```
```   160   from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
```
```   161     unfolding det_def by (simp add: sign_id)
```
```   162 qed
```
```   163
```
```   164 lemma  det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
```
```   165   by (simp add: det_diagonal mat_def)
```
```   166
```
```   167 lemma  det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
```
```   168   by (simp add: det_def prod_zero power_0_left)
```
```   169
```
```   170 lemma  det_permute_rows:
```
```   171   fixes A :: "'a::comm_ring_1^'n^'n"
```
```   172   assumes p: "p permutes (UNIV :: 'n::finite set)"
```
```   173   shows "det (\<chi> i. A\$p i :: 'a^'n^'n) = of_int (sign p) * det A"
```
```   174 proof -
```
```   175   let ?U = "UNIV :: 'n set"
```
```   176   let ?PU = "{p. p permutes ?U}"
```
```   177   have *: "(\<Sum>q\<in>?PU. of_int (sign (q \<circ> p)) * (\<Prod>i\<in>?U. A \$ p i \$ (q \<circ> p) i)) =
```
```   178            (\<Sum>n\<in>?PU. of_int (sign p) * of_int (sign n) * (\<Prod>i\<in>?U. A \$ i \$ n i))"
```
```   179   proof (rule sum.cong)
```
```   180     fix q
```
```   181     assume qPU: "q \<in> ?PU"
```
```   182     have fU: "finite ?U"
```
```   183       by simp
```
```   184     from qPU have q: "q permutes ?U"
```
```   185       by blast
```
```   186     have "prod (\<lambda>i. A\$p i\$ (q \<circ> p) i) ?U = prod ((\<lambda>i. A\$p i\$(q \<circ> p) i) \<circ> inv p) ?U"
```
```   187       by (simp only: prod.permute[OF permutes_inv[OF p], symmetric])
```
```   188     also have "\<dots> = prod (\<lambda>i. A \$ (p \<circ> inv p) i \$ (q \<circ> (p \<circ> inv p)) i) ?U"
```
```   189       by (simp only: o_def)
```
```   190     also have "\<dots> = prod (\<lambda>i. A\$i\$q i) ?U"
```
```   191       by (simp only: o_def permutes_inverses[OF p])
```
```   192     finally have thp: "prod (\<lambda>i. A\$p i\$ (q \<circ> p) i) ?U = prod (\<lambda>i. A\$i\$q i) ?U"
```
```   193       by blast
```
```   194     from p q have pp: "permutation p" and qp: "permutation q"
```
```   195       by (metis fU permutation_permutes)+
```
```   196     show "of_int (sign (q \<circ> p)) * prod (\<lambda>i. A\$ p i\$ (q \<circ> p) i) ?U =
```
```   197           of_int (sign p) * of_int (sign q) * prod (\<lambda>i. A\$i\$q i) ?U"
```
```   198       by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult)
```
```   199   qed auto
```
```   200   show ?thesis
```
```   201     apply (simp add: det_def sum_distrib_left mult.assoc[symmetric])
```
```   202     apply (subst sum_permutations_compose_right[OF p])
```
```   203     apply (rule *)
```
```   204     done
```
```   205 qed
```
```   206
```
```   207 lemma  det_permute_columns:
```
```   208   fixes A :: "'a::comm_ring_1^'n^'n"
```
```   209   assumes p: "p permutes (UNIV :: 'n set)"
```
```   210   shows "det(\<chi> i j. A\$i\$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
```
```   211 proof -
```
```   212   let ?Ap = "\<chi> i j. A\$i\$ p j :: 'a^'n^'n"
```
```   213   let ?At = "transpose A"
```
```   214   have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A \$ p i))"
```
```   215     unfolding det_permute_rows[OF p, of ?At] det_transpose ..
```
```   216   moreover
```
```   217   have "?Ap = transpose (\<chi> i. transpose A \$ p i)"
```
```   218     by (simp add: transpose_def vec_eq_iff)
```
```   219   ultimately show ?thesis
```
```   220     by simp
```
```   221 qed
```
```   222
```
```   223 lemma  det_identical_columns:
```
```   224   fixes A :: "'a::comm_ring_1^'n^'n"
```
```   225   assumes jk: "j \<noteq> k"
```
```   226     and r: "column j A = column k A"
```
```   227   shows "det A = 0"
```
```   228 proof -
```
```   229   let ?U="UNIV::'n set"
```
```   230   let ?t_jk="Fun.swap j k id"
```
```   231   let ?PU="{p. p permutes ?U}"
```
```   232   let ?S1="{p. p\<in>?PU \<and> evenperm p}"
```
```   233   let ?S2="{(?t_jk \<circ> p) |p. p \<in>?S1}"
```
```   234   let ?f="\<lambda>p. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i)"
```
```   235   let ?g="\<lambda>p. ?t_jk \<circ> p"
```
```   236   have g_S1: "?S2 = ?g` ?S1" by auto
```
```   237   have inj_g: "inj_on ?g ?S1"
```
```   238   proof (unfold inj_on_def, auto)
```
```   239     fix x y assume x: "x permutes ?U" and even_x: "evenperm x"
```
```   240       and y: "y permutes ?U" and even_y: "evenperm y" and eq: "?t_jk \<circ> x = ?t_jk \<circ> y"
```
```   241     show "x = y" by (metis (hide_lams, no_types) comp_assoc eq id_comp swap_id_idempotent)
```
```   242   qed
```
```   243   have tjk_permutes: "?t_jk permutes ?U" unfolding permutes_def swap_id_eq by (auto,metis)
```
```   244   have tjk_eq: "\<forall>i l. A \$ i \$ ?t_jk l  =  A \$ i \$ l"
```
```   245     using r jk
```
```   246     unfolding column_def vec_eq_iff swap_id_eq by fastforce
```
```   247   have sign_tjk: "sign ?t_jk = -1" using sign_swap_id[of j k] jk by auto
```
```   248   {fix x
```
```   249     assume x: "x\<in> ?S1"
```
```   250     have "sign (?t_jk \<circ> x) = sign (?t_jk) * sign x"
```
```   251       by (metis (lifting) finite_class.finite_UNIV mem_Collect_eq
```
```   252           permutation_permutes permutation_swap_id sign_compose x)
```
```   253     also have "\<dots> = - sign x" using sign_tjk by simp
```
```   254     also have "\<dots> \<noteq> sign x" unfolding sign_def by simp
```
```   255     finally have "sign (?t_jk \<circ> x) \<noteq> sign x" and "(?t_jk \<circ> x) \<in> ?S2"
```
```   256       using x by force+
```
```   257   }
```
```   258   hence disjoint: "?S1 \<inter> ?S2 = {}"
```
```   259     by (force simp: sign_def)
```
```   260   have PU_decomposition: "?PU = ?S1 \<union> ?S2"
```
```   261   proof (auto)
```
```   262     fix x
```
```   263     assume x: "x permutes ?U" and "\<forall>p. p permutes ?U \<longrightarrow> x = Fun.swap j k id \<circ> p \<longrightarrow> \<not> evenperm p"
