diff --git a/thys/Irrational_Series_Erdos_Straus/Irrational_Series_Erdos_Straus.thy b/thys/Irrational_Series_Erdos_Straus/Irrational_Series_Erdos_Straus.thy
--- a/thys/Irrational_Series_Erdos_Straus/Irrational_Series_Erdos_Straus.thy
+++ b/thys/Irrational_Series_Erdos_Straus/Irrational_Series_Erdos_Straus.thy
@@ -1,1984 +1,1968 @@
 (* Title:      Irrational_Series_Erdos_Straus.thy
    Author:     Angeliki Koutsoukou-Argyraki and Wenda Li, University of Cambridge, UK.
 
 We formalise certain irrationality criteria for infinite series by P. Erdos and E.G. Straus.
 In particular, we formalise Theorem 2.1, Corollary 2.10 and Theorem 3.1 in [1]. The latter is an 
 application of Theorem 2.1 involving the prime numbers.
 
 References:
 [1]  P. Erdos and E.G. Straus, On the irrationality of certain series, Pacific Journal of 
 Mathematics, Vol. 55, No 1, 1974  https://projecteuclid.org/euclid.pjm/1102911140
 *)
 
 theory "Irrational_Series_Erdos_Straus" imports
   Prime_Number_Theorem.Prime_Number_Theorem
   Prime_Distribution_Elementary.PNT_Consequences
 begin
 
 section \<open>Miscellaneous\<close>
 
 lemma suminf_comparison:
   assumes "summable f" and gf: "\<And>n. norm (g n) \<le> f n"
   shows "suminf g \<le> suminf f"
 proof (rule suminf_le)
   show "g n \<le> f n" for n
     using gf[of n] by auto
   show "summable g"
     using assms summable_comparison_test' by blast 
   show "summable f" using assms(1) .
 qed
 
 lemma tendsto_of_int_diff_0:
   assumes "(\<lambda>n. f n - of_int(g n)) \<longlonglongrightarrow> (0::real)" "\<forall>\<^sub>F n in sequentially. f n > 0"
   shows "\<forall>\<^sub>F n in sequentially. 0 \<le> g n"
 proof -
   have "\<forall>\<^sub>F n in sequentially. \<bar>f n - of_int(g n)\<bar> < 1 / 2" 
     using assms(1)[unfolded tendsto_iff,rule_format,of "1/2"] by auto
   then show ?thesis using assms(2)
     by eventually_elim linarith
 qed
 
 lemma eventually_mono_sequentially:
   assumes "eventually P sequentially"
   assumes "\<And>x. P (x+k) \<Longrightarrow> Q (x+k)"
   shows "eventually Q sequentially"
   using sequentially_offset[OF assms(1),of k]
   apply (subst eventually_sequentially_seg[symmetric,of _ k])
   apply (elim eventually_mono)
   by fact
 
 lemma frequently_eventually_at_top:
   fixes P Q::"'a::linorder \<Rightarrow> bool"
   assumes "frequently P at_top" "eventually Q at_top"
   shows "frequently (\<lambda>x. P x \<and> (\<forall>y\<ge>x. Q y) ) at_top"
   using assms
   unfolding frequently_def eventually_at_top_linorder 
   by (metis (mono_tags, opaque_lifting) le_cases order_trans)
 
 lemma eventually_at_top_mono:
   fixes P Q::"'a::linorder \<Rightarrow> bool"
   assumes event_P:"eventually P at_top"
   assumes PQ_imp:"\<And>x. x\<ge>z \<Longrightarrow> \<forall>y\<ge>x. P y \<Longrightarrow> Q x"
   shows "eventually Q at_top"
 proof -
   obtain N where "\<forall>n\<ge>N. P n"
     by (meson event_P eventually_at_top_linorder)
   then have "Q x" when "x \<ge> max N z" for x
     using PQ_imp that by auto 
   then show ?thesis unfolding eventually_at_top_linorder
     by blast
 qed
 
 lemma frequently_at_top_elim:
   fixes P Q::"'a::linorder \<Rightarrow> bool"
   assumes "\<exists>\<^sub>Fx in at_top. P x"
   assumes "\<And>i. P i \<Longrightarrow> \<exists>j>i. Q j"
   shows "\<exists>\<^sub>Fx in at_top. Q x"
   using assms unfolding frequently_def eventually_at_top_linorder 
   by (meson leD le_cases less_le_trans)
 
 lemma less_Liminf_iff:
   fixes X :: "_ \<Rightarrow> _ :: complete_linorder"
   shows "Liminf F X < C \<longleftrightarrow> (\<exists>y<C. frequently (\<lambda>x. y \<ge> X x) F)"
   by (force simp: not_less not_frequently not_le le_Liminf_iff simp flip: Not_eq_iff)
 
 lemma sequentially_even_odd_imp:
   assumes "\<forall>\<^sub>F N in sequentially. P (2*N)" "\<forall>\<^sub>F N in sequentially. P (2*N+1)"
   shows "\<forall>\<^sub>F n in sequentially. P n"
 proof -
   obtain N where N_P:"\<forall>x\<ge>N.  P (2 * x) \<and> P (2 * x + 1)"
     using eventually_conj[OF assms] 
     unfolding eventually_at_top_linorder by auto
   have "P n" when "n \<ge> 2*N" for n
   proof -
     define n' where "n'= n div 2"
     then have "n' \<ge> N" using that by auto
     then have "P (2 * n') \<and> P (2 * n' + 1)"
       using N_P by auto
     then show ?thesis unfolding n'_def
       by (cases "even n") auto
   qed
   then show ?thesis unfolding eventually_at_top_linorder by auto
 qed
 
 
 section \<open>Theorem 2.1 and Corollary 2.10\<close>
 
 context
   fixes a b ::"nat\<Rightarrow>int "
   assumes a_pos: "\<forall> n. a n >0 " and a_large: "\<forall>\<^sub>F n in sequentially. a n > 1" 
     and ab_tendsto: "(\<lambda>n. \<bar>b n\<bar> / (a (n-1) * a n)) \<longlonglongrightarrow> 0"
 begin
 
 private lemma aux_series_summable: "summable (\<lambda>n. b n / (\<Prod>k\<le>n. a k))" 
 proof -
   have "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F x in sequentially. \<bar>b x\<bar> / (a (x-1) * a x) < e"
     using ab_tendsto[unfolded tendsto_iff] 
     apply (simp add: abs_mult flip: of_int_abs)
     by (subst (asm) (2) abs_of_pos,use \<open>\<forall> n. a n > 0\<close> in auto)+              
   from this[of 1]
   have "\<forall>\<^sub>F x in sequentially. \<bar>real_of_int(b x)\<bar> < (a (x-1) * a x)"
     using \<open>\<forall> n. a n > 0\<close> by auto
   moreover have "\<forall>n. (\<Prod>k\<le>n. real_of_int (a k)) > 0" 
     using a_pos by (auto intro!:linordered_semidom_class.prod_pos)
   ultimately have "\<forall>\<^sub>F n in sequentially. \<bar>b n\<bar> / (\<Prod>k\<le>n. a k) 
                         < (a (n-1) * a n) / (\<Prod>k\<le>n. a k)"
     apply (elim eventually_mono)
-    by (auto simp add:field_simps)
+    by (auto simp:field_simps)
   moreover have "\<bar>b n\<bar> / (\<Prod>k\<le>n. a k) = norm (b n / (\<Prod>k\<le>n. a k))" for n 
     using \<open>\<forall>n. (\<Prod>k\<le>n. real_of_int (a k)) > 0\<close>[rule_format,of n] by auto
   ultimately have "\<forall>\<^sub>F n in sequentially. norm (b n / (\<Prod>k\<le>n. a k)) 
                         < (a (n-1) * a n) / (\<Prod>k\<le>n. a k)"
     by algebra
   moreover have "summable (\<lambda>n. (a (n-1) * a n) / (\<Prod>k\<le>n. a k))" 
   proof -
     obtain s where a_gt_1:"\<forall> n\<ge>s. a n >1"
       using a_large[unfolded eventually_at_top_linorder] by auto
     define cc where "cc= (\<Prod>k<s. a k)"
     have "cc>0" 
       unfolding cc_def by (meson a_pos prod_pos)
     have "(\<Prod>k\<le>n+s. a k) \<ge> cc * 2^n" for n
     proof -
       have "prod a {s..<Suc (s + n)} \<ge> 2^n"
       proof (induct n)
         case 0
         then show ?case using a_gt_1 by auto
       next
         case (Suc n)
-        moreover have "a (s + Suc n) \<ge> 2" 
-          using a_gt_1 by (smt le_add1)
+        moreover have "a (s + Suc n) \<ge> 2"
+          by (smt (verit, ccfv_threshold) a_gt_1 le_add1)
         ultimately show ?case 
           apply (subst prod.atLeastLessThan_Suc,simp)
           using mult_mono'[of 2 "a (Suc (s + n))" " 2 ^ n" "prod a {s..<Suc (s + n)}"] 
           by (simp add: mult.commute)
       qed
       moreover have "prod a {0..(n + s)} = prod a {..<s} * prod a {s..<Suc (s + n)} "
         using prod.atLeastLessThan_concat[of 0 s "s+n+1" a]
         by (simp add: add.commute lessThan_atLeast0 prod.atLeastLessThan_concat prod.head_if)
       ultimately show ?thesis
         using \<open>cc>0\<close> unfolding cc_def by (simp add: atLeast0AtMost)
     qed
     then have "1/(\<Prod>k\<le>n+s. a k) \<le> 1/(cc * 2^n)" for n
     proof -
       assume asm:"\<And>n. cc * 2 ^ n \<le> prod a {..n + s}"
       then have "real_of_int (cc * 2 ^ n) \<le> prod a {..n + s}" using of_int_le_iff by blast
       moreover have "prod a {..n + s} >0" using \<open>cc>0\<close> by (simp add: a_pos prod_pos)
       ultimately show ?thesis using \<open>cc>0\<close> 
         by (auto simp:field_simps simp del:of_int_prod)
     qed
     moreover have "summable (\<lambda>n. 1/(cc * 2^n))"
     proof -
       have "summable (\<lambda>n. 1/(2::int)^n)"
         using summable_geometric[of "1/(2::int)"] by (simp add:power_one_over)
       from summable_mult[OF this,of "1/cc"] show ?thesis by auto
     qed
     ultimately have "summable (\<lambda>n. 1 / (\<Prod>k\<le>n+s. a k))"
       apply (elim summable_comparison_test'[where N=0])
       apply (unfold real_norm_def, subst abs_of_pos)
-      by (auto simp add: \<open>\<forall>n. 0 < (\<Prod>k\<le>n. real_of_int (a k))\<close>)
+      by (auto simp: \<open>\<forall>n. 0 < (\<Prod>k\<le>n. real_of_int (a k))\<close>)
     then have "summable (\<lambda>n. 1 / (\<Prod>k\<le>n. a k))"
       apply (subst summable_iff_shift[where k=s,symmetric])
       by simp
     then have "summable (\<lambda>n. (a (n+1) * a (n+2)) / (\<Prod>k\<le>n+2. a k))"
     proof -
       assume asm:"summable (\<lambda>n. 1 / real_of_int (prod a {..n}))"
       have "1 / real_of_int (prod a {..n}) = (a (n+1) * a (n+2)) / (\<Prod>k\<le>n+2. a k)" for n 
       proof -
         have "a (Suc (Suc n)) \<noteq> 0" "a (Suc n) \<noteq>0" 
           using a_pos by (metis less_irrefl)+
         then show ?thesis 
           by (simp add: atLeast0_atMost_Suc atMost_atLeast0)
       qed
       then show ?thesis using asm by auto
     qed
     then show "summable (\<lambda>n. (a (n-1) * a n) / (\<Prod>k\<le>n. a k))"
       apply (subst summable_iff_shift[symmetric,of _ 2])
       by auto
   qed
   ultimately show ?thesis 
     apply (elim summable_comparison_test_ev[rotated])
     by (simp add: eventually_mono)
 qed
 
 private fun get_c::"(nat \<Rightarrow> int) \<Rightarrow> (nat \<Rightarrow> int) \<Rightarrow> int \<Rightarrow> nat \<Rightarrow> (nat \<Rightarrow> int)" where
   "get_c a' b' B N 0 = round (B * b' N / a' N)"|
   "get_c a' b' B N (Suc n) = get_c a' b' B N n * a' (n+N) - B * b' (n+N)"
 
 lemma ab_rationality_imp:
   assumes ab_rational:"(\<Sum>n. (b n / (\<Prod>i \<le> n. a i))) \<in> \<rat>"
   shows "\<exists> (B::int)>0. \<exists> c::nat\<Rightarrow> int.
             (\<forall>\<^sub>F n in sequentially. B*b n = c n * a n - c(n+1) \<and> \<bar>c(n+1)\<bar><a n/2)
             \<and> (\<lambda>n. c (Suc n) / a n) \<longlonglongrightarrow> 0"
 proof -
   have [simp]:"a n \<noteq> 0" for n using a_pos by (metis less_numeral_extra(3))
   obtain A::int and B::int where 
     AB_eq:"(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) = A / B" and "B>0"
   proof -
     obtain q::rat where "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) = real_of_rat q"
       using ab_rational by (rule Rats_cases) simp
     moreover obtain A::int and B::int where "q = Rat.Fract A B" "B > 0" "coprime A B"
       by (rule Rat_cases) auto
     ultimately show ?thesis by (auto intro!: that[of A B] simp:of_rat_rat)
   qed  
   define f where "f \<equiv> (\<lambda>n. b n / real_of_int (prod a {..n}))"
   define R where "R \<equiv> (\<lambda>N. (\<Sum>n. B*b (n+N+1) / prod a {N..n+N+1}))"
   have all_e_ubound:"\<forall>e>0. \<forall>\<^sub>F M in sequentially. \<forall>n. \<bar>B*b (n+M+1) / prod a {M..n+M+1}\<bar> < e/4 * 1/2^n"
   proof safe
     fix e::real assume "e>0"
     obtain N where N_a2: "\<forall>n \<ge> N. a n \<ge> 2" 
       and N_ba: "\<forall>n \<ge> N. \<bar>b n\<bar> / (a (n-1) * a n)  < e/(4*B)"
     proof -
       have "\<forall>\<^sub>F n in sequentially. \<bar>b n\<bar> / (a (n - 1) * a n) < e/(4*B)" 
         using order_topology_class.order_tendstoD[OF ab_tendsto,of "e/(4*B)"] \<open>B>0\<close> \<open>e>0\<close>
         by auto
       moreover have "\<forall>\<^sub>F n in sequentially. a n \<ge> 2"
         using a_large by (auto elim: eventually_mono)
       ultimately have "\<forall>\<^sub>F n in sequentially. \<bar>b n\<bar> / (a (n - 1) * a n) < e/(4*B) \<and> a n \<ge> 2" 
         by eventually_elim auto
       then show ?thesis unfolding eventually_at_top_linorder using that
         by auto
     qed
     have geq_N_bound:"\<bar>B*b (n+M+1) / prod a {M..n+M+1}\<bar> < e/4 * 1/2^n" when "M\<ge>N" for n M   
     proof -
       define D where "D = B*b (n+M+1)/ (a (n+M) * a (n+M+1))"
       have "\<bar>B*b (n+M+1) / prod a {M..n+M+1}\<bar> = \<bar>D / prod a {M..<n+M}\<bar>"
       proof -
         have "{M..n+M+1} = {M..<n+M} \<union> {n+M,n+M+1}" by auto
         then have "prod a {M..n+M+1} = a (n+M) * a (n+M+1)* prod a {M..<n+M}" by simp
         then show ?thesis unfolding D_def by (simp add:algebra_simps)
       qed
       also have "... <  \<bar>e/4 * (1/prod a {M..<n+M})\<bar>"
       proof -
         have "\<bar>D\<bar> < e/4" 
           unfolding D_def using N_ba[rule_format, of "n+M+1"] \<open>B>0\<close> \<open>M \<ge> N\<close> \<open>e>0\<close> a_pos
           by (auto simp:field_simps abs_mult abs_of_pos)
         from mult_strict_right_mono[OF this,of "1/prod a {M..<n+M}"] a_pos \<open>e>0\<close>
         show ?thesis 
           apply (auto simp:abs_prod abs_mult prod_pos)
           by (subst (2) abs_of_pos,auto)+
       qed
       also have "... \<le> e/4 * 1/2^n"
       proof -
         have "prod a {M..<n+M} \<ge> 2^n"
         proof (induct n)
           case 0
           then show ?case by simp
         next
           case (Suc n)
           then show ?case 
             using \<open>M\<ge>N\<close> by (simp add: N_a2 mult.commute mult_mono' prod.atLeastLessThan_Suc)
         qed
         then have "real_of_int (prod a {M..<n+M}) \<ge> 2^n" 
           using numeral_power_le_of_int_cancel_iff by blast
-        then show ?thesis using \<open>e>0\<close> by (auto simp add:divide_simps)
+        then show ?thesis using \<open>e>0\<close> by (auto simp:divide_simps)
       qed
       finally show ?thesis .
     qed
     show "\<forall>\<^sub>F M in sequentially. \<forall>n. \<bar>real_of_int (B * b (n + M + 1)) 
                 / real_of_int (prod a {M..n + M + 1})\<bar> < e / 4 * 1 / 2 ^ n"
       apply (rule eventually_sequentiallyI[of N])
       using geq_N_bound by blast
   qed
   have R_tendsto_0:"R \<longlonglongrightarrow> 0"
   proof (rule tendstoI)
     fix e::real assume "e>0"
     show "\<forall>\<^sub>F x in sequentially. dist (R x) 0 < e" using all_e_ubound[rule_format,OF \<open>e>0\<close>]
     proof eventually_elim
       case (elim M)
       define g where "g = (\<lambda>n. B*b (n+M+1) / prod a {M..n+M+1})"
       have g_lt:"\<bar>g n\<bar> < e/4 * 1/2^n" for n
         using elim unfolding g_def by auto
       have \<section>: "summable (\<lambda>n. (e/4) * (1/2)^n)"
         by simp 
       then have g_abs_summable:"summable (\<lambda>n. \<bar>g n\<bar>)"
         apply (elim summable_comparison_test')
         by (metis abs_idempotent g_lt less_eq_real_def power_one_over real_norm_def times_divide_eq_right)
       have "\<bar>\<Sum>n. g n\<bar> \<le> (\<Sum>n. \<bar>g n\<bar>)" by (rule summable_rabs[OF g_abs_summable])
       also have "... \<le>(\<Sum>n. e/4 * 1/2^n)"
       proof (rule suminf_comparison)
         show "summable (\<lambda>n. e/4 * 1/2^n)" 
           using \<section> unfolding power_divide by simp
         show "\<And>n. norm \<bar>g n\<bar> \<le> e / 4 * 1 / 2 ^ n" using g_lt less_eq_real_def by auto
       qed
       also have "... = (e/4) * (\<Sum>n. (1/2)^n)"
         apply (subst suminf_mult[symmetric])
          by (auto simp: algebra_simps power_divide)
       also have "... = e/2" by (simp add:suminf_geometric[of "1/2"])
       finally have "\<bar>\<Sum>n. g n\<bar> \<le> e / 2" .
       then show "dist (R M) 0 < e" unfolding R_def g_def using \<open>e>0\<close> by auto
     qed
   qed
 
   obtain N where R_N_bound:"\<forall>M \<ge> N. \<bar>R M\<bar> \<le>  1 / 4"
     and N_geometric:"\<forall>M\<ge>N. \<forall>n. \<bar>real_of_int (B * b (n + M + 1)) / (prod a {M..n + M + 1})\<bar> < 1 / 2 ^ n"
   proof -
     obtain N1 where N1:"\<forall>M \<ge> N1. \<bar>R M\<bar> \<le>  1 / 4"
       using metric_LIMSEQ_D[OF R_tendsto_0,of "1/4"] all_e_ubound[rule_format,of 4,unfolded eventually_sequentially]
       by (auto simp:less_eq_real_def)
     obtain N2 where N2:"\<forall>M\<ge>N2. \<forall>n. \<bar>real_of_int (B * b (n + M + 1)) 
                           / (prod a {M..n + M + 1})\<bar> < 1 / 2 ^ n"
       using all_e_ubound[rule_format,of 4,unfolded eventually_sequentially]
       by (auto simp:less_eq_real_def)
     define N where "N=max N1 N2"
     show ?thesis using that[of N] N1 N2 unfolding N_def by simp
   qed
 
   define C where "C = B * prod a {..<N} * (\<Sum>n<N. f n)"
   have "summable f"
     unfolding f_def using aux_series_summable .
   have "A * prod a {..<N} = C + B * b N / a N  + R N" 
   proof -
     have "A * prod a {..<N} = B * prod a {..<N} * (\<Sum>n. f n)"
       unfolding AB_eq f_def using \<open>B>0\<close> by auto
     also have "... = B * prod a {..<N} * ((\<Sum>n<N+1. f n) + (\<Sum>n. f (n+N+1)))"
       using suminf_split_initial_segment[OF \<open>summable f\<close>, of "N+1"] by auto
     also have "... = B * prod a {..<N} * ((\<Sum>n<N. f n) + f N + (\<Sum>n. f (n+N+1)))"
       using sum.atLeast0_lessThan_Suc by simp
     also have "... = C + B * b N / a N + (\<Sum>n. B*b (n+N+1) / prod a {N..n+N+1})"
     proof -
       have "B * prod a {..<N} * f N = B * b N / a N" 
       proof -
         have "{..N} =  {..<N} \<union> {N}" using ivl_disj_un_singleton(2) by blast
         then show ?thesis unfolding f_def by auto
       qed
       moreover have "B * prod a {..<N} * (\<Sum>n. f (n+N+1)) = (\<Sum>n. B*b (n+N+1) / prod a {N..n+N+1})"
       proof -
         have "summable (\<lambda>n. f (n + N + 1))" 
           using \<open>summable f\<close> summable_iff_shift[of f "N+1"] by auto
         moreover have "prod a {..<N} * f (n + N + 1) = b (n + N + 1) / prod a {N..n + N + 1}" for n
         proof -
           have "{..n + N + 1} = {..<N} \<union> {N..n + N + 1}" by auto
           then show ?thesis 
             unfolding f_def
             apply simp
             apply (subst prod.union_disjoint)
             by auto
         qed
         ultimately show ?thesis 
           apply (subst suminf_mult[symmetric])
-          by (auto simp add: mult.commute mult.left_commute)
+          by (auto simp: mult.commute mult.left_commute)
       qed
       ultimately show ?thesis unfolding C_def by (auto simp:algebra_simps)
     qed
     also have "... = C +B * b N / a N  + R N"
       unfolding R_def by simp
     finally show ?thesis .
   qed
   have R_bound:"\<bar>R M\<bar> \<le> 1 / 4" and R_Suc:"R (Suc M) = a M * R M - B * b (Suc M) / a (Suc M)" 
     when "M \<ge> N" for M
   proof -
     define g where "g = (\<lambda>n. B*b (n+M+1) / prod a {M..n+M+1})"
     have g_abs_summable:"summable (\<lambda>n. \<bar>g n\<bar>)"
     proof -
       have "summable (\<lambda>n. (1/2::real) ^ n)" 
         by simp
       moreover have "\<bar>g n\<bar> < 1/2^n" for n
         using N_geometric[rule_format,OF that] unfolding g_def by simp
       ultimately show ?thesis 
         apply (elim summable_comparison_test')
         by (simp add: less_eq_real_def power_one_over)
     qed
     show "\<bar>R M\<bar> \<le> 1 / 4" using R_N_bound[rule_format,OF that] .
     have "R M = (\<Sum>n. g n)" unfolding R_def g_def by simp
     also have "... = g 0 + (\<Sum>n. g (Suc n))"
       apply (subst suminf_split_head)
       using summable_rabs_cancel[OF g_abs_summable] by auto
     also have "... = g 0 + 1/a M * (\<Sum>n. a M * g (Suc n))"
       apply (subst suminf_mult)
-      by (auto simp add: g_abs_summable summable_Suc_iff summable_rabs_cancel)
+      by (auto simp: g_abs_summable summable_Suc_iff summable_rabs_cancel)
     also have "... = g 0 + 1/a M * R (Suc M)"
     proof -
       have "a M * g (Suc n) = B * b (n + M + 2) / prod a {Suc M..n + M + 2}" for n
       proof -
         have "{M..Suc (Suc (M + n))} = {M} \<union> {Suc M..Suc (Suc (M + n))}" by auto
         then show ?thesis 
-          unfolding g_def using \<open>B>0\<close> by (auto simp add:algebra_simps)
+          unfolding g_def using \<open>B>0\<close> by (auto simp:algebra_simps)
       qed
       then have "(\<Sum>n. a M * g (Suc n)) = R (Suc M)"
         unfolding R_def by auto
       then show ?thesis by auto
     qed
     finally have "R M = g 0 + 1 / a M * R (Suc M)" .
     then have "R (Suc M) = a M * R M - g 0 * a M" 
-      by (auto simp add:algebra_simps)
+      by (auto simp:algebra_simps)
     moreover have "{M..Suc M} = {M,Suc M}" by auto
     ultimately show "R (Suc M) = a M * R M - B * b (Suc M) / a (Suc M)" 
       unfolding g_def by auto
   qed
 
   define c where "c = (\<lambda>n. if n\<ge>N then get_c a b B N (n-N) else undefined)"
   have c_rec:"c (n+1) = c n * a n -  B * b n" when "n \<ge> N" for n
     unfolding c_def using that by (auto simp:Suc_diff_le)
   have c_R:"c (Suc n) / a n = R n" when "n \<ge> N" for n
     using that
   proof (induct rule:nat_induct_at_least)
     case base
     have "\<bar> c (N+1) / a N \<bar> \<le> 1/2" 
     proof -
       have "c N = round (B * b N / a N)" unfolding c_def by simp
       moreover have "c (N+1) / a N = c N - B * b N / a N"
         using a_pos[rule_format,of N]
-        by (auto simp add:c_rec[of N,simplified] divide_simps)
+        by (auto simp:c_rec[of N,simplified] divide_simps)
       ultimately show ?thesis using of_int_round_abs_le by auto
     qed        
     moreover have "\<bar>R N\<bar> \<le> 1 / 4" using R_bound[of N] by simp
     ultimately have "\<bar>c (N+1) / a N - R N \<bar> < 1" by linarith
     moreover have "c (N+1) / a N - R N \<in> \<int>"
     proof -
       have "c (N+1) / a N = c N - B * b N / a N"
         using a_pos[rule_format,of N]
-        by (auto simp add:c_rec[of N,simplified] divide_simps)
+        by (auto simp:c_rec[of N,simplified] divide_simps)
       moreover have " B * b N / a N + R N \<in> \<int>" 
       proof -
         have "C = B * (\<Sum>n<N. prod a {..<N} * (b n / prod a {..n}))"
           unfolding C_def f_def by (simp add:sum_distrib_left algebra_simps)
         also have "... = B * (\<Sum>n<N. prod a {n<..<N} * b n)"
         proof -
           have "{..<N} = {n<..<N} \<union> {..n}" if "n<N" for n 
             by (simp add: ivl_disj_un_one(1) sup_commute that)
           then show ?thesis
             using \<open>B>0\<close> 
             apply simp
             apply (subst prod.union_disjoint)
             by auto
         qed
         finally have "C = real_of_int (B * (\<Sum>n<N. prod a {n<..<N} * b n))" .
         then have "C \<in> \<int>" using Ints_of_int by blast
         moreover note \<open>A * prod a {..<N} = C + B * b N / a N  + R N\<close>
         ultimately show ?thesis 
           by (metis Ints_diff Ints_of_int add.assoc add_diff_cancel_left')
       qed
       ultimately show ?thesis by (simp add: diff_diff_add)
     qed
     ultimately have "c (N+1) / a N - R N = 0"
       by (metis Ints_cases less_irrefl of_int_0 of_int_lessD)
     then show ?case by simp
   next
     case (Suc n)
     have "c (Suc (Suc n)) / a (Suc n) = c (Suc n) - B * b (Suc n) / a (Suc n)"
       apply (subst c_rec[of "Suc n",simplified])
-      using \<open>N \<le> n\<close> by (auto simp add: divide_simps)
+      using \<open>N \<le> n\<close> by (auto simp: divide_simps)
     also have "... = a n * R n - B * b (Suc n) / a (Suc n)"  
       using Suc by (auto simp: divide_simps)
     also have "... = R (Suc n)"
       using R_Suc[OF \<open>N \<le> n\<close>] by simp
     finally  show ?case .
   qed
   have ca_tendsto_zero:"(\<lambda>n. c (Suc n) / a n) \<longlonglongrightarrow> 0"
     using R_tendsto_0 
     apply (elim filterlim_mono_eventually)
     using c_R by (auto intro!:eventually_sequentiallyI[of N])
   have ca_bound:"\<bar>c (n + 1)\<bar> < a n / 2" when "n \<ge> N" for n
   proof -
     have "\<bar>c (Suc n)\<bar> / a n  = \<bar>c (Suc n) / a n\<bar>" using a_pos[rule_format,of n] by auto
     also have "... = \<bar>R n\<bar>" using c_R[OF that] by auto
     also have "... < 1/2" using R_bound[OF that] by auto
     finally have "\<bar>c (Suc n)\<bar> / a n < 1/2" .
     then show ?thesis using a_pos[rule_format,of n] by auto
   qed
 
