diff --git a/thys/Smith_Normal_Form/Cauchy_Binet.thy b/thys/Smith_Normal_Form/Cauchy_Binet.thy
--- a/thys/Smith_Normal_Form/Cauchy_Binet.thy
+++ b/thys/Smith_Normal_Form/Cauchy_Binet.thy
@@ -1,1426 +1,1425 @@
 (*
     Author:      Jose Divasón
-    Email:       jose.divason@unirioja.es           
+    Email:       jose.divason@unirioja.es
 *)
 
 section \<open>The Cauchy--Binet formula\<close>
 
 theory Cauchy_Binet
   imports
   Diagonal_To_Smith
-  SNF_Missing_Lemmas  
+  SNF_Missing_Lemmas
 begin
 
 subsection \<open>Previous missing results about @{text "pick"} and @{text "insert"}\<close>
 
 lemma pick_insert:
   assumes a_notin_I: "a \<notin> I" and i2: "i < card I"
     and a_def: "pick (insert a I) a' = a" (*Alternative: index (insort a (sorted_list_of_set I)) a = a'*)
     and ia': "i < a'" (*Case 1*)
     and a'_card: "a' < card I + 1"
   shows "pick (insert a I) i = pick I i"
 proof -
   have finI: "finite I"
     using i2
     using card_infinite by force
   have "pick (insert a I) i = sorted_list_of_set (insert a I) ! i"
   proof (rule sorted_list_of_set_eq_pick[symmetric])
     have "finite (insert a I)"
       using card_infinite i2 by force
     thus "i < length (sorted_list_of_set (insert a I))"
-      by (metis a_notin_I card_insert_disjoint distinct_card finite_insert 
+      by (metis a_notin_I card_insert_disjoint distinct_card finite_insert
           i2 less_Suc_eq sorted_list_of_set(1) sorted_list_of_set(3))
   qed
   also have "... = insort a (sorted_list_of_set I) ! i"
     using sorted_list_of_set.insert
     by (metis a_notin_I card_infinite i2 not_less0)
   also have "... = (sorted_list_of_set I) ! i"
   proof (rule insort_nth[OF])
      show "sorted (sorted_list_of_set I)" by auto
      show "a \<notin> set (sorted_list_of_set I)" using a_notin_I
        by (metis card_infinite i2 not_less_zero set_sorted_list_of_set)
      have "index (sorted_list_of_set (insert a I)) a = a'"
        using pick_index a_def
        using a'_card a_notin_I finI by auto
      hence "index (insort a (sorted_list_of_set I)) a = a'"
        by (simp add: a_notin_I finI)
      thus "i < index (insort a (sorted_list_of_set I)) a" using ia' by auto
-     show "sorted_list_of_set I \<noteq> []" using finI i2 by fastforce                  
+     show "sorted_list_of_set I \<noteq> []" using finI i2 by fastforce
    qed
   also have "... = pick I i"
   proof (rule sorted_list_of_set_eq_pick)
     have "finite I" using card_infinite i2 by fastforce
-    thus "i < length (sorted_list_of_set I)" 
+    thus "i < length (sorted_list_of_set I)"
       by (metis distinct_card distinct_sorted_list_of_set i2 set_sorted_list_of_set)
   qed
   finally show ?thesis .
 qed
 
 
 lemma pick_insert2:
   assumes a_notin_I: "a \<notin> I" and i2: "i < card I"
     and a_def: "pick (insert a I) a' = a" (*Alternative: index (sorted_list_of_set (insert a I)) a = a'*)
     and ia': "i \<ge> a'" (*Case 2*)
     and a'_card: "a' < card I + 1"
   shows "pick (insert a I) i < pick I i"
 proof (cases "i = 0")
   case True
-  then show ?thesis 
-    by (auto, metis (mono_tags, lifting) DL_Missing_Sublist.pick.simps(1) Least_le a_def a_notin_I 
+  then show ?thesis
+    by (auto, metis (mono_tags, lifting) DL_Missing_Sublist.pick.simps(1) Least_le a_def a_notin_I
         dual_order.order_iff_strict i2 ia' insertCI le_zero_eq not_less_Least pick_in_set_le)
 next
   case False
   hence i0: "i = Suc (i - 1)" using a'_card ia' by auto
   have finI: "finite I"
     using i2 card_infinite by force
   have index_a'1: "index (sorted_list_of_set (insert a I)) a = a'"
     using pick_index
     using a'_card a_def a_notin_I finI by auto
-  hence index_a': "index (insort a (sorted_list_of_set I)) a = a'" 
+  hence index_a': "index (insort a (sorted_list_of_set I)) a = a'"
     by (simp add: a_notin_I finI)
   have i1_length: "i - 1 < length (sorted_list_of_set I)" using i2
-    by (metis distinct_card distinct_sorted_list_of_set finI 
+    by (metis distinct_card distinct_sorted_list_of_set finI
         less_imp_diff_less set_sorted_list_of_set)
   have 1: "pick (insert a I) i = sorted_list_of_set (insert a I) ! i"
   proof (rule sorted_list_of_set_eq_pick[symmetric])
     have "finite (insert a I)"
       using card_infinite i2 by force
     thus "i < length (sorted_list_of_set (insert a I))"
-      by (metis a_notin_I card_insert_disjoint distinct_card finite_insert 
+      by (metis a_notin_I card_insert_disjoint distinct_card finite_insert
           i2 less_Suc_eq sorted_list_of_set(1) sorted_list_of_set(3))
   qed
   also have 2: "... = insort a (sorted_list_of_set I) ! i"
     using sorted_list_of_set.insert
     by (metis a_notin_I card_infinite i2 not_less0)
   also have "... = insort a (sorted_list_of_set I) ! Suc (i-1)" using i0 by auto
   also have "... < pick I i"
   proof (cases "i = a'")
     case True
     have "(sorted_list_of_set I) ! i > a"
       by (smt "1" Suc_less_eq True a_def a_notin_I distinct_card distinct_sorted_list_of_set finI i2
-          ia' index_a' insort_nth2 length_insort lessI list.size(3) nat_less_le not_less_zero 
-          pick_in_set_le set_sorted_list_of_set sorted_list_of_set(2) sorted_list_of_set.insert 
+          ia' index_a' insort_nth2 length_insort lessI list.size(3) nat_less_le not_less_zero
+          pick_in_set_le set_sorted_list_of_set sorted_list_of_set(2) sorted_list_of_set.insert
           sorted_list_of_set_eq_pick sorted_sorted_wrt sorted_wrt_nth_less)
-    moreover have "a = insort a (sorted_list_of_set I) ! i" using True 1 2 a_def by auto        
+    moreover have "a = insort a (sorted_list_of_set I) ! i" using True 1 2 a_def by auto
     ultimately show ?thesis using 1 2
-      by (metis distinct_card finI i0 i2 set_sorted_list_of_set 
+      by (metis distinct_card finI i0 i2 set_sorted_list_of_set
           sorted_list_of_set(3) sorted_list_of_set_eq_pick)
   next
-    case False   
+    case False
     have "insort a (sorted_list_of_set I) ! Suc (i-1) = (sorted_list_of_set I) ! (i-1)"
       by (rule insort_nth2, insert i1_length False ia' index_a', auto simp add: a_notin_I finI)
     also have "... = pick I (i-1)"
       by (rule sorted_list_of_set_eq_pick[OF i1_length])
     also have "... < pick I i" using i0 i2 pick_mono_le by auto
     finally show ?thesis .
   qed
   finally show ?thesis .
 qed
 
 lemma pick_insert3:
   assumes a_notin_I: "a \<notin> I" and i2: "i < card I"
     and a_def: "pick (insert a I) a' = a" (*Alternative: index (sorted_list_of_set (insert a I)) a = a'.*)
     and ia': "i \<ge> a'" (*Case 2*)
     and a'_card: "a' < card I + 1"
   shows "pick (insert a I) (Suc i) = pick I i"
 proof (cases "i = 0")
   case True
   have a_LEAST: "a < (LEAST aa. aa\<in>I)"
     using True a_def a_notin_I i2 ia' pick_insert2 by fastforce
-  have Least_rw: "(LEAST aa. aa = a \<or> aa \<in> I) = a"  
-    by (rule Least_equality, insert a_notin_I, auto, 
+  have Least_rw: "(LEAST aa. aa = a \<or> aa \<in> I) = a"
+    by (rule Least_equality, insert a_notin_I, auto,
         metis a_LEAST le_less_trans nat_le_linear not_less_Least)
   let ?P = "\<lambda>aa. (aa = a \<or> aa \<in> I) \<and> (LEAST aa. aa = a \<or> aa \<in> I) < aa"
-  let ?Q = "\<lambda>aa. aa \<in> I" 
+  let ?Q = "\<lambda>aa. aa \<in> I"
   have "?P = ?Q" unfolding Least_rw fun_eq_iff
-    by (auto, metis a_LEAST le_less_trans not_le not_less_Least)  
-  thus ?thesis using True by auto    
+    by (auto, metis a_LEAST le_less_trans not_le not_less_Least)
+  thus ?thesis using True by auto
 next
   case False
   have finI: "finite I"
     using i2 card_infinite by force
   have index_a'1: "index (sorted_list_of_set (insert a I)) a = a'"
     using pick_index
     using a'_card a_def a_notin_I finI by auto
-  hence index_a': "index (insort a (sorted_list_of_set I)) a = a'" 
+  hence index_a': "index (insort a (sorted_list_of_set I)) a = a'"
     by (simp add: a_notin_I finI)
   have i1_length: "i < length (sorted_list_of_set I)" using i2
     by (metis distinct_card distinct_sorted_list_of_set finI set_sorted_list_of_set)
   have 1: "pick (insert a I) (Suc i) = sorted_list_of_set (insert a I) ! (Suc i)"
   proof (rule sorted_list_of_set_eq_pick[symmetric])
     have "finite (insert a I)"
       using card_infinite i2 by force
     thus "Suc i < length (sorted_list_of_set (insert a I))"
-      by (metis Suc_mono a_notin_I card_insert_disjoint distinct_card distinct_sorted_list_of_set 
+      by (metis Suc_mono a_notin_I card_insert_disjoint distinct_card distinct_sorted_list_of_set
           finI i2 set_sorted_list_of_set)
   qed
   also have 2: "... = insort a (sorted_list_of_set I) ! Suc i"
     using sorted_list_of_set.insert
     by (metis a_notin_I card_infinite i2 not_less0)
   also have "... = pick I i"
   proof (cases "i = a'")
     case True
     show ?thesis
-      by (metis True a_notin_I finI i1_length index_a' insort_nth2 le_refl list.size(3) not_less0 
+      by (metis True a_notin_I finI i1_length index_a' insort_nth2 le_refl list.size(3) not_less0
           set_sorted_list_of_set sorted_list_of_set(2) sorted_list_of_set_eq_pick)
   next
-    case False   
+    case False
     have "insort a (sorted_list_of_set I) ! Suc i = (sorted_list_of_set I) ! i"
       by (rule insort_nth2, insert i1_length False ia' index_a', auto simp add: a_notin_I finI)
     also have "... = pick I i"
       by (rule sorted_list_of_set_eq_pick[OF i1_length])
     finally show ?thesis .
   qed
   finally show ?thesis .
 qed
 
 
 lemma pick_insert_index:
   assumes Ik: "card I = k"
   and a_notin_I: "a \<notin> I"
   and ik: "i < k"
   and a_def: "pick (insert a I) a' = a"
   and a'k: "a' < card I + 1"
 shows "pick (insert a I) (insert_index a' i) = pick I i"
 proof (cases "i<a'")
   case True
   have "pick (insert a I) i = pick I i"
     by (rule pick_insert[OF a_notin_I _ a_def _ a'k], auto simp add: Ik ik True)
   thus ?thesis using True unfolding insert_index_def by auto
 next
   case False note i_ge_a' = False
   have fin_aI: "finite (insert a I)"
     using Ik finite_insert ik by fastforce
   let ?P = "\<lambda>aa. (aa = a \<or> aa \<in> I) \<and> pick (insert a I) i < aa"
   let ?Q = "\<lambda>aa. aa \<in> I \<and> pick (insert a I) i < aa"
   have "?P = ?Q" using a_notin_I unfolding fun_eq_iff
-    by (auto, metis False Ik a_def card_infinite card_insert_disjoint ik less_SucI 
+    by (auto, metis False Ik a_def card_infinite card_insert_disjoint ik less_SucI
         linorder_neqE_nat not_less_zero order.asym pick_mono_le)
   hence "Least ?P = Least ?Q" by simp
   also have "... = pick I i"
   proof (rule Least_equality, rule conjI)
     show "pick I i \<in> I"
       by (simp add: Ik ik pick_in_set_le)
     show "pick (insert a I) i < pick I i"
       by (rule pick_insert2[OF a_notin_I _ a_def _ a'k], insert False, auto simp add: Ik ik)
     fix y assume y: "y \<in> I \<and> pick (insert a I) i < y"
     let ?xs = "sorted_list_of_set (insert a I)"
     have "y \<in> set ?xs" using y by (metis fin_aI insertI2 set_sorted_list_of_set y)
     from this obtain j where xs_j_y: "?xs ! j = y" and j: "j < length ?xs"
       using in_set_conv_nth by metis
     have ij: "i<j"
-      by (metis (no_types, lifting) Ik a_notin_I card_infinite card_insert_disjoint ik j less_SucI 
+      by (metis (no_types, lifting) Ik a_notin_I card_infinite card_insert_disjoint ik j less_SucI
           linorder_neqE_nat not_less_zero order.asym pick_mono_le sorted_list_of_set_eq_pick xs_j_y y)
     have "pick I i = pick (insert a I) (Suc i)"
       by (rule pick_insert3[symmetric, OF a_notin_I _ a_def _ a'k], insert False Ik ik, auto)
-    also have "... \<le> pick (insert a I) j" 
-      by (metis Ik Suc_lessI card_infinite distinct_card distinct_sorted_list_of_set eq_iff 
+    also have "... \<le> pick (insert a I) j"
+      by (metis Ik Suc_lessI card_infinite distinct_card distinct_sorted_list_of_set eq_iff
           finite_insert ij ik j less_imp_le_nat not_less_zero pick_mono_le set_sorted_list_of_set)
     also have "... = ?xs ! j" by (rule sorted_list_of_set_eq_pick[symmetric, OF j])
     also have "... = y" by (rule xs_j_y)
     finally show "pick I i \<le> y" .
   qed
   finally show ?thesis unfolding insert_index_def using False by auto
 qed
 
 
 subsection\<open>Start of the proof\<close>
 
 definition "strict_from_inj n f = (\<lambda>i. if i\<in>{0..<n} then (sorted_list_of_set (f`{0..<n})) ! i else i)"
 
 lemma strict_strict_from_inj:
   fixes f::"nat \<Rightarrow> nat"
   assumes "inj_on f {0..<n}" shows "strict_mono_on (strict_from_inj n f) {0..<n}"
 proof -
   let ?I="f`{0..<n}"
-  have "strict_from_inj n f x < strict_from_inj n f y" 
+  have "strict_from_inj n f x < strict_from_inj n f y"
     if xy: "x < y" and x: "x \<in> {0..<n}" and y: "y \<in> {0..<n}" for x y
   proof -
     let ?xs = "(sorted_list_of_set ?I)"
     have sorted_xs: "sorted ?xs" by (rule sorted_sorted_list_of_set)
-    have "strict_from_inj n f x = (sorted_list_of_set ?I) ! x" 
+    have "strict_from_inj n f x = (sorted_list_of_set ?I) ! x"
       unfolding strict_from_inj_def using x by auto
-    also have "... < (sorted_list_of_set ?I) ! y" 
-    proof (rule sorted_nth_strict_mono, auto)
-      show "y < length (sorted_list_of_set ?I)"
-        by (metis assms atLeastLessThan_iff card_atLeastLessThan card_image diff_zero distinct_card 
-            distinct_sorted_list_of_set finite_atLeastLessThan finite_imageI set_sorted_list_of_set y)
+    also have "... < (sorted_list_of_set ?I) ! y"
+    proof (rule sorted_nth_strict_mono; clarsimp?)
+      show "y < card (f ` {0..<n})"
+        by (metis assms atLeastLessThan_iff card_atLeastLessThan card_image diff_zero y)
     qed (simp add: xy)
     also have "... = strict_from_inj n f y" using y unfolding strict_from_inj_def by simp
     finally show ?thesis .
   qed
   thus ?thesis unfolding strict_mono_on_def by simp
 qed
 
 
 
 
 lemma strict_from_inj_image':
   assumes f: "inj_on f {0..<n}"
   shows "strict_from_inj n f ` {0..<n} = f`{0..<n}"
 proof (auto)
   let ?I = "f ` {0..<n}"
-  fix xa assume xa: "xa < n" 
+  fix xa assume xa: "xa < n"
   have inj_on: "inj_on f {0..<n}" using f  by auto
   have length_I: "length (sorted_list_of_set ?I) = n"
-    by (metis card_atLeastLessThan card_image diff_zero distinct_card distinct_sorted_list_of_set 
+    by (metis card_atLeastLessThan card_image diff_zero distinct_card distinct_sorted_list_of_set
         finite_atLeastLessThan finite_imageI inj_on sorted_list_of_set(1))
- 
+
   have "strict_from_inj n f xa = sorted_list_of_set ?I ! xa"
     using xa unfolding strict_from_inj_def by auto
-  also have "... = pick ?I xa" 
+  also have "... = pick ?I xa"
     by (rule sorted_list_of_set_eq_pick, unfold length_I, auto simp add: xa)
   also have "... \<in> f ` {0..<n}" by (rule pick_in_set_le, simp add: card_image inj_on xa)
   finally show "strict_from_inj n f xa \<in> f ` {0..<n}" .
   obtain i where "sorted_list_of_set (f`{0..<n}) ! i = f xa" and "i<n"
-    by (metis atLeast0LessThan finite_atLeastLessThan finite_imageI imageI 
+    by (metis atLeast0LessThan finite_atLeastLessThan finite_imageI imageI
         in_set_conv_nth length_I lessThan_iff sorted_list_of_set(1) xa)
   thus "f xa \<in> strict_from_inj n f ` {0..<n}"
-    by (metis atLeast0LessThan imageI lessThan_iff strict_from_inj_def)    
+    by (metis atLeast0LessThan imageI lessThan_iff strict_from_inj_def)
 qed
 
 
 
-definition "Z (n::nat) (m::nat) = {(f,\<pi>)|f \<pi>. f \<in> {0..<n} \<rightarrow> {0..<m} 
-  \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+definition "Z (n::nat) (m::nat) = {(f,\<pi>)|f \<pi>. f \<in> {0..<n} \<rightarrow> {0..<m}
+  \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> \<pi> permutes {0..<n}}"
 
 lemma Z_alt_def: "Z n m = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} \<times> {\<pi>. \<pi> permutes {0..<n}}"
   unfolding Z_def by auto
 
 lemma det_mul_finsum_alt:
   assumes A: "A \<in> carrier_mat n m"
     and B: "B \<in> carrier_mat m n"
-  shows "det (A*B) = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a::comm_ring_1) n 
+  shows "det (A*B) = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a::comm_ring_1) n
   (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v Matrix.col A k) {0..<m}))"
 proof -
   have AT: "A\<^sup>T \<in> carrier_mat m n" using A by auto
   have BT: "B\<^sup>T \<in> carrier_mat n m" using B by auto
   let ?f = "(\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B\<^sup>T $$ (i, k) \<cdot>\<^sub>v Matrix.row A\<^sup>T k) {0..<m})"
   let ?g = "(\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v Matrix.col A k) {0..<m})"
   let ?lhs = "mat\<^sub>r n n ?f"
   let ?rhs = "mat\<^sub>r n n ?g"
   have lhs_rhs: "?lhs = ?rhs"
   proof (rule eq_matI)
     show "dim_row ?lhs = dim_row ?rhs" by auto
     show "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i < dim_row ?rhs" and j: "j < dim_col ?rhs"
     have j_n: "j<n" using j by auto
     have "?lhs $$ (i, j) = ?f i $v j" by (rule index_mat, insert i j, auto)
-    also have "... = (\<Sum>k\<in>{0..<m}. (B\<^sup>T $$ (i, k) \<cdot>\<^sub>v row A\<^sup>T k) $ j)" 
+    also have "... = (\<Sum>k\<in>{0..<m}. (B\<^sup>T $$ (i, k) \<cdot>\<^sub>v row A\<^sup>T k) $ j)"
       by (rule index_finsum_vec[OF _ j_n], auto simp add: A)
     also have "... = (\<Sum>k\<in>{0..<m}. (B $$ (k, i) \<cdot>\<^sub>v col A k) $ j)"
     proof (rule sum.cong, auto)
       fix x assume x: "x<m"
       have row_rw: "Matrix.row A\<^sup>T x = col A x" by (rule row_transpose, insert A x, auto)
-      have B_rw: "B\<^sup>T $$ (i,x) = B $$ (x, i)" 
+      have B_rw: "B\<^sup>T $$ (i,x) = B $$ (x, i)"
         by (rule index_transpose_mat, insert x i B, auto)
-      have "(B\<^sup>T $$ (i, x) \<cdot>\<^sub>v Matrix.row A\<^sup>T x) $v j = B\<^sup>T $$ (i, x) * Matrix.row A\<^sup>T x $v j" 
+      have "(B\<^sup>T $$ (i, x) \<cdot>\<^sub>v Matrix.row A\<^sup>T x) $v j = B\<^sup>T $$ (i, x) * Matrix.row A\<^sup>T x $v j"
         by (rule index_smult_vec, insert A j_n, auto)
       also have "... = B $$ (x, i) * col A x $v j" unfolding row_rw B_rw by simp
       also have "... = (B $$ (x, i) \<cdot>\<^sub>v col A x) $v j"
         by (rule index_smult_vec[symmetric], insert A j_n, auto)
       finally show " (B\<^sup>T $$ (i, x) \<cdot>\<^sub>v Matrix.row A\<^sup>T x) $v j = (B $$ (x, i) \<cdot>\<^sub>v col A x) $v j" .
-    qed      
-    also have "... = ?g i $v j" 
+    qed
+    also have "... = ?g i $v j"
       by (rule index_finsum_vec[symmetric, OF _ j_n], auto simp add: A)
     also have "... = ?rhs $$ (i, j)" by (rule index_mat[symmetric], insert i j, auto)
     finally show "?lhs $$ (i, j) = ?rhs $$ (i, j)" .
   qed
   have "det (A*B) = det (B\<^sup>T*A\<^sup>T)"
     using det_transpose
     by (metis A B Matrix.transpose_mult mult_carrier_mat)
-  also have "... =  det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B\<^sup>T $$ (i, k) \<cdot>\<^sub>v Matrix.row A\<^sup>T k) {0..<m}))" 
+  also have "... =  det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B\<^sup>T $$ (i, k) \<cdot>\<^sub>v Matrix.row A\<^sup>T k) {0..<m}))"
     using mat_mul_finsum_alt[OF BT AT] by auto
   also have "... = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a) n (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v Matrix.col A k) {0..<m}))"
     by (rule arg_cong[of _ _ det], rule lhs_rhs)
   finally show ?thesis .
 qed
 
 
 lemma det_cols_mul:
   assumes A: "A \<in> carrier_mat n m"
     and B: "B \<in> carrier_mat m n"
   shows "det (A*B) = (\<Sum>f | (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i).
        (\<Prod>i = 0..<n. B $$ (f i, i)) * Determinant.det (mat\<^sub>r n n (\<lambda>i. col A (f i))))"
 proof -
   let ?V="{0..<n}"
   let ?U = "{0..<m}"
   let ?F = " {f. (\<forall>i\<in> {0..<n}. f i \<in> ?U) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   let ?g = "\<lambda>f. det (mat\<^sub>r n n (\<lambda> i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i)))"
   have fm: "finite {0..<m}" by auto
   have fn: "finite {0..<n}" by auto
-  have det_rw: "det (mat\<^sub>r n n (\<lambda>i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i))) = 
-    (prod (\<lambda>i. B $$ (f i, i)) {0..<n}) * det (mat\<^sub>r n n (\<lambda>i. col A (f i)))" 
+  have det_rw: "det (mat\<^sub>r n n (\<lambda>i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i))) =
+    (prod (\<lambda>i. B $$ (f i, i)) {0..<n}) * det (mat\<^sub>r n n (\<lambda>i. col A (f i)))"
     if  f: "(\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)" for f
-    by (rule det_rows_mul, insert A col_dim, auto) 
+    by (rule det_rows_mul, insert A col_dim, auto)
   have "det (A*B) = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a::comm_ring_1) n (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v Matrix.col A k) ?U))"
     by (rule det_mul_finsum_alt[OF A B])
   also have "... = sum ?g ?F" by (rule det_linear_rows_sum[OF fm], auto simp add: A)
-  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} * det (mat\<^sub>r n n (\<lambda>i. col A (f i))))" 
+  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} * det (mat\<^sub>r n n (\<lambda>i. col A (f i))))"
     using det_rw by auto
   finally show ?thesis .
 qed
 
 lemma det_cols_mul':
   assumes A: "A \<in> carrier_mat n m"
     and B: "B \<in> carrier_mat m n"
   shows "det (A*B) = (\<Sum>f | (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i).
        (\<Prod>i = 0..<n. A $$ (i, f i)) * det (mat\<^sub>r n n (\<lambda>i. row B (f i))))"
 proof -
   let ?F="{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have t: "A * B = (B\<^sup>T*A\<^sup>T)\<^sup>T" using transpose_mult[OF A B] transpose_transpose by metis
-  have "det (B\<^sup>T*A\<^sup>T) = (\<Sum>f\<in>?F. (\<Prod>i = 0..<n. A\<^sup>T $$ (f i, i)) * det (mat\<^sub>r n n (\<lambda>i. col B\<^sup>T (f i))))" 
+  have "det (B\<^sup>T*A\<^sup>T) = (\<Sum>f\<in>?F. (\<Prod>i = 0..<n. A\<^sup>T $$ (f i, i)) * det (mat\<^sub>r n n (\<lambda>i. col B\<^sup>T (f i))))"
     by (rule det_cols_mul, auto simp add: A B)
   also have "... = (\<Sum>f \<in>?F. (\<Prod>i = 0..<n. A $$ (i, f i)) * det (mat\<^sub>r n n (\<lambda>i. row B (f i))))"
   proof (rule sum.cong, rule refl)
     fix f assume f: "f \<in> ?F"
-    have "(\<Prod>i = 0..<n. A\<^sup>T $$ (f i, i)) = (\<Prod>i = 0..<n. A $$ (i, f i))" 
+    have "(\<Prod>i = 0..<n. A\<^sup>T $$ (f i, i)) = (\<Prod>i = 0..<n. A $$ (i, f i))"
     proof (rule prod.cong, rule refl)
-      fix x assume x: "x \<in> {0..<n}" 
-      show "A\<^sup>T $$ (f x, x) = A $$ (x, f x)" 
+      fix x assume x: "x \<in> {0..<n}"
+      show "A\<^sup>T $$ (f x, x) = A $$ (x, f x)"
         by (rule index_transpose_mat(1), insert f A x, auto)
     qed
     moreover have "det (mat\<^sub>r n n (\<lambda>i. col B\<^sup>T (f i))) = det (mat\<^sub>r n n (\<lambda>i. row B (f i)))"
     proof -
       have row_eq_colT: "row B (f i) $v j = col B\<^sup>T (f i) $v j" if i: "i < n" and j: "j < n" for i j
       proof -
         have fi_m: "f i < m" using f i by auto
         have "col B\<^sup>T (f i) $v j = B\<^sup>T $$(j, f i)" by (rule index_col, insert B fi_m j, auto)
         also have "... = B $$ (f i, j)" using B fi_m j by auto
         also have "... = row B (f i) $v j" by (rule index_row[symmetric], insert B fi_m j, auto)
         finally show ?thesis ..
-      qed      
-      show ?thesis by (rule arg_cong[of _ _ det], rule eq_matI, insert row_eq_colT, auto)      
-    qed            
+      qed
+      show ?thesis by (rule arg_cong[of _ _ det], rule eq_matI, insert row_eq_colT, auto)
+    qed
     ultimately show "(\<Prod>i = 0..<n. A\<^sup>T $$ (f i, i)) * det (mat\<^sub>r n n (\<lambda>i. col B\<^sup>T (f i))) =
-         (\<Prod>i = 0..<n. A $$ (i, f i)) * det (mat\<^sub>r n n (\<lambda>i. row B (f i)))" by simp    
+         (\<Prod>i = 0..<n. A $$ (i, f i)) * det (mat\<^sub>r n n (\<lambda>i. row B (f i)))" by simp
   qed
-  finally show ?thesis 
+  finally show ?thesis
     by (metis (no_types, lifting) A B det_transpose transpose_mult mult_carrier_mat)
 qed
 
 (*We need a more general version of this lemma*)
 lemma
   assumes F: "F= {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   and p: " \<pi> permutes {0..<n}"
   shows " (\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))) = (\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, i)))"
-proof - 
+proof -
   let ?h = "(\<lambda>f. f \<circ> \<pi>)"
   have inj_on_F: "inj_on ?h F"
   proof (rule inj_onI)
-    fix f g assume fop: "f \<circ> \<pi> = g \<circ> \<pi>"     
+    fix f g assume fop: "f \<circ> \<pi> = g \<circ> \<pi>"
     have "f x = g x" for x
     proof (cases "x \<in> {0..<n}")
       case True
-      then show ?thesis 
+      then show ?thesis
         by (metis fop comp_apply p permutes_def)
     next
       case False
-      then show ?thesis 
+      then show ?thesis
         by (metis fop comp_eq_elim p permutes_def)
     qed
     thus "f = g" by auto
   qed
-  have hF: "?h` F = F" 
-    unfolding image_def 
+  have hF: "?h` F = F"
+    unfolding image_def
   proof auto
-    fix xa assume xa: "xa \<in> F" show "xa \<circ> \<pi> \<in> F" 
-      unfolding o_def F 
-      using F PiE p xa 
+    fix xa assume xa: "xa \<in> F" show "xa \<circ> \<pi> \<in> F"
+      unfolding o_def F
+      using F PiE p xa
       by (auto, smt F atLeastLessThan_iff mem_Collect_eq p permutes_def xa)
-    show "\<exists>x\<in>F. xa = x \<circ> \<pi>" 
+    show "\<exists>x\<in>F. xa = x \<circ> \<pi>"
     proof (rule bexI[of _ "xa \<circ> Hilbert_Choice.inv \<pi>"])
       show "xa = xa \<circ> Hilbert_Choice.inv \<pi> \<circ> \<pi>"
-        using p by auto 
-      show "xa \<circ> Hilbert_Choice.inv \<pi> \<in> F" 
-        unfolding o_def F 
-        using F PiE p xa       
+        using p by auto
+      show "xa \<circ> Hilbert_Choice.inv \<pi> \<in> F"
+        unfolding o_def F
+        using F PiE p xa
         by (auto, smt atLeastLessThan_iff permutes_def permutes_less(3))
     qed
   qed
   have prod_rw: "(\<Prod>i = 0..<n. B $$ (f i, i)) = (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i))" if "f\<in>F" for f
   using prod.permute[OF p] by auto
   let ?g = "\<lambda>f. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))"
-  have "(\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, i))) = (\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i)))" 
+  have "(\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, i))) = (\<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i)))"
     using prod_rw by auto
   also have "... = (\<Sum>f\<in>(?h`F). \<Prod>i = 0..<n. B $$ (f i, \<pi> i))"
     using sum.reindex[OF inj_on_F, of ?g] unfolding hF by auto
   also have "... = (\<Sum>f\<in>F. \<Prod>i = 0..<n. B $$ (f i, \<pi> i))" unfolding hF by auto
-  finally show ?thesis ..    
+  finally show ?thesis ..
 qed
 
 
 lemma detAB_Znm_aux:
   assumes F: "F= {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
-  shows"(\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Sum>f\<in>F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} 
-        * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i))))) 
-    = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) 
+  shows"(\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Sum>f\<in>F. prod (\<lambda>i. B $$ (f i, i)) {0..<n}
+        * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i)))))
+    = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))
         * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))"
 proof -
-  have "(\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Sum>f\<in>F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} 
-      * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i))))) = 
+  have "(\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Sum>f\<in>F. prod (\<lambda>i. B $$ (f i, i)) {0..<n}
+      * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i))))) =
     (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (\<pi> i, f i)))"
     by (smt mult.left_commute prod.cong prod.distrib sum.cong)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof (Hilbert_Choice.inv \<pi>) 
-    * (\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (Hilbert_Choice.inv \<pi> i, f i)))" 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof (Hilbert_Choice.inv \<pi>)
+    * (\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (Hilbert_Choice.inv \<pi> i, f i)))"
     by (rule sum_permutations_inverse)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof (Hilbert_Choice.inv \<pi>) 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof (Hilbert_Choice.inv \<pi>)
     * (\<Prod>i = 0..<n. B $$ (f (\<pi> i), (\<pi> i)) * A $$ (Hilbert_Choice.inv \<pi> (\<pi> i), f (\<pi> i))))"
   proof (rule sum.cong)
     fix x assume x: "x \<in> {\<pi>. \<pi> permutes {0..<n}}"
     let ?inv_x = "Hilbert_Choice.inv x"
     have p: "x permutes {0..<n}" using x by simp
-    have prod_rw: "(\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (?inv_x i, f i)) 
+    have prod_rw: "(\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (?inv_x i, f i))
         = (\<Prod>i = 0..<n. B $$ (f (x i), x i) * A $$ (?inv_x (x i), f (x i)))" if "f \<in> F" for f
       using prod.permute[OF p] by auto
     then show "(\<Sum>f\<in>F. signof ?inv_x * (\<Prod>i = 0..<n. B $$ (f i, i) * A $$ (?inv_x i, f i))) =
          (\<Sum>f\<in>F. signof ?inv_x * (\<Prod>i = 0..<n. B $$ (f (x i), x i) * A $$ (?inv_x (x i), f (x i))))"
       by auto
   qed (simp)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof \<pi> 
-    * (\<Prod>i = 0..<n. B $$ (f (\<pi> i), (\<pi> i)) * A $$ (i, f (\<pi> i))))" 
-    by (rule sum.cong, auto, rule sum.cong, auto) 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof \<pi>
+    * (\<Prod>i = 0..<n. B $$ (f (\<pi> i), (\<pi> i)) * A $$ (i, f (\<pi> i))))"
+    by (rule sum.cong, auto, rule sum.cong, auto)
         (metis (no_types, lifting) finite_atLeastLessThan signof_inv)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof \<pi> 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. signof \<pi>
     * (\<Prod>i = 0..<n. B $$ (f i, (\<pi> i)) * A $$ (i, f i)))"
   proof (rule sum.cong)
     fix \<pi> assume p: "\<pi> \<in> {\<pi>. \<pi> permutes {0..<n}}"
     hence p: "\<pi> permutes {0..<n}" by auto
     let ?inv_pi = "(Hilbert_Choice.inv \<pi>)"
     let ?h = "(\<lambda>f. f \<circ> (Hilbert_Choice.inv \<pi>))"
   have inj_on_F: "inj_on ?h F"
   proof (rule inj_onI)
-    fix f g assume fop: "f \<circ> ?inv_pi = g \<circ> ?inv_pi"     
+    fix f g assume fop: "f \<circ> ?inv_pi = g \<circ> ?inv_pi"
     have "f x = g x" for x
     proof (cases "x \<in> {0..<n}")
       case True
       then show ?thesis
         by (metis fop o_inv_o_cancel p permutes_inj)
     next
       case False
       then show ?thesis
         by (metis fop o_inv_o_cancel p permutes_inj)
     qed
     thus "f = g" by auto
   qed
-  have hF: "?h` F = F" 
-    unfolding image_def 
+  have hF: "?h` F = F"
+    unfolding image_def
   proof auto
-    fix xa assume xa: "xa \<in> F" show "xa \<circ> ?inv_pi \<in> F" 
-      unfolding o_def F 
+    fix xa assume xa: "xa \<in> F" show "xa \<circ> ?inv_pi \<in> F"
+      unfolding o_def F
       using F PiE p xa
       by (auto, smt atLeastLessThan_iff permutes_def permutes_less(3))
-    show "\<exists>x\<in>F. xa = x \<circ> ?inv_pi" 
+    show "\<exists>x\<in>F. xa = x \<circ> ?inv_pi"
     proof (rule bexI[of _ "xa \<circ> \<pi>"])
       show "xa = xa \<circ> \<pi> \<circ> Hilbert_Choice.inv \<pi> "
-        using p by auto 
-      show "xa \<circ> \<pi> \<in> F" 
-        unfolding o_def F 
-        using F PiE p xa       
+        using p by auto
+      show "xa \<circ> \<pi> \<in> F"
+        unfolding o_def F
+        using F PiE p xa
         by (auto, smt atLeastLessThan_iff permutes_def permutes_less(3))
     qed
   qed
   let ?g = "\<lambda>f. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i) * A $$ (i, f (\<pi> i)))"
     show "(\<Sum>f\<in>F. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f (\<pi> i), \<pi> i) * A $$ (i, f (\<pi> i)))) =
          (\<Sum>f\<in>F. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i) * A $$ (i, f i)))"
       using sum.reindex[OF inj_on_F, of "?g"] p unfolding hF unfolding o_def by auto
   qed (simp)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))
   * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))"
     by (smt mult.left_commute prod.cong prod.distrib sum.cong)
   finally show ?thesis .
 qed
 
 
 lemma detAB_Znm:
   assumes A: "A \<in> carrier_mat n m"
     and B: "B \<in> carrier_mat m n"
-  shows "det (A*B) = (\<Sum>(f, \<pi>)\<in>Z n m. signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))"  
+  shows "det (A*B) = (\<Sum>(f, \<pi>)\<in>Z n m. signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))"
 proof -
   let ?V="{0..<n}"
   let ?U = "{0..<m}"
   let ?PU = "{p. p permutes ?U}"
   let ?F = " {f. (\<forall>i\<in> {0..<n}. f i \<in> ?U) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   let ?f = "\<lambda>f. det (mat\<^sub>r n n (\<lambda> i. A $$ (i, f i) \<cdot>\<^sub>v row B (f i)))"
   let ?g = "\<lambda>f. det (mat\<^sub>r n n (\<lambda> i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i)))"
   have fm: "finite {0..<m}" by auto
   have fn: "finite {0..<n}" by auto
   have F: "?F= {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}" by auto
-  have det_rw: "det (mat\<^sub>r n n (\<lambda>i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i))) = 
-    (prod (\<lambda>i. B $$ (f i, i)) {0..<n}) * det (mat\<^sub>r n n (\<lambda>i. col A (f i)))" 
+  have det_rw: "det (mat\<^sub>r n n (\<lambda>i. B $$ (f i, i) \<cdot>\<^sub>v col A (f i))) =
+    (prod (\<lambda>i. B $$ (f i, i)) {0..<n}) * det (mat\<^sub>r n n (\<lambda>i. col A (f i)))"
     if  f: "(\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)" for f
     by (rule det_rows_mul, insert A col_dim, auto)
-  have det_rw2: "det (mat\<^sub>r n n (\<lambda>i. col A (f i))) 
+  have det_rw2: "det (mat\<^sub>r n n (\<lambda>i. col A (f i)))
   = (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i)))"
     if f: "f \<in> ?F" for f
   proof (unfold Determinant.det_def, auto, rule sum.cong, auto)
     fix x assume x: "x permutes {0..<n}"
-    have "(\<Prod>i = 0..<n. col A (f i) $v x i) = (\<Prod>i = 0..<n. A $$ (x i, f i))" 
+    have "(\<Prod>i = 0..<n. col A (f i) $v x i) = (\<Prod>i = 0..<n. A $$ (x i, f i))"
     proof (rule prod.cong)
       fix xa assume xa: "xa \<in> {0..<n}" show "col A (f xa) $v x xa = A $$ (x xa, f  xa)"
         by (metis A atLeastLessThan_iff carrier_matD(1) col_def index_vec permutes_less(1) x xa)
     qed (auto)
-    then show "signof x * (\<Prod>i = 0..<n. col A (f i) $v x i) 
+    then show "signof x * (\<Prod>i = 0..<n. col A (f i) $v x i)
       = signof x * (\<Prod>i = 0..<n. A $$ (x i, f i))" by auto
   qed
-  have fin_n: "finite {0..<n}" and fin_m: "finite {0..<m}" by auto  
-  have "det (A*B) = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a::comm_ring_1) n 
+  have fin_n: "finite {0..<n}" and fin_m: "finite {0..<m}" by auto
+  have "det (A*B) = det (mat\<^sub>r n n (\<lambda>i. finsum_vec TYPE('a::comm_ring_1) n
     (\<lambda>k. B $$ (k, i) \<cdot>\<^sub>v Matrix.col A k) {0..<m}))"
     by (rule det_mul_finsum_alt[OF A B])
   also have "... = sum ?g ?F" by (rule det_linear_rows_sum[OF fm], auto simp add: A)
-  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} * det (mat\<^sub>r n n (\<lambda>i. col A (f i))))" 
+  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} * det (mat\<^sub>r n n (\<lambda>i. col A (f i))))"
     using det_rw by auto
-  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} * 
-  (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f (i)))))" 
+  also have "... = (\<Sum>f\<in>?F. prod (\<lambda>i. B $$ (f i, i)) {0..<n} *
+  (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f (i)))))"
     by (rule sum.cong, auto simp add: det_rw2)
-  also have "... = 
-  (\<Sum>f\<in>?F. \<Sum>\<pi> | \<pi> permutes {0..<n}. prod (\<lambda>i. B $$ (f i, i)) {0..<n} 
-    * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f (i)))))"    
+  also have "... =
+  (\<Sum>f\<in>?F. \<Sum>\<pi> | \<pi> permutes {0..<n}. prod (\<lambda>i. B $$ (f i, i)) {0..<n}
+    * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f (i)))))"
     by (simp add: mult_hom.hom_sum)
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>?F.prod (\<lambda>i. B $$ (f i, i)) {0..<n} 
-    * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i))))" 
-    by (rule VS_Connect.class_semiring.finsum_finsum_swap, 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>?F.prod (\<lambda>i. B $$ (f i, i)) {0..<n}
+    * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (\<pi> i, f i))))"
+    by (rule VS_Connect.class_semiring.finsum_finsum_swap,
       insert finite_permutations finite_bounded_functions[OF fin_m fin_n], auto)
   thm detAB_Znm_aux
-  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>?F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) 
+  also have "... = (\<Sum>\<pi> | \<pi> permutes {0..<n}. \<Sum>f\<in>?F. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))
   * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))" by (rule detAB_Znm_aux, auto)
-  also have "... = (\<Sum>f\<in>?F.\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) 
-  * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))" 
-    by (rule VS_Connect.class_semiring.finsum_finsum_swap, 
+  also have "... = (\<Sum>f\<in>?F.\<Sum>\<pi> | \<pi> permutes {0..<n}. (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))
+  * (signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i))))"
+    by (rule VS_Connect.class_semiring.finsum_finsum_swap,
       insert finite_permutations finite_bounded_functions[OF fin_m fin_n], auto)
-  also have "... =  (\<Sum>f\<in>?F.\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> 
+  also have "... =  (\<Sum>f\<in>?F.\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi>
     * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))"
     unfolding prod.distrib by (rule sum.cong, auto, rule sum.cong, auto)
-  also have "... = sum (\<lambda>(f,\<pi>). (signof \<pi>) 
+  also have "... = sum (\<lambda>(f,\<pi>). (signof \<pi>)
     * (prod (\<lambda>i. A$$(i,f i) * B $$ (f i, \<pi> i)) {0..<n})) (Z n m)"
     unfolding Z_alt_def unfolding sum.cartesian_product[symmetric] F by auto
   finally show ?thesis .
 qed
 