```
```   264     then obtain p where p: "p permutes UNIV" and x_eq: "x = Fun.swap j k id \<circ> p"
```
```   265       and odd_p: "\<not> evenperm p"
```
```   266       by (metis (mono_tags) id_o o_assoc permutes_compose swap_id_idempotent tjk_permutes)
```
```   267     thus "evenperm x"
```
```   268       by (meson evenperm_comp evenperm_swap finite_class.finite_UNIV
```
```   269           jk permutation_permutes permutation_swap_id)
```
```   270   next
```
```   271     fix p assume p: "p permutes ?U"
```
```   272     show "Fun.swap j k id \<circ> p permutes UNIV" by (metis p permutes_compose tjk_permutes)
```
```   273   qed
```
```   274   have "sum ?f ?S2 = sum ((\<lambda>p. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i))
```
```   275   \<circ> (\<circ>) (Fun.swap j k id)) {p \<in> {p. p permutes UNIV}. evenperm p}"
```
```   276     unfolding g_S1 by (rule sum.reindex[OF inj_g])
```
```   277   also have "\<dots> = sum (\<lambda>p. of_int (sign (?t_jk \<circ> p)) * (\<Prod>i\<in>UNIV. A \$ i \$ p i)) ?S1"
```
```   278     unfolding o_def by (rule sum.cong, auto simp: tjk_eq)
```
```   279   also have "\<dots> = sum (\<lambda>p. - ?f p) ?S1"
```
```   280   proof (rule sum.cong, auto)
```
```   281     fix x assume x: "x permutes ?U"
```
```   282       and even_x: "evenperm x"
```
```   283     hence perm_x: "permutation x" and perm_tjk: "permutation ?t_jk"
```
```   284       using permutation_permutes[of x] permutation_permutes[of ?t_jk] permutation_swap_id
```
```   285       by (metis finite_code)+
```
```   286     have "(sign (?t_jk \<circ> x)) = - (sign x)"
```
```   287       unfolding sign_compose[OF perm_tjk perm_x] sign_tjk by auto
```
```   288     thus "of_int (sign (?t_jk \<circ> x)) * (\<Prod>i\<in>UNIV. A \$ i \$ x i)
```
```   289       = - (of_int (sign x) * (\<Prod>i\<in>UNIV. A \$ i \$ x i))"
```
```   290       by auto
```
```   291   qed
```
```   292   also have "\<dots>= - sum ?f ?S1" unfolding sum_negf ..
```
```   293   finally have *: "sum ?f ?S2 = - sum ?f ?S1" .
```
```   294   have "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i))"
```
```   295     unfolding det_def ..
```
```   296   also have "\<dots>= sum ?f ?S1 + sum ?f ?S2"
```
```   297     by (subst PU_decomposition, rule sum.union_disjoint[OF _ _ disjoint], auto)
```
```   298   also have "\<dots>= sum ?f ?S1 - sum ?f ?S1 " unfolding * by auto
```
```   299   also have "\<dots>= 0" by simp
```
```   300   finally show "det A = 0" by simp
```
```   301 qed
```
```   302
```
```   303 lemma  det_identical_rows:
```
```   304   fixes A :: "'a::comm_ring_1^'n^'n"
```
```   305   assumes ij: "i \<noteq> j" and r: "row i A = row j A"
```
```   306   shows "det A = 0"
```
```   307   by (metis column_transpose det_identical_columns det_transpose ij r)
```
```   308
```
```   309 lemma  det_zero_row:
```
```   310   fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
```
```   311   shows "row i A = 0 \<Longrightarrow> det A = 0" and "row j F = 0 \<Longrightarrow> det F = 0"
```
```   312   by (force simp: row_def det_def vec_eq_iff sign_nz intro!: sum.neutral)+
```
```   313
```
```   314 lemma  det_zero_column:
```
```   315   fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
```
```   316   shows "column i A = 0 \<Longrightarrow> det A = 0" and "column j F = 0 \<Longrightarrow> det F = 0"
```
```   317   unfolding atomize_conj atomize_imp
```
```   318   by (metis det_transpose det_zero_row row_transpose)
```
```   319
```
```   320 lemma  det_row_add:
```
```   321   fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
```
```   322   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
```
```   323     det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
```
```   324     det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
```
```   325   unfolding det_def vec_lambda_beta sum.distrib[symmetric]
```
```   326 proof (rule sum.cong)
```
```   327   let ?U = "UNIV :: 'n set"
```
```   328   let ?pU = "{p. p permutes ?U}"
```
```   329   let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
```
```   330   let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
```
```   331   let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
```
```   332   fix p
```
```   333   assume p: "p \<in> ?pU"
```
```   334   let ?Uk = "?U - {k}"
```
```   335   from p have pU: "p permutes ?U"
```
```   336     by blast
```
```   337   have kU: "?U = insert k ?Uk"
```
```   338     by blast
```
```   339   have eq: "prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?g i \$ p i) ?Uk"
```
```   340            "prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?h i \$ p i) ?Uk"
```
```   341     by auto
```
```   342   have Uk: "finite ?Uk" "k \<notin> ?Uk"
```
```   343     by auto
```
```   344   have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
```
```   345     unfolding kU[symmetric] ..
```
```   346   also have "\<dots> = ?f k \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk"
```
```   347     by (rule prod.insert) auto
```
```   348   also have "\<dots> = (a k \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk) + (b k\$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk)"
```
```   349     by (simp add: field_simps)
```
```   350   also have "\<dots> = (a k \$ p k * prod (\<lambda>i. ?g i \$ p i) ?Uk) + (b k\$ p k * prod (\<lambda>i. ?h i \$ p i) ?Uk)"
```
```   351     by (metis eq)
```
```   352   also have "\<dots> = prod (\<lambda>i. ?g i \$ p i) (insert k ?Uk) + prod (\<lambda>i. ?h i \$ p i) (insert k ?Uk)"
```
```   353     unfolding  prod.insert[OF Uk] by simp
```
```   354   finally have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?g i \$ p i) ?U + prod (\<lambda>i. ?h i \$ p i) ?U"
```
```   355     unfolding kU[symmetric] .