 (* (* the following part corresponds to (2.7) (2.8) in the original paper, but turns out to be
         not necessary. *)
   have c_round:"c n = round (B * b n / a n)" when "n \<ge> N" for n
   proof (cases "n=N")
     case True
     then show ?thesis unfolding c_def by simp
   next
     case False
     with \<open>n\<ge>N\<close> obtain n' where n_Suc:"n=Suc n'" and "n' \<ge> N" 
       by (metis le_eq_less_or_eq lessE less_imp_le_nat)
     have "B * b n / a n = c n - R n"
     proof -
       have "R n = c n - B * b n / a n"
         using c_R[OF \<open>n'\<ge>N\<close>,symmetric,folded n_Suc] R_Suc[OF \<open>n'\<ge>N\<close>,folded n_Suc]
         by (auto simp:field_simps)
       then show ?thesis by (auto simp:field_simps)
     qed
     then have "\<bar>B * b n / a n - c n\<bar> = \<bar>R n\<bar>" by auto
     then have "\<bar>B * b n / a n - c n\<bar> < 1/2" using R_bound[OF \<open>n \<ge> N\<close>] by auto
     from round_unique'[OF this] show ?thesis by auto
   qed
   *)
   show "\<exists>B>0. \<exists>c. (\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) 
             \<and> real_of_int \<bar>c (n + 1)\<bar> < a n / 2) \<and> (\<lambda>n. c (Suc n) / a n) \<longlonglongrightarrow> 0"
     unfolding eventually_at_top_linorder
     apply (rule exI[of _ B],use \<open>B>0\<close> in simp)
     apply (intro exI[of _c] exI[of _ N])
     using c_rec ca_bound ca_tendsto_zero 
     by fastforce
 qed
 
 private lemma imp_ab_rational:
   assumes "\<exists> (B::int)>0. \<exists> c::nat\<Rightarrow> int.
               (\<forall>\<^sub>F n in sequentially. B*b n = c n * a n - c(n+1) \<and> \<bar>c(n+1)\<bar><a n/2)"
   shows "(\<Sum>n. (b n / (\<Prod>i \<le> n. a i))) \<in> \<rat>"
 proof -
   obtain B::int and c::"nat\<Rightarrow>int" and N::nat where "B>0" and  
     large_n:"\<forall>n\<ge>N. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < a n / 2
                       \<and> a n\<ge>2"
   proof -
     obtain B c where "B>0" and event1:"\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) 
               \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2"
       using assms by auto
     from eventually_conj[OF event1 a_large,unfolded eventually_at_top_linorder]
     obtain N where "\<forall>n\<ge>N. (B * b n = c n * a n - c (n + 1) 
                         \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2) \<and> 2 \<le> a n"
       by fastforce
     then show ?thesis using that[of B N c] \<open>B>0\<close> by auto
   qed
   define f where "f=(\<lambda>n. real_of_int (b n) / real_of_int (prod a {..n}))"
   define S where "S = (\<Sum>n. f n)"
   have "summable f"
     unfolding f_def by (rule aux_series_summable)
   define C where "C=B*prod a {..<N} * (\<Sum>n<N. f n)"
   have "B*prod a {..<N} * S = C + real_of_int (c N)" 
   proof -
     define h1 where "h1 \<equiv> (\<lambda>n. (c (n+N) * a (n+N)) / prod a {N..n+N})"
     define h2 where "h2 \<equiv> (\<lambda>n. c (n+N+1) / prod a {N..n+N})"
     have f_h12: "B * prod a {..<N}*f (n+N) = h1 n - h2 n" for n 
     proof -
       define g1 where "g1 \<equiv> (\<lambda>n. B * b (n+N))"
       define g2 where "g2 \<equiv> (\<lambda>n. prod a {..<N} / prod a {..n + N})"
       have "B * prod a {..<N}*f (n+N) = (g1 n * g2 n)"
         unfolding f_def g1_def g2_def by (auto simp:algebra_simps)
       moreover have "g1 n = c (n+N) * a (n+N) - c (n+N+1)" 
         using large_n[rule_format,of "n+N"] unfolding g1_def by auto
       moreover have "g2 n = (1/prod a {N..n+N})" 
       proof -
         have "prod a ({..<N} \<union> {N..n + N}) = prod a {..<N} * prod a {N..n + N}"
           apply (rule prod.union_disjoint[of "{..<N}" "{N..n+N}" a])
           by auto
         moreover have "prod a ({..<N} \<union> {N..n + N}) = prod a {..n+N}"
           by (simp add: ivl_disj_un_one(4))
         ultimately show ?thesis 
           unfolding g2_def
           apply simp
           using a_pos by (metis less_irrefl)
       qed
       ultimately have "B*prod a {..<N}*f (n+N) = (c (n+N) * a (n+N) - c (n+N+1)) / prod a {N..n+N}"
         by auto
       also have "... = h1 n - h2 n"
         unfolding h1_def h2_def by (auto simp:algebra_simps diff_divide_distrib)
       finally show ?thesis by simp
     qed
     have "B*prod a {..<N} * S = B*prod a {..<N} * ((\<Sum>n<N. f n) + (\<Sum>n. f (n+N)))"
       using suminf_split_initial_segment[OF \<open>summable f\<close>,of N] 
       unfolding S_def by (auto simp:algebra_simps)
     also have "... = C + B*prod a {..<N}*(\<Sum>n. f (n+N))"
       unfolding C_def by (auto simp:algebra_simps)
     also have "... = C + (\<Sum>n. h1 n - h2 n)"
       apply (subst suminf_mult[symmetric])
       using \<open>summable f\<close> f_h12 by auto
     also have "... = C + h1 0"
     proof -
       have "(\<lambda>n. \<Sum>i\<le>n. h1 i - h2 i) \<longlonglongrightarrow> (\<Sum>i. h1 i - h2 i)"
       proof (rule summable_LIMSEQ')
         have "(\<lambda>i. h1 i - h2 i) = (\<lambda>i.  real_of_int (B * prod a {..<N}) * f (i + N))"
           using f_h12 by auto
         then show "summable (\<lambda>i. h1 i - h2 i)"
           using \<open>summable f\<close> by (simp add: summable_mult)
       qed
       moreover have "(\<Sum>i\<le>n. h1 i - h2 i)  = h1 0 - h2 n" for n
       proof (induct n)
         case 0
         then show ?case by simp
       next
         case (Suc n)
         have "(\<Sum>i\<le>Suc n. h1 i - h2 i) = (\<Sum>i\<le>n. h1 i - h2 i) + h1 (n+1) - h2 (n+1)"
           by auto
         also have "... =  h1 0 - h2 n + h1 (n+1) - h2 (n+1)" using Suc by auto
         also have "... = h1 0 - h2 (n+1)"
         proof -
           have "h2 n = h1 (n+1)"
             unfolding h2_def h1_def 
             apply (auto simp:prod.nat_ivl_Suc')
             using a_pos by (metis less_numeral_extra(3))
           then show ?thesis by auto
         qed
         finally show ?case by simp
       qed
       ultimately have "(\<lambda>n. h1 0 - h2 n) \<longlonglongrightarrow> (\<Sum>i. h1 i - h2 i)" by simp
       then have "h2 \<longlonglongrightarrow> (h1 0 -  (\<Sum>i. h1 i - h2 i))"
         apply (elim metric_tendsto_imp_tendsto)
         by (auto intro!:eventuallyI simp add:dist_real_def)
       moreover have "h2 \<longlonglongrightarrow> 0"
       proof -
         have h2_n:"\<bar>h2 n\<bar> < (1 / 2)^(n+1)" for n 
         proof -
           have "\<bar>h2 n\<bar> = \<bar>c (n + N + 1)\<bar> / prod a {N..n + N}"
             unfolding h2_def abs_divide
             using a_pos by (simp add: abs_of_pos prod_pos)
           also have "... < (a (N+n) / 2) / prod a {N..n + N}"
             unfolding h2_def
             apply (rule divide_strict_right_mono)
-            subgoal using large_n[rule_format,of "N+n"] by (auto simp add:algebra_simps)
+            subgoal using large_n[rule_format,of "N+n"] by (auto simp:algebra_simps)
             subgoal using a_pos by (simp add: prod_pos)
             done
           also have "... = 1 / (2*prod a {N..<n + N})"
             apply (subst ivl_disj_un(6)[of N "n+N",symmetric])
-            using a_pos[rule_format,of "N+n"] by (auto simp add:algebra_simps)
+            using a_pos[rule_format,of "N+n"] by (auto simp:algebra_simps)
           also have "... \<le> (1/2)^(n+1)"
           proof (induct n)
             case 0
             then show ?case by auto
           next
             case (Suc n)
             define P where "P=1 / real_of_int (2 * prod a {N..<n + N})"
             have "1 / real_of_int (2 * prod a {N..<Suc n + N}) = P / a (n+N)"
-              unfolding P_def by (auto simp add: prod.atLeastLessThan_Suc)
+              unfolding P_def by (auto simp: prod.atLeastLessThan_Suc)
             also have "... \<le>  ( (1 / 2) ^ (n + 1) ) / a (n+N) "
               apply (rule divide_right_mono)
               subgoal unfolding P_def using Suc by auto
               subgoal by (simp add: a_pos less_imp_le)
               done
             also have "... \<le> ( (1 / 2) ^ (n + 1) ) / 2 "
               apply (rule divide_left_mono)
               using large_n[rule_format,of "n+N",simplified] by auto
             also have "... = (1 / 2) ^ (n + 2)" by auto
             finally show ?case by simp
           qed
           finally show ?thesis .
         qed
         have "(\<lambda>n. (1 / 2)^(n+1)) \<longlonglongrightarrow> (0::real)" 
           using tendsto_mult_right_zero[OF LIMSEQ_abs_realpow_zero2[of "1/2",simplified],of "1/2"]
           by auto
         then show ?thesis 
           apply (elim Lim_null_comparison[rotated])
           using h2_n less_eq_real_def by (auto intro!:eventuallyI)
       qed
       ultimately have "(\<Sum>i. h1 i - h2 i) = h1 0"
         using LIMSEQ_unique by fastforce
       then show ?thesis by simp
     qed
     also have "... = C + c N"
       unfolding h1_def using a_pos 
       by auto (metis less_irrefl)
     finally show ?thesis . 
   qed
   then have "S = (C + real_of_int (c N)) / (B*prod a {..<N})" 
     by (metis \<open>0 < B\<close> a_pos less_irrefl mult.commute mult_pos_pos 
         nonzero_mult_div_cancel_right of_int_eq_0_iff prod_pos)
   moreover have "... \<in> \<rat>"
     unfolding C_def f_def by (intro Rats_divide Rats_add Rats_mult Rats_of_int Rats_sum)
   ultimately show "S \<in> \<rat>" by auto
 qed
 
 theorem theorem_2_1_Erdos_Straus :
   "(\<Sum>n. (b n / (\<Prod>i \<le> n. a i))) \<in> \<rat> \<longleftrightarrow> (\<exists> (B::int)>0. \<exists> c::nat\<Rightarrow> int.
             (\<forall>\<^sub>F n in sequentially. B*b n = c n * a n - c(n+1) \<and> \<bar>c(n+1)\<bar><a n/2))"
   using ab_rationality_imp imp_ab_rational by auto
 
 text\<open>The following is a Corollary to Theorem 2.1.  \<close>
 
 corollary corollary_2_10_Erdos_Straus:
   assumes ab_event:"\<forall>\<^sub>F n in sequentially. b n > 0 \<and> a (n+1) \<ge> a n" 
     and ba_lim_leq:"lim (\<lambda>n. (b(n+1) - b n )/a n) \<le> 0" 
     and ba_lim_exist:"convergent (\<lambda>n. (b(n+1) - b n )/a n)"
     and "liminf (\<lambda>n. a n / b n) = 0 "
   shows "(\<Sum>n. (b n / (\<Prod>i \<le> n. a i))) \<notin> \<rat>"
 proof 
   assume "(\<Sum>n. (b n / (\<Prod>i \<le> n. a i))) \<in> \<rat>"
   then obtain B c where "B>0" and abc_event:"\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) 
           \<and> \<bar>c (n + 1)\<bar> < a n / 2" and ca_vanish: "(\<lambda>n. c (Suc n) / a n) \<longlonglongrightarrow> 0"
     using ab_rationality_imp by auto
 
   have bac_close:"(\<lambda>n. B * b n / a n - c n) \<longlonglongrightarrow> 0"
   proof -
     have "\<forall>\<^sub>F n in sequentially. B * b n - c n * a n + c (n + 1) = 0"
       using abc_event by (auto elim!:eventually_mono)
     then have "\<forall>\<^sub>F n in sequentially. (B * b n - c n * a n + c (n+1)) / a n = 0 "
       apply eventually_elim
       by auto
     then have "\<forall>\<^sub>F n in sequentially. B * b n / a n - c n  + c (n + 1) / a n = 0"
       apply eventually_elim
       using a_pos by (auto simp:divide_simps) (metis less_irrefl)
     then have "(\<lambda>n. B * b n / a n - c n  + c (n + 1) / a n) \<longlonglongrightarrow> 0"
       by (simp add: eventually_mono tendsto_iff)
     from tendsto_diff[OF this ca_vanish]  
     show ?thesis by auto
   qed
 
   have c_pos:"\<forall>\<^sub>F n in sequentially. c n > 0" 
   proof -
     from bac_close have *:"\<forall>\<^sub>F n in sequentially. c n \<ge> 0"
       apply (elim tendsto_of_int_diff_0)
       using ab_event a_large apply (eventually_elim) 
       using \<open>B>0\<close> by auto 
     show ?thesis
     proof (rule ccontr)
       assume "\<not> (\<forall>\<^sub>F n in sequentially. c n > 0)"
       moreover have "\<forall>\<^sub>F n in sequentially. c (Suc n) \<ge> 0 \<and> c n\<ge>0"
         using * eventually_sequentially_Suc[of "\<lambda>n. c n\<ge>0"] 
         by (metis (mono_tags, lifting) eventually_at_top_linorder le_Suc_eq)
       ultimately have "\<exists>\<^sub>F n in sequentially. c n = 0 \<and> c (Suc n) \<ge> 0"
         using eventually_elim2 frequently_def by fastforce
       moreover have "\<forall>\<^sub>F n in sequentially. b n > 0 \<and> B * b n = c n * a n - c (n + 1)"
         using ab_event abc_event by eventually_elim auto
       ultimately have "\<exists>\<^sub>F n in sequentially. c n = 0 \<and> c (Suc n) \<ge> 0 \<and> b n > 0 
                           \<and> B * b n = c n * a n - c (n + 1)"
         using frequently_eventually_frequently by fastforce
       from frequently_ex[OF this] 
       obtain n where "c n = 0" "c (Suc n) \<ge> 0" "b n > 0" 
         "B * b n = c n * a n - c (n + 1)"
         by auto
       then have "B * b n \<le> 0" by auto
       then show False using \<open>b n>0\<close> \<open>B > 0\<close> using mult_pos_pos not_le by blast
     qed
   qed
 
   have bc_epsilon:"\<forall>\<^sub>F n in sequentially. b (n+1) / b n >  (c (n+1) - \<epsilon>) / c n" when "\<epsilon>>0" "\<epsilon><1" for \<epsilon>::real
   proof -
     have "\<forall>\<^sub>F x in sequentially. \<bar>c (Suc x) / a x\<bar> < \<epsilon> / 2"
       using ca_vanish[unfolded tendsto_iff,rule_format, of "\<epsilon>/2"] \<open>\<epsilon>>0\<close> by auto
     moreover then have "\<forall>\<^sub>F x in sequentially. \<bar>c (x+2) / a (x+1)\<bar> < \<epsilon> / 2"
       apply (subst (asm) eventually_sequentially_Suc[symmetric])
       by simp
     moreover have "\<forall>\<^sub>F n in sequentially. B * b (n+1) = c (n+1) * a (n+1) - c (n + 2)"
       using abc_event
       apply (subst (asm) eventually_sequentially_Suc[symmetric])
       by (auto elim:eventually_mono)
     moreover have "\<forall>\<^sub>F n in sequentially. c n > 0 \<and> c (n+1) > 0 \<and> c (n+2) > 0"
     proof -
       have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc n)"
         using c_pos by (subst eventually_sequentially_Suc) simp
       moreover then have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc (Suc n))"
         using c_pos by (subst eventually_sequentially_Suc) simp
       ultimately show ?thesis using c_pos by eventually_elim auto
     qed
     ultimately show ?thesis using ab_event abc_event 
     proof eventually_elim
       case (elim n)
       define \<epsilon>\<^sub>0 \<epsilon>\<^sub>1 where "\<epsilon>\<^sub>0 = c (n+1) / a n" and "\<epsilon>\<^sub>1 = c (n+2) / a (n+1)"
-      have "\<epsilon>\<^sub>0 > 0" "\<epsilon>\<^sub>1 > 0" "\<epsilon>\<^sub>0 < \<epsilon>/2" "\<epsilon>\<^sub>1 < \<epsilon>/2" using a_pos elim by (auto simp add: \<epsilon>\<^sub>0_def \<epsilon>\<^sub>1_def)
+      have "\<epsilon>\<^sub>0 > 0" "\<epsilon>\<^sub>1 > 0" "\<epsilon>\<^sub>0 < \<epsilon>/2" "\<epsilon>\<^sub>1 < \<epsilon>/2" using a_pos elim by (auto simp: \<epsilon>\<^sub>0_def \<epsilon>\<^sub>1_def)
       have "(\<epsilon> - \<epsilon>\<^sub>1) * c n > 0"
-        apply (rule mult_pos_pos)
-        using \<open>\<epsilon>\<^sub>1 > 0\<close> \<open>\<epsilon>\<^sub>1 < \<epsilon>/2\<close> \<open>\<epsilon>>0\<close> elim by auto
+        using \<open>\<epsilon>\<^sub>1 < \<epsilon> / 2\<close> elim(4) that(1) by auto
       moreover have "\<epsilon>\<^sub>0 * (c (n+1) - \<epsilon>) > 0"
-        apply (rule mult_pos_pos[OF \<open>\<epsilon>\<^sub>0 > 0\<close>])
-        using elim(4) that(2) by linarith
+        using \<open>0 < \<epsilon>\<^sub>0\<close> elim(4) that(2) by auto
       ultimately have "(\<epsilon> - \<epsilon>\<^sub>1) * c n + \<epsilon>\<^sub>0 * (c (n+1) - \<epsilon>) > 0" by auto
-      moreover have "c n - \<epsilon>\<^sub>0 > 0" using \<open>\<epsilon>\<^sub>0 < \<epsilon> / 2\<close> elim(4) that(2) by linarith
+      moreover have gt0: "c n - \<epsilon>\<^sub>0 > 0" using \<open>\<epsilon>\<^sub>0 < \<epsilon> / 2\<close> elim(4) that(2) by linarith
       moreover have "c n > 0" by (simp add: elim(4))
       ultimately have "(c (n+1) - \<epsilon>) / c n < (c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0)"
-        by (auto simp add: field_simps)
+        by (auto simp: field_simps)
       also have "... \<le> (c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) * (a (n+1) / a n)"
       proof -
-        have "(c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) > 0" 
-          by (smt \<open>0 < (\<epsilon> - \<epsilon>\<^sub>1) * real_of_int (c n)\<close> \<open>0 < real_of_int (c n) - \<epsilon>\<^sub>0\<close> 
-              divide_pos_pos elim(4) mult_le_0_iff of_int_less_1_iff that(2))
+        have "(c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) > 0"
+          using gt0 \<open>\<epsilon>\<^sub>1 < \<epsilon> / 2\<close> elim(4) that(2) by force 
         moreover have "(a (n+1) / a n) \<ge> 1" 
           using a_pos elim(5) by auto
         ultimately show ?thesis by (metis mult_cancel_left1 mult_le_cancel_left_pos)
       qed
       also have "... = (B * b (n+1)) / (B * b n)"
       proof -
         have "B * b n = c n * a n - c (n + 1)"
           using elim by auto
         also have "... = a n * (c n - \<epsilon>\<^sub>0)"
           using a_pos[rule_format,of n] unfolding \<epsilon>\<^sub>0_def by (auto simp:field_simps)
         finally have "B * b n = a n * (c n - \<epsilon>\<^sub>0)" .
         moreover have "B * b (n+1) = a (n+1) * (c (n+1) - \<epsilon>\<^sub>1)" 
           unfolding \<epsilon>\<^sub>1_def 
           using a_pos[rule_format,of "n+1"] 
           apply (subst \<open>B * b (n + 1) = c (n + 1) * a (n + 1) - c (n + 2)\<close>)
           by (auto simp:field_simps)
         ultimately show ?thesis by (simp add: mult.commute)
       qed
       also have "... = b (n+1) / b n" 
         using \<open>B>0\<close> by auto
       finally show ?case .
     qed
   qed
 
   have eq_2_11:"\<exists>\<^sub>F n in sequentially. b (n+1) > b n + (1 - \<epsilon>)^2 * a n / B" 
     when "\<epsilon>>0" "\<epsilon><1" "\<not> (\<forall>\<^sub>F n in sequentially. c (n+1) \<le> c n)"  for \<epsilon>::real
   proof -
     have "\<exists>\<^sub>F x in sequentially. c x < c (Suc x) " using that(3) 
       by (simp add:not_eventually frequently_elim1)
     moreover have "\<forall>\<^sub>F x in sequentially. \<bar>c (Suc x) / a x\<bar> < \<epsilon>"
       using ca_vanish[unfolded tendsto_iff,rule_format, of \<epsilon>] \<open>\<epsilon>>0\<close> by auto
     moreover have "\<forall>\<^sub>F n in sequentially. c n > 0 \<and> c (n+1) > 0"
     proof -
       have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc n)"
         using c_pos by (subst eventually_sequentially_Suc) simp
       then show ?thesis using c_pos by eventually_elim auto
     qed
     ultimately show ?thesis  using ab_event abc_event bc_epsilon[OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><1\<close>] 
     proof (elim frequently_rev_mp,eventually_elim)
       case (elim n)
       then have "c (n+1) / a n < \<epsilon>" 
         using a_pos[rule_format,of n] by auto
       also have "... \<le> \<epsilon> * c n" using elim(7) that(1) by auto
       finally have "c (n+1) / a n < \<epsilon> * c n" .
       then have "c (n+1) / c n < \<epsilon> * a n" 
         using a_pos[rule_format,of n] elim by (auto simp:field_simps)
       then have "(1 - \<epsilon>) * a n < a n - c (n+1) / c n"
         by (auto simp:algebra_simps)
       then have "(1 - \<epsilon>)^2 * a n / B < (1 - \<epsilon>) * (a n - c (n+1) / c n) / B"
         apply (subst (asm) mult_less_cancel_right_pos[symmetric, of "(1-\<epsilon>)/B"])
         using \<open>\<epsilon><1\<close> \<open>B>0\<close> by (auto simp: divide_simps power2_eq_square mult_less_cancel_right_pos)
       then have "b n + (1 - \<epsilon>)^2 * a n / B < b n + (1 - \<epsilon>) * (a n - c (n+1) / c n) / B"
         using \<open>B>0\<close> by auto
       also have "... = b n + (1 - \<epsilon>) * ((c n *a n - c (n+1)) / c n) / B"
         using elim by (auto simp:field_simps)
       also have "... = b n + (1 - \<epsilon>) * (b n / c n)"
       proof -
         have "B * b n = c n * a n - c (n + 1)" using elim by auto
         from this[symmetric] show ?thesis
           using \<open>B>0\<close> by simp
       qed
       also have "... = (1+(1-\<epsilon>)/c n) * b n"
         by (auto simp:algebra_simps)
       also have "... = ((c n+1-\<epsilon>)/c n) * b n"
         using elim by (auto simp:divide_simps)
       also have "... \<le> ((c (n+1) -\<epsilon>)/c n) * b n"
       proof -
         define cp where "cp = c n+1"
         have "c (n+1) \<ge> cp" unfolding cp_def using \<open>c n < c (Suc n)\<close> by auto
         moreover have "c n>0" "b n>0" using elim by auto
         ultimately show ?thesis 
           apply (fold cp_def)
           by (auto simp:divide_simps)
       qed
       also have "... < b (n+1)"
         using elim by (auto simp:divide_simps)
       finally show ?case .
     qed
   qed
 
   have "\<forall>\<^sub>F n in sequentially. c (n+1) \<le> c n"
   proof (rule ccontr)
     assume "\<not> (\<forall>\<^sub>F n in sequentially. c (n + 1) \<le> c n)"
     from eq_2_11[OF _ _ this,of "1/2"]
     have "\<exists>\<^sub>F n in sequentially. b (n+1) > b n + 1/4 * a n / B" 
       by (auto simp:algebra_simps power2_eq_square)
     then have *:"\<exists>\<^sub>F n in sequentially. (b (n+1) - b n) / a n > 1 / (B * 4)" 
       apply (elim frequently_elim1)
       subgoal for n
         using a_pos[rule_format,of n] by (auto simp:field_simps)
       done
     define f where "f = (\<lambda>n. (b (n+1) - b n) / a n)"
     have "f \<longlonglongrightarrow> lim f"
       using convergent_LIMSEQ_iff ba_lim_exist unfolding f_def by auto
     from this[unfolded tendsto_iff,rule_format, of "1 / (B*4)"] 
     have "\<forall>\<^sub>F x in sequentially. \<bar>f x - lim f\<bar> < 1 / (B * 4)"
       using \<open>B>0\<close> by (auto simp:dist_real_def)
     moreover have "\<exists>\<^sub>F n in sequentially. f n > 1 / (B * 4)" 
       using * unfolding f_def by auto
     ultimately have "\<exists>\<^sub>F n in sequentially. f n > 1 / (B * 4) \<and> \<bar>f n - lim f\<bar> < 1 / (B * 4)"
       by (auto elim:frequently_eventually_frequently[rotated])
     from frequently_ex[OF this] 
     obtain n where "f n > 1 / (B * 4)" "\<bar>f n - lim f\<bar> < 1 / (B * 4)"
       by auto
     moreover have "lim f \<le> 0" using ba_lim_leq unfolding f_def by auto
     ultimately show False by linarith
   qed
   then obtain N where N_dec:"\<forall>n\<ge>N. c (n+1) \<le> c n" by (meson eventually_at_top_linorder)
   define max_c where "max_c = (MAX n \<in> {..N}. c n)"
   have max_c:"c n \<le> max_c" for n
   proof (cases "n\<le>N")
     case True
     then show ?thesis unfolding max_c_def by simp
   next
     case False
     then have "n\<ge>N" by auto
     then have "c n\<le>c N" 
     proof (induct rule:nat_induct_at_least)
       case base
       then show ?case by simp
     next
       case (Suc n)
       then have "c (n+1) \<le> c n" using N_dec by auto
       then show ?case using \<open>c n \<le> c N\<close> by auto
     qed
     moreover have "c N \<le> max_c" unfolding max_c_def by auto
     ultimately show ?thesis by auto
   qed
   have "max_c > 0 " 
   proof -
     obtain N where "\<forall>n\<ge>N. 0 < c n" 
       using c_pos[unfolded eventually_at_top_linorder] by auto
     then have "c N > 0" by auto
     then show ?thesis using max_c[of N] by simp
   qed
   have ba_limsup_bound:"1/(B*(B+1)) \<le> limsup (\<lambda>n. b n/a n)" 
     "limsup (\<lambda>n. b n/a n) \<le> max_c / B + 1 / (B+1)"
   proof -
     define f where "f = (\<lambda>n. b n/a n)"
     from tendsto_mult_right_zero[OF bac_close,of "1/B"] 
     have "(\<lambda>n. f n - c n / B) \<longlonglongrightarrow> 0"
       unfolding f_def using  \<open>B>0\<close> by (auto simp:algebra_simps)
     from this[unfolded tendsto_iff,rule_format,of "1/(B+1)"]
     have "\<forall>\<^sub>F x in sequentially. \<bar>f x - c x / B\<bar> < 1 / (B+1)"
       using \<open>B>0\<close> by auto
     then have *:"\<forall>\<^sub>F n in sequentially. 1/(B*(B+1)) \<le> ereal (f n) \<and> ereal (f n) \<le> max_c / B + 1 / (B+1)" 
       using c_pos
     proof eventually_elim
       case (elim n)
       then have "f n - c n / B < 1 / (B+1)" by auto
       then have "f n < c n / B + 1 / (B+1)" by simp
       also have "... \<le> max_c / B + 1 / (B+1)"
         using max_c[of n] using \<open>B>0\<close> by (auto simp:divide_simps)
       finally have *:"f n < max_c / B + 1 / (B+1)" .
 