 (*Several lemmas here can be moved outside the context*)
 context
   fixes n m and A B::"'a::comm_ring_1 mat"
   assumes A: "A \<in> carrier_mat n m"
     and B: "B \<in> carrier_mat m n"
 begin
 
-private definition "Z_inj = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+private definition "Z_inj = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> inj_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}})"
 
-private definition "Z_not_inj = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+private definition "Z_not_inj = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> \<not> inj_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}})"
 
-private definition "Z_strict = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+private definition "Z_strict = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> strict_mono_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}})"
 
-private definition "Z_not_strict = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+private definition "Z_not_strict = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> \<not> strict_mono_on f {0..<n}} \<times> {\<pi>. \<pi> permutes {0..<n}})"
 
-private definition "weight f \<pi> 
+private definition "weight f \<pi>
   = (signof \<pi>) * (prod (\<lambda>i. A$$(i,f i) * B $$ (f i, \<pi> i)) {0..<n})"
 
-private definition "Z_good g = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+private definition "Z_good g = ({f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> inj_on f {0..<n} \<and> (f`{0..<n} = g`{0..<n})} \<times> {\<pi>. \<pi> permutes {0..<n}})"
 
-private definition "F_strict = {f. f \<in> {0..<n} \<rightarrow> {0..<m} 
+private definition "F_strict = {f. f \<in> {0..<n} \<rightarrow> {0..<m}
   \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> strict_mono_on f {0..<n}}"
 
-private definition "F_inj = {f. f \<in> {0..<n} \<rightarrow> {0..<m} 
+private definition "F_inj = {f. f \<in> {0..<n} \<rightarrow> {0..<m}
   \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n}}"
 
-private definition "F_not_inj = {f. f \<in> {0..<n} \<rightarrow> {0..<m} 
+private definition "F_not_inj = {f. f \<in> {0..<n} \<rightarrow> {0..<m}
   \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<not> inj_on f {0..<n}}"
 
 private definition "F = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
 
 text\<open>The Cauchy--Binet formula is proven in \url{https://core.ac.uk/download/pdf/82475020.pdf}
-  In that work, they define @{text "\<sigma> \<equiv> inv \<phi> \<circ> \<pi>"}. I had problems following this proof 
-  in Isabelle, since I was demanded to show that such permutations commute, which is false. 
+  In that work, they define @{text "\<sigma> \<equiv> inv \<phi> \<circ> \<pi>"}. I had problems following this proof
+  in Isabelle, since I was demanded to show that such permutations commute, which is false.
   It is a notation problem of the  @{text "\<circ>"} operator, the author means @{text "\<sigma> \<equiv> \<pi> \<circ> inv \<phi>"} using
   the Isabelle notation (i.e., @{text "\<sigma> x = \<pi> ((inv \<phi>) x)"}).
 \<close>
 
 lemma step_weight:
   fixes \<phi> \<pi>
   defines "\<sigma> \<equiv> \<pi> \<circ> Hilbert_Choice.inv \<phi>"
   assumes f_inj: "f \<in> F_inj" and gF: "g \<in> F" and pi: "\<pi> permutes {0..<n}"
-  and phi: "\<phi> permutes {0..<n}" and fg_phi: "\<forall>x \<in> {0..<n}. f x = g (\<phi> x)" 
-shows "weight f \<pi> = (signof \<phi>) * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) 
+  and phi: "\<phi> permutes {0..<n}" and fg_phi: "\<forall>x \<in> {0..<n}. f x = g (\<phi> x)"
+shows "weight f \<pi> = (signof \<phi>) * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i)))
   * (signof \<sigma>) * (\<Prod>i = 0..<n. B $$ (g i, \<sigma> i))"
 proof -
   let ?A = "(\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) "
   let ?B = "(\<Prod>i = 0..<n. B $$ (g i, \<sigma> i))"
-  have sigma: "\<sigma> permutes {0..<n}" unfolding \<sigma>_def 
-    by (rule permutes_compose[OF permutes_inv[OF phi] pi]) 
+  have sigma: "\<sigma> permutes {0..<n}" unfolding \<sigma>_def
+    by (rule permutes_compose[OF permutes_inv[OF phi] pi])
   have A_rw: "?A = (\<Prod>i = 0..<n. A $$ (i, f i))" using fg_phi by auto
   have "?B = (\<Prod>i = 0..<n. B $$ (g (\<phi> i), \<sigma> (\<phi> i)))"
-    by (rule prod.permute[unfolded o_def, OF phi])  
-  also have "... = (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))" 
+    by (rule prod.permute[unfolded o_def, OF phi])
+  also have "... = (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))"
     using fg_phi
     unfolding \<sigma>_def unfolding o_def unfolding permutes_inverses(2)[OF phi] by auto
   finally have B_rw: "?B = (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))" .
   have "(signof \<phi>) * ?A * (signof \<sigma>) * ?B = (signof \<phi>) * (signof \<sigma>) * ?A * ?B" by auto
   also have "... = signof (\<phi> \<circ> \<sigma>) * ?A * ?B" unfolding signof_compose[OF phi sigma] by simp
   also have "... = signof \<pi> * ?A * ?B"
-    by (metis (no_types, lifting) \<sigma>_def mult.commute o_inv_o_cancel permutes_inj 
+    by (metis (no_types, lifting) \<sigma>_def mult.commute o_inv_o_cancel permutes_inj
         phi sigma signof_compose)
   also have "... = signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i)) * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))"
     using A_rw B_rw by auto
   also have "... = signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i))" by auto
   also have "... = weight f \<pi>" unfolding weight_def by simp
   finally show ?thesis ..
 qed
 
 
 lemma Z_good_fun_alt_sum:
-  fixes g  
-  defines "Z_good_fun \<equiv> {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+  fixes g
+  defines "Z_good_fun \<equiv> {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
     \<and> inj_on f {0..<n} \<and> (f`{0..<n} = g`{0..<n})}"
   assumes g: "g \<in> F_inj"
   shows "(\<Sum>f\<in>Z_good_fun. P f)= (\<Sum>\<pi>\<in>{\<pi>. \<pi> permutes {0..<n}}. P (g \<circ> \<pi>))"
 proof -
   let ?f = "\<lambda>\<pi>. g \<circ> \<pi>"
   let ?P = "{\<pi>. \<pi> permutes {0..<n}}"
-  have fP: "?f`?P = Z_good_fun"    
+  have fP: "?f`?P = Z_good_fun"
   proof (unfold Z_good_fun_def, auto)
-    fix xa xb assume "xa permutes {0..<n}" and "xb < n" 
+    fix xa xb assume "xa permutes {0..<n}" and "xb < n"
     hence "xa xb < n" by auto
     thus "g (xa xb) < m" using g unfolding F_inj_def by fastforce
   next
-    fix xa i assume "xa permutes {0..<n}" and i_ge_n: "\<not> i < n" 
+    fix xa i assume "xa permutes {0..<n}" and i_ge_n: "\<not> i < n"
     hence "xa i = i" unfolding permutes_def by auto
     thus "g (xa i) = i" using g i_ge_n unfolding F_inj_def by auto
   next
     fix xa assume "xa permutes {0..<n}" thus "inj_on (g \<circ> xa) {0..<n}"
-      by (metis (mono_tags, lifting) F_inj_def atLeast0LessThan comp_inj_on g 
+      by (metis (mono_tags, lifting) F_inj_def atLeast0LessThan comp_inj_on g
           mem_Collect_eq permutes_image permutes_inj_on)
   next
     fix \<pi> xb assume "\<pi> permutes {0..<n}" and "xb < n" thus " g xb \<in> (\<lambda>x. g (\<pi> x)) ` {0..<n}"
       by (metis (full_types) atLeast0LessThan imageI image_image lessThan_iff permutes_image)
   next
    fix x assume x1: "x \<in> {0..<n} \<rightarrow> {0..<m}" and x2: "\<forall>i. \<not> i < n \<longrightarrow> x i = i"
-     and inj_on_x: "inj_on x {0..<n}" and xg: "x ` {0..<n} = g ` {0..<n}" 
+     and inj_on_x: "inj_on x {0..<n}" and xg: "x ` {0..<n} = g ` {0..<n}"
    let ?\<tau> = "\<lambda>i. if i<n then (THE j. j<n \<and> x i = g j) else i"
    show "x \<in> (\<circ>) g ` {\<pi>. \<pi> permutes {0..<n}}"
    proof (unfold image_def, auto, rule exI[of _ ?\<tau>], rule conjI)
      have "?\<tau> i = i" if i: "i \<notin> {0..<n}" for i
        using i by auto
      moreover have "\<exists>!j. ?\<tau> j = i" for i
      proof (cases "i<n")
        case True
        obtain a where xa_gi: "x a = g i" and a: "a < n" using xg True
-         by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff)        
+         by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff)
        show ?thesis
        proof (rule ex1I[of _ a])
          have the_ai: "(THE j. j < n \<and> x a = g j) = i"
          proof (rule theI2)
            show "i < n \<and> x a = g i" using xa_gi True by auto
-           fix xa assume "xa < n \<and> x a = g xa" thus "xa = i" 
-             by (metis (mono_tags, lifting) F_inj_def True atLeast0LessThan 
+           fix xa assume "xa < n \<and> x a = g xa" thus "xa = i"
+             by (metis (mono_tags, lifting) F_inj_def True atLeast0LessThan
                 g inj_onD lessThan_iff mem_Collect_eq xa_gi)
            thus "xa = i" .
          qed
          thus ta: "?\<tau> a = i" using a by auto
-         fix j assume tj: "?\<tau> j = i"          
+         fix j assume tj: "?\<tau> j = i"
          show "j = a"
          proof (cases "j<n")
-           case True   
+           case True
            obtain b where xj_gb: "x j = g b" and b: "b < n" using xg True
-             by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff)      
+             by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff)
            let ?P = "\<lambda>ja. ja < n \<and> x j = g ja"
-           have the_ji: "(THE ja. ja < n \<and> x j = g ja) = i" using tj True by auto         
-           have "?P (THE ja. ?P ja)" 
+           have the_ji: "(THE ja. ja < n \<and> x j = g ja) = i" using tj True by auto
+           have "?P (THE ja. ?P ja)"
            proof (rule theI)
             show "b < n \<and> x j = g b" using xj_gb b by auto
-            fix xa assume "xa < n \<and> x j = g xa" thus "xa = b" 
-              by (metis (mono_tags, lifting) F_inj_def b atLeast0LessThan 
+            fix xa assume "xa < n \<and> x j = g xa" thus "xa = b"
+              by (metis (mono_tags, lifting) F_inj_def b atLeast0LessThan
                   g inj_onD lessThan_iff mem_Collect_eq xj_gb)
            qed
            hence "x j = g i" unfolding the_ji by auto
            hence "x j = x a" using xa_gi by auto
            then show ?thesis using inj_on_x a True unfolding inj_on_def by auto
          next
            case False
            then show ?thesis using tj True by auto
-         qed      
+         qed
        qed
      next
        case False note i_ge_n = False
-       show ?thesis 
+       show ?thesis
        proof (rule ex1I[of _ i])
          show "?\<tau> i = i" using False by simp
-         fix j assume tj: "?\<tau> j = i"          
-         show "j = i" 
+         fix j assume tj: "?\<tau> j = i"
+         show "j = i"
          proof (cases "j<n")
            case True
            obtain a where xj_ga: "x j = g a" and a: "a < n" using xg True
              by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff)
            have "(THE ja. ja < n \<and> x j = g ja) < n"
            proof (rule theI2)
              show "a < n \<and> x j = g a" using xj_ga a by auto
              fix xa assume a1: "xa < n \<and> x j = g xa" thus "xa = a"
                using F_inj_def  a atLeast0LessThan g inj_on_eq_iff xj_ga by fastforce
              show "xa < n" by (simp add: a1)
-           qed            
+           qed
             then show ?thesis using tj i_ge_n by auto
           next
             case False
             then show ?thesis using tj  by auto
           qed
        qed
      qed
      ultimately show "?\<tau> permutes {0..<n}" unfolding permutes_def by auto
      show "x = g \<circ> ?\<tau>"
-     proof - 
-       have "x xa = g (THE j. j < n \<and> x xa = g j)" if xa: "xa < n" for xa 
+     proof -
+       have "x xa = g (THE j. j < n \<and> x xa = g j)" if xa: "xa < n" for xa
        proof -
          obtain c where c: "c < n" and xxa_gc: "x xa = g c"
            by (metis (mono_tags, hide_lams) atLeast0LessThan imageE imageI lessThan_iff xa xg)
          show ?thesis
          proof (rule theI2)
            show c1: "c < n \<and> x xa = g c" using c xxa_gc by auto
            fix xb assume c2: "xb < n \<and> x xa = g xb" thus "xb = c"
-             by (metis (mono_tags, lifting) F_inj_def c1 atLeast0LessThan 
+             by (metis (mono_tags, lifting) F_inj_def c1 atLeast0LessThan
                  g inj_onD lessThan_iff mem_Collect_eq)
            show "x xa = g xb" using c1 c2 by simp
          qed
        qed
-       moreover have "x xa = g xa" if xa: "\<not> xa < n" for xa 
+       moreover have "x xa = g xa" if xa: "\<not> xa < n" for xa
          using g x1 x2 xa unfolding F_inj_def by simp
        ultimately show ?thesis unfolding o_def fun_eq_iff by auto
-     qed 
+     qed
    qed
  qed
-  have inj: "inj_on ?f ?P" 
+  have inj: "inj_on ?f ?P"
   proof (rule inj_onI)
     fix x y assume x: "x \<in> ?P" and y: "y \<in> ?P" and gx_gy: "g \<circ> x = g \<circ> y"
     have "x i = y i" for i
     proof (cases "i<n")
       case True
-      hence xi: "x i \<in> {0..<n}" and yi: "y i \<in> {0..<n}" using x y by auto 
+      hence xi: "x i \<in> {0..<n}" and yi: "y i \<in> {0..<n}" using x y by auto
       have "g (x i) = g (y i)" using gx_gy unfolding o_def by meson
-      thus ?thesis using xi yi using g unfolding F_inj_def inj_on_def by blast      
+      thus ?thesis using xi yi using g unfolding F_inj_def inj_on_def by blast
     next
       case False
       then show ?thesis using x y unfolding permutes_def by auto
     qed
     thus "x = y" unfolding fun_eq_iff by auto
   qed
   have "(\<Sum>f\<in>Z_good_fun. P f) = (\<Sum>f\<in>?f`?P. P f)" using fP by simp
-  also have "... =  sum (P \<circ> (\<circ>) g) {\<pi>. \<pi> permutes {0..<n}}" 
+  also have "... =  sum (P \<circ> (\<circ>) g) {\<pi>. \<pi> permutes {0..<n}}"
     by (rule sum.reindex[OF inj])
   also have "... =  (\<Sum>\<pi> | \<pi> permutes {0..<n}. P (g \<circ> \<pi>))" by auto
   finally show ?thesis .
 qed
 
 
-lemma F_injI: 
-  assumes "f \<in> {0..<n} \<rightarrow> {0..<m}" 
+lemma F_injI:
+  assumes "f \<in> {0..<n} \<rightarrow> {0..<m}"
   and "(\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)" and "inj_on f {0..<n}"
   shows "f \<in> F_inj" using assms unfolding F_inj_def by simp
 
 lemma F_inj_composition_permutation:
   assumes phi: "\<phi> permutes {0..<n}"
   and g: "g \<in> F_inj"
   shows "g \<circ> \<phi> \<in> F_inj"
 proof (rule F_injI)
-  show "g \<circ> \<phi> \<in> {0..<n} \<rightarrow> {0..<m}" 
+  show "g \<circ> \<phi> \<in> {0..<n} \<rightarrow> {0..<m}"
     using g unfolding permutes_def F_inj_def
     by (simp add: Pi_iff phi)
-  show "\<forall>i. i \<notin> {0..<n} \<longrightarrow> (g \<circ> \<phi>) i = i" 
+  show "\<forall>i. i \<notin> {0..<n} \<longrightarrow> (g \<circ> \<phi>) i = i"
     using g phi unfolding permutes_def F_inj_def by simp
-  show "inj_on (g \<circ> \<phi>) {0..<n}" 
-    by (rule comp_inj_on, insert g permutes_inj_on[OF phi] permutes_image[OF phi]) 
+  show "inj_on (g \<circ> \<phi>) {0..<n}"
+    by (rule comp_inj_on, insert g permutes_inj_on[OF phi] permutes_image[OF phi])
        (auto simp add: F_inj_def)
 qed
 
 
-lemma F_strict_imp_F_inj: 
-  assumes f: "f \<in> F_strict" 
+lemma F_strict_imp_F_inj:
+  assumes f: "f \<in> F_strict"
   shows "f \<in> F_inj"
   using f strict_mono_on_imp_inj_on
   unfolding F_strict_def F_inj_def by auto
 
 
 lemma one_step:
   assumes g1: "g \<in> F_strict"
   shows "det (submatrix A UNIV (g`{0..<n})) * det (submatrix B (g`{0..<n}) UNIV)
     = (\<Sum>(x, y) \<in> Z_good g. weight x y)" (is "?lhs = ?rhs")
 proof -
-  define Z_good_fun where "Z_good_fun = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+  define Z_good_fun where "Z_good_fun = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
     \<and> inj_on f {0..<n} \<and> (f`{0..<n} = g`{0..<n})}"
   let ?Perm = "{\<pi>. \<pi> permutes {0..<n}}"
   let ?P = "(\<lambda>f. \<Sum>\<pi> \<in> ?Perm. weight f \<pi>)"
   let ?inv = "Hilbert_Choice.inv"
   have g: "g \<in> F_inj" by (rule F_strict_imp_F_inj[OF g1])
-  have detA: "(\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i)))) 
+  have detA: "(\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))))
     = det (submatrix A UNIV (g`{0..<n}))"
   proof -
-    have "{j. j < dim_col A \<and> j \<in> g ` {0..<n}} = {j. j \<in> g ` {0..<n}}" 
+    have "{j. j < dim_col A \<and> j \<in> g ` {0..<n}} = {j. j \<in> g ` {0..<n}}"
       using g A unfolding F_inj_def by fastforce
     also have "card ... = n" using F_inj_def card_image g by force
     finally have card_J: "card {j. j < dim_col A \<and> j \<in> g ` {0..<n}} = n" by simp
-    have subA_carrier: "submatrix A UNIV (g ` {0..<n}) \<in> carrier_mat n n" 
-      unfolding submatrix_def card_J using A by auto   
+    have subA_carrier: "submatrix A UNIV (g ` {0..<n}) \<in> carrier_mat n n"
+      unfolding submatrix_def card_J using A by auto
     have "det (submatrix A UNIV (g`{0..<n})) = (\<Sum>p | p permutes {0..<n}.
             signof p * (\<Prod>i = 0..<n. submatrix A UNIV (g ` {0..<n}) $$ (i, p i)))"
       using subA_carrier unfolding Determinant.det_def by auto
-    also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))))" 
+    also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))))"
     proof (rule sum.cong)
       fix x assume x: "x \<in> {\<pi>. \<pi> permutes {0..<n}}"
-      have "(\<Prod>i = 0..<n. submatrix A UNIV (g ` {0..<n}) $$ (i, x i)) 
+      have "(\<Prod>i = 0..<n. submatrix A UNIV (g ` {0..<n}) $$ (i, x i))
           = (\<Prod>i = 0..<n. A $$ (i, g (x i)))"
       proof (rule prod.cong, rule refl)
-        fix i assume i: "i \<in> {0..<n}" 
+        fix i assume i: "i \<in> {0..<n}"
         have pick_rw: "pick (g ` {0..<n}) (x i) = g (x i)"
         proof -
           have "index (sorted_list_of_set (g ` {0..<n})) (g (x i)) = x i"
           proof -
-            have rw: "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]" 
+            have rw: "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]"
               by (rule sorted_list_of_set_map_strict, insert g1, simp add: F_strict_def)
-            have "index (sorted_list_of_set (g`{0..<n})) (g (x i)) = index (map g [0..<n]) (g (x i))" 
+            have "index (sorted_list_of_set (g`{0..<n})) (g (x i)) = index (map g [0..<n]) (g (x i))"
               unfolding rw by auto
-            also have "... = index [0..<n] (x i)" 
+            also have "... = index [0..<n] (x i)"
               by (rule index_map_inj_on[of _ "{0..<n}"], insert x i g, auto simp add: F_inj_def)
             also have "... = x i" using x i by auto
-            finally show ?thesis .            
-          qed                        
+            finally show ?thesis .
+          qed
           moreover have "(g (x i))  \<in> (g ` {0..<n})" using x g i unfolding F_inj_def by auto
           moreover have "x i < card (g ` {0..<n})" using x i g by (simp add: F_inj_def card_image)
-          ultimately show ?thesis using pick_index by auto            
+          ultimately show ?thesis using pick_index by auto
         qed
-        have "submatrix A UNIV (g`{0..<n}) $$ (i, x i) = A $$ (pick UNIV i, pick (g`{0..<n}) (x i))" 
+        have "submatrix A UNIV (g`{0..<n}) $$ (i, x i) = A $$ (pick UNIV i, pick (g`{0..<n}) (x i))"
           by (rule submatrix_index, insert i A card_J x, auto)
-        also have "... = A $$ (i, g (x i))" using pick_rw pick_UNIV by auto 
+        also have "... = A $$ (i, g (x i))" using pick_rw pick_UNIV by auto
         finally show "submatrix A UNIV (g ` {0..<n}) $$ (i, x i) = A $$ (i, g (x i))" .
       qed
-      thus "signof x * (\<Prod>i = 0..<n. submatrix A UNIV (g ` {0..<n}) $$ (i, x i)) 
+      thus "signof x * (\<Prod>i = 0..<n. submatrix A UNIV (g ` {0..<n}) $$ (i, x i))
         = signof x * (\<Prod>i = 0..<n. A $$ (i, g (x i)))" by auto
     qed (simp)
     finally show ?thesis by simp
   qed
   have detB_rw: "(\<Sum>\<pi> \<in> ?Perm. signof (\<pi> \<circ> ?inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i)))
-   = (\<Sum>\<pi> \<in> ?Perm. signof (\<pi>) * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))" 
+   = (\<Sum>\<pi> \<in> ?Perm. signof (\<pi>) * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
     if phi: "\<phi> permutes {0..<n}" for \<phi>
   proof -
     let ?h="\<lambda>\<pi>. \<pi> \<circ> ?inv \<phi>"
     let ?g = "\<lambda>\<pi>. signof (\<pi>) * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i))"
-    have "?h`?Perm = ?Perm" 
+    have "?h`?Perm = ?Perm"
     proof -
       have "\<pi> \<circ> ?inv \<phi> permutes {0..<n}" if pi: "\<pi> permutes {0..<n}" for \<pi>
         using permutes_compose permutes_inv phi that by blast
       moreover have "x \<in> (\<lambda>\<pi>. \<pi> \<circ> ?inv \<phi>) ` ?Perm" if "x permutes {0..<n}" for x
       proof -
         have "x \<circ> \<phi> permutes {0..<n}"
           using permutes_compose phi that by blast
         moreover have "x = x \<circ> \<phi> \<circ> ?inv \<phi>" using phi by auto
         ultimately show ?thesis unfolding image_def by auto
       qed
       ultimately show ?thesis by auto
     qed
     hence "(\<Sum>\<pi> \<in> ?Perm. ?g \<pi>) = (\<Sum>\<pi> \<in> ?h`?Perm. ?g \<pi>)" by simp
-    also have "... = sum (?g \<circ> ?h) ?Perm" 
+    also have "... = sum (?g \<circ> ?h) ?Perm"
     proof (rule sum.reindex)
       show "inj_on (\<lambda>\<pi>. \<pi> \<circ> ?inv \<phi>) {\<pi>. \<pi> permutes {0..<n}}"
         by (metis (no_types, lifting) inj_onI o_inv_o_cancel permutes_inj phi)
     qed
     also have "... = (\<Sum>\<pi> \<in> ?Perm. signof (\<pi> \<circ> ?inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i)))"
       unfolding o_def by auto
     finally show ?thesis by simp
   qed
 
-  have detB: "det (submatrix B (g`{0..<n}) UNIV) 
+  have detB: "det (submatrix B (g`{0..<n}) UNIV)
     = (\<Sum>\<pi> \<in> ?Perm. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
   proof -
     have "{i. i < dim_row B \<and> i \<in> g ` {0..<n}} = {i. i \<in> g ` {0..<n}}"
      using g B unfolding F_inj_def by fastforce
     also have "card ... = n" using F_inj_def card_image g by force
     finally have card_I: "card {j. j < dim_row B \<and> j \<in> g ` {0..<n}} = n" by simp
     have subB_carrier: "submatrix B (g ` {0..<n}) UNIV \<in> carrier_mat n n"
       unfolding submatrix_def using card_I B by auto
-    have "det (submatrix B (g`{0..<n}) UNIV) = (\<Sum>p \<in> ?Perm. signof p 
-      * (\<Prod>i=0..<n. submatrix B (g ` {0..<n}) UNIV $$ (i, p i)))" 
+    have "det (submatrix B (g`{0..<n}) UNIV) = (\<Sum>p \<in> ?Perm. signof p
+      * (\<Prod>i=0..<n. submatrix B (g ` {0..<n}) UNIV $$ (i, p i)))"
       unfolding Determinant.det_def using subB_carrier by auto
     also have "... = (\<Sum>\<pi> \<in> ?Perm. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
     proof (rule sum.cong, rule refl)
       fix x assume x: "x \<in> {\<pi>. \<pi> permutes {0..<n}}"
       have "(\<Prod>i=0..<n. submatrix B (g`{0..<n}) UNIV $$ (i, x i)) = (\<Prod>i=0..<n. B $$ (g i, x i))"
       proof (rule prod.cong, rule refl)
         fix i assume i: "i \<in> {0..<n}"
         have pick_rw: "pick (g ` {0..<n}) i = g i"
         proof -
           have "index (sorted_list_of_set (g ` {0..<n})) (g i) = i"
           proof -
-            have rw: "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]" 
+            have rw: "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]"
               by (rule sorted_list_of_set_map_strict, insert g1, simp add: F_strict_def)
-            have "index (sorted_list_of_set (g`{0..<n})) (g i) = index (map g [0..<n]) (g i)" 
+            have "index (sorted_list_of_set (g`{0..<n})) (g i) = index (map g [0..<n]) (g i)"
               unfolding rw by auto
-            also have "... = index [0..<n] (i)" 
+            also have "... = index [0..<n] (i)"
               by (rule index_map_inj_on[of _ "{0..<n}"], insert x i g, auto simp add: F_inj_def)
             also have "... = i" using i by auto
-            finally show ?thesis .            
-          qed                        
+            finally show ?thesis .
+          qed
           moreover have "(g i)  \<in> (g ` {0..<n})" using x g i unfolding F_inj_def by auto
           moreover have "i < card (g ` {0..<n})" using x i g by (simp add: F_inj_def card_image)
-          ultimately show ?thesis using pick_index by auto            
+          ultimately show ?thesis using pick_index by auto
         qed
         have "submatrix B (g`{0..<n}) UNIV $$ (i, x i) = B $$ (pick (g`{0..<n}) i, pick UNIV (x i))"
           by (rule submatrix_index, insert i B card_I x, auto)
         also have "... = B $$ (g i, x i)" using pick_rw pick_UNIV by auto
         finally show "submatrix B (g ` {0..<n}) UNIV $$ (i, x i) = B $$ (g i, x i)" .
       qed
-      thus "signof x * (\<Prod>i = 0..<n. submatrix B (g ` {0..<n}) UNIV $$ (i, x i)) 
+      thus "signof x * (\<Prod>i = 0..<n. submatrix B (g ` {0..<n}) UNIV $$ (i, x i))
           = signof x * (\<Prod>i = 0..<n. B $$ (g i, x i))" by simp
     qed
     finally show ?thesis .
   qed
 
-  have "?rhs = (\<Sum>f\<in>Z_good_fun. \<Sum>\<pi>\<in>?Perm. weight f \<pi>)" 
+  have "?rhs = (\<Sum>f\<in>Z_good_fun. \<Sum>\<pi>\<in>?Perm. weight f \<pi>)"
     unfolding Z_good_def sum.cartesian_product Z_good_fun_def by blast
-  also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. ?P (g \<circ> \<phi>))" unfolding Z_good_fun_def 
-    by (rule Z_good_fun_alt_sum[OF g]) 
-  also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. \<Sum>\<pi>\<in>{\<pi>. \<pi> permutes {0..<n}}. 
-    signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) * signof (\<pi> \<circ> ?inv \<phi>) 
+  also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. ?P (g \<circ> \<phi>))" unfolding Z_good_fun_def
+    by (rule Z_good_fun_alt_sum[OF g])
+  also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. \<Sum>\<pi>\<in>{\<pi>. \<pi> permutes {0..<n}}.
+    signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) * signof (\<pi> \<circ> ?inv \<phi>)
     * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i)))"
   proof (rule sum.cong, simp, rule sum.cong, simp)
     fix \<phi> \<pi> assume phi: "\<phi> \<in> ?Perm" and pi: "\<pi> \<in> ?Perm"
-    show "weight (g \<circ> \<phi>) \<pi> = signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) * 
-      signof (\<pi> \<circ> ?inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i))" 
+    show "weight (g \<circ> \<phi>) \<pi> = signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) *
+      signof (\<pi> \<circ> ?inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i))"
     proof (rule step_weight)
       show "g \<circ> \<phi> \<in> F_inj" by (rule F_inj_composition_permutation[OF _ g], insert phi, auto)
-      show "g \<in> F" using g unfolding F_def F_inj_def by simp      
+      show "g \<in> F" using g unfolding F_def F_inj_def by simp
     qed (insert phi pi, auto)
   qed
   also have "... = (\<Sum>\<phi>\<in>{\<pi>. \<pi> permutes {0..<n}}. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) *
      (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof (\<pi> \<circ> ?inv \<phi>) * (\<Prod>i = 0..<n. B $$ (g i, (\<pi> \<circ> ?inv \<phi>) i))))"
     by (metis (mono_tags, lifting) Groups.mult_ac(1) semiring_0_class.sum_distrib_left sum.cong)
-  also have "... = (\<Sum>\<phi> \<in> ?Perm. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) * 
+  also have "... = (\<Sum>\<phi> \<in> ?Perm. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i))) *
     (\<Sum>\<pi> \<in> ?Perm. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i))))" using detB_rw by auto
-  also have "... = (\<Sum>\<phi> \<in> ?Perm. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i)))) * 
+  also have "... = (\<Sum>\<phi> \<in> ?Perm. signof \<phi> * (\<Prod>i = 0..<n. A $$ (i, g (\<phi> i)))) *
     (\<Sum>\<pi> \<in> ?Perm. signof \<pi> * (\<Prod>i = 0..<n. B $$ (g i, \<pi> i)))"
     by (simp add: semiring_0_class.sum_distrib_right)
   also have "... = ?lhs" unfolding detA detB ..
   finally show ?thesis ..
 qed
 
 
 lemma gather_by_strictness:
-"sum (\<lambda>g. sum (\<lambda>(f,\<pi>). weight f \<pi>) (Z_good g)) F_strict 
+"sum (\<lambda>g. sum (\<lambda>(f,\<pi>). weight f \<pi>) (Z_good g)) F_strict
   = sum (\<lambda>g. det (submatrix A UNIV (g`{0..<n})) * det (submatrix B (g`{0..<n}) UNIV)) F_strict"
 proof (rule sum.cong)
   fix f assume f: "f \<in> F_strict"
-  show "(\<Sum>(x, y)\<in>Z_good f. weight x y) 
+  show "(\<Sum>(x, y)\<in>Z_good f. weight x y)
     = det (submatrix A UNIV (f ` {0..<n})) * det (submatrix B (f ` {0..<n}) UNIV)"
     by (rule one_step[symmetric], rule f)
 qed (simp)
 
-lemma finite_Z_strict[simp]: "finite Z_strict" 
+lemma finite_Z_strict[simp]: "finite Z_strict"
 proof (unfold Z_strict_def, rule finite_cartesian_product)
   have finN: "finite {0..<n}" and finM: "finite {0..<m}" by auto
   let ?A="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> strict_mono_on f {0..<n}}"
   let ?B="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have B: "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = ?B" by auto
   have "?A\<subseteq>?B" by auto
   moreover have "finite ?B" using B finite_bounded_functions[OF finM finN] by auto
   ultimately show "finite ?A" using rev_finite_subset by blast
   show "finite {\<pi>. \<pi> permutes {0..<n}}" using finite_permutations by blast
 qed
-  
+
 lemma finite_Z_not_strict[simp]: "finite Z_not_strict"
 proof (unfold Z_not_strict_def, rule finite_cartesian_product)
   have finN: "finite {0..<n}" and finM: "finite {0..<m}" by auto
   let ?A="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> \<not> strict_mono_on f {0..<n}}"
   let ?B="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have B: "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = ?B" by auto
   have "?A\<subseteq>?B" by auto
   moreover have "finite ?B" using B finite_bounded_functions[OF finM finN] by auto
   ultimately show "finite ?A" using rev_finite_subset by blast
   show "finite {\<pi>. \<pi> permutes {0..<n}}" using finite_permutations by blast
 qed
 
 lemma finite_Znm[simp]: "finite (Z n m)"
 proof (unfold Z_alt_def, rule finite_cartesian_product)
   have finN: "finite {0..<n}" and finM: "finite {0..<m}" by auto
   let ?A="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) }"
   let ?B="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have B: "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = ?B" by auto
   have "?A\<subseteq>?B" by auto
   moreover have "finite ?B" using B finite_bounded_functions[OF finM finN] by auto
   ultimately show "finite ?A" using rev_finite_subset by blast
   show "finite {\<pi>. \<pi> permutes {0..<n}}" using finite_permutations by blast
 qed
 
-lemma finite_F_inj[simp]: "finite F_inj" 
+lemma finite_F_inj[simp]: "finite F_inj"
 proof -
   have finN: "finite {0..<n}" and finM: "finite {0..<m}" by auto
   let ?A="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n}}"
   let ?B="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have B: "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = ?B" by auto
   have "?A\<subseteq>?B" by auto
   moreover have "finite ?B" using B finite_bounded_functions[OF finM finN] by auto
   ultimately show "finite F_inj" unfolding F_inj_def using rev_finite_subset by blast
 qed
 
 lemma finite_F_strict[simp]: "finite F_strict"
 proof -
  have finN: "finite {0..<n}" and finM: "finite {0..<m}" by auto
   let ?A="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> strict_mono_on f {0..<n}}"
   let ?B="{f \<in> {0..<n} \<rightarrow> {0..<m}. (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   have B: "{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)} = ?B" by auto
   have "?A\<subseteq>?B" by auto
   moreover have "finite ?B" using B finite_bounded_functions[OF finM finN] by auto
   ultimately show "finite F_strict" unfolding F_strict_def using rev_finite_subset by blast
 qed
 
 lemma nth_strict_mono:
   fixes f::"nat \<Rightarrow> nat"
   assumes  strictf: "strict_mono f" and i: "i<n"
 shows "f i = (sorted_list_of_set (f`{0..<n})) ! i"
 proof -
   let ?I = "f`{0..<n}"
   have "length (sorted_list_of_set (f ` {0..<n})) = card ?I"
-    by (metis distinct_card finite_atLeastLessThan finite_imageI 
+    by (metis distinct_card finite_atLeastLessThan finite_imageI
         sorted_list_of_set(1) sorted_list_of_set(3))
   also have "... = n"
     by (simp add: card_image strict_mono_imp_inj_on strictf)
-  finally have length_I: "length (sorted_list_of_set ?I) = n" .    
+  finally have length_I: "length (sorted_list_of_set ?I) = n" .
   have card_eq: "card {a \<in> ?I. a < f i} = i"
     using i
   proof (induct i)
     case 0
     then show ?case
       by (auto simp add: strict_mono_less strictf)
   next
     case (Suc i)
     have i: "i < n" using Suc.prems by auto
     let ?J'="{a \<in> f ` {0..<n}. a < f i}"
     let ?J = "{a \<in> f ` {0..<n}. a < f (Suc i)}"
     have cardJ': "card ?J' = i" by (rule Suc.hyps[OF i])
     have J: "?J = insert (f i) ?J'"
     proof (auto)
-      fix xa assume 1: "f xa \<noteq> f i" and 2: "f xa < f (Suc i)" 
+      fix xa assume 1: "f xa \<noteq> f i" and 2: "f xa < f (Suc i)"
       show "f xa < f i"
         using 1 2 not_less_less_Suc_eq strict_mono_less strictf by fastforce
     next
       fix xa assume "f xa < f i" thus "f xa < f (Suc i)"
         using less_SucI strict_mono_less strictf by blast
     next
       show "f i \<in> f ` {0..<n}" using i by auto
       show "f i < f (Suc i)" using strictf strict_mono_less by auto
     qed
     have "card ?J = Suc (card ?J')" by (unfold J, rule card_insert_disjoint, auto)
     then show ?case using cardJ' by auto
   qed
   have "sorted_list_of_set ?I ! i = pick ?I i"
-    by (rule sorted_list_of_set_eq_pick, auto simp add: length_I i)
+    by (rule sorted_list_of_set_eq_pick, simp add: \<open>card (f ` {0..<n}) = n\<close> i)
   also have "... =  pick ?I (card {a \<in> ?I. a < f i})" unfolding card_eq by simp
   also have "... = f i" by (rule pick_card_in_set, simp add: i)
   finally show ?thesis ..
 qed
 
 lemma nth_strict_mono_on:
   fixes f::"nat \<Rightarrow> nat"
   assumes  strictf: "strict_mono_on f {0..<n}" and i: "i<n"
 shows "f i = (sorted_list_of_set (f`{0..<n})) ! i"
 proof -
   let ?I = "f`{0..<n}"
   have "length (sorted_list_of_set (f ` {0..<n})) = card ?I"
-    by (metis distinct_card finite_atLeastLessThan finite_imageI 
+    by (metis distinct_card finite_atLeastLessThan finite_imageI
         sorted_list_of_set(1) sorted_list_of_set(3))
   also have "... = n"
-    by (metis (mono_tags, lifting) card_atLeastLessThan card_image diff_zero 
-        inj_on_def strict_mono_on_eqD strictf)    
-  finally have length_I: "length (sorted_list_of_set ?I) = n" .    
+    by (metis (mono_tags, lifting) card_atLeastLessThan card_image diff_zero
+        inj_on_def strict_mono_on_eqD strictf)
+  finally have length_I: "length (sorted_list_of_set ?I) = n" .
   have card_eq: "card {a \<in> ?I. a < f i} = i"
     using i
   proof (induct i)
     case 0
     then show ?case
-      by (auto, metis (no_types, lifting) atLeast0LessThan lessThan_iff less_Suc_eq 
+      by (auto, metis (no_types, lifting) atLeast0LessThan lessThan_iff less_Suc_eq
           not_less0 not_less_eq strict_mono_on_def strictf)
   next
     case (Suc i)
     have i: "i < n" using Suc.prems by auto
     let ?J'="{a \<in> f ` {0..<n}. a < f i}"
     let ?J = "{a \<in> f ` {0..<n}. a < f (Suc i)}"
     have cardJ': "card ?J' = i" by (rule Suc.hyps[OF i])
     have J: "?J = insert (f i) ?J'"
     proof (auto)
-      fix xa assume 1: "f xa \<noteq> f i" and 2: "f xa < f (Suc i)" and 3: "xa < n" 
+      fix xa assume 1: "f xa \<noteq> f i" and 2: "f xa < f (Suc i)" and 3: "xa < n"
       show "f xa < f i"
-        by (metis (full_types) 1 2 3 antisym_conv3 atLeast0LessThan i lessThan_iff 
-            less_SucE order.asym strict_mono_onD strictf)     
+        by (metis (full_types) 1 2 3 antisym_conv3 atLeast0LessThan i lessThan_iff
+            less_SucE order.asym strict_mono_onD strictf)
     next
       fix xa assume "f xa < f i" and "xa < n" thus "f xa < f (Suc i)"
         using less_SucI strictf
-        by (metis (no_types, lifting) Suc.prems atLeast0LessThan 
+        by (metis (no_types, lifting) Suc.prems atLeast0LessThan
             lessI lessThan_iff less_trans strict_mono_onD)
     next
       show "f i \<in> f ` {0..<n}" using i by auto
       show "f i < f (Suc i)"
         using Suc.prems strict_mono_onD strictf by fastforce
     qed
     have "card ?J = Suc (card ?J')" by (unfold J, rule card_insert_disjoint, auto)
     then show ?case using cardJ' by auto
   qed
   have "sorted_list_of_set ?I ! i = pick ?I i"
-    by (rule sorted_list_of_set_eq_pick, auto simp add: length_I i)
+    by (rule sorted_list_of_set_eq_pick, simp add: \<open>card (f ` {0..<n}) = n\<close> i)
   also have "... =  pick ?I (card {a \<in> ?I. a < f i})" unfolding card_eq by simp
   also have "... = f i" by (rule pick_card_in_set, simp add: i)
   finally show ?thesis ..
 qed
 
-lemma strict_fun_eq: 
+lemma strict_fun_eq:
   assumes f: "f \<in> F_strict" and g: "g \<in> F_strict" and fg: "f`{0..<n} = g`{0..<n}"
-  shows "f = g" 
+  shows "f = g"
 proof (unfold fun_eq_iff, auto)
   fix x
   show "f x = g x"
   proof (cases "x<n")
     case True
-    have strictf: "strict_mono_on f {0..<n}" and strictg: "strict_mono_on g {0..<n}" 
+    have strictf: "strict_mono_on f {0..<n}" and strictg: "strict_mono_on g {0..<n}"
       using f g unfolding F_strict_def by auto
     have "f x = (sorted_list_of_set (f`{0..<n})) ! x" by (rule nth_strict_mono_on[OF strictf True])
     also have "... = (sorted_list_of_set (g`{0..<n})) ! x" unfolding fg by simp
     also have "... = g x" by (rule nth_strict_mono_on[symmetric, OF strictg True])
     finally show ?thesis .
   next
     case False
     then show ?thesis using f g unfolding F_strict_def by auto
   qed
 qed
 
 
 lemma strict_from_inj_preserves_F:
   assumes f: "f \<in> F_inj"
   shows "strict_from_inj n f \<in> F"
 proof -
-  {  
+  {
     fix x assume x: "x < n"
     have inj_on: "inj_on f {0..<n}" using f unfolding F_inj_def by auto
     have "{a. a < m \<and> a \<in> f ` {0..<n}} = f`{0..<n}" using f unfolding F_inj_def by auto
     hence card_eq: "card {a. a < m \<and> a \<in> f ` {0..<n}} = n"
       by (simp add: card_image inj_on)
     let ?I = "f`{0..<n}"
     have "length (sorted_list_of_set (f ` {0..<n})) = card ?I"
-      by (metis distinct_card finite_atLeastLessThan finite_imageI 
+      by (metis distinct_card finite_atLeastLessThan finite_imageI
           sorted_list_of_set(1) sorted_list_of_set(3))
     also have "... = n"
       by (simp add: card_image strict_mono_imp_inj_on inj_on)
-    finally have length_I: "length (sorted_list_of_set ?I) = n" .  
+    finally have length_I: "length (sorted_list_of_set ?I) = n" .
     have "sorted_list_of_set (f ` {0..<n}) ! x = pick (f ` {0..<n}) x"
       by (rule sorted_list_of_set_eq_pick, unfold length_I, auto simp add: x)
     also have "... < m" by (rule pick_le, unfold card_eq, rule x)
     finally have "sorted_list_of_set (f ` {0..<n}) ! x < m" .
   }
   thus ?thesis unfolding strict_from_inj_def F_def by auto
 qed
 