```
```   356   then show "of_int (sign p) * prod (\<lambda>i. ?f i \$ p i) ?U =
```
```   357     of_int (sign p) * prod (\<lambda>i. ?g i \$ p i) ?U + of_int (sign p) * prod (\<lambda>i. ?h i \$ p i) ?U"
```
```   358     by (simp add: field_simps)
```
```   359 qed auto
```
```   360
```
```   361 lemma  det_row_mul:
```
```   362   fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
```
```   363   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
```
```   364     c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
```
```   365   unfolding det_def vec_lambda_beta sum_distrib_left
```
```   366 proof (rule sum.cong)
```
```   367   let ?U = "UNIV :: 'n set"
```
```   368   let ?pU = "{p. p permutes ?U}"
```
```   369   let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
```
```   370   let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
```
```   371   fix p
```
```   372   assume p: "p \<in> ?pU"
```
```   373   let ?Uk = "?U - {k}"
```
```   374   from p have pU: "p permutes ?U"
```
```   375     by blast
```
```   376   have kU: "?U = insert k ?Uk"
```
```   377     by blast
```
```   378   have eq: "prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?g i \$ p i) ?Uk"
```
```   379     by auto
```
```   380   have Uk: "finite ?Uk" "k \<notin> ?Uk"
```
```   381     by auto
```
```   382   have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
```
```   383     unfolding kU[symmetric] ..
```
```   384   also have "\<dots> = ?f k \$ p k  * prod (\<lambda>i. ?f i \$ p i) ?Uk"
```
```   385     by (rule prod.insert) auto
```
```   386   also have "\<dots> = (c*s a k) \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk"
```
```   387     by (simp add: field_simps)
```
```   388   also have "\<dots> = c* (a k \$ p k * prod (\<lambda>i. ?g i \$ p i) ?Uk)"
```
```   389     unfolding eq by (simp add: ac_simps)
```
```   390   also have "\<dots> = c* (prod (\<lambda>i. ?g i \$ p i) (insert k ?Uk))"
```
```   391     unfolding prod.insert[OF Uk] by simp
```
```   392   finally have "prod (\<lambda>i. ?f i \$ p i) ?U = c* (prod (\<lambda>i. ?g i \$ p i) ?U)"
```
```   393     unfolding kU[symmetric] .
```
```   394   then show "of_int (sign p) * prod (\<lambda>i. ?f i \$ p i) ?U = c * (of_int (sign p) * prod (\<lambda>i. ?g i \$ p i) ?U)"
```
```   395     by (simp add: field_simps)
```
```   396 qed auto
```
```   397
```
```   398 lemma  det_row_0:
```
```   399   fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
```
```   400   shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
```
```   401   using det_row_mul[of k 0 "\<lambda>i. 1" b]
```
```   402   apply simp
```
```   403   apply (simp only: vector_smult_lzero)
```
```   404   done
```
```   405
```
```   406 lemma  det_row_operation:
```
```   407   fixes A :: "'a::{comm_ring_1}^'n^'n"
```
```   408   assumes ij: "i \<noteq> j"
```
```   409   shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
```
```   410 proof -
```
```   411   let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
```
```   412   have th: "row i ?Z = row j ?Z" by (vector row_def)
```
```   413   have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
```
```   414     by (vector row_def)
```
```   415   show ?thesis
```
```   416     unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2
```
```   417     by simp
```
```   418 qed
```
```   419
```
```   420 lemma  det_row_span:
```
```   421   fixes A :: "'a::{field}^'n^'n"
```
```   422   assumes x: "x \<in> vec.span {row j A |j. j \<noteq> i}"
```
```   423   shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
```
```   424   using x
```
```   425 proof (induction rule: vec.span_induct_alt)
```
```   426   case base
```
```   427   have "(if k = i then row i A + 0 else row k A) = row k A" for k
```
```   428     by simp
```
```   429   then show ?case
```
```   430     by (simp add: row_def)
```
```   431 next
```
```   432   case (step c z y)
```
```   433   then obtain j where j: "z = row j A" "i \<noteq> j"
```
```   434     by blast
```
```   435   let ?w = "row i A + y"
```
```   436   have th0: "row i A + (c*s z + y) = ?w + c*s z"
```
```   437     by vector
```
```   438   let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
```
```   439   have thz: "?d z = 0"
```
```   440     apply (rule det_identical_rows[OF j(2)])
```
```   441     using j
```
```   442     apply (vector row_def)
```
```   443     done
```
```   444   have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)"
```
```   445     unfolding th0 ..
```
```   446   then have "?d (row i A + (c*s z + y)) = det A"
```
```   447     unfolding thz step.IH det_row_mul[of i] det_row_add[of i] by simp
```
```   448   then show ?case
```
```   449     unfolding scalar_mult_eq_scaleR .
```
```   450 qed
```
```   451
```
```   452 lemma  matrix_id [simp]: "det (matrix id) = 1"
```
```   453   by (simp add: matrix_id_mat_1)
```
```   454
```
```   455 proposition  det_matrix_scaleR [simp]: "det (matrix (((*\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
```
```   456   apply (subst det_diagonal)
```
```   457    apply (auto simp: matrix_def mat_def)
```
```   458   apply (simp add: cart_eq_inner_axis inner_axis_axis)
```
```   459   done
```
```   460
```
```   461 text \<open>
```
```   462   May as well do this, though it's a bit unsatisfactory since it ignores
```
```   463   exact duplicates by considering the rows/columns as a set.
```
```   464 \<close>
```
```   465
```
```   466 lemma  det_dependent_rows:
```
```   467   fixes A:: "'a::{field}^'n^'n"
```
```   468   assumes d: "vec.dependent (rows A)"
```
```   469   shows "det A = 0"
```
```   470 proof -
```
```   471   let ?U = "UNIV :: 'n set"
```
```   472   from d obtain i where i: "row i A \<in> vec.span (rows A - {row i A})"
```
```   473     unfolding vec.dependent_def rows_def by blast
```
```   474   show ?thesis
```
```   475   proof (cases "\<forall>i j. i \<noteq> j \<longrightarrow> row i A \<noteq> row j A")
```
```   476     case True
```
```   477     with i have "vec.span (rows A - {row i A}) \<subseteq> vec.span {row j A |j. j \<noteq> i}"
```
```   478       by (auto simp: rows_def intro!: vec.span_mono)
```
```   479     then have "- row i A \<in> vec.span {row j A|j. j \<noteq> i}"
```
```   480       by (meson i subsetCE vec.span_neg)
```
```   481     from det_row_span[OF this]
```
```   482     have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
```
```   483       unfolding right_minus vector_smult_lzero ..
```
```   484     with det_row_mul[of i 0 "\<lambda>i. 1"]
```
```   485     show ?thesis by simp
```
```   486   next
```
```   487     case False
```
```   488     then obtain j k where jk: "j \<noteq> k" "row j A = row k A"
```
```   489       by auto
```
```   490     from det_identical_rows[OF jk] show ?thesis .