       have "1/(B*(B+1)) = 1/B - 1 / (B+1)" 
         using \<open>B>0\<close> by (auto simp:divide_simps)
       also have "... \<le> c n/B - 1 / (B+1)" 
         using \<open>0 < c n\<close> \<open>B>0\<close> by (auto,auto simp:divide_simps)
       also have "... < f n" using elim by auto
       finally have "1/(B*(B+1)) < f n" .
       with * show ?case by simp
     qed
     show "limsup f \<le> max_c / B + 1 / (B+1)"
       apply (rule Limsup_bounded)
       using * by (auto elim:eventually_mono)
     have "1/(B*(B+1)) \<le> liminf f"
       apply (rule Liminf_bounded)
       using * by (auto elim:eventually_mono)
     also have "liminf f \<le> limsup f" by (simp add: Liminf_le_Limsup)
     finally show "1/(B*(B+1)) \<le> limsup f" .
   qed
 
   have "0 < inverse (ereal (max_c / B + 1 / (B+1)))"
     using \<open>max_c > 0\<close> \<open>B>0\<close> 
     by (simp add: pos_add_strict)
   also have "... \<le> inverse (limsup (\<lambda>n. b n/a n))"
   proof (rule ereal_inverse_antimono[OF _ ba_limsup_bound(2)])
     have "0<1/(B*(B+1))" using \<open>B>0\<close> by auto
     also have "... \<le> limsup (\<lambda>n. b n/a n)" using ba_limsup_bound(1) .
     finally show "0\<le>limsup (\<lambda>n. b n/a n)" using zero_ereal_def by auto
   qed 
   also have "... = liminf (\<lambda>n. inverse (ereal ( b n/a n)))"
     apply (subst Liminf_inverse_ereal[symmetric])
     using a_pos ab_event by (auto elim!:eventually_mono simp:divide_simps)
   also have "... = liminf (\<lambda>n. ( a n/b n))"
     apply (rule Liminf_eq)
     using a_pos ab_event 
     apply (auto elim!:eventually_mono) 
     by (metis less_int_code(1))
   finally have "liminf (\<lambda>n. ( a n/b n)) > 0" .
   then show False using \<open>liminf (\<lambda>n. a n / b n) = 0\<close> by simp
 qed
 
 end
 
 section\<open>Some auxiliary results on the prime numbers. \<close>
 
 lemma nth_prime_nonzero[simp]:"nth_prime n \<noteq> 0"
   by (simp add: prime_gt_0_nat prime_nth_prime)
 
 lemma nth_prime_gt_zero[simp]:"nth_prime n >0"
   by (simp add: prime_gt_0_nat prime_nth_prime)
 
 lemma ratio_of_consecutive_primes:
   "(\<lambda>n. nth_prime (n+1)/nth_prime n) \<longlonglongrightarrow>1"
 proof -
   define f where "f=(\<lambda>x. real (nth_prime (Suc x)) /real (nth_prime x))"
   define g where "g=(\<lambda>x. (real x * ln (real x)) 
                               / (real (Suc x) * ln (real (Suc x))))"
   have p_n:"(\<lambda>x. real (nth_prime x) / (real x * ln (real x))) \<longlonglongrightarrow> 1"
     using nth_prime_asymptotics[unfolded asymp_equiv_def,simplified] .
   moreover have p_sn:"(\<lambda>n. real (nth_prime (Suc n)) 
                         / (real (Suc n) * ln (real (Suc n)))) \<longlonglongrightarrow> 1"
     using nth_prime_asymptotics[unfolded asymp_equiv_def,simplified
         ,THEN LIMSEQ_Suc] .
   ultimately have "(\<lambda>x. f x * g x) \<longlonglongrightarrow> 1"
     using tendsto_divide[OF p_sn p_n] 
     unfolding f_def g_def by (auto simp:algebra_simps)
   moreover have "g \<longlonglongrightarrow> 1" unfolding g_def
     by real_asymp
   ultimately have "(\<lambda>x. if g x = 0 then 0 else f x) \<longlonglongrightarrow> 1"
     apply (drule_tac tendsto_divide[OF _ \<open>g \<longlonglongrightarrow> 1\<close>])
     by auto
   then have "f \<longlonglongrightarrow> 1"
   proof (elim filterlim_mono_eventually)
     have "\<forall>\<^sub>F x in sequentially. (if g (x+3) = 0 then 0 
                 else f (x+3)) = f (x+3)" 
       unfolding g_def by auto
     then show "\<forall>\<^sub>F x in sequentially. (if g x = 0 then 0 else f x) = f x"
       apply (subst (asm) eventually_sequentially_seg)
       by simp
   qed auto
   then show ?thesis unfolding f_def by auto
 qed
 
 lemma nth_prime_double_sqrt_less:
   assumes "\<epsilon> > 0"
   shows "\<forall>\<^sub>F n in sequentially. (nth_prime (2*n) - nth_prime n) 
             / sqrt (nth_prime n) < n powr (1/2+\<epsilon>)"
 proof -
   define pp ll where 
     "pp=(\<lambda>n. (nth_prime (2*n) - nth_prime n) / sqrt (nth_prime n))" and
     "ll=(\<lambda>x::nat. x * ln x)"
   have pp_pos:"pp (n+1) > 0" for n
     unfolding pp_def by simp
 
   have "(\<lambda>x. nth_prime (2 * x)) \<sim>[sequentially] (\<lambda>x. (2 * x) * ln (2 * x))"
     using nth_prime_asymptotics[THEN asymp_equiv_compose
         ,of "(*) 2" sequentially,unfolded comp_def]
     using mult_nat_left_at_top pos2 by blast
   also have "... \<sim>[sequentially] (\<lambda>x. 2 *x * ln x)"
     by real_asymp
   finally have "(\<lambda>x. nth_prime (2 * x)) \<sim>[sequentially] (\<lambda>x. 2 *x * ln x)" .
   from this[unfolded asymp_equiv_def, THEN tendsto_mult_left,of 2]
   have "(\<lambda>x. nth_prime (2 * x) / (x * ln x)) \<longlonglongrightarrow> 2"
     unfolding asymp_equiv_def by auto
   moreover have *:"(\<lambda>x. nth_prime x / (x * ln x)) \<longlonglongrightarrow> 1"
     using nth_prime_asymptotics unfolding asymp_equiv_def by auto
   ultimately    
   have "(\<lambda>x. (nth_prime (2 * x) - nth_prime x) / ll x) \<longlonglongrightarrow> 1"
     unfolding ll_def
     apply -
     apply (drule (1) tendsto_diff)
     apply (subst of_nat_diff,simp)
     by (subst diff_divide_distrib,simp)
   moreover have "(\<lambda>x. sqrt (nth_prime x) / sqrt (ll x)) \<longlonglongrightarrow> 1"
     unfolding ll_def
     using tendsto_real_sqrt[OF *] 
     by (auto simp: real_sqrt_divide)
   ultimately have "(\<lambda>x. pp x * (sqrt (ll x) / (ll x))) \<longlonglongrightarrow> 1"
     apply -
     apply (drule (1) tendsto_divide,simp)
     by (auto simp:field_simps of_nat_diff pp_def)
   moreover have "\<forall>\<^sub>F x in sequentially. sqrt (ll x) / ll x = 1/sqrt (ll x)"
     apply (subst eventually_sequentially_Suc[symmetric])
     by (auto intro!:eventuallyI simp:ll_def divide_simps)
   ultimately have "(\<lambda>x. pp x / sqrt (ll x)) \<longlonglongrightarrow> 1"
     apply (elim filterlim_mono_eventually)
     by (auto elim!:eventually_mono) (metis mult.right_neutral times_divide_eq_right)
   moreover have "(\<lambda>x. sqrt (ll x) / x powr (1/2+\<epsilon>)) \<longlonglongrightarrow> 0"
     unfolding ll_def using \<open>\<epsilon>>0\<close> by real_asymp
   ultimately have "(\<lambda>x. pp x / x powr (1/2+\<epsilon>) * 
                       (sqrt (ll x) / sqrt (ll x))) \<longlonglongrightarrow> 0"
     apply -
     apply (drule (1) tendsto_mult)
     by (auto elim:filterlim_mono_eventually)
   moreover have "\<forall>\<^sub>F x in sequentially. sqrt (ll x) / sqrt (ll x) = 1"
     apply (subst eventually_sequentially_Suc[symmetric])
     by (auto intro!:eventuallyI simp:ll_def )
   ultimately have "(\<lambda>x. pp x / x powr (1/2+\<epsilon>)) \<longlonglongrightarrow> 0"
     apply (elim filterlim_mono_eventually)
     by (auto elim:eventually_mono)
   from tendstoD[OF this, of 1,simplified]
   show "\<forall>\<^sub>F x in sequentially. pp x < x powr (1 / 2 + \<epsilon>)"
     apply (elim eventually_mono_sequentially[of _ 1])
     using pp_pos by auto
 qed
 
 
 section \<open>Theorem 3.1\<close>
 
 text\<open>Theorem 3.1 is an application of Theorem 2.1 with the sequences considered involving 
 the prime numbers.\<close>
 
 theorem theorem_3_10_Erdos_Straus:
   fixes a::"nat \<Rightarrow> int"
   assumes a_pos:"\<forall> n. a n >0" and "mono a"
     and nth_1:"(\<lambda>n. nth_prime n / (a n)^2) \<longlonglongrightarrow> 0"
     and nth_2:"liminf (\<lambda>n. a n / nth_prime n) = 0"
   shows "(\<Sum>n. (nth_prime n / (\<Prod>i \<le> n. a i))) \<notin> \<rat>"
 proof
   assume asm:"(\<Sum>n. (nth_prime n / (\<Prod>i \<le> n. a i))) \<in> \<rat>"
 
   have a2_omega:"(\<lambda>n. (a n)^2) \<in> \<omega>(\<lambda>x. x * ln x)"
   proof -
     have "(\<lambda>n. real (nth_prime n)) \<in> o(\<lambda>n. real_of_int ((a n)\<^sup>2))"
       apply (rule smalloI_tendsto[OF nth_1])
       using a_pos by (metis (mono_tags, lifting) less_int_code(1) 
           not_eventuallyD of_int_0_eq_iff zero_eq_power2)
     moreover have "(\<lambda>x. real (nth_prime x)) \<in> \<Omega>(\<lambda>x. real x * ln (real x))"
       using nth_prime_bigtheta
       by blast
     ultimately show ?thesis 
       using landau_omega.small_big_trans smallo_imp_smallomega by blast
   qed
 
   have a_gt_1:"\<forall>\<^sub>F n in sequentially. 1 < a n" 
   proof -
     have "\<forall>\<^sub>F x in sequentially. \<bar>x * ln x\<bar> \<le> (a x)\<^sup>2"
       using a2_omega[unfolded smallomega_def,simplified,rule_format,of 1]
       by auto
     then have "\<forall>\<^sub>F x in sequentially. \<bar>(x+3) * ln (x+3)\<bar> \<le> (a (x+3))\<^sup>2" 
       apply (subst (asm) eventually_sequentially_seg[symmetric, of _ 3])
       by simp
     then have "\<forall>\<^sub>F n in sequentially. 1 < a ( n+3)" 
     proof (elim eventually_mono)
       fix x
       assume "\<bar>real (x + 3) * ln (real (x + 3))\<bar> \<le> real_of_int ((a (x + 3))\<^sup>2)"
       moreover have "\<bar>real (x + 3) * ln (real (x + 3))\<bar> > 3"
       proof -
         have "ln (real (x + 3)) > 1"
-          apply simp using ln3_gt_1 ln_gt_1 by force
+          using ln3_gt_1 ln_gt_1 by force
         moreover have "real(x+3) \<ge> 3" by simp
-        ultimately have "(x+3)*ln (real (x + 3)) > 3*1 "
-          apply (rule_tac mult_le_less_imp_less)
-          by auto
+        ultimately have "(x+3)*ln (real (x + 3)) > 3*1"
+          by (smt (verit, best) mult_less_cancel_left1)
         then show ?thesis by auto
       qed
-      ultimately have "real_of_int ((a (x + 3))\<^sup>2) > 3"
-        by auto
-      then show "1 < a (x + 3)" 
-        by (smt Suc3_eq_add_3 a_pos add.commute of_int_1 one_power2)
+      ultimately have "(a (x + 3))\<^sup>2 > 3"
+        by linarith
+      then show "1 < a (x + 3)"
+        by (smt (verit) assms(1) one_power2)
     qed
     then show ?thesis 
-      apply (subst eventually_sequentially_seg[symmetric, of _ 3])
-      by auto
+      using eventually_sequentially_seg[symmetric, of _ 3]
+      by blast
   qed
 
   obtain B::int and c where 
     "B>0" and Bc_large:"\<forall>\<^sub>F n in sequentially. B * nth_prime n 
                 = c n * a n - c (n + 1) \<and> \<bar>c (n + 1)\<bar> <  a n / 2"
     and ca_vanish: "(\<lambda>n. c (Suc n) / real_of_int (a n)) \<longlonglongrightarrow> 0"
   proof -
     note a_gt_1
     moreover have "(\<lambda>n. real_of_int \<bar>int (nth_prime n)\<bar> 
                     / real_of_int (a (n - 1) * a n)) \<longlonglongrightarrow> 0" 
     proof -
       define f where "f=(\<lambda>n. nth_prime (n+1) / (a n * a (n+1)))"
       define g where "g=(\<lambda>n. 2*nth_prime n / (a n)^2)"
       have "\<forall>\<^sub>F x in sequentially. norm (f x) \<le> g x"
       proof -
         have "\<forall>\<^sub>F n in sequentially. nth_prime (n+1) < 2*nth_prime n" 
           using ratio_of_consecutive_primes[unfolded tendsto_iff
               ,rule_format,of 1,simplified]
           apply (elim eventually_mono)
           by (auto simp :divide_simps dist_norm)
         moreover have "\<forall>\<^sub>F n in sequentially. real_of_int (a n * a (n+1))
                            \<ge> (a n)^2"
           apply (rule eventuallyI)
           using \<open>mono a\<close> by (auto simp:power2_eq_square a_pos incseq_SucD)
         ultimately show ?thesis unfolding f_def g_def
           apply eventually_elim
           apply (subst norm_divide)
           apply (rule_tac linordered_field_class.frac_le)
           using a_pos[rule_format, THEN order.strict_implies_not_eq ]
           by auto
       qed
       moreover have "g \<longlonglongrightarrow> 0 " 
         using nth_1[THEN tendsto_mult_right_zero,of 2] unfolding g_def 
         by auto
       ultimately have "f \<longlonglongrightarrow> 0" 
         using Lim_null_comparison[of f g sequentially]
         by auto
       then show ?thesis
         unfolding f_def
         by (rule_tac LIMSEQ_imp_Suc) auto
     qed
     moreover have "(\<Sum>n. real_of_int (int (nth_prime n)) 
                     / real_of_int (prod a {..n})) \<in> \<rat>" 
       using asm by simp
     ultimately have "\<exists>B>0. \<exists>c. (\<forall>\<^sub>F n in sequentially.
               B * int (nth_prime n) = c n * a n - c (n + 1) \<and>
               real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2) \<and>
           (\<lambda>n. real_of_int (c (Suc n)) / real_of_int (a n)) \<longlonglongrightarrow> 0"
       using ab_rationality_imp[OF a_pos,of nth_prime] by fast
     then show thesis 
       apply clarify
       apply (rule_tac c=c and B=B in that)
       by auto
   qed
 
   have bac_close:"(\<lambda>n. B * nth_prime n / a n - c n) \<longlonglongrightarrow> 0"
   proof -
     have "\<forall>\<^sub>F n in sequentially. B * nth_prime n - c n * a n + c (n + 1) = 0"
       using Bc_large by (auto elim!:eventually_mono)
     then have "\<forall>\<^sub>F n in sequentially. (B * nth_prime n - c n * a n + c (n+1)) / a n = 0 "
       by eventually_elim auto
     then have "\<forall>\<^sub>F n in sequentially. B * nth_prime n / a n - c n  + c (n + 1) / a n = 0"
       apply eventually_elim
       using a_pos by (auto simp:divide_simps) (metis less_irrefl)
     then have "(\<lambda>n. B * nth_prime n / a n - c n  + c (n + 1) / a n) \<longlonglongrightarrow> 0"
       by (simp add: eventually_mono tendsto_iff)
     from tendsto_diff[OF this ca_vanish]  
     show ?thesis by auto
   qed
 
   have c_pos:"\<forall>\<^sub>F n in sequentially. c n > 0" 
   proof -
     from bac_close have *:"\<forall>\<^sub>F n in sequentially. c n \<ge> 0"
       apply (elim tendsto_of_int_diff_0)
       using a_gt_1 apply (eventually_elim) 
       using \<open>B>0\<close> by auto 
     show ?thesis
     proof (rule ccontr)
       assume "\<not> (\<forall>\<^sub>F n in sequentially. c n > 0)"
       moreover have "\<forall>\<^sub>F n in sequentially. c (Suc n) \<ge> 0 \<and> c n\<ge>0"
         using * eventually_sequentially_Suc[of "\<lambda>n. c n\<ge>0"] 
         by (metis (mono_tags, lifting) eventually_at_top_linorder le_Suc_eq)
       ultimately have "\<exists>\<^sub>F n in sequentially. c n = 0 \<and> c (Suc n) \<ge> 0"
         using eventually_elim2 frequently_def by fastforce
       moreover have "\<forall>\<^sub>F n in sequentially. nth_prime n > 0 
                         \<and> B * nth_prime n = c n * a n - c (n + 1)"
         using Bc_large by eventually_elim auto
       ultimately have "\<exists>\<^sub>F n in sequentially. c n = 0 \<and> c (Suc n) \<ge> 0 
           \<and> B * nth_prime n = c n * a n - c (n + 1)"
         using frequently_eventually_frequently by fastforce
       from frequently_ex[OF this] 
       obtain n where "c n = 0" "c (Suc n) \<ge> 0"
         "B * nth_prime n = c n * a n - c (n + 1)"
         by auto
       then have "B * nth_prime n \<le> 0" by auto
       then show False using \<open>B > 0\<close> 
         by (simp add: mult_le_0_iff)
     qed
   qed
 
   have B_nth_prime:"\<forall>\<^sub>F n in sequentially. nth_prime n > B" 
   proof -
     have "\<forall>\<^sub>F x in sequentially. B+1 \<le> nth_prime x"
       using nth_prime_at_top[unfolded filterlim_at_top_ge[where c="nat B+1"]
           ,rule_format,of "nat B + 1",simplified] 
 
       apply (elim eventually_mono)
       using \<open>B>0\<close> by auto
     then show ?thesis 
       by (auto elim: eventually_mono)
   qed
 
   have bc_epsilon:"\<forall>\<^sub>F n in sequentially. nth_prime (n+1) 
             / nth_prime n >  (c (n+1) - \<epsilon>) / c n" when "\<epsilon>>0" "\<epsilon><1" for \<epsilon>::real
   proof -
     have "\<forall>\<^sub>F x in sequentially. \<bar>c (Suc x) / a x\<bar> < \<epsilon> / 2"
       using ca_vanish[unfolded tendsto_iff,rule_format, of "\<epsilon>/2"] \<open>\<epsilon>>0\<close> by auto
     moreover then have "\<forall>\<^sub>F x in sequentially. \<bar>c (x+2) / a (x+1)\<bar> < \<epsilon> / 2"
       apply (subst (asm) eventually_sequentially_Suc[symmetric])
       by simp
     moreover have "\<forall>\<^sub>F n in sequentially. B * nth_prime (n+1) = c (n+1) * a (n+1) - c (n + 2)"
       using Bc_large
       apply (subst (asm) eventually_sequentially_Suc[symmetric])
       by (auto elim:eventually_mono)
     moreover have "\<forall>\<^sub>F n in sequentially. c n > 0 \<and> c (n+1) > 0 \<and> c (n+2) > 0"
     proof -
       have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc n)"
         using c_pos by (subst eventually_sequentially_Suc) simp
       moreover then have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc (Suc n))"
         using c_pos by (subst eventually_sequentially_Suc) simp
       ultimately show ?thesis using c_pos by eventually_elim auto
     qed
     ultimately show ?thesis using Bc_large
     proof eventually_elim
       case (elim n)
       define \<epsilon>\<^sub>0 \<epsilon>\<^sub>1 where "\<epsilon>\<^sub>0 = c (n+1) / a n" and "\<epsilon>\<^sub>1 = c (n+2) / a (n+1)"
       have "\<epsilon>\<^sub>0 > 0" "\<epsilon>\<^sub>1 > 0" "\<epsilon>\<^sub>0 < \<epsilon>/2" "\<epsilon>\<^sub>1 < \<epsilon>/2" 
         using a_pos elim \<open>mono a\<close>
-        by (auto simp add: \<epsilon>\<^sub>0_def \<epsilon>\<^sub>1_def abs_of_pos)
+        by (auto simp: \<epsilon>\<^sub>0_def \<epsilon>\<^sub>1_def abs_of_pos)
       have "(\<epsilon> - \<epsilon>\<^sub>1) * c n > 0"
         using \<open>\<epsilon>\<^sub>1 > 0\<close> \<open>\<epsilon>\<^sub>1 < \<epsilon>/2\<close> \<open>\<epsilon>>0\<close> elim by auto
-      moreover have "\<epsilon>\<^sub>0 * (c (n+1) - \<epsilon>) > 0"
+      moreover have A: "\<epsilon>\<^sub>0 * (c (n+1) - \<epsilon>) > 0"
         using \<open>\<epsilon>\<^sub>0 > 0\<close> elim(4) that(2) by force
       ultimately have "(\<epsilon> - \<epsilon>\<^sub>1) * c n + \<epsilon>\<^sub>0 * (c (n+1) - \<epsilon>) > 0" by auto
-      moreover have "c n - \<epsilon>\<^sub>0 > 0" using \<open>\<epsilon>\<^sub>0 < \<epsilon> / 2\<close> elim(4) that(2) by linarith
+      moreover have B: "c n - \<epsilon>\<^sub>0 > 0" using \<open>\<epsilon>\<^sub>0 < \<epsilon> / 2\<close> elim(4) that(2) by linarith
       moreover have "c n > 0" by (simp add: elim(4))
       ultimately have "(c (n+1) - \<epsilon>) / c n < (c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0)"
-        by (auto simp add:field_simps)
+        by (auto simp:field_simps)
       also have "... \<le> (c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) * (a (n+1) / a n)"
       proof -
-        have "(c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) > 0" 
-          by (smt \<open>0 < (\<epsilon> - \<epsilon>\<^sub>1) * real_of_int (c n)\<close> \<open>0 < real_of_int (c n) - \<epsilon>\<^sub>0\<close> 
-              divide_pos_pos elim(4) mult_le_0_iff of_int_less_1_iff that(2))
+        have "(c (n+1) - \<epsilon>\<^sub>1) / (c n -  \<epsilon>\<^sub>0) > 0"
+          using A \<open>0 < \<epsilon>\<^sub>0\<close> B \<open>\<epsilon>\<^sub>1 < \<epsilon> / 2\<close> divide_pos_pos that(1) by force 
         moreover have "(a (n+1) / a n) \<ge> 1" 
           using a_pos \<open>mono a\<close> by (simp add: mono_def)
         ultimately show ?thesis by (metis mult_cancel_left1 mult_le_cancel_left_pos)
       qed
       also have "... = (B * nth_prime (n+1)) / (B * nth_prime n)"
       proof -
         have "B * nth_prime n = c n * a n - c (n + 1)"
           using elim by auto
         also have "... = a n * (c n - \<epsilon>\<^sub>0)"
           using a_pos[rule_format,of n] unfolding \<epsilon>\<^sub>0_def by (auto simp:field_simps)
         finally have "B * nth_prime n = a n * (c n - \<epsilon>\<^sub>0)" .
         moreover have "B * nth_prime (n+1) = a (n+1) * (c (n+1) - \<epsilon>\<^sub>1)" 
           unfolding \<epsilon>\<^sub>1_def 
           using a_pos[rule_format,of "n+1"] 
           apply (subst \<open>B * nth_prime (n + 1) = c (n + 1) * a (n + 1) - c (n + 2)\<close>)
           by (auto simp:field_simps)
         ultimately show ?thesis by (simp add: mult.commute)
       qed
       also have "... = nth_prime (n+1) / nth_prime n" 
         using \<open>B>0\<close> by auto
       finally show ?case .
     qed
   qed
 
 
   have c_ubound:"\<forall>x. \<exists>n. c n > x"
   proof (rule ccontr)
     assume " \<not> (\<forall>x. \<exists>n. x < c n)"
     then obtain ub where "\<forall>n. c n \<le> ub" "ub > 0"
       by (meson dual_order.trans int_one_le_iff_zero_less le_cases not_le)
     define pa where "pa = (\<lambda>n. nth_prime n / a n)"
     have pa_pos:"\<And>n. pa n > 0" unfolding pa_def by (simp add: a_pos)
     have "liminf (\<lambda>n. 1 / pa n) = 0"
       using nth_2 unfolding pa_def by auto
     then have "(\<exists>y<ereal (real_of_int B / real_of_int (ub + 1)). 
       \<exists>\<^sub>F x in sequentially. ereal (1 / pa x) \<le> y)"
       apply (subst less_Liminf_iff[symmetric])
       using \<open>0 < B\<close> \<open>0 < ub\<close> by auto
     then have "\<exists>\<^sub>F x in sequentially. 1 / pa x < B/(ub+1)"
       by (meson frequently_mono le_less_trans less_ereal.simps(1))
     then have "\<exists>\<^sub>F x in sequentially. B*pa x > (ub+1)"
       apply (elim frequently_elim1)
       by (metis \<open>0 < ub\<close> mult.left_neutral of_int_0_less_iff pa_pos pos_divide_less_eq 
           pos_less_divide_eq times_divide_eq_left zless_add1_eq)
     moreover have "\<forall>\<^sub>F x in sequentially. c x \<le> ub"
       using \<open>\<forall>n. c n \<le> ub\<close> by simp
     ultimately have "\<exists>\<^sub>F x in sequentially. B*pa x - c x > 1"
       by (elim frequently_rev_mp eventually_mono) linarith
     moreover have "(\<lambda>n. B * pa n - c n) \<longlonglongrightarrow>0" 
       unfolding pa_def using bac_close by auto
     from tendstoD[OF this,of 1] 
     have "\<forall>\<^sub>F n in sequentially.  \<bar>B * pa n - c n\<bar> < 1"
       by auto
     ultimately have "\<exists>\<^sub>F x in sequentially. B*pa x - c x > 1 \<and> \<bar>B * pa x - c x\<bar> < 1"
       using frequently_eventually_frequently by blast
     then show False 
       by (simp add: frequently_def)
   qed
 
   have eq_2_11:"\<forall>\<^sub>F n in sequentially. c (n+1)>c n \<longrightarrow> 
                   nth_prime (n+1) > nth_prime n + (1 - \<epsilon>)^2 * a n / B" 
     when "\<epsilon>>0" "\<epsilon><1" for \<epsilon>::real
   proof -
     have "\<forall>\<^sub>F x in sequentially. \<bar>c (Suc x) / a x\<bar> < \<epsilon>"
       using ca_vanish[unfolded tendsto_iff,rule_format, of \<epsilon>] \<open>\<epsilon>>0\<close> by auto
     moreover have "\<forall>\<^sub>F n in sequentially. c n > 0 \<and> c (n+1) > 0"
     proof -
       have "\<forall>\<^sub>F n in sequentially. 0 < c (Suc n)"
         using c_pos by (subst eventually_sequentially_Suc) simp
       then show ?thesis using c_pos by eventually_elim auto
     qed
     ultimately show ?thesis  using Bc_large bc_epsilon[OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><1\<close>] 
     proof (eventually_elim, rule_tac impI)
       case (elim n)
       assume "c n < c (n + 1)"
       have "c (n+1) / a n < \<epsilon>" 
         using a_pos[rule_format,of n] using elim(1,2) by auto
       also have "... \<le> \<epsilon> * c n" using elim(2) that(1) by auto
       finally have "c (n+1) / a n < \<epsilon> * c n" .
       then have "c (n+1) / c n < \<epsilon> * a n" 
         using a_pos[rule_format,of n] elim by (auto simp:field_simps)
       then have "(1 - \<epsilon>) * a n < a n - c (n+1) / c n"
         by (auto simp:algebra_simps)
       then have "(1 - \<epsilon>)^2 * a n / B < (1 - \<epsilon>) * (a n - c (n+1) / c n) / B"
         apply (subst (asm) mult_less_cancel_right_pos[symmetric, of "(1-\<epsilon>)/B"])
         using \<open>\<epsilon><1\<close> \<open>B>0\<close> by (auto simp: divide_simps power2_eq_square mult_less_cancel_right_pos)
       then have "nth_prime n + (1 - \<epsilon>)^2 * a n / B < nth_prime n + (1 - \<epsilon>) * (a n - c (n+1) / c n) / B"
         using \<open>B>0\<close> by auto
       also have "... = nth_prime n + (1 - \<epsilon>) * ((c n *a n - c (n+1)) / c n) / B"
         using elim by (auto simp:field_simps)
       also have "... = nth_prime n + (1 - \<epsilon>) * (nth_prime n / c n)"
       proof -
         have "B * nth_prime n = c n * a n - c (n + 1)" using elim by auto
         from this[symmetric] show ?thesis
           using \<open>B>0\<close> by simp
       qed
       also have "... = (1+(1-\<epsilon>)/c n) * nth_prime n"
         by (auto simp:algebra_simps)
       also have "... = ((c n+1-\<epsilon>)/c n) * nth_prime n"
         using elim by (auto simp:divide_simps)
       also have "... \<le> ((c (n+1) -\<epsilon>)/c n) * nth_prime n"
       proof -
         define cp where "cp = c n+1"
         have "c (n+1) \<ge> cp" unfolding cp_def using \<open>c n < c (n + 1)\<close> by auto
         moreover have "c n>0" "nth_prime n>0" using elim by auto
         ultimately show ?thesis 
           apply (fold cp_def)
           by (auto simp:divide_simps)
       qed
       also have "... < nth_prime (n+1)"
         using elim by (auto simp:divide_simps)
       finally show "real (nth_prime n) + (1 - \<epsilon>)\<^sup>2 * real_of_int (a n) 
           / real_of_int B < real (nth_prime (n + 1))" .
     qed
   qed
 