-lemma strict_from_inj_F_strict: "strict_from_inj n xa \<in> F_strict" 
-  if xa: "xa \<in> F_inj" for xa  
+lemma strict_from_inj_F_strict: "strict_from_inj n xa \<in> F_strict"
+  if xa: "xa \<in> F_inj" for xa
 proof -
   have "strict_mono_on (strict_from_inj n xa) {0..<n}"
     by (rule strict_strict_from_inj, insert xa, simp add: F_inj_def)
   thus ?thesis using strict_from_inj_preserves_F[OF xa] unfolding F_def F_strict_def by auto
 qed
 
 lemma strict_from_inj_image:
   assumes f: "f\<in> F_inj"
   shows "strict_from_inj n f ` {0..<n} = f`{0..<n}"
 proof (auto)
   let ?I = "f ` {0..<n}"
-  fix xa assume xa: "xa < n" 
+  fix xa assume xa: "xa < n"
   have inj_on: "inj_on f {0..<n}" using f unfolding F_inj_def by auto
     have "{a. a < m \<and> a \<in> f ` {0..<n}} = f`{0..<n}" using f unfolding F_inj_def by auto
     hence card_eq: "card {a. a < m \<and> a \<in> f ` {0..<n}} = n"
       by (simp add: card_image inj_on)
     let ?I = "f`{0..<n}"
     have "length (sorted_list_of_set (f ` {0..<n})) = card ?I"
-      by (metis distinct_card finite_atLeastLessThan finite_imageI 
+      by (metis distinct_card finite_atLeastLessThan finite_imageI
           sorted_list_of_set(1) sorted_list_of_set(3))
     also have "... = n"
       by (simp add: card_image strict_mono_imp_inj_on inj_on)
-    finally have length_I: "length (sorted_list_of_set ?I) = n" .  
+    finally have length_I: "length (sorted_list_of_set ?I) = n" .
   have "strict_from_inj n f xa = sorted_list_of_set ?I ! xa"
     using xa unfolding strict_from_inj_def by auto
-  also have "... = pick ?I xa" 
+  also have "... = pick ?I xa"
     by (rule sorted_list_of_set_eq_pick, unfold length_I, auto simp add: xa)
   also have "... \<in> f ` {0..<n}" by (rule pick_in_set_le, simp add: \<open>card (f ` {0..<n}) = n\<close> xa)
   finally show "strict_from_inj n f xa \<in> f ` {0..<n}" .
   obtain i where "sorted_list_of_set (f`{0..<n}) ! i = f xa" and "i<n"
-    by (metis atLeast0LessThan finite_atLeastLessThan finite_imageI imageI 
+    by (metis atLeast0LessThan finite_atLeastLessThan finite_imageI imageI
         in_set_conv_nth length_I lessThan_iff sorted_list_of_set(1) xa)
   thus "f xa \<in> strict_from_inj n f ` {0..<n}"
-    by (metis atLeast0LessThan imageI lessThan_iff strict_from_inj_def)    
+    by (metis atLeast0LessThan imageI lessThan_iff strict_from_inj_def)
 qed
 
 
 lemma Z_good_alt:
   assumes g: "g \<in> F_strict"
   shows "Z_good g = {x \<in> F_inj. strict_from_inj n x = g} \<times> {\<pi>. \<pi> permutes {0..<n}}"
 proof -
-  define Z_good_fun where "Z_good_fun = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) 
+  define Z_good_fun where "Z_good_fun = {f. f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)
   \<and> inj_on f {0..<n} \<and> (f`{0..<n} = g`{0..<n})}"
   have "Z_good_fun = {x \<in> F_inj. strict_from_inj n x = g}"
   proof (auto)
     fix f assume f: "f \<in> Z_good_fun" thus f_inj: "f \<in> F_inj" unfolding F_inj_def Z_good_fun_def by auto
     show "strict_from_inj n f = g"
     proof (rule strict_fun_eq[OF _ g])
-      show "strict_from_inj n f ` {0..<n} = g ` {0..<n}" 
-        using f_inj f strict_from_inj_image 
+      show "strict_from_inj n f ` {0..<n} = g ` {0..<n}"
+        using f_inj f strict_from_inj_image
         unfolding Z_good_fun_def F_inj_def by auto
-      show "strict_from_inj n f \<in> F_strict" 
+      show "strict_from_inj n f \<in> F_strict"
         using F_strict_def f_inj strict_from_inj_F_strict by blast
     qed
   next
-    fix f assume f_inj: "f \<in> F_inj" and g_strict_f: "g = strict_from_inj n f" 
+    fix f assume f_inj: "f \<in> F_inj" and g_strict_f: "g = strict_from_inj n f"
     have "f xa \<in> g ` {0..<n}" if "xa < n" for xa
       using f_inj g_strict_f strict_from_inj_image that by auto
     moreover have "g xa \<in> f ` {0..<n}" if "xa < n" for xa
       by (metis f_inj g_strict_f imageI lessThan_atLeast0 lessThan_iff strict_from_inj_image that)
-    ultimately show "f \<in> Z_good_fun" 
+    ultimately show "f \<in> Z_good_fun"
       using f_inj g_strict_f unfolding Z_good_fun_def F_inj_def
       by auto
   qed
   thus ?thesis unfolding Z_good_fun_def Z_good_def by simp
 qed
 
 
 lemma weight_0: "(\<Sum>(f, \<pi>) \<in> Z_not_inj. weight f \<pi>) = 0"
 proof -
   let ?F="{f. (\<forall>i\<in>{0..<n}. f i \<in> {0..<m}) \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i)}"
   let ?Perm = "{\<pi>. \<pi> permutes {0..<n}}"
-  have "(\<Sum>(f, \<pi>)\<in>Z_not_inj. weight f \<pi>) 
+  have "(\<Sum>(f, \<pi>)\<in>Z_not_inj. weight f \<pi>)
     = (\<Sum>f \<in> F_not_inj. (\<Prod>i = 0..<n. A $$ (i, f i)) * det (mat\<^sub>r n n (\<lambda>i. row B (f i))))"
   proof -
     have dim_row_rw: "dim_row (mat\<^sub>r n n (\<lambda>i. col A (f i))) = n" for f by auto
     have dim_row_rw2: "dim_row (mat\<^sub>r n n (\<lambda>i. Matrix.row B (f i))) = n" for f by auto
-    have prod_rw: "(\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) =  (\<Prod>i = 0..<n. row B (f i) $v \<pi> i)" 
-      if f: "f \<in> F_not_inj" and pi: "\<pi> \<in> ?Perm" for f \<pi> 
+    have prod_rw: "(\<Prod>i = 0..<n. B $$ (f i, \<pi> i)) =  (\<Prod>i = 0..<n. row B (f i) $v \<pi> i)"
+      if f: "f \<in> F_not_inj" and pi: "\<pi> \<in> ?Perm" for f \<pi>
     proof (rule prod.cong, rule refl)
-      fix x assume x: "x \<in> {0..<n}" 
+      fix x assume x: "x \<in> {0..<n}"
       have "f x < dim_row B" using f B x unfolding F_not_inj_def by fastforce
       moreover have "\<pi> x < dim_col B" using x pi B by auto
       ultimately show "B $$ (f x, \<pi> x) = Matrix.row B (f x) $v \<pi> x" by (rule index_row[symmetric])
     qed
-    have sum_rw: "(\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))) 
+    have sum_rw: "(\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i)))
       = det (mat\<^sub>r n n (\<lambda>i. row B (f i)))" if f: "f \<in> F_not_inj" for f
       unfolding Determinant.det_def using dim_row_rw2 prod_rw f by auto
-    have "(\<Sum>(f, \<pi>)\<in>Z_not_inj. weight f \<pi>) = (\<Sum>f\<in>F_not_inj.\<Sum>\<pi> \<in> ?Perm. weight f \<pi>)" 
+    have "(\<Sum>(f, \<pi>)\<in>Z_not_inj. weight f \<pi>) = (\<Sum>f\<in>F_not_inj.\<Sum>\<pi> \<in> ?Perm. weight f \<pi>)"
       unfolding Z_not_inj_def unfolding sum.cartesian_product
       unfolding F_not_inj_def by simp
-    also have "... = (\<Sum>f\<in>F_not_inj. \<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> 
+    also have "... = (\<Sum>f\<in>F_not_inj. \<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi>
       * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))"
       unfolding weight_def by simp
-    also have "... = (\<Sum>f\<in>F_not_inj. (\<Prod>i = 0..<n. A $$ (i, f i)) 
-      * (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))))" 
+    also have "... = (\<Sum>f\<in>F_not_inj. (\<Prod>i = 0..<n. A $$ (i, f i))
+      * (\<Sum>\<pi> | \<pi> permutes {0..<n}. signof \<pi> * (\<Prod>i = 0..<n. B $$ (f i, \<pi> i))))"
       by (rule sum.cong, rule refl, auto)
          (metis (no_types, lifting) mult.left_commute mult_hom.hom_sum sum.cong)
-    also have "... = (\<Sum>f \<in> F_not_inj. (\<Prod>i = 0..<n. A $$ (i, f i)) 
+    also have "... = (\<Sum>f \<in> F_not_inj. (\<Prod>i = 0..<n. A $$ (i, f i))
       * det (mat\<^sub>r n n (\<lambda>i. row B (f i))))" using sum_rw by auto
     finally show ?thesis by auto
   qed
-  also have "... = 0" 
+  also have "... = 0"
     by (rule sum.neutral, insert det_not_inj_on[of _ n B], auto simp add: F_not_inj_def)
   finally show ?thesis .
 qed
 
 subsection \<open>Final theorem\<close>
 
 lemma Cauchy_Binet1:
-  shows "det (A*B) = 
-  sum (\<lambda>f. det (submatrix A UNIV (f`{0..<n})) * det (submatrix B (f`{0..<n}) UNIV)) F_strict" 
+  shows "det (A*B) =
+  sum (\<lambda>f. det (submatrix A UNIV (f`{0..<n})) * det (submatrix B (f`{0..<n}) UNIV)) F_strict"
 (is "?lhs = ?rhs")
 proof -
   have sum0: "(\<Sum>(f, \<pi>) \<in> Z_not_inj. weight f \<pi>) = 0" by (rule weight_0)
-  let ?f = "strict_from_inj n"  
-  have sum_rw: "sum g F_inj = (\<Sum>y \<in> F_strict. sum g {x \<in> F_inj. ?f x = y})" for g     
-    by (rule sum.group[symmetric], insert strict_from_inj_F_strict, auto)    
+  let ?f = "strict_from_inj n"
+  have sum_rw: "sum g F_inj = (\<Sum>y \<in> F_strict. sum g {x \<in> F_inj. ?f x = y})" for g
+    by (rule sum.group[symmetric], insert strict_from_inj_F_strict, auto)
   have Z_Union: "Z_inj \<union> Z_not_inj = Z n m"
     unfolding Z_def Z_not_inj_def Z_inj_def by auto
   have Z_Inter: "Z_inj \<inter> Z_not_inj = {}"
     unfolding Z_def Z_not_inj_def Z_inj_def by auto
   have "det (A*B) = (\<Sum>(f, \<pi>)\<in>Z n m. weight f \<pi>)"
     using detAB_Znm[OF A B] unfolding weight_def by auto
   also have "... = (\<Sum>(f, \<pi>)\<in>Z_inj. weight f \<pi>) + (\<Sum>(f, \<pi>)\<in>Z_not_inj. weight f \<pi>)"
     by (metis Z_Inter Z_Union finite_Un finite_Znm sum.union_disjoint)
   also have "... = (\<Sum>(f, \<pi>)\<in>Z_inj. weight f \<pi>)" using sum0 by force
-  also have "... = (\<Sum>f \<in> F_inj. \<Sum>\<pi>\<in>{\<pi>. \<pi> permutes {0..<n}}. weight f \<pi>)" 
+  also have "... = (\<Sum>f \<in> F_inj. \<Sum>\<pi>\<in>{\<pi>. \<pi> permutes {0..<n}}. weight f \<pi>)"
     unfolding Z_inj_def unfolding F_inj_def sum.cartesian_product ..
-  also have "... =  (\<Sum>y\<in>F_strict. \<Sum>f\<in>{x \<in> F_inj. strict_from_inj n x = y}. 
+  also have "... =  (\<Sum>y\<in>F_strict. \<Sum>f\<in>{x \<in> F_inj. strict_from_inj n x = y}.
     sum (weight f) {\<pi>. \<pi> permutes {0..<n}})" unfolding sum_rw ..
-  also have "... =  (\<Sum>y\<in>F_strict. \<Sum>(f,\<pi>)\<in>({x \<in> F_inj. strict_from_inj n x = y} 
+  also have "... =  (\<Sum>y\<in>F_strict. \<Sum>(f,\<pi>)\<in>({x \<in> F_inj. strict_from_inj n x = y}
   \<times> {\<pi>. \<pi> permutes {0..<n}}). weight f \<pi>)"
     unfolding F_inj_def sum.cartesian_product ..
-  also have "... = sum (\<lambda>g. sum (\<lambda>(f,\<pi>). weight f \<pi>) (Z_good g)) F_strict" 
-    using Z_good_alt by auto  
+  also have "... = sum (\<lambda>g. sum (\<lambda>(f,\<pi>). weight f \<pi>) (Z_good g)) F_strict"
+    using Z_good_alt by auto
   also have "... = ?rhs" unfolding gather_by_strictness by simp
   finally show ?thesis .
 qed
 
 
 lemma Cauchy_Binet:
   "det (A*B) = (\<Sum>I\<in>{I. I\<subseteq>{0..<m} \<and> card I=n}. det (submatrix A UNIV I) * det (submatrix B I UNIV))"
 proof -
   let ?f="(\<lambda>I. (\<lambda>i. if i<n then sorted_list_of_set I ! i else i))"
   let ?setI = "{I. I \<subseteq> {0..<m} \<and> card I = n}"
   have inj_on: "inj_on ?f ?setI"
   proof (rule inj_onI)
-    fix I J assume I: "I \<in> ?setI" and J: "J \<in> ?setI" and fI_fJ: "?f I = ?f J" 
+    fix I J assume I: "I \<in> ?setI" and J: "J \<in> ?setI" and fI_fJ: "?f I = ?f J"
     have "x \<in> J" if x: "x \<in> I" for x
-      by (metis (mono_tags) fI_fJ I J distinct_card in_set_conv_nth mem_Collect_eq 
+      by (metis (mono_tags) fI_fJ I J distinct_card in_set_conv_nth mem_Collect_eq
           sorted_list_of_set(1) sorted_list_of_set(3) subset_eq_atLeast0_lessThan_finite x)
     moreover have "x \<in> I" if x: "x \<in> J" for x
-      by (metis (mono_tags) fI_fJ I J distinct_card in_set_conv_nth mem_Collect_eq 
+      by (metis (mono_tags) fI_fJ I J distinct_card in_set_conv_nth mem_Collect_eq
           sorted_list_of_set(1) sorted_list_of_set(3) subset_eq_atLeast0_lessThan_finite x)
     ultimately show "I = J" by auto
   qed
   have rw: "?f I ` {0..<n} = I" if I: "I \<in> ?setI" for I
   proof -
-    have "sorted_list_of_set I ! xa \<in> I" if "xa < n" for xa 
-      by (metis (mono_tags, lifting) I distinct_card distinct_sorted_list_of_set mem_Collect_eq 
+    have "sorted_list_of_set I ! xa \<in> I" if "xa < n" for xa
+      by (metis (mono_tags, lifting) I distinct_card distinct_sorted_list_of_set mem_Collect_eq
           nth_mem set_sorted_list_of_set subset_eq_atLeast0_lessThan_finite that)
-    moreover have "\<exists>xa\<in>{0..<n}. x = sorted_list_of_set I ! xa" if x: "x\<in>I" for x 
-      by (metis (full_types) x I atLeast0LessThan distinct_card in_set_conv_nth mem_Collect_eq  
+    moreover have "\<exists>xa\<in>{0..<n}. x = sorted_list_of_set I ! xa" if x: "x\<in>I" for x
+      by (metis (full_types) x I atLeast0LessThan distinct_card in_set_conv_nth mem_Collect_eq
          lessThan_iff sorted_list_of_set(1) sorted_list_of_set(3) subset_eq_atLeast0_lessThan_finite)
     ultimately show ?thesis unfolding image_def by auto
   qed
   have f_setI: "?f` ?setI = F_strict"
   proof -
-    have "sorted_list_of_set I ! xa < m" if I: "I \<subseteq> {0..<m}" and "n = card I" and "xa < card I" 
+    have "sorted_list_of_set I ! xa < m" if I: "I \<subseteq> {0..<m}" and "n = card I" and "xa < card I"
         for I xa
-      by (metis I \<open>xa < card I\<close> atLeast0LessThan distinct_card finite_atLeastLessThan lessThan_iff 
-          pick_in_set_le rev_finite_subset sorted_list_of_set(1) 
-          sorted_list_of_set(3) sorted_list_of_set_eq_pick subsetCE)      
+      by (metis I \<open>xa < card I\<close> atLeast0LessThan distinct_card finite_atLeastLessThan lessThan_iff
+          pick_in_set_le rev_finite_subset sorted_list_of_set(1)
+          sorted_list_of_set(3) sorted_list_of_set_eq_pick subsetCE)
     moreover have "strict_mono_on (\<lambda>i. if i < card I then sorted_list_of_set I ! i else i) {0..<card I}"
       if "I \<subseteq> {0..<m}" and "n = card I" for I
-      by (smt \<open>I \<subseteq> {0..<m}\<close> atLeastLessThan_iff distinct_card finite_atLeastLessThan pick_mono_le 
-          rev_finite_subset sorted_list_of_set(1) sorted_list_of_set(3) 
+      by (smt \<open>I \<subseteq> {0..<m}\<close> atLeastLessThan_iff distinct_card finite_atLeastLessThan pick_mono_le
+          rev_finite_subset sorted_list_of_set(1) sorted_list_of_set(3)
           sorted_list_of_set_eq_pick strict_mono_on_def)
-    moreover have "x \<in> ?f ` {I. I \<subseteq> {0..<m} \<and> card I = n}" 
+    moreover have "x \<in> ?f ` {I. I \<subseteq> {0..<m} \<and> card I = n}"
       if x1: "x \<in> {0..<n} \<rightarrow> {0..<m}" and x2: "\<forall>i. \<not> i < n \<longrightarrow> x i = i"
       and s: "strict_mono_on x {0..<n}" for x
     proof -
-      have inj_x: "inj_on x {0..<n}" 
+      have inj_x: "inj_on x {0..<n}"
         using s strict_mono_on_imp_inj_on by blast
       hence card_xn: "card (x ` {0..<n}) = n" by (simp add: card_image)
       have x_eq: "x = (\<lambda>i. if i < n then sorted_list_of_set (x ` {0..<n}) ! i else i)"
         unfolding fun_eq_iff
         using nth_strict_mono_on s using x2 by auto
       show ?thesis
         unfolding image_def by (auto, rule exI[of _"x`{0..<n}"], insert card_xn x1 x_eq, auto)
     qed
     ultimately show ?thesis unfolding F_strict_def by auto
   qed
   let ?g = "(\<lambda>f. det (submatrix A UNIV (f`{0..<n})) * det(submatrix B (f`{0..<n}) UNIV))"
-  have "det (A*B) = sum ((\<lambda>f. det (submatrix A UNIV (f ` {0..<n})) 
-    * det (submatrix B (f ` {0..<n}) UNIV)) \<circ> ?f) {I. I \<subseteq> {0..<m} \<and> card I = n}" 
+  have "det (A*B) = sum ((\<lambda>f. det (submatrix A UNIV (f ` {0..<n}))
+    * det (submatrix B (f ` {0..<n}) UNIV)) \<circ> ?f) {I. I \<subseteq> {0..<m} \<and> card I = n}"
     unfolding Cauchy_Binet1 f_setI[symmetric] by (rule sum.reindex[OF inj_on])
   also have "... = (\<Sum>I\<in>{I. I\<subseteq>{0..<m} \<and> card I=n}.det(submatrix A UNIV I)*det(submatrix B I UNIV))"
     by (rule sum.cong, insert rw, auto)
   finally show ?thesis .
 qed
 end
 
 end
diff --git a/thys/Smith_Normal_Form/Cauchy_Binet_HOL_Analysis.thy b/thys/Smith_Normal_Form/Cauchy_Binet_HOL_Analysis.thy
--- a/thys/Smith_Normal_Form/Cauchy_Binet_HOL_Analysis.thy
+++ b/thys/Smith_Normal_Form/Cauchy_Binet_HOL_Analysis.thy
@@ -1,119 +1,119 @@
 (*
     Author:      Jose Divasón
-    Email:       jose.divason@unirioja.es           
+    Email:       jose.divason@unirioja.es
 *)
 
 section \<open>The Cauchy--Binet formula in HOL Analysis\<close>
 
 theory Cauchy_Binet_HOL_Analysis
-  imports 
+  imports
     Cauchy_Binet
     Perron_Frobenius.HMA_Connect
 begin
 
 subsection \<open>Definition of submatrices in HOL Analysis\<close>
 
 definition submatrix_hma :: "'a^'nc^'nr\<Rightarrow>nat set\<Rightarrow>nat set\<Rightarrow>('a^'nc2^'nr2)"
   where "submatrix_hma A I J = (\<chi> a b. A $h (from_nat (pick I (to_nat a))) $h (from_nat (pick J (to_nat b))))"
 
 context includes lifting_syntax
 begin
 
-context 
+context
   fixes I::"nat set" and J::"nat set"
   assumes I: "card {i. i < CARD('nr::finite) \<and> i \<in> I} = CARD('nr2::finite)"
   assumes J: "card {i. i < CARD('nc::finite) \<and> i \<in> J} = CARD('nc2::finite)"
 begin
 
 lemma HMA_submatrix[transfer_rule]: "(HMA_M ===> HMA_M) (\<lambda>A. submatrix A I J)
   ((\<lambda>A. submatrix_hma A I J):: 'a^ 'nc ^ 'nr \<Rightarrow> 'a ^ 'nc2 ^ 'nr2)"
 proof (intro rel_funI, goal_cases)
   case (1 A B)
-  note relAB[transfer_rule] = this  
-  show ?case unfolding  HMA_M_def 
+  note relAB[transfer_rule] = this
+  show ?case unfolding  HMA_M_def
   proof (rule eq_matI, auto)
-    show "dim_row (submatrix A I J) = CARD('nr2)" 
+    show "dim_row (submatrix A I J) = CARD('nr2)"
       unfolding submatrix_def
       using I dim_row_transfer_rule relAB by force
     show "dim_col (submatrix A I J) = CARD('nc2)"
       unfolding submatrix_def
       using J dim_col_transfer_rule relAB by force
     let ?B="(submatrix_hma B I J)::'a ^ 'nc2 ^ 'nr2"
     fix i j assume i: "i < CARD('nr2)" and
            j: "j < CARD('nc2)"
     have i2: "i < card {i. i < dim_row A \<and> i \<in> I}"
       using I dim_row_transfer_rule i relAB by fastforce
     have j2: "j < card {j. j < dim_col A \<and> j \<in> J}"
       using J dim_col_transfer_rule j relAB by fastforce
     let ?i = "(from_nat (pick I i))::'nr"
     let ?j = "(from_nat (pick J j))::'nc"
     let ?i' = "Bij_Nat.to_nat ((Bij_Nat.from_nat i)::'nr2)"
     let ?j' = "Bij_Nat.to_nat ((Bij_Nat.from_nat j)::'nc2)"
     have i': "?i' = i" by (rule to_nat_from_nat_id[OF i])
     have j': "?j' = j" by (rule to_nat_from_nat_id[OF j])
     let ?f = "(\<lambda>(i, j).
          B $h Bij_Nat.from_nat (pick I (Bij_Nat.to_nat ((Bij_Nat.from_nat i)::'nr2))) $h
          Bij_Nat.from_nat (pick J (Bij_Nat.to_nat ((Bij_Nat.from_nat j)::'nc2))))"
     have [transfer_rule]: "HMA_I (pick I i) ?i"
       by (simp add: Bij_Nat.to_nat_from_nat_id I i pick_le HMA_I_def)
     have [transfer_rule]: "HMA_I (pick J j) ?j"
       by (simp add: Bij_Nat.to_nat_from_nat_id J j pick_le HMA_I_def)
     have "submatrix A I J $$ (i, j) = A $$ (pick I i, pick J j)" by (rule submatrix_index[OF i2 j2])
     also have "... = index_hma B ?i ?j" by (transfer, simp)
     also have "... =  B $h Bij_Nat.from_nat (pick I (Bij_Nat.to_nat ((Bij_Nat.from_nat i)::'nr2))) $h
-         Bij_Nat.from_nat (pick J (Bij_Nat.to_nat ((Bij_Nat.from_nat j)::'nc2)))" 
+         Bij_Nat.from_nat (pick J (Bij_Nat.to_nat ((Bij_Nat.from_nat j)::'nc2)))"
       unfolding i' j' index_hma_def by auto
     also have "... = ?f (i,j)" by auto
-    also have "... = Matrix.mat CARD('nr2) CARD('nc2) ?f $$ (i, j)" 
+    also have "... = Matrix.mat CARD('nr2) CARD('nc2) ?f $$ (i, j)"
       by (rule index_mat[symmetric, OF i j])
-    also have "... = from_hma\<^sub>m ?B $$ (i, j)" 
+    also have "... = from_hma\<^sub>m ?B $$ (i, j)"
       unfolding from_hma\<^sub>m_def submatrix_hma_def by auto
     finally show "submatrix A I J $$ (i, j) = from_hma\<^sub>m ?B $$ (i, j)" .
   qed
 qed
-  
+
 end
 end
 
 
 subsection \<open>Transferring the proof from JNF to HOL Analysis\<close>
 
 lemma Cauchy_Binet_HOL_Analysis:
   fixes A::"'a::comm_ring_1^'m^'n" and B::"'a^'n^'m"
-  shows "Determinants.det (A**B) = (\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I=nrows A}. 
-         Determinants.det ((submatrix_hma A UNIV I)::'a^'n^'n) * 
+  shows "Determinants.det (A**B) = (\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I=nrows A}.
+         Determinants.det ((submatrix_hma A UNIV I)::'a^'n^'n) *
          Determinants.det ((submatrix_hma B I UNIV)::'a^'n^'n))"
-proof -  
+proof -
   let ?A = "(from_hma\<^sub>m A)"
   let ?B = "(from_hma\<^sub>m B)"
   have relA[transfer_rule]: "HMA_M ?A A" unfolding HMA_M_def by simp
-  have relB[transfer_rule]: "HMA_M ?B B" unfolding HMA_M_def by simp  
-  have "(\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I = nrows A}. 
-         Determinants.det ((submatrix_hma A UNIV I)::'a^'n^'n) * 
-         Determinants.det ((submatrix_hma B I UNIV)::'a^'n^'n)) = 
-          (\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I=nrows A}. det (submatrix ?A UNIV I) 
+  have relB[transfer_rule]: "HMA_M ?B B" unfolding HMA_M_def by simp
+  have "(\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I = nrows A}.
+         Determinants.det ((submatrix_hma A UNIV I)::'a^'n^'n) *
+         Determinants.det ((submatrix_hma B I UNIV)::'a^'n^'n)) =
+          (\<Sum>I\<in>{I. I\<subseteq>{0..<ncols A} \<and> card I=nrows A}. det (submatrix ?A UNIV I)
         * det (submatrix ?B I UNIV))"
   proof (rule sum.cong)
     fix I assume I: "I \<in>{I. I\<subseteq>{0..<ncols A} \<and> card I=nrows A}"
     let ?sub_A= "((submatrix_hma A UNIV I)::'a^'n^'n)"
     let ?sub_B= "((submatrix_hma B I UNIV)::'a^'n^'n)"
     have c1: "card {i. i < CARD('n) \<and> i \<in> UNIV} = CARD('n)" using I by auto
     have c2: "card {i. i < CARD('m) \<and> i \<in> I} = CARD('n)"
     proof -
       have "I = {i. i < CARD('m) \<and> i \<in> I}" using I unfolding nrows_def ncols_def by auto
       thus ?thesis using I nrows_def by auto
     qed
     have [transfer_rule]: "HMA_M (submatrix ?A UNIV I) ?sub_A"
       using HMA_submatrix[OF c1 c2] relA unfolding rel_fun_def by auto
     have [transfer_rule]: "HMA_M (submatrix ?B I UNIV) ?sub_B"
-      using HMA_submatrix[OF c2 c1] relB unfolding rel_fun_def by auto  
-    show "Determinants.det ?sub_A * Determinants.det ?sub_B 
+      using HMA_submatrix[OF c2 c1] relB unfolding rel_fun_def by auto
+    show "Determinants.det ?sub_A * Determinants.det ?sub_B
       = det (submatrix ?A UNIV I) * det (submatrix ?B I UNIV)" by (transfer', auto)
-  qed (auto)  
-  also have "... = det (?A*?B)" 
+  qed (auto)
+  also have "... = det (?A*?B)"
     by (rule Cauchy_Binet[symmetric], unfold nrows_def ncols_def, auto)
   also have "... = Determinants.det (A**B)" by (transfer', auto)
   finally show ?thesis ..
 qed
 
 end
\ No newline at end of file
diff --git a/thys/Smith_Normal_Form/Rings2_Extended.thy b/thys/Smith_Normal_Form/Rings2_Extended.thy
--- a/thys/Smith_Normal_Form/Rings2_Extended.thy
+++ b/thys/Smith_Normal_Form/Rings2_Extended.thy
@@ -1,809 +1,809 @@
 (*
     Author:      Jose Divasón
-    Email:       jose.divason@unirioja.es           
+    Email:       jose.divason@unirioja.es
 *)
 
 section \<open>Some theorems about rings and ideals\<close>
 
 theory Rings2_Extended
-  imports 
+  imports
     Echelon_Form.Rings2
     "HOL-Types_To_Sets.Types_To_Sets"
 begin
 
 subsection \<open>Missing properties on ideals\<close>
 
 lemma ideal_generated_subset2:
   assumes  "\<forall>b\<in>B. b \<in> ideal_generated A"
   shows "ideal_generated B \<subseteq> ideal_generated A"
-  by (metis (mono_tags, lifting) InterE assms ideal_generated_def 
+  by (metis (mono_tags, lifting) InterE assms ideal_generated_def
 ideal_ideal_generated mem_Collect_eq subsetI)
 
 context comm_ring_1
 begin
 
-lemma ideal_explicit: "ideal_generated S 
+lemma ideal_explicit: "ideal_generated S
       = {y. \<exists>f U. finite U \<and> U \<subseteq> S \<and> (\<Sum>i\<in>U. f i * i) = y}"
   by (simp add: ideal_generated_eq_left_ideal left_ideal_explicit)
 end
 
 lemma ideal_generated_minus:
-  assumes a: "a \<in> ideal_generated (S-{a})" 
+  assumes a: "a \<in> ideal_generated (S-{a})"
   shows "ideal_generated S = ideal_generated (S-{a})"
 proof (cases "a \<in> S")
   case True note a_in_S = True
-  show ?thesis 
+  show ?thesis
   proof
     show "ideal_generated S \<subseteq> ideal_generated (S - {a})"
     proof (rule ideal_generated_subset2, auto)
-      fix b assume b: "b \<in> S" show "b \<in> ideal_generated (S - {a})" 
+      fix b assume b: "b \<in> S" show "b \<in> ideal_generated (S - {a})"
       proof (cases "b = a")
         case True
         then show ?thesis using a by auto
       next
         case False
         then show ?thesis using b
           by (simp add: ideal_generated_in)
-      qed      
+      qed
     qed
-    show "ideal_generated (S - {a}) \<subseteq> ideal_generated S" 
+    show "ideal_generated (S - {a}) \<subseteq> ideal_generated S"
       by (rule ideal_generated_subset, auto)
   qed
 next
   case False
   then show ?thesis by simp
 qed
 
 lemma ideal_generated_dvd_eq:
   assumes a_dvd_b: "a dvd b"
   and a: "a \<in> S"
   and a_not_b: "a \<noteq> b"
   shows "ideal_generated S = ideal_generated (S - {b})"
 proof
   show "ideal_generated S \<subseteq> ideal_generated (S - {b})"
   proof (rule ideal_generated_subset2, auto)
-    fix x assume x: "x \<in> S" 
+    fix x assume x: "x \<in> S"
     show "x \<in> ideal_generated (S - {b})"
     proof (cases "x = b")
       case True
       obtain k where b_ak: "b = a * k" using a_dvd_b unfolding dvd_def by blast
       let ?f = "\<lambda>c. k"
       have "(\<Sum>i\<in>{a}. i * ?f i) = x" using True b_ak by auto
       moreover have "{a} \<subseteq> S - {b}" using a_not_b a by auto
       moreover have "finite {a}" by auto
-      ultimately show ?thesis 
+      ultimately show ?thesis
         unfolding ideal_def
         by (metis True b_ak ideal_def ideal_generated_in ideal_ideal_generated insert_subset right_ideal_def)
     next
       case False
       then show ?thesis by (simp add: ideal_generated_in x)
     qed
   qed
-  show "ideal_generated (S - {b}) \<subseteq> ideal_generated S" by (rule ideal_generated_subset, auto)    
+  show "ideal_generated (S - {b}) \<subseteq> ideal_generated S" by (rule ideal_generated_subset, auto)
 qed
 
 lemma ideal_generated_dvd_eq_diff_set:
   assumes i_in_I: "i\<in>I" and i_in_J: "i \<notin> J" and i_dvd_j: "\<forall>j\<in>J. i dvd j"
   and f: "finite J"
   shows "ideal_generated I = ideal_generated (I - J)"
   using f i_in_J i_dvd_j i_in_I
   proof (induct J arbitrary: I)
   case empty
     then show ?case by auto
   next
     case (insert x J)
     have "ideal_generated I = ideal_generated (I-{x})"
       by (rule ideal_generated_dvd_eq[of i], insert insert.prems , auto)
     also have "... = ideal_generated ((I-{x}) - J)"
       by (rule insert.hyps, insert insert.prems insert.hyps, auto)
     also have "... = ideal_generated (I - insert x J)"
       using Diff_insert2[of I x J] by auto
     finally show ?case .
   qed
 
 
 context comm_ring_1
 begin
 
 lemma ideal_generated_singleton_subset:
   assumes d: "d \<in> ideal_generated S" and fin_S: "finite S"
   shows "ideal_generated {d} \<subseteq> ideal_generated S"
-proof 
+proof
   fix x assume x: "x \<in> ideal_generated {d}"
   obtain k where x_kd: "x = k*d " using x using obtain_sum_ideal_generated[OF x]
     by (metis finite.emptyI finite.insertI sum_singleton)
   show "x \<in> ideal_generated S"
     using d ideal_eq_right_ideal ideal_ideal_generated right_ideal_def mult_commute x_kd by auto
 qed
 
 lemma ideal_generated_singleton_dvd:
   assumes i: "ideal_generated S = ideal_generated {d}" and x: "x \<in> S"
   shows "d dvd x"
-  by (metis i x finite.intros dvd_ideal_generated_singleton 
+  by (metis i x finite.intros dvd_ideal_generated_singleton
       ideal_generated_in ideal_generated_singleton_subset)
 
 lemma ideal_generated_UNIV_insert:
   assumes "ideal_generated S = UNIV"
   shows "ideal_generated (insert a S) = UNIV" using assms
   using local.ideal_generated_subset by blast
 
 lemma ideal_generated_UNIV_union:
   assumes "ideal_generated S = UNIV"
-  shows "ideal_generated (A \<union> S) = UNIV" 
+  shows "ideal_generated (A \<union> S) = UNIV"
   using assms local.ideal_generated_subset
   by (metis UNIV_I Un_subset_iff equalityI subsetI)
 
-lemma ideal_explicit2: 
+lemma ideal_explicit2:
   assumes "finite S"
   shows "ideal_generated S = {y. \<exists>f. (\<Sum>i\<in>S. f i * i) = y}"
   by (smt Collect_cong assms ideal_explicit obtain_sum_ideal_generated mem_Collect_eq subsetI)
 
 lemma ideal_generated_unit:
   assumes u: "u dvd 1"
   shows "ideal_generated {u} = UNIV"
 proof -
   have "x \<in> ideal_generated {u}" for x
   proof -
     obtain inv_u where inv_u: "inv_u * u = 1" using u unfolding dvd_def
-      using local.mult_ac(2) by blast    
+      using local.mult_ac(2) by blast
     have "x = x * inv_u * u" using inv_u by (simp add: local.mult_ac(1))
     also have "... \<in> {k * u |k. k \<in> UNIV}" by auto
     also have "... = ideal_generated {u}" unfolding ideal_generated_singleton by simp
     finally show ?thesis .
   qed
   thus ?thesis by auto
 qed
 
 
 lemma ideal_generated_dvd_subset:
   assumes x: "\<forall>x \<in> S. d dvd x" and S: "finite S"
   shows "ideal_generated S \<subseteq> ideal_generated {d}"
-proof 
-  fix x assume "x\<in> ideal_generated S" 
+proof
+  fix x assume "x\<in> ideal_generated S"
   from this obtain f where f: "(\<Sum>i\<in>S. f i * i) = x" using ideal_explicit2[OF S] by auto
   have "d dvd (\<Sum>i\<in>S. f i * i)" by (rule dvd_sum, insert x, auto)
   thus "x \<in> ideal_generated {d}"
     using f dvd_ideal_generated_singleton' ideal_generated_in singletonI by blast
 qed
 
 
 lemma ideal_generated_mult_unit:
   assumes f: "finite S" and u: "u dvd 1"
   shows "ideal_generated ((\<lambda>x. u*x)` S) = ideal_generated S"
   using f
 proof (induct S)
   case empty
   then show ?case by auto
 next
   case (insert x S)
   obtain inv_u where inv_u: "inv_u * u = 1" using u unfolding dvd_def
     using mult_ac by blast
   have f: "finite (insert (u*x) ((\<lambda>x. u*x)` S))" using insert.hyps by auto
   have f2: "finite (insert x S)" by (simp add: insert(1))
   have f3: "finite S" by (simp add: insert)
   have f4: "finite ((*) u ` S)" by (simp add: insert)
   have inj_ux: "inj_on (\<lambda>x. u*x) S" unfolding inj_on_def
-    by (auto, metis inv_u local.mult_1_left local.semiring_normalization_rules(18))    
+    by (auto, metis inv_u local.mult_1_left local.semiring_normalization_rules(18))
   have "ideal_generated ((\<lambda>x. u*x)` (insert x S)) = ideal_generated (insert (u*x) ((\<lambda>x. u*x)` S))"
     by auto
   also have "... = {y. \<exists>f. (\<Sum>i\<in>insert (u*x) ((\<lambda>x. u*x)` S). f i * i) = y}"
     using ideal_explicit2[OF f] by auto
   also have "... = {y. \<exists>f. (\<Sum>i\<in>(insert x S). f i * i) = y}" (is "?L = ?R")
   proof -
     have "a \<in> ?L"  if a: "a \<in> ?R" for a
     proof -
       obtain f where sum_rw: "(\<Sum>i\<in>(insert x S). f i * i) = a" using a by auto
       define b where "b=(\<Sum>i\<in>S. f i * i)"
       have "b \<in> ideal_generated S" unfolding b_def ideal_explicit2[OF f3] by auto
       hence "b \<in> ideal_generated ((*) u ` S)" using insert.hyps(3) by auto
-      from this obtain g where "(\<Sum>i\<in>((*) u ` S). g i * i) = b" 
+      from this obtain g where "(\<Sum>i\<in>((*) u ` S). g i * i) = b"
         unfolding ideal_explicit2[OF f4] by auto
       hence sum_rw2: "(\<Sum>i\<in>S. f i * i) = (\<Sum>i\<in>((*) u ` S). g i * i)" unfolding b_def by auto
-      let ?g = "\<lambda>i. if i = u*x then f x * inv_u else g i"  
+      let ?g = "\<lambda>i. if i = u*x then f x * inv_u else g i"
        have sum_rw3: "sum ((\<lambda>i. g i * i) \<circ> (\<lambda>x. u*x)) S = sum ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) S"
-        by (rule sum.cong, auto, metis inv_u local.insert(2) local.mult_1_right 
+        by (rule sum.cong, auto, metis inv_u local.insert(2) local.mult_1_right
               local.mult_ac(2) local.semiring_normalization_rules(18))
-      have sum_rw4: "(\<Sum>i\<in>(\<lambda>x. u*x)` S. g i * i) = sum ((\<lambda>i. g i * i) \<circ> (\<lambda>x. u*x)) S" 
+      have sum_rw4: "(\<Sum>i\<in>(\<lambda>x. u*x)` S. g i * i) = sum ((\<lambda>i. g i * i) \<circ> (\<lambda>x. u*x)) S"
         by (rule sum.reindex[OF inj_ux])
       have "a = f x * x + (\<Sum>i\<in>S. f i * i)"
         using sum_rw local.insert(1) local.insert(2) by auto
       also have "... = f x * x + (\<Sum>i\<in>(\<lambda>x. u*x)` S. g i * i)" using sum_rw2 by auto
-      also have "... = ?g (u * x) * (u * x) + (\<Sum>i\<in>(\<lambda>x. u*x)` S. g i * i)" 
-        using inv_u by (smt local.mult_1_right local.mult_ac(1)) 
+      also have "... = ?g (u * x) * (u * x) + (\<Sum>i\<in>(\<lambda>x. u*x)` S. g i * i)"
+        using inv_u by (smt local.mult_1_right local.mult_ac(1))
       also have "... =  ?g (u * x) * (u * x) + sum ((\<lambda>i. g i * i) \<circ> (\<lambda>x. u*x)) S"
         using sum_rw4 by auto
       also have "... = ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) x + sum ((\<lambda>i. g i * i) \<circ> (\<lambda>x. u*x)) S" by auto
-      also have "... = ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) x + sum ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) S" 
+      also have "... = ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) x + sum ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) S"
         using sum_rw3 by auto
       also have "... = sum ((\<lambda>i. ?g i * i) \<circ> (\<lambda>x. u*x)) (insert x S)"
         by (rule sum.insert[symmetric], auto simp add: insert)
       also have "... = (\<Sum>i\<in>insert (u * x) ((\<lambda>x. u*x)` S). ?g i * i)"
-        by (smt abel_semigroup.commute f2 image_insert inv_u mult.abel_semigroup_axioms mult_1_right 
+        by (smt abel_semigroup.commute f2 image_insert inv_u mult.abel_semigroup_axioms mult_1_right
             semiring_normalization_rules(18) sum.reindex_nontrivial)
       also have "... = (\<Sum>i\<in>(\<lambda>x. u*x)` (insert x S). ?g i * i)" by auto
-      finally show ?thesis by auto      
+      finally show ?thesis by auto
     qed
     moreover have "a \<in> ?R" if a: "a \<in> ?L" for a
     proof -
       obtain f where sum_rw: "(\<Sum>i\<in>(insert (u * x) ((*) u ` S)). f i * i) = a" using a by auto
       have ux_notin: "u*x \<notin> ((*) u ` S)"
-        by (metis UNIV_I inj_on_image_mem_iff inj_on_inverseI inv_u local.insert(2) local.mult_1_left 
-            local.semiring_normalization_rules(18) subsetI)      
+        by (metis UNIV_I inj_on_image_mem_iff inj_on_inverseI inv_u local.insert(2) local.mult_1_left
+            local.semiring_normalization_rules(18) subsetI)
       let ?f = "(\<lambda>x. f x * x)"
       have "sum ?f ((*) u ` S) \<in> ideal_generated ((*) u ` S)"
         unfolding ideal_explicit2[OF f4] by auto
-      from this obtain g where sum_rw1: "sum (\<lambda>i. g i * i) S = sum ?f (((*) u ` S))" 
+      from this obtain g where sum_rw1: "sum (\<lambda>i. g i * i) S = sum ?f (((*) u ` S))"
         using insert.hyps(3) unfolding ideal_explicit2[OF f3] by blast
       let ?g = "(\<lambda>i. if i = x  then (f (u*x) *u) * x else g i * i)"
       let ?g' = "\<lambda>i. if i = x  then f (u*x) * u else g i"
-      have sum_rw2: "sum (\<lambda>i. g i * i) S = sum ?g S" by (rule sum.cong, insert inj_ux ux_notin, auto)  
+      have sum_rw2: "sum (\<lambda>i. g i * i) S = sum ?g S" by (rule sum.cong, insert inj_ux ux_notin, auto)
       have "a = (\<Sum>i\<in>(insert (u * x) ((*) u ` S)). f i * i)" using sum_rw by simp
-      also have "... = ?f (u*x) +  sum ?f (((*) u ` S))" 
-        by (rule sum.insert[OF f4], insert inj_ux) (metis UNIV_I inj_on_image_mem_iff inj_on_inverseI 
+      also have "... = ?f (u*x) +  sum ?f (((*) u ` S))"
+        by (rule sum.insert[OF f4], insert inj_ux) (metis UNIV_I inj_on_image_mem_iff inj_on_inverseI
             inv_u local.insert(2) local.mult_1_left local.semiring_normalization_rules(18) subsetI)
       also have "... = ?f (u*x) + sum (\<lambda>i. g i * i) S" unfolding sum_rw1 by auto
       also have "... = ?g x + sum ?g S" unfolding sum_rw2 using mult.assoc by auto
       also have "... = sum ?g (insert x S)" by (rule sum.insert[symmetric, OF f3 insert.hyps(2)])
       also have "... = sum (\<lambda>i. ?g' i * i) (insert x S)" by (rule sum.cong, auto)
       finally show ?thesis by fast
     qed
     ultimately show ?thesis by blast
   qed
   also have "... = ideal_generated (insert x S)" using ideal_explicit2[OF f2] by auto
   finally show ?case by auto
 qed
 