```
```   491   qed
```
```   492 qed
```
```   493
```
```   494 lemma  det_dependent_columns:
```
```   495   assumes d: "vec.dependent (columns (A::real^'n^'n))"
```
```   496   shows "det A = 0"
```
```   497   by (metis d det_dependent_rows rows_transpose det_transpose)
```
```   498
```
```   499 text \<open>Multilinearity and the multiplication formula\<close>
```
```   500
```
```   501 lemma  Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
```
```   502   by auto
```
```   503
```
```   504 lemma  det_linear_row_sum:
```
```   505   assumes fS: "finite S"
```
```   506   shows "det ((\<chi> i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
```
```   507     sum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
```
```   508   using fS  by (induct rule: finite_induct; simp add: det_row_0 det_row_add cong: if_cong)
```
```   509
```
```   510 lemma  finite_bounded_functions:
```
```   511   assumes fS: "finite S"
```
```   512   shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
```
```   513 proof (induct k)
```
```   514   case 0
```
```   515   have *: "{f. \<forall>i. f i = i} = {id}"
```
```   516     by auto
```
```   517   show ?case
```
```   518     by (auto simp: *)
```
```   519 next
```
```   520   case (Suc k)
```
```   521   let ?f = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
```
```   522   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)})"
```
```   523   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)}"
```
```   524     apply (auto simp: image_iff)
```
```   525     apply (rename_tac f)
```
```   526     apply (rule_tac x="f (Suc k)" in bexI)
```
```   527     apply (rule_tac x = "\<lambda>i. if i = Suc k then i else f i" in exI, auto)
```
```   528     done
```
```   529   with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
```
```   530   show ?case
```
```   531     by metis
```
```   532 qed
```
```   533
```
```   534
```
```   535 lemma  det_linear_rows_sum_lemma:
```
```   536   assumes fS: "finite S"
```
```   537     and fT: "finite T"
```
```   538   shows "det ((\<chi> i. if i \<in> T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
```
```   539     sum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
```
```   540       {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
```
```   541   using fT
```
```   542 proof (induct T arbitrary: a c set: finite)
```
```   543   case empty
```
```   544   have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
```
```   545     by vector
```
```   546   from empty.prems show ?case
```
```   547     unfolding th0 by (simp add: eq_id_iff)
```
```   548 next
```
```   549   case (insert z T a c)
```
```   550   let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
```
```   551   let ?h = "\<lambda>(y,g) i. if i = z then y else g i"
```
```   552   let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
```
```   553   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)"
```
```   554   let ?c = "\<lambda>j i. if i = z then a i j else c i"
```
```   555   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)"
```
```   556     by simp
```
```   557   have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
```
```   558      (if c then (if a then b else d) else (if a then b else e))"
```
```   559     by simp
```
```   560   from \<open>z \<notin> T\<close> have nz: "\<And>i. i \<in> T \<Longrightarrow> i \<noteq> z"
```
```   561     by auto
```
```   562   have "det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
```
```   563     det (\<chi> i. if i = z then sum (a i) S else if i \<in> T then sum (a i) S else c i)"
```
```   564     unfolding insert_iff thif ..
```
```   565   also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then sum (a i) S else if i = z then a i j else c i))"
```
```   566     unfolding det_linear_row_sum[OF fS]
```
```   567     by (subst thif2) (simp add: nz cong: if_cong)
```
```   568   finally have tha:
```
```   569     "det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
```
```   570      (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
```
```   571                                 else if i = z then a i j
```
```   572                                 else c i))"
```
```   573     unfolding insert.hyps unfolding sum.cartesian_product by blast
```
```   574   show ?case unfolding tha
```
```   575     using \<open>z \<notin> T\<close>
```
```   576     by (intro sum.reindex_bij_witness[where i="?k" and j="?h"])
```
```   577        (auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
```
```   578 qed
```
```   579
```
```   580 lemma  det_linear_rows_sum:
```
```   581   fixes S :: "'n::finite set"
```
```   582   assumes fS: "finite S"
```
```   583   shows "det (\<chi> i. sum (a i) S) =
```
```   584     sum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
```
```   585 proof -
```
```   586   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)"
```
```   587     by vector
```
```   588   from det_linear_rows_sum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
```
```   589   show ?thesis by simp
```
```   590 qed
```
```   591
```
```   592 lemma  matrix_mul_sum_alt:
```
```   593   fixes A B :: "'a::comm_ring_1^'n^'n"
```
```   594   shows "A ** B = (\<chi> i. sum (\<lambda>k. A\$i\$k *s B \$ k) (UNIV :: 'n set))"
```
```   595   by (vector matrix_matrix_mult_def sum_component)
```
```   596
```
```   597 lemma  det_rows_mul:
```
```   598   "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
```
```   599     prod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
```
```   600 proof (simp add: det_def sum_distrib_left cong add: prod.cong, rule sum.cong)
```
```   601   let ?U = "UNIV :: 'n set"
```
```   602   let ?PU = "{p. p permutes ?U}"
```
```   603   fix p
```
```   604   assume pU: "p \<in> ?PU"
```
```   605   let ?s = "of_int (sign p)"
```
```   606   from pU have p: "p permutes ?U"
```
```   607     by blast
```
```   608   have "prod (\<lambda>i. c i * a i \$ p i) ?U = prod c ?U * prod (\<lambda>i. a i \$ p i) ?U"
```
```   609     unfolding prod.distrib ..