   have c_neq_large:"\<forall>\<^sub>F n in sequentially. c (n+1) \<noteq> c n"
   proof (rule ccontr)
     assume "\<not> (\<forall>\<^sub>F n in sequentially. c (n + 1) \<noteq> c n)"
     then have that:"\<exists>\<^sub>F n in sequentially. c (n + 1) = c n"
       unfolding frequently_def .
     have "\<forall>\<^sub>F x in sequentially. (B * int (nth_prime x) = c x * a x - c (x + 1) 
         \<and> \<bar>real_of_int (c (x + 1))\<bar> < real_of_int (a x) / 2) \<and> 0 < c x \<and> B < int (nth_prime x)
         \<and> (c (x+1)>c x \<longrightarrow> nth_prime (x+1) > nth_prime x +  a x / (2* B))"
       using Bc_large c_pos B_nth_prime eq_2_11[of "1-1/ sqrt 2",simplified]
       by eventually_elim (auto simp:divide_simps)
     then have "\<exists>\<^sub>F m in sequentially. nth_prime (m+1) > (1+1/(2*B))*nth_prime m"
     proof (elim frequently_eventually_at_top[OF that, THEN frequently_at_top_elim])
       fix n
       assume "c (n + 1) = c n \<and>
            (\<forall>y\<ge>n. (B * int (nth_prime y) = c y * a y - c (y + 1) \<and>
                     \<bar>real_of_int (c (y + 1))\<bar> < real_of_int (a y) / 2) \<and>
                    0 < c y \<and> B < int (nth_prime y) \<and> (c y < c (y + 1) \<longrightarrow>
                    real (nth_prime y) + real_of_int (a y) / real_of_int (2 * B)
                    < real (nth_prime (y + 1))))"
       then have "c (n + 1) = c n" 
         and Bc_eq:"\<forall>y\<ge>n. B * int (nth_prime y) = c y * a y - c (y + 1) \<and> 0 < c y 
                             \<and> \<bar>real_of_int (c (y + 1))\<bar> < real_of_int (a y) / 2 
                             \<and> B < int (nth_prime y)
                             \<and> (c y < c (y + 1) \<longrightarrow>
                    real (nth_prime y) + real_of_int (a y) / real_of_int (2 * B)
                    < real (nth_prime (y + 1)))"
         by auto
       obtain m where "n<m" "c m \<le> c n" "c n<c (m+1)" 
       proof -
         have "\<exists>N. N > n \<and> c N > c n"
           using c_ubound[rule_format, of "MAX x\<in>{..n}. c x"] 
           by (metis Max_ge atMost_iff dual_order.trans finite_atMost finite_imageI image_eqI 
               linorder_not_le order_refl)
         then obtain N where "N>n" "c N>c n" by auto
         define A m where "A={m. n<m \<and> (m+1)\<le>N \<and> c (m+1) > c n}" and "m = Min A"
         have "finite A" unfolding A_def
           by (metis (no_types, lifting) A_def add_leE finite_nat_set_iff_bounded_le mem_Collect_eq)
         moreover have "N-1\<in>A" unfolding A_def 
-          using \<open>c n < c N\<close> \<open>n < N\<close> \<open>c (n + 1) = c n\<close>
-          by (smt Suc_diff_Suc Suc_eq_plus1 Suc_leI Suc_pred  add.commute 
-              add_diff_inverse_nat add_leD1 diff_is_0_eq' mem_Collect_eq nat_add_left_cancel_less
-              zero_less_one)
+          using \<open>c n < c N\<close> \<open>n < N\<close> \<open>c (n + 1) = c n\<close> nat_less_le by force
         ultimately have "m\<in>A"
           using Min_in unfolding m_def by auto
         then have "n<m"  "c n<c (m+1)" "m>0"
           unfolding m_def A_def by auto
         moreover have "c m \<le> c n"
         proof (rule ccontr)
-          assume " \<not> c m \<le> c n"
-          then have "m-1\<in>A" using \<open>m\<in>A\<close> \<open>c (n + 1) = c n\<close>
-            unfolding A_def 
-            by auto (smt One_nat_def Suc_eq_plus1 Suc_lessI less_diff_conv)
-          from  Min_le[OF \<open>finite A\<close> this,folded m_def] \<open>m>0\<close> show False by auto
+          assume "\<not> c m \<le> c n"
+          then have "m-1\<in>A" 
+            using \<open>m\<in>A\<close> \<open>c (n + 1) = c n\<close> le_eq_less_or_eq less_diff_conv by (fastforce simp: A_def)
+          from Min_le[OF \<open>finite A\<close> this,folded m_def] \<open>m>0\<close> show False by auto
         qed
         ultimately show ?thesis using that[of m] by auto
       qed
       have "(1 + 1 / (2 * B)) * nth_prime m < nth_prime m + a m / (2*B)"
       proof -
         have "nth_prime m < a m"
         proof -
           have "B * int (nth_prime m) < c m * (a m - 1)"
             using Bc_eq[rule_format,of m] \<open>c m \<le> c n\<close> \<open>c n < c (m + 1)\<close> \<open>n < m\<close> 
             by (auto simp:algebra_simps)
           also have "... \<le> c n * (a m - 1)"
             by (simp add: \<open>c m \<le> c n\<close> a_pos mult_right_mono)
           finally have "B * int (nth_prime m) < c n * (a m - 1)" .
           moreover have "c n\<le>B" 
           proof -
-            have " B * int (nth_prime n) = c n * (a n - 1)" "B < int (nth_prime n)"
-              and c_a:"\<bar>real_of_int (c (n + 1))\<bar> < real_of_int (a n) / 2"
+            have B: "B * int (nth_prime n) = c n * (a n - 1)" "B < int (nth_prime n)"
+              and c_a: "\<bar>real_of_int (c (n + 1))\<bar> < real_of_int (a n) / 2"
               using Bc_eq[rule_format,of n] \<open>c (n + 1) = c n\<close> by (auto simp:algebra_simps)
             from this(1) have " c n dvd (B * int (nth_prime n))"
               by simp
             moreover have "coprime (c n) (int (nth_prime n))"
             proof -
               have "c n < int (nth_prime n)" 
               proof (rule ccontr)
                 assume "\<not> c n < int (nth_prime n)"
                 then have asm:"c n \<ge> int (nth_prime n)" by auto
                 then have "a n > 2 * nth_prime n"
                   using c_a \<open>c (n + 1) = c n\<close> by auto
                 then have "a n -1 \<ge> 2 * nth_prime n"
                   by simp
                 then have "a n - 1 > 2 * B"
                   using \<open>B < int (nth_prime n)\<close> by auto
                 from mult_le_less_imp_less[OF asm this] \<open>B>0\<close>
                 have "int (nth_prime n) * (2 * B) < c n * (a n - 1)"
                   by auto
-                then show False using \<open>B * int (nth_prime n) = c n * (a n - 1)\<close>
-                  by (smt \<open>0 < B\<close> \<open>B < int (nth_prime n)\<close> combine_common_factor 
-                      mult.commute mult_pos_pos)
+                then show False using B
+                  by (smt (verit, best) \<open>0 < B\<close> mult.commute mult_right_mono)
               qed
               then have "\<not> nth_prime n dvd c n" 
                 by (simp add: Bc_eq zdvd_not_zless)
               then have "coprime (int (nth_prime n)) (c n)" 
                 by (auto intro!:prime_imp_coprime_int)
               then show ?thesis using coprime_commute by blast
             qed
             ultimately have "c n dvd B"
               using coprime_dvd_mult_left_iff by auto
             then show ?thesis using \<open>0 < B\<close> zdvd_imp_le by blast
           qed
           moreover have "c n > 0 " using Bc_eq by blast
           ultimately show ?thesis
-            using \<open>B>0\<close> by (smt a_pos mult_mono)
+            using \<open>B>0\<close> by (smt (verit) a_pos mult_mono)
         qed
         then show ?thesis using \<open>B>0\<close> by (auto simp:field_simps)
       qed
       also have "... < nth_prime (m+1)"
         using Bc_eq[rule_format, of m] \<open>n<m\<close> \<open>c m \<le> c n\<close> \<open>c n < c (m+1)\<close>
         by linarith
       finally show "\<exists>j>n. (1 + 1 / real_of_int (2 * B)) * real (nth_prime j)
                        < real (nth_prime (j + 1))" using \<open>m>n\<close> by auto
     qed
     then have "\<exists>\<^sub>F m in sequentially. nth_prime (m+1)/nth_prime m > (1+1/(2*B))"
       by (auto elim:frequently_elim1 simp:field_simps)
     moreover have "\<forall>\<^sub>F m in sequentially. nth_prime (m+1)/nth_prime m < (1+1/(2*B))"
       using ratio_of_consecutive_primes[unfolded tendsto_iff,rule_format,of "1/(2*B)"]
         \<open>B>0\<close>
       unfolding dist_real_def
       by (auto elim!:eventually_mono simp:algebra_simps)
     ultimately show False by (simp add: eventually_mono frequently_def)
   qed
 
   have c_gt_half:"\<forall>\<^sub>F N in sequentially. card {n\<in>{N..<2*N}.  c n > c (n+1)} > N / 2"
   proof -
     define h where "h=(\<lambda>n. (nth_prime (2*n) - nth_prime n) 
                               /  sqrt (nth_prime n))"
     have "\<forall>\<^sub>F n in sequentially. h n < n / 2"
     proof -
       have "\<forall>\<^sub>F n in sequentially. h n < n powr (5/6)"
         using nth_prime_double_sqrt_less[of "1/3"]
         unfolding h_def by auto
       moreover have "\<forall>\<^sub>F n in sequentially. n powr (5/6) < (n /2)"
         by real_asymp
       ultimately show ?thesis 
         by eventually_elim auto 
     qed
     moreover have "\<forall>\<^sub>F n in sequentially. sqrt (nth_prime n) / a n < 1 / (2*B)"
       using nth_1[THEN tendsto_real_sqrt,unfolded tendsto_iff
           ,rule_format,of "1/(2*B)"] \<open>B>0\<close> a_pos
       by (auto simp:real_sqrt_divide abs_of_pos)
     ultimately have "\<forall>\<^sub>F x in sequentially. c (x+1) \<noteq> c x
         \<and> sqrt (nth_prime x) / a x < 1 / (2*B)
         \<and>  h x < x / 2
         \<and> (c (x+1)>c x \<longrightarrow> nth_prime (x+1) > nth_prime x +  a x / (2* B))"
       using c_neq_large B_nth_prime eq_2_11[of "1-1/ sqrt 2",simplified]
       by eventually_elim (auto simp:divide_simps)
     then show ?thesis
     proof (elim eventually_at_top_mono)
       fix N assume "N\<ge>1" and N_asm:"\<forall>y\<ge>N. c (y + 1) \<noteq> c y \<and>
                sqrt (real (nth_prime y)) / real_of_int (a y)
                < 1 / real_of_int (2 * B) \<and>  h y < y / 2 \<and>
                (c y < c (y + 1) \<longrightarrow>
                 real (nth_prime y) + real_of_int (a y) / real_of_int (2 * B)
                 < real (nth_prime (y + 1)))"
 
       define S where "S={n \<in> {N..<2 * N}. c n < c (n + 1)}"
       define g where "g=(\<lambda>n. (nth_prime (n+1) - nth_prime n) 
                               /  sqrt (nth_prime n))"
       define f where "f=(\<lambda>n. nth_prime (n+1) - nth_prime n)"
       have g_gt_1:"g n>1" when "n\<ge>N" "c n < c (n + 1)" for n
       proof -
         have "nth_prime n + sqrt (nth_prime n) < nth_prime (n+1)"
         proof -
           have "nth_prime n + sqrt (nth_prime n) < nth_prime n + a n / (2*B)"
             using N_asm[rule_format,OF \<open>n\<ge>N\<close>] a_pos
             by (auto simp:field_simps)
           also have "... < nth_prime (n+1)"
             using N_asm[rule_format,OF \<open>n\<ge>N\<close>] \<open>c n < c (n + 1)\<close> by auto
           finally show ?thesis .
         qed
         then show ?thesis unfolding g_def 
           using \<open>c n < c (n + 1)\<close> by auto
       qed
       have g_geq_0:"g n \<ge> 0" for n 
         unfolding g_def  by auto
 
       have "finite S" "\<forall>x\<in>S. x\<ge>N \<and> c x<c (x+1)"
         unfolding S_def by auto
       then have "card S \<le> sum g S" 
       proof (induct S)
         case empty
         then show ?case by auto
       next
         case (insert x F)
         moreover have "g x>1" 
         proof -
           have "c x < c (x+1)" "x\<ge>N" using insert(4) by auto
           then show ?thesis using g_gt_1 by auto
         qed
         ultimately show ?case by simp
       qed
       also have "... \<le> sum g {N..<2*N}"
         apply (rule sum_mono2)
         unfolding S_def using g_geq_0 by auto
       also have "... \<le> sum (\<lambda>n. f n/sqrt (nth_prime N)) {N..<2*N}"
         unfolding f_def g_def by (auto intro!:sum_mono divide_left_mono)
       also have "... = sum f {N..<2*N} / sqrt (nth_prime N)"
         unfolding sum_divide_distrib[symmetric] by auto
       also have "... = (nth_prime (2*N) - nth_prime N) / sqrt (nth_prime N)"
       proof -
         have "sum f {N..<2 * N} = nth_prime (2 * N) - nth_prime N"   
         proof (induct N)
           case 0
           then show ?case by simp
         next
           case (Suc N)
           have ?case if "N=0"
           proof -
             have "sum f {Suc N..<2 * Suc N} = sum f {1}"
               using that by (simp add: numeral_2_eq_2)
             also have "... = nth_prime 2 - nth_prime 1"
               unfolding f_def by (simp add:numeral_2_eq_2)
             also have "... = nth_prime (2 * Suc N) - nth_prime (Suc N)"
               using that by auto
             finally show ?thesis .
           qed
           moreover have ?case if "N\<noteq>0"
           proof -
             have "sum f {Suc N..<2 * Suc N} = sum f {N..<2 * Suc N} - f N"
               apply (subst (2) sum.atLeast_Suc_lessThan)
               using that by auto
             also have "... = sum f {N..<2 * N}+ f (2*N) + f(2*N+1) - f N"
               by auto
             also have "... = nth_prime (2 * Suc N) - nth_prime (Suc N)"
               using Suc unfolding f_def by auto
             finally show ?thesis .
           qed
           ultimately show ?case by blast
         qed
         then show ?thesis by auto
       qed
       also have "... = h N"
         unfolding h_def by auto
       also have "... < N/2"
         using N_asm by auto
       finally have "card S < N/2" .
 
       define T where "T={n \<in> {N..<2 * N}. c n > c (n + 1)}"
       have "T \<union> S = {N..<2 * N}" "T \<inter> S = {}" "finite T"
         unfolding T_def S_def using N_asm by fastforce+
 
       then have "card T + card S = card {N..<2 * N}"
         using card_Un_disjoint \<open>finite S\<close> by metis
       also have "... = N"
         by simp
       finally have "card T + card S = N" .
       with \<open>card S < N/2\<close> 
       show "card T > N/2" by linarith
     qed
   qed
 
   text\<open>Inequality (3.5) in the original paper required a slight modification: \<close>
 
   have a_gt_plus:"\<forall>\<^sub>F n in sequentially. c n > c (n+1) \<longrightarrow>a (n+1) > a n + (a n - c(n+1) - 1) / c (n+1)"
   proof -
     note a_gt_1[THEN eventually_all_ge_at_top] c_pos[THEN eventually_all_ge_at_top]
     moreover have "\<forall>\<^sub>F n in sequentially. 
               B * int (nth_prime (n+1)) = c (n+1) * a (n+1) - c (n + 2)"
       using Bc_large 
       apply (subst (asm) eventually_sequentially_Suc[symmetric])
       by (auto elim:eventually_mono)
     moreover have "\<forall>\<^sub>F n in sequentially. 
                         B * int (nth_prime n) = c n * a n - c (n + 1) \<and> \<bar>c (n + 1)\<bar> <  a n / 2"
       using Bc_large by (auto elim:eventually_mono)
     ultimately show ?thesis 
       apply (eventually_elim)
     proof (rule impI)
       fix n 
       assume "\<forall>y\<ge>n. 1 < a y" "\<forall>y\<ge>n. 0 < c y"
         and
         Suc_n_eq:"B * int (nth_prime (n + 1)) = c (n + 1) * a (n + 1) - c (n + 2)" and
         "B * int (nth_prime n) = c n * a n - c (n + 1) \<and>
                 real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2" 
         and "c (n + 1) < c n"
       then have n_eq:"B * int (nth_prime n) = c n * a n - c (n + 1)" and 
         c_less_a: "real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2" 
         by auto
       from \<open>\<forall>y\<ge>n. 1 < a y\<close> \<open>\<forall>y\<ge>n. 0 < c y\<close> 
       have *:"a n>1" "a (n+1) > 1" "c n > 0" 
         "c (n+1) > 0"  "c (n+2) > 0"
         by auto
       then have "(1+1/c (n+1))* (a n - 1)/a (n+1) = (c (n+1)+1) * ((a n - 1) / (c (n+1) * a (n+1)))"
         by (auto simp:field_simps)
       also have "... \<le> c n * ((a n - 1) / (c (n+1) * a (n+1)))"
-        apply (rule mult_right_mono)
-        subgoal using \<open>c (n + 1) < c n\<close> by auto
-        subgoal by (smt \<open>0 < c (n + 1)\<close> a_pos divide_nonneg_pos mult_pos_pos of_int_0_le_iff 
-              of_int_0_less_iff)
-        done
+        by (smt (verit) "*"(4) \<open>c (n + 1) < c n\<close> a_pos divide_nonneg_nonneg mult_mono mult_nonneg_nonneg of_int_0_le_iff of_int_le_iff)
       also have "... = (c n * (a n - 1)) / (c (n+1) * a (n+1))" by auto
       also have "... < (c n * (a n - 1)) / (c (n+1) * a (n+1)  - c (n+2))"
         apply (rule divide_strict_left_mono)
         subgoal using \<open>c (n+2) > 0\<close> by auto
         unfolding Suc_n_eq[symmetric] using * \<open>B>0\<close> by auto
       also have "... < (c n * a n - c (n+1)) / (c (n+1) * a (n+1)  - c (n+2))"
         apply (rule frac_less)
         unfolding Suc_n_eq[symmetric] using * \<open>B>0\<close> \<open>c (n + 1) < c n\<close> 
         by (auto simp:algebra_simps)
       also have "... = nth_prime n / nth_prime (n+1)"
         unfolding Suc_n_eq[symmetric] n_eq[symmetric] using \<open>B>0\<close> by auto
       also have "... < 1" by auto
       finally have "(1 + 1 / real_of_int (c (n + 1))) * real_of_int (a n - 1) 
         / real_of_int (a (n + 1)) < 1 " .
       then show "a n + (a n - c (n + 1) - 1) /  (c (n + 1)) < (a (n + 1))"
         using * by (auto simp:field_simps)
     qed
   qed
   have a_gt_1:"\<forall>\<^sub>F n in sequentially. c n > c (n+1) \<longrightarrow> a (n+1) > a n + 1"
     using Bc_large a_gt_plus c_pos[THEN eventually_all_ge_at_top]
     apply eventually_elim
   proof (rule impI)
     fix n assume  
       "c (n + 1) < c n \<longrightarrow> a n +  (a n - c (n + 1) - 1) / c (n + 1) <  a (n + 1)" 
       "c (n + 1) < c n" and B_eq:"B * int (nth_prime n) = c n * a n - c (n + 1) \<and>
          \<bar>real_of_int (c (n + 1))\<bar> < real_of_int (a n) / 2" and c_pos:"\<forall>y\<ge>n. 0 < c y"
     from this(1,2) 
     have "a n +  (a n - c (n + 1) - 1) / c (n + 1) <  a (n + 1)" by auto
     moreover have "a n - 2 * c (n+1) > 0"
       using B_eq c_pos[rule_format,of "n+1"] by auto
     then have "a n - 2 * c (n+1) \<ge> 1" by simp
     then have "(a n - c (n + 1) - 1) / c (n + 1) \<ge> 1"
       using c_pos[rule_format,of "n+1"] by (auto simp:field_simps)
     ultimately show "a n + 1 < a (n + 1)" by auto
   qed
 
   text\<open>The following corresponds to inequality (3.6) in the paper, which had to be
         slightly corrected:  \<close>
 
   have a_gt_sqrt:"\<forall>\<^sub>F n in sequentially. c n > c (n+1) \<longrightarrow> a (n+1) > a n + (sqrt n - 2)"
   proof -
     have a_2N:"\<forall>\<^sub>F N in sequentially. a (2*N) \<ge> N /2 +1" 
       using c_gt_half a_gt_1[THEN eventually_all_ge_at_top] 
     proof eventually_elim
       case (elim N)
       define S where "S={n \<in> {N..<2 * N}. c (n + 1) < c n}"
       define f where "f = (\<lambda>n. a (Suc n) - a n)"
 
       have f_1:"\<forall>x\<in>S. f x\<ge>1" and f_0:"\<forall>x. f x\<ge>0"
         subgoal using elim unfolding S_def f_def by auto
         subgoal using \<open>mono a\<close>[THEN incseq_SucD] unfolding f_def by auto
         done
       have "N / 2 < card S" 
         using elim unfolding S_def by auto
       also have "... \<le> sum f S" 
         unfolding of_int_sum
         apply (rule sum_bounded_below[of _ 1,simplified])
         using f_1 by auto
       also have "... \<le> sum f {N..<2 * N}" 
         unfolding of_int_sum
         apply (rule sum_mono2)
         unfolding S_def using f_0 by auto
       also have "... = a (2*N) - a N" 
         unfolding of_int_sum f_def of_int_diff
         apply (rule sum_Suc_diff')
         by auto
       finally have "N / 2 < a (2*N) - a N" .
       then show ?case using a_pos[rule_format,of N] by linarith
     qed
 
     have a_n4:"\<forall>\<^sub>F n in sequentially. a n > n/4" 
     proof -
       obtain N where a_N:"\<forall>n\<ge>N. a (2*n) \<ge> n /2+1"
         using a_2N unfolding eventually_at_top_linorder by auto
       have "a n>n/4" when "n\<ge>2*N" for n 
       proof -
         define n' where "n'=n div 2"
         have "n'\<ge>N" unfolding n'_def using that by auto
         have "n/4 < n' /2+1"
           unfolding n'_def by auto
         also have "... \<le> a (2*n')"
           using a_N \<open>n'\<ge>N\<close> by auto
         also have "... \<le>a n" unfolding n'_def
           apply (cases "even n")
           subgoal by simp
           subgoal by (simp add: assms(2) incseqD)
           done
         finally show ?thesis .
       qed
       then show ?thesis
         unfolding eventually_at_top_linorder by auto
     qed
 
     have c_sqrt:"\<forall>\<^sub>F n in sequentially. c n < sqrt n / 4"
     proof -
       have "\<forall>\<^sub>F x in sequentially. x>1" by simp
       moreover have "\<forall>\<^sub>F x in sequentially. real (nth_prime x) / (real x * ln (real x)) < 2"
         using nth_prime_asymptotics[unfolded asymp_equiv_def,THEN order_tendstoD(2),of 2]
         by simp
       ultimately have "\<forall>\<^sub>F n in sequentially. c n < B*8 *ln n + 1" using a_n4 Bc_large
       proof eventually_elim
         case (elim n)
         from this(4) have "c n=(B*nth_prime n+c (n+1))/a n"
           using a_pos[rule_format,of n] 
           by (auto simp:divide_simps)
         also have "... = (B*nth_prime n)/a n+c (n+1)/a n"
           by (auto simp:divide_simps)
         also have "... < (B*nth_prime n)/a n + 1"
         proof -
           have "c (n+1)/a n < 1" using elim(4) by auto
           then show ?thesis by auto
         qed
         also have "... < B*8 * ln n + 1"
         proof -
           have "B*nth_prime n < 2*B*n*ln n"
             using \<open>real (nth_prime n) / (real n * ln (real n)) < 2\<close> \<open>B>0\<close> \<open> 1 < n\<close> 
             by (auto simp:divide_simps)
           moreover have "real n / 4 < real_of_int (a n)" by fact
           ultimately have "(B*nth_prime n) / a n < (2*B*n*ln n) / (n/4)"
             apply (rule_tac frac_less)
             using \<open>B>0\<close> \<open> 1 < n\<close>  by auto
           also have "... = B*8 * ln n"
             using \<open> 1 < n\<close> by auto
           finally show ?thesis by auto
         qed
         finally show ?case .
       qed
       moreover have "\<forall>\<^sub>F n in sequentially. B*8 *ln n + 1 < sqrt n / 4"
         by real_asymp
       ultimately show ?thesis 
         by eventually_elim auto
     qed
 
     have 
       "\<forall>\<^sub>F n in sequentially. 0 < c (n+1)"
       "\<forall>\<^sub>F n in sequentially. c (n+1) < sqrt (n+1) / 4"
       "\<forall>\<^sub>F n in sequentially. n > 4"
       "\<forall>\<^sub>F n in sequentially. (n - 4) / sqrt (n + 1) + 1 > sqrt n"
       subgoal using c_pos[THEN eventually_all_ge_at_top]    
         by eventually_elim auto
       subgoal using  c_sqrt[THEN eventually_all_ge_at_top] 
         by eventually_elim (use le_add1 in blast)
       subgoal by simp
       subgoal 
         by real_asymp
       done
     then show ?thesis using a_gt_plus a_n4
       apply eventually_elim
     proof (rule impI)
       fix n assume asm:"0 < c (n + 1)" "c (n + 1) < sqrt (real (n + 1)) / 4" and
         a_ineq:"c (n + 1) < c n \<longrightarrow> a n +  (a n - c (n + 1) - 1) /  c (n + 1) < a (n + 1)" 
         "c (n + 1) < c n"  and "n / 4 < a n" "n > 4" 
         and n_neq:" sqrt (real n) < real (n - 4) / sqrt (real (n + 1)) + 1"
 
       have "(n-4) / sqrt(n+1) = (n/4 - 1)/ (sqrt (real (n + 1)) / 4)"
         using \<open>n>4\<close> by (auto simp:divide_simps)
       also have "... < (a n - 1) /  c (n + 1)" 
         apply (rule frac_less)
         using \<open>n > 4\<close> \<open>n / 4 < a n\<close> \<open>0 < c (n + 1)\<close> \<open>c (n + 1) < sqrt (real (n + 1)) / 4\<close>
         by auto
       also have "... - 1 = (a n - c (n + 1) - 1) /  c (n + 1)"
         using \<open>0 < c (n + 1)\<close> by (auto simp:field_simps)
       also have "a n + ... < a (n+1)"
         using a_ineq by auto
       finally have "a n + ((n - 4) / sqrt (n + 1) - 1) < a (n + 1)" by simp
       moreover have "(n - 4) / sqrt (n + 1) - 1 > sqrt n - 2"
         using n_neq[THEN diff_strict_right_mono,of 2] \<open>n>4\<close> 
         by (auto simp:algebra_simps of_nat_diff) 
       ultimately show "real_of_int (a n) + (sqrt (real n) - 2) < real_of_int (a (n + 1))"
         by argo
     qed
   qed
 
   text\<open>The following corresponds to inequality $ a_{2N} > N^{3/2}/2$ in the paper,
  which had to be slightly corrected: \<close>
 
   have a_2N_sqrt:"\<forall>\<^sub>F N in sequentially. a (2*N) >  real N * (sqrt (real N)/2 - 1)"
     using c_gt_half a_gt_sqrt[THEN eventually_all_ge_at_top] eventually_gt_at_top[of 4]
   proof eventually_elim
     case (elim N)
     define S where "S={n \<in> {N..<2 * N}. c (n + 1) < c n}"
     define f where "f = (\<lambda>n. a (Suc n) - a n)"
 
     have f_N:"\<forall>x\<in>S. f x\<ge>sqrt N - 2"
     proof 
       fix x assume "x\<in>S"
       then have "sqrt (real x) - 2 < f x" "x\<ge>N"
         using elim unfolding S_def f_def by auto
       moreover have "sqrt x - 2 \<ge> sqrt N - 2"
         using \<open>x\<ge>N\<close> by simp
       ultimately show  "sqrt (real N) - 2 \<le> real_of_int (f x)" by argo
     qed
     have f_0:"\<forall>x. f x\<ge>0"
       using \<open>mono a\<close>[THEN incseq_SucD] unfolding f_def by auto
 