 corollary ideal_generated_mult_unit2:
   assumes u: "u dvd 1"
   shows "ideal_generated {u*a,u*b} = ideal_generated {a,b}"
 proof -
   let ?S = "{a,b}"
   have "ideal_generated {u*a,u*b} = ideal_generated ((\<lambda>x. u*x)` {a,b})" by auto
   also have "... = ideal_generated {a,b}" by (rule ideal_generated_mult_unit[OF _ u], simp)
   finally show ?thesis .
 qed
 
 lemma ideal_generated_1[simp]: "ideal_generated {1} = UNIV"
   by (metis ideal_generated_unit dvd_ideal_generated_singleton order_refl)
 
 lemma ideal_generated_pair: "ideal_generated {a,b} = {p*a+q*b | p q. True}"
 proof -
   have i: "ideal_generated {a,b} = {y. \<exists>f. (\<Sum>i\<in>{a,b}. f i * i) = y}" using ideal_explicit2 by auto
   show ?thesis
   proof (cases "a=b")
     case True
     show ?thesis using True i
       by (auto, metis mult_ac(2) semiring_normalization_rules)
       (metis (no_types, hide_lams) add_minus_cancel mult_ac ring_distribs semiring_normalization_rules)
   next
-    case False  
+    case False
     have 1: "\<exists>p q. (\<Sum>i\<in>{a, b}. f i * i) = p * a + q * b" for f
       by (rule exI[of _ "f a"], rule exI[of _ "f b"], rule sum_two_elements[OF False])
     moreover have "\<exists>f. (\<Sum>i\<in>{a, b}. f i * i) = p * a + q * b" for p q
-      by (rule exI[of _ "\<lambda>i. if i=a then p else q"], 
+      by (rule exI[of _ "\<lambda>i. if i=a then p else q"],
           unfold sum_two_elements[OF False], insert False, auto)
-    ultimately show ?thesis using i by auto  
+    ultimately show ?thesis using i by auto
   qed
 qed
 
 lemma ideal_generated_pair_exists_pq1:
   assumes i: "ideal_generated {a,b} = (UNIV::'a set)"
   shows "\<exists>p q. p*a + q*b = 1"
   using i unfolding ideal_generated_pair
   by (smt iso_tuple_UNIV_I mem_Collect_eq)
 
 lemma ideal_generated_pair_UNIV:
   assumes sa_tb_u: "s*a+t*b = u" and u: "u dvd 1"
   shows "ideal_generated {a,b} = UNIV"
 proof -
   have f: "finite {a,b}" by simp
   obtain inv_u where inv_u: "inv_u * u = 1" using u unfolding dvd_def
     by (metis mult.commute)
   have "x \<in> ideal_generated {a,b}" for x
   proof (cases "a = b")
     case True
     then show ?thesis
-      by (metis UNIV_I dvd_def dvd_ideal_generated_singleton' ideal_generated_unit insert_absorb2 
+      by (metis UNIV_I dvd_def dvd_ideal_generated_singleton' ideal_generated_unit insert_absorb2
           mult.commute sa_tb_u semiring_normalization_rules(34) subsetI subset_antisym u)
   next
-    case False note a_not_b = False    
+    case False note a_not_b = False
     let ?f = "\<lambda>y. if y = a then inv_u * x * s else inv_u * x * t"
     have "(\<Sum>i\<in>{a,b}. ?f i * i) = ?f a * a + ?f b * b" by (rule sum_two_elements[OF a_not_b])
-    also have "... = x" using a_not_b sa_tb_u inv_u      
+    also have "... = x" using a_not_b sa_tb_u inv_u
       by (auto, metis mult_ac(1) mult_ac(2) ring_distribs(1) semiring_normalization_rules(12))
     finally show ?thesis unfolding ideal_explicit2[OF f] by auto
   qed
   thus ?thesis by auto
 qed
 
 
 lemma ideal_generated_pair_exists:
   assumes l: "(ideal_generated {a,b} = ideal_generated {d})"
   shows "(\<exists> p q. p*a+q*b = d)"
 proof -
   have d: "d \<in> ideal_generated {d}" by (simp add: ideal_generated_in)
   hence "d \<in> ideal_generated {a,b}" using l by auto
-  from this obtain p q where "d = p*a+q*b" using ideal_generated_pair[of a b] by auto  
+  from this obtain p q where "d = p*a+q*b" using ideal_generated_pair[of a b] by auto
   thus ?thesis by auto
 qed
 
 
 lemma obtain_ideal_generated_pair:
   assumes "c \<in> ideal_generated {a,b}"
   obtains p q where "p*a+q*b=c"
 proof -
   have "c \<in> {p * a + q * b |p q. True}" using assms ideal_generated_pair by auto
   thus ?thesis using that by auto
 qed
 
 lemma ideal_generated_pair_exists_UNIV:
   shows "(ideal_generated {a,b} = ideal_generated {1}) = (\<exists>p q. p*a+q*b = 1)" (is "?lhs = ?rhs")
-proof   
-  assume r: ?rhs 
+proof
+  assume r: ?rhs
   have "x \<in> ideal_generated {a,b}" for x
   proof (cases "a=b")
     case True
-    then show ?thesis 
+    then show ?thesis
       by (metis UNIV_I r dvd_ideal_generated_singleton finite.intros ideal_generated_1
           ideal_generated_pair_UNIV ideal_generated_singleton_subset)
   next
     case False
     have f: "finite {a,b}" by simp
     have 1: "1 \<in> ideal_generated {a,b}"
       using ideal_generated_pair_UNIV local.one_dvd r by blast
     hence i: "ideal_generated {a,b} = {y. \<exists>f. (\<Sum>i\<in>{a,b}. f i * i) = y}"
-      using ideal_explicit2[of "{a,b}"] by auto    
+      using ideal_explicit2[of "{a,b}"] by auto
     from this obtain f where f: "f a * a + f b * b = 1" using sum_two_elements 1 False by auto
     let ?f = "\<lambda>y. if y = a then x * f a else x * f b"
-    have "(\<Sum>i\<in>{a,b}. ?f i * i) = x" unfolding sum_two_elements[OF False] using f False      
+    have "(\<Sum>i\<in>{a,b}. ?f i * i) = x" unfolding sum_two_elements[OF False] using f False
       using mult_ac(1) ring_distribs(1) semiring_normalization_rules(12) by force
     thus ?thesis unfolding i by auto
   qed
   thus ?lhs by auto
 next
   assume ?lhs thus ?rhs using ideal_generated_pair_exists[of a b 1] by auto
 qed
 
 corollary ideal_generated_UNIV_obtain_pair:
   assumes "ideal_generated {a,b} = ideal_generated {1}"
   shows " (\<exists>p q. p*a+q*b = d)"
 proof -
   obtain x y where "x*a+y*b = 1" using ideal_generated_pair_exists_UNIV assms by auto
   hence "d*x*a+d*y*b=d"
     using local.mult_ac(1) local.ring_distribs(1) local.semiring_normalization_rules(12) by force
   thus ?thesis by auto
 qed
 
 
 
 lemma sum_three_elements:
   shows "\<exists>x y z::'a. (\<Sum>i\<in>{a,b,c}. f i * i) = x * a + y * b + z * c"
 proof (cases "a \<noteq> b \<and> b \<noteq> c \<and> a \<noteq> c")
   case True
   then show ?thesis by (auto, metis add.assoc)
 next
   case False
   have 1: "\<exists>x y z. f c * c = x * c + y * c + z * c"
     by (rule exI[of _ 0],rule exI[of _ 0], rule exI[of _ "f c"], auto)
   have 2: "\<exists>x y z. f b * b + f c * c = x * b + y * b + z * c"
     by (rule exI[of _ 0],rule exI[of _ "f b"], rule exI[of _ "f c"], auto)
   have 3: "\<exists>x y z. f a * a + f c * c = x * a + y * c + z * c"
     by (rule exI[of _ "f a"],rule exI[of _ 0], rule exI[of _ "f c"], auto)
   have 4: "\<exists>x y z. (\<Sum>i\<in>{c, b, c}. f i * i) = x * c + y * b + z * c" if a: "a = c" and b: "b \<noteq> c"
-    by (rule exI[of _ 0],rule exI[of _ "f b"], rule exI[of _ "f c"], insert a b, 
+    by (rule exI[of _ 0],rule exI[of _ "f b"], rule exI[of _ "f c"], insert a b,
         auto simp add: insert_commute)
-  show ?thesis using False 
+  show ?thesis using False
     by (cases "b=c", cases "a=c", auto simp add: 1 2 3 4)
 qed
 
 lemma sum_three_elements':
   shows "\<exists>f::'a\<Rightarrow>'a. (\<Sum>i\<in>{a,b,c}. f i * i) = x * a + y * b + z * c"
 proof (cases "a \<noteq> b \<and> b \<noteq> c \<and> a \<noteq> c")
   case True
   let ?f = "\<lambda>i. if i = a then x else if i = b then y else if i = c then z else 0"
   show ?thesis by (rule exI[of _ "?f"], insert True mult.assoc, auto simp add: local.add_ac)
 next
-  case False  
+  case False
   have 1: "\<exists>f. f c * c = x * c + y * c + z * c"
     by (rule exI[of _ "\<lambda>i. if i = c then x+y+z else 0"], auto simp add: local.ring_distribs)
   have 2: "\<exists>f. f a * a + f c * c = x * a + y * c + z * c" if bc: " b = c" and ac: "a \<noteq> c"
     by (rule exI[of _ "\<lambda>i. if i = a then x else y+z"], insert ac bc add_ac ring_distribs, auto)
   have 3: "\<exists>f. f b * b + f c * c = x * b + y * b + z * c" if bc: " b \<noteq> c" and ac: "a = b"
     by (rule exI[of _ "\<lambda>i. if i = a then x+y else z"], insert ac bc add_ac ring_distribs, auto)
   have 4: "\<exists>f. (\<Sum>i\<in>{c, b, c}. f i * i) = x * c + y * b + z * c" if a: "a = c" and b: "b \<noteq> c"
-    by (rule exI[of _ "\<lambda>i. if i = c then x+z else y"], insert a b add_ac ring_distribs, 
+    by (rule exI[of _ "\<lambda>i. if i = c then x+z else y"], insert a b add_ac ring_distribs,
         auto simp add: insert_commute)
-  show ?thesis using False 
+  show ?thesis using False
     by (cases "b=c", cases "a=c", auto simp add: 1 2 3 4)
 qed
 
 
 (*This is generalizable to arbitrary sets.*)
 lemma ideal_generated_triple_pair_rewrite:
   assumes i1: "ideal_generated {a, b, c} = ideal_generated {d}"
     and i2: "ideal_generated {a, b} = ideal_generated {d'}"
   shows "ideal_generated{d',c} = ideal_generated {d}"
-proof 
+proof
   have d': "d' \<in> ideal_generated {a,b}" using i2 by (simp add: ideal_generated_in)
   show "ideal_generated {d', c} \<subseteq> ideal_generated {d}"
-  proof 
+  proof
     fix x assume x: "x \<in> ideal_generated {d', c}"
     obtain f1 f2 where f: "f1*d' + f2*c = x" using obtain_ideal_generated_pair[OF x] by auto
     obtain g1 g2 where g: "g1*a + g2*b = d'" using obtain_ideal_generated_pair[OF d'] by blast
     have 1: "f1*g1*a + f1*g2*b + f2*c = x"
       using f g local.ring_distribs(1) local.semiring_normalization_rules(18) by auto
     have "x \<in> ideal_generated {a, b, c}"
     proof -
-      obtain f where "(\<Sum>i\<in>{a,b,c}. f i * i) = f1*g1*a + f1*g2*b + f2*c" 
+      obtain f where "(\<Sum>i\<in>{a,b,c}. f i * i) = f1*g1*a + f1*g2*b + f2*c"
         using sum_three_elements' 1 by blast
-      moreover have "ideal_generated {a,b,c} = {y. \<exists>f. (\<Sum>i\<in>{a,b,c}. f i * i) = y}" 
+      moreover have "ideal_generated {a,b,c} = {y. \<exists>f. (\<Sum>i\<in>{a,b,c}. f i * i) = y}"
         using ideal_explicit2[of "{a,b,c}"] by simp
       ultimately show ?thesis using 1 by auto
     qed
     thus "x \<in> ideal_generated {d}" using i1 by auto
   qed
   show "ideal_generated {d} \<subseteq> ideal_generated {d', c}"
   proof (rule ideal_generated_singleton_subset)
     obtain f1 f2 f3 where f: "f1*a + f2*b + f3*c = d"
     proof -
       have "d \<in> ideal_generated {a,b,c}" using i1  by (simp add: ideal_generated_in)
       from this obtain f where d: "(\<Sum>i\<in>{a,b,c}. f i * i) = d"
         using ideal_explicit2[of "{a,b,c}"] by auto
-      obtain x y z where "(\<Sum>i\<in>{a,b,c}. f i * i) = x * a + y * b + z * c" 
+      obtain x y z where "(\<Sum>i\<in>{a,b,c}. f i * i) = x * a + y * b + z * c"
         using sum_three_elements by blast
       thus ?thesis using d that by auto
     qed
-    obtain k where k: "f1*a + f2*b = k*d'" 
+    obtain k where k: "f1*a + f2*b = k*d'"
     proof -
       have "f1*a + f2*b \<in> ideal_generated{a,b}" using ideal_generated_pair by blast
       also have "... = ideal_generated {d'}" using i2 by simp
       also have "... = {k*d' |k. k\<in>UNIV}" using ideal_generated_singleton by auto
       finally show ?thesis using that by auto
     qed
     have "k*d'+f3*c=d" using f k by auto
     thus "d \<in> ideal_generated {d', c}"
       using ideal_generated_pair by blast
   qed (simp)
 qed
 
 lemma ideal_generated_dvd:
   assumes i: "ideal_generated {a,b::'a} = ideal_generated{d} "
   and a: "d' dvd a" and b: "d' dvd b"
 shows "d' dvd d"
 proof -
   obtain p q where "p*a+q*b = d"
     using i ideal_generated_pair_exists by blast
   thus ?thesis using a b by auto
 qed
 
 lemma ideal_generated_dvd2:
   assumes i: "ideal_generated S = ideal_generated{d::'a} "
   and "finite S"
   and x: "\<forall>x\<in>S. d' dvd x"
-shows "d' dvd d"                                                                                   
+shows "d' dvd d"
   by (metis assms dvd_ideal_generated_singleton ideal_generated_dvd_subset)
 
 end
 
 
 subsection \<open>An equivalent characterization of B\'ezout rings\<close>
 
 text \<open>The goal of this subsection is to prove that a ring is B\'ezout ring if and only if every
   finitely generated ideal is principal.\<close>
 
 definition "finitely_generated_ideal I = (ideal I \<and> (\<exists>S. finite S \<and> ideal_generated S = I))"
 
-context 
+context
   assumes "SORT_CONSTRAINT('a::comm_ring_1)"
 begin
 
 
-lemma sum_two_elements': 
+lemma sum_two_elements':
   fixes d::'a
   assumes s: "(\<Sum>i\<in>{a,b}. f i * i) = d"
   obtains p and q where "d = p * a + q * b"
 proof (cases "a=b")
   case True
   then show ?thesis
-    by (metis (no_types, lifting) add_diff_cancel_left' emptyE finite.emptyI insert_absorb2 
+    by (metis (no_types, lifting) add_diff_cancel_left' emptyE finite.emptyI insert_absorb2
         left_diff_distrib' s sum.insert sum_singleton that)
 next
   case False
   show ?thesis using s unfolding sum_two_elements[OF False]
-    using that by auto    
+    using that by auto
 qed
 
 text \<open>This proof follows Theorem 6-3 in "First Course in Rings and Ideals" by Burton\<close>
 
 lemma all_fin_gen_ideals_are_principal_imp_bezout:
   assumes all: "\<forall>I::'a set. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
   shows "OFCLASS ('a, bezout_ring_class)"
-proof (intro_classes)  
+proof (intro_classes)
   fix a b::'a
   obtain d where ideal_d: "ideal_generated {a,b} = ideal_generated {d}"
     using all unfolding finitely_generated_ideal_def
     by (metis finite.emptyI finite_insert ideal_ideal_generated principal_ideal_def)
   have a_in_d: "a \<in> ideal_generated {d}"
+    using ideal_d ideal_generated_subset_generator by blast
+  have b_in_d: "b \<in> ideal_generated {d}"
+    using ideal_d ideal_generated_subset_generator by blast
+  have d_in_ab: "d \<in> ideal_generated {a,b}"
     using ideal_d ideal_generated_subset_generator by auto
-  have b_in_d: "b \<in> ideal_generated {d}" 
-    using ideal_d ideal_generated_subset_generator by auto
-  have d_in_ab: "d \<in> ideal_generated {a,b}" 
-    using ideal_d ideal_generated_subset_generator by auto  
   obtain f where "(\<Sum>i\<in>{a,b}. f i * i) = d" using obtain_sum_ideal_generated[OF d_in_ab] by auto
   from this obtain p q where d_eq: "d = p*a + q*b" using sum_two_elements' by blast
   moreover have d_dvd_a: "d dvd a"
-    by (metis dvd_ideal_generated_singleton ideal_d ideal_generated_subset insert_commute 
+    by (metis dvd_ideal_generated_singleton ideal_d ideal_generated_subset insert_commute
         subset_insertI)
   moreover have "d dvd b"
     by (metis dvd_ideal_generated_singleton ideal_d ideal_generated_subset subset_insertI)
-  moreover have "d' dvd d" if d'_dvd: "d' dvd a \<and> d' dvd b" for d'  
+  moreover have "d' dvd d" if d'_dvd: "d' dvd a \<and> d' dvd b" for d'
   proof -
-    obtain s1 s2 where s1_dvd: "a = s1*d'" and s2_dvd: "b = s2*d'" 
+    obtain s1 s2 where s1_dvd: "a = s1*d'" and s2_dvd: "b = s2*d'"
       using mult.commute d'_dvd unfolding dvd_def by auto
     have "d = p*a + q*b" using d_eq .
     also have "...= p * s1 * d' + q * s2 *d'" unfolding s1_dvd s2_dvd by auto
     also have "... = (p * s1 + q * s2) * d'" by (simp add: ring_class.ring_distribs(2))
     finally show "d' dvd d" using mult.commute unfolding dvd_def by auto
   qed
-  ultimately show "\<exists>p q d. p * a + q * b = d \<and> d dvd a \<and> d dvd b 
+  ultimately show "\<exists>p q d. p * a + q * b = d \<and> d dvd a \<and> d dvd b
   \<and> (\<forall>d'. d' dvd a \<and> d' dvd b \<longrightarrow> d' dvd d)" by auto
 qed
 end
 
 
 context bezout_ring
 begin
 
 lemma exists_bezout_extended:
   assumes S: "finite S" and ne: "S \<noteq> {}"
   shows "\<exists>f d. (\<Sum>a\<in>S. f a * a) = d \<and> (\<forall>a\<in>S. d dvd a) \<and> (\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
   using S ne
 proof (induct S)
   case empty
   then show ?case by auto
 next
   case (insert x S)
-  show ?case 
+  show ?case
   proof (cases  "S={}")
     case True
     let ?f = "\<lambda>x. 1"
     show ?thesis by (rule exI[of _ ?f], insert True, auto)
   next
-    case False note ne = False    
-    note x_notin_S = insert.hyps(2)  
-    obtain f d where sum_eq_d: "(\<Sum>a\<in>S. f a * a) = d" 
-      and d_dvd_each_a: "(\<forall>a\<in>S. d dvd a)" 
+    case False note ne = False
+    note x_notin_S = insert.hyps(2)
+    obtain f d where sum_eq_d: "(\<Sum>a\<in>S. f a * a) = d"
+      and d_dvd_each_a: "(\<forall>a\<in>S. d dvd a)"
       and d_is_gcd: "(\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
-      using insert.hyps(3)[OF ne] by auto  
+      using insert.hyps(3)[OF ne] by auto
     have "\<exists>p q d'. p * d + q * x = d' \<and> d' dvd d \<and> d' dvd x \<and> (\<forall>c. c dvd d \<and> c dvd x \<longrightarrow> c dvd d')"
       using exists_bezout by auto
     from this obtain p q d' where pd_qx_d': "p*d + q*x = d'"
       and d'_dvd_d: "d' dvd d" and d'_dvd_x: "d' dvd x"
-      and d'_dvd: "\<forall>c. (c dvd d \<and> c dvd x) \<longrightarrow> c dvd d'" by blast  
+      and d'_dvd: "\<forall>c. (c dvd d \<and> c dvd x) \<longrightarrow> c dvd d'" by blast
     let ?f = "\<lambda>a. if a = x then q else p * f a"
     have "(\<Sum>a\<in>insert x S. ?f a * a) = d'"
     proof -
       have "(\<Sum>a\<in>insert x S. ?f a * a) = (\<Sum>a\<in>S. ?f a * a) + ?f x * x"
         by (simp add: add_commute insert.hyps(1) insert.hyps(2))
-      also have "... = p * (\<Sum>a\<in>S. f a * a) + q * x" 
+      also have "... = p * (\<Sum>a\<in>S. f a * a) + q * x"
         unfolding sum_distrib_left
-        by (auto, rule sum.cong, insert x_notin_S, 
+        by (auto, rule sum.cong, insert x_notin_S,
             auto simp add: mult.semigroup_axioms semigroup.assoc)
-      finally show ?thesis using pd_qx_d' sum_eq_d by auto    
+      finally show ?thesis using pd_qx_d' sum_eq_d by auto
     qed
     moreover have "(\<forall>a\<in>insert x S. d' dvd a)"
       by (metis d'_dvd_d d'_dvd_x d_dvd_each_a insert_iff local.dvdE local.dvd_mult_left)
     moreover have " (\<forall>c. (\<forall>a\<in>insert x S. c dvd a) \<longrightarrow> c dvd d')"
       by (simp add: d'_dvd d_is_gcd)
     ultimately show ?thesis by auto
   qed
 qed
 
 end
 
 lemma ideal_generated_empty: "ideal_generated {} = {0}"
   unfolding ideal_generated_def using ideal_generated_0
-  by (metis empty_subsetI ideal_generated_def ideal_generated_subset ideal_ideal_generated 
+  by (metis empty_subsetI ideal_generated_def ideal_generated_subset ideal_ideal_generated
       ideal_not_empty subset_singletonD)
 
 
 lemma bezout_imp_all_fin_gen_ideals_are_principal:
   fixes I::"'a :: bezout_ring set"
   assumes fin: "finitely_generated_ideal I"
   shows "principal_ideal I"
 proof -
   obtain S where fin_S: "finite S" and ideal_gen_S: "ideal_generated S = I"
     using fin unfolding finitely_generated_ideal_def by auto
-  show ?thesis 
+  show ?thesis
   proof (cases "S = {}")
     case True
-    then show ?thesis 
-      using ideal_gen_S unfolding True 
+    then show ?thesis
+      using ideal_gen_S unfolding True
       using ideal_generated_empty ideal_generated_0 principal_ideal_def by fastforce
   next
     case False note ne = False
     obtain d f where sum_S_d: "(\<Sum>i\<in>S. f i * i) = d"
     and d_dvd_a: "(\<forall>a\<in>S. d dvd a)" and d_is_gcd: "(\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
       using exists_bezout_extended[OF fin_S ne] by auto
     have d_in_S: "d \<in> ideal_generated S"
-      by (metis fin_S ideal_def ideal_generated_subset_generator 
+      by (metis fin_S ideal_def ideal_generated_subset_generator
           ideal_ideal_generated sum_S_d sum_left_ideal)
-    have "ideal_generated {d} \<subseteq> ideal_generated S" 
-      by (rule ideal_generated_singleton_subset[OF d_in_S fin_S])  
+    have "ideal_generated {d} \<subseteq> ideal_generated S"
+      by (rule ideal_generated_singleton_subset[OF d_in_S fin_S])
     moreover have "ideal_generated S \<subseteq> ideal_generated {d}"
-    proof 
+    proof
       fix x assume x_in_S: "x \<in> ideal_generated S"
       obtain f where sum_S_x: "(\<Sum>a\<in>S. f a * a) = x"
         using fin_S obtain_sum_ideal_generated x_in_S by blast
       have d_dvd_each_a: "\<exists>k. a = k * d" if "a \<in> S" for a
         by (metis d_dvd_a dvdE mult.commute that)
       let ?g = "\<lambda>a. SOME k. a = k*d"
       have "x = (\<Sum>a\<in>S. f a * a)" using sum_S_x by simp
-      also have "... = (\<Sum>a\<in>S. f a * (?g a * d))" 
+      also have "... = (\<Sum>a\<in>S. f a * (?g a * d))"
       proof (rule sum.cong)
-        fix a assume a_in_S: "a \<in> S" 
+        fix a assume a_in_S: "a \<in> S"
         obtain k where a_kd: "a = k * d" using d_dvd_each_a a_in_S by auto
-        have "a = ((SOME k. a = k * d) * d)" by (rule someI_ex, auto simp add: a_kd)        
+        have "a = ((SOME k. a = k * d) * d)" by (rule someI_ex, auto simp add: a_kd)
         thus "f a * a = f a * ((SOME k. a = k * d) * d)" by auto
       qed (simp)
-      also have "... = (\<Sum>a\<in>S. f a * ?g a * d)" by (rule sum.cong, auto)     
+      also have "... = (\<Sum>a\<in>S. f a * ?g a * d)" by (rule sum.cong, auto)
       also have "... = (\<Sum>a\<in>S. f a * ?g a)*d" using sum_distrib_right[of _ S d] by auto
       finally show "x \<in> ideal_generated {d}"
-        by (meson contra_subsetD dvd_ideal_generated_singleton' dvd_triv_right 
+        by (meson contra_subsetD dvd_ideal_generated_singleton' dvd_triv_right
             ideal_generated_in singletonI)
     qed
     ultimately show ?thesis unfolding principal_ideal_def using ideal_gen_S by auto
   qed
 qed
 
-text \<open>Now we have the required lemmas to prove the theorem that states that 
+text \<open>Now we have the required lemmas to prove the theorem that states that
   a ring is B\'ezout ring if and only if every
   finitely generated ideal is principal. They are the following ones.
 
 \begin{itemize}
 \item @{text "all_fin_gen_ideals_are_principal_imp_bezout"}
 \item @{text "bezout_imp_all_fin_gen_ideals_are_principal"}
 \end{itemize}
 
-However, in order to prove the final lemma, we need the lemmas with no type restrictions. 
+However, in order to prove the final lemma, we need the lemmas with no type restrictions.
 For instance, we need a version of theorem @{text "bezout_imp_all_fin_gen_ideals_are_principal"}
 as
 
-@{text "OFCLASS('a,bezout_ring) \<Longrightarrow>"} the theorem with generic types 
+@{text "OFCLASS('a,bezout_ring) \<Longrightarrow>"} the theorem with generic types
   (i.e., @{text "'a"} with no type restrictions)
-  
+
 
 or as
 
-@{text "class.bezout_ring _ _ _ _ \<Longrightarrow>"} the theorem with generic 
+@{text "class.bezout_ring _ _ _ _ \<Longrightarrow>"} the theorem with generic
   types (i.e., @{text "'a"} with no type restrictions)
 \<close>
 
 (*A possible workaround is to adapt the proof*)
 (*
 lemma bezout_imp_all_fin_gen_ideals_are_principal_unsatisfactory:
   assumes a1: "class.bezout_ring ( * ) (1::'a::comm_ring_1) (+) 0 (-) uminus" (*Me da igual esto que OFCLASS*)
   shows "\<forall>I::'a set. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
 proof (rule allI, rule impI)
   fix I::"'a set" assume fin: "finitely_generated_ideal I"
   interpret a: bezout_ring "( * )" "(1::'a)" "(+)" 0 "(-)" uminus using a1 .
   interpret dvd "( * )::'a\<Rightarrow>'a\<Rightarrow>'a" .
   interpret b: comm_monoid_add "(+)" "(0::'a)" using a1 by intro_locales
   have c: " class.comm_monoid_add (+) (0::'a)"  using a1 by intro_locales
-  have [simp]: "(dvd.dvd ( * ) d a) = (d dvd a)" for d a::'a 
+  have [simp]: "(dvd.dvd ( * ) d a) = (d dvd a)" for d a::'a
     by (auto simp add: dvd.dvd_def dvd_def)
-  have [simp]: "comm_monoid_add.sum (+) 0 (\<lambda>a. f a * a) S = sum (\<lambda>a. f a * a) S" 
-    for f and S::"'a set"     
-    unfolding sum_def unfolding comm_monoid_add.sum_def[OF c] ..        
+  have [simp]: "comm_monoid_add.sum (+) 0 (\<lambda>a. f a * a) S = sum (\<lambda>a. f a * a) S"
+    for f and S::"'a set"
+    unfolding sum_def unfolding comm_monoid_add.sum_def[OF c] ..
   obtain S where fin_S: "finite S" and ideal_gen_S: "ideal_generated S = I"
     using fin unfolding finitely_generated_ideal_def by auto
   show "principal_ideal I"
   proof (cases "S = {}")
     case True
-    then show ?thesis 
-      using ideal_gen_S unfolding True 
+    then show ?thesis
+      using ideal_gen_S unfolding True
       using ideal_generated_empty ideal_generated_0 principal_ideal_def by fastforce
   next
     case False note ne = False
     obtain d f where sum_S_d: "(\<Sum>i\<in>S. f i * i) = d"
     and d_dvd_a: "(\<forall>a\<in>S. d dvd a)" and d_is_gcd: "(\<forall>d'. (\<forall>a\<in>S. d' dvd a) \<longrightarrow> d' dvd d)"
       using a.exists_bezout_extended[OF fin_S ne] by auto
     have d_in_S: "d \<in> ideal_generated S"
-      by (metis fin_S ideal_def ideal_generated_subset_generator 
+      by (metis fin_S ideal_def ideal_generated_subset_generator
           ideal_ideal_generated sum_S_d sum_left_ideal)
-    have "ideal_generated {d} \<subseteq> ideal_generated S" 
-      by (rule ideal_generated_singleton_subset[OF d_in_S fin_S])  
+    have "ideal_generated {d} \<subseteq> ideal_generated S"
+      by (rule ideal_generated_singleton_subset[OF d_in_S fin_S])
     moreover have "ideal_generated S \<subseteq> ideal_generated {d}"
-    proof 
+    proof
       fix x assume x_in_S: "x \<in> ideal_generated S"
       obtain f where sum_S_x: "(\<Sum>a\<in>S. f a * a) = x"
         using fin_S obtain_sum_ideal_generated x_in_S by blast
       have d_dvd_each_a: "\<exists>k. a = k * d" if "a \<in> S" for a
         by (metis d_dvd_a dvdE mult.commute that)
       let ?g = "\<lambda>a. SOME k. a = k*d"
       have "x = (\<Sum>a\<in>S. f a * a)" using sum_S_x by simp
-      also have "... = (\<Sum>a\<in>S. f a * (?g a * d))" 
+      also have "... = (\<Sum>a\<in>S. f a * (?g a * d))"
       proof (rule sum.cong)
-        fix a assume a_in_S: "a \<in> S" 
+        fix a assume a_in_S: "a \<in> S"
         obtain k where a_kd: "a = k * d" using d_dvd_each_a a_in_S by auto
-        have "a = ((SOME k. a = k * d) * d)" by (rule someI_ex, auto simp add: a_kd)        
+        have "a = ((SOME k. a = k * d) * d)" by (rule someI_ex, auto simp add: a_kd)
         thus "f a * a = f a * ((SOME k. a = k * d) * d)" by auto
       qed (simp)
-      also have "... = (\<Sum>a\<in>S. f a * ?g a * d)" by (rule sum.cong, auto)     
+      also have "... = (\<Sum>a\<in>S. f a * ?g a * d)" by (rule sum.cong, auto)
       also have "... = (\<Sum>a\<in>S. f a * ?g a)*d" using sum_distrib_right[of _ S d] by auto
       finally show "x \<in> ideal_generated {d}"
-        by (meson contra_subsetD dvd_ideal_generated_singleton' dvd_triv_right 
+        by (meson contra_subsetD dvd_ideal_generated_singleton' dvd_triv_right
             ideal_generated_in singletonI)
     qed
     ultimately show ?thesis unfolding principal_ideal_def using ideal_gen_S by auto
   qed
 qed
 *)
 
-text \<open>Thanks to local type definitions, we can obtain it automatically by means 
+text \<open>Thanks to local type definitions, we can obtain it automatically by means
   of @{text "internalize-sort"}.\<close>
 
 lemma bezout_imp_all_fin_gen_ideals_are_principal_unsatisfactory:
   assumes a1: "class.bezout_ring (*) (1::'b::comm_ring_1) (+) 0 (-) uminus" (*It is algo possible to prove it using OFCLASS*)
   shows "\<forall>I::'b set. finitely_generated_ideal I \<longrightarrow> principal_ideal I"
-  using bezout_imp_all_fin_gen_ideals_are_principal[internalize_sort "'a::bezout_ring"] 
+  using bezout_imp_all_fin_gen_ideals_are_principal[internalize_sort "'a::bezout_ring"]
   using a1 by auto
 
 
-text \<open>The standard library does not connect @{text "OFCLASS"} and @{text "class.bezout_ring"} 
+text \<open>The standard library does not connect @{text "OFCLASS"} and @{text "class.bezout_ring"}
 in both directions. Here we show that @{text "OFCLASS \<Longrightarrow> class.bezout_ring"}. \<close>
 
 lemma OFCLASS_bezout_ring_imp_class_bezout_ring:
   assumes "OFCLASS('a::comm_ring_1,bezout_ring_class)"
   shows "class.bezout_ring ((*)::'a\<Rightarrow>'a\<Rightarrow>'a) 1 (+) 0 (-) uminus"
   using assms
-  unfolding bezout_ring_class_def class.bezout_ring_def 
-  using conjunctionD2[of "OFCLASS('a, comm_ring_1_class)" 
+  unfolding bezout_ring_class_def class.bezout_ring_def
+  using conjunctionD2[of "OFCLASS('a, comm_ring_1_class)"
                          "class.bezout_ring_axioms ((*)::'a\<Rightarrow>'a\<Rightarrow>'a) (+)"]
   by (auto, intro_locales)
 
-text \<open>The other implication can be obtained 
+text \<open>The other implication can be obtained
   by thm @{text "Rings2.class.Rings2.bezout_ring.of_class.intro"} \<close>
 thm Rings2.class.Rings2.bezout_ring.of_class.intro
 
 
 (*OFCLASS is a proposition (Prop), and then the following statement is not valid.*)
 
 (*
 lemma
-  shows "(\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I) 
+  shows "(\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I)
     = OFCLASS('a, bezout_ring_class)"
 *)
 
 (*Thus, we use the meta-equality and the meta universal quantifier.*)
 text \<open>Final theorem (with OFCLASS)\<close>
 lemma bezout_ring_iff_fin_gen_principal_ideal:
-    "(\<And>I::'a::comm_ring_1 set. finitely_generated_ideal I \<Longrightarrow> principal_ideal I) 
-    \<equiv> OFCLASS('a, bezout_ring_class)"  
+    "(\<And>I::'a::comm_ring_1 set. finitely_generated_ideal I \<Longrightarrow> principal_ideal I)
+    \<equiv> OFCLASS('a, bezout_ring_class)"
 proof
-  show "(\<And>I::'a::comm_ring_1 set. finitely_generated_ideal I \<Longrightarrow> principal_ideal I) 
+  show "(\<And>I::'a::comm_ring_1 set. finitely_generated_ideal I \<Longrightarrow> principal_ideal I)
     \<Longrightarrow> OFCLASS('a, bezout_ring_class)"
     using all_fin_gen_ideals_are_principal_imp_bezout [where ?'a='a] by auto
-  show "\<And>I::'a::comm_ring_1 set. OFCLASS('a, bezout_ring_class) 
+  show "\<And>I::'a::comm_ring_1 set. OFCLASS('a, bezout_ring_class)
     \<Longrightarrow> finitely_generated_ideal I \<Longrightarrow> principal_ideal I"
     using bezout_imp_all_fin_gen_ideals_are_principal_unsatisfactory[where ?'b='a]
     using OFCLASS_bezout_ring_imp_class_bezout_ring[where ?'a='a] by auto
 qed
 
 text \<open>Final theorem (with @{text "class.bezout_ring"})\<close>
 
 lemma bezout_ring_iff_fin_gen_principal_ideal2:
-    "(\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I) 
-    = (class.bezout_ring ((*)::'a\<Rightarrow>'a\<Rightarrow>'a) 1 (+) 0 (-) uminus)"  
+    "(\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I)
+    = (class.bezout_ring ((*)::'a\<Rightarrow>'a\<Rightarrow>'a) 1 (+) 0 (-) uminus)"
 proof
-  show "\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I 
+  show "\<forall>I::'a::comm_ring_1 set. finitely_generated_ideal I \<longrightarrow> principal_ideal I
       \<Longrightarrow> class.bezout_ring (*) 1 (+) (0::'a) (-) uminus"
-    using all_fin_gen_ideals_are_principal_imp_bezout[where ?'a='a] 
+    using all_fin_gen_ideals_are_principal_imp_bezout[where ?'a='a]
     using OFCLASS_bezout_ring_imp_class_bezout_ring[where ?'a='a]
     by auto
-  show "class.bezout_ring (*) 1 (+) (0::'a) (-) uminus \<Longrightarrow> \<forall>I::'a set. 
-    finitely_generated_ideal I \<longrightarrow> principal_ideal I"  
+  show "class.bezout_ring (*) 1 (+) (0::'a) (-) uminus \<Longrightarrow> \<forall>I::'a set.
+    finitely_generated_ideal I \<longrightarrow> principal_ideal I"
     using bezout_imp_all_fin_gen_ideals_are_principal_unsatisfactory by auto
 qed
 
 end
diff --git a/thys/Smith_Normal_Form/SNF_Missing_Lemmas.thy b/thys/Smith_Normal_Form/SNF_Missing_Lemmas.thy
--- a/thys/Smith_Normal_Form/SNF_Missing_Lemmas.thy
+++ b/thys/Smith_Normal_Form/SNF_Missing_Lemmas.thy
@@ -1,1168 +1,1168 @@
 (*
   Author: Jose Divasón
   Email:  jose.divason@unirioja.es
 *)
 
 section \<open>Missing results\<close>
 
 theory SNF_Missing_Lemmas
   imports
     Hermite.Hermite
-    Mod_Type_Connect    
+    Mod_Type_Connect
     Jordan_Normal_Form.DL_Rank_Submatrix
     "List-Index.List_Index"
 begin
 
-text \<open>This theory presents some missing lemmas that are required for the Smith normal form 
-development. Some of them could be added to different AFP entries, such as the Jordan Normal 
+text \<open>This theory presents some missing lemmas that are required for the Smith normal form
+development. Some of them could be added to different AFP entries, such as the Jordan Normal
 Form AFP entry by Ren\'e Thiemann and Akihisa Yamada.
 