```
```   610   then show "?s * (\<Prod>xa\<in>?U. c xa * a xa \$ p xa) =
```
```   611     prod c ?U * (?s* (\<Prod>xa\<in>?U. a xa \$ p xa))"
```
```   612     by (simp add: field_simps)
```
```   613 qed rule
```
```   614
```
```   615 proposition  det_mul:
```
```   616   fixes A B :: "'a::comm_ring_1^'n^'n"
```
```   617   shows "det (A ** B) = det A * det B"
```
```   618 proof -
```
```   619   let ?U = "UNIV :: 'n set"
```
```   620   let ?F = "{f. (\<forall>i \<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
```
```   621   let ?PU = "{p. p permutes ?U}"
```
```   622   have "p \<in> ?F" if "p permutes ?U" for p
```
```   623     by simp
```
```   624   then have PUF: "?PU \<subseteq> ?F" by blast
```
```   625   {
```
```   626     fix f
```
```   627     assume fPU: "f \<in> ?F - ?PU"
```
```   628     have fUU: "f ` ?U \<subseteq> ?U"
```
```   629       using fPU by auto
```
```   630     from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U" "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)"
```
```   631       unfolding permutes_def by auto
```
```   632
```
```   633     let ?A = "(\<chi> i. A\$i\$f i *s B\$f i) :: 'a^'n^'n"
```
```   634     let ?B = "(\<chi> i. B\$f i) :: 'a^'n^'n"
```
```   635     {
```
```   636       assume fni: "\<not> inj_on f ?U"
```
```   637       then obtain i j where ij: "f i = f j" "i \<noteq> j"
```
```   638         unfolding inj_on_def by blast
```
```   639       then have "row i ?B = row j ?B"
```
```   640         by (vector row_def)
```
```   641       with det_identical_rows[OF ij(2)]
```
```   642       have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
```
```   643         unfolding det_rows_mul by force
```
```   644     }
```
```   645     moreover
```
```   646     {
```
```   647       assume fi: "inj_on f ?U"
```
```   648       from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
```
```   649         unfolding inj_on_def by metis
```
```   650       note fs = fi[unfolded surjective_iff_injective_gen[OF finite finite refl fUU, symmetric]]
```
```   651       have "\<exists>!x. f x = y" for y
```
```   652         using fith fs by blast
```
```   653       with f(3) have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
```
```   654         by blast
```
```   655     }
```
```   656     ultimately have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
```
```   657       by blast
```
```   658   }
```
```   659   then have zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
```
```   660     by simp
```
```   661   {
```
```   662     fix p
```
```   663     assume pU: "p \<in> ?PU"
```
```   664     from pU have p: "p permutes ?U"
```
```   665       by blast
```
```   666     let ?s = "\<lambda>p. of_int (sign p)"
```
```   667     let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))"
```
```   668     have "(sum (\<lambda>q. ?s q *
```
```   669         (\<Prod>i\<in> ?U. (\<chi> i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
```
```   670       (sum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))) ?PU)"
```
```   671       unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
```
```   672     proof (rule sum.cong)
```
```   673       fix q
```
```   674       assume qU: "q \<in> ?PU"
```
```   675       then have q: "q permutes ?U"
```
```   676         by blast
```
```   677       from p q have pp: "permutation p" and pq: "permutation q"
```
```   678         unfolding permutation_permutes by auto
```
```   679       have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
```
```   680         "\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
```
```   681         unfolding mult.assoc[symmetric]
```
```   682         unfolding of_int_mult[symmetric]
```
```   683         by (simp_all add: sign_idempotent)
```
```   684       have ths: "?s q = ?s p * ?s (q \<circ> inv p)"
```
```   685         using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
```
```   686         by (simp add: th00 ac_simps sign_idempotent sign_compose)
```
```   687       have th001: "prod (\<lambda>i. B\$i\$ q (inv p i)) ?U = prod ((\<lambda>i. B\$i\$ q (inv p i)) \<circ> p) ?U"
```
```   688         by (rule prod.permute[OF p])
```
```   689       have thp: "prod (\<lambda>i. (\<chi> i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U =
```
```   690         prod (\<lambda>i. A\$i\$p i) ?U * prod (\<lambda>i. B\$i\$ q (inv p i)) ?U"
```
```   691         unfolding th001 prod.distrib[symmetric] o_def permutes_inverses[OF p]
```
```   692         apply (rule prod.cong[OF refl])
```
```   693         using permutes_in_image[OF q]
```
```   694         apply vector
```
```   695         done
```
```   696       show "?s q * prod (\<lambda>i. (((\<chi> i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U =
```
```   697         ?s p * (prod (\<lambda>i. A\$i\$p i) ?U) * (?s (q \<circ> inv p) * prod (\<lambda>i. B\$i\$(q \<circ> inv p) i) ?U)"
```
```   698         using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
```
```   699         by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
```
```   700     qed rule
```
```   701   }
```
```   702   then have th2: "sum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU = det A * det B"
```
```   703     unfolding det_def sum_product
```
```   704     by (rule sum.cong [OF refl])
```
```   705   have "det (A**B) = sum (\<lambda>f.  det (\<chi> i. A \$ i \$ f i *s B \$ f i)) ?F"
```
```   706     unfolding matrix_mul_sum_alt det_linear_rows_sum[OF finite]
```
```   707     by simp
```
```   708   also have "\<dots> = sum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU"
```
```   709     using sum.mono_neutral_cong_left[OF finite PUF zth, symmetric]
```
```   710     unfolding det_rows_mul by auto
```
```   711   finally show ?thesis unfolding th2 .
```
```   712 qed
```
```   713
```
```   714
```
```   715 subsection \<open>Relation to invertibility\<close>
```
```   716
```
```   717 proposition  invertible_det_nz:
```
```   718   fixes A::"'a::{field}^'n^'n"
```
```   719   shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
```
```   720 proof (cases "invertible A")
```
```   721   case True
```
```   722   then obtain B :: "'a^'n^'n" where B: "A ** B = mat 1"
```
```   723     unfolding invertible_right_inverse by blast
```
```   724   then have "det (A ** B) = det (mat 1 :: 'a^'n^'n)"
```
```   725     by simp
```
```   726   then show ?thesis
```
```   727     by (metis True det_I det_mul mult_zero_left one_neq_zero)
```
```   728 next
```
```   729   case False
```
```   730   let ?U = "UNIV :: 'n set"
```
```   731   have fU: "finite ?U"
```
```   732     by simp
```
```   733   from False obtain c i where c: "sum (\<lambda>i. c i *s row i A) ?U = 0" and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
```
```   734     unfolding invertible_right_inverse matrix_right_invertible_independent_rows
```
```   735     by blast
```
```   736   have thr0: "- row i A = sum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
```
```   737     unfolding sum_cmul  using c ci
```
```   738     by (auto simp: sum.remove[OF fU iU] eq_vector_fraction_iff add_eq_0_iff)
```
```   739   have thr: "- row i A \<in> vec.span {row j A| j. j \<noteq> i}"
```
```   740     unfolding thr0 by (auto intro: vec.span_base vec.span_scale vec.span_sum)
```
```   741   let ?B = "(\<chi> k. if k = i then 0 else row k A) :: 'a^'n^'n"
```
```   742   have thrb: "row i ?B = 0" using iU by (vector row_def)
```
```   743   have "det A = 0"
```
```   744     unfolding det_row_span[OF thr, symmetric] right_minus
```
```   745     unfolding det_zero_row(2)[OF thrb] ..