     have "(N / 2)  * (sqrt N - 2) < card S * (sqrt N - 2)" 
       apply (rule mult_strict_right_mono)
       subgoal using elim unfolding S_def by auto
       subgoal using \<open>N>4\<close> 
         by (metis diff_gt_0_iff_gt numeral_less_real_of_nat_iff real_sqrt_four real_sqrt_less_iff)
       done
     also have "...  \<le> sum f S" 
       unfolding of_int_sum
       apply (rule sum_bounded_below)
       using f_N by auto
     also have "... \<le> sum f {N..<2 * N}" 
       unfolding of_int_sum
       apply (rule sum_mono2)
       unfolding S_def using f_0 by auto
     also have "... = a (2*N) - a N" 
       unfolding of_int_sum f_def of_int_diff
       apply (rule sum_Suc_diff')
       by auto
     finally have "real N / 2 * (sqrt (real N) - 2) < real_of_int (a (2 * N) - a N)" 
       .              
     then have "real N / 2 * (sqrt (real N) - 2) < a (2 * N)"
       using a_pos[rule_format,of N] by linarith
     then show ?case by (auto simp:field_simps)
   qed
 
   text\<open>The following part is required to derive the final contradiction of the proof.\<close>
 
   have a_n_sqrt:"\<forall>\<^sub>F n in sequentially. a n > (((n-1)/2) powr (3/2) - (n-1)) /2"
   proof (rule sequentially_even_odd_imp)
     define f where "f=(\<lambda>N. ((real (2 * N - 1) / 2) powr (3 / 2) - real (2 * N - 1)) / 2)"
     define g where "g=(\<lambda>N. real N * (sqrt (real N) / 2 - 1))"
     have "\<forall>\<^sub>F N in sequentially. g N > f N"
       unfolding f_def g_def
       by real_asymp
     moreover have "\<forall>\<^sub>F N in sequentially. a (2 * N) > g N"
       unfolding g_def using a_2N_sqrt .
     ultimately show "\<forall>\<^sub>F N in sequentially. f N < a (2 * N)"
       by eventually_elim auto
   next
     define f where "f=(\<lambda>N. ((real (2 * N + 1 - 1) / 2) powr (3 / 2) 
                                   - real (2 * N + 1 - 1)) / 2)"
     define g where "g=(\<lambda>N. real N * (sqrt (real N) / 2 - 1))"
     have "\<forall>\<^sub>F N in sequentially. g N = f N"
       using eventually_gt_at_top[of 0]
       apply eventually_elim
       unfolding f_def g_def
       by (auto simp:algebra_simps powr_half_sqrt[symmetric] powr_mult_base)
     moreover have "\<forall>\<^sub>F N in sequentially. a (2 * N) > g N"
       unfolding g_def using a_2N_sqrt .
     moreover have "\<forall>\<^sub>F N in sequentially. a (2 * N + 1) \<ge> a (2*N)"
       apply (rule eventuallyI)
       using \<open>mono a\<close> by (simp add: incseqD)
     ultimately show "\<forall>\<^sub>F N in sequentially. f N < (a (2 * N + 1))"
       by eventually_elim auto
   qed
 
   have a_nth_prime_gt:"\<forall>\<^sub>F n in sequentially. a n / nth_prime n > 1"
   proof -
     define f where "f=(\<lambda>n::nat. (((n-1)/2) powr (3/2) - (n-1)) /2)"
     have "\<forall>\<^sub>F x in sequentially. real (nth_prime x) / (real x * ln (real x)) < 2"
       using nth_prime_asymptotics[unfolded asymp_equiv_def,THEN order_tendstoD(2),of 2]
       by simp
     from this eventually_gt_at_top[of 1]
     have "\<forall>\<^sub>F n in sequentially. real (nth_prime n)   < 2*(real n * ln n)"
       by eventually_elim (auto simp:field_simps)
     moreover have *:"\<forall>\<^sub>F N in sequentially. f N >0 "
       unfolding f_def
       by real_asymp
     moreover have " \<forall>\<^sub>F n in sequentially. f n < a n"
       using a_n_sqrt unfolding f_def .
     ultimately have "\<forall>\<^sub>F n in sequentially. a n / nth_prime n > f n / (2*(real n * ln n))"
     proof eventually_elim
       case (elim n)
       then show ?case
         by (auto intro: frac_less2)
     qed
     moreover have "\<forall>\<^sub>F n in sequentially. (f n)/ (2*(real n * ln n)) > 1"
       unfolding f_def by real_asymp
     ultimately show ?thesis 
       by eventually_elim argo
   qed
 
   have a_nth_prime_lt:"\<exists>\<^sub>F n in sequentially. a n / nth_prime n < 1"
   proof -
     have "liminf (\<lambda>x. a x / nth_prime x) < 1"
       using nth_2 by auto
     from this[unfolded less_Liminf_iff]
     show ?thesis
-      apply (auto elim!:frequently_elim1)
-      by (meson divide_less_eq_1 ereal_less_eq(7) leD leI 
-          nth_prime_nonzero of_nat_eq_0_iff of_nat_less_0_iff order.trans)
+      by (smt (verit) ereal_less(3) frequently_elim1 le_less_trans)
   qed
 
   from a_nth_prime_gt a_nth_prime_lt show False 
     by (simp add: eventually_mono frequently_def) 
 qed
 
 section\<open>Acknowledgements\<close>
 
 text\<open>A.K.-A. and W.L. were supported by the ERC Advanced Grant ALEXANDRIA (Project 742178)
  funded by the European Research Council and led by Professor Lawrence Paulson
  at the University of Cambridge, UK.\<close>
 
 end
\ No newline at end of file
diff --git a/thys/Lovasz_Local/Lovasz_Local_Lemma.thy b/thys/Lovasz_Local/Lovasz_Local_Lemma.thy
--- a/thys/Lovasz_Local/Lovasz_Local_Lemma.thy
+++ b/thys/Lovasz_Local/Lovasz_Local_Lemma.thy
@@ -1,706 +1,706 @@
 (* Theory: Lovasz_Local_Lemma
    Author: Chelsea Edmonds *)
 
 section \<open>Lovasz Local Lemma \<close>
 
 theory Lovasz_Local_Lemma 
   imports 
     Basic_Method 
     "HOL-Real_Asymp.Real_Asymp" 
     Indep_Events
     Digraph_Extensions
 begin
 
 subsection \<open>Random Lemmas on Product Operator \<close>
 
 lemma prod_constant_ge: 
   fixes y :: "'b :: {comm_monoid_mult, linordered_semidom}"
   assumes "card A \<le> k"
   assumes "y \<ge> 0" and "y < 1"
   shows "(\<Prod>x\<in>A. y) \<ge> y ^ k"
   using assms power_decreasing by fastforce
 
 lemma (in linordered_idom) prod_mono3:
   assumes "finite J" "I \<subseteq> J" "\<And>i. i \<in> J \<Longrightarrow> 0 \<le> f i"  "(\<And>i. i \<in> J \<Longrightarrow> f i \<le> 1)"
   shows "prod f J \<le> prod f I"
 proof -
   have "prod f J \<le> (\<Prod>i\<in>J. if i \<in> I then f i else 1)"
     using assms by (intro prod_mono) auto
   also have "\<dots> = prod f I"
     using \<open>finite J\<close> \<open>I \<subseteq> J\<close> by (simp add: prod.If_cases Int_absorb1)
   finally show ?thesis .
 qed
 
 lemma bij_on_ss_image: 
   assumes "A \<subseteq> B"
   assumes "bij_betw g B B'"
   shows "g ` A \<subseteq> B'"
   using assms by (auto simp add: bij_betw_apply subsetD)
 
 lemma bij_on_ss_proper_image: 
   assumes "A \<subset> B"
   assumes "bij_betw g B B'"
   shows "g ` A \<subset> B'"
   by (smt (verit, ccfv_SIG) assms bij_betw_iff_bijections bij_betw_subset leD psubsetD psubsetI subsetI)
 
 
 subsection \<open>Dependency Graph Concept \<close>
 
 text \<open>Uses directed graphs. The pair\_digraph locale was sufficient as multi-edges are irrelevant \<close>
 
 locale dependency_digraph = pair_digraph "G :: nat pair_pre_digraph" + prob_space "M :: 'a measure" 
   for G M + fixes F :: "nat \<Rightarrow> 'a set"
   assumes vss: "F ` (pverts G) \<subseteq> events"
   assumes mis: "\<And> i. i \<in> (pverts G) \<Longrightarrow> mutual_indep_events (F i) F ((pverts G) - ({i} \<union> neighborhood i))" 
 begin
 
 lemma dep_graph_indiv_nh_indep: 
   assumes "A \<in> pverts G" "B \<in> pverts G"
   assumes "B \<notin> neighborhood A"
   assumes "A \<noteq> B"
   assumes "prob (F B) \<noteq> 0"
   shows "\<P>((F A) | (F B)) = prob (F A)"
 proof-
   have "B \<notin> {A} \<union> neighborhood A" using assms(3) assms(4) by auto
   then have "B \<in> (pverts G - ({A} \<union> neighborhood A))" using assms(2) by auto
   moreover have "mutual_indep_events (F A) F (pverts G - ({A} \<union> neighborhood A))" using mis assms by auto
   ultimately show ?thesis using 
       assms(5) assms(1) assms(2) vss mutual_indep_ev_cond_single by auto
 qed
 
 lemma mis_subset: 
   assumes "i \<in> pverts G"
   assumes "A \<subseteq> pverts G"
   shows "mutual_indep_events (F i) F (A - ({i} \<union> neighborhood i))"
 proof (cases "A \<subseteq> ({i} \<union> neighborhood i)")
   case True
   then have "A - ({i} \<union> neighborhood i) = {}" by auto
   then show ?thesis using mutual_indep_ev_empty vss assms(1) by blast
 next
   case False
   then have "A - ({i} \<union> neighborhood i) \<subseteq> pverts G - ({i} \<union> neighborhood i)" using assms(2) by auto
   then show ?thesis using mutual_indep_ev_subset mis assms(1) by blast
 qed
 
 lemma dep_graph_indep_events:
   assumes "A \<subseteq> pverts G"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> out_degree G Ai = 0"
   shows "indep_events F A"
 proof -
   have "\<And> Ai. Ai \<in> A \<Longrightarrow> (mutual_indep_events (F Ai) F (A - {Ai}))"
   proof -
     fix Ai assume ain: "Ai \<in> A"
     then have "(neighborhood Ai) = {}" using assms(2) neighborhood_empty_iff by simp
     moreover have "mutual_indep_events (F Ai) F (A - ({Ai} \<union> neighborhood Ai))" 
       using mis_subset[of Ai A] ain assms(1) by auto
     ultimately show "mutual_indep_events (F Ai) F (A - {Ai})" by simp
   qed
   then show ?thesis using mutual_indep_ev_set_all[of F A] vss by auto 
 qed
 
 end
 
 subsection \<open>Lovasz Local General Lemma \<close>
 context prob_space
 begin
 
 lemma compl_sets_index: 
   assumes "F ` A \<subseteq> events"
   shows "(\<lambda> i. space M - F i) ` A \<subseteq> events"
 proof (intro subsetI)
   fix x assume "x \<in> (\<lambda>i. space M - F i) ` A"
   then obtain i where xeq: "x = space M - F i" and "i \<in> A" by blast
   then have "F i \<in> events" using assms by auto
   thus "x \<in> events" using sets.compl_sets xeq by simp
 qed
 
 lemma lovasz_inductive_base: 
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai . Ai \<in> A \<Longrightarrow> g Ai \<ge> 0 \<and> g Ai < 1"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> (prob (F Ai) \<le> (g Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai. (1 - (g Aj))))"
   assumes "Ai \<in> A"
   assumes "pverts G = A"
   shows "prob (F Ai) \<le> g Ai"
 proof -
   have genprod: "\<And> S. S \<subseteq> A \<Longrightarrow> (\<Prod>Aj \<in> S . (1 - (g Aj))) \<le> 1" using assms(2)
     by (smt (verit) prod_le_1 subsetD) 
   interpret dg: dependency_digraph G M F using assms(1) by simp
   have "dg.neighborhood Ai \<subseteq> A" using assms(3) dg.neighborhood_wf assms(5) by simp
   then  show ?thesis 
     using genprod assms mult_left_le by (smt (verit)) 
 qed
 
 lemma lovasz_inductive_base_set: 
   assumes "N \<subseteq> A"
   assumes "\<And> Ai . Ai \<in> A \<Longrightarrow> g Ai \<ge> 0 \<and> g Ai < 1"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> (prob (F Ai) \<le> (g Ai) * (\<Prod> Aj \<in> N. (1 - (g Aj))))"
   assumes "Ai \<in> A"
   shows "prob (F Ai) \<le> g Ai"
 proof -
   have genprod: "\<And> S. S \<subseteq> A \<Longrightarrow> (\<Prod>Aj \<in> S . (1 - (g Aj))) \<le> 1" using assms(2)
     by (smt (verit) prod_le_1 subsetD) 
   then  show ?thesis 
     using genprod assms mult_left_le by (smt (verit)) 
 qed
 
 lemma split_prob_lt_helper:
   assumes dep_graph: "dependency_digraph G M F"
   assumes dep_graph_verts: "pverts G = A"
   assumes fbounds: "\<And> i . i \<in> A \<Longrightarrow> f i \<ge> 0 \<and> f i < 1"
   assumes prob_Ai: "\<And> Ai.  Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> 
     (f Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai . (1 - (f Aj)))"
   assumes aiin: "Ai \<in> A"
   assumes "N \<subseteq> pre_digraph.neighborhood G Ai"
   assumes "\<exists> P1 P2. \<P>(F Ai | \<Inter>Aj\<in>S. space M - F Aj) = P1/P2 \<and> 
     P1 \<le> prob (F Ai)\<and> P2 \<ge> (\<Prod> Aj \<in> N . (1 - (f Aj)))"
   shows "\<P>(F Ai | \<Inter>Aj\<in>S. space M - F Aj) \<le> f Ai"
 proof -
   interpret dg: dependency_digraph G M F using assms(1) by simp
   have lt1: "\<And> Aj. Aj \<in> A \<Longrightarrow>  (1 - (f Aj)) \<le> 1"
     using assms(3) by auto
   have gt0: "\<And> Aj. Aj \<in> A \<Longrightarrow> (1 - (f Aj)) > 0" using assms(3) by auto
   then have prodgt0: "\<And> S'. S' \<subseteq> A \<Longrightarrow>  (\<Prod> Aj \<in> S' . (1 - f Aj)) > 0"
     using prod_pos by (metis subsetD) 
   obtain P1 P2 where peq: "\<P>(F Ai | \<Inter>Aj\<in>S. space M - F Aj) = P1/P2" and  
    "P1 \<le> prob (F Ai)"
     and p2gt: "P2 \<ge> (\<Prod> Aj \<in> N . (1 - (f Aj)))" using assms(7) by auto
   then have "P1 \<le> (f Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai . (1 - (f Aj)))" 
     using prob_Ai aiin by fastforce
   moreover have "P2 \<ge> (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))" using assms(6) 
       gt0 dg.neighborhood_wf dep_graph_verts subset_iff lt1 dg.neighborhood_finite p2gt
     by (smt (verit, ccfv_threshold) prod_mono3) 
   ultimately have "P1/P2 \<le> ((f Ai) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))/(\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj))))"
     using frac_le[of "(f Ai) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))" "P1" "(\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))"] 
       prodgt0[of "dg.neighborhood Ai"] assms(3) dg.neighborhood_wf[of Ai]
     by (simp add: assms(2) bounded_measure finite_measure_compl assms(5))
   then show ?thesis using prodgt0[of "dg.neighborhood Ai"] dg.neighborhood_wf[of Ai] assms(2) peq
     by (metis divide_eq_imp rel_simps(70)) 
 qed
 
 lemma lovasz_inequality: 
   assumes finS: "finite S"
   assumes sevents: "F ` S \<subseteq> events"
   assumes S_subset: "S \<subseteq> A - {Ai}"
   assumes prob2: "prob (\<Inter> Aj \<in> S . (space M - (F Aj))) > 0"
   assumes irange: "i \<in> {0..<card S1}"
   assumes bb: "bij_betw g {0..<card S1} S1"
   assumes s1_def: "S1 = (S \<inter> N)"
   assumes s2_def: "S2 = S - S1"
   assumes ne_cond: "i > 0 \<or> S2 \<noteq> {}"
   assumes hyps: "\<And> B. B \<subset> S \<Longrightarrow> g i \<in> A \<Longrightarrow> B \<subseteq> A - {g i} \<Longrightarrow> B \<noteq> {} \<Longrightarrow> 
     0 < prob (\<Inter>Aj\<in>B. space M - F Aj) \<Longrightarrow> \<P>(F (g i) | \<Inter>Aj\<in>B. space M - F Aj) \<le> f (g i)"
   shows "\<P>((space M - F (g i)) | (\<Inter> ((\<lambda> i. space M - F i) ` g ` {0..<i} \<union> ((\<lambda> i. space M - F i) ` S2)))) 
     \<ge> (1 - f (g i))"
 proof -
   let ?c = "(\<lambda> i. space M - F i)"
   define S1ss where "S1ss = g ` {0..<i}" 
   have "i \<notin> {0..<i}" by simp
   moreover have "{0..<i} \<subseteq> {0..<card S1}" using irange by simp
   ultimately have ginotin1: "g i \<notin> S1ss" using bb S1ss_def irange
     by (smt (verit, best) bij_betw_iff_bijections image_iff subset_eq) 
   have ginotin2: "g i \<notin> S2" unfolding s2_def using irange bb by (simp add: bij_betwE) 
   have giS: "g i \<in> S" using irange bij_betw_imp_surj_on imageI Int_iff s1_def bb 
     by blast
   have "{0..<i} \<subset> {0..<card S1}" using irange by auto
   then have "S1ss \<subset> S1" unfolding S1ss_def using irange bb bij_on_ss_proper_image by meson
   then have sss: "S1ss \<union> S2 \<subset> S" using s1_def s2_def by blast 
   moreover have xsiin: "g i \<in> A"using irange  
     using giS S_subset by (metis DiffE in_mono)
   moreover have ne: "S1ss \<union> S2 \<noteq> {}" using ne_cond S1ss_def by auto
   moreover have "S1ss \<union> S2 \<subseteq> A - {g i}"  using S_subset sss ginotin1 ginotin2 by auto
   moreover have gt02: "0 < prob (\<Inter> (?c ` (S1ss \<union> S2)))" using finS prob2 sevents
     prob_inter_ss_lt_index[of S ?c "S1ss \<union> S2"] ne sss compl_sets_index[of F S]  by fastforce
   ultimately have ltfAi: "\<P>(F (g i) | \<Inter> (?c ` (S1ss \<union> S2))) \<le> f (g i)"
     using hyps[of "S1ss \<union> S2"] by blast 
   have "?c ` (S1ss \<union> S2) \<subseteq> events" using sss \<open>S1ss \<subset> S1\<close> compl_subset_in_events sevents s1_def s2_def
     by fastforce
   then have "\<Inter> (?c ` (S1ss \<union> S2)) \<in> events" using Inter_event_ss sss
     by (meson \<open>S1ss \<union> S2 \<noteq> {}\<close> finite_imageI finite_subset image_is_empty finS subset_iff_psubset_eq)
   moreover have "F (g i) \<in> events" using xsiin giS sevents by auto
   ultimately have "\<P>(?c (g i) | \<Inter> (?c ` (S1ss \<union> S2))) \<ge> 1 - f (g i)" 
     using cond_prob_neg[of "\<Inter> (?c ` (S1ss \<union> S2))" "F (g i)"] gt02 xsiin ltfAi by simp
   then show "\<P>(?c (g i) | (\<Inter> (?c ` g ` {0..<i} \<union> (?c ` S2)))) \<ge> (1 - f (g i))"
     by (simp add: S1ss_def image_Un)
 qed
 
 text \<open>The main helper lemma \<close>
 lemma lovasz_inductive: 
   assumes finA: "finite A"
   assumes Aevents: "F ` A \<subseteq> events"
   assumes fbounds: "\<And> i . i \<in> A \<Longrightarrow> f i \<ge> 0 \<and> f i < 1"
   assumes dep_graph: "dependency_digraph G M F"
   assumes dep_graph_verts: "pverts G = A"
   assumes prob_Ai: "\<And> Ai.  Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> 
     (f Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai . (1 - (f Aj)))"
   assumes Ai_in: "Ai \<in> A"
   assumes S_subset: "S \<subseteq> A - {Ai}"
   assumes S_nempty: "S \<noteq> {}"
   assumes prob2: "prob (\<Inter> Aj \<in> S . (space M - (F Aj))) > 0"
   shows "\<P>((F Ai) | (\<Inter> Aj \<in> S . (space M - (F Aj)))) \<le> f Ai"
 proof -
   let ?c = "\<lambda> i. space M - F i"
   have ceq: "\<And> A. ?c ` A = ((-) (space M)) ` (F ` A)"  by auto
   interpret dg: dependency_digraph G M F using assms(4) by simp
   have finS: "finite S" using assms finite_subset by (metis finite_Diff) 
   show "\<P>(( F Ai) | (\<Inter> Aj \<in> S . (space M - (F Aj)))) \<le> f Ai" 
     using finS Ai_in S_subset S_nempty prob2
   proof (induct S arbitrary: Ai rule: finite_psubset_induct )
     case (psubset S)
     define S1 where "S1 = (S \<inter> dg.neighborhood Ai)"
     define S2 where "S2 = S - S1"
     have "\<And> s . s \<in> S2 \<Longrightarrow> s \<in> A - ({Ai} \<union> dg.neighborhood Ai)" 
       using S1_def S2_def psubset.prems(2) by blast 
     then have s2ssmis: "S2 \<subseteq> A - ({Ai} \<union> dg.neighborhood Ai)" by auto
     have sevents: "F ` S \<subseteq> events" using assms(2) psubset.prems(2) by auto
     then have s1events: "F ` S1 \<subseteq> events" using S1_def by auto
     have finS2: "finite S2" and finS1: "finite S1" using S2_def S1_def by (simp_all add: psubset(1))
     have "mutual_indep_set (F Ai) (F ` S2)" using dg.mis[of Ai] mutual_indep_ev_subset s2ssmis 
       psubset.prems(1) dep_graph_verts mutual_indep_iff by auto
     then have mis2: "mutual_indep_set (F Ai) (?c ` S2)" 
       using mutual_indep_events_compl[of "F ` S2" "F Ai"] finS2 ceq[of S2] by simp
     have scompl_ev: "?c ` S \<subseteq> events"
       using compl_sets_index sevents by simp
     then have s2cev: "?c ` S2 \<subseteq> events" using S2_def scompl_ev by blast
     have "(\<Inter> Aj \<in> S . space M - (F Aj)) \<subseteq> (\<Inter> Aj \<in> S2 . space M - (F Aj))"
       unfolding S2_def using Diff_subset image_mono Inter_anti_mono by blast
     then have "S2 \<noteq> {} \<Longrightarrow> prob (\<Inter> Aj \<in> S2 . space M - (F Aj)) \<noteq> 0" using psubset.prems(4) s2cev 
       finS2 Inter_event_ss[of "?c ` S2"] finite_measure_mono[of "\<Inter> (?c ` S)" "\<Inter>(?c ` S2)"] by simp
     then have s2prob_eq: "S2 \<noteq> {} \<Longrightarrow> \<P>((F Ai) | (\<Inter> (?c ` S2))) = prob (F Ai)" using assms(2)
       mutual_indep_cond_full[of " F Ai" "?c ` S2"]  psubset.prems(1) s2cev finS2 mis2 by simp
     show ?case 
     proof (cases "S1 = {}")
       case True
       then show ?thesis using lovasz_inductive_base[of G F A f Ai]  psubset.prems(3) S2_def
           assms(3) assms(4) psubset.prems(1) prob_Ai s2prob_eq dep_graph_verts by (simp) 
     next
       case s1F: False
       then have csgt0: "card S1 > 0" using s1F finS1 card_gt_0_iff by blast
       obtain g where bb: "bij_betw g {0..<card S1} S1" using finS1 ex_bij_betw_nat_finite by auto
       have igt0: "\<And>i. i \<in> {0..<card S1} \<Longrightarrow> 1 - f (g i) \<ge> 0"
         using S1_def psubset.prems(2) bb bij_betw_apply assms(3) by fastforce
       have s1ss: "S1 \<subseteq> dg.neighborhood Ai" using S1_def by auto
       moreover have "\<exists> P1 P2. \<P>(F Ai | \<Inter>Aj\<in>S. space M - F Aj) = P1/P2 \<and>  P1 \<le> prob (F Ai) 
         \<and> P2 \<ge> (\<Prod> Aj \<in> S1 . (1 - (f Aj)))"
       proof (cases "S2 = {}")
         case True
         then have Seq: "S1 = S" using S1_def S2_def by auto
         have inter_eventsS: "(\<Inter> Aj \<in> S . (space M - (F Aj))) \<in> events" using  psubset.prems assms
           by (meson measure_notin_sets zero_less_measure_iff)
         then have peq: "\<P>((F Ai) | (\<Inter> Aj \<in> S1 . ?c Aj)) = 
             prob ((\<Inter> Aj \<in> S1 . ?c Aj) \<inter> (F Ai))/prob ((\<Inter> (?c ` S1)))" 
             (is "\<P>((F Ai) | (\<Inter> Aj \<in> S1 . ?c Aj)) = ?Num/?Den")
           using cond_prob_ev_def[of "(\<Inter> Aj \<in> S1 . (space M - (F Aj)))" "F Ai"] 
           using Seq psubset.prems(1) assms(2) by blast
         have "?Num \<le> prob (F Ai)" using finite_measure_mono assms(2) psubset.prems(1) by simp
         moreover have "?Den \<ge> (\<Prod> Aj \<in> S1 . (1 - (f Aj)))"
         proof -
           have pcond: "prob (\<Inter>(?c ` S1)) = 
               prob (?c (g 0)) * (\<Prod>i \<in> {1..<card S1} . \<P>(?c (g i) | (\<Inter>(?c ` g ` {0..<i}))))"
             using prob_cond_inter_index_fn_compl[of "S1" F] Seq s1events psubset(1) s1F bb by auto
           have ineq: "\<And> i. i \<in> {1..<card S1} \<Longrightarrow> \<P>(?c (g i) | (\<Inter>(?c ` g ` {0..<i}))) \<ge> (1 - (f (g i)))" 
             using lovasz_inequality[of S1 F A Ai _ S1 g S1 "{}" f] sevents finS psubset.prems(2) 
               psubset.prems(4)  bb psubset.hyps(2)[of _ "g _"] Seq by fastforce
           have "(\<And>i. i \<in> {1..<card S1} \<Longrightarrow> 1 - f (g i) \<ge> 0)" using igt0 by simp
           then have "(\<Prod>i \<in> {1..<(card S1)} . \<P>(?c (g i) | (\<Inter>(?c ` g ` {0..<i})))) 
             \<ge> (\<Prod>i \<in> {1..<(card S1)} . (1 - (f (g i))))" 
             using ineq prod_mono by (smt(verit, ccfv_threshold))
           moreover have "prob (?c (g 0)) \<ge> (1 - f (g 0))" 
           proof -
             have g0in: "g 0 \<in> A" using bb csgt0 using psubset.prems(2) bij_betwE Seq by fastforce
             then have "prob (?c (g 0)) = 1 - prob (F (g 0))" using Aevents by (simp add: prob_compl) 
             then show ?thesis using lovasz_inductive_base[of G F A f "g 0"] 
                 prob_Ai assms(4) dep_graph_verts fbounds g0in by auto
           qed
-          moreover have "0 \<le> (\<Prod>i = 1..<card S1. 1 - f (g i))" using igt0 by (simp add: prod_nonneg) 
+          moreover have "0 \<le> (\<Prod>i = 1..<card S1. 1 - f (g i))" using igt0 by (force intro: prod_nonneg) 
           ultimately have  "prob (\<Inter>(?c ` S1)) \<ge> (1 - (f (g 0))) * (\<Prod>i \<in> {1..<(card S1)} . (1 - (f (g i))))" 
             using pcond igt0 mult_mono'[of "(1 - (f (g 0)))" ] by fastforce
           moreover have "{0..<card S1} = {0} \<union> {1..<card S1}" using csgt0 by auto
           ultimately have "prob (\<Inter>(?c ` S1)) \<ge> (\<Prod>i \<in> {0..<(card S1)} . (1 - (f (g i))))" by auto     
           moreover have "(\<Prod>i \<in> {0..<(card S1)} . (1 - (f (g i)))) = (\<Prod>i \<in> S1 . (1 - (f (i))))" 
             using prod.reindex_bij_betw bb by simp
           ultimately show ?thesis by simp
         qed
         ultimately show ?thesis using peq Seq by blast 
       next
         case s2F: False
         have s2inter: "\<Inter> (?c ` S2) \<in> events" 
           using s2F finS2 s2cev Inter_event_ss[of "?c ` S2"]  by auto
         have split: "(\<Inter> Aj \<in> S . (?c Aj)) = (\<Inter> (?c `S1)) \<inter> (\<Inter> (?c ` S2))" 
           using S1_def S2_def by auto
         then have "\<P>(F Ai | (\<Inter> Aj \<in> S . (?c Aj))) = \<P>(F Ai | (\<Inter> (?c `S1)) \<inter> (\<Inter> (?c ` S2)))" by simp
         moreover have s2n0: "prob (\<Inter> (?c ` S2)) \<noteq> 0" using psubset.prems(4) S2_def
           by (metis Int_lower2 split finite_measure_mono measure_le_0_iff s2inter semiring_norm(137)) 
         moreover have "\<Inter> (?c ` S1) \<in> events" 
           using finS1 S1_def scompl_ev s1F Inter_event_ss[of "(?c ` S1)"] by auto
         ultimately have peq: "\<P>(F Ai | (\<Inter> Aj \<in> S . (?c Aj))) = \<P>(F Ai \<inter> (\<Inter> (?c `S1)) | \<Inter> (?c ` S2))/ 
           \<P>(\<Inter> (?c `S1) | \<Inter> (?c `S2))" (is "\<P>(F Ai | (\<Inter> Aj \<in> S . (?c Aj))) = ?Num/?Den")
           using cond_prob_dual_intersect[of "F Ai" "\<Inter> (?c `S1)" "\<Inter> (?c `S2)"]  assms(2) 
             psubset.prems(1) s2inter by fastforce
         have "?Num \<le> \<P>(F Ai | \<Inter> (?c `S2))" using cond_prob_inter_set_lt[of "F Ai" "\<Inter> (?c `S2)" "?c ` S1"]
           using s1events finS1 psubset.prems(1) assms(2) s2inter finite_imageI[of S1 F] by blast
         then have "?Num \<le> prob (F Ai)" using s2F s2prob_eq by auto
         moreover have "?Den \<ge> (\<Prod> Aj \<in> S1 . (1 - (f Aj)))"  using psubset.hyps
         proof -
           have "prob (\<Inter>(?c ` S2)) > 0" using s2n0 by (meson zero_less_measure_iff) 
           then have pcond: "\<P>(\<Inter> (?c `S1) | \<Inter> (?c `S2)) = 
               (\<Prod>i = 0..<card S1 . \<P>(?c (g i) | (\<Inter> (?c ` g ` {0..<i} \<union> (?c ` S2)))))"
             using prob_cond_Inter_index_cond_compl_fn[of S1 "?c ` S2" F] s1F finS1 s2cev finS2 s2F 
               s1events bb by auto  
           have  "\<And> i. i \<in> {0..<card S1} \<Longrightarrow> \<P>(?c (g i) | (\<Inter> (?c ` g ` {0..<i} \<union> (?c ` S2)))) \<ge> (1 - f (g i))"
             using lovasz_inequality[of S F A Ai _ S1 g "dg.neighborhood Ai" S2 f] S1_def S2_def sevents 
               finS psubset.prems(2) psubset.prems(4)  bb psubset.hyps(2)[of _ "g _"] psubset(1) s2F by meson
           then have c1: "\<P>(\<Inter> (?c `S1) | \<Inter> (?c `S2)) \<ge> (\<Prod>i = 0..<card S1 . (1 - f (g i)))" 
             using prod_mono igt0 pcond bb  by (smt(verit, ccfv_threshold))
           then have "\<P>(\<Inter> (?c `S1) | \<Inter> (?c `S2)) \<ge> (\<Prod>i \<in> {0..<card S1} . (1 - f (g i)))" by blast
           moreover have "(\<Prod>i \<in> {0..<card S1} . (1 - f (g i))) = (\<Prod>x \<in> S1 . (1 - f x))" using bb
             using prod.reindex_bij_betw by fastforce 
           ultimately show ?thesis by simp
         qed 
         ultimately show ?thesis using peq by blast
       qed
       ultimately show ?thesis  by (intro split_prob_lt_helper[of G F A])
         (simp_all add: dep_graph dep_graph_verts fbounds psubset.prems(1) prob_Ai)
     qed
   qed
 qed
 