 However, not all the lemmas can be added directly, since some imports are required.\<close>
 
 hide_const (open) C
 hide_const (open) measure
 
 subsection \<open>Miscellaneous lemmas\<close>
 
 lemma sum_two_rw: "(\<Sum>i = 0..<2. f i) = (\<Sum>i \<in> {0,1::nat}. f i)"
   by (rule sum.cong, auto)
 
 lemma sum_common_left:
   fixes f::"'a \<Rightarrow> 'b::comm_ring_1"
   assumes "finite A"
-  shows "sum (\<lambda>i. c * f i) A = c * sum f A" 
+  shows "sum (\<lambda>i. c * f i) A = c * sum f A"
   by (simp add: mult_hom.hom_sum)
 
-lemma prod3_intro: 
+lemma prod3_intro:
   assumes "fst A = a" and "fst (snd A) = b" and "snd (snd A) = c"
   shows "A = (a,b,c)" using assms by auto
 
 
 subsection \<open>Transfer rules for the HMA\_Connect file of the Perron-Frobenius development\<close>
 
 hide_const (open) HMA_M HMA_I to_hma\<^sub>m from_hma\<^sub>m
-hide_fact (open) from_hma\<^sub>m_def from_hma_to_hma\<^sub>m HMA_M_def HMA_I_def dim_row_transfer_rule 
+hide_fact (open) from_hma\<^sub>m_def from_hma_to_hma\<^sub>m HMA_M_def HMA_I_def dim_row_transfer_rule
                   dim_col_transfer_rule
 
-context 
+context
   includes lifting_syntax
 begin
 
-lemma HMA_invertible_matrix[transfer_rule]: 
-  "((HMA_Connect.HMA_M :: _ \<Rightarrow> 'a :: comm_ring_1 ^ 'n ^ 'n \<Rightarrow> _) ===> (=)) invertible_mat invertible" 
+lemma HMA_invertible_matrix[transfer_rule]:
+  "((HMA_Connect.HMA_M :: _ \<Rightarrow> 'a :: comm_ring_1 ^ 'n ^ 'n \<Rightarrow> _) ===> (=)) invertible_mat invertible"
 proof (intro rel_funI, goal_cases)
   case (1 x y)
   note rel_xy[transfer_rule] = "1"
-  have eq_dim: "dim_col x = dim_row x" 
-    using HMA_Connect.dim_col_transfer_rule HMA_Connect.dim_row_transfer_rule rel_xy 
+  have eq_dim: "dim_col x = dim_row x"
+    using HMA_Connect.dim_col_transfer_rule HMA_Connect.dim_row_transfer_rule rel_xy
     by fastforce
-  moreover have "\<exists>A'. y ** A' = Finite_Cartesian_Product.mat 1 \<and> A' ** y = Finite_Cartesian_Product.mat 1" 
+  moreover have "\<exists>A'. y ** A' = Finite_Cartesian_Product.mat 1 \<and> A' ** y = Finite_Cartesian_Product.mat 1"
     if xB: "x * B = 1\<^sub>m (dim_row x)" and Bx: "B * x = 1\<^sub>m (dim_row B)" for B
   proof -
-    let ?A' = "HMA_Connect.to_hma\<^sub>m B:: 'a :: comm_ring_1 ^ 'n ^ 'n"       
+    let ?A' = "HMA_Connect.to_hma\<^sub>m B:: 'a :: comm_ring_1 ^ 'n ^ 'n"
     have rel_BA[transfer_rule]: "HMA_M B ?A'"
       by (metis (no_types, lifting) Bx HMA_M_def eq_dim carrier_mat_triv dim_col_mat(1)
           from_hma\<^sub>m_def from_hma_to_hma\<^sub>m index_mult_mat(3) index_one_mat(3) rel_xy xB)
     have [simp]: "dim_row B = CARD('n)" using dim_row_transfer_rule rel_BA by blast
     have [simp]: "dim_row x = CARD('n)" using dim_row_transfer_rule rel_xy by blast
     have "y ** ?A' = Finite_Cartesian_Product.mat 1" using xB by (transfer, simp)
     moreover have "?A' ** y  = Finite_Cartesian_Product.mat 1" using Bx by (transfer, simp)
     ultimately show ?thesis by blast
   qed
   moreover have "\<exists>B. x * B = 1\<^sub>m (dim_row x) \<and> B * x = 1\<^sub>m (dim_row B)"
     if yA: "y ** A' = Finite_Cartesian_Product.mat 1" and Ay: "A' ** y = Finite_Cartesian_Product.mat 1" for A'
   proof -
     let ?B = "(from_hma\<^sub>m A')"
     have [simp]: "dim_row x = CARD('n)" using dim_row_transfer_rule rel_xy by blast
     have [transfer_rule]: "HMA_M ?B A'" by (simp add: HMA_M_def)
     hence [simp]: "dim_row ?B = CARD('n)" using dim_row_transfer_rule by auto
     have "x * ?B = 1\<^sub>m (dim_row x)" using yA by (transfer', auto)
     moreover have "?B * x = 1\<^sub>m (dim_row ?B)" using Ay by (transfer', auto)
     ultimately show ?thesis by auto
   qed
   ultimately show ?case unfolding invertible_mat_def invertible_def inverts_mat_def by auto
 qed
 end
 
 subsection \<open>Lemmas obtained from HOL Analysis using local type definitions\<close>
 
 thm Cartesian_Space.invertible_mult (*In HOL Analysis*)
-thm invertible_iff_is_unit (*In HOL Analysis*) 
+thm invertible_iff_is_unit (*In HOL Analysis*)
 thm det_non_zero_imp_unit (*In JNF, but only for fields*)
 thm mat_mult_left_right_inverse (*In JNF, but only for fields*)
-      
+
 lemma invertible_mat_zero:
   assumes A: "A \<in> carrier_mat 0 0"
-  shows "invertible_mat A"  
+  shows "invertible_mat A"
   using A unfolding invertible_mat_def inverts_mat_def one_mat_def times_mat_def scalar_prod_def
     Matrix.row_def col_def carrier_mat_def
   by (auto, metis (no_types, lifting) cong_mat not_less_zero)
 
 lemma invertible_mult_JNF:
   fixes A::"'a::comm_ring_1 mat"
   assumes A: "A\<in>carrier_mat n n" and B: "B\<in>carrier_mat n n"
-    and inv_A: "invertible_mat A" and inv_B: "invertible_mat B" 
+    and inv_A: "invertible_mat A" and inv_B: "invertible_mat B"
 shows "invertible_mat (A*B)"
 proof (cases "n = 0")
   case True
   then show ?thesis using assms
     by (simp add: invertible_mat_zero)
 next
   case False
-  then show ?thesis using 
-      invertible_mult[where ?'a="'a::comm_ring_1", where ?'b="'n::finite", where ?'c="'n::finite", 
+  then show ?thesis using
+      invertible_mult[where ?'a="'a::comm_ring_1", where ?'b="'n::finite", where ?'c="'n::finite",
       where ?'d="'n::finite", untransferred, cancel_card_constraint, OF assms] by auto
 qed
 
 lemma invertible_iff_is_unit_JNF:
   assumes A: "A \<in> carrier_mat n n"
   shows "invertible_mat A \<longleftrightarrow> (Determinant.det A) dvd 1"
 proof (cases "n=0")
   case True
   then show ?thesis using det_dim_zero invertible_mat_zero A by auto
 next
   case False
   then show ?thesis using invertible_iff_is_unit[untransferred, cancel_card_constraint] A by auto
 qed
 
 
 subsection \<open>Lemmas about matrices, submatrices and determinants\<close>
 
 (*This is a generalization of thm mat_mult_left_right_inverse*)
 thm mat_mult_left_right_inverse
 lemma mat_mult_left_right_inverse:
   fixes A :: "'a::comm_ring_1 mat"
   assumes A: "A \<in> carrier_mat n n"
     and B: "B \<in> carrier_mat n n" and AB: "A * B = 1\<^sub>m n"
   shows "B * A = 1\<^sub>m n"
 proof -
   have "Determinant.det (A * B) = Determinant.det (1\<^sub>m n)" using AB by auto
-  hence "Determinant.det A * Determinant.det B = 1" 
+  hence "Determinant.det A * Determinant.det B = 1"
     using Determinant.det_mult[OF A B] det_one by auto
   hence det_A: "(Determinant.det A) dvd 1" and det_B: "(Determinant.det B) dvd 1"
     using dvd_triv_left dvd_triv_right by metis+
   hence inv_A: "invertible_mat A" and inv_B: "invertible_mat B"
     using A B invertible_iff_is_unit_JNF by blast+
-  obtain B' where inv_BB': "inverts_mat B B'" and inv_B'B: "inverts_mat B' B" 
+  obtain B' where inv_BB': "inverts_mat B B'" and inv_B'B: "inverts_mat B' B"
     using inv_B unfolding invertible_mat_def by auto
-  have B'_carrier: "B' \<in> carrier_mat n n" 
-    by (metis B inv_B'B inv_BB' carrier_matD(1) carrier_matD(2) carrier_mat_triv 
+  have B'_carrier: "B' \<in> carrier_mat n n"
+    by (metis B inv_B'B inv_BB' carrier_matD(1) carrier_matD(2) carrier_mat_triv
         index_mult_mat(3) index_one_mat(3) inverts_mat_def)
   have "B * A * B = B" using A AB B by auto
   hence "B * A * (B * B') = B * B'"
     by (smt A AB B B'_carrier assoc_mult_mat carrier_matD(1) inv_BB' inverts_mat_def one_carrier_mat)
   thus ?thesis
-    by (metis A B carrier_matD(1) carrier_matD(2) index_mult_mat(3) inv_BB' 
+    by (metis A B carrier_matD(1) carrier_matD(2) index_mult_mat(3) inv_BB'
         inverts_mat_def right_mult_one_mat')
 qed
 
 context comm_ring_1
 begin
 
 lemma col_submatrix_UNIV:
 assumes "j < card {i. i < dim_col A \<and> i \<in> J}"
 shows "col (submatrix A UNIV J) j = col A (pick J j)"
 proof (rule eq_vecI)
   show dim_eq:"dim_vec (col (submatrix A UNIV J) j) = dim_vec (col A (pick J j))"
     by (simp add: dim_submatrix(1))
   fix i assume "i < dim_vec (col A (pick J j))"
   show "col (submatrix A UNIV J) j $v i = col A (pick J j) $v i"
-    by (smt Collect_cong assms col_def dim_col dim_eq dim_submatrix(1) 
+    by (smt Collect_cong assms col_def dim_col dim_eq dim_submatrix(1)
         eq_vecI index_vec pick_UNIV submatrix_index)
 qed
 
 lemma submatrix_split2: "submatrix A I J = submatrix (submatrix A I UNIV) UNIV J" (is "?lhs = ?rhs")
 proof (rule eq_matI)
   show dr: "dim_row ?lhs = dim_row ?rhs"
     by (simp add: dim_submatrix(1))
   show dc: "dim_col ?lhs = dim_col ?rhs"
     by (simp add: dim_submatrix(2))
   fix i j assume i: "i < dim_row ?rhs"
-    and j: "j < dim_col ?rhs" 
-  have "?rhs $$ (i, j) = (submatrix A I UNIV) $$ (pick UNIV i, pick J j)" 
+    and j: "j < dim_col ?rhs"
+  have "?rhs $$ (i, j) = (submatrix A I UNIV) $$ (pick UNIV i, pick J j)"
   proof (rule submatrix_index)
-    show "i < card {i. i < dim_row (submatrix A I UNIV) \<and> i \<in> UNIV}" 
+    show "i < card {i. i < dim_row (submatrix A I UNIV) \<and> i \<in> UNIV}"
       by (metis (full_types) dim_submatrix(1) i)
     show "j < card {j. j < dim_col (submatrix A I UNIV) \<and> j \<in> J}"
       by (metis (full_types) dim_submatrix(2) j)
   qed
   also have "... = A $$ (pick I (pick UNIV i), pick UNIV (pick J j))"
   proof (rule submatrix_index)
     show "pick UNIV i < card {i. i < dim_row A \<and> i \<in> I}"
       by (metis (full_types) dr dim_submatrix(1) i pick_UNIV)
     show "pick J j < card {j. j < dim_col A \<and> j \<in> UNIV}"
       by (metis (full_types) dim_submatrix(2) j pick_le)
   qed
-  also have "... = ?lhs $$ (i,j)" 
+  also have "... = ?lhs $$ (i,j)"
   proof (unfold pick_UNIV, rule submatrix_index[symmetric])
     show "i < card {i. i < dim_row A \<and> i \<in> I}"
       by (metis (full_types) dim_submatrix(1) dr i)
     show "j < card {j. j < dim_col A \<and> j \<in> J}"
       by (metis (full_types) dim_submatrix(2) dc j)
   qed
   finally show "?lhs $$ (i, j) = ?rhs $$ (i, j)" ..
 qed
 
 lemma submatrix_mult:
   "submatrix (A*B) I J = submatrix A I UNIV * submatrix B UNIV J" (is "?lhs = ?rhs")
 proof (rule eq_matI)
   show "dim_row ?lhs = dim_row ?rhs" unfolding submatrix_def by auto
   show "dim_col ?lhs = dim_col ?rhs" unfolding submatrix_def by auto
   fix i j assume i: "i < dim_row ?rhs" and j: "j < dim_col ?rhs"
   have i1: "i < card {i. i < dim_row (A * B) \<and> i \<in> I}"
     by (metis (full_types) dim_submatrix(1) i index_mult_mat(2))
   have j1: "j < card {j. j < dim_col (A * B) \<and> j \<in> J}"
     by (metis dim_submatrix(2) index_mult_mat(3) j)
   have pi: "pick I i < dim_row A" using i1 pick_le by auto
   have pj: "pick J j < dim_col B" using j1 pick_le by auto
   have row_rw: "Matrix.row (submatrix A I UNIV) i = Matrix.row A (pick I i)"
     using i1 row_submatrix_UNIV by auto
   have col_rw: "col (submatrix B UNIV J) j = col B (pick J j)" using j1 col_submatrix_UNIV by auto
   have "?lhs $$ (i,j) =  (A*B) $$ (pick I i, pick J j)" by (rule submatrix_index[OF i1 j1])
   also have "... = Matrix.row A (pick I i) \<bullet> col B (pick J j)" by (rule index_mult_mat(1)[OF pi pj])
-  also have "... = Matrix.row (submatrix A I UNIV) i \<bullet> col (submatrix B UNIV J) j" 
-    using row_rw col_rw by simp  
+  also have "... = Matrix.row (submatrix A I UNIV) i \<bullet> col (submatrix B UNIV J) j"
+    using row_rw col_rw by simp
   also have "... = (?rhs) $$ (i,j)" by (rule index_mult_mat[symmetric], insert i j, auto)
   finally show "?lhs $$ (i, j) = ?rhs $$ (i, j)" .
 qed
 
 lemma det_singleton:
   assumes A: "A \<in> carrier_mat 1 1"
-  shows "det A = A $$ (0,0)" 
+  shows "det A = A $$ (0,0)"
   using A unfolding carrier_mat_def Determinant.det_def by auto
 
 lemma submatrix_singleton_index:
   assumes A: "A \<in> carrier_mat n m"
     and an: "a < n" and bm: "b < m"
     shows "submatrix A {a} {b} $$ (0,0) = A $$ (a,b)"
 proof -
   have a: "{i. i = a \<and> i < dim_row A} = {a}" using an A unfolding carrier_mat_def by auto
   have b: "{i. i = b \<and> i < dim_col A} = {b}" using bm A unfolding carrier_mat_def by auto
   have "submatrix A {a} {b} $$ (0,0) = A $$ (pick {a} 0,pick {b} 0)"
     by (rule submatrix_index, insert a b, auto)
   moreover have "pick {a} 0 = a" by (auto, metis (full_types) LeastI)
   moreover have "pick {b} 0 = b" by (auto, metis (full_types) LeastI)
   ultimately show ?thesis by simp
 qed
 end
 
 lemma det_not_inj_on:
   assumes not_inj_on: "\<not> inj_on f {0..<n}"
   shows "det (mat\<^sub>r n n (\<lambda>i. Matrix.row B (f i))) = 0"
 proof -
-  obtain i j where i: "i<n" and j: "j<n" and fi_fj: "f i = f j" and ij: "i\<noteq>j" 
-    using not_inj_on unfolding inj_on_def by auto  
-  show ?thesis 
+  obtain i j where i: "i<n" and j: "j<n" and fi_fj: "f i = f j" and ij: "i\<noteq>j"
+    using not_inj_on unfolding inj_on_def by auto
+  show ?thesis
   proof (rule det_identical_rows[OF _ ij i j])
     let ?B="(mat\<^sub>r n n (\<lambda>i. row B (f i)))"
-    show "row ?B i = row ?B j" 
+    show "row ?B i = row ?B j"
     proof (rule eq_vecI, auto)
-      fix ia assume ia: "ia < n" 
+      fix ia assume ia: "ia < n"
       have "row ?B i $ ia = ?B $$ (i, ia)" by (rule index_row(1), insert i ia, auto)
       also have "... = ?B $$ (j, ia)" by (simp add: fi_fj i ia j)
       also have "... = row ?B j $ ia" by (rule index_row(1)[symmetric], insert j ia, auto)
       finally show "row ?B i $ ia = row (mat\<^sub>r n n (\<lambda>i. row B (f i))) j $ ia" by simp
     qed
     show "mat\<^sub>r n n (\<lambda>i. Matrix.row B (f i)) \<in> carrier_mat n n" by auto
   qed
 qed
 
 
 
 lemma mat_row_transpose: "(mat\<^sub>r nr nc f)\<^sup>T = mat nc nr (\<lambda>(i,j). vec_index (f j) i)"
-  by (rule eq_matI, auto) 
+  by (rule eq_matI, auto)
 
 
 lemma obtain_inverse_matrix:
-  assumes A: "A \<in> carrier_mat n n" and i: "invertible_mat A" 
+  assumes A: "A \<in> carrier_mat n n" and i: "invertible_mat A"
   obtains B where "inverts_mat A B" and "inverts_mat B A" and "B \<in> carrier_mat n n"
 proof -
   have "(\<exists>B. inverts_mat A B \<and> inverts_mat B A)" using i unfolding invertible_mat_def by auto
   from this obtain B where AB: "inverts_mat A B" and BA: "inverts_mat B A" by auto
   moreover have "B \<in> carrier_mat n n" using A AB BA unfolding carrier_mat_def inverts_mat_def
     by (auto, metis index_mult_mat(3) index_one_mat(3))+
-  ultimately show ?thesis using that by blast  
+  ultimately show ?thesis using that by blast
 qed
 
 
 lemma invertible_mat_smult_mat:
   fixes A :: "'a::comm_ring_1 mat"
   assumes inv_A: "invertible_mat A" and k: "k dvd 1"
-  shows "invertible_mat (k \<cdot>\<^sub>m A)" 
+  shows "invertible_mat (k \<cdot>\<^sub>m A)"
 proof -
   obtain n where A: "A \<in> carrier_mat n n" using inv_A unfolding invertible_mat_def by auto
   have det_dvd_1: "Determinant.det A dvd 1" using inv_A invertible_iff_is_unit_JNF[OF A] by auto
   have "Determinant.det (k \<cdot>\<^sub>m A) = k ^ dim_col A * Determinant.det A" by simp
   also have "... dvd 1" by (rule unit_prod, insert k det_dvd_1 dvd_power_same, force+)
   finally show ?thesis using invertible_iff_is_unit_JNF by (metis A smult_carrier_mat)
 qed
 
-lemma invertible_mat_one[simp]: "invertible_mat (1\<^sub>m n)" 
+lemma invertible_mat_one[simp]: "invertible_mat (1\<^sub>m n)"
   unfolding invertible_mat_def using inverts_mat_def by fastforce
 
 lemma four_block_mat_dim0:
   assumes A: "A \<in> carrier_mat n n"
   and B: "B \<in> carrier_mat n 0"
   and C: "C \<in> carrier_mat 0 n"
   and D: "D \<in> carrier_mat 0 0"
 shows "four_block_mat A B C D = A"
   unfolding four_block_mat_def using assms by auto
 
 
 lemma det_four_block_mat_lower_right_id:
   assumes A: "A \<in> carrier_mat m m"
 and B: "B = 0\<^sub>m m (n-m)"
 and C: "C = 0\<^sub>m (n-m) m"
 and D: "D = 1\<^sub>m (n-m)"
 and "n>m"
 shows "Determinant.det (four_block_mat A B C D) = Determinant.det A"
   using assms
 proof (induct n arbitrary: A B C D)
   case 0
-  then show ?case by auto 
+  then show ?case by auto
 next
-  case (Suc n) 
+  case (Suc n)
   let ?block = "(four_block_mat A B C D)"
   let ?B = "Matrix.mat m (n-m) (\<lambda>(i,j). 0)"
   let ?C = "Matrix.mat (n-m) m (\<lambda>(i,j). 0)"
-  let ?D = "1\<^sub>m (n-m)" 
+  let ?D = "1\<^sub>m (n-m)"
   have mat_eq: "(mat_delete ?block n n) = four_block_mat A ?B ?C ?D" (is "?lhs = ?rhs")
   proof (rule eq_matI)
-    fix i j assume i: "i < dim_row (four_block_mat A ?B ?C ?D)" 
+    fix i j assume i: "i < dim_row (four_block_mat A ?B ?C ?D)"
       and j: "j < dim_col (four_block_mat A ?B ?C ?D)"
     let ?f = " (if i < dim_row A then if j < dim_col A then A $$ (i, j) else B $$ (i, j - dim_col A)
      else if j < dim_col A then C $$ (i - dim_row A, j) else D $$ (i - dim_row A, j - dim_col A))"
     let ?g = "(if i < dim_row A then if j < dim_col A then A $$ (i, j) else ?B $$ (i, j - dim_col A)
      else if j < dim_col A then ?C $$ (i - dim_row A, j) else ?D $$ (i - dim_row A, j - dim_col A))"
-    have "(mat_delete ?block n n) $$ (i,j) = ?block $$ (i,j)" 
+    have "(mat_delete ?block n n) $$ (i,j) = ?block $$ (i,j)"
       using i j Suc.prems unfolding mat_delete_def by auto
-    also have "... = ?f" 
+    also have "... = ?f"
       by (rule index_mat_four_block, insert Suc.prems i j, auto)
-    also have "... = ?g" using i j Suc.prems by auto    
+    also have "... = ?g" using i j Suc.prems by auto
     also have "... = four_block_mat A ?B ?C ?D $$ (i,j)"
       by (rule index_mat_four_block[symmetric], insert Suc.prems i j, auto)
-    finally show "?lhs $$ (i,j) = ?rhs $$ (i,j)" .  
+    finally show "?lhs $$ (i,j) = ?rhs $$ (i,j)" .
   qed (insert Suc.prems, auto)
   have nn_1: "?block $$ (n, n) = 1" using Suc.prems by auto
   have rw0: "(\<Sum>i<n. ?block $$ (i,n) * Determinant.cofactor ?block i n) = 0"
   proof (rule sum.neutral, rule)
     fix x assume x: "x \<in> {..<n}"
-    have block_index: "?block $$ (x,n) = (if x < dim_row A then if n < dim_col A then A $$ (x, n) 
-      else B $$ (x, n - dim_col A) else if n < dim_col A then C $$ (x - dim_row A, n) 
+    have block_index: "?block $$ (x,n) = (if x < dim_row A then if n < dim_col A then A $$ (x, n)
+      else B $$ (x, n - dim_col A) else if n < dim_col A then C $$ (x - dim_row A, n)
       else D $$ (x - dim_row A, n - dim_col A))"
       by (rule index_mat_four_block, insert Suc.prems x, auto)
-    have "four_block_mat A B C D $$ (x,n) = 0" using x Suc.prems by auto   
-    thus "four_block_mat A B C D $$ (x, n) * Determinant.cofactor (four_block_mat A B C D) x n = 0" 
+    have "four_block_mat A B C D $$ (x,n) = 0" using x Suc.prems by auto
+    thus "four_block_mat A B C D $$ (x, n) * Determinant.cofactor (four_block_mat A B C D) x n = 0"
       by simp
   qed
   have "Determinant.det ?block = (\<Sum>i<Suc n. ?block $$ (i, n) * Determinant.cofactor ?block i n)"
     by (rule laplace_expansion_column, insert Suc.prems, auto)
-  also have "... = ?block $$ (n, n) * Determinant.cofactor ?block n n 
+  also have "... = ?block $$ (n, n) * Determinant.cofactor ?block n n
     + (\<Sum>i<n. ?block $$ (i,n) * Determinant.cofactor ?block i n)"
     by simp
   also have "... = ?block $$ (n, n) * Determinant.cofactor ?block n n" using rw0 by auto
   also have "... = Determinant.cofactor ?block n n" using nn_1 by simp
   also have "... = Determinant.det (mat_delete ?block n n)" unfolding cofactor_def by auto
   also have "... = Determinant.det (four_block_mat A ?B ?C ?D)" using mat_eq by simp
   also have "... = Determinant.det A" (is "Determinant.det ?lhs = Determinant.det ?rhs")
   proof (cases "n = m")
     case True
     have "?lhs = ?rhs" by (rule four_block_mat_dim0, insert Suc.prems True, auto)
     then show ?thesis by simp
   next
     case False
     show ?thesis by (rule Suc.hyps, insert Suc.prems False, auto)
-  qed   
+  qed
   finally show ?case .
 qed
 
 
 lemma mult_eq_first_row:
   assumes A: "A \<in> carrier_mat 1 n"
   and B: "B \<in> carrier_mat m n"
   and m0: "m \<noteq> 0"
   and r: "Matrix.row A 0 = Matrix.row B 0"
 shows "Matrix.row (A * V) 0 = Matrix.row (B * V) 0"
-proof (rule eq_vecI)  
+proof (rule eq_vecI)
   show "dim_vec (Matrix.row (A * V) 0) = dim_vec (Matrix.row (B * V) 0)" using A B r by auto
   fix i assume i: "i < dim_vec (Matrix.row (B * V) 0)"
   have "Matrix.row (A * V) 0 $v i = (A * V) $$ (0,i)" by (rule index_row, insert i A, auto)
   also have "... = Matrix.row A 0 \<bullet> col V i" by (rule index_mult_mat, insert A i, auto)
   also have "... = Matrix.row B 0 \<bullet> col V i" using r by auto
   also have "... = (B * V) $$ (0,i)" by (rule index_mult_mat[symmetric], insert m0 B i, auto)
   also have "... = Matrix.row (B * V) 0 $v i" by (rule index_row[symmetric], insert i B m0, auto)
   finally show "Matrix.row (A * V) 0 $v i = Matrix.row (B * V) 0 $v i" .
 qed
 
 
 lemma smult_mat_mat_one_element:
   assumes A: "A \<in> carrier_mat 1 1" and B: "B \<in> carrier_mat 1 n"
-  shows "A * B = A $$ (0,0) \<cdot>\<^sub>m B" 
+  shows "A * B = A $$ (0,0) \<cdot>\<^sub>m B"
 proof (rule eq_matI)
   fix i j assume i: "i < dim_row (A $$ (0, 0) \<cdot>\<^sub>m B)" and j: "j < dim_col (A $$ (0, 0) \<cdot>\<^sub>m B)"
   have i0: "i = 0" using A B i by auto
-  have "(A * B) $$ (i, j) =  Matrix.row A i \<bullet> col B j" 
+  have "(A * B) $$ (i, j) =  Matrix.row A i \<bullet> col B j"
     by (rule index_mult_mat, insert i j A B, auto)
-  also have "... =  Matrix.row A i $v 0 * col B j $v 0" unfolding scalar_prod_def using B by auto 
+  also have "... =  Matrix.row A i $v 0 * col B j $v 0" unfolding scalar_prod_def using B by auto
   also have "... = A$$(i,i) * B$$(i,j)" using A i i0 j by auto
-  also have "... = (A $$ (i, i) \<cdot>\<^sub>m B) $$ (i, j)" 
+  also have "... = (A $$ (i, i) \<cdot>\<^sub>m B) $$ (i, j)"
     unfolding i by (rule index_smult_mat[symmetric], insert i j B, auto)
-  finally show "(A * B) $$ (i, j) = (A $$ (0, 0) \<cdot>\<^sub>m B) $$ (i, j)" using i0 by simp  
+  finally show "(A * B) $$ (i, j) = (A $$ (0, 0) \<cdot>\<^sub>m B) $$ (i, j)" using i0 by simp
 qed (insert A B, auto)
 
 lemma determinant_one_element:
   assumes A: "A \<in> carrier_mat 1 1" shows "Determinant.det A = A $$ (0,0)"
-proof - 
+proof -
   have "Determinant.det A = prod_list (diag_mat A)"
     by (rule det_upper_triangular[OF _ A], insert A, unfold upper_triangular_def, auto)
   also have "... = A $$ (0,0)" using A unfolding diag_mat_def by auto
   finally show ?thesis .
 qed
 
 
 
 lemma invertible_mat_transpose:
   assumes inv_A: "invertible_mat (A::'a::comm_ring_1 mat)"
   shows "invertible_mat A\<^sup>T"
 proof -
-  obtain n where A: "A \<in> carrier_mat n n" 
+  obtain n where A: "A \<in> carrier_mat n n"
     using inv_A unfolding invertible_mat_def square_mat.simps by auto
   hence At: "A\<^sup>T \<in> carrier_mat n n" by simp
   have "Determinant.det A\<^sup>T = Determinant.det A"
-    by (metis Determinant.det_def Determinant.det_transpose carrier_matI 
+    by (metis Determinant.det_def Determinant.det_transpose carrier_matI
         index_transpose_mat(2) index_transpose_mat(3))
   also have "... dvd 1" using invertible_iff_is_unit_JNF[OF A] inv_A by simp
   finally show ?thesis  using invertible_iff_is_unit_JNF[OF At] by auto
 qed
 
 lemma dvd_elements_mult_matrix_left:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat m n"
   and P: "P \<in> carrier_mat m m"
   and x: "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd A$$(i,j))"
   shows "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd (P*A)$$(i,j))"
 proof -
   have "x dvd (P * A) $$ (i, j)" if i: "i < m" and j: "j < n" for i j
   proof -
-    have "(P * A) $$ (i, j) =  (\<Sum>ia = 0..<m. Matrix.row P i $v ia * col A j $v ia)" 
+    have "(P * A) $$ (i, j) =  (\<Sum>ia = 0..<m. Matrix.row P i $v ia * col A j $v ia)"
       unfolding times_mat_def scalar_prod_def using A P j i by auto
     also have "... = (\<Sum>ia = 0..<m. Matrix.row P i $v ia *  A $$ (ia,j))"
       by (rule sum.cong, insert A j, auto)
     also have "x dvd ..." using x by (meson atLeastLessThan_iff dvd_mult dvd_sum j)
     finally show ?thesis .
   qed
   thus ?thesis by auto
 qed
 
 
 lemma dvd_elements_mult_matrix_right:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat m n"
   and Q: "Q \<in> carrier_mat n n"
   and x: "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd A$$(i,j))"
   shows "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd (A*Q)$$(i,j))"
 proof -
   have "x dvd (A*Q) $$ (i, j)" if i: "i < m" and j: "j < n" for i j
   proof -
-    have "(A*Q) $$ (i, j) =  (\<Sum>ia = 0..<n. Matrix.row A i $v ia * col Q j $v ia)" 
+    have "(A*Q) $$ (i, j) =  (\<Sum>ia = 0..<n. Matrix.row A i $v ia * col Q j $v ia)"
       unfolding times_mat_def scalar_prod_def using A Q j i by auto
     also have "... = (\<Sum>ia = 0..<n. A $$ (i, ia) * col Q j $v ia)"
       by (rule sum.cong, insert A Q i, auto)
     also have "x dvd ..." using x
       by (meson atLeastLessThan_iff dvd_mult2 dvd_sum i)
     finally show ?thesis .
   qed
   thus ?thesis by auto
 qed
 
 
 lemma dvd_elements_mult_matrix_left_right:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat m n"
   and P: "P \<in> carrier_mat m m"
   and Q: "Q \<in> carrier_mat n n"
   and x: "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd A$$(i,j))"
 shows "(\<forall>i j. i<m \<and> j<n \<longrightarrow> x dvd (P*A*Q)$$(i,j))"
-  using dvd_elements_mult_matrix_left[OF A P x]  
+  using dvd_elements_mult_matrix_left[OF A P x]
   by (meson P A Q dvd_elements_mult_matrix_right mult_carrier_mat)
 
 
 definition append_cols :: "'a :: zero mat \<Rightarrow> 'a mat \<Rightarrow> 'a mat" (infixr "@\<^sub>c" 65)where
   "A @\<^sub>c B = four_block_mat A B (0\<^sub>m 0 (dim_col A)) (0\<^sub>m 0 (dim_col B))"
 
 lemma append_cols_carrier[simp,intro]:
-  "A \<in> carrier_mat n a \<Longrightarrow> B \<in> carrier_mat n b \<Longrightarrow> (A @\<^sub>c B) \<in> carrier_mat n (a+b)" 
+  "A \<in> carrier_mat n a \<Longrightarrow> B \<in> carrier_mat n b \<Longrightarrow> (A @\<^sub>c B) \<in> carrier_mat n (a+b)"
   unfolding append_cols_def by auto
 
 lemma append_cols_mult_left:
   assumes A: "A \<in> carrier_mat n a"
   and B: "B \<in> carrier_mat n b"
   and P: "P \<in> carrier_mat n n"
 shows "P * (A @\<^sub>c B) = (P*A) @\<^sub>c (P*B)"
 proof -
   let ?P = "four_block_mat P (0\<^sub>m n 0) (0\<^sub>m 0 n) (0\<^sub>m 0 0)"
   have "P = ?P" by (rule eq_matI, auto)
   hence "P * (A @\<^sub>c B) = ?P * (A @\<^sub>c B)" by simp
-  also have "?P * (A @\<^sub>c B) = four_block_mat (P * A + 0\<^sub>m n 0 * 0\<^sub>m 0 (dim_col A)) 
+  also have "?P * (A @\<^sub>c B) = four_block_mat (P * A + 0\<^sub>m n 0 * 0\<^sub>m 0 (dim_col A))
   (P * B + 0\<^sub>m n 0 * 0\<^sub>m 0 (dim_col B)) (0\<^sub>m 0 n * A + 0\<^sub>m 0 0 * 0\<^sub>m 0 (dim_col A))
-  (0\<^sub>m 0 n * B + 0\<^sub>m 0 0 * 0\<^sub>m 0 (dim_col B))" unfolding append_cols_def 
+  (0\<^sub>m 0 n * B + 0\<^sub>m 0 0 * 0\<^sub>m 0 (dim_col B))" unfolding append_cols_def
     by (rule mult_four_block_mat, insert A B P, auto)
   also have "... = four_block_mat (P * A) (P * B) (0\<^sub>m 0 (dim_col (P*A))) (0\<^sub>m 0 (dim_col (P*B)))"
     by (rule cong_four_block_mat, insert P, auto)
   also have "... = (P*A) @\<^sub>c (P*B)" unfolding append_cols_def by auto
   finally show ?thesis .
 qed
 
 lemma append_cols_mult_right_id:
   assumes A: "(A::'a::semiring_1 mat) \<in> carrier_mat n 1"
   and B: "B \<in> carrier_mat n (m-1)"
   and C: "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"
   and D: "D \<in> carrier_mat (m-1) (m-1)"
 shows "(A @\<^sub>c B) * C = A @\<^sub>c (B * D)"
 proof -
   let ?C = "four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"
   have "(A @\<^sub>c B) * C = (A @\<^sub>c B) * ?C" unfolding C by auto
   also have "... = four_block_mat A B (0\<^sub>m 0 (dim_col A)) (0\<^sub>m 0 (dim_col B)) * ?C"
     unfolding append_cols_def by auto
   also have "... = four_block_mat (A * 1\<^sub>m 1 + B * 0\<^sub>m (m - 1) 1) (A * 0\<^sub>m 1 (m - 1) + B * D)
-    (0\<^sub>m 0 (dim_col A) * 1\<^sub>m 1 + 0\<^sub>m 0 (dim_col B) * 0\<^sub>m (m - 1) 1) 
+    (0\<^sub>m 0 (dim_col A) * 1\<^sub>m 1 + 0\<^sub>m 0 (dim_col B) * 0\<^sub>m (m - 1) 1)
     (0\<^sub>m 0 (dim_col A) * 0\<^sub>m 1 (m - 1) + 0\<^sub>m 0 (dim_col B) * D)"
     by (rule mult_four_block_mat, insert assms, auto)
   also have "... = four_block_mat A (B * D) (0\<^sub>m 0 (dim_col A)) (0\<^sub>m 0 (dim_col (B*D)))"
     by (rule cong_four_block_mat, insert assms, auto)
   also have "... = A @\<^sub>c (B * D)" unfolding append_cols_def by auto
   finally show ?thesis .
 qed
 
 
 lemma append_cols_mult_right_id2:
  assumes A: "(A::'a::semiring_1 mat) \<in> carrier_mat n a"
    and B: "B \<in> carrier_mat n b"
    and C: "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"
    and D: "D \<in> carrier_mat a a"
 shows "(A @\<^sub>c B) * C = (A * D) @\<^sub>c B"
 proof -
   let ?C = "four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"
   have "(A @\<^sub>c B) * C = (A @\<^sub>c B) * ?C" unfolding C by auto
-  also have "... = four_block_mat A B (0\<^sub>m 0 a) (0\<^sub>m 0 b) * ?C" 
+  also have "... = four_block_mat A B (0\<^sub>m 0 a) (0\<^sub>m 0 b) * ?C"
     unfolding append_cols_def using A B by auto
-  also have "... = four_block_mat (A * D + B * 0\<^sub>m b a) (A * 0\<^sub>m a b + B * 1\<^sub>m b) 
+  also have "... = four_block_mat (A * D + B * 0\<^sub>m b a) (A * 0\<^sub>m a b + B * 1\<^sub>m b)
     (0\<^sub>m 0 a * D + 0\<^sub>m 0 b * 0\<^sub>m b a) (0\<^sub>m 0 a * 0\<^sub>m a b + 0\<^sub>m 0 b * 1\<^sub>m b)"
     by (rule mult_four_block_mat, insert A B C D, auto)
   also have "... = four_block_mat (A * D) B (0\<^sub>m 0 (dim_col (A*D))) (0\<^sub>m 0 (dim_col B))"
     by (rule cong_four_block_mat, insert assms, auto)
   also have "... = (A * D) @\<^sub>c B" unfolding append_cols_def by auto
   finally show ?thesis .
 qed
 
 
 lemma append_cols_nth:
   assumes A: "A \<in> carrier_mat n a"
   and B: "B \<in> carrier_mat n b"
   and i: "i<n" and j: "j < a + b"
 shows "(A @\<^sub>c B) $$ (i, j) = (if j < dim_col A then A $$(i,j) else B$$(i,j-a))" (is "?lhs = ?rhs")
 proof -
   let ?C = "(0\<^sub>m 0 (dim_col A))"
   let ?D = "(0\<^sub>m 0 (dim_col B))"
   have i2: "i < dim_row A + dim_row ?D" using i A by auto
   have j2: "j < dim_col A + dim_col (0\<^sub>m 0 (dim_col B))" using j B A by auto
   have "(A @\<^sub>c B) $$ (i, j) = four_block_mat A B ?C ?D $$ (i, j)"
     unfolding append_cols_def by auto
-  also have "... = (if i < dim_row A then if j < dim_col A then A $$ (i, j) 
-  else B $$ (i, j - dim_col A) else if j < dim_col A then ?C $$ (i - dim_row A, j) 
+  also have "... = (if i < dim_row A then if j < dim_col A then A $$ (i, j)
+  else B $$ (i, j - dim_col A) else if j < dim_col A then ?C $$ (i - dim_row A, j)
   else 0\<^sub>m 0 (dim_col B) $$ (i - dim_row A, j - dim_col A))"
     by (rule index_mat_four_block(1)[OF i2 j2])
   also have "... = ?rhs" using i A by auto
   finally show ?thesis .
 qed
 
 lemma append_cols_split:
   assumes d: "dim_col A > 0"
-  shows "A = mat_of_cols (dim_row A) [col A 0] @\<^sub>c 
+  shows "A = mat_of_cols (dim_row A) [col A 0] @\<^sub>c
              mat_of_cols (dim_row A) (map (col A) [1..<dim_col A])" (is "?lhs = ?A1 @\<^sub>c ?A2")
-proof (rule eq_matI) 
+proof (rule eq_matI)
   fix i j assume i: "i < dim_row (?A1 @\<^sub>c ?A2)" and j: "j < dim_col (?A1 @\<^sub>c ?A2)"
   have "(?A1 @\<^sub>c ?A2) $$ (i, j) = (if j < dim_col ?A1 then ?A1 $$(i,j) else ?A2$$(i,j-(dim_col ?A1)))"
-    by (rule append_cols_nth, insert i j, auto simp add: append_cols_def) 
+    by (rule append_cols_nth, insert i j, auto simp add: append_cols_def)
   also have "... = A $$ (i,j)"
   proof (cases "j< dim_col ?A1")
     case True
     then show ?thesis
-      by (metis One_nat_def Suc_eq_plus1 add.right_neutral append_cols_def col_def i 
-          index_mat_four_block(2) index_vec index_zero_mat(2) less_one list.size(3) list.size(4) 
+      by (metis One_nat_def Suc_eq_plus1 add.right_neutral append_cols_def col_def i
+          index_mat_four_block(2) index_vec index_zero_mat(2) less_one list.size(3) list.size(4)
           mat_of_cols_Cons_index_0 mat_of_cols_carrier(2) mat_of_cols_carrier(3))
   next
     case False
     then show ?thesis
-      by (metis (no_types, lifting) Suc_eq_plus1 Suc_less_eq Suc_pred add_diff_cancel_right' append_cols_def 
-          diff_zero i index_col index_mat_four_block(2) index_mat_four_block(3) index_zero_mat(2) 
-          index_zero_mat(3) j length_map length_upt linordered_semidom_class.add_diff_inverse list.size(3) 
-          list.size(4) mat_of_cols_carrier(2) mat_of_cols_carrier(3) mat_of_cols_index nth_map_upt 
+      by (metis (no_types, lifting) Suc_eq_plus1 Suc_less_eq Suc_pred add_diff_cancel_right' append_cols_def
+          diff_zero i index_col index_mat_four_block(2) index_mat_four_block(3) index_zero_mat(2)
+          index_zero_mat(3) j length_map length_upt linordered_semidom_class.add_diff_inverse list.size(3)
+          list.size(4) mat_of_cols_carrier(2) mat_of_cols_carrier(3) mat_of_cols_index nth_map_upt
           plus_1_eq_Suc upt_0)
   qed
   finally show "A $$ (i, j) = (?A1 @\<^sub>c ?A2) $$ (i, j)" ..
 qed (auto simp add: append_cols_def d)
 
 
 lemma append_rows_nth:
   assumes A: "A \<in> carrier_mat a n"
   and B: "B \<in> carrier_mat b n"
   and i: "i<a+b" and j: "j < n"
 shows "(A @\<^sub>r B) $$ (i, j) = (if i < dim_row A then A $$(i,j) else B$$(i-a,j))" (is "?lhs = ?rhs")
 proof -
   let ?C = "(0\<^sub>m (dim_row A) 0)"
   let ?D = "(0\<^sub>m (dim_row B) 0)"
   have i2: "i < dim_row A + dim_row ?D" using i j A B by auto
   have j2: "j < dim_col A + dim_col ?D" using i j A B by auto
   have "(A @\<^sub>r B) $$ (i, j) = four_block_mat A ?C B ?D $$ (i, j)"
     unfolding append_rows_def by auto
   also have "... =  (if i < dim_row A then if j < dim_col A then A $$ (i, j) else ?C $$ (i, j - dim_col A)
-   else if j < dim_col A then B $$ (i - dim_row A, j) else ?D $$ (i - dim_row A, j - dim_col A))" 
+   else if j < dim_col A then B $$ (i - dim_row A, j) else ?D $$ (i - dim_row A, j - dim_col A))"
     by (rule index_mat_four_block(1)[OF i2 j2])
   also have "... = ?rhs" using i A j B by auto
   finally show ?thesis .
 qed
 
 lemma append_rows_split:
   assumes k: "k\<le>dim_row A"
-  shows "A = (mat_of_rows (dim_col A) [Matrix.row A i. i \<leftarrow> [0..<k]]) @\<^sub>r 
+  shows "A = (mat_of_rows (dim_col A) [Matrix.row A i. i \<leftarrow> [0..<k]]) @\<^sub>r
              (mat_of_rows (dim_col A) [Matrix.row A i. i \<leftarrow> [k..<dim_row A]])" (is "?lhs = ?A1 @\<^sub>r ?A2")
 proof (rule eq_matI)
-  have "(?A1 @\<^sub>r ?A2) \<in> carrier_mat (k + (dim_row A-k)) (dim_col A)" 
+  have "(?A1 @\<^sub>r ?A2) \<in> carrier_mat (k + (dim_row A-k)) (dim_col A)"
     by (rule carrier_append_rows, insert k, auto)
   hence A1_A2: "(?A1 @\<^sub>r ?A2) \<in> carrier_mat (dim_row A) (dim_col A)" using k by simp
   thus "dim_row A = dim_row (?A1 @\<^sub>r ?A2)" and "dim_col A = dim_col (?A1 @\<^sub>r ?A2)" by auto
   fix i j assume i: "i < dim_row (?A1 @\<^sub>r ?A2)" and j: "j < dim_col (?A1 @\<^sub>r ?A2)"
   have "(?A1 @\<^sub>r ?A2) $$ (i, j) = (if i < dim_row ?A1 then ?A1 $$(i,j) else ?A2$$(i-(dim_row ?A1),j))"
-    by (rule append_rows_nth, insert k i j, auto simp add: append_rows_def) 
+    by (rule append_rows_nth, insert k i j, auto simp add: append_rows_def)
   also have "... = A $$ (i,j)"
   proof (cases "i<dim_row ?A1")
     case True
     then show ?thesis
-      by (metis (no_types, lifting) Matrix.row_def add.left_neutral add.right_neutral append_rows_def 
-          index_mat(1) index_mat_four_block(3) index_vec index_zero_mat(3) j length_map length_upt 
+      by (metis (no_types, lifting) Matrix.row_def add.left_neutral add.right_neutral append_rows_def
+          index_mat(1) index_mat_four_block(3) index_vec index_zero_mat(3) j length_map length_upt
           mat_of_rows_carrier(2,3) mat_of_rows_def nth_map_upt prod.simps(2))
   next
     case False
     let ?xs = "(map (Matrix.row A) [k..<dim_row A])"
     have dim_row_A1: "dim_row ?A1 = k" by auto
-    have "?A2 $$ (i-k,j) = ?xs ! (i-k) $v j" 
+    have "?A2 $$ (i-k,j) = ?xs ! (i-k) $v j"
       by (rule mat_of_rows_index, insert i k False A1_A2 j, auto)
     also have "... = A $$ (i,j)" using A1_A2 False i j by auto
     finally show ?thesis using A1_A2 False i j by auto
   qed
   finally show " A $$ (i, j) = (?A1 @\<^sub>r ?A2) $$ (i,j)" by simp
 qed
 