```
```   746   then show ?thesis
```
```   747     by (simp add: False)
```
```   748 qed
```
```   749
```
```   750
```
```   751 lemma  det_nz_iff_inj_gen:
```
```   752   fixes f :: "'a::field^'n \<Rightarrow> 'a::field^'n"
```
```   753   assumes "Vector_Spaces.linear (*s) (*s) f"
```
```   754   shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
```
```   755 proof
```
```   756   assume "det (matrix f) \<noteq> 0"
```
```   757   then show "inj f"
```
```   758     using assms invertible_det_nz inj_matrix_vector_mult by force
```
```   759 next
```
```   760   assume "inj f"
```
```   761   show "det (matrix f) \<noteq> 0"
```
```   762     using vec.linear_injective_left_inverse [OF assms \<open>inj f\<close>]
```
```   763     by (metis assms invertible_det_nz invertible_left_inverse matrix_compose_gen matrix_id_mat_1)
```
```   764 qed
```
```   765
```
```   766 lemma  det_nz_iff_inj:
```
```   767   fixes f :: "real^'n \<Rightarrow> real^'n"
```
```   768   assumes "linear f"
```
```   769   shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
```
```   770   using det_nz_iff_inj_gen[of f] assms
```
```   771   unfolding linear_matrix_vector_mul_eq .
```
```   772
```
```   773 lemma  det_eq_0_rank:
```
```   774   fixes A :: "real^'n^'n"
```
```   775   shows "det A = 0 \<longleftrightarrow> rank A < CARD('n)"
```
```   776   using invertible_det_nz [of A]
```
```   777   by (auto simp: matrix_left_invertible_injective invertible_left_inverse less_rank_noninjective)
```
```   778
```
```   779 subsubsection%important  \<open>Invertibility of matrices and corresponding linear functions\<close>
```
```   780
```
```   781 lemma  matrix_left_invertible_gen:
```
```   782   fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
```
```   783   assumes "Vector_Spaces.linear (*s) (*s) f"
```
```   784   shows "((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id))"
```
```   785 proof safe
```
```   786   fix B
```
```   787   assume 1: "B ** matrix f = mat 1"
```
```   788   show "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id"
```
```   789   proof (intro exI conjI)
```
```   790     show "Vector_Spaces.linear (*s) (*s) (\<lambda>y. B *v y)"
```
```   791       by simp
```
```   792     show "((*v) B) \<circ> f = id"
```
```   793       unfolding o_def
```
```   794       by (metis assms 1 eq_id_iff matrix_vector_mul(1) matrix_vector_mul_assoc matrix_vector_mul_lid)
```
```   795   qed
```
```   796 next
```
```   797   fix g
```
```   798   assume "Vector_Spaces.linear (*s) (*s) g" "g \<circ> f = id"
```
```   799   then have "matrix g ** matrix f = mat 1"
```
```   800     by (metis assms matrix_compose_gen matrix_id_mat_1)
```
```   801   then show "\<exists>B. B ** matrix f = mat 1" ..
```
```   802 qed
```
```   803
```
```   804 lemma  matrix_left_invertible:
```
```   805   "linear f \<Longrightarrow> ((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> g \<circ> f = id))" for f::"real^'m \<Rightarrow> real^'n"
```
```   806   using matrix_left_invertible_gen[of f]
```
```   807   by (auto simp: linear_matrix_vector_mul_eq)
```
```   808
```
```   809 lemma  matrix_right_invertible_gen:
```
```   810   fixes f :: "'a::field^'m \<Rightarrow> 'a^'n"
```
```   811   assumes "Vector_Spaces.linear (*s) (*s) f"
```
```   812   shows "((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id))"
```
```   813 proof safe
```
```   814   fix B
```
```   815   assume 1: "matrix f ** B = mat 1"
```
```   816   show "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id"
```
```   817   proof (intro exI conjI)
```
```   818     show "Vector_Spaces.linear (*s) (*s) ((*v) B)"
```
```   819       by simp
```
```   820     show "f \<circ> (*v) B = id"
```
```   821       using 1 assms comp_apply eq_id_iff vec.linear_id matrix_id_mat_1 matrix_vector_mul_assoc matrix_works
```
```   822       by (metis (no_types, hide_lams))
```
```   823   qed
```
```   824 next
```
```   825   fix g
```
```   826   assume "Vector_Spaces.linear (*s) (*s) g" and "f \<circ> g = id"
```
```   827   then have "matrix f ** matrix g = mat 1"
```
```   828     by (metis assms matrix_compose_gen matrix_id_mat_1)
```
```   829   then show "\<exists>B. matrix f ** B = mat 1" ..
```
```   830 qed
```
```   831
```
```   832 lemma  matrix_right_invertible:
```
```   833   "linear f \<Longrightarrow> ((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id))" for f::"real^'m \<Rightarrow> real^'n"
```
```   834   using matrix_right_invertible_gen[of f]
```
```   835   by (auto simp: linear_matrix_vector_mul_eq)
```
```   836
```
```   837 lemma  matrix_invertible_gen:
```
```   838   fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
```
```   839   assumes "Vector_Spaces.linear (*s) (*s) f"
```
```   840   shows  "invertible (matrix f) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
```
```   841     (is "?lhs = ?rhs")
```
```   842 proof
```
```   843   assume ?lhs then show ?rhs
```
```   844     by (metis assms invertible_def left_right_inverse_eq matrix_left_invertible_gen matrix_right_invertible_gen)
```
```   845 next
```
```   846   assume ?rhs then show ?lhs
```
```   847     by (metis assms invertible_def matrix_compose_gen matrix_id_mat_1)
```
```   848 qed
```
```   849
```
```   850 lemma  matrix_invertible:
```
```   851   "linear f \<Longrightarrow> invertible (matrix f) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
```
```   852   for f::"real^'m \<Rightarrow> real^'n"
```
```   853   using matrix_invertible_gen[of f]
```
```   854   by (auto simp: linear_matrix_vector_mul_eq)
```
```   855
```
```   856 lemma  invertible_eq_bij:
```
```   857   fixes m :: "'a::field^'m^'n"
```
```   858   shows "invertible m \<longleftrightarrow> bij ((*v) m)"
```
```   859   using matrix_invertible_gen[OF matrix_vector_mul_linear_gen, of m, simplified matrix_of_matrix_vector_mul]
```
```   860   by (metis bij_betw_def left_right_inverse_eq matrix_vector_mul_linear_gen o_bij
```
```   861       vec.linear_injective_left_inverse vec.linear_surjective_right_inverse)
```
```   862
```
```   863
```
```   864 subsection \<open>Cramer's rule\<close>
```
```   865
```
```   866 lemma  cramer_lemma_transpose:
```
```   867   fixes A:: "'a::{field}^'n^'n"
```