 text \<open>The main lemma \<close>
 theorem lovasz_local_general: 
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai . Ai \<in> A \<Longrightarrow>  f Ai \<ge> 0 \<and> f Ai < 1"
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> (prob (F Ai) \<le> (f Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai. (1 - (f Aj))))"
   assumes "pverts G = A"
   shows "prob (\<Inter> Ai \<in> A . (space M - (F Ai))) \<ge> (\<Prod> Ai \<in> A . (1 - f Ai))" "(\<Prod> Ai \<in> A . (1 - f Ai)) > 0"
 proof -
   show gt0: "(\<Prod> Ai \<in> A . (1 - f Ai)) > 0" using assms(4) by (simp add: prod_pos)
   let ?c = "\<lambda> i. space M - F i"
   interpret dg: dependency_digraph G M F using assms(5) by simp
   have general: "\<And>Ai S. Ai \<in> A \<Longrightarrow> S \<subseteq> A - {Ai} \<Longrightarrow> S \<noteq> {} \<Longrightarrow>  prob (\<Inter> Aj \<in> S . (?c Aj)) > 0 
     \<Longrightarrow>  \<P>(F Ai | (\<Inter> Aj \<in> S . (?c Aj))) \<le> f Ai" 
     using assms lovasz_inductive[of A F f G] by simp
   have base: "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> f Ai"
     using lovasz_inductive_base assms(4) assms(6) assms(5) assms(7) by blast
   show "prob (\<Inter> Ai \<in> A . (?c Ai)) \<ge> (\<Prod> Ai \<in> A . (1 - f Ai))"
     using assms(3) assms(1) assms(2) assms(4) general base
   proof (induct A rule: finite_ne_induct)
     case (singleton x)
     then show ?case using singleton.prems singleton prob_compl by auto
   next
     case (insert x X)
     define Ax where "Ax = ?c ` (insert x X)"
     have xie: "F x \<in> events" using insert.prems by simp
     have A'ie: "\<Inter>(?c ` X) \<in> events" using insert.prems insert.hyps by auto
     have "(\<And>Ai S. Ai \<in> insert x X \<Longrightarrow> S \<subseteq> insert x X - {Ai} \<Longrightarrow> S \<noteq> {} \<Longrightarrow> prob (\<Inter> Aj \<in> S . (?c Aj)) > 0 
       \<Longrightarrow> \<P>(F Ai | \<Inter> (?c ` S)) \<le> f Ai)" using insert.prems by simp
     then have  "(\<And>Ai S. Ai \<in> X \<Longrightarrow> S \<subseteq> X - {Ai} \<Longrightarrow> S \<noteq> {} \<Longrightarrow>  prob (\<Inter> Aj \<in> S . (?c Aj)) > 0 
       \<Longrightarrow> \<P>(F Ai | \<Inter> (?c ` S)) \<le> f Ai)" by auto
     then have A'gt: "(\<Prod>Ai\<in>X. 1 - f Ai) \<le> prob (\<Inter> (?c ` X))"
       using insert.hyps(4) insert.prems(2) insert.prems(1) insert.prems(4) by auto
     then have "prob (\<Inter>(?c ` X)) > 0" using insert.hyps insert.prems prod_pos basic_trans_rules(22) 
       diff_gt_0_iff_gt by (metis (no_types, lifting) insert_Diff insert_subset subset_insertI) 
     then have "\<P>((?c x) | (\<Inter>(?c ` X))) = 1 - \<P>(F x | (\<Inter>(?c ` X)))" 
       using cond_prob_neg[of "\<Inter>(?c ` X)" "F x"] xie A'ie by simp
     moreover have "\<P>(F x | (\<Inter>(?c ` X))) \<le> f x" using insert.prems(3)[of x X]  insert.hyps(2) insert(3) 
         A'gt \<open>0 < prob (\<Inter> (?c ` X))\<close> by fastforce
     ultimately have pnxgt: "\<P>((?c x) | (\<Inter>(?c ` X))) \<ge> 1 - f x" by simp
     have xgt0: "1 - f x \<ge> 0" using insert.prems(2)[of x] by auto
     have "prob (\<Inter> Ax) = prob ((?c x) \<inter> \<Inter>(?c ` X))" using Ax_def by simp
     also have "... = prob (\<Inter>(?c ` X)) * \<P>((?c x) | (\<Inter>(?c ` X)))" 
       using prob_intersect_B xie A'ie by simp
     also have "... \<ge> (\<Prod>Ai\<in>X. 1 - f Ai) * (1 - f x)" using A'gt pnxgt mult_left_le 
         \<open>0 < prob (\<Inter>(?c ` X))\<close> xgt0 mult_mono by (smt(verit))
     finally have "prob (\<Inter> Ax) \<ge> (\<Prod>Ai\<in>insert x X. 1 - f Ai)"
       by (simp add: local.insert(1) local.insert(3) mult.commute) 
     then show ?case using Ax_def by auto
   qed
 qed
 
 subsection \<open>Lovasz Corollaries and Variations \<close>
 
 corollary lovasz_local_general_positive: 
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai . Ai \<in> A \<Longrightarrow>  f Ai \<ge> 0 \<and> f Ai < 1"
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> (prob (F Ai) \<le> 
     (f Ai) * (\<Prod> Aj \<in> pre_digraph.neighborhood G Ai. (1 - (f Aj))))"
   assumes "pverts G = A"
   shows "prob (\<Inter> Ai \<in> A . (space M - (F Ai))) > 0"
   using assms lovasz_local_general(1)[of A F f G] lovasz_local_general(2)[of A F f G] by simp
 
 theorem lovasz_local_symmetric_dep_graph: 
   fixes e :: real
   fixes d :: nat
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d" 
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> p"
   assumes "exp(1)* p * (d + 1) \<le> 1" (* e should be euler's number \<Rightarrow> using exponential function? *)
   assumes "pverts G = A"
   shows "prob (\<Inter> Ai \<in> A . (space M - (F Ai))) > 0"
 proof (cases "d = 0")
   case True
   interpret g: dependency_digraph G M F using assms(4) by simp
   (* Because all have mutual independence \<Longrightarrow> complete independence *)
   have "indep_events F A" using g.dep_graph_indep_events[of A] assms(8) assms(5) True by simp
   moreover have "p < 1" 
   proof -
     have "exp (1) * p \<le> 1" using assms(7) True by simp
     then show ?thesis using exp_gt_one  less_1_mult linorder_neqE_linordered_idom rel_simps(68) 
       verit_prod_simplify(2) by (smt (verit) mult_le_cancel_left1)
   qed
   ultimately show ?thesis 
     using complete_indep_bound3[of A F] assms(2) assms(1) assms(3) assms(6) by force
 next
   case False
   define f :: "nat \<Rightarrow> real" where "f \<equiv> (\<lambda> Ai . 1 /(d + 1))"
   then have fbounds: "\<And> Ai. f Ai \<ge> 0 \<and> f Ai < 1" using f_def False by simp
   interpret dg: dependency_digraph G M F using assms(4) by auto
 (* Showing bound is the most work *)
   have "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> (f Ai) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))"
   proof -
     fix Ai assume ain: "Ai \<in> A"
     have d_boundslt1: "(1/(d + 1)) < 1" and d_boundsgt0: "(1/(d + 1))> 0" using False by fastforce+ 
     have d_bounds2: "(1 - (1 /(d + 1)))^d < 1" using False 
       by(simp add: field_simps) (smt (verit) of_nat_0_le_iff power_mono_iff)
     have d_bounds0: "(1 - (1 /(d + 1)))^d > 0" using False by (simp)
     have "exp(1) > (1 + 1/d) powr d" using exp_1_gt_powr False by simp
     then have "exp(1) > (1 + 1/d)^d" using False by (simp add: powr_realpow zero_compare_simps(2))
     moreover have "1/(1+ 1/d)^d = (1 - (1/(d+1)))^d"
     proof -
       have "1/(1+ 1/d)^d = 1/((d/d) + 1/d)^d" by (simp add: field_simps)
       then show ?thesis by (simp add: field_simps)
     qed
     ultimately have exp_lt: "1/exp(1) < (1 - (1 /(d + 1)))^d"
       by (metis d_bounds0 frac_less2 less_eq_real_def of_nat_zero_less_power_iff power_eq_if zero_less_divide_1_iff) 
     then have "(1 /(d + 1))* (1 - (1 /(d + 1)))^d > (1 /(d + 1))*(1/exp(1))"
       using exp_lt mult_strict_left_mono[of "1/exp(1)" "(1 - (1 /(d + 1)))^d" "(1/(d+1))"] d_boundslt1 
       by simp
     then have "(1 /(d + 1))* (1 - (1 /(d + 1)))^d > (1/((d+1)*exp(1)))" by auto 
     then have gtp: "(1 /(d + 1))* (1 - (1 /(d + 1)))^d > p"
       by (smt (verit, ccfv_SIG) d_boundslt1 d_boundsgt0 assms(7) divide_divide_eq_left divide_less_cancel 
           divide_less_eq divide_nonneg_nonpos nonzero_mult_div_cancel_left not_exp_le_zero) 
     have "card (dg.neighborhood Ai) \<le> d" using assms(5) dg.out_degree_neighborhood ain by auto
     then have "(\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) \<ge> (1 - (1 /(d + 1)))^d"
       using prod_constant_ge[of  "dg.neighborhood Ai" "d" "1 - (1/d+1)"] using d_boundslt1 by auto
     then have "(1 /(d + 1)) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) \<ge> (1 /(d + 1))* (1 - (1 /(d + 1)))^d"
       by (simp add: divide_right_mono) 
     then have "(1 /(d + 1)) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) > p" 
       using gtp by simp
     then show "prob (F Ai) \<le> f Ai * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - f Aj))"
       using assms(6) \<open>Ai \<in> A\<close> f_def by force 
   qed
   then show ?thesis using lovasz_local_general_positive[of A F f G] 
       assms(4) assms(1) assms(2) assms(3) assms(8) fbounds by auto
 qed
 
 corollary lovasz_local_symmetric4gt: 
   fixes e :: real
   fixes d :: nat
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d" 
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> p"
   assumes "4 * p * d \<le> 1" (* only works if d \ge 3 *)
   assumes "d \<ge> 3"
   assumes "pverts G = A"
   shows "prob (\<Inter> Ai \<in> A . (space M - F Ai)) > 0"
 proof -
   have "exp(1)* p * (d + 1) \<le> 1"
   proof (cases "p = 0")
     case True
     then show ?thesis by simp
   next
     case False
     then have pgt: "p > 0" using assms(1) assms(6) assms(3) ex_min_if_finite less_eq_real_def
       by (meson basic_trans_rules(23) basic_trans_rules(24) linorder_neqE_linordered_idom measure_nonneg)
     have "3 * (d + 1) \<le> 4 * d" by (simp add: field_simps assms(8))
     then have "exp(1) * (d + 1) \<le> 4 *d"
       using exp_le exp_gt_one[of 1] assms(8)
       by (smt (verit, del_insts) Num.of_nat_simps(2) Num.of_nat_simps(5) le_add2 le_eq_less_or_eq 
           mult_right_mono nat_less_real_le numeral.simps(3) numerals(1) of_nat_numeral) 
     then have "exp(1) * (d + 1) * p \<le> 4 *d *p" using pgt by simp 
     then show ?thesis using assms(7) by (simp add: field_simps) 
   qed
   then show ?thesis using assms lovasz_local_symmetric_dep_graph[of A F G d p] by auto
 qed
 
 
 lemma lovasz_local_symmetric4: 
   fixes e :: real
   fixes d :: nat
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "dependency_digraph G M F"
   assumes "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d" 
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> p"
   assumes "4 * p * d \<le> 1"
   assumes "d \<ge> 1"
   assumes "pverts G = A"
   shows "prob (\<Inter> Ai \<in> A . (space M - F Ai)) > 0"
 proof (cases "d \<ge> 3")
   case True
   then show ?thesis using lovasz_local_symmetric4gt assms
     by presburger
 next
   case d3: False
   define f :: "nat \<Rightarrow> real" where "f \<equiv> (\<lambda> Ai . 1 /(d + 1))"
   then have fbounds: "\<And> Ai. f Ai \<ge> 0 \<and> f Ai < 1" using f_def assms(8) by simp
   interpret dg: dependency_digraph G M F using assms(4) by auto
   have "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> (f Ai) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (f Aj)))"
   proof -
     fix Ai assume ain: "Ai \<in> A"
     have d_boundslt1: "(1/(d + 1)) < 1" and d_boundsgt0: "(1/(d + 1))> 0" using assms by fastforce+ 
     have plt: "1/(4*d) \<ge> p" using assms(7) assms(8)
       by (metis (mono_tags, opaque_lifting) Num.of_nat_simps(5) bot_nat_0.not_eq_extremum le_numeral_extra(2) 
           more_arith_simps(11) mult_of_nat_commute nat_0_less_mult_iff of_nat_0_less_iff of_nat_numeral 
           pos_divide_less_eq rel_simps(51) verit_comp_simplify(3)) 
     then have gtp: "(1 /(d + 1))* (1 - (1 /(d + 1)))^d \<ge> p"
     proof (cases "d = 1")
       case False
       then have "d = 2" using d3 assms(8) by auto 
       then show ?thesis using plt by (simp add: field_simps)
     qed (simp)
     have "card (dg.neighborhood Ai) \<le> d" using assms(5) dg.out_degree_neighborhood ain by auto
     then have "(\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) \<ge> (1 - (1 /(d + 1)))^d"
       using prod_constant_ge[of  "dg.neighborhood Ai" "d" "1 - (1/d+1)"] using d_boundslt1 by auto
     then have "(1 /(d + 1)) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) \<ge> (1 /(d + 1))* (1 - (1 /(d + 1)))^d"
       by (simp add: divide_right_mono) 
     then have "(1 /(d + 1)) * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - (1 /(d + 1)))) \<ge> p" 
       using gtp by simp
     then show "prob (F Ai) \<le> f Ai * (\<Prod> Aj \<in> dg.neighborhood Ai . (1 - f Aj))"
       using assms(6) \<open>Ai \<in> A\<close> f_def by force 
   qed
   then show ?thesis 
     using lovasz_local_general_positive[of A F f G] assms(4) assms(1) assms(2) assms(3) assms(9) fbounds by auto
 qed
 
 text \<open>Converting between dependency graph and indexed set representation of mutual independence \<close>
 
 lemma (in pair_digraph) g_Ai_simplification:
   assumes "Ai \<in> A"
   assumes "g Ai \<subseteq> A - {Ai}"
   assumes "pverts G = A"
   assumes "parcs G = {e \<in> A \<times> A . snd e \<in> (A - ({fst e} \<union> (g (fst e))))}"
   shows "g Ai = A - ({Ai} \<union> neighborhood Ai)"
 proof -
   have "g Ai = A - ({Ai} \<union> {v \<in> A . v \<in> (A - ({Ai} \<union> (g (Ai))))})"  using assms(2) by auto
   then have "g Ai = A - ({Ai} \<union> {v \<in> A . (Ai, v) \<in> parcs G})"
     using Collect_cong assms(1) mem_Collect_eq assms(3) assms(4) by auto
   then show "g Ai = A - ({Ai} \<union> neighborhood Ai)" unfolding neighborhood_def using assms(3) by simp
 qed
 
 lemma define_dep_graph_set:
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> g Ai \<subseteq> A - {Ai} \<and> mutual_indep_events (F Ai) F (g Ai)"
   shows "dependency_digraph \<lparr> pverts = A, parcs = {e \<in> A \<times> A . snd e \<in> (A - ({fst e} \<union> (g (fst e))))} \<rparr> M F"
     (is "dependency_digraph ?G M F")
 proof -
   interpret pd: pair_digraph ?G 
     using assms(3)by (unfold_locales) auto 
   have "\<And> Ai. Ai \<in> A \<Longrightarrow> g Ai \<subseteq> A - {Ai}" using assms(4) by simp
   then have "\<And> i. i \<in> A \<Longrightarrow> g i = A - ({i} \<union> pd.neighborhood i)"
     using pd.g_Ai_simplification[of _ A g] pd.pair_digraph by auto
   then have "dependency_digraph ?G M F" using assms(2) assms(4) by (unfold_locales) auto
   then show ?thesis by simp
 qed
 
 lemma define_dep_graph_deg_bound:
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> g Ai \<subseteq> A - {Ai} \<and> card (g Ai) \<ge> card A - d - 1 \<and> 
     mutual_indep_events (F Ai) F (g Ai)"
   shows "\<And> Ai.  Ai \<in> A \<Longrightarrow> 
     out_degree \<lparr> pverts = A, parcs = {e \<in> A \<times> A . snd e \<in> (A - ({fst e} \<union> (g (fst e))))} \<rparr> Ai \<le> d"
     (is "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree (with_proj ?G) Ai \<le> d")
 proof -
   interpret pd: dependency_digraph ?G M F using assms define_dep_graph_set by simp
   show "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree ?G Ai \<le> d"
   proof -
     fix Ai assume a: "Ai \<in> A"
     then have geq: "g Ai = A - ({Ai} \<union> pd.neighborhood Ai)" 
       using assms(4)[of Ai] pd.pair_digraph pd.g_Ai_simplification[of Ai A g ]  by simp
     then have pss: "g Ai \<subset> A" using a by auto
     then have "card (g Ai) = card (A - ({Ai} \<union> pd.neighborhood Ai))" using assms(4) geq by argo
     moreover have ss: "({Ai} \<union> pd.neighborhood Ai) \<subseteq> A" using pd.neighborhood_wf a  by simp
     moreover have "finite ({Ai} \<union> pd.neighborhood Ai)" 
       using calculation(2) assms(3) finite_subset by auto
     moreover have "Ai \<notin> pd.neighborhood Ai" using pd.neighborhood_self_not by simp
     moreover have "card {Ai} = 1" using is_singleton_altdef by auto
     moreover have cardss: "card ({Ai} \<union> pd.neighborhood Ai) = 1 + card (pd.neighborhood Ai)" 
       using calculation(5) calculation(4) card_Un_disjoint pd.neighborhood_finite by auto
     ultimately have eq: "card (g Ai) = card A - 1 - card (pd.neighborhood Ai)" 
       using card_Diff_subset[of "({Ai} \<union> pd.neighborhood Ai)" A] assms(3) by presburger
     have ggt: "\<And> Ai. Ai \<in> A \<Longrightarrow> card (g Ai) \<ge> int (card A) - int d - 1"
       using assms(4) by fastforce 
     have "card (pd.neighborhood Ai) = card A - 1 - card (g Ai)" 
       using cardss assms(3) card_mono diff_add_inverse diff_diff_cancel diff_le_mono ss eq
       by (metis (no_types, lifting)) 
     moreover have "card A \<ge> (1 + card (g Ai))" using pss assms(3) card_seteq not_less_eq_eq by auto 
     ultimately have "card (pd.neighborhood Ai) = int (card A) - 1 - int (card (g Ai))"  by auto
     moreover have "int (card (g Ai)) \<ge> (card A) - (int d) - 1" using ggt a by simp
     ultimately show "out_degree ?G Ai \<le> d" using pd.out_degree_neighborhood by simp
   qed
 qed
 
 lemma obtain_dependency_graph:
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> 
     (\<exists> S . S \<subseteq> A - {Ai} \<and> card S \<ge> card A - d - 1 \<and> mutual_indep_events (F Ai) F S)"
   obtains G where "dependency_digraph G M F" "pverts G = A" "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d"
 proof -
   obtain g where gdef: "\<And> Ai. Ai \<in> A \<Longrightarrow> g Ai \<subseteq> A - {Ai} \<and> card (g Ai) \<ge> card A - d - 1 \<and> 
     mutual_indep_events (F Ai) F (g Ai)" using assms(4) by metis
   then show ?thesis 
     using define_dep_graph_set[of A F g] define_dep_graph_deg_bound[of A F g d]that assms by auto 
 qed
 
 text \<open>This is the variation of the symmetric version most commonly in use \<close>
 
 theorem lovasz_local_symmetric: 
   fixes d :: nat
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> (\<exists> S . S \<subseteq> A - {Ai} \<and> card S \<ge> card A - d - 1 \<and> mutual_indep_events (F Ai) F S)" 
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> p"
   assumes "exp(1)* p * (d + 1) \<le> 1"
   shows "prob (\<Inter> Ai \<in> A . (space M - (F Ai))) > 0"
 proof -
   obtain G where odg: "dependency_digraph G M F" "pverts G = A" "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d"
     using assms obtain_dependency_graph by metis
   then show ?thesis using odg assms lovasz_local_symmetric_dep_graph[of A F G d p] by auto
 qed
 
 lemma lovasz_local_symmetric4_set: 
   fixes d :: nat
   assumes "A \<noteq> {}"
   assumes "F ` A \<subseteq> events"
   assumes "finite A"
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> (\<exists> S . S \<subseteq> A - {Ai} \<and> card S \<ge> card A - d - 1 \<and> mutual_indep_events (F Ai) F S)" 
   assumes "\<And> Ai. Ai \<in> A \<Longrightarrow> prob (F Ai) \<le> p"
   assumes "4 * p * d \<le> 1"
   assumes "d \<ge> 1"
   shows "prob (\<Inter> Ai \<in> A . (space M - F Ai)) > 0"
 proof -
   obtain G where odg: "dependency_digraph G M F" "pverts G = A" "\<And> Ai.  Ai \<in> A \<Longrightarrow> out_degree G Ai \<le> d"
     using assms obtain_dependency_graph by metis
   then show ?thesis using odg assms lovasz_local_symmetric4[of A F G d p] by auto
 qed
 end
 
 end
\ No newline at end of file
diff --git a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Algorithm.thy b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Algorithm.thy
--- a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Algorithm.thy
+++ b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Algorithm.thy
@@ -1,381 +1,381 @@
 
 section \<open>Formalization of an efficient Hermite normal form algorithm\<close>
 
 text \<open>We formalize a version of the Hermite normal form algorithm based on reductions modulo
 the determinant. This avoids the growth of the intermediate coefficients.\<close>
 
 subsection \<open>Implementation of the algorithm using generic modulo operation\<close>
 
 text \<open>Exception on generic modulo: currently in Hermite-reduce-above, ordinary div/mod is used,
   since that is our choice for the complete set of residues.\<close>
 
 theory HNF_Mod_Det_Algorithm
   imports
     Jordan_Normal_Form.Gauss_Jordan_IArray_Impl
     Show.Show_Instances
     Jordan_Normal_Form.Determinant_Impl
     Jordan_Normal_Form.Show_Matrix
     LLL_Basis_Reduction.LLL_Certification   
     Smith_Normal_Form.SNF_Algorithm_Euclidean_Domain
     Smith_Normal_Form.SNF_Missing_Lemmas
     Uniqueness_Hermite_JNF
     Matrix_Change_Row
 begin
 
 subsubsection \<open>Echelon form algorithm\<close>
 
 fun make_first_column_positive :: "int mat \<Rightarrow> int mat" where
   "make_first_column_positive A = (
        Matrix.mat (dim_row A) (dim_col A) \<comment> \<open> Create a matrix of the same dimensions \<close>
           (\<lambda>(i,j). if A $$(i,0) < 0 then - A $$(i,j) else A $$(i,j)
             )
   )"
 
 
 locale mod_operation =
   fixes generic_mod :: "int \<Rightarrow> int \<Rightarrow> int" (infixl "gmod" 70)
     and generic_div :: "int \<Rightarrow> int \<Rightarrow> int" (infixl "gdiv" 70)
 begin
 
 text \<open>Version for reducing all elements\<close>
 
 fun reduce :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat" where
   "reduce a b D A = (let Aaj = A$$(a,0); Abj = A $$ (b,0)     
   in
   if Aaj = 0 then A else
   case euclid_ext2 Aaj Abj of (p,q,u,v,d) \<Rightarrow> \<comment> \<open> p*Aaj + q * Abj = d, u = - Abj/d, v = Aaj/d \<close>
        Matrix.mat (dim_row A) (dim_col A) \<comment> \<open> Create a matrix of the same dimensions \<close>
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                               if k = 0 then if D dvd r then D else r else r gmod D \<comment> \<open> Row a is multiplied by p and added row b multiplied by q, modulo D\<close>
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                                if k = 0 then r else r gmod D \<comment> \<open> Row b is multiplied by v and added row a multiplied by u, modulo D\<close>
                    else A$$(i,k)  \<comment> \<open> All the other rows remain unchanged\<close>
             )
   )"
 
 text \<open>Version for reducing, with abs-checking\<close>
 
 fun reduce_abs :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat" where
   "reduce_abs a b D A = (let Aaj = A$$(a,0); Abj = A $$ (b,0)     
   in
   if Aaj = 0 then A else
   case euclid_ext2 Aaj Abj of (p,q,u,v,d) \<Rightarrow> \<comment> \<open> p*Aaj + q * Abj = d, u = - Abj/d, v = Aaj/d \<close>
        Matrix.mat (dim_row A) (dim_col A) \<comment> \<open> Create a matrix of the same dimensions \<close>
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                             if abs r > D then if k = 0 \<and> D dvd r then D else r gmod D else r 
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                               if abs r > D then r gmod D else r
                    else A$$(i,k)  \<comment> \<open> All the other rows remain unchanged\<close>
             )
   )"
 
 definition reduce_impl :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat" where
   "reduce_impl a b D A = (let 
     row_a = Matrix.row A a; 
     Aaj = row_a $v 0     
   in
   if Aaj = 0 then A else let 
     row_b = Matrix.row A b;
     Abj = row_b $v 0 in
   case euclid_ext2 Aaj Abj of (p,q,u,v,d) \<Rightarrow> 
     let row_a' = (\<lambda> k ak. let r = (p * ak + q * row_b $v k) in
               if k = 0 then if D dvd r then D else r else r gmod D);
         row_b' = (\<lambda> k bk. let r = u * row_a $v k + v * bk in
                                if k = 0 then r else r gmod D)
      in change_row a row_a' (change_row b row_b' A)
   )"
 
 definition reduce_abs_impl :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat" where
   "reduce_abs_impl a b D A = (let 
     row_a = Matrix.row A a; 
     Aaj = row_a $v 0     
   in
   if Aaj = 0 then A else let 
     row_b = Matrix.row A b;
     Abj = row_b $v 0 in
   case euclid_ext2 Aaj Abj of (p,q,u,v,d) \<Rightarrow> 
     let row_a' = (\<lambda> k ak. let r = (p * ak + q * row_b $v k) in
               if abs r > D then if k = 0 \<and> D dvd r then D else r gmod D else r);
         row_b' = (\<lambda> k bk. let r = u * row_a $v k + v * bk in
                                 if abs r > D then r gmod D else r)
      in change_row a row_a' (change_row b row_b' A)
   )"
 
 lemma reduce_impl: "a < nr \<Longrightarrow> b < nr \<Longrightarrow> 0 < nc \<Longrightarrow> a \<noteq> b \<Longrightarrow> A \<in> carrier_mat nr nc
   \<Longrightarrow> reduce_impl a b D A = reduce a b D A" 
   unfolding reduce_impl_def reduce.simps Let_def
   apply (intro if_cong[OF _ refl], force)
   apply (intro prod.case_cong refl, force)
   apply (intro eq_matI, auto)
   done
 
 
 lemma reduce_abs_impl: "a < nr \<Longrightarrow> b < nr \<Longrightarrow> 0 < nc \<Longrightarrow> a \<noteq> b \<Longrightarrow> A \<in> carrier_mat nr nc
   \<Longrightarrow> reduce_abs_impl a b D A = reduce_abs a b D A" 
   unfolding reduce_abs_impl_def reduce_abs.simps Let_def
   apply (intro if_cong[OF _ refl], force)
   apply (intro prod.case_cong refl, force)
   apply (intro eq_matI, auto)
   done
   