 
 
 lemma transpose_mat_append_rows:
  assumes A: "A \<in> carrier_mat a n" and B: "B \<in> carrier_mat b n"
  shows "(A @\<^sub>r B)\<^sup>T = A\<^sup>T @\<^sub>c B\<^sup>T"
   by (smt append_cols_def append_rows_def A B carrier_matD(1) index_transpose_mat(3)
       transpose_four_block_mat zero_carrier_mat zero_transpose_mat)
 
 lemma transpose_mat_append_cols:
  assumes A: "A \<in> carrier_mat n a" and B: "B \<in> carrier_mat n b"
  shows "(A @\<^sub>c B)\<^sup>T = A\<^sup>T @\<^sub>r B\<^sup>T"
-  by (metis Matrix.transpose_transpose A B carrier_matD(1) carrier_mat_triv 
+  by (metis Matrix.transpose_transpose A B carrier_matD(1) carrier_mat_triv
       index_transpose_mat(3) transpose_mat_append_rows)
 
 
 lemma append_rows_mult_right:
-  assumes A: "(A::'a::comm_semiring_1 mat) \<in> carrier_mat a n" and B: "B \<in> carrier_mat b n" 
+  assumes A: "(A::'a::comm_semiring_1 mat) \<in> carrier_mat a n" and B: "B \<in> carrier_mat b n"
     and Q: "Q\<in> carrier_mat n n"
   shows "(A @\<^sub>r B) * Q = (A * Q) @\<^sub>r (B*Q)"
 proof -
-  have "transpose_mat ((A @\<^sub>r B) * Q) = Q\<^sup>T * (A @\<^sub>r B)\<^sup>T" 
+  have "transpose_mat ((A @\<^sub>r B) * Q) = Q\<^sup>T * (A @\<^sub>r B)\<^sup>T"
     by (rule transpose_mult, insert A B Q, auto)
   also have "... = Q\<^sup>T * (A\<^sup>T @\<^sub>c B\<^sup>T)" using transpose_mat_append_rows assms by metis
-  also have "... = Q\<^sup>T * A\<^sup>T @\<^sub>c Q\<^sup>T * B\<^sup>T" 
+  also have "... = Q\<^sup>T * A\<^sup>T @\<^sub>c Q\<^sup>T * B\<^sup>T"
     using append_cols_mult_left assms by (metis transpose_carrier_mat)
   also have "transpose_mat ... = (A * Q) @\<^sub>r (B*Q)"
-    by (smt A B Matrix.transpose_mult Matrix.transpose_transpose append_cols_def append_rows_def Q 
-        carrier_mat_triv index_mult_mat(2) index_transpose_mat(2) transpose_four_block_mat 
+    by (smt A B Matrix.transpose_mult Matrix.transpose_transpose append_cols_def append_rows_def Q
+        carrier_mat_triv index_mult_mat(2) index_transpose_mat(2) transpose_four_block_mat
         zero_carrier_mat zero_transpose_mat)
   finally show ?thesis by simp
 qed
 
 lemma append_rows_mult_left_id:
   assumes A: "(A::'a::comm_semiring_1 mat) \<in> carrier_mat 1 n"
   and B: "B \<in> carrier_mat (m-1) n"
   and C: "C = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) D"
   and D: "D \<in> carrier_mat (m-1) (m-1)"
 shows "C * (A @\<^sub>r B) = A @\<^sub>r (D * B)"
 proof -
   have "transpose_mat (C * (A @\<^sub>r B)) = (A @\<^sub>r B)\<^sup>T * C\<^sup>T"
-    by (metis (no_types, lifting) B C D Matrix.transpose_mult append_rows_def A carrier_matD 
+    by (metis (no_types, lifting) B C D Matrix.transpose_mult append_rows_def A carrier_matD
         carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2) one_carrier_mat)
   also have "... = (A\<^sup>T @\<^sub>c B\<^sup>T) * C\<^sup>T" using transpose_mat_append_rows[OF A B] by auto
   also have "... = A\<^sup>T @\<^sub>c (B\<^sup>T * D\<^sup>T)" by (rule append_cols_mult_right_id, insert A B C D, auto)
-  also have "transpose_mat ... = A @\<^sub>r (D * B)" 
-    by (smt B D Matrix.transpose_mult Matrix.transpose_transpose append_cols_def append_rows_def A 
+  also have "transpose_mat ... = A @\<^sub>r (D * B)"
+    by (smt B D Matrix.transpose_mult Matrix.transpose_transpose append_cols_def append_rows_def A
         carrier_matD(2) carrier_mat_triv index_mult_mat(3) index_transpose_mat(3)
         transpose_four_block_mat zero_carrier_mat zero_transpose_mat)
   finally show ?thesis by auto
 qed
 
 lemma append_rows_mult_left_id2:
  assumes A: "(A::'a::comm_semiring_1 mat) \<in> carrier_mat a n"
    and B: "B \<in> carrier_mat b n"
    and C: "C = four_block_mat D (0\<^sub>m a b) (0\<^sub>m b a) (1\<^sub>m b)"
    and D: "D \<in> carrier_mat a a"
  shows "C * (A @\<^sub>r B) = (D * A) @\<^sub>r B"
 proof -
   have "(C * (A @\<^sub>r B))\<^sup>T = (A @\<^sub>r B)\<^sup>T * C\<^sup>T" by (rule transpose_mult, insert assms, auto)
   also have "... = (A\<^sup>T @\<^sub>c B\<^sup>T) * C\<^sup>T" by (metis A B transpose_mat_append_rows)
   also have "... = (A\<^sup>T * D\<^sup>T @\<^sub>c B\<^sup>T)" by (rule append_cols_mult_right_id2, insert assms, auto)
   also have "...\<^sup>T = (D * A) @\<^sub>r B"
     by (metis A B D transpose_mult transpose_transpose mult_carrier_mat transpose_mat_append_rows)
   finally show ?thesis by simp
 qed
 
 lemma four_block_mat_preserves_column:
   assumes A: "(A::'a::semiring_1 mat) \<in> carrier_mat n m"
     and B: "B = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) C"
-  and C: "C \<in> carrier_mat (m-1) (m-1)"  
+  and C: "C \<in> carrier_mat (m-1) (m-1)"
   and i: "i<n" and m: "0<m"
 shows "(A*B) $$ (i,0) = A $$ (i,0)"
 proof -
-  let ?A1 = "mat_of_cols n [col A 0]" 
+  let ?A1 = "mat_of_cols n [col A 0]"
   let ?A2 = "mat_of_cols n (map (col A) [1..<dim_col A])"
   have n2: "dim_row A = n" using A by auto
   have "A = ?A1 @\<^sub>c ?A2" by (rule append_cols_split[of A, unfolded n2], insert m A, auto)
   hence "A * B = (?A1 @\<^sub>c ?A2) * B" by simp
   also have "... = ?A1 @\<^sub>c (?A2 * C)" by (rule append_cols_mult_right_id[OF _ _ B C], insert A, auto)
   also have "... $$ (i,0) = ?A1 $$ (i,0)" using append_cols_nth by (simp add: append_cols_def i)
   also have "... = A $$ (i,0)"
     by (metis A i carrier_matD(1) col_def index_vec mat_of_cols_Cons_index_0)
   finally show ?thesis .
 qed
 
 
 definition "lower_triangular A = (\<forall>i j. i < j \<and> i < dim_row A \<and> j < dim_col A \<longrightarrow> A $$ (i,j) = 0)"
 
 lemma lower_triangular_index:
   assumes "lower_triangular A" "i<j" "i < dim_row A" "j < dim_col A"
   shows "A $$ (i,j) = 0"
   using assms unfolding lower_triangular_def by auto
 
 lemma commute_multiples_identity:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat n n"
   shows "A * (k \<cdot>\<^sub>m (1\<^sub>m n)) = (k \<cdot>\<^sub>m (1\<^sub>m n)) * A"
 proof -
-  have "(\<Sum>ia = 0..<n. A $$ (i, ia) * (k * (if ia = j then 1 else 0))) 
+  have "(\<Sum>ia = 0..<n. A $$ (i, ia) * (k * (if ia = j then 1 else 0)))
     = (\<Sum>ia = 0..<n. k * (if i = ia then 1 else 0) * A $$ (ia, j))" (is "?lhs=?rhs")
     if i: "i<n" and j: "j<n" for i j
   proof -
     let ?f = "\<lambda>ia. A $$ (i, ia) * (k * (if ia = j then 1 else 0))"
     let ?g = "\<lambda>ia. k * (if i = ia then 1 else 0) * A $$ (ia, j)"
     have rw0: "(\<Sum>ia \<in> ({0..<n}-{j}). ?f ia) = 0" by (rule sum.neutral, auto)
     have rw0': "(\<Sum>ia \<in> ({0..<n}-{i}). ?g ia) = 0" by (rule sum.neutral, auto)
     have "?lhs = ?f j + (\<Sum>ia \<in> ({0..<n}-{j}). ?f ia)"
       by (smt atLeast0LessThan finite_atLeastLessThan lessThan_iff sum.remove j)
     also have "... = A $$ (i, j) * k" using rw0 by auto
     also have "... = ?g i + (\<Sum>ia \<in> ({0..<n}-{i}). ?g ia)" using rw0' by auto
-    also have "... = ?rhs" 
+    also have "... = ?rhs"
       by (smt atLeast0LessThan finite_atLeastLessThan lessThan_iff sum.remove i)
-    finally show ?thesis . 
+    finally show ?thesis .
   qed
   thus ?thesis using A
-  unfolding times_mat_def scalar_prod_def 
+  unfolding times_mat_def scalar_prod_def
   by auto (rule eq_matI, auto, smt sum.cong)
 qed
 
 (*This lemma is obtained using Mod_Type_Connect, otherwise we cannot prove HMA_I 0 0.
 The lelma could also be obtained with no use of transfer rules, proving it directly.*)
-lemma det_2: 
+lemma det_2:
   assumes A: "A \<in> carrier_mat 2 2"
   shows "Determinant.det A = A$$(0,0) * A $$ (1,1) - A$$(0,1)*A$$(1,0)"
 proof -
   let ?A = "(Mod_Type_Connect.to_hma\<^sub>m A)::'a^2^2"
-  have [transfer_rule]: "Mod_Type_Connect.HMA_M A ?A" 
+  have [transfer_rule]: "Mod_Type_Connect.HMA_M A ?A"
     unfolding Mod_Type_Connect.HMA_M_def using from_hma_to_hma\<^sub>m A by auto
   have [transfer_rule]: "Mod_Type_Connect.HMA_I 0 0"
     unfolding Mod_Type_Connect.HMA_I_def by (simp add: to_nat_0)
   have [transfer_rule]: "Mod_Type_Connect.HMA_I 1 1"
     unfolding Mod_Type_Connect.HMA_I_def by (simp add: to_nat_1)
   have "Determinant.det A = Determinants.det ?A" by (transfer, simp)
   also have "... = ?A $h 1 $h 1 * ?A $h 2 $h 2 - ?A $h 1 $h 2 * ?A $h 2 $h 1" unfolding det_2 by simp
   also have "... = ?A $h 0 $h 0 * ?A $h 1 $h 1 - ?A $h 0 $h 1 * ?A $h 1 $h 0"
     by (smt Groups.mult_ac(2) exhaust_2 semiring_norm(160))
-  also have "... = A$$(0,0) * A $$ (1,1) - A$$(0,1)*A$$(1,0)" 
+  also have "... = A$$(0,0) * A $$ (1,1) - A$$(0,1)*A$$(1,0)"
     unfolding index_hma_def[symmetric] by (transfer, auto)
   finally show ?thesis .
 qed
 
 lemma mat_diag_smult: "mat_diag n (\<lambda> x. (k::'a::comm_ring_1)) = (k \<cdot>\<^sub>m 1\<^sub>m n)"
 proof -
   have "mat_diag n (\<lambda> x. k) = mat_diag n (\<lambda> x. k * 1)" by auto
   also have "... = mat_diag n (\<lambda> x. k) * mat_diag n (\<lambda> x. 1)" using mat_diag_diag
     by (simp add: mat_diag_def)
   also have "... = mat_diag n (\<lambda> x. k) * (1\<^sub>m n)" by auto  thm mat_diag_mult_left
   also have "... = Matrix.mat n n (\<lambda>(i, j). k * (1\<^sub>m n) $$ (i, j))" by (rule mat_diag_mult_left, auto)
   also have "... = (k \<cdot>\<^sub>m 1\<^sub>m n)" unfolding smult_mat_def by auto
   finally show ?thesis .
 qed
 
 lemma invertible_mat_four_block_mat_lower_right:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat n n" and inv_A: "invertible_mat A"
   shows "invertible_mat (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 n) (0\<^sub>m n 1) A)"
 proof -
   let ?I = "(four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 n) (0\<^sub>m n 1) A)"
-  have "Determinant.det ?I = Determinant.det (1\<^sub>m 1) * Determinant.det A" 
+  have "Determinant.det ?I = Determinant.det (1\<^sub>m 1) * Determinant.det A"
     by (rule det_four_block_mat_lower_left_zero_col, insert assms, auto)
   also have "... = Determinant.det A" by auto
   finally have "Determinant.det ?I = Determinant.det A" .
   thus ?thesis
-    by (metis (no_types, lifting) assms carrier_matD(1) carrier_matD(2) carrier_mat_triv 
-        index_mat_four_block(2) index_mat_four_block(3) index_one_mat(2) index_one_mat(3) 
+    by (metis (no_types, lifting) assms carrier_matD(1) carrier_matD(2) carrier_mat_triv
+        index_mat_four_block(2) index_mat_four_block(3) index_one_mat(2) index_one_mat(3)
         invertible_iff_is_unit_JNF)
 qed
 
 
 lemma invertible_mat_four_block_mat_lower_right_id:
   assumes A: "(A::'a::comm_ring_1 mat) \<in> carrier_mat m m" and B: "B = 0\<^sub>m m (n-m)" and C: "C = 0\<^sub>m (n-m) m"
     and D: "D = 1\<^sub>m (n-m)" and "n>m" and inv_A: "invertible_mat A"
   shows "invertible_mat (four_block_mat A B C D)"
 proof -
   have "Determinant.det (four_block_mat A B C D) = Determinant.det A"
     by (rule det_four_block_mat_lower_right_id, insert assms, auto)
   thus ?thesis using inv_A
-    by (metis (no_types, lifting) assms(1) assms(4) carrier_matD(1) carrier_matD(2) carrier_mat_triv 
-        index_mat_four_block(2) index_mat_four_block(3) index_one_mat(2) index_one_mat(3) 
+    by (metis (no_types, lifting) assms(1) assms(4) carrier_matD(1) carrier_matD(2) carrier_mat_triv
+        index_mat_four_block(2) index_mat_four_block(3) index_one_mat(2) index_one_mat(3)
         invertible_iff_is_unit_JNF)
 qed
 
 lemma split_block4_decreases_dim_row:
-  assumes E: "(A,B,C,D) = split_block E 1 1" 
+  assumes E: "(A,B,C,D) = split_block E 1 1"
     and E1: "dim_row E > 1" and E2: "dim_col E > 1"
   shows "dim_row D < dim_row E"
 proof -
   have "D \<in> carrier_mat (1 + (dim_row E - 2)) (1 + (dim_col E - 2))"
     by (rule split_block(4)[OF E[symmetric]], insert E1 E2, auto)
   hence "D \<in> carrier_mat (dim_row E - 1) (dim_col E - 1)" using E1 E2 by auto
   thus ?thesis using E1 by auto
 qed
 
 
 lemma inv_P'PAQQ':
-  assumes A: "A \<in> carrier_mat n n" 
-    and P: "P \<in> carrier_mat n n" 
-    and inv_P: "inverts_mat P' P" 
+  assumes A: "A \<in> carrier_mat n n"
+    and P: "P \<in> carrier_mat n n"
+    and inv_P: "inverts_mat P' P"
     and inv_Q: "inverts_mat Q Q'"
-    and Q: "Q \<in> carrier_mat n n" 
-    and P': "P' \<in> carrier_mat n n" 
+    and Q: "Q \<in> carrier_mat n n"
+    and P': "P' \<in> carrier_mat n n"
     and Q': "Q' \<in> carrier_mat n n"
 shows  "(P'*(P*A*Q)*Q') = A"
 proof -
   have "(P'*(P*A*Q)*Q') = (P'*(P*A*Q*Q'))"
-    by (smt P P' Q Q' assoc_mult_mat carrier_matD(1) carrier_matD(2) carrier_mat_triv 
+    by (smt P P' Q Q' assoc_mult_mat carrier_matD(1) carrier_matD(2) carrier_mat_triv
         index_mult_mat(2) index_mult_mat(3))
   also have "... = ((P'*P)*A*(Q*Q'))"
-    by (smt A P P' Q Q' assoc_mult_mat carrier_matD(1) carrier_matD(2) carrier_mat_triv 
-        index_mult_mat(3) inv_Q inverts_mat_def right_mult_one_mat') 
-  finally show ?thesis 
-    by (metis P' Q A inv_P inv_Q carrier_matD(1) inverts_mat_def 
+    by (smt A P P' Q Q' assoc_mult_mat carrier_matD(1) carrier_matD(2) carrier_mat_triv
+        index_mult_mat(3) inv_Q inverts_mat_def right_mult_one_mat')
+  finally show ?thesis
+    by (metis P' Q A inv_P inv_Q carrier_matD(1) inverts_mat_def
         left_mult_one_mat right_mult_one_mat)
 qed
 
-lemma 
+lemma
   assumes "U \<in> carrier_mat 2 2" and "V \<in> carrier_mat 2 2" and "A = U * V"
 shows mat_mult2_00: "A $$ (0,0) = U $$ (0,0)*V $$ (0,0) + U $$ (0,1)*V $$ (1,0)"
   and mat_mult2_01: "A $$ (0,1) = U $$ (0,0)*V $$ (0,1) + U $$ (0,1)*V $$ (1,1)"
   and mat_mult2_10: "A $$ (1,0) = U $$ (1,0)*V $$ (0,0) + U $$ (1,1)*V $$ (1,0)"
   and mat_mult2_11: "A $$ (1,1) = U $$ (1,0)*V $$ (0,1) + U $$ (1,1)*V $$ (1,1)"
     using assms unfolding times_mat_def Matrix.row_def col_def scalar_prod_def
     using sum_two_rw by auto
 
 
 subsection\<open>Lemmas about @{text "sorted lists"}, @{text "insort"} and @{text "pick"}\<close>
 
 
 lemma sorted_distinct_imp_sorted_wrt:
   assumes "sorted xs" and "distinct xs"
-  shows "sorted_wrt (<) xs" 
-  using assms 
+  shows "sorted_wrt (<) xs"
+  using assms
   by (induct xs, insert le_neq_trans, auto)
 
 
 lemma sorted_map_strict:
   assumes "strict_mono_on g {0..<n}"
   shows "sorted (map g [0..<n])"
-  using assms 
+  using assms
   by (induct n, auto simp add: sorted_append strict_mono_on_def less_imp_le)
-  
+
 
 lemma sorted_list_of_set_map_strict:
   assumes "strict_mono_on g {0..<n}"
   shows "sorted_list_of_set (g ` {0..<n}) = map g [0..<n]"
-  using assms 
+  using assms
   proof (induct n)
   case 0
   then show ?case by auto
 next
   case (Suc n)
-  note sg = Suc.prems  
+  note sg = Suc.prems
   have sg_n: "strict_mono_on g {0..<n}" using sg unfolding strict_mono_on_def by auto
-  have g_image_rw: "g ` {0..<Suc n} = insert (g n) (g ` {0..<n})" 
+  have g_image_rw: "g ` {0..<Suc n} = insert (g n) (g ` {0..<n})"
     by (simp add: set_upt_Suc)
   have "sorted_list_of_set (g ` {0..<Suc n}) = sorted_list_of_set (insert (g n) (g ` {0..<n}))"
     using g_image_rw by simp
   also have "... = insort (g n) (sorted_list_of_set (g ` {0..<n}))"
-  proof (rule sorted_list_of_set.insert) 
+  proof (rule sorted_list_of_set.insert)
     have "inj_on g {0..<Suc n}" using sg strict_mono_on_imp_inj_on by blast
     thus "g n \<notin> g ` {0..<n}" unfolding inj_on_def by fastforce
   qed (simp)
-  also have "... = insort (g n) (map g [0..<n])" 
+  also have "... = insort (g n) (map g [0..<n])"
     using Suc.hyps sg unfolding strict_mono_on_def by auto
   also have "... = map g [0..<Suc n]"
   proof (simp, rule sorted_insort_is_snoc)
     show "sorted (map g [0..<n])" by (rule sorted_map_strict[OF sg_n])
-    show "\<forall>x\<in>set (map g [0..<n]). x \<le> g n" using sg unfolding strict_mono_on_def 
+    show "\<forall>x\<in>set (map g [0..<n]). x \<le> g n" using sg unfolding strict_mono_on_def
       by (simp add: less_imp_le)
   qed
   finally show ?case .
 qed
 
 
 lemma sorted_nth_strict_mono:
   "sorted xs \<Longrightarrow> distinct xs \<Longrightarrow>i < j \<Longrightarrow> j < length xs \<Longrightarrow> xs!i < xs!j"
   by (simp add: less_le nth_eq_iff_index_eq sorted_iff_nth_mono_less)
 
 
 lemma sorted_list_of_set_0_LEAST:
   assumes finI: "finite I" and I: "I \<noteq> {}"
   shows "sorted_list_of_set I ! 0 = (LEAST n. n\<in>I)"
 proof (rule Least_equality[symmetric])
   show "sorted_list_of_set I ! 0 \<in> I"
     by (metis I Max_in finI gr_zeroI in_set_conv_nth not_less_zero set_sorted_list_of_set)
   fix y assume "y \<in> I"
   thus "sorted_list_of_set I ! 0 \<le> y"
-    by (metis eq_iff finI in_set_conv_nth neq0_conv sorted_iff_nth_mono_less 
+    by (metis eq_iff finI in_set_conv_nth neq0_conv sorted_iff_nth_mono_less
         sorted_list_of_set(1) sorted_sorted_list_of_set)
 qed
 
 lemma sorted_list_of_set_eq_pick:
-  assumes i: "i < length (sorted_list_of_set I)" 
+  assumes i: "i < length (sorted_list_of_set I)"
   shows "sorted_list_of_set I ! i = pick I i"
-proof - 
-  have finI: "finite I" 
+proof -
+  have finI: "finite I"
   proof (rule ccontr)
     assume "infinite I"
     hence "length (sorted_list_of_set I) = 0" using sorted_list_of_set.infinite by auto
     thus False using i by simp
   qed
-  show ?thesis 
+  show ?thesis
   using i
 proof (induct i)
   case 0
   have I: "I \<noteq> {}" using "0.prems" sorted_list_of_set_empty by blast
   show ?case unfolding pick.simps by (rule sorted_list_of_set_0_LEAST[OF finI I])
 next
   case (Suc i)
   note x_less = Suc.prems
   show ?case
   proof (unfold pick.simps, rule Least_equality[symmetric], rule conjI)
     show 1: "pick I i < sorted_list_of_set I ! Suc i"
       by (metis Suc.hyps Suc.prems Suc_lessD distinct_sorted_list_of_set find_first_unique lessI
           nat_less_le sorted_sorted_list_of_set sorted_sorted_wrt sorted_wrt_nth_less)
     show "sorted_list_of_set I ! Suc i \<in> I"
       using Suc.prems finI nth_mem set_sorted_list_of_set by blast
-    have rw: "sorted_list_of_set I ! i = pick I i" 
-      by (rule Suc.hyps, simp add: Suc.prems Suc_lessD)
-    have sorted_less: "sorted_list_of_set I ! i < sorted_list_of_set I ! Suc i"      
+    have rw: "sorted_list_of_set I ! i = pick I i"
+      using Suc.hyps Suc_lessD x_less by blast
+    have sorted_less: "sorted_list_of_set I ! i < sorted_list_of_set I ! Suc i"
       by (simp add: 1 rw)
     fix y assume y: "y \<in> I \<and> pick I i < y"
-    show "sorted_list_of_set I ! Suc i \<le> y" 
-      by (smt antisym_conv finI in_set_conv_nth less_Suc_eq less_Suc_eq_le nat_neq_iff rw 
+    show "sorted_list_of_set I ! Suc i \<le> y"
+      by (smt antisym_conv finI in_set_conv_nth less_Suc_eq less_Suc_eq_le nat_neq_iff rw
           sorted_iff_nth_mono_less sorted_list_of_set(1) sorted_sorted_list_of_set x_less y)
-  qed 
+  qed
 qed
 qed
 
-text\<open>$b$ is the position where we add, $a$ the element to be added and $i$ the position 
+text\<open>$b$ is the position where we add, $a$ the element to be added and $i$ the position
   that is checked\<close>
 
 lemma insort_nth':
-  assumes "\<forall>j<b. xs ! j < a" and "sorted xs" and "a \<notin> set xs" 
-    and "i < length xs + 1" and "i < b" 
+  assumes "\<forall>j<b. xs ! j < a" and "sorted xs" and "a \<notin> set xs"
+    and "i < length xs + 1" and "i < b"
     and "xs \<noteq> []" and "b < length xs"
   shows "insort a xs ! i = xs ! i"
   using assms
 proof (induct xs arbitrary: a b i)
   case Nil
   then show ?case by auto
 next
   case (Cons x xs)
   note less = Cons.prems(1)
   note sorted = Cons.prems(2)
   note a_notin = Cons.prems(3)
   note i_length = Cons.prems(4)
   note i_b = Cons.prems(5)
   note b_length = Cons.prems(7)
   show ?case
   proof (cases "a \<le> x")
     case True
     have "insort a (x # xs) ! i = (a # x # xs) ! i" using True by simp
     also have "... =  (x # xs) ! i"
       using Cons.prems(1) Cons.prems(5) True by force
     finally show ?thesis .
   next
     case False note x_less_a = False
     have "insort a (x # xs) ! i = (x # insort a xs) ! i" using False by simp
     also have "... = (x # xs) ! i"
     proof (cases "i = 0")
       case True
       then show ?thesis by auto
     next
       case False
       have "(x # insort a xs) ! i = (insort a xs) ! (i-1)"
         by (simp add: False nth_Cons')
       also have "... = xs ! (i-1)"
       proof (rule Cons.hyps)
         show "sorted xs" using sorted by simp
         show "a \<notin> set xs" using a_notin by simp
         show "i - 1 < length xs + 1" using i_length False by auto
         show "xs \<noteq> []" using i_b b_length by force
         show "i - 1 < b - 1" by (simp add: False diff_less_mono i_b leI)
         show "b - 1 < length xs" using b_length i_b by auto
         show "\<forall>j<b - 1. xs ! j < a" using less less_diff_conv by auto
       qed
       also have "... = (x # xs) ! i" by (simp add: False nth_Cons')
       finally show ?thesis .
     qed
     finally show ?thesis .
   qed
 qed
 
 
 lemma insort_nth:
-  assumes  "sorted xs" and "a \<notin> set xs" 
-    and "i < index (insort a xs) a" 
+  assumes  "sorted xs" and "a \<notin> set xs"
+    and "i < index (insort a xs) a"
     and "xs \<noteq> []"
   shows "insort a xs ! i = xs ! i"
   using assms
 proof (induct xs arbitrary: a i)
 case Nil
   then show ?case by auto
 next
   case (Cons x xs)
   note sorted = Cons.prems(1)
   note a_notin = Cons.prems(2)
-  note i_index = Cons.prems(3)  
+  note i_index = Cons.prems(3)
   show ?case
   proof (cases "a \<le> x")
     case True
     have "insort a (x # xs) ! i = (a # x # xs) ! i" using True by simp
     also have "... = (x # xs) ! i"
       using Cons.prems(1) Cons.prems(3) True by force
     finally show ?thesis .
   next
     case False note x_less_a = False
-    show ?thesis 
+    show ?thesis
     proof (cases "xs = []")
       case True
       have "x \<noteq> a" using False by auto
       then show ?thesis  using True i_index False by auto
     next
-      case False note xs_not_empty = False     
+      case False note xs_not_empty = False
       have "insort a (x # xs) ! i = (x # insort a xs) ! i" using x_less_a by simp
       also have "... = (x # xs) ! i"
       proof (cases "i = 0")
         case True
         then show ?thesis by auto
       next
         case False note i0 = False
         have "(x # insort a xs) ! i = (insort a xs) ! (i-1)"
           by (simp add: False nth_Cons')
         also have "... = xs ! (i-1)"
         proof (rule Cons.hyps[OF _ _ _ xs_not_empty])
           show "sorted xs" using sorted by simp
           show "a \<notin> set xs" using a_notin by simp
-          have "index (insort a (x # xs)) a = index ((x # insort a xs)) a" 
+          have "index (insort a (x # xs)) a = index ((x # insort a xs)) a"
             using x_less_a by auto
-          also have "... = index (insort a xs) a + 1" 
+          also have "... = index (insort a xs) a + 1"
             unfolding index_Cons using x_less_a by simp
           finally show "i - 1 < index (insort a xs) a" using False i_index by linarith
         qed
         also have "... = (x # xs) ! i" by (simp add: False nth_Cons')
         finally show ?thesis .
       qed
       finally show ?thesis .
     qed
   qed
 qed
 
 lemma insort_nth2:
-  assumes "sorted xs" and "a \<notin> set xs" 
-    and "i < length xs" and "i \<ge> index (insort a xs) a" 
+  assumes "sorted xs" and "a \<notin> set xs"
+    and "i < length xs" and "i \<ge> index (insort a xs) a"
     and "xs \<noteq> []"
   shows "insort a xs ! (Suc i) = xs ! i"
   using assms
 proof (induct xs arbitrary: a i)
   case Nil
   then show ?case by auto
 next
   case (Cons x xs)
   note sorted = Cons.prems(1)
   note a_notin = Cons.prems(2)
   note i_length = Cons.prems(3)
   note index_i = Cons.prems(4)
   show ?case
   proof (cases "a \<le> x")
     case True
     have "insort a (x # xs) ! (Suc i) = (a # x # xs) ! (Suc i)" using True by simp
     also have "... =  (x # xs) ! i"
       using Cons.prems(1) Cons.prems(5) True by force
     finally show ?thesis .
   next
     case False note x_less_a = False
     have "insort a (x # xs) ! (Suc i) = (x # insort a xs) ! (Suc i)" using False by simp
     also have "... = (x # xs) ! i"
     proof (cases "i = 0")
         case True
         then show ?thesis using index_i linear x_less_a by fastforce
       next
         case False note i0 = False
         show ?thesis
-        proof -     
+        proof -
           have Suc_i: "Suc (i - 1) = i"
             using i0 by auto
           have "(x # insort a xs) ! (Suc i) = (insort a xs) ! i"
             by (simp add: nth_Cons')
           also have "... = (insort a xs) ! Suc (i - 1)" using Suc_i by simp
           also have "... = xs ! (i - 1)"
           proof (rule Cons.hyps)
             show "sorted xs" using sorted by simp
             show "a \<notin> set xs" using a_notin by simp
             show "i - 1 < length xs" using i_length using Suc_i by auto
-            thus "xs \<noteq> []" by auto        
+            thus "xs \<noteq> []" by auto
             have "index (insort a (x # xs)) a = index ((x # insort a xs)) a" using x_less_a by simp
             also have "... = index (insort a xs) a + 1" unfolding index_Cons using x_less_a by simp
             finally show "index (insort a xs) a \<le> i - 1" using index_i i0 by auto
           qed
           also have "... = (x # xs) ! i" using Suc_i by auto
           finally show ?thesis .
         qed
       qed
       finally show ?thesis .
     qed
 qed
 
 lemma pick_index:
   assumes a: "a \<in> I" and a'_card: "a' < card I"
   shows "(pick I a' = a) = (index (sorted_list_of_set I) a = a')"
 proof -
   have finI: "finite I" using a'_card card_infinite by force
   have length_I: "length (sorted_list_of_set I) = card I"
     by (metis a'_card card_infinite distinct_card distinct_sorted_list_of_set
         not_less_zero set_sorted_list_of_set)
   let ?i = "index (sorted_list_of_set I) a"
   have "(sorted_list_of_set I) ! a' = pick I a'"
     by (rule sorted_list_of_set_eq_pick, auto simp add: finI a'_card length_I)
   moreover have "(sorted_list_of_set I) ! ?i = a"
     by (rule nth_index, simp add: a finI)
   ultimately show ?thesis
     by (metis a'_card distinct_sorted_list_of_set index_nth_id length_I)
 qed
 
 end
 
diff --git a/thys/Smith_Normal_Form/SNF_Uniqueness.thy b/thys/Smith_Normal_Form/SNF_Uniqueness.thy
--- a/thys/Smith_Normal_Form/SNF_Uniqueness.thy
+++ b/thys/Smith_Normal_Form/SNF_Uniqueness.thy
@@ -1,1180 +1,1180 @@
 (*
     Author:      Jose Divasón
-    Email:       jose.divason@unirioja.es           
+    Email:       jose.divason@unirioja.es
 *)
 
 section \<open>Uniqueness of the Smith normal form\<close>
 
 theory SNF_Uniqueness
 imports
   Cauchy_Binet
-  Smith_Normal_Form_JNF  
+  Smith_Normal_Form_JNF
   Admits_SNF_From_Diagonal_Iff_Bezout_Ring
-begin         
+begin
 
 lemma dvd_associated1:
   fixes a::"'a::comm_ring_1"
   assumes "\<exists>u. u dvd 1 \<and> a = u*b"
   shows "a dvd b \<and> b dvd a"
   using assms by auto
 
 
-text \<open>This is a key lemma. It demands the type class to be an integral domain. This means that 
+text \<open>This is a key lemma. It demands the type class to be an integral domain. This means that
 the uniqueness result will be obtained for GCD domains, instead of rings.\<close>
 lemma dvd_associated2:
   fixes a::"'a::idom"
   assumes ab: "a dvd b" and ba: "b dvd a" and a: "a\<noteq>0"
   shows "\<exists>u. u dvd 1 \<and> a = u*b"
 proof -
   obtain k where a_kb: "a = k*b" using ab unfolding dvd_def
     by (metis Groups.mult_ac(2) ba dvdE)
-  obtain q where b_qa: "b = q*a" using ba unfolding dvd_def    
+  obtain q where b_qa: "b = q*a" using ba unfolding dvd_def
     by (metis Groups.mult_ac(2) ab dvdE)
-  have 1: "a = k*q*a" using a_kb b_qa by auto  
+  have 1: "a = k*q*a" using a_kb b_qa by auto
   hence "k*q = 1" using a by simp
   thus ?thesis using 1 by (metis a_kb dvd_triv_left)
 qed
 
 corollary dvd_associated:
   fixes a::"'a::idom"
   assumes "a\<noteq>0"
   shows "(a dvd b \<and> b dvd a) = (\<exists>u. u dvd 1 \<and> a = u*b)"
   using assms dvd_associated1 dvd_associated2 by metis
 
 
 lemma exists_inj_ge_index:
   assumes S: "S \<subseteq> {0..<n}" and Sk: "card S = k"
   shows "\<exists>f. inj_on f {0..<k} \<and> f`{0..<k} = S \<and> (\<forall>i\<in>{0..<k}. i \<le> f i)"
 proof -
   have "\<exists>h. bij_betw h {0..<k} S"
     using S Sk ex_bij_betw_nat_finite subset_eq_atLeast0_lessThan_finite by blast
   from this obtain g where inj_on_g: "inj_on g {0..<k}" and gk_S: "g`{0..<k} = S"
     unfolding bij_betw_def by blast
   let ?f = "strict_from_inj k g"
   have "strict_mono_on ?f {0..<k}" by (rule strict_strict_from_inj[OF inj_on_g])
   hence 1: "inj_on ?f {0..<k}" using strict_mono_on_imp_inj_on by blast
-  have 2: "?f`{0..<k} = S" by (simp add: strict_from_inj_image' inj_on_g gk_S)  
+  have 2: "?f`{0..<k} = S" by (simp add: strict_from_inj_image' inj_on_g gk_S)
   have 3: "\<forall>i\<in>{0..<k}. i \<le> ?f i"
   proof
     fix i assume i: "i \<in> {0..<k}"
     let ?xs = "sorted_list_of_set (g`{0..<k})"
     have "strict_from_inj k g i = ?xs ! i" unfolding strict_from_inj_def using i by auto
-    moreover have "i \<le> ?xs ! i" 
+    moreover have "i \<le> ?xs ! i"
     proof (rule sorted_wrt_less_idx, rule sorted_distinct_imp_sorted_wrt)
       show "sorted ?xs"
         using sorted_sorted_list_of_set by blast
       show "distinct ?xs" using distinct_sorted_list_of_set by blast
       show "i < length ?xs"
         by (metis S Sk atLeast0LessThan distinct_card distinct_sorted_list_of_set gk_S i
             lessThan_iff set_sorted_list_of_set subset_eq_atLeast0_lessThan_finite)
     qed
     ultimately show "i \<le> ?f i" by auto
   qed
   show ?thesis using 1 2 3 by auto
 qed
 
 
 subsection \<open>More specific results about submatrices\<close>
 
 
 lemma diagonal_imp_submatrix0:
   assumes dA: "diagonal_mat A" and A_carrier: "A\<in> carrier_mat n m"
-  and Ik: "card I = k" and Jk: "card J = k" 
+  and Ik: "card I = k" and Jk: "card J = k"
   and r: "\<forall>row_index \<in> I. row_index < n" (*I \<subseteq> {0..<n}*)
   and c: "\<forall>col_index \<in> J. col_index < m"
   and a: "a<k" and b: "b<k"
 shows "submatrix A I J $$ (a, b) = 0 \<or> submatrix A I J $$ (a,b) = A $$(pick I a, pick I a)"
 proof (cases "submatrix A I J $$ (a, b) = 0")
   case True
   then show ?thesis by auto
 next
   case False note not0 = False
   have aux: "submatrix A I J $$ (a, b) = A $$(pick I a, pick J b)"
   proof (rule submatrix_index)
     have "card {i. i < dim_row A \<and> i \<in> I} = k"
       by (smt A_carrier Ik carrier_matD(1) equalityI mem_Collect_eq r subsetI)
-    moreover have "card {i. i < dim_col A \<and> i \<in> J} = k"      
-      by (metis (no_types, lifting) A_carrier Jk c carrier_matD(2) carrier_mat_def 
+    moreover have "card {i. i < dim_col A \<and> i \<in> J} = k"
+      by (metis (no_types, lifting) A_carrier Jk c carrier_matD(2) carrier_mat_def
          equalityI mem_Collect_eq subsetI)
-    ultimately show " a < card {i. i < dim_row A \<and> i \<in> I}" 
+    ultimately show " a < card {i. i < dim_row A \<and> i \<in> I}"
       and "b < card {i. i < dim_col A \<and> i \<in> J}" using a b by auto
   qed
   thus ?thesis
   proof (cases "pick I a = pick J b")
     case True
     then show ?thesis using aux by auto
   next
     case False
-    then show ?thesis 
+    then show ?thesis
       by (metis aux A_carrier Ik Jk a b c carrier_matD dA diagonal_mat_def pick_in_set_le r)
   qed
 qed
 
 
 
 lemma diagonal_imp_submatrix_element_not0:
   assumes dA: "diagonal_mat A"
   and A_carrier: "A \<in> carrier_mat n m"
-  and Ik: "card I = k" and Jk: "card J = k" 
+  and Ik: "card I = k" and Jk: "card J = k"
   and I: "I \<subseteq> {0..<n}"
   and J: "J \<subseteq> {0..<m}"
   and b: "b < k"
   and ex_not0: "\<exists>i. i<k \<and> submatrix A I J $$ (i, b) \<noteq> 0"
 shows "\<exists>!i. i<k \<and> submatrix A I J $$ (i, b) \<noteq> 0"
 proof -
   have I_eq: "I = {i. i < dim_row A \<and> i \<in> I}" using I A_carrier unfolding carrier_mat_def by auto
   have J_eq: "J = {i. i < dim_col A \<and> i \<in> J}" using J A_carrier unfolding carrier_mat_def by auto
-  obtain a where sub_ab: "submatrix A I J $$ (a, b) \<noteq> 0" and ak: "a < k" using ex_not0 by auto  
+  obtain a where sub_ab: "submatrix A I J $$ (a, b) \<noteq> 0" and ak: "a < k" using ex_not0 by auto
   moreover have "i = a" if sub_ib: "submatrix A I J $$ (i, b) \<noteq> 0" and ik: "i < k" for i
   proof -
     have 1: "pick I i < dim_row A"
       using I_eq Ik ik pick_in_set_le by auto
     have 2: "pick J b < dim_col A"
       using J_eq Jk b pick_le by auto
     have 3: "pick I a < dim_row A"
       using I_eq Ik calculation(2) pick_le by auto
     have "submatrix A I J $$ (i, b) = A $$ (pick I i, pick J b)"
       by (rule submatrix_index, insert I_eq Ik ik J_eq Jk b, auto)
     hence pick_Ii_Jb: "pick I i = pick J b" using dA sub_ib 1 2 unfolding diagonal_mat_def by auto
     have "submatrix A I J $$ (a, b) = A $$ (pick I a, pick J b)"
       by (rule submatrix_index, insert I_eq Ik ak J_eq Jk b, auto)
     hence pick_Ia_Jb: "pick I a = pick J b" using dA sub_ab 3 2 unfolding diagonal_mat_def by auto
     have pick_Ia_Ii: "pick I a = pick I i" using pick_Ii_Jb pick_Ia_Jb by simp
-    thus ?thesis by (metis Ik ak ik nat_neq_iff pick_mono_le)        
+    thus ?thesis by (metis Ik ak ik nat_neq_iff pick_mono_le)
   qed
   ultimately show ?thesis by auto
 qed
 