```   868     and x :: "'a::{field}^'n"
```
```   869   shows "det ((\<chi> i. if i = k then sum (\<lambda>i. x\$i *s row i A) (UNIV::'n set)
```
```   870                              else row i A)::'a::{field}^'n^'n) = x\$k * det A"
```
```   871   (is "?lhs = ?rhs")
```
```   872 proof -
```
```   873   let ?U = "UNIV :: 'n set"
```
```   874   let ?Uk = "?U - {k}"
```
```   875   have U: "?U = insert k ?Uk"
```
```   876     by blast
```
```   877   have kUk: "k \<notin> ?Uk"
```
```   878     by simp
```
```   879   have th00: "\<And>k s. x\$k *s row k A + s = (x\$k - 1) *s row k A + row k A + s"
```
```   880     by (vector field_simps)
```
```   881   have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f"
```
```   882     by auto
```
```   883   have "(\<chi> i. row i A) = A" by (vector row_def)
```
```   884   then have thd1: "det (\<chi> i. row i A) = det A"
```
```   885     by simp
```
```   886   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"
```
```   887     by (force intro: det_row_span vec.span_sum vec.span_scale vec.span_base)
```
```   888   show "?lhs = x\$k * det A"
```
```   889     apply (subst U)
```
```   890     unfolding sum.insert[OF finite kUk]
```
```   891     apply (subst th00)
```
```   892     unfolding add.assoc
```
```   893     apply (subst det_row_add)
```
```   894     unfolding thd0
```
```   895     unfolding det_row_mul
```
```   896     unfolding th001[of k "\<lambda>i. row i A"]
```
```   897     unfolding thd1
```
```   898     apply (simp add: field_simps)
```
```   899     done
```
```   900 qed
```
```   901
```
```   902 proposition  cramer_lemma:
```
```   903   fixes A :: "'a::{field}^'n^'n"
```
```   904   shows "det((\<chi> i j. if j = k then (A *v x)\$i else A\$i\$j):: 'a::{field}^'n^'n) = x\$k * det A"
```
```   905 proof -
```
```   906   let ?U = "UNIV :: 'n set"
```
```   907   have *: "\<And>c. sum (\<lambda>i. c i *s row i (transpose A)) ?U = sum (\<lambda>i. c i *s column i A) ?U"
```
```   908     by (auto intro: sum.cong)
```
```   909   show ?thesis
```
```   910     unfolding matrix_mult_sum
```
```   911     unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
```
```   912     unfolding *[of "\<lambda>i. x\$i"]
```
```   913     apply (subst det_transpose[symmetric])
```
```   914     apply (rule cong[OF refl[of det]])
```
```   915     apply (vector transpose_def column_def row_def)
```
```   916     done
```
```   917 qed
```
```   918
```
```   919 proposition  cramer:
```
```   920   fixes A ::"'a::{field}^'n^'n"
```
```   921   assumes d0: "det A \<noteq> 0"
```
```   922   shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j) / det A)"
```
```   923 proof -
```
```   924   from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
```
```   925     unfolding invertible_det_nz[symmetric] invertible_def
```
```   926     by blast
```
```   927   have "(A ** B) *v b = b"
```
```   928     by (simp add: B)
```
```   929   then have "A *v (B *v b) = b"
```
```   930     by (simp add: matrix_vector_mul_assoc)
```
```   931   then have xe: "\<exists>x. A *v x = b"
```
```   932     by blast
```
```   933   {
```
```   934     fix x
```
```   935     assume x: "A *v x = b"
```
```   936     have "x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j) / det A)"
```
```   937       unfolding x[symmetric]
```
```   938       using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)
```
```   939   }
```
```   940   with xe show ?thesis
```
```   941     by auto
```
```   942 qed
```
```   943
```
```   944 lemma  det_1: "det (A::'a::comm_ring_1^1^1) = A\$1\$1"
```
```   945   by (simp add: det_def sign_id)
```
```   946
```
```   947 lemma  det_2: "det (A::'a::comm_ring_1^2^2) = A\$1\$1 * A\$2\$2 - A\$1\$2 * A\$2\$1"
```
```   948 proof -
```
```   949   have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
```
```   950   show ?thesis
```
```   951     unfolding det_def UNIV_2
```
```   952     unfolding sum_over_permutations_insert[OF f12]
```
```   953     unfolding permutes_sing
```
```   954     by (simp add: sign_swap_id sign_id swap_id_eq)
```
```   955 qed
```
```   956
```
```   957 lemma  det_3:
```
```   958   "det (A::'a::comm_ring_1^3^3) =
```
```   959     A\$1\$1 * A\$2\$2 * A\$3\$3 +
```
```   960     A\$1\$2 * A\$2\$3 * A\$3\$1 +
```
```   961     A\$1\$3 * A\$2\$1 * A\$3\$2 -
```
```   962     A\$1\$1 * A\$2\$3 * A\$3\$2 -
```
```   963     A\$1\$2 * A\$2\$1 * A\$3\$3 -
```
```   964     A\$1\$3 * A\$2\$2 * A\$3\$1"
```
```   965 proof -
```
```   966   have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}"
```
```   967     by auto
```
```   968   have f23: "finite {3::3}" "2 \<notin> {3::3}"
```
```   969     by auto
```
```   970
```
```   971   show ?thesis
```
```   972     unfolding det_def UNIV_3
```
```   973     unfolding sum_over_permutations_insert[OF f123]
```
```   974     unfolding sum_over_permutations_insert[OF f23]
```
```   975     unfolding permutes_sing
```
```   976     by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
```
```   977 qed
```
```   978
```
```   979 proposition  det_orthogonal_matrix:
```
```   980   fixes Q:: "'a::linordered_idom^'n^'n"
```
```   981   assumes oQ: "orthogonal_matrix Q"
```
```   982   shows "det Q = 1 \<or> det Q = - 1"
```
```   983 proof -
```
```   984   have "Q ** transpose Q = mat 1"
```
```   985     by (metis oQ orthogonal_matrix_def)
```
```   986   then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
```
```   987     by simp
```
```   988   then have "det Q * det Q = 1"
```
```   989     by (simp add: det_mul)
```
```   990   then show ?thesis
```
```   991     by (simp add: square_eq_1_iff)
```
```   992 qed
```
```   993
```
```   994 proposition  orthogonal_transformation_det [simp]:
```
```   995   fixes f :: "real^'n \<Rightarrow> real^'n"
```
```   996   shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
```
```   997   using%unimportant det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
```
```   998
```
```   999 subsection  \<open>Rotation, reflection, rotoinversion\<close>
```
```  1000
```
```  1001 definition%important  "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
```
```  1002 definition%important  "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
```
```  1003
```