 
 (* This functions reduce the elements below the position (a,0), given a list of positions 
    of non-zero positions as input*)
 fun reduce_below :: "nat \<Rightarrow> nat list \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat"
 where "reduce_below a [] D A = A"
   | "reduce_below a (x # xs) D A = reduce_below a xs D (reduce a x D A)"
 
 fun reduce_below_impl :: "nat \<Rightarrow> nat list \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat"
 where "reduce_below_impl a [] D A = A"
   | "reduce_below_impl a (x # xs) D A = reduce_below_impl a xs D (reduce_impl a x D A)"
 
 lemma reduce_impl_carrier[simp,intro]: "A \<in> carrier_mat m n \<Longrightarrow> reduce_impl a b D A \<in> carrier_mat m n" 
   unfolding reduce_impl_def Let_def by (auto split: prod.splits)
 
 lemma reduce_below_impl: "a < nr \<Longrightarrow> 0 < nc \<Longrightarrow> (\<And> b. b \<in> set bs \<Longrightarrow> b < nr) \<Longrightarrow> a \<notin> set bs 
   \<Longrightarrow> A \<in> carrier_mat nr nc \<Longrightarrow> reduce_below_impl a bs D A = reduce_below a bs D A" 
 proof (induct bs arbitrary: A)
   case (Cons b bs A)
   show ?case by (simp del: reduce.simps, 
       subst reduce_impl[of _ nr _ nc], 
       (insert Cons, auto simp del: reduce.simps)[5],
       rule Cons(1), insert Cons(2-), auto simp: Let_def split: prod.splits)
 qed simp
 
 
 
 fun reduce_below_abs :: "nat \<Rightarrow> nat list \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat"
 where "reduce_below_abs a [] D A = A"
   | "reduce_below_abs a (x # xs) D A = reduce_below_abs a xs D (reduce_abs a x D A)"
 
 fun reduce_below_abs_impl :: "nat \<Rightarrow> nat list \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat"
 where "reduce_below_abs_impl a [] D A = A"
   | "reduce_below_abs_impl a (x # xs) D A = reduce_below_abs_impl a xs D (reduce_abs_impl a x D A)"
 
 lemma reduce_abs_impl_carrier[simp,intro]: "A \<in> carrier_mat m n \<Longrightarrow> reduce_abs_impl a b D A \<in> carrier_mat m n" 
   unfolding reduce_abs_impl_def Let_def by (auto split: prod.splits)
 
 lemma reduce_abs_below_impl: "a < nr \<Longrightarrow> 0 < nc \<Longrightarrow> (\<And> b. b \<in> set bs \<Longrightarrow> b < nr) \<Longrightarrow> a \<notin> set bs 
   \<Longrightarrow> A \<in> carrier_mat nr nc \<Longrightarrow> reduce_below_abs_impl a bs D A = reduce_below_abs a bs D A" 
 proof (induct bs arbitrary: A)
   case (Cons b bs A)
   show ?case by (simp del: reduce_abs.simps, 
       subst reduce_abs_impl[of _ nr _ nc], 
       (insert Cons, auto simp del: reduce_abs.simps)[5],
       rule Cons(1), insert Cons(2-), auto simp: Let_def split: prod.splits)
 qed simp
 
 text \<open>This function outputs a matrix in echelon form via reductions modulo the determinant\<close>
 
 
 function FindPreHNF :: "bool \<Rightarrow> int \<Rightarrow> int mat \<Rightarrow> int mat"
   where "FindPreHNF abs_flag D A = 
   (let m = dim_row A; n = dim_col A in 
   if m < 2 \<or> n = 0 then A else \<comment> \<open> No operations are carried out if m = 1 \<close>
   let non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A];
          A' = (if A$$(0,0) \<noteq> 0 then A
               else let i = non_zero_positions ! 0  \<comment> \<open> Select the first non-zero position below the first element\<close>
                    in swaprows 0 i A  
               );
         Reduce = (if abs_flag then reduce_below_abs else reduce_below)
     in
       if n < 2 then Reduce 0 non_zero_positions D A'  \<comment> \<open> If n = 1, then we have to reduce the column \<close>   
     else 
       let         
         (A_UL,A_UR,A_DL,A_DR) = split_block (Reduce 0 non_zero_positions D (make_first_column_positive A')) 1 1; 
         sub_PreHNF = FindPreHNF abs_flag D A_DR in       
         four_block_mat A_UL A_UR A_DL sub_PreHNF)"                 
   by pat_completeness auto 
 
 termination
 proof (relation "Wellfounded.measure (\<lambda>(abs_flag,D,A). dim_col A)")
   show "wf (Wellfounded.measure (\<lambda>(abs_flag,D, A). dim_col A))" by auto
   fix abs_flag D A m n nz A' R xd A'_UL y A'_UR ya A'_DL A'_DR
   assume m: "m = dim_row A" and n:"n = dim_col A"
     and m2: "\<not> (m < 2 \<or> n = 0)" and nz_def: "nz = filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A] "
     and A'_def: "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = nz ! 0 in swaprows 0 i A)"
     and R_def: "R = (if abs_flag then reduce_below_abs else reduce_below)"
     and n2: "\<not> n < 2" and "xd = split_block (R 0 nz D (make_first_column_positive A')) 1 1"
     and "(A'_UL, y) = xd" and "(A'_UR, ya) = y" and "(A'_DL, A'_DR) = ya" 
   hence A'_split: "(A'_UL, A'_UR, A'_DL, A'_DR) 
         = split_block (R 0 nz D (make_first_column_positive A')) 1 1" by force
   have dr_mk1: "dim_row (make_first_column_positive A) = dim_row A" for A by auto
   have dr_mk2: "dim_col (make_first_column_positive A) = dim_col A" for A by auto  
   have r1: "reduce_below a xs D A \<in> carrier_mat m n" if "A \<in> carrier_mat m n" for A a xs
     using that by (induct a xs D A rule: reduce_below.induct, auto simp add: Let_def euclid_ext2_def) 
   hence R: "(reduce_below 0 nz D (make_first_column_positive A')) \<in> carrier_mat m n"
     using A'_def m n 
     by (metis carrier_matI index_mat_swaprows(2,3) dr_mk1 dr_mk2)
   have "reduce_below_abs a xs D A \<in> carrier_mat m n" if "A \<in> carrier_mat m n" for A a xs
     using that by (induct a xs D A rule: reduce_below_abs.induct, auto simp add: Let_def euclid_ext2_def) 
   hence R2: "(reduce_below_abs 0 nz D (make_first_column_positive A')) \<in> carrier_mat m n"
     using A'_def m n 
     by (metis carrier_matI index_mat_swaprows(2,3) dr_mk1 dr_mk2)
  
   have "A'_DR \<in> carrier_mat (m-1) (n-1)"
     by (cases abs_flag; rule split_block(4)[OF A'_split[symmetric]],insert m2 n2 m n R_def R R2, auto)
   thus "((abs_flag, D, A'_DR),abs_flag, D, A) \<in> Wellfounded.measure (\<lambda>(abs_flag,D, A). dim_col A)" using n2 m2 n m by auto
 qed
 
 lemma FindPreHNF_code: "FindPreHNF abs_flag D A = 
   (let m = dim_row A; n = dim_col A in 
   if m < 2 \<or> n = 0 then A else 
   let non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A];
          A' = (if A$$(0,0) \<noteq> 0 then A
               else let i = non_zero_positions ! 0 in swaprows 0 i A  
               );
          Reduce_impl = (if abs_flag then reduce_below_abs_impl else reduce_below_impl)
     in
       if n < 2 then Reduce_impl 0 non_zero_positions D A'   
     else 
       let         
         (A_UL,A_UR,A_DL,A_DR) = split_block (Reduce_impl 0 non_zero_positions D (make_first_column_positive A')) 1 1; 
         sub_PreHNF = FindPreHNF abs_flag D A_DR in       
         four_block_mat A_UL A_UR A_DL sub_PreHNF)"  (is "?lhs = ?rhs")
 proof -
   let ?f = "\<lambda>R. (if dim_row A < 2 \<or> dim_col A = 0 then A else if dim_col A < 2
           then R 0 (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A]) D
   (if A $$ (0, 0) \<noteq> 0 then A else swaprows 0 (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A] ! 0) A)
   else case split_block (R 0 (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A]) D
     (make_first_column_positive (if A $$ (0, 0) \<noteq> 0 then A else 
       swaprows 0 (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A] ! 0) A))) 1 1 of
   (A_UL, A_UR, A_DL, A_DR) \<Rightarrow> four_block_mat A_UL A_UR A_DL (FindPreHNF abs_flag D A_DR))"
   have M_carrier: "make_first_column_positive (if A $$ (0, 0) \<noteq> 0 then A 
     else swaprows 0 (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A] ! 0) A) 
     \<in> carrier_mat (dim_row A) (dim_col A)"
-    by (smt (z3) index_mat_swaprows(2) index_mat_swaprows(3) make_first_column_positive.simps mat_carrier)
+    by (smt (verit) index_mat_swaprows(2) index_mat_swaprows(3) make_first_column_positive.simps mat_carrier)
   have *: "0 \<notin> set (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A])" by simp
   have "?lhs = ?f (if abs_flag then reduce_below_abs else reduce_below)"
     unfolding FindPreHNF.simps[of abs_flag D A] Let_def by presburger
   also have "... = ?rhs"
   proof (cases abs_flag)
     case True
     have "?f (if abs_flag then reduce_below_abs else reduce_below) = ?f reduce_below_abs" 
       using True by presburger
     also have "... = ?f reduce_below_abs_impl" 
       by ((intro if_cong refl prod.case_cong arg_cong[of _ _ "\<lambda> x. split_block x 1 1"];
        (subst reduce_abs_below_impl[where nr = "dim_row A" and nc = "dim_col A"])), (auto)[9])
        (insert M_carrier *, blast+)
     also have "... = ?f (if abs_flag then reduce_below_abs_impl else reduce_below_impl)" 
       using True by presburger
     finally show ?thesis using True unfolding FindPreHNF.simps[of abs_flag D A] Let_def by blast
   next
     case False
     have "?f (if abs_flag then reduce_below_abs else reduce_below) = ?f reduce_below" 
       using False by presburger
     also have "... = ?f reduce_below_impl" 
       by ((intro if_cong refl prod.case_cong arg_cong[of _ _ "\<lambda> x. split_block x 1 1"];
        (subst reduce_below_impl[where nr = "dim_row A" and nc = "dim_col A"])), (auto)[9])
        (insert M_carrier *, blast+)
     also have "... = ?f (if abs_flag then reduce_below_abs_impl else reduce_below_impl)" 
       using False by presburger
     finally show ?thesis using False unfolding FindPreHNF.simps[of abs_flag D A] Let_def by blast
   qed
   finally show ?thesis by blast
 qed
 end
 
 declare mod_operation.FindPreHNF_code[code]
 declare mod_operation.reduce_below_impl.simps[code]
 declare mod_operation.reduce_impl_def[code]
 declare mod_operation.reduce_below_abs_impl.simps[code]
 declare mod_operation.reduce_abs_impl_def[code]
 
 subsubsection \<open>From echelon form to Hermite normal form\<close>
 
 text \<open>From here on, we define functions to transform a matrix in echelon form into its Hermite
 normal form. Essentially, we are defining the functions that are available in the AFP entry Hermite
 (which uses HOL Analysis + mod-type) in the JNF matrix representation.\<close>
 
 (*Find the first nonzero element of row l (A is upper triangular)*)
 definition find_fst_non0_in_row :: "nat \<Rightarrow> int mat \<Rightarrow> nat option" where
   "find_fst_non0_in_row l A = (let is = [l ..< dim_col A];
     Ais = filter (\<lambda>j. A $$ (l, j) \<noteq> 0) is
     in case Ais of [] \<Rightarrow> None | _ \<Rightarrow> Some (Ais!0))"
 
 primrec Hermite_reduce_above
 where "Hermite_reduce_above (A::int mat) 0 i j = A"
     | "Hermite_reduce_above A (Suc n) i j = (let
     Aij = A $$ (i,j);
     Anj = A $$ (n,j)
     in 
     Hermite_reduce_above (addrow (- (Anj div Aij)) n i A) n i j)"
 
 definition Hermite_of_row_i :: "int mat \<Rightarrow> nat \<Rightarrow> int mat" 
   where "Hermite_of_row_i A i = (
   case find_fst_non0_in_row i A of None \<Rightarrow> A | Some j \<Rightarrow> 
     let Aij = A $$(i,j) in
     if Aij < 0 then Hermite_reduce_above (multrow i (-1) A) i i j
     else Hermite_reduce_above A i i j)"
 
 
 primrec Hermite_of_list_of_rows 
   where
  "Hermite_of_list_of_rows A [] = A" | 
  "Hermite_of_list_of_rows A (a#xs) = Hermite_of_list_of_rows (Hermite_of_row_i A a) xs"
 
 text \<open>We combine the previous functions to assemble the algorithm\<close>
 
 definition (in mod_operation) "Hermite_mod_det abs_flag A = 
   (let m = dim_row A; n = dim_col A; 
    D = abs(det_int A); 
    A' = A @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n;
    E = FindPreHNF abs_flag D A';
    H = Hermite_of_list_of_rows E [0..<m+n]
   in mat_of_rows n (map (Matrix.row H) [0..<m]))"
 
 subsubsection \<open>Some examples of execution\<close>
 
 declare mod_operation.Hermite_mod_det_def[code]
 
 value "let B = mat_of_rows_list 4 ([[0,3,1,4],[7,1,0,0],[8,0,19,16],[2,0,0,3::int]]) in
   show (mod_operation.Hermite_mod_det (mod) True B)"
 
 (*
 sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
 sage: A = matrix(ZZ, [[0,3,1,4],[7,1,0,0],[8,0,19,16],[2,0,0,3]])
 sage: A
 [ 0  3  1  4]
 [ 7  1  0  0]
 [ 8  0 19 16]
 [ 2  0  0  3]
 sage:  H, U = matrix_integer_dense_hnf.hnf_with_transformation(A); H
 [   1    0    0  672]
 [   0    1    0  660]
 [   0    0    1  706]
 [   0    0    0 1341]
 sage: 
 *)
 
 
 value "let B = mat_of_rows_list 7 ([
 [  1,  17, -41,  -1,   1,   0,  0],
 [  0,  -1,   2,   0,  -6,   2,   1],
 [  9,   2,   1,   1,  -2,   2,  -5],
 [ -1,  -3,  -1,   0,  -9,   0,   0],
 [  9,  -1,  -9,   0,   0,   0,   1],
 [  1,  -1,   1,   0,   1,  -8,   0],
 [  1,  -1,   0,  -2,  -1,  -1,   0::int]]) in 
   show (mod_operation.Hermite_mod_det (mod) True B)"
 
 (*
 sage: import sage.matrix.matrix_integer_dense_hnf as matrix_integer_dense_hnf
 sage: A = random_matrix(ZZ,7,7); A
 [  1  17 -41  -1   1   0   0]
 [  0  -1   2   0  -6   2   1]
 [  9   2   1   1  -2   2  -5]
 [ -1  -3  -1   0  -9   0   0]
 [  9  -1  -9   0   0   0   1]
 [  1  -1   1   0   1  -8   0]
 [  1  -1   0  -2  -1  -1   0]
 sage: H, U = matrix_integer_dense_hnf.hnf_with_transformation(A); H
 [     1      0      0      0      0      1 191934]
 [     0      1      0      0      0      0 435767]
 [     0      0      1      0      0      1 331950]
 [     0      0      0      1      0      0 185641]
 [     0      0      0      0      1      0  38022]
 [     0      0      0      0      0      2 477471]
 [     0      0      0      0      0      0 565304]
 *)
 
 end
\ No newline at end of file
diff --git a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Soundness.thy b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Soundness.thy
--- a/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Soundness.thy
+++ b/thys/Modular_arithmetic_LLL_and_HNF_algorithms/HNF_Mod_Det_Soundness.thy
@@ -1,11568 +1,11572 @@
 
 subsection \<open>Soundness of the algorithm\<close>
 
 theory HNF_Mod_Det_Soundness
   imports
     HNF_Mod_Det_Algorithm
     Signed_Modulo
 begin
 
 hide_const(open) Determinants.det Determinants2.upper_triangular
   Finite_Cartesian_Product.row Finite_Cartesian_Product.rows
   Finite_Cartesian_Product.vec
 
 subsubsection \<open>Results connecting lattices and Hermite normal form\<close>
 
 text \<open>The following results will also be useful for proving the soundness of the certification 
 approach.\<close>
 
 lemma of_int_mat_hom_int_id[simp]:
   fixes A::"int mat"
   shows "of_int_hom.mat_hom A = A" unfolding map_mat_def by auto
 
 
 definition "is_sound_HNF algorithm associates res 
     = (\<forall>A. let (P,H) = algorithm A; m = dim_row A; n = dim_col A in 
         P \<in> carrier_mat m m \<and> H \<in> carrier_mat m n \<and> invertible_mat P \<and> A = P * H 
         \<and> Hermite_JNF associates res H)"
 
 lemma HNF_A_eq_HNF_PA:
   fixes A::"'a::{bezout_ring_div,normalization_euclidean_semiring,unique_euclidean_ring} mat"
   assumes A: "A \<in> carrier_mat n n" and inv_A: "invertible_mat A" 
     and inv_P: "invertible_mat P" and P: "P \<in> carrier_mat n n"
     and sound_HNF: "is_sound_HNF HNF associates res"
     and P1_H1: "(P1,H1) = HNF (P*A)"
     and P2_H2: "(P2,H2) = HNF A"
   shows "H1 = H2"
 proof -
   obtain inv_P where P_inv_P: "inverts_mat P inv_P" and inv_P_P: "inverts_mat inv_P P"
     and inv_P: "inv_P \<in> carrier_mat n n"
     using P inv_P obtain_inverse_matrix by blast
   have P1: "P1 \<in> carrier_mat n n"
       using P1_H1 sound_HNF unfolding is_sound_HNF_def Let_def
       by (metis (no_types, lifting) P carrier_matD(1) index_mult_mat(2) old.prod.case)
     have H1: "H1 \<in> carrier_mat n n" using P1_H1 sound_HNF unfolding is_sound_HNF_def Let_def
   by (metis (no_types, lifting) A P carrier_matD(1) carrier_matD(2) case_prodD index_mult_mat(2,3))
   have invertible_inv_P: "invertible_mat inv_P"
       using P_inv_P inv_P inv_P_P invertible_mat_def square_mat.simps by blast
   have P_A_P1_H1: "P * A = P1 * H1" using P1_H1 sound_HNF unfolding is_sound_HNF_def Let_def
     by (metis (mono_tags, lifting) case_prod_conv)
   hence "A = inv_P * (P1 * H1)"
-    by (smt A P inv_P_P inv_P assoc_mult_mat carrier_matD(1) inverts_mat_def left_mult_one_mat)
+    by (smt (verit) A P inv_P_P inv_P assoc_mult_mat carrier_matD(1) inverts_mat_def left_mult_one_mat)
   hence A_inv_P_P1_H1: "A = (inv_P * P1) * H1"
-    by (smt P P1_H1 assoc_mult_mat carrier_matD(1) fst_conv index_mult_mat(2) inv_P 
-        is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
+    using H1 P1 inv_P by fastforce
   have A_P2_H2: "A = P2 * H2" using P2_H2 sound_HNF unfolding is_sound_HNF_def Let_def
     by (metis (mono_tags, lifting) case_prod_conv)
   have invertible_inv_P_P1: "invertible_mat (inv_P * P1)"
   proof (rule invertible_mult_JNF[OF inv_P P1 invertible_inv_P])   
     show "invertible_mat P1"
-      by (smt P1_H1 is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
+      by (smt (verit) P1_H1 is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
   qed
   show ?thesis
   proof (rule Hermite_unique_JNF[OF A _ H1 _ _ A_inv_P_P1_H1 A_P2_H2 inv_A invertible_inv_P_P1])
     show "inv_P * P1 \<in> carrier_mat n n"
       by (metis carrier_matD(1) carrier_matI index_mult_mat(2) inv_P
           invertible_inv_P_P1 invertible_mat_def square_mat.simps)
     show "P2 \<in> carrier_mat n n" 
-      by (smt A P2_H2 carrier_matD(1) is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
+      by (smt (verit) A P2_H2 carrier_matD(1) is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
     show "H2 \<in> carrier_mat n n"
-      by (smt A P2_H2 carrier_matD(1) carrier_matD(2) is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
+      by (smt (verit) A P2_H2 carrier_matD(1) carrier_matD(2) is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
     show "invertible_mat P2"
-      by (smt P2_H2 is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
+      by (smt (verit) P2_H2 is_sound_HNF_def prod.sel(1) sound_HNF split_beta)
     show "Hermite_JNF associates res H1" 
-      by (smt P1_H1 is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
+      by (smt (verit) P1_H1 is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
     show "Hermite_JNF associates res H2"
-      by (smt P2_H2 is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
+      by (smt (verit) P2_H2 is_sound_HNF_def prod.sel(2) sound_HNF split_beta)
   qed
 qed
 
 
 context vec_module
 begin
 
 lemma mat_mult_invertible_lattice_eq: 
   assumes fs: "set fs \<subseteq> carrier_vec n"
   and gs: "set gs \<subseteq> carrier_vec n"  
   and P: "P \<in> carrier_mat m m" and invertible_P: "invertible_mat P"
   and length_fs: "length fs = m" and length_gs: "length gs = m"
   and prod: "mat_of_rows n fs = (map_mat of_int P) * mat_of_rows n gs" 
   shows "lattice_of fs = lattice_of gs" 
 proof thm mat_mult_sub_lattice
   show "lattice_of fs \<subseteq> lattice_of gs"
     by (rule mat_mult_sub_lattice[OF fs gs _ prod],simp add: length_fs length_gs P)
 next
   obtain inv_P where P_inv_P: "inverts_mat P inv_P" and inv_P_P: "inverts_mat inv_P P"
     and inv_P: "inv_P \<in> carrier_mat m m"
     using P invertible_P obtain_inverse_matrix by blast
   have "of_int_hom.mat_hom (inv_P) * mat_of_rows n fs 
       = of_int_hom.mat_hom (inv_P) * ((map_mat of_int P) * mat_of_rows n gs)" 
     using prod by auto
   also have "... = of_int_hom.mat_hom (inv_P) * (map_mat of_int P) * mat_of_rows n gs"
-    by (smt P assoc_mult_mat inv_P length_gs map_carrier_mat mat_of_rows_carrier(1))
+    by (smt (verit) P assoc_mult_mat inv_P length_gs map_carrier_mat mat_of_rows_carrier(1))
   also have "... = of_int_hom.mat_hom (inv_P * P) * mat_of_rows n gs"
     by (metis P inv_P of_int_hom.mat_hom_mult)
   also have "... = mat_of_rows n gs"
     by (metis carrier_matD(1) inv_P inv_P_P inverts_mat_def left_mult_one_mat' 
         length_gs mat_of_rows_carrier(2) of_int_hom.mat_hom_one)    
   finally have prod: "mat_of_rows n gs = of_int_hom.mat_hom (inv_P) * mat_of_rows n fs" ..
   show "lattice_of gs \<subseteq> lattice_of fs"
     by (rule mat_mult_sub_lattice[OF gs fs _ prod], simp add: length_fs length_gs inv_P)
 qed                     
 
 end
 
 
 context
   fixes n :: nat
 begin
 
 interpretation vec_module "TYPE(int)" .
 