 
-lemma submatrix_index_exists: 
+lemma submatrix_index_exists:
   assumes A_carrier: "A\<in> carrier_mat n m"
-  and Ik: "card I = k" and Jk: "card J = k" 
+  and Ik: "card I = k" and Jk: "card J = k"
   and a: "a \<in> I" and b: "b \<in> J" and k: "k > 0"
   and I: "I \<subseteq> {0..<n}" and J: "J \<subseteq> {0..<m}"
-shows "\<exists>a' b'. a' < k \<and> b' < k \<and> submatrix A I J $$ (a',b') = A $$ (a,b) 
+shows "\<exists>a' b'. a' < k \<and> b' < k \<and> submatrix A I J $$ (a',b') = A $$ (a,b)
         \<and> a = pick I a' \<and> b = pick J b'"
 proof -
   let ?xs = "sorted_list_of_set I"
   let ?ys = "sorted_list_of_set J"
   have finI: "finite I" and finJ: "finite J" using k Ik Jk card_ge_0_finite by metis+
-  have set_xs: "set ?xs = I" by (rule set_sorted_list_of_set[OF finI]) 
+  have set_xs: "set ?xs = I" by (rule set_sorted_list_of_set[OF finI])
   have set_ys: "set ?ys = J" by (rule set_sorted_list_of_set[OF finJ])
   have a_in_xs: "a \<in> set ?xs" and b_in_ys: "b \<in> set ?ys" using set_xs a set_ys b by auto
   have length_xs: "length ?xs = k" by (metis Ik distinct_card set_xs sorted_list_of_set(3))
   have length_ys: "length ?ys = k" by (metis Jk distinct_card set_ys sorted_list_of_set(3))
-  obtain a' where a': "?xs ! a' = a" and a'_length: "a' < length ?xs" 
+  obtain a' where a': "?xs ! a' = a" and a'_length: "a' < length ?xs"
     by (meson a_in_xs in_set_conv_nth)
-  obtain b' where b': "?ys ! b' = b" and b'_length: "b' < length ?ys" 
+  obtain b' where b': "?ys ! b' = b" and b'_length: "b' < length ?ys"
     by (meson b_in_ys in_set_conv_nth)
   have pick_a: "a = pick I a'" using a' a'_length finI sorted_list_of_set_eq_pick by auto
   have pick_b: "b = pick J b'" using b' b'_length finJ sorted_list_of_set_eq_pick by auto
-  have I_rw: "I = {i. i < dim_row A \<and> i \<in> I}" and J_rw: "J = {i. i < dim_col A \<and> i \<in> J}" 
+  have I_rw: "I = {i. i < dim_row A \<and> i \<in> I}" and J_rw: "J = {i. i < dim_col A \<and> i \<in> J}"
     using I A_carrier J by auto
   have a'k: "a' < k" using a'_length length_xs by auto
   moreover have b'k: "b'<k" using b'_length length_ys by auto
-  moreover have sub_eq: "submatrix A I J $$ (a', b') = A $$ (a, b)" 
-    unfolding pick_a pick_b 
-    by (rule submatrix_index, insert J_rw I_rw Ik Jk a'_length length_xs b'_length length_ys, auto) 
-  ultimately show ?thesis using pick_a pick_b by auto  
+  moreover have sub_eq: "submatrix A I J $$ (a', b') = A $$ (a, b)"
+    unfolding pick_a pick_b
+    by (rule submatrix_index, insert J_rw I_rw Ik Jk a'_length length_xs b'_length length_ys, auto)
+  ultimately show ?thesis using pick_a pick_b by auto
 qed
 
 
 lemma mat_delete_submatrix_insert:
   assumes A_carrier: "A \<in> carrier_mat n m"
-  and Ik: "card I = k" and Jk: "card J = k" 
+  and Ik: "card I = k" and Jk: "card J = k"
   and I: "I \<subseteq> {0..<n}" and J: "J \<subseteq> {0..<m}"
   and a: "a < n" and b: "b < m"
   and k: "k < min n m"
   and a_notin_I: "a \<notin> I" and b_notin_J: "b \<notin> J"
-  and a'k: "a' < Suc k" and b'k:  "b' < Suc k" 
+  and a'k: "a' < Suc k" and b'k:  "b' < Suc k"
   and a_def: "pick (insert a I) a' = a"
   and b_def: "pick (insert b J) b' = b"
 shows "mat_delete (submatrix A (insert a I) (insert b J)) a' b' = submatrix A I J" (is "?lhs = ?rhs")
 proof (rule eq_matI)
   have I_eq: "I = {i. i < dim_row A \<and> i \<in> I}"
     using I A_carrier unfolding carrier_mat_def by auto
   have J_eq: "J = {i. i < dim_col A \<and> i \<in> J}"
     using J A_carrier unfolding carrier_mat_def by auto
-  have insert_I_eq: "insert a I = {i. i < dim_row A \<and> i \<in> insert a I}" 
+  have insert_I_eq: "insert a I = {i. i < dim_row A \<and> i \<in> insert a I}"
     using I A_carrier a k unfolding carrier_mat_def by auto
-  have card_Suc_k: "card {i. i < dim_row A \<and> i \<in> insert a I} = Suc k" 
+  have card_Suc_k: "card {i. i < dim_row A \<and> i \<in> insert a I} = Suc k"
     using insert_I_eq Ik a_notin_I
     by (metis I card_insert_disjoint finite_atLeastLessThan finite_subset)
-  have insert_J_eq: "insert b J = {i. i < dim_col A \<and> i \<in> insert b J}" 
+  have insert_J_eq: "insert b J = {i. i < dim_col A \<and> i \<in> insert b J}"
     using J A_carrier b k unfolding carrier_mat_def by auto
-  have card_Suc_k': "card {i. i < dim_col A \<and> i \<in> insert b J} = Suc k" 
+  have card_Suc_k': "card {i. i < dim_col A \<and> i \<in> insert b J} = Suc k"
     using insert_J_eq Jk b_notin_J
     by (metis J card_insert_disjoint finite_atLeastLessThan finite_subset)
   show "dim_row ?lhs = dim_row ?rhs"
     unfolding mat_delete_dim unfolding dim_submatrix using card_Suc_k I_eq Ik by auto
   show "dim_col ?lhs = dim_col ?rhs"
     unfolding mat_delete_dim unfolding dim_submatrix using card_Suc_k' J_eq Jk by auto
   fix i j assume i: "i < dim_row (submatrix A I J)"
     and j: "j < dim_col (submatrix A I J)"
   have ik: "i < k" by (metis I_eq Ik dim_submatrix(1) i)
   have jk: "j < k" by (metis J_eq Jk dim_submatrix(2) j)
   show "?lhs $$ (i, j) = ?rhs $$ (i, j)"
-  proof -    
+  proof -
     have index_eq1: "pick (insert a I) (insert_index a' i) = pick I i"
       by (rule pick_insert_index[OF Ik a_notin_I ik a_def], simp add: Ik a'k)
     have index_eq2: "pick (insert b J) (insert_index b' j) = pick J j"
       by (rule pick_insert_index[OF Jk b_notin_J jk b_def], simp add: Jk b'k)
-    have "?lhs $$ (i,j) 
+    have "?lhs $$ (i,j)
         = (submatrix A (insert a I) (insert b J)) $$ (insert_index a' i, insert_index b' j)"
     proof (rule mat_delete_index[symmetric, OF _ a'k b'k ik jk])
       show "submatrix A (insert a I) (insert b J) \<in> carrier_mat (Suc k) (Suc k)"
         by (metis card_Suc_k card_Suc_k' carrier_matI dim_submatrix(1) dim_submatrix(2))
     qed
-    also have "... = A $$ (pick (insert a I) (insert_index a' i), pick (insert b J) (insert_index b' j))" 
+    also have "... = A $$ (pick (insert a I) (insert_index a' i), pick (insert b J) (insert_index b' j))"
     proof (rule submatrix_index)
       show "insert_index a' i < card {i. i < dim_row A \<and> i \<in> insert a I}"
         using card_Suc_k ik insert_index_def by auto
       show "insert_index b' j < card {j. j < dim_col A \<and> j \<in> insert b J}"
         using card_Suc_k' insert_index_def jk by auto
     qed
     also have "... = A $$ (pick I i, pick J j)" unfolding index_eq1 index_eq2 by auto
     also have "... = submatrix A I J $$ (i,j)"
-      by (rule submatrix_index[symmetric], insert ik I_eq Ik Jk J_eq jk, auto)        
+      by (rule submatrix_index[symmetric], insert ik I_eq Ik Jk J_eq jk, auto)
     finally show ?thesis .
   qed
 qed
 
 
 
 subsection \<open>On the minors of a diagonal matrix\<close>
 
 lemma det_minors_diagonal:
   assumes dA: "diagonal_mat A" and A_carrier: "A \<in> carrier_mat n m"
-    and Ik: "card I = k" and Jk: "card J = k" 
-    and r: "I \<subseteq> {0..<n}" 
+    and Ik: "card I = k" and Jk: "card J = k"
+    and r: "I \<subseteq> {0..<n}"
     and c: "J \<subseteq> {0..<m}" and k: "k>0"
-  shows "det (submatrix A I J) = 0 
-  \<or> (\<exists>xs. (det (submatrix A I J) = prod_list xs \<or> det (submatrix A I J) = - prod_list xs) 
+  shows "det (submatrix A I J) = 0
+  \<or> (\<exists>xs. (det (submatrix A I J) = prod_list xs \<or> det (submatrix A I J) = - prod_list xs)
   \<and> set xs \<subseteq> {A$$(i,i)|i. i<min n m \<and> A$$(i,i)\<noteq> 0} \<and> length xs = k)"
   using Ik Jk r c k
 proof (induct k arbitrary: I J)
   case 0
   then show ?case by auto
 next
   case (Suc k)
   note cardI = Suc.prems(1)
   note cardJ = Suc.prems(2)
   note I = Suc.prems(3)
   note J = Suc.prems(4)
   have *: "{i. i < dim_row A \<and> i \<in> I} = I" using I Ik A_carrier carrier_mat_def by auto
   have **: "{j. j < dim_col A \<and> j \<in> J} = J" using J Jk A_carrier carrier_mat_def by auto
   show ?case
   proof (cases "k = 0")
     case True note k0 = True
     from this obtain a where aI: "I = {a}" using True cardI card_1_singletonE by auto
     from this obtain b where bJ: "J = {b}" using True cardJ card_1_singletonE by auto
     have an: "a<n" using aI I by auto
     have bm: "b<m" using bJ J by auto
-    have sub_carrier: "submatrix A {a} {b} \<in> carrier_mat 1 1" 
-      unfolding carrier_mat_def submatrix_def 
+    have sub_carrier: "submatrix A {a} {b} \<in> carrier_mat 1 1"
+      unfolding carrier_mat_def submatrix_def
       using * ** aI bJ by auto
-    have 1: "det (submatrix A {a} {b}) = (submatrix A {a} {b}) $$ (0,0)" 
+    have 1: "det (submatrix A {a} {b}) = (submatrix A {a} {b}) $$ (0,0)"
       by (rule det_singleton[OF sub_carrier])
-    have 2: "... = A $$ (a,b)" 
+    have 2: "... = A $$ (a,b)"
       by (rule submatrix_singleton_index[OF A_carrier an bm])
-    show ?thesis 
+    show ?thesis
     proof (cases "A $$ (a,b) \<noteq> 0")
       let ?xs = "[submatrix A {a} {b} $$ (0,0)]"
       case True
       hence "a = b" using dA A_carrier an bm unfolding diagonal_mat_def carrier_mat_def by auto
-      hence "set ?xs \<subseteq> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}" 
+      hence "set ?xs \<subseteq> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}"
         using 2 True an bm by auto
       moreover have "det (submatrix A {a} {b}) = prod_list ?xs" using 1 by auto
       moreover have "length ?xs = Suc k" using k0 by auto
       ultimately show ?thesis using an bm unfolding aI bJ by blast
     next
       case False
       then show ?thesis using 1 2 aI bJ by auto
-    qed   
+    qed
   next
-    case False   
+    case False
     hence k0: "0 < k" by simp
     have k: "k < min n m"
-      by (metis I J cardI cardJ le_imp_less_Suc less_Suc_eq_le min.commute 
-          min_def not_less subset_eq_atLeast0_lessThan_card)    
-    have subIJ_carrier: "(submatrix A I J) \<in> carrier_mat (Suc k) (Suc k)" 
+      by (metis I J cardI cardJ le_imp_less_Suc less_Suc_eq_le min.commute
+          min_def not_less subset_eq_atLeast0_lessThan_card)
+    have subIJ_carrier: "(submatrix A I J) \<in> carrier_mat (Suc k) (Suc k)"
       unfolding carrier_mat_def using * ** cardI cardJ
-      unfolding submatrix_def by auto  
+      unfolding submatrix_def by auto
     obtain b' where b'k: "b' < Suc k" by auto
-    let ?f="\<lambda>i. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b'"  
+    let ?f="\<lambda>i. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b'"
     have det_rw: "det (submatrix A I J)
         = (\<Sum>i<Suc k. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b')"
-      by (rule laplace_expansion_column[OF subIJ_carrier b'k])        
+      by (rule laplace_expansion_column[OF subIJ_carrier b'k])
     show ?thesis
     proof (cases "\<exists>a'<Suc k. submatrix A I J $$ (a',b') \<noteq> 0")
       case True
-      obtain a' where sub_IJ_0: "submatrix A I J $$ (a',b') \<noteq> 0" 
+      obtain a' where sub_IJ_0: "submatrix A I J $$ (a',b') \<noteq> 0"
         and a'k: "a' < Suc k"
         and unique: "\<forall>j. j<Suc k \<and> submatrix A I J $$ (j,b') \<noteq> 0 \<longrightarrow> j = a'"
         using diagonal_imp_submatrix_element_not0[OF dA A_carrier cardI cardJ I J b'k True] by auto
       have "submatrix A I J $$ (a', b') = A $$ (pick I a', pick J b')"
         by (rule submatrix_index, auto simp add: "*" a'k cardI "**" b'k cardJ)
-      from this obtain a b where an: "a < n" and bm:  "b < m" 
-        and sub_index: "submatrix A I J $$ (a', b') = A $$ (a, b)" 
+      from this obtain a b where an: "a < n" and bm:  "b < m"
+        and sub_index: "submatrix A I J $$ (a', b') = A $$ (a, b)"
         and pick_a: "pick I a' = a" and pick_b: "pick J b' = b"
         using * ** A_carrier a'k b'k cardI cardJ pick_le by fastforce
       obtain I' where aI': "I = insert a I'" and a_notin: "a \<notin> I'"
         by (metis Set.set_insert a'k cardI pick_a pick_in_set_le)
       obtain J' where bJ': "J = insert b J'" and b_notin: "b \<notin> J'"
-        by (metis Set.set_insert b'k cardJ pick_b pick_in_set_le)    
+        by (metis Set.set_insert b'k cardJ pick_b pick_in_set_le)
       have Suc_k0: "0 < Suc k" by simp
       have aI: "a \<in> I" using aI' by auto
       have bJ: "b \<in> J" using bJ' by auto
       have cardI':  "card I' = k"
-        by (metis aI' a_notin cardI card_infinite card_insert_disjoint 
+        by (metis aI' a_notin cardI card_infinite card_insert_disjoint
             finite_insert nat.inject nat.simps(3))
       have cardJ':  "card J' = k"
-        by (metis bJ' b_notin cardJ card_infinite card_insert_disjoint 
+        by (metis bJ' b_notin cardJ card_infinite card_insert_disjoint
             finite_insert nat.inject nat.simps(3))
       have I': "I' \<subseteq> {0..<n}" using I aI' by blast
       have J': "J' \<subseteq> {0..<m}" using J bJ' by blast
       have det_sub_I'J': "Determinant.det (submatrix A I' J') = 0 \<or>
-      (\<exists>xs. (det (submatrix A I' J') = prod_list xs \<or> det (submatrix A I' J') = - prod_list xs) 
+      (\<exists>xs. (det (submatrix A I' J') = prod_list xs \<or> det (submatrix A I' J') = - prod_list xs)
       \<and> set xs \<subseteq> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0} \<and> length xs = k)"
       proof (rule Suc.hyps[OF cardI' cardJ' _ _ k0])
         show "I' \<subseteq> {0..<n}" using I aI' by blast
         show "J' \<subseteq> {0..<m}" using J bJ' by blast
-      qed  
-      have mat_delete_sub: 
+      qed
+      have mat_delete_sub:
         "mat_delete (submatrix A (insert a I') (insert b J')) a' b' = submatrix A I' J'"
-        by (rule mat_delete_submatrix_insert[OF A_carrier cardI' cardJ' I' J' an bm k 
-              a_notin b_notin a'k b'k],insert pick_a pick_b aI' bJ', auto)        
+        by (rule mat_delete_submatrix_insert[OF A_carrier cardI' cardJ' I' J' an bm k
+              a_notin b_notin a'k b'k],insert pick_a pick_b aI' bJ', auto)
       have set_rw: "{0..<Suc k} = insert a' ({0..<Suc k}-{a'})"
         by (simp add: a'k insert_absorb)
       have rw0: "sum ?f ({0..<Suc k}-{a'}) = 0" by (rule sum.neutral, insert unique, auto)
       have "det (submatrix A I J)
         = (\<Sum>i<Suc k. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b')"
         by (rule laplace_expansion_column[OF subIJ_carrier b'k])
       also have "... = ?f a' + sum ?f ({0..<Suc k}-{a'})"
-        by (metis (no_types, lifting) Diff_iff atLeast0LessThan finite_atLeastLessThan 
+        by (metis (no_types, lifting) Diff_iff atLeast0LessThan finite_atLeastLessThan
             finite_insert set_rw singletonI sum.insert)
       also have "... = ?f a'" using rw0 unfolding cofactor_def by auto
-      also have "... = submatrix A I J $$ (a', b') * ((-1) ^ (a' + b') * det (submatrix A I' J'))" 
+      also have "... = submatrix A I J $$ (a', b') * ((-1) ^ (a' + b') * det (submatrix A I' J'))"
         unfolding cofactor_def using mat_delete_sub aI' bJ' by simp
-      finally have det_submatrix_IJ: "det (submatrix A I J) 
-         = A $$ (a, b) * ((- 1) ^ (a' + b') * det (submatrix A I' J'))" unfolding sub_index .       
+      finally have det_submatrix_IJ: "det (submatrix A I J)
+         = A $$ (a, b) * ((- 1) ^ (a' + b') * det (submatrix A I' J'))" unfolding sub_index .
       show ?thesis
       proof (cases "det (submatrix A I' J') = 0")
-        case True 
+        case True
         then show ?thesis using det_submatrix_IJ by auto
       next
         case False note det_not0 = False
-        from this obtain xs where prod_list_xs: "det (submatrix A I' J') = prod_list xs  
-          \<or> det (submatrix A I' J') = - prod_list xs" 
+        from this obtain xs where prod_list_xs: "det (submatrix A I' J') = prod_list xs
+          \<or> det (submatrix A I' J') = - prod_list xs"
           and xs: "set xs \<subseteq> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}"
           and length_xs: "length xs = k"
           using det_sub_I'J' by blast
         let ?ys = "A$$(a,b) # xs"
         have length_ys: "length ?ys = Suc k" using length_xs by auto
         have a_eq_b: "a=b"
           using A_carrier an bm sub_IJ_0 sub_index dA unfolding diagonal_mat_def by auto
         have A_aa_in: "A$$(a,a) \<in> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}"
           using a_eq_b an bm sub_IJ_0 sub_index by auto
         have ys: "set ?ys \<subseteq> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}"
           using xs A_aa_in a_eq_b by auto
         show ?thesis
         proof (cases "even (a'+b')")
           case True
           have det_submatrix_IJ: "det (submatrix A I J) = A $$ (a, b) * det (submatrix A I' J')"
             using det_submatrix_IJ True by auto
           show ?thesis
           proof (cases "det (submatrix A I' J') = prod_list xs")
             case True
-            have "det (submatrix A I J) = prod_list ?ys" 
+            have "det (submatrix A I J) = prod_list ?ys"
               using det_submatrix_IJ unfolding True by auto
             then show ?thesis using ys length_ys by blast
           next
             case False
             hence "det (submatrix A I' J') = - prod_list xs" using prod_list_xs by simp
             hence "det (submatrix A I J) = - prod_list ?ys" using det_submatrix_IJ by auto
             then show ?thesis using ys length_ys by blast
-          qed  
+          qed
         next
           case False
           have det_submatrix_IJ: "det (submatrix A I J) = A $$ (a, b) * - det (submatrix A I' J')"
             using det_submatrix_IJ False by auto
           show ?thesis
           proof (cases "det (submatrix A I' J') = prod_list xs")
             case True
-            have "det (submatrix A I J) = - prod_list ?ys" 
+            have "det (submatrix A I J) = - prod_list ?ys"
               using det_submatrix_IJ unfolding True by auto
             then show ?thesis using ys length_ys by blast
           next
             case False
             hence "det (submatrix A I' J') = - prod_list xs" using prod_list_xs by simp
             hence "det (submatrix A I J) = prod_list ?ys" using det_submatrix_IJ by auto
             then show ?thesis using ys length_ys by blast
           qed
         qed
-      qed  
+      qed
     next
-      case False    
+      case False
       have "sum ?f {0..<Suc k} = 0" by (rule sum.neutral, insert False, auto)
       thus ?thesis using det_rw
         by (simp add: atLeast0LessThan)
     qed
   qed
 qed
 
 
-definition "minors A k = {det (submatrix A I J)| I J. I \<subseteq> {0..<dim_row A} 
+definition "minors A k = {det (submatrix A I J)| I J. I \<subseteq> {0..<dim_row A}
   \<and> J \<subseteq> {0..<dim_col A} \<and> card I = k \<and> card J = k}"
 
 
 lemma Gcd_minors_dvd:
   fixes A::"'a::{semiring_Gcd,comm_ring_1} mat"
   assumes PAQ_B: "P * A * Q = B"
-  and P: "P \<in> carrier_mat m m"  
+  and P: "P \<in> carrier_mat m m"
   and A: "A \<in> carrier_mat m n"
-  and Q: "Q \<in> carrier_mat n n"  
+  and Q: "Q \<in> carrier_mat n n"
   and I: "I \<subseteq> {0..<dim_row A}" and J: "J \<subseteq> {0..<dim_col A}"
   and Ik: "card I = k" and Jk: "card J = k"
   shows "Gcd (minors A k) dvd det (submatrix B I J)"
 proof -
   let ?subPA = "submatrix (P * A) I UNIV"
   let ?subQ = "submatrix Q UNIV J"
   have subPA: "?subPA \<in> carrier_mat k n"
   proof -
     have "I = {i. i < dim_row P \<and> i \<in> I}" using P I A by auto
     hence "card {i. i < dim_row P \<and> i \<in> I} = k" using Ik by auto
     thus ?thesis using A unfolding submatrix_def by auto
-  qed 
+  qed
   have subQ: "submatrix Q UNIV J \<in> carrier_mat n k"
   proof -
     have J_eq: "J = {j. j < dim_col Q \<and> j \<in> J}" using Q J A by auto
     hence "card {j. j < dim_col Q \<and> j \<in> J} = k" using Jk by auto
-    moreover have "card {i. i < dim_row Q \<and> i \<in> UNIV} = n" using Q by auto    
+    moreover have "card {i. i < dim_row Q \<and> i \<in> UNIV} = n" using Q by auto
     ultimately show ?thesis unfolding submatrix_def by auto
   qed
   have sub_sub_PA: "(submatrix ?subPA UNIV I') = submatrix (P * A) I I'" for I'
     using submatrix_split2[symmetric] by auto
   have det_subPA_rw: "det (submatrix (P * A) I I') =
     (\<Sum>J' | J' \<subseteq> {0..<m} \<and> card J' = k. det ((submatrix P I J')) * det (submatrix A J' I'))"
     if I'1: "I' \<subseteq> {0..<n}" and I'2: "card I' = k" for I'
   proof -
     have "submatrix (P * A) I I' = submatrix P I UNIV * submatrix A UNIV I'"
       unfolding submatrix_mult ..
     also have "det ... = (\<Sum>C | C \<subseteq> {0..<m} \<and> card C = k.
      det (submatrix (submatrix P I UNIV) UNIV C) * det (submatrix (submatrix A UNIV I') C UNIV))"
     proof (rule Cauchy_Binet)
       have "I = {i. i < dim_row P \<and> i \<in> I}" using P I A by auto
       thus "submatrix P I UNIV \<in> carrier_mat k m" using Ik P unfolding submatrix_def by auto
       have "I' = {j. j < dim_col A \<and> j \<in> I'}" using I'1 A by auto
       thus "submatrix A UNIV I' \<in> carrier_mat m k" using I'2 A unfolding submatrix_def by auto
     qed
     also have "... =  (\<Sum>J' | J' \<subseteq> {0..<m} \<and> card J' = k.
       det (submatrix P I J') * det (submatrix A J' I'))"
       unfolding submatrix_split2[symmetric] submatrix_split[symmetric] by simp
     finally show ?thesis .
   qed
   have "det (submatrix B I J) = det (submatrix (P*A*Q) I J)" using PAQ_B by simp
   also have "... = det (?subPA * ?subQ)" unfolding submatrix_mult by auto
-  also have "... = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k. det (submatrix ?subPA UNIV I') 
+  also have "... = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k. det (submatrix ?subPA UNIV I')
     * det (submatrix ?subQ I' UNIV))"
     by (rule Cauchy_Binet[OF subPA subQ])
   also have "... = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k.
     det (submatrix (P * A) I I') * det (submatrix Q I' J))"
     using submatrix_split[symmetric, of Q] submatrix_split2[symmetric, of "P*A"] by presburger
   also have "... = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k. \<Sum>J' | J' \<subseteq> {0..<m} \<and> card J' = k.
     det (submatrix P I J') * det (submatrix A J' I') * det (submatrix Q I' J))"
     using det_subPA_rw by (simp add: semiring_0_class.sum_distrib_right)
-  finally have det_rw: "det (submatrix B I J) = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k. 
+  finally have det_rw: "det (submatrix B I J) = (\<Sum>I' | I' \<subseteq> {0..<n} \<and> card I' = k.
     \<Sum>J' | J' \<subseteq> {0..<m} \<and> card J' = k.
     det (submatrix P I J') * det (submatrix A J' I') * det (submatrix Q I' J))" .
   show ?thesis
   proof (unfold det_rw, (rule dvd_sum)+)
     fix I' J'
     assume I': "I' \<in> {I'. I' \<subseteq> {0..<n} \<and> card I' = k}"
       and J': "J' \<in> {J'. J' \<subseteq> {0..<m} \<and> card J' = k}"
-    have "Gcd (minors A k) dvd  det (submatrix A J' I')" 
+    have "Gcd (minors A k) dvd  det (submatrix A J' I')"
       by (rule Gcd_dvd, unfold minors_def, insert A I' J', auto)
-    then show "Gcd (minors A k) dvd det (submatrix P I J') * det (submatrix A J' I') 
+    then show "Gcd (minors A k) dvd det (submatrix P I J') * det (submatrix A J' I')
           * det (submatrix Q I' J)" by auto
   qed
 qed
 
 (*The conclusion could be simplified since we have S = I.*)
 lemma det_minors_diagonal2:
   assumes dA: "diagonal_mat A" and A_carrier: "A \<in> carrier_mat n m"
-    and Ik: "card I = k" and Jk: "card J = k" 
-    and r: "I \<subseteq> {0..<n}" 
+    and Ik: "card I = k" and Jk: "card J = k"
+    and r: "I \<subseteq> {0..<n}"
     and c: "J \<subseteq> {0..<m}" and k: "k>0"
   shows "det (submatrix A I J) = 0 \<or> (\<exists>S. S \<subseteq> {0..<min n m} \<and> card S = k \<and> S=I \<and>
    (det (submatrix A I J) = (\<Prod>i\<in>S. A $$ (i,i)) \<or> det (submatrix A I J) = - (\<Prod>i\<in>S. A $$ (i,i))))"
   using Ik Jk r c k
 proof (induct k arbitrary: I J)
  case 0
   then show ?case by auto
 next
   case (Suc k)
   note cardI = Suc.prems(1)
   note cardJ = Suc.prems(2)
   note I = Suc.prems(3)
   note J = Suc.prems(4)
   have *: "{i. i < dim_row A \<and> i \<in> I} = I" using I Ik A_carrier carrier_mat_def by auto
   have **: "{j. j < dim_col A \<and> j \<in> J} = J" using J Jk A_carrier carrier_mat_def by auto
   show ?case
   proof (cases "k = 0")
     case True note k0 = True
     from this obtain a where aI: "I = {a}" using True cardI card_1_singletonE by auto
     from this obtain b where bJ: "J = {b}" using True cardJ card_1_singletonE by auto
     have an: "a<n" using aI I by auto
     have bm: "b<m" using bJ J by auto
-    have sub_carrier: "submatrix A {a} {b} \<in> carrier_mat 1 1" 
-      unfolding carrier_mat_def submatrix_def 
+    have sub_carrier: "submatrix A {a} {b} \<in> carrier_mat 1 1"
+      unfolding carrier_mat_def submatrix_def
       using * ** aI bJ by auto
-    have 1: "det (submatrix A {a} {b}) = (submatrix A {a} {b}) $$ (0,0)" 
+    have 1: "det (submatrix A {a} {b}) = (submatrix A {a} {b}) $$ (0,0)"
       by (rule det_singleton[OF sub_carrier])
-    have 2: "... = A $$ (a,b)" 
+    have 2: "... = A $$ (a,b)"
       by (rule submatrix_singleton_index[OF A_carrier an bm])
-    show ?thesis 
+    show ?thesis
     proof (cases "A $$ (a,b) \<noteq> 0")
       let ?S="{a}"
       case True
       hence ab: "a = b" using dA A_carrier an bm unfolding diagonal_mat_def carrier_mat_def by auto
       hence "?S \<subseteq> {0..<min n m}" using an bm by auto
       moreover have "det (submatrix A {a} {b}) = (\<Prod>i\<in>?S. A $$ (i, i))" using 1 2 ab by auto
       moreover have "card ?S = Suc k" using k0 by auto
       ultimately show ?thesis using an bm unfolding aI bJ by blast
     next
       case False
       then show ?thesis using 1 2 aI bJ by auto
-    qed   
+    qed
   next
     case False
     hence k0: "0 < k" by simp
     have k: "k < min n m"
-      by (metis I J cardI cardJ le_imp_less_Suc less_Suc_eq_le min.commute 
-          min_def not_less subset_eq_atLeast0_lessThan_card)    
-    have subIJ_carrier: "(submatrix A I J) \<in> carrier_mat (Suc k) (Suc k)" 
+      by (metis I J cardI cardJ le_imp_less_Suc less_Suc_eq_le min.commute
+          min_def not_less subset_eq_atLeast0_lessThan_card)
+    have subIJ_carrier: "(submatrix A I J) \<in> carrier_mat (Suc k) (Suc k)"
       unfolding carrier_mat_def using * ** cardI cardJ
-      unfolding submatrix_def by auto  
+      unfolding submatrix_def by auto
     obtain b' where b'k: "b' < Suc k" by auto
-    let ?f="\<lambda>i. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b'"  
+    let ?f="\<lambda>i. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b'"
     have det_rw: "det (submatrix A I J)
         = (\<Sum>i<Suc k. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b')"
-      by (rule laplace_expansion_column[OF subIJ_carrier b'k])        
+      by (rule laplace_expansion_column[OF subIJ_carrier b'k])
     show ?thesis
     proof (cases "\<exists>a'<Suc k. submatrix A I J $$ (a',b') \<noteq> 0")
       case True
-      obtain a' where sub_IJ_0: "submatrix A I J $$ (a',b') \<noteq> 0" 
+      obtain a' where sub_IJ_0: "submatrix A I J $$ (a',b') \<noteq> 0"
         and a'k: "a' < Suc k"
         and unique: "\<forall>j. j<Suc k \<and> submatrix A I J $$ (j,b') \<noteq> 0 \<longrightarrow> j = a'"
         using diagonal_imp_submatrix_element_not0[OF dA A_carrier cardI cardJ I J b'k True] by auto
       have "submatrix A I J $$ (a', b') = A $$ (pick I a', pick J b')"
         by (rule submatrix_index, auto simp add: "*" a'k cardI "**" b'k cardJ)
-      from this obtain a b where an: "a < n" and bm:  "b < m" 
-        and sub_index: "submatrix A I J $$ (a', b') = A $$ (a, b)" 
+      from this obtain a b where an: "a < n" and bm:  "b < m"
+        and sub_index: "submatrix A I J $$ (a', b') = A $$ (a, b)"
         and pick_a: "pick I a' = a" and pick_b: "pick J b' = b"
         using * ** A_carrier a'k b'k cardI cardJ pick_le by fastforce
       obtain I' where aI': "I = insert a I'" and a_notin: "a \<notin> I'"
         by (metis Set.set_insert a'k cardI pick_a pick_in_set_le)
       obtain J' where bJ': "J = insert b J'" and b_notin: "b \<notin> J'"
-        by (metis Set.set_insert b'k cardJ pick_b pick_in_set_le)    
+        by (metis Set.set_insert b'k cardJ pick_b pick_in_set_le)
       have Suc_k0: "0 < Suc k" by simp
       have aI: "a \<in> I" using aI' by auto
       have bJ: "b \<in> J" using bJ' by auto
       have cardI':  "card I' = k"
-        by (metis aI' a_notin cardI card_infinite card_insert_disjoint 
+        by (metis aI' a_notin cardI card_infinite card_insert_disjoint
             finite_insert nat.inject nat.simps(3))
       have cardJ':  "card J' = k"
-        by (metis bJ' b_notin cardJ card_infinite card_insert_disjoint 
+        by (metis bJ' b_notin cardJ card_infinite card_insert_disjoint
             finite_insert nat.inject nat.simps(3))
       have I': "I' \<subseteq> {0..<n}" using I aI' by blast
       have J': "J' \<subseteq> {0..<m}" using J bJ' by blast
       have det_sub_I'J': "det (submatrix A I' J') = 0 \<or> (\<exists>S\<subseteq>{0..<min n m}. card S = k \<and> S=I'
-        \<and> (det (submatrix A I' J') = (\<Prod>i\<in>S. A $$ (i, i)) 
+        \<and> (det (submatrix A I' J') = (\<Prod>i\<in>S. A $$ (i, i))
         \<or> det (submatrix A I' J') = - (\<Prod>i\<in>S. A $$ (i, i))))"
       proof (rule Suc.hyps[OF cardI' cardJ' _ _ k0])
         show "I' \<subseteq> {0..<n}" using I aI' by blast
         show "J' \<subseteq> {0..<m}" using J bJ' by blast
-      qed  
-      have mat_delete_sub: 
+      qed
+      have mat_delete_sub:
         "mat_delete (submatrix A (insert a I') (insert b J')) a' b' = submatrix A I' J'"
-        by (rule mat_delete_submatrix_insert[OF A_carrier cardI' cardJ' I' J' an bm k 
-              a_notin b_notin a'k b'k],insert pick_a pick_b aI' bJ', auto)        
+        by (rule mat_delete_submatrix_insert[OF A_carrier cardI' cardJ' I' J' an bm k
+              a_notin b_notin a'k b'k],insert pick_a pick_b aI' bJ', auto)
       have set_rw: "{0..<Suc k} = insert a' ({0..<Suc k}-{a'})"
         by (simp add: a'k insert_absorb)
       have rw0: "sum ?f ({0..<Suc k}-{a'}) = 0" by (rule sum.neutral, insert unique, auto)
       have "det (submatrix A I J)
         = (\<Sum>i<Suc k. submatrix A I J $$ (i, b') * cofactor (submatrix A I J) i b')"
         by (rule laplace_expansion_column[OF subIJ_carrier b'k])
       also have "... = ?f a' + sum ?f ({0..<Suc k}-{a'})"
-        by (metis (no_types, lifting) Diff_iff atLeast0LessThan finite_atLeastLessThan 
+        by (metis (no_types, lifting) Diff_iff atLeast0LessThan finite_atLeastLessThan
             finite_insert set_rw singletonI sum.insert)
       also have "... = ?f a'" using rw0 unfolding cofactor_def by auto
-      also have "... = submatrix A I J $$ (a', b') * ((-1) ^ (a' + b') * det (submatrix A I' J'))" 
+      also have "... = submatrix A I J $$ (a', b') * ((-1) ^ (a' + b') * det (submatrix A I' J'))"
         unfolding cofactor_def using mat_delete_sub aI' bJ' by simp
-      finally have det_submatrix_IJ: "det (submatrix A I J) 
-         = A $$ (a, b) * ((- 1) ^ (a' + b') * det (submatrix A I' J'))" unfolding sub_index .       
+      finally have det_submatrix_IJ: "det (submatrix A I J)
+         = A $$ (a, b) * ((- 1) ^ (a' + b') * det (submatrix A I' J'))" unfolding sub_index .
       show ?thesis
       proof (cases "det (submatrix A I' J') = 0")
-        case True 
+        case True
         then show ?thesis using det_submatrix_IJ by auto
       next
         case False note det_not0 = False
-        from this obtain xs where prod_list_xs: "det (submatrix A I' J') = (\<Prod>i\<in>xs. A $$ (i, i))  
-          \<or> det (submatrix A I' J') = - (\<Prod>i\<in>xs. A $$ (i, i))" 
+        from this obtain xs where prod_list_xs: "det (submatrix A I' J') = (\<Prod>i\<in>xs. A $$ (i, i))
+          \<or> det (submatrix A I' J') = - (\<Prod>i\<in>xs. A $$ (i, i))"
           and xs: "xs\<subseteq>{0..<min n m}"
           and length_xs: "card xs = k"
           and xs_I': "xs=I'"
           using det_sub_I'J' by blast
         let ?ys = "insert a xs"
         have a_notin_xs: "a \<notin> xs"
           by (simp add: xs_I' a_notin)
         have length_ys: "card ?ys = Suc k"
           using length_xs a_notin_xs by (simp add: card_ge_0_finite k0)
         have a_eq_b: "a=b"
           using A_carrier an bm sub_IJ_0 sub_index dA unfolding diagonal_mat_def by auto
         have A_aa_in: "A$$(a,a) \<in> {A $$ (i, i) |i. i < min n m \<and> A $$ (i, i) \<noteq> 0}"
           using a_eq_b an bm sub_IJ_0 sub_index by auto
         show ?thesis
         proof (cases "even (a'+b')")
           case True
           have det_submatrix_IJ: "det (submatrix A I J) = A $$ (a, b) * det (submatrix A I' J')"
             using det_submatrix_IJ True by auto
           show ?thesis
           proof (cases "det (submatrix A I' J') = (\<Prod>i\<in>xs. A $$ (i, i))")
             case True
-            have "det (submatrix A I J) = (\<Prod>i\<in>?ys. A $$ (i, i))" 
+            have "det (submatrix A I J) = (\<Prod>i\<in>?ys. A $$ (i, i))"
               using det_submatrix_IJ unfolding True a_eq_b
-              by (metis (no_types, lifting) a_notin_xs a_eq_b 
+              by (metis (no_types, lifting) a_notin_xs a_eq_b
                   card_ge_0_finite k0 length_xs prod.insert)
             then show ?thesis using length_ys
               using a_eq_b an bm xs xs_I'
               by (simp add: aI')
           next
             case False
             hence "det (submatrix A I' J') = - (\<Prod>i\<in>xs. A $$ (i, i))" using prod_list_xs by simp
             hence "det (submatrix A I J) = -(\<Prod>i\<in>?ys. A $$ (i, i))" using det_submatrix_IJ a_eq_b
-              by (metis (no_types, lifting) a_notin_xs card_ge_0_finite k0 
+              by (metis (no_types, lifting) a_notin_xs card_ge_0_finite k0
                   length_xs mult_minus_right prod.insert)
             then show ?thesis using length_ys
               using a_eq_b an bm xs aI' xs_I' by force
-          qed  
+          qed
         next
           case False
           have det_submatrix_IJ: "det (submatrix A I J) = A $$ (a, b) * - det (submatrix A I' J')"
             using det_submatrix_IJ False by auto
           show ?thesis
           proof (cases "det (submatrix A I' J') = (\<Prod>i\<in>xs. A $$ (i, i))")
             case True
-            have "det (submatrix A I J) = - (\<Prod>i\<in>?ys. A $$ (i, i))" 
+            have "det (submatrix A I J) = - (\<Prod>i\<in>?ys. A $$ (i, i))"
               using det_submatrix_IJ unfolding True
-              by (metis (no_types, lifting) a_eq_b a_notin_xs card_ge_0_finite k0 
+              by (metis (no_types, lifting) a_eq_b a_notin_xs card_ge_0_finite k0
                   length_xs mult_minus_right prod.insert)
             then show ?thesis using length_ys
               using a_eq_b an bm xs aI' xs_I' by force
           next
             case False
             hence "det (submatrix A I' J') = - (\<Prod>i\<in>xs. A $$ (i, i))" using prod_list_xs by simp
             hence "det (submatrix A I J) = (\<Prod>i\<in>?ys. A $$ (i, i))" using det_submatrix_IJ
-              by (metis (mono_tags, lifting) a_eq_b a_notin_xs card_ge_0_finite 
+              by (metis (mono_tags, lifting) a_eq_b a_notin_xs card_ge_0_finite
                   equation_minus_iff k0 length_xs prod.insert)
-            then show ?thesis using length_ys 
+            then show ?thesis using length_ys
               using a_eq_b an bm xs aI' xs_I' by force
           qed
         qed
-      qed  
+      qed
     next
-      case False    
+      case False
       have "sum ?f {0..<Suc k} = 0" by (rule sum.neutral, insert False, auto)
       thus ?thesis using det_rw
         by (simp add: atLeast0LessThan)
     qed
   qed
 qed
 
 
 subsection \<open>Relating minors and GCD\<close>
 
 lemma diagonal_dvd_Gcd_minors:
   fixes A::"'a::{semiring_Gcd,comm_ring_1} mat"
-  assumes A: "A \<in> carrier_mat n m" 
+  assumes A: "A \<in> carrier_mat n m"
     and SNF_A: "Smith_normal_form_mat A"
 shows "(\<Prod>i=0..<k. A $$ (i,i)) dvd Gcd (minors A k)"
 proof (cases "k=0")
   case True
   then show ?thesis by auto
 next
   case False
   hence k: "0<k" by simp
   show ?thesis
   proof (rule Gcd_greatest)
-    have diag_A: "diagonal_mat A" 
+    have diag_A: "diagonal_mat A"
       using SNF_A unfolding Smith_normal_form_mat_def isDiagonal_mat_def diagonal_mat_def by auto
     fix b assume b_in_minors: "b \<in> minors A k"
     show "(\<Prod>i = 0..<k. A $$ (i, i)) dvd b"
     proof (cases "b=0")
       case True
       then show ?thesis by auto
     next
       case False
      obtain I J where b: "b = det (submatrix A I J)" and I: "I \<subseteq> {0..<dim_row A} "
-    and J: "J \<subseteq> {0..<dim_col A}" and Ik: "card I = k" and Jk: "card J = k" 
+    and J: "J \<subseteq> {0..<dim_col A}" and Ik: "card I = k" and Jk: "card J = k"
        using b_in_minors  unfolding minors_def by blast
     obtain S where S: "S \<subseteq> {0..<min n m}" and Sk: "card S = k"
-      and det_subS: "det (submatrix A I J) = (\<Prod>i\<in>S. A $$ (i,i)) 
+      and det_subS: "det (submatrix A I J) = (\<Prod>i\<in>S. A $$ (i,i))
         \<or> det (submatrix A I J) = -(\<Prod>i\<in>S. A $$ (i,i))"
       using det_minors_diagonal2[OF diag_A A Ik Jk _ _ k] I J A False b by auto
-    obtain f where inj_f: "inj_on f {0..<k}" and fk_S: "f`{0..<k} = S" 
+    obtain f where inj_f: "inj_on f {0..<k}" and fk_S: "f`{0..<k} = S"
       and i_fi: " (\<forall>i\<in>{0..<k}. i \<le> f i)" using exists_inj_ge_index[OF S Sk] by blast
-    have "(\<Prod>i = 0..<k. A $$ (i, i)) dvd (\<Prod>i\<in>{0..<k}. A $$ (f i,f i))" 
-      by (rule prod_dvd_prod, rule SNF_divides_diagonal[OF A SNF_A], insert fk_S S i_fi, force+) 
-    also have "... = (\<Prod>i\<in>f`{0..<k}. A $$ (i,i))" 
+    have "(\<Prod>i = 0..<k. A $$ (i, i)) dvd (\<Prod>i\<in>{0..<k}. A $$ (f i,f i))"
+      by (rule prod_dvd_prod, rule SNF_divides_diagonal[OF A SNF_A], insert fk_S S i_fi, force+)
+    also have "... = (\<Prod>i\<in>f`{0..<k}. A $$ (i,i))"
       by (rule prod.reindex[symmetric, unfolded o_def, OF inj_f])
     also have "... = (\<Prod>i\<in>S. A $$ (i, i))" using fk_S by auto
-    finally have *: "(\<Prod>i = 0..<k. A $$ (i, i)) dvd (\<Prod>i\<in>S. A $$ (i, i))" .  
+    finally have *: "(\<Prod>i = 0..<k. A $$ (i, i)) dvd (\<Prod>i\<in>S. A $$ (i, i))" .
     show "(\<Prod>i = 0..<k. A $$ (i, i)) dvd b" using det_subS b * by auto
   qed
-qed 
+qed
 qed
 
 
 lemma Gcd_minors_dvd_diagonal:
   fixes A::"'a::{semiring_Gcd,comm_ring_1} mat"
-  assumes A: "A \<in> carrier_mat n m" 
+  assumes A: "A \<in> carrier_mat n m"
     and SNF_A: "Smith_normal_form_mat A"
     and k: "k \<le> min n m"
   shows "Gcd (minors A k) dvd (\<Prod>i=0..<k. A $$ (i,i))"
 proof (rule Gcd_dvd)
-  define I where "I = {0..<k}"  
+  define I where "I = {0..<k}"
   have "(\<Prod>i = 0..<k. A $$ (i, i)) = det (submatrix A I I)"
   proof -
     have sub_eq: "submatrix A I I = mat k k (\<lambda>(i, j). A $$ (i, j))"
     proof (rule eq_matI, auto)
       have "I = {i. i < dim_row A \<and> i \<in> I}" unfolding I_def using A k by auto
       hence ck: "card {i. i < dim_row A \<and> i \<in> I} = k"
         unfolding I_def using card_atLeastLessThan by presburger
       have "I = {i. i < dim_col A \<and> i \<in> I}" unfolding I_def using A k by auto
-      hence ck2: "card {j. j < dim_col A \<and> j \<in> I} = k" 
+      hence ck2: "card {j. j < dim_col A \<and> j \<in> I} = k"
         unfolding I_def using card_atLeastLessThan by presburger
       show dr: "dim_row (submatrix A I I) = k" using ck unfolding submatrix_def by auto
       show dc: "dim_col (submatrix A I I) = k" using ck2 unfolding submatrix_def by auto
       fix i j assume i: "i < k" and j: "j < k"
       have p1: "pick I i = i"
       proof -
         have "{0..<i} = {a \<in> I. a < i}" using I_def i by auto
         hence i_eq: "i = card {a \<in> I. a < i}"
           by (metis card_atLeastLessThan diff_zero)
         have "pick I i = pick I (card {a \<in> I. a < i})" using i_eq by simp
         also have "... = i" by (rule pick_card_in_set, insert i I_def, simp)
         finally show ?thesis .
       qed
       have p2: "pick I j = j"
       proof -
         have "{0..<j} = {a \<in> I. a < j}" using I_def j by auto
         hence j_eq: "j = card {a \<in> I. a < j}"
           by (metis card_atLeastLessThan diff_zero)
         have "pick I j = pick I (card {a \<in> I. a < j})" using j_eq by simp
         also have "... = j" by (rule pick_card_in_set, insert j I_def, simp)
         finally show ?thesis .
       qed
       have "submatrix A I I $$ (i, j) = A $$ (pick I i, pick I j)"
       proof (rule submatrix_index)
         show "i < card {i. i < dim_row A \<and> i \<in> I}" by (metis dim_submatrix(1) dr i)
         show "j < card {j. j < dim_col A \<and> j \<in> I}" by (metis dim_submatrix(2) dc j)
       qed
       also have "... = A $$ (i,j)" using p1 p2 by simp
       finally show "submatrix A I I $$ (i, j) = A $$ (i, j)" .
     qed
     hence "det (submatrix A I I) = det (mat k k (\<lambda>(i, j). A $$ (i, j)))" by simp
     also have "... = prod_list (diag_mat (mat k k (\<lambda>(i, j). A $$ (i, j))))"
     proof (rule det_upper_triangular)
       show "mat k k (\<lambda>(i, j). A $$ (i, j)) \<in> carrier_mat k k" by auto
-      show "upper_triangular (Matrix.mat k k (\<lambda>(i, j). A $$ (i, j)))" 
+      show "upper_triangular (Matrix.mat k k (\<lambda>(i, j). A $$ (i, j)))"
         using SNF_A A k unfolding Smith_normal_form_mat_def isDiagonal_mat_def by auto
     qed
     also have "... = (\<Prod>i = 0..<k. A $$ (i, i))"
-      by (metis (mono_tags, lifting) atLeastLessThan_iff dim_row_mat(1) index_mat(1) 
-          prod.cong prod_list_diag_prod split_conv)    
+      by (metis (mono_tags, lifting) atLeastLessThan_iff dim_row_mat(1) index_mat(1)
+          prod.cong prod_list_diag_prod split_conv)
     finally show ?thesis ..
   qed
   moreover have "I \<subseteq> {0..<dim_row A}" using k A I_def by auto
   moreover have "I \<subseteq> {0..<dim_col A}" using k A I_def by auto
   moreover have "card I = k" using I_def by auto
-  ultimately show "(\<Prod>i = 0..<k. A $$ (i, i)) \<in> minors A k" unfolding minors_def by auto  
+  ultimately show "(\<Prod>i = 0..<k. A $$ (i, i)) \<in> minors A k" unfolding minors_def by auto
 qed
 