```  1004 lemma  orthogonal_rotation_or_rotoinversion:
```
```  1005   fixes Q :: "'a::linordered_idom^'n^'n"
```
```  1006   shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
```
```  1007   by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
```
```  1008
```
```  1009 text\<open> Slightly stronger results giving rotation, but only in two or more dimensions\<close>
```
```  1010
```
```  1011 lemma  rotation_matrix_exists_basis:
```
```  1012   fixes a :: "real^'n"
```
```  1013   assumes 2: "2 \<le> CARD('n)" and "norm a = 1"
```
```  1014   obtains A where "rotation_matrix A" "A *v (axis k 1) = a"
```
```  1015 proof -
```
```  1016   obtain A where "orthogonal_matrix A" and A: "A *v (axis k 1) = a"
```
```  1017     using orthogonal_matrix_exists_basis assms by metis
```
```  1018   with orthogonal_rotation_or_rotoinversion
```
```  1019   consider "rotation_matrix A" | "rotoinversion_matrix A"
```
```  1020     by metis
```
```  1021   then show thesis
```
```  1022   proof cases
```
```  1023     assume "rotation_matrix A"
```
```  1024     then show ?thesis
```
```  1025       using \<open>A *v axis k 1 = a\<close> that by auto
```
```  1026   next
```
```  1027     from ex_card[OF 2] obtain h i::'n where "h \<noteq> i"
```
```  1028       by (auto simp add: eval_nat_numeral card_Suc_eq)
```
```  1029     then obtain j where "j \<noteq> k"
```
```  1030       by (metis (full_types))
```
```  1031     let ?TA = "transpose A"
```
```  1032     let ?A = "\<chi> i. if i = j then - 1 *\<^sub>R (?TA \$ i) else ?TA \$i"
```
```  1033     assume "rotoinversion_matrix A"
```
```  1034     then have [simp]: "det A = -1"
```
```  1035       by (simp add: rotoinversion_matrix_def)
```
```  1036     show ?thesis
```
```  1037     proof
```
```  1038       have [simp]: "row i (\<chi> i. if i = j then - 1 *\<^sub>R ?TA \$ i else ?TA \$ i) = (if i = j then - row i ?TA else row i ?TA)" for i
```
```  1039         by (auto simp: row_def)
```
```  1040       have "orthogonal_matrix ?A"
```
```  1041         unfolding orthogonal_matrix_orthonormal_rows
```
```  1042         using \<open>orthogonal_matrix A\<close> by (auto simp: orthogonal_matrix_orthonormal_columns orthogonal_clauses)
```
```  1043       then show "rotation_matrix (transpose ?A)"
```
```  1044         unfolding rotation_matrix_def
```
```  1045         by (simp add: det_row_mul[of j _ "\<lambda>i. ?TA \$ i", unfolded scalar_mult_eq_scaleR])
```
```  1046       show "transpose ?A *v axis k 1 = a"
```
```  1047         using \<open>j \<noteq> k\<close> A by (simp add: matrix_vector_column axis_def scalar_mult_eq_scaleR if_distrib [of "\<lambda>z. z *\<^sub>R c" for c] cong: if_cong)
```
```  1048     qed
```
```  1049   qed
```
```  1050 qed
```
```  1051
```
```  1052 lemma  rotation_exists_1:
```
```  1053   fixes a :: "real^'n"
```
```  1054   assumes "2 \<le> CARD('n)" "norm a = 1" "norm b = 1"
```
```  1055   obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
```
```  1056 proof -
```
```  1057   obtain k::'n where True
```
```  1058     by simp
```
```  1059   obtain A B where AB: "rotation_matrix A" "rotation_matrix B"
```
```  1060                and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
```
```  1061     using rotation_matrix_exists_basis assms by metis
```
```  1062   let ?f = "\<lambda>x. (B ** transpose A) *v x"
```
```  1063   show thesis
```
```  1064   proof
```
```  1065     show "orthogonal_transformation ?f"
```
```  1066       using AB orthogonal_matrix_mul orthogonal_transformation_matrix rotation_matrix_def matrix_vector_mul_linear by force
```
```  1067     show "det (matrix ?f) = 1"
```
```  1068       using AB by (auto simp: det_mul rotation_matrix_def)
```
```  1069     show "?f a = b"
```
```  1070       using AB unfolding orthogonal_matrix_def rotation_matrix_def
```
```  1071       by (metis eq matrix_mul_rid matrix_vector_mul_assoc)
```
```  1072   qed
```
```  1073 qed
```
```  1074
```
```  1075 lemma  rotation_exists:
```
```  1076   fixes a :: "real^'n"
```
```  1077   assumes 2: "2 \<le> CARD('n)" and eq: "norm a = norm b"
```
```  1078   obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
```
```  1079 proof (cases "a = 0 \<or> b = 0")
```
```  1080   case True
```
```  1081   with assms have "a = 0" "b = 0"
```
```  1082     by auto
```
```  1083   then show ?thesis
```
```  1084     by (metis eq_id_iff matrix_id orthogonal_transformation_id that)
```
```  1085 next
```
```  1086   case False
```
```  1087   then obtain f where f: "orthogonal_transformation f" "det (matrix f) = 1"
```
```  1088     and f': "f (a /\<^sub>R norm a) = b /\<^sub>R norm b"
```
```  1089     using rotation_exists_1 [of "a /\<^sub>R norm a" "b /\<^sub>R norm b", OF 2] by auto
```
```  1090   then interpret linear f by (simp add: orthogonal_transformation)
```
```  1091   have "f a = b"
```
```  1092     using f' False
```
```  1093     by (simp add: eq scale)
```
```  1094   with f show thesis ..
```
```  1095 qed
```
```  1096
```
```  1097 lemma  rotation_rightward_line:
```
```  1098   fixes a :: "real^'n"
```
```  1099   obtains f where "orthogonal_transformation f" "2 \<le> CARD('n) \<Longrightarrow> det(matrix f) = 1"
```
```  1100                   "f(norm a *\<^sub>R axis k 1) = a"
```
```  1101 proof (cases "CARD('n) = 1")
```
```  1102   case True
```
```  1103   obtain f where "orthogonal_transformation f" "f (norm a *\<^sub>R axis k (1::real)) = a"
```
```  1104   proof (rule orthogonal_transformation_exists)
```
```  1105     show "norm (norm a *\<^sub>R axis k (1::real)) = norm a"
```
```  1106       by simp
```
```  1107   qed auto
```
```  1108   then show thesis
```
```  1109     using True that by auto
```
```  1110 next
```
```  1111   case False
```
```  1112   obtain f where "orthogonal_transformation f" "det(matrix f) = 1" "f (norm a *\<^sub>R axis k 1) = a"
```
```  1113   proof (rule rotation_exists)
```
```  1114     show "2 \<le> CARD('n)"
```
```  1115       using False one_le_card_finite [where 'a='n] by linarith
```
```  1116     show "norm (norm a *\<^sub>R axis k (1::real)) = norm a"
```
```  1117       by simp
```
```  1118   qed auto
```
```  1119   then show thesis
```
```  1120     using that by blast
```
```  1121 qed
```
```  1122
```
```  1123 end
```