 lemma lattice_of_HNF:
   assumes sound_HNF: "is_sound_HNF HNF associates res"
     and P1_H1: "(P,H) = HNF (mat_of_rows n fs)"
     and fs: "set fs \<subseteq> carrier_vec n" and len: "length fs = m"
   shows "lattice_of fs = lattice_of (rows H)"
 proof (rule mat_mult_invertible_lattice_eq[OF fs])
   have H: "H \<in> carrier_mat m n" using sound_HNF P1_H1 unfolding is_sound_HNF_def Let_def
     by (metis (mono_tags, lifting) assms(4) mat_of_rows_carrier(2) mat_of_rows_carrier(3) prod.sel(2) split_beta)
   have H_rw: "mat_of_rows n (Matrix.rows H) = H" using mat_of_rows_rows H by fast
   have PH_fs_init: "mat_of_rows n fs = P * H" using sound_HNF P1_H1 unfolding is_sound_HNF_def Let_def
     by (metis (mono_tags, lifting) case_prodD)
   show "mat_of_rows n fs = of_int_hom.mat_hom P * mat_of_rows n (Matrix.rows H)"
     unfolding H_rw of_int_mat_hom_int_id using PH_fs_init by simp  
   show "set (Matrix.rows H) \<subseteq> carrier_vec n" using H rows_carrier by blast
   show "P \<in> carrier_mat m m" using sound_HNF P1_H1 unfolding is_sound_HNF_def Let_def
     by (metis (no_types, lifting) len case_prodD mat_of_rows_carrier(2))    
   show "invertible_mat P" using sound_HNF P1_H1 unfolding is_sound_HNF_def Let_def
     by (metis (no_types, lifting) case_prodD)
   show "length fs = m" using len by simp
   show "length (Matrix.rows H) = m" using H by auto
 qed
 end
 
 
 context LLL_with_assms 
 begin            
 
 (*For this proof, it seems that is not necessary fs_init to be a list of independent vectors. 
 The context assumes it, though.*)
 lemma certification_via_eq_HNF:
   assumes sound_HNF: "is_sound_HNF HNF associates res"
     and P1_H1: "(P1,H1) = HNF (mat_of_rows n fs_init)"
     and P2_H2: "(P2,H2) = HNF (mat_of_rows n gs)"
     and H1_H2: "H1 = H2" (*The HNF are equal*)
     and gs: "set gs \<subseteq> carrier_vec n" and len_gs: "length gs = m"
   shows "lattice_of gs = lattice_of fs_init" "LLL_with_assms n m gs \<alpha>"
 proof -                                           
   have "lattice_of fs_init = lattice_of (rows H1)"
     by (rule lattice_of_HNF[OF sound_HNF P1_H1 fs_init], simp add: len)
   also have "... = lattice_of (rows H2)" using H1_H2 by auto
   also have "... = lattice_of gs" 
     by (rule lattice_of_HNF[symmetric, OF sound_HNF P2_H2 gs len_gs])
   finally show "lattice_of gs = lattice_of fs_init" ..
     have invertible_P1: "invertible_mat P1" 
       using sound_HNF P1_H1 unfolding is_sound_HNF_def
       by (metis (mono_tags, lifting) case_prodD)
   have invertible_P2: "invertible_mat P2"
       using sound_HNF P2_H2 unfolding is_sound_HNF_def
       by (metis (mono_tags, lifting) case_prodD)
     have P2: "P2 \<in> carrier_mat m m" 
       using sound_HNF P2_H2 unfolding is_sound_HNF_def
       by (metis (no_types, lifting) len_gs case_prodD mat_of_rows_carrier(2))
     obtain inv_P2 where P2_inv_P2: "inverts_mat P2 inv_P2" and inv_P2_P2: "inverts_mat inv_P2 P2"
     and inv_P2: "inv_P2 \<in> carrier_mat m m"
       using P2 invertible_P2 obtain_inverse_matrix by blast
     have P1: "P1 \<in> carrier_mat m m" 
       using sound_HNF P1_H1 unfolding is_sound_HNF_def
       by (metis (no_types, lifting) len case_prodD mat_of_rows_carrier(2))
     have H1: "H1 \<in> carrier_mat m n" 
       using sound_HNF P1_H1 unfolding is_sound_HNF_def
       by (metis (no_types, lifting) case_prodD len mat_of_rows_carrier(2) mat_of_rows_carrier(3))
     have H2: "H2 \<in> carrier_mat m n" 
       using sound_HNF P2_H2 unfolding is_sound_HNF_def
       by (metis (no_types, lifting) len_gs case_prodD mat_of_rows_carrier(2) mat_of_rows_carrier(3))
     have P2_H2: "P2 * H2 = mat_of_rows n gs"
-      by (smt P2_H2 sound_HNF case_prodD is_sound_HNF_def)
+      by (smt (verit) P2_H2 sound_HNF case_prodD is_sound_HNF_def)
     have P1_H1_fs: "P1 * H1 = mat_of_rows n fs_init"
-      by (smt P1_H1 sound_HNF case_prodD is_sound_HNF_def)
+      by (smt (verit) P1_H1 sound_HNF case_prodD is_sound_HNF_def)
     obtain inv_P1 where P1_inv_P1: "inverts_mat P1 inv_P1" and inv_P1_P1: "inverts_mat inv_P1 P1"
     and inv_P1: "inv_P1 \<in> carrier_mat m m"
       using P1 invertible_P1 obtain_inverse_matrix by blast
   show "LLL_with_assms n m gs \<alpha>"
   proof (rule LLL_change_basis(2)[OF gs len_gs])
     show "P1 * inv_P2 \<in> carrier_mat m m" using P1 inv_P2 by auto
     have "mat_of_rows n fs_init = P1 * H1" using sound_HNF P2_H2 unfolding is_sound_HNF_def
       by (metis (mono_tags, lifting) P1_H1 case_prodD)
     also have "... = P1 * inv_P2 * P2 * H1"
-      by (smt P1 P2 assoc_mult_mat carrier_matD(1) inv_P2 inv_P2_P2 inverts_mat_def right_mult_one_mat)
+      by (smt (verit) P1 P2 assoc_mult_mat carrier_matD(1) inv_P2 inv_P2_P2 inverts_mat_def right_mult_one_mat)
     also have "... = P1 * inv_P2 * P2 * H2" using H1_H2 by blast
     also have "... = P1 * inv_P2 * (P2 * H2)" 
       using H2 P2 \<open>P1 * inv_P2 \<in> carrier_mat m m\<close> assoc_mult_mat by blast
     also have "... = P1 * (inv_P2 * P2 * H2)"
       by (metis H2 \<open>P1 * H1 = P1 * inv_P2 * P2 * H1\<close> \<open>P1 * inv_P2 * P2 * H2 = P1 * inv_P2 * (P2 * H2)\<close> 
           H1_H2 carrier_matD(1) inv_P2 inv_P2_P2 inverts_mat_def left_mult_one_mat)
     also have "... = P1 * (inv_P2 * (P2 * H2))" using H2 P2 inv_P2 by auto
     also have "... =  P1 * inv_P2 * mat_of_rows n gs"
       using P2_H2 \<open>P1 * (inv_P2 * P2 * H2) = P1 * (inv_P2 * (P2 * H2))\<close> 
         \<open>P1 * inv_P2 * (P2 * H2) = P1 * (inv_P2 * P2 * H2)\<close> by auto
     finally show "mat_of_rows n fs_init = P1 * inv_P2 * mat_of_rows n gs" .
     show "P2 * inv_P1 \<in> carrier_mat m m" 
       using P2 inv_P1 by auto
     have "mat_of_rows n gs = P2 * H2" using sound_HNF P2_H2 unfolding is_sound_HNF_def by metis
     also have "... = P2 * inv_P1 * P1 * H2"
-      by (smt P1 P2 assoc_mult_mat carrier_matD(1) inv_P1 inv_P1_P1 inverts_mat_def right_mult_one_mat)
+      by (smt (verit) P1 P2 assoc_mult_mat carrier_matD(1) inv_P1 inv_P1_P1 inverts_mat_def right_mult_one_mat)
     also have "... = P2 * inv_P1 * P1 * H1" using H1_H2 by blast
     also have "... = P2 * inv_P1 * (P1 * H1)" 
       using H1 P1 \<open>P2 * inv_P1 \<in> carrier_mat m m\<close> assoc_mult_mat by blast
     also have "... = P2 * (inv_P1 * P1 * H1)"
       by (metis H2 \<open>P2 * H2 = P2 * inv_P1 * P1 * H2\<close> \<open>P2 * inv_P1 * P1 * H1 = P2 * inv_P1 * (P1 * H1)\<close> 
           H1_H2 carrier_matD(1) inv_P1 inv_P1_P1 inverts_mat_def left_mult_one_mat)
     also have "... = P2 * (inv_P1 * (P1 * H1))" using H1 P1 inv_P1 by auto
     also have "... =  P2 * inv_P1 * mat_of_rows n fs_init"
       using P1_H1_fs \<open>P2 * (inv_P1 * P1 * H1) = P2 * (inv_P1 * (P1 * H1))\<close> 
         \<open>P2 * inv_P1 * (P1 * H1) = P2 * (inv_P1 * P1 * H1)\<close> by auto
     finally show "mat_of_rows n gs = P2 * inv_P1 * mat_of_rows n fs_init" .
   qed
 qed
 
 end
 
 text \<open>Now, we need to generalize some lemmas.\<close>
 
 context vec_module
 begin
 
 (*Generalized version of thm vec_space.finsum_index, now in vec_module*)
 lemma finsum_index:
   assumes i: "i < n"
     and f: "f \<in> A \<rightarrow> carrier_vec n"
     and A: "A \<subseteq> carrier_vec n"
   shows "finsum V f A $ i = sum (\<lambda>x. f x $ i) A"
   using A f
 proof (induct A rule: infinite_finite_induct)
   case empty
     then show ?case using i by simp next
   case (insert x X)
     then have Xf: "finite X"
       and xX: "x \<notin> X"
       and x: "x \<in> carrier_vec n"
       and X: "X \<subseteq> carrier_vec n"
       and fx: "f x \<in> carrier_vec n"
       and f: "f \<in> X \<rightarrow> carrier_vec n" by auto
     have i2: "i < dim_vec (finsum V f X)"
       using i finsum_closed[OF f] by auto
     have ix: "i < dim_vec x" using x i by auto
     show ?case
       unfolding finsum_insert[OF Xf xX f fx]
       unfolding sum.insert[OF Xf xX]
       unfolding index_add_vec(1)[OF i2]
       using insert lincomb_def
       by auto
 qed (insert i, auto)
 
 (*Generalized version of thm vec_space.mat_of_rows_mult_as_finsum, now in vec_module*)
 lemma mat_of_rows_mult_as_finsum:
   assumes "v \<in> carrier_vec (length lst)" "\<And> i. i < length lst \<Longrightarrow> lst ! i \<in> carrier_vec n"
   defines "f l \<equiv> sum (\<lambda> i. if l = lst ! i then v $ i else 0) {0..<length lst}"
   shows mat_of_cols_mult_as_finsum:"mat_of_cols n lst *\<^sub>v v = lincomb f (set lst)"
 proof -
   from assms have "\<forall> i < length lst. lst ! i \<in> carrier_vec n" by blast
   note an = all_nth_imp_all_set[OF this] hence slc:"set lst \<subseteq> carrier_vec n" by auto
   hence dn [simp]:"\<And> x. x \<in> set lst \<Longrightarrow> dim_vec x = n" by auto
   have dl [simp]:"dim_vec (lincomb f (set lst)) = n" using an
     by (simp add: slc)
   show ?thesis proof
     show "dim_vec (mat_of_cols n lst *\<^sub>v v) = dim_vec (lincomb f (set lst))" using assms(1,2) by auto
     fix i assume i:"i < dim_vec (lincomb f (set lst))" hence i':"i < n" by auto
     with an have fcarr:"(\<lambda>v. f v \<cdot>\<^sub>v v) \<in> set lst \<rightarrow> carrier_vec n" by auto
     from i' have "(mat_of_cols n lst *\<^sub>v v) $ i = row (mat_of_cols n lst) i \<bullet> v" by auto
     also have "\<dots> = (\<Sum>ia = 0..<dim_vec v. lst ! ia $ i * v $ ia)"
       unfolding mat_of_cols_def row_def scalar_prod_def
       apply(rule sum.cong[OF refl]) using i an assms(1) by auto
     also have "\<dots> = (\<Sum>ia = 0..<length lst. lst ! ia $ i * v $ ia)" using assms(1) by auto
     also have "\<dots> = (\<Sum>x\<in>set lst. f x * x $ i)"
       unfolding f_def sum_distrib_right apply (subst sum.swap)
       apply(rule sum.cong[OF refl])
       unfolding if_distrib if_distribR mult_zero_left sum.delta[OF finite_set] by auto
     also have "\<dots> = (\<Sum>x\<in>set lst. (f x \<cdot>\<^sub>v x) $ i)"
       apply(rule sum.cong[OF refl],subst index_smult_vec) using i slc by auto
     also have "\<dots> = (\<Oplus>\<^bsub>V\<^esub>v\<in>set lst. f v \<cdot>\<^sub>v v) $ i" 
       unfolding finsum_index[OF i' fcarr slc] by auto
     finally show "(mat_of_cols n lst *\<^sub>v v) $ i = lincomb f (set lst) $ i"
       by (auto simp:lincomb_def)
   qed
 qed
 
 
 lemma lattice_of_altdef_lincomb:
   assumes "set fs \<subseteq> carrier_vec n"
   shows "lattice_of fs = {y. \<exists>f. lincomb (of_int \<circ> f) (set fs) = y}"
   unfolding lincomb_def lattice_of_altdef[OF assms] image_def by auto
 
 end
 
 context vec_module
 begin
 
 (*Generalized version of thm idom_vec.lincomb_as_lincomb_list, now in vec_module*)
 lemma lincomb_as_lincomb_list:
   fixes ws f
   assumes s: "set ws \<subseteq> carrier_vec n"
   shows "lincomb f (set ws) = lincomb_list (\<lambda>i. if \<exists>j<i. ws!i = ws!j then 0 else f (ws ! i)) ws"
   using assms
 proof (induct ws rule: rev_induct)
   case (snoc a ws)
   let ?f = "\<lambda>i. if \<exists>j<i. ws ! i = ws ! j then 0 else f (ws ! i)"
   let ?g = "\<lambda>i. (if \<exists>j<i. (ws @ [a]) ! i = (ws @ [a]) ! j then 0 else f ((ws @ [a]) ! i)) \<cdot>\<^sub>v (ws @ [a]) ! i"
   let ?g2= "(\<lambda>i. (if \<exists>j<i. ws ! i = ws ! j then 0 else f (ws ! i)) \<cdot>\<^sub>v ws ! i)"
   have [simp]: "\<And>v. v \<in> set ws \<Longrightarrow> v \<in> carrier_vec n" using snoc.prems(1) by auto
   then have ws: "set ws \<subseteq> carrier_vec n" by auto
   have hyp: "lincomb f (set ws) = lincomb_list ?f ws"
     by (intro snoc.hyps ws)  
   show ?case
   proof (cases "a\<in>set ws")
     case True    
     have g_length: "?g (length ws) = 0\<^sub>v n" using True
       by (auto, metis in_set_conv_nth nth_append)
     have "(map ?g [0..<length (ws @ [a])]) = (map ?g [0..<length ws]) @ [?g (length ws)]"
        by auto
     also have "... = (map ?g [0..<length ws]) @ [0\<^sub>v n]" using g_length by simp
     finally have map_rw: "(map ?g [0..<length (ws @ [a])]) = (map ?g [0..<length ws]) @ [0\<^sub>v n]" .
     have "M.sumlist (map ?g2 [0..<length ws]) = M.sumlist (map ?g [0..<length ws])"
       by (rule arg_cong[of _ _ "M.sumlist"], intro nth_equalityI, auto simp add: nth_append)
     also have "... =  M.sumlist (map ?g [0..<length ws]) + 0\<^sub>v n "
       by (metis M.r_zero calculation hyp lincomb_closed lincomb_list_def ws)
     also have "... = M.sumlist (map ?g [0..<length ws] @ [0\<^sub>v n])" 
       by (rule M.sumlist_snoc[symmetric], auto simp add: nth_append)
     finally have summlist_rw: "M.sumlist (map ?g2 [0..<length ws]) 
       = M.sumlist (map ?g [0..<length ws] @ [0\<^sub>v n])" .
     have "lincomb f (set (ws @ [a])) = lincomb f (set ws)" using True unfolding lincomb_def
       by (simp add: insert_absorb)
     thus ?thesis 
       unfolding hyp lincomb_list_def map_rw summlist_rw
       by auto
   next
     case False    
     have g_length: "?g (length ws) = f a \<cdot>\<^sub>v a" using False by (auto simp add: nth_append)
     have "(map ?g [0..<length (ws @ [a])]) = (map ?g [0..<length ws]) @ [?g (length ws)]"
        by auto
     also have "... = (map ?g [0..<length ws]) @ [(f a \<cdot>\<^sub>v a)]" using g_length by simp
     finally have map_rw: "(map ?g [0..<length (ws @ [a])]) = (map ?g [0..<length ws]) @ [(f a \<cdot>\<^sub>v a)]" .
     have summlist_rw: "M.sumlist (map ?g2 [0..<length ws]) = M.sumlist (map ?g [0..<length ws])"
       by (rule arg_cong[of _ _ "M.sumlist"], intro nth_equalityI, auto simp add: nth_append)
     have "lincomb f (set (ws @ [a])) = lincomb f (set (a # ws))" by auto
     also have "... = (\<Oplus>\<^bsub>V\<^esub>v\<in>set (a # ws). f v \<cdot>\<^sub>v v)" unfolding lincomb_def ..
     also have "... = (\<Oplus>\<^bsub>V\<^esub>v\<in> insert a (set ws). f v \<cdot>\<^sub>v v)" by simp    
     also have "... = (f a \<cdot>\<^sub>v a) + (\<Oplus>\<^bsub>V\<^esub>v\<in> (set ws). f v \<cdot>\<^sub>v v)"
     proof (rule finsum_insert)
       show "finite (set ws)" by auto
       show "a \<notin> set ws" using False by auto
       show "(\<lambda>v. f v \<cdot>\<^sub>v v) \<in> set ws \<rightarrow> carrier_vec n"
         using snoc.prems(1) by auto
       show "f a \<cdot>\<^sub>v a \<in> carrier_vec n" using snoc.prems by auto
     qed
     also have "... = (f a \<cdot>\<^sub>v a) + lincomb f (set ws)" unfolding lincomb_def ..
     also have "... = (f a \<cdot>\<^sub>v a) + lincomb_list ?f ws" using hyp by auto
     also have "... =  lincomb_list ?f ws  + (f a \<cdot>\<^sub>v a)"
       using M.add.m_comm lincomb_list_carrier snoc.prems by auto
     also have "... = lincomb_list (\<lambda>i. if \<exists>j<i. (ws @ [a]) ! i 
       = (ws @ [a]) ! j then 0 else f ((ws @ [a]) ! i)) (ws @ [a])" 
     proof (unfold lincomb_list_def map_rw summlist_rw, rule M.sumlist_snoc[symmetric])
       show "set (map ?g [0..<length ws]) \<subseteq> carrier_vec n" using snoc.prems
         by (auto simp add: nth_append)
       show "f a \<cdot>\<^sub>v a \<in> carrier_vec n"
         using snoc.prems by auto
     qed
     finally show ?thesis .
   qed
 qed auto
 end
 
 context 
 begin
 
 interpretation vec_module "TYPE(int)" .
 
 lemma lattice_of_cols_as_mat_mult:
   assumes A: "A \<in> carrier_mat n nc" (*Integer matrix*)
   shows "lattice_of (cols A) = {y\<in>carrier_vec (dim_row A). \<exists>x\<in>carrier_vec (dim_col A). A *\<^sub>v x = y}"
 proof -
   let ?ws = "cols A"
   have set_cols_in: "set (cols A) \<subseteq> carrier_vec n" using A unfolding cols_def by auto
   have "lincomb (of_int \<circ> f)(set  ?ws) \<in> carrier_vec (dim_row A)" for f 
     using lincomb_closed A
     by (metis (full_types) carrier_matD(1) cols_dim lincomb_closed)
   moreover have "\<exists>x\<in>carrier_vec (dim_col A). A *\<^sub>v x = lincomb (of_int \<circ> f) (set (cols A))" for f
   proof -    
     let ?g = "(\<lambda>v. of_int (f v))"
     let ?g' = "(\<lambda>i. if \<exists>j<i. ?ws ! i = ?ws ! j then 0 else ?g (?ws ! i))"           
     have "lincomb (of_int \<circ> f) (set (cols A)) = lincomb ?g (set ?ws)" unfolding o_def by auto
     also have "... = lincomb_list ?g' ?ws" 
       by (rule lincomb_as_lincomb_list[OF set_cols_in])
     also have "... = mat_of_cols n ?ws *\<^sub>v vec (length ?ws) ?g'" 
       by (rule lincomb_list_as_mat_mult, insert set_cols_in A, auto)
     also have "... = A *\<^sub>v (vec (length ?ws) ?g')" using mat_of_cols_cols A by auto
     finally show ?thesis by auto
   qed 
   moreover have "\<exists>f. A *\<^sub>v x = lincomb (of_int \<circ> f) (set (cols A))" 
     if Ax: "A *\<^sub>v x \<in> carrier_vec (dim_row A)" and x: "x \<in> carrier_vec (dim_col A)" for x 
   proof -
     let ?c = "\<lambda>i. x $ i"
     have x_vec: "vec (length ?ws) ?c = x" using x by auto
     have "A *\<^sub>v x = mat_of_cols n ?ws *\<^sub>v vec (length ?ws) ?c" using mat_of_cols_cols A x_vec by auto
     also have "... = lincomb_list ?c ?ws"
       by (rule lincomb_list_as_mat_mult[symmetric], insert set_cols_in A, auto)
     also have "... = lincomb (mk_coeff ?ws ?c) (set ?ws)" 
       by (rule lincomb_list_as_lincomb, insert set_cols_in A, auto)    
     finally show ?thesis by auto
   qed
   ultimately show ?thesis unfolding lattice_of_altdef_lincomb[OF set_cols_in]
     by (metis (mono_tags, opaque_lifting))
 qed
 
 
 corollary lattice_of_as_mat_mult:
   assumes fs: "set fs \<subseteq> carrier_vec n"
   shows "lattice_of fs = {y\<in>carrier_vec n. \<exists>x\<in>carrier_vec (length fs). (mat_of_cols n fs) *\<^sub>v x = y}"
 proof -
   have cols_eq: "cols (mat_of_cols n fs) = fs" using cols_mat_of_cols[OF fs] by simp
   have m: "(mat_of_cols n fs) \<in> carrier_mat n (length fs)" using mat_of_cols_carrier(1) by auto
   show ?thesis using lattice_of_cols_as_mat_mult[OF m] unfolding cols_eq using m by auto
 qed
 end
 
 context vec_space
 begin
 
 lemma lin_indpt_cols_imp_det_not_0:
   fixes A::"'a mat"
   assumes A: "A \<in> carrier_mat n n" and li: "lin_indpt (set (cols A))" and d: "distinct (cols A)" 
   shows "det A \<noteq> 0"  
   using A li d det_rank_iff lin_indpt_full_rank by blast
 
 corollary lin_indpt_rows_imp_det_not_0:
   fixes A::"'a mat"
   assumes A: "A \<in> carrier_mat n n" and li: "lin_indpt (set (rows A))" and d: "distinct (rows A)" 
   shows "det A \<noteq> 0"  
   using A li d det_rank_iff lin_indpt_full_rank
   by (metis (full_types) Determinant.det_transpose cols_transpose transpose_carrier_mat)
 end
 
 context LLL
 begin
 
 lemma eq_lattice_imp_mat_mult_invertible_cols:
   assumes fs: "set fs \<subseteq> carrier_vec n"
   and gs: "set gs \<subseteq> carrier_vec n"  and ind_fs: "lin_indep fs" (*fs is a basis*)
   and length_fs: "length fs = n" and length_gs: "length gs = n" (*For the moment, only valid for square matrices*)
   and l: "lattice_of fs = lattice_of gs" 
 shows "\<exists>Q \<in> carrier_mat n n. invertible_mat Q \<and> mat_of_cols n fs = mat_of_cols n gs * Q"
 proof (cases "n=0")
   case True
   show ?thesis
     by (rule bexI[of _ "1\<^sub>m 0"], insert True assms, auto) 
 next
   case False
   hence n: "0<n" by simp
   have ind_RAT_fs: "gs.lin_indpt (set (RAT fs))" using ind_fs
     by (simp add: cof_vec_space.lin_indpt_list_def)
   have fs_carrier: "mat_of_cols n fs \<in> carrier_mat n n" by (simp add: length_fs carrier_matI)
   let ?f = "(\<lambda>i. SOME x. x\<in>carrier_vec (length gs) \<and> (mat_of_cols n gs) *\<^sub>v x = fs ! i)"
   let ?cols_Q = "map ?f [0..<length fs]"
   let ?Q = "mat_of_cols n ?cols_Q"
   show ?thesis
   proof (rule bexI[of _ ?Q], rule conjI)
     show Q: "?Q \<in> carrier_mat n n" using length_fs by auto
     show fs_gs_Q: "mat_of_cols n fs = mat_of_cols n gs * ?Q"
     proof (rule mat_col_eqI)
       fix j assume j: "j < dim_col (mat_of_cols n gs * ?Q)"
       have j2: "j<n" using j Q length_gs by auto
       have fs_j_in_gs: "fs ! j \<in> lattice_of gs" using fs l basis_in_latticeI j by auto
       have fs_j_carrier_vec: "fs ! j \<in> carrier_vec n"
         using fs_j_in_gs gs lattice_of_as_mat_mult by blast      
       let ?x = "SOME x. x\<in>carrier_vec (length gs) \<and> (mat_of_cols n gs) *\<^sub>v x = fs ! j"
       have "?x\<in>carrier_vec (length gs) \<and> (mat_of_cols n gs) *\<^sub>v ?x = fs ! j"
         by (rule someI_ex, insert fs_j_in_gs lattice_of_as_mat_mult[OF gs], auto)
       hence x: "?x \<in> carrier_vec (length gs)"
         and gs_x: "(mat_of_cols n gs) *\<^sub>v ?x = fs ! j" by blast+
       have "col ?Q j = ?cols_Q ! j"
       proof (rule col_mat_of_cols)
         show "j < length (map ?f [0..<length fs])" using length_fs j2 by auto
         have "map ?f [0..<length fs] ! j = ?f ([0..<length fs] ! j)" 
           by (rule nth_map, insert j2 length_fs, auto) 
         also have "... = ?f j" by (simp add: length_fs j2)
         also have "... \<in> carrier_vec n" using x length_gs by auto        
         finally show "map ?f [0..<length fs] ! j \<in> carrier_vec n" .
       qed
       also have "... = ?f ([0..<length fs] ! j)" 
         by (rule nth_map, insert j2 length_fs, auto)
       also have "... = ?x" by (simp add: length_fs j2)
       finally have col_Qj_x: "col ?Q j = ?x" .
       have "col (mat_of_cols n fs) j = fs ! j"
         by (metis (no_types, lifting) j Q fs length_fs carrier_matD(2) cols_mat_of_cols cols_nth
             index_mult_mat(3) mat_of_cols_carrier(3))
       also have "... = (mat_of_cols n gs) *\<^sub>v ?x" using gs_x by auto
       also have "... = (mat_of_cols n gs) *\<^sub>v (col ?Q j)" unfolding col_Qj_x by simp
       also have "... = col (mat_of_cols n gs * ?Q) j"
         by (rule col_mult2[symmetric, OF _ Q j2], insert length_gs mat_of_cols_def, auto)
       finally show "col (mat_of_cols n fs) j = col (mat_of_cols n gs * ?Q) j" .      
     qed (insert length_gs gs, auto)
     show "invertible_mat ?Q"
     (* Sketch of the proof:
       1) fs = gs * Q, proved previously
       2) gs = fs * Q', similar proof as the previous one.
       3) fs = fs * Q' * Q
       4) fs * (?Q' * ?Q - 1\<^sub>m n) = 0\<^sub>m n n and hence (?Q' * ?Q - 1\<^sub>m n) = 0 since fs independent
       5) det ?Q' = det ?Q = det 1 = 1, then det ?Q = \<plusminus>1 and ?Q invertible since the determinant 
          divides a unit.
     *)
     proof -
       let ?f' = "(\<lambda>i. SOME x. x\<in>carrier_vec (length fs) \<and> (mat_of_cols n fs) *\<^sub>v x = gs ! i)"
       let ?cols_Q' = "map ?f' [0..<length gs]"
       let ?Q' = "mat_of_cols n ?cols_Q'"
       have Q': "?Q' \<in> carrier_mat n n" using length_gs by auto
       have gs_fs_Q': "mat_of_cols n gs = mat_of_cols n fs * ?Q'"
       proof (rule mat_col_eqI)
         fix j assume j: "j < dim_col (mat_of_cols n fs * ?Q')"
         have j2: "j<n" using j Q length_gs by auto
         have gs_j_in_fs: "gs ! j \<in> lattice_of fs" using gs l basis_in_latticeI j by auto
         have gs_j_carrier_vec: "gs ! j \<in> carrier_vec n"
           using gs_j_in_fs fs lattice_of_as_mat_mult by blast      
         let ?x = "SOME x. x\<in>carrier_vec (length fs) \<and> (mat_of_cols n fs) *\<^sub>v x = gs ! j"
         have "?x\<in>carrier_vec (length fs) \<and> (mat_of_cols n fs) *\<^sub>v ?x = gs ! j"
           by (rule someI_ex, insert gs_j_in_fs lattice_of_as_mat_mult[OF fs], auto)
         hence x: "?x \<in> carrier_vec (length fs)"
           and fs_x: "(mat_of_cols n fs) *\<^sub>v ?x = gs ! j" by blast+
         have "col ?Q' j = ?cols_Q' ! j"
         proof (rule col_mat_of_cols)
           show "j < length (map ?f' [0..<length gs])" using length_gs j2 by auto
           have "map ?f' [0..<length gs] ! j = ?f' ([0..<length gs] ! j)" 
             by (rule nth_map, insert j2 length_gs, auto) 
           also have "... = ?f' j" by (simp add: length_gs j2)
           also have "... \<in> carrier_vec n" using x length_fs by auto        
           finally show "map ?f' [0..<length gs] ! j \<in> carrier_vec n" .
         qed
         also have "... = ?f' ([0..<length gs] ! j)" 
           by (rule nth_map, insert j2 length_gs, auto)
         also have "... = ?x" by (simp add: length_gs j2)
         finally have col_Qj_x: "col ?Q' j = ?x" .
         have "col (mat_of_cols n gs) j = gs ! j" by (simp add: length_gs gs_j_carrier_vec j2)
         also have "... = (mat_of_cols n fs) *\<^sub>v ?x" using fs_x by auto
         also have "... = (mat_of_cols n fs) *\<^sub>v (col ?Q' j)" unfolding col_Qj_x by simp
         also have "... = col (mat_of_cols n fs * ?Q') j"
           by (rule col_mult2[symmetric, OF _ Q' j2], insert length_fs mat_of_cols_def, auto)
         finally show "col (mat_of_cols n gs) j = col (mat_of_cols n fs * ?Q') j" .      
       qed (insert length_fs fs, auto)
       
       have det_fs_not_zero: "rat_of_int (det (mat_of_cols n fs)) \<noteq> 0"
       proof -
         let ?A = "(of_int_hom.mat_hom (mat_of_cols n fs)):: rat mat"
         have "rat_of_int (det (mat_of_cols n fs)) = det ?A"
           by simp
         moreover have "det ?A \<noteq> 0"
         proof (rule gs.lin_indpt_cols_imp_det_not_0[of ?A])
           have c_eq: "(set (cols ?A)) = set (RAT fs)"
             by (metis assms(3) cof_vec_space.lin_indpt_list_def cols_mat_of_cols fs mat_of_cols_map)
           show "?A \<in> carrier_mat n n" by (simp add: fs_carrier)
           show "gs.lin_indpt (set (cols ?A))" using ind_RAT_fs c_eq by auto
           show "distinct (cols ?A)"
             by (metis ind_fs cof_vec_space.lin_indpt_list_def cols_mat_of_cols fs mat_of_cols_map)
         qed
         ultimately show ?thesis by auto
       qed
       have Q'Q: "?Q' * ?Q \<in> carrier_mat n n" using Q Q' mult_carrier_mat by blast
       have fs_fs_Q'Q: "mat_of_cols n fs = mat_of_cols n fs * ?Q' * ?Q" using gs_fs_Q' fs_gs_