 
 
 lemma Gcd_minors_A_dvd_Gcd_minors_PAQ:
  fixes A::"'a::{semiring_Gcd,comm_ring_1} mat"
- assumes A: "A \<in> carrier_mat m n" 
+ assumes A: "A \<in> carrier_mat m n"
     and P: "P \<in> carrier_mat m m" and Q: "Q \<in> carrier_mat n n"
   shows "Gcd (minors A k) dvd Gcd (minors (P*A*Q) k)"
 proof (rule Gcd_greatest)
   let ?B="(P * A * Q)"
   fix b assume "b \<in> minors ?B k"
   from this  obtain I J where b: "b = det (submatrix ?B I J)" and I: "I \<subseteq> {0..<dim_row ?B}"
-    and J: "J \<subseteq> {0..<dim_col ?B}" and Ik: "card I = k" and Jk: "card J = k" 
+    and J: "J \<subseteq> {0..<dim_col ?B}" and Ik: "card I = k" and Jk: "card J = k"
     unfolding minors_def by blast
   have "Gcd (minors A k) dvd det (submatrix ?B I J)"
-    by (rule Gcd_minors_dvd[OF _ P A Q _ _ Ik Jk], insert A I J P Q, auto)    
-  thus "Gcd (minors A k) dvd b" using b by simp  
+    by (rule Gcd_minors_dvd[OF _ P A Q _ _ Ik Jk], insert A I J P Q, auto)
+  thus "Gcd (minors A k) dvd b" using b by simp
 qed
 
 
 lemma Gcd_minors_PAQ_dvd_Gcd_minors_A:
  fixes A::"'a::{semiring_Gcd,comm_ring_1} mat"
- assumes A: "A \<in> carrier_mat m n" 
-    and P: "P \<in> carrier_mat m m" 
+ assumes A: "A \<in> carrier_mat m n"
+    and P: "P \<in> carrier_mat m m"
     and Q: "Q \<in> carrier_mat n n"
-    and inv_P: "invertible_mat P" 
+    and inv_P: "invertible_mat P"
     and inv_Q: "invertible_mat Q"
   shows "Gcd (minors (P*A*Q) k) dvd Gcd (minors A k)"
 proof (rule Gcd_greatest)
   let ?B = "P * A * Q"
   fix b assume "b \<in> minors A k"
   from this obtain I J where b: "b = det (submatrix A I J)" and I: "I \<subseteq> {0..<dim_row A} "
-    and J: "J \<subseteq> {0..<dim_col A}" and Ik: "card I = k" and Jk: "card J = k" 
+    and J: "J \<subseteq> {0..<dim_col A}" and Ik: "card I = k" and Jk: "card J = k"
     unfolding minors_def by blast
-  obtain P' where PP': "inverts_mat P P'" and P'P: "inverts_mat P' P" 
-    using inv_P unfolding invertible_mat_def by auto 
-  obtain Q' where QQ': "inverts_mat Q Q'" and Q'Q: "inverts_mat Q' Q" 
+  obtain P' where PP': "inverts_mat P P'" and P'P: "inverts_mat P' P"
+    using inv_P unfolding invertible_mat_def by auto
+  obtain Q' where QQ': "inverts_mat Q Q'" and Q'Q: "inverts_mat Q' Q"
     using inv_Q unfolding invertible_mat_def by auto
   have P': "P' \<in> carrier_mat m m" using PP' P'P unfolding inverts_mat_def
     by (metis P carrier_matD(1) carrier_matD(2) carrier_matI index_mult_mat(3) index_one_mat(3))
   have Q': "Q' \<in> carrier_mat n n"
     using QQ' Q'Q unfolding inverts_mat_def
     by (metis Q carrier_matD(1) carrier_matD(2) carrier_matI index_mult_mat(3) index_one_mat(3))
   have rw: "P' *?B *Q' = A"
   proof -
     have f1: "P' * P = 1\<^sub>m m"
       by (metis (no_types) P' P'P carrier_matD(1) inverts_mat_def)
     have *: "P' * P * A = P' * (P * A)"
       by (meson A P P' assoc_mult_mat)
     have " P' * (P * A * Q) * Q' =  P' * P * A * Q * Q'"
       by (smt A P P' Q assoc_mult_mat mult_carrier_mat)
-    also have "... =  P' * P * (A * Q * Q')" 
+    also have "... =  P' * P * (A * Q * Q')"
       using A P P' Q Q' f1 * by auto
     also have "... = A * Q * Q'" using P'P A P' unfolding inverts_mat_def by auto
     also have "... = A" using QQ' A Q' Q unfolding inverts_mat_def by auto
     finally show ?thesis .
   qed
   have "Gcd (minors ?B k) dvd det (submatrix (P'*?B*Q') I J)"
-    by (rule Gcd_minors_dvd[OF _ P' _ Q' _ _ Ik Jk], insert P A Q I J, auto)    
+    by (rule Gcd_minors_dvd[OF _ P' _ Q' _ _ Ik Jk], insert P A Q I J, auto)
   also have "... = det (submatrix A I J)" using rw by simp
   finally show "Gcd (minors ?B k) dvd b" using b by simp
 qed
 
 lemma Gcd_minors_dvd_diag_PAQ:
-  fixes P A Q::"'a::{semiring_Gcd,comm_ring_1} mat" 
- assumes A: "A \<in> carrier_mat m n" 
-    and P: "P \<in> carrier_mat m m" 
+  fixes P A Q::"'a::{semiring_Gcd,comm_ring_1} mat"
+ assumes A: "A \<in> carrier_mat m n"
+    and P: "P \<in> carrier_mat m m"
     and Q: "Q \<in> carrier_mat n n"
     and SNF: "Smith_normal_form_mat (P*A*Q)"
     and k: "k\<le>min m n"
   shows "Gcd (minors A k) dvd (\<Prod>i=0..<k. (P * A * Q) $$ (i,i))"
 proof -
   have "Gcd (minors A k) dvd Gcd (minors (P * A * Q) k)"
     by (rule Gcd_minors_A_dvd_Gcd_minors_PAQ[OF A P Q])
   also have "... dvd (\<Prod>i=0..<k. (P*A*Q) $$ (i,i))"
     by (rule Gcd_minors_dvd_diagonal[OF _ SNF k], insert P A Q, auto)
   finally show ?thesis .
 qed
 
 
 lemma diag_PAQ_dvd_Gcd_minors:
-  fixes P A Q::"'a::{semiring_Gcd,comm_ring_1} mat" 
- assumes A: "A \<in> carrier_mat m n" 
-    and P: "P \<in> carrier_mat m m" 
+  fixes P A Q::"'a::{semiring_Gcd,comm_ring_1} mat"
+ assumes A: "A \<in> carrier_mat m n"
+    and P: "P \<in> carrier_mat m m"
     and Q: "Q \<in> carrier_mat n n"
-    and inv_P: "invertible_mat P" 
+    and inv_P: "invertible_mat P"
     and inv_Q: "invertible_mat Q"
     and SNF: "Smith_normal_form_mat (P*A*Q)"
   shows "(\<Prod>i=0..<k. (P * A * Q) $$ (i,i)) dvd Gcd (minors A k)"
 proof -
   have "(\<Prod>i=0..<k. (P*A*Q) $$ (i,i)) dvd Gcd (minors (P * A * Q) k)"
     by (rule diagonal_dvd_Gcd_minors[OF _ SNF], auto)
   also have "... dvd Gcd (minors A k)"
     by (rule Gcd_minors_PAQ_dvd_Gcd_minors_A[OF _ _ _ inv_P inv_Q], insert P A Q, auto)
   finally show ?thesis .
 qed
 
 
 (*This lemma requires semidom in order to apply prod_zero_iff*)
 lemma Smith_prod_zero_imp_last_zero:
   fixes A::"'a::{semidom,comm_ring_1} mat"
   assumes  A: "A \<in> carrier_mat m n"
     and SNF: "Smith_normal_form_mat A"
     and prod_0: "(\<Prod>j=0..<Suc i. A $$ (j,j)) = 0"
   and i: "i<min m n"
   shows "A $$(i,i) = 0"
 proof -
   obtain j where Ajj: "A$$(j,j) = 0" and j: "j<Suc i" using prod_0 prod_zero_iff by auto
   show "A $$(i,i) = 0" by (rule Smith_zero_imp_zero[OF A SNF Ajj i], insert j, auto)
 qed
 
 subsection \<open>Final theorem\<close>
 
 lemma Smith_normal_form_uniqueness_aux:
-  fixes P A Q::"'a::{idom,semiring_Gcd} mat" 
-  assumes A: "A \<in> carrier_mat m n" 
+  fixes P A Q::"'a::{idom,semiring_Gcd} mat"
+  assumes A: "A \<in> carrier_mat m n"
   (*PAQ = B with B in SNF and P,Q invertible matrices*)
   and P: "P \<in> carrier_mat m m"
   and Q: "Q \<in> carrier_mat n n"
-  and inv_P: "invertible_mat P" 
+  and inv_P: "invertible_mat P"
   and inv_Q: "invertible_mat Q"
   and PAQ_B: "P*A*Q = B"
   and SNF: "Smith_normal_form_mat B"
   (*P'AQ' = B' with B' in SNF and P',Q' invertible matrices*)
-  and P': "P' \<in> carrier_mat m m" 
+  and P': "P' \<in> carrier_mat m m"
   and Q': "Q' \<in> carrier_mat n n"
-  and inv_P': "invertible_mat P'" 
+  and inv_P': "invertible_mat P'"
   and inv_Q': "invertible_mat Q'"
   and P'AQ'_B': "P'*A*Q' = B'"
-  and SNF_B': "Smith_normal_form_mat B'"    
-  and k: "k<min m n"    
+  and SNF_B': "Smith_normal_form_mat B'"
+  and k: "k<min m n"
 shows "\<forall>i\<le>k. B$$(i,i) dvd B'$$(i,i) \<and> B'$$(i,i) dvd B$$(i,i)"
 proof (rule allI, rule impI)
   fix i assume ik: "i \<le> k"
-  show " B $$ (i, i) dvd B' $$ (i, i) \<and> B' $$ (i, i) dvd B $$ (i, i)" 
+  show " B $$ (i, i) dvd B' $$ (i, i) \<and> B' $$ (i, i) dvd B $$ (i, i)"
   proof -
     let ?\<Pi>Bi = "(\<Prod>i=0..<i. B $$ (i,i))"
     let ?\<Pi>B'i = "(\<Prod>i=0..<i. B' $$ (i,i))"
-    have "?\<Pi>B'i dvd Gcd (minors A i)" 
-      by (unfold P'AQ'_B'[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P' Q' inv_P' inv_Q'], 
+    have "?\<Pi>B'i dvd Gcd (minors A i)"
+      by (unfold P'AQ'_B'[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P' Q' inv_P' inv_Q'],
           insert P'AQ'_B' SNF_B' ik k, auto )
     also have "... dvd ?\<Pi>Bi"
-      by (unfold PAQ_B[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P Q], 
+      by (unfold PAQ_B[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P Q],
           insert PAQ_B SNF ik k, auto)
     finally have B'_i_dvd_B_i: "?\<Pi>B'i dvd ?\<Pi>Bi" .
-    have "?\<Pi>Bi dvd Gcd (minors A i)" 
-      by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q], 
+    have "?\<Pi>Bi dvd Gcd (minors A i)"
+      by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q],
           insert PAQ_B SNF ik k, auto )
     also have "... dvd ?\<Pi>B'i"
-      by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'], 
+      by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'],
           insert P'AQ'_B' SNF_B' ik k, auto)
     finally have B_i_dvd_B'_i: "?\<Pi>Bi dvd ?\<Pi>B'i" .
     let ?\<Pi>B_Suc = "(\<Prod>i=0..<Suc i. B $$ (i,i))"
     let ?\<Pi>B'_Suc = "(\<Prod>i=0..<Suc i. B' $$ (i,i))"
-    have "?\<Pi>B'_Suc dvd Gcd (minors A (Suc i))" 
-      by (unfold P'AQ'_B'[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P' Q' inv_P' inv_Q'], 
+    have "?\<Pi>B'_Suc dvd Gcd (minors A (Suc i))"
+      by (unfold P'AQ'_B'[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P' Q' inv_P' inv_Q'],
           insert P'AQ'_B' SNF_B' ik k, auto )
     also have "... dvd ?\<Pi>B_Suc"
-      by (unfold PAQ_B[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P Q], 
+      by (unfold PAQ_B[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P Q],
           insert PAQ_B SNF ik k, auto)
     finally have 3: "?\<Pi>B'_Suc dvd ?\<Pi>B_Suc" .
     have "?\<Pi>B_Suc dvd Gcd (minors A (Suc i))"
-      by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q], 
+      by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q],
           insert PAQ_B SNF ik k, auto )
     also have "... dvd ?\<Pi>B'_Suc"
-      by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'], 
+      by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'],
           insert P'AQ'_B' SNF_B' ik k, auto)
-    finally have 4: "?\<Pi>B_Suc dvd ?\<Pi>B'_Suc" .  
+    finally have 4: "?\<Pi>B_Suc dvd ?\<Pi>B'_Suc" .
     show ?thesis
     proof (cases "?\<Pi>B_Suc = 0")
       case True
       have True2: "?\<Pi>B'_Suc = 0" using 4 True by fastforce
-      have "B$$(i,i) = 0" 
+      have "B$$(i,i) = 0"
         by (rule Smith_prod_zero_imp_last_zero[OF _ SNF True], insert ik k PAQ_B P Q, auto)
       moreover have "B'$$(i,i) = 0"
-        by (rule Smith_prod_zero_imp_last_zero[OF _ SNF_B' True2], 
+        by (rule Smith_prod_zero_imp_last_zero[OF _ SNF_B' True2],
             insert ik k P'AQ'_B' P' Q', auto)
       ultimately show ?thesis by auto
     next
       case False
       have "\<exists>u. u dvd 1 \<and> ?\<Pi>B'i = u * ?\<Pi>Bi"
         by (rule dvd_associated2[OF B'_i_dvd_B_i B_i_dvd_B'_i], insert False B'_i_dvd_B_i, force)
-      from this obtain u where eq1: "(\<Prod>i=0..<i. B' $$ (i,i)) = u * (\<Prod>i=0..<i. B $$ (i,i))" 
+      from this obtain u where eq1: "(\<Prod>i=0..<i. B' $$ (i,i)) = u * (\<Prod>i=0..<i. B $$ (i,i))"
         and u_dvd_1: "u dvd 1" by blast
       have "\<exists>u. u dvd 1 \<and> ?\<Pi>B_Suc = u * ?\<Pi>B'_Suc"
         by (rule dvd_associated2[OF 4 3 False])
       from this obtain w where eq2: "(\<Prod>i=0..<Suc i. B $$ (i,i)) = w * (\<Prod>i=0..<Suc i. B' $$ (i,i))"
-        and w_dvd_1: "w dvd 1" by blast      
+        and w_dvd_1: "w dvd 1" by blast
       have "B $$ (i, i) * (\<Prod>i=0..<i. B $$ (i,i)) = (\<Prod>i=0..<Suc i. B $$ (i,i))"
         by (simp add: prod.atLeast0_lessThan_Suc ik)
       also have "... = w * (\<Prod>i=0..<Suc i. B' $$ (i,i))" unfolding eq2 by auto
-      also have "... = w * (B' $$ (i,i) * (\<Prod>i=0..<i. B' $$ (i,i)))" 
+      also have "... = w * (B' $$ (i,i) * (\<Prod>i=0..<i. B' $$ (i,i)))"
         by (simp add: prod.atLeast0_lessThan_Suc ik)
-      also have "... = w * (B' $$ (i,i) * u * (\<Prod>i=0..<i. B $$ (i,i)))" 
+      also have "... = w * (B' $$ (i,i) * u * (\<Prod>i=0..<i. B $$ (i,i)))"
         unfolding eq1 by auto
-      finally have "B $$ (i,i) = w * u * B' $$ (i,i)" 
+      finally have "B $$ (i,i) = w * u * B' $$ (i,i)"
         using False by auto
       moreover have "w*u dvd 1" using u_dvd_1 w_dvd_1 by auto
       ultimately have "\<exists>u. is_unit u \<and> B $$ (i, i) = u * B' $$ (i, i)" by auto
       thus ?thesis using dvd_associated2 by force
     qed
   qed
 qed
 
 
 lemma Smith_normal_form_uniqueness:
-  fixes P A Q::"'a::{idom,semiring_Gcd} mat" 
-  assumes A: "A \<in> carrier_mat m n" 
+  fixes P A Q::"'a::{idom,semiring_Gcd} mat"
+  assumes A: "A \<in> carrier_mat m n"
     (*PAQ = B with B in SNF and P,Q invertible matrices*)
     and P: "P \<in> carrier_mat m m"
     and Q: "Q \<in> carrier_mat n n"
-    and inv_P: "invertible_mat P" 
+    and inv_P: "invertible_mat P"
     and inv_Q: "invertible_mat Q"
     and PAQ_B: "P*A*Q = B"
     and SNF: "Smith_normal_form_mat B"
     (*P'AQ' = B' with B' in SNF and P',Q' invertible matrices*)
-    and P': "P' \<in> carrier_mat m m" 
+    and P': "P' \<in> carrier_mat m m"
     and Q': "Q' \<in> carrier_mat n n"
-    and inv_P': "invertible_mat P'" 
+    and inv_P': "invertible_mat P'"
     and inv_Q': "invertible_mat Q'"
     and P'AQ'_B': "P'*A*Q' = B'"
-    and SNF_B': "Smith_normal_form_mat B'"    
+    and SNF_B': "Smith_normal_form_mat B'"
     and i: "i < min m n"
   shows "\<exists>u. u dvd 1 \<and> B $$ (i,i) = u * B' $$ (i,i)"
 proof (cases "B $$ (i,i) = 0")
   case True
   let ?\<Pi>B_Suc = "(\<Prod>i=0..<Suc i. B $$ (i,i))"
   let ?\<Pi>B'_Suc = "(\<Prod>i=0..<Suc i. B' $$ (i,i))"
   have "?\<Pi>B_Suc dvd Gcd (minors A (Suc i))"
-    by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q], 
+    by (unfold PAQ_B[symmetric], rule diag_PAQ_dvd_Gcd_minors[OF A P Q inv_P inv_Q],
         insert PAQ_B SNF i, auto)
   also have "... dvd ?\<Pi>B'_Suc"
-    by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'], 
+    by (unfold P'AQ'_B'[symmetric], rule Gcd_minors_dvd_diag_PAQ[OF A P' Q'],
         insert P'AQ'_B' SNF_B' i, auto)
-  finally have 4: "?\<Pi>B_Suc dvd ?\<Pi>B'_Suc" .  
+  finally have 4: "?\<Pi>B_Suc dvd ?\<Pi>B'_Suc" .
   have prod0: "?\<Pi>B_Suc=0" using True by auto
   have True2: "?\<Pi>B'_Suc = 0" using 4 by (metis dvd_0_left_iff prod0)
   have "B'$$(i,i) = 0"
-    by (rule Smith_prod_zero_imp_last_zero[OF _ SNF_B' True2], 
-        insert i P'AQ'_B' P' Q', auto) 
+    by (rule Smith_prod_zero_imp_last_zero[OF _ SNF_B' True2],
+        insert i P'AQ'_B' P' Q', auto)
   thus ?thesis using True by auto
 next
   case False
-  have "\<forall>a\<le>i. B$$(a,a) dvd B'$$(a,a) \<and> B'$$(a,a) dvd B$$(a,a)" 
+  have "\<forall>a\<le>i. B$$(a,a) dvd B'$$(a,a) \<and> B'$$(a,a) dvd B$$(a,a)"
     by (rule Smith_normal_form_uniqueness_aux[OF assms])
   hence "B$$(i,i) dvd B'$$(i,i) \<and> B'$$(i,i) dvd B$$(i,i)" using i by auto
   thus ?thesis using dvd_associated2 False by blast
 qed
 
 text \<open>The final theorem, moved to HOL Analysis\<close>
 
 lemma Smith_normal_form_uniqueness_HOL_Analysis:
   fixes A::"'a::{idom,semiring_Gcd}^'m::mod_type^'n::mod_type"
   and P P'::"'a^'n::mod_type^'n::mod_type"
   and Q Q'::"'a^'m::mod_type^'m::mod_type"
   assumes
-    (*PAQ = B with B in SNF and P,Q invertible matrices*)        
-    inv_P: "invertible P" 
+    (*PAQ = B with B in SNF and P,Q invertible matrices*)
+    inv_P: "invertible P"
     and inv_Q: "invertible Q"
     and PAQ_B: "P**A**Q = B"
     and SNF: "Smith_normal_form B"
     (*P'AQ' = B' with B' in SNF and P',Q' invertible matrices*)
-    and inv_P': "invertible P'" 
+    and inv_P': "invertible P'"
     and inv_Q': "invertible Q'"
     and P'AQ'_B': "P'**A**Q' = B'"
-    and SNF_B': "Smith_normal_form B'"    
+    and SNF_B': "Smith_normal_form B'"
     and i: "i < min (nrows A) (ncols A)"
-  shows "\<exists>u. u dvd 1 \<and> B $h Mod_Type.from_nat i $h Mod_Type.from_nat i 
+  shows "\<exists>u. u dvd 1 \<and> B $h Mod_Type.from_nat i $h Mod_Type.from_nat i
   = u * B' $h Mod_Type.from_nat i $h Mod_Type.from_nat i"
 proof -
   let ?P = "Mod_Type_Connect.from_hma\<^sub>m P"
   let ?A = "Mod_Type_Connect.from_hma\<^sub>m A"
   let ?Q = "Mod_Type_Connect.from_hma\<^sub>m Q"
   let ?B = "Mod_Type_Connect.from_hma\<^sub>m B"
   let ?P' = "Mod_Type_Connect.from_hma\<^sub>m P'"
   let ?Q' = "Mod_Type_Connect.from_hma\<^sub>m Q'"
   let ?B' = "Mod_Type_Connect.from_hma\<^sub>m B'"
   let ?i = "(Mod_Type.from_nat i)::'n"
   let ?i' = "(Mod_Type.from_nat i)::'m"
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?P P" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?A A" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?Q Q" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?B B" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?P' P'" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?Q' Q'" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_M ?B' B'" by (simp add: Mod_Type_Connect.HMA_M_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_I i ?i"
-    by (metis Mod_Type_Connect.HMA_I_def i min.strict_boundedE 
+    by (metis Mod_Type_Connect.HMA_I_def i min.strict_boundedE
         mod_type_class.to_nat_from_nat_id nrows_def)
   have [transfer_rule]: "Mod_Type_Connect.HMA_I i ?i'"
-       by (metis Mod_Type_Connect.HMA_I_def i min.strict_boundedE 
+       by (metis Mod_Type_Connect.HMA_I_def i min.strict_boundedE
         mod_type_class.to_nat_from_nat_id ncols_def)
   have i2: "i < min CARD('m) CARD('n)" using i unfolding nrows_def ncols_def by auto
-  have "\<exists>u. u dvd 1 \<and> ?B $$(i,i) = u * ?B' $$ (i,i)" 
+  have "\<exists>u. u dvd 1 \<and> ?B $$(i,i) = u * ?B' $$ (i,i)"
   proof (rule Smith_normal_form_uniqueness[of _ "CARD('n)" "CARD('m)"])
     show "?P*?A*?Q=?B" using PAQ_B by (transfer', auto)
     show "Smith_normal_form_mat ?B" using SNF by (transfer', auto)
     show "?P'*?A*?Q'=?B'" using P'AQ'_B' by (transfer', auto)
     show "Smith_normal_form_mat ?B'" using SNF_B' by (transfer', auto)
     show "invertible_mat ?P" using inv_P by (transfer, auto)
     show "invertible_mat ?P'" using inv_P' by (transfer, auto)
     show "invertible_mat ?Q" using inv_Q by (transfer, auto)
-    show "invertible_mat ?Q'" using inv_Q' by (transfer, auto)    
+    show "invertible_mat ?Q'" using inv_Q' by (transfer, auto)
   qed (insert i2, auto)
   hence "\<exists>u. u dvd 1 \<and> (index_hma B ?i ?i') = u * (index_hma B' ?i ?i')" by (transfer', rule)
   thus ?thesis unfolding index_hma_def by simp
 qed
 
 subsection \<open>Uniqueness fixing a complete set of non-associates\<close>
 
 definition "Smith_normal_form_wrt A \<Q> = (
-    (\<forall>a b. Mod_Type.to_nat a = Mod_Type.to_nat b \<and> Mod_Type.to_nat a + 1 < nrows A 
+    (\<forall>a b. Mod_Type.to_nat a = Mod_Type.to_nat b \<and> Mod_Type.to_nat a + 1 < nrows A
             \<and> Mod_Type.to_nat b + 1 < ncols A \<longrightarrow> A $h a $h b dvd A $h (a+1) $h (b+1))
     \<and> isDiagonal A \<and> Complete_set_non_associates \<Q>
-    \<and> (\<forall>a b. Mod_Type.to_nat a = Mod_Type.to_nat b \<and> Mod_Type.to_nat a < min (nrows A) (ncols A) 
-       \<and> Mod_Type.to_nat b < min (nrows A) (ncols A) \<longrightarrow> A $h a $h b \<in> \<Q>)  
+    \<and> (\<forall>a b. Mod_Type.to_nat a = Mod_Type.to_nat b \<and> Mod_Type.to_nat a < min (nrows A) (ncols A)
+       \<and> Mod_Type.to_nat b < min (nrows A) (ncols A) \<longrightarrow> A $h a $h b \<in> \<Q>)
   )"
 
 lemma Smith_normal_form_wrt_uniqueness_HOL_Analysis:
   fixes A::"'a::{idom,semiring_Gcd}^'m::mod_type^'n::mod_type"
   and P P'::"'a^'n::mod_type^'n::mod_type"
   and Q Q'::"'a^'m::mod_type^'m::mod_type"
   assumes
-    (*PAQ = S with S in SNF and P,Q invertible matrices*)        
-    P: "invertible P" 
+    (*PAQ = S with S in SNF and P,Q invertible matrices*)
+    P: "invertible P"
     and Q: "invertible Q"
     and PAQ_S: "P**A**Q = S"
     and SNF: "Smith_normal_form_wrt S \<Q>"
     (*P'AQ' = S' with S' in SNF and P',Q' invertible matrices*)
-    and P': "invertible P'" 
+    and P': "invertible P'"
     and Q': "invertible Q'"
     and P'AQ'_S': "P'**A**Q' = S'"
-    and SNF_S': "Smith_normal_form_wrt S' \<Q>"    
+    and SNF_S': "Smith_normal_form_wrt S' \<Q>"
   shows "S = S'"
 proof -
   have "S $h i $h j = S' $h i $h j" for i j
   proof (cases "Mod_Type.to_nat i \<noteq> Mod_Type.to_nat j")
   case True
     then show ?thesis using SNF SNF_S' unfolding Smith_normal_form_wrt_def isDiagonal_def by auto
   next
     case False
     let ?i = "Mod_Type.to_nat i"
     let ?j = "Mod_Type.to_nat j"
     have complete_set: "Complete_set_non_associates \<Q>"
       using SNF_S' unfolding Smith_normal_form_wrt_def by simp
     have ij: "?i = ?j" using False by auto
     show ?thesis
     proof (rule ccontr)
       assume d: "S $h i $h j \<noteq> S' $h i $h j"
       have n: "normalize (S $h i $h j) \<noteq> normalize (S' $h i $h j)"
       proof (rule in_Ass_not_associated[OF complete_set _ _ d])
         show "S $h i $h j \<in> \<Q>" using SNF unfolding Smith_normal_form_wrt_def
           by (metis False min_less_iff_conj mod_type_class.to_nat_less_card ncols_def nrows_def)
         show "S' $h i $h j \<in> \<Q>" using SNF_S' unfolding Smith_normal_form_wrt_def
-          by (metis False min_less_iff_conj mod_type_class.to_nat_less_card ncols_def nrows_def)        
-      qed      
+          by (metis False min_less_iff_conj mod_type_class.to_nat_less_card ncols_def nrows_def)
+      qed
       have "\<exists>u. u dvd 1 \<and> S $h i $h j = u * S' $h i $h j"
       proof -
-        have "\<exists>u. u dvd 1 \<and> S $h Mod_Type.from_nat ?i $h Mod_Type.from_nat ?i 
+        have "\<exists>u. u dvd 1 \<and> S $h Mod_Type.from_nat ?i $h Mod_Type.from_nat ?i
           = u * S' $h Mod_Type.from_nat ?i $h Mod_Type.from_nat ?i"
         proof (rule Smith_normal_form_uniqueness_HOL_Analysis[OF P Q PAQ_S _ P' Q' P'AQ'_S' _])
           show "Smith_normal_form S" and "Smith_normal_form S'"
             using SNF SNF_S' Smith_normal_form_def Smith_normal_form_wrt_def by blast+
           show "?i < min (nrows A) (ncols A)"
             by (metis ij min_less_iff_conj mod_type_class.to_nat_less_card ncols_def nrows_def)
         qed
         thus ?thesis using False by auto
       qed
       from this obtain u where "is_unit u" and "S $h i $h j = u * S' $h i $h j" by auto
       thus False using n
         by (simp add: normalize_1_iff normalize_mult)
     qed
   qed
   thus ?thesis by vector
 qed
 
 
 end
\ No newline at end of file
diff --git a/thys/Smith_Normal_Form/Smith_Certified.thy b/thys/Smith_Normal_Form/Smith_Certified.thy
--- a/thys/Smith_Normal_Form/Smith_Certified.thy
+++ b/thys/Smith_Normal_Form/Smith_Certified.thy
@@ -1,125 +1,125 @@
 (*
     Author:      Jose Divasón
-    Email:       jose.divason@unirioja.es           
+    Email:       jose.divason@unirioja.es
 *)
 
 section \<open>A certified checker based on an external algorithm to compute Smith normal form\<close>
 
 theory Smith_Certified
-  imports        
+  imports
     SNF_Algorithm_Euclidean_Domain
 begin
 
-text\<open>This (unspecified) function takes as input the matrix $A$ and returns five matrices 
-$(P,S,Q,P',Q')$, which must satisfy $S = PAQ$, $S$ is in Smith normal form, $P'$ and $Q'$ 
+text\<open>This (unspecified) function takes as input the matrix $A$ and returns five matrices
+$(P,S,Q,P',Q')$, which must satisfy $S = PAQ$, $S$ is in Smith normal form, $P'$ and $Q'$
 are the inverse matrices of $P$ and $Q$ respectively\<close>
 
 text\<open>The matrices are given in terms of lists for the sake of simplicity when connecting the
 function to external solvers, like Mathematica or Sage.\<close>
 
-consts external_SNF :: 
-  "int list list \<Rightarrow> int list list \<times> int list list \<times> int list list \<times> int list list \<times> int list list" 
+consts external_SNF ::
+  "int list list \<Rightarrow> int list list \<times> int list list \<times> int list list \<times> int list list \<times> int list list"
 
 
 text \<open>We implement the checker by means of the following definition. The checker is implemented
 in the JNF representation of matrices to make use of the Strassen matrix multiplication algorithm.
 In case that the certification fails, then the verified Smith normal form algorithm is executed.
 Thus, we will always get a verified result.\<close>
 
 definition "checker_SNF A = (
     let A' = mat_to_list A; m = dim_row A; n = dim_col A in
       case external_SNF A' of (P_ext,S_ext,Q_ext,P'_ext,Q'_ext) \<Rightarrow> let
           P = mat_of_rows_list m P_ext;
           S = mat_of_rows_list m S_ext;
           Q = mat_of_rows_list m Q_ext;
           P' = mat_of_rows_list m P'_ext;
           Q' = mat_of_rows_list m Q'_ext in
             (if dim_row P = m \<and> dim_col P = m
               \<and> dim_row S = m \<and> dim_col S = n
               \<and> dim_row Q = n \<and> dim_col Q = n
               \<and> dim_row P' = m \<and> dim_col P' = m
               \<and> dim_row Q' = n \<and> dim_col Q' = n
               \<and> P * P' = 1\<^sub>m m \<and> Q * Q' = 1\<^sub>m n
               \<and> Smith_normal_form_mat S \<and> (S = P*A*Q) then
       (P,S,Q) else Code.abort (STR ''Certification failed'') (\<lambda> _. Smith_ED_mxn A))
 )"
 
 theorem checker_SNF_soudness:
   assumes A: "A \<in> carrier_mat m n"
     and c: "checker_SNF A = (P,S,Q)"
   shows "is_SNF A (P,S,Q)"
 proof -
-  let ?ext = "external_SNF (mat_to_list A)" 
-  obtain P_ext S_ext Q_ext P'_ext Q'_ext where ext: "?ext = (P_ext,S_ext,Q_ext,P'_ext,Q'_ext)" 
+  let ?ext = "external_SNF (mat_to_list A)"
+  obtain P_ext S_ext Q_ext P'_ext Q'_ext where ext: "?ext = (P_ext,S_ext,Q_ext,P'_ext,Q'_ext)"
     by (cases "?ext", auto)
   let ?case_external = "let
           P = mat_of_rows_list m P_ext;
           S = mat_of_rows_list m S_ext;
           Q = mat_of_rows_list n Q_ext;
           P' = mat_of_rows_list m P'_ext;
           Q' = mat_of_rows_list n Q'_ext in
             (dim_row P = m \<and> dim_col P = m
               \<and> dim_row S = m \<and> dim_col S = n
               \<and> dim_row Q = n \<and> dim_col Q = n
               \<and> dim_row P' = m \<and> dim_col P' = m
               \<and> dim_row Q' = n \<and> dim_col Q' = n
               \<and> P * P' = 1\<^sub>m m \<and> Q * Q' = 1\<^sub>m n
               \<and> Smith_normal_form_mat S \<and> (S = P*A*Q))"
-  show ?thesis 
+  show ?thesis
   proof (cases ?case_external)
-    case True    
+    case True
     define P' where "P' = mat_of_rows_list m P'_ext"
     define Q' where "Q' = mat_of_rows_list m Q'_ext"
-    have S_PAQ: "S = P * A * Q " 
+    have S_PAQ: "S = P * A * Q "
       and SNF_S: "Smith_normal_form_mat S" and PP'_1: "P * P' = 1\<^sub>m m" and QQ'_1: "Q * Q' = 1\<^sub>m n"
       and sm_P: "square_mat P" and sm_Q: "square_mat Q"
-      using ext True c A 
-      unfolding checker_SNF_def Let_def mat_of_rows_list_def P'_def Q'_def 
+      using ext True c A
+      unfolding checker_SNF_def Let_def mat_of_rows_list_def P'_def Q'_def
       by (auto split: if_splits)
     have inv_P: "invertible_mat P"
-    proof (unfold invertible_mat_def, rule conjI, rule sm_P, 
+    proof (unfold invertible_mat_def, rule conjI, rule sm_P,
         unfold inverts_mat_def, rule exI[of _ P'], rule conjI)
-      show *: "P * P' = 1\<^sub>m (dim_row P)" 
-        by (metis PP'_1 True index_mult_mat(2))  
+      show *: "P * P' = 1\<^sub>m (dim_row P)"
+        by (metis PP'_1 True index_mult_mat(2))
       show "P' * P = 1\<^sub>m (dim_row P')"
       proof (rule mat_mult_left_right_inverse)
         show "P \<in> carrier_mat (dim_row P') (dim_row P')"
           by (metis * P'_def PP'_1 True carrier_mat_triv index_one_mat(2) sm_P square_mat.elims(2))
-        show "P' \<in> carrier_mat (dim_row P') (dim_row P')" 
+        show "P' \<in> carrier_mat (dim_row P') (dim_row P')"
           by (metis P'_def True carrier_mat_triv)
         show "P * P' = 1\<^sub>m (dim_row P')"
           by (metis P'_def PP'_1 True)
       qed
     qed
     have inv_Q: "invertible_mat Q"
-    proof (unfold invertible_mat_def, rule conjI, rule sm_Q, 
+    proof (unfold invertible_mat_def, rule conjI, rule sm_Q,
         unfold inverts_mat_def, rule exI[of _ Q'], rule conjI)
-      show *: "Q * Q' = 1\<^sub>m (dim_row Q)" 
+      show *: "Q * Q' = 1\<^sub>m (dim_row Q)"
         by (metis QQ'_1 True index_mult_mat(2))
       show "Q' * Q = 1\<^sub>m (dim_row Q')"
       proof (rule mat_mult_left_right_inverse)
-        show 1: "Q \<in> carrier_mat (dim_row Q') (dim_row Q')" 
-          by (metis Q'_def QQ'_1 True carrier_mat_triv dim_row_mat(1) index_mult_mat(2) 
+        show 1: "Q \<in> carrier_mat (dim_row Q') (dim_row Q')"
+          by (metis Q'_def QQ'_1 True carrier_mat_triv dim_row_mat(1) index_mult_mat(2)
               mat_of_rows_list_def sm_Q square_mat.simps)
         thus "Q' \<in> carrier_mat (dim_row Q') (dim_row Q')"
           by (metis * carrier_matD(1) carrier_mat_triv index_mult_mat(3) index_one_mat(3))
         show "Q * Q' = 1\<^sub>m (dim_row Q')" using * 1 by auto
       qed
     qed
     have "P \<in> carrier_mat m m"
       by (metis PP'_1 True carrier_matI index_mult_mat(2) sm_P square_mat.simps)
     moreover have  "Q \<in> carrier_mat n n"
       by (metis QQ'_1 True carrier_matI index_mult_mat(2) sm_Q square_mat.simps)
     ultimately show ?thesis unfolding is_SNF_def using inv_P inv_Q SNF_S S_PAQ A by auto
   next
     case False
-    hence "checker_SNF A = Smith_ED_mxn A" 
-      using ext False c A 
+    hence "checker_SNF A = Smith_ED_mxn A"
+      using ext False c A
       unfolding checker_SNF_def Let_def Code.abort_def
       by (smt carrier_matD case_prod_conv dim_col_mat(1) mat_of_rows_list_def)
-    then show ?thesis using Smith_ED.is_SNF_Smith_mxn[OF A] c unfolding is_SNF_def        
+    then show ?thesis using Smith_ED.is_SNF_Smith_mxn[OF A] c unfolding is_SNF_def
       by auto
   qed
 qed
 
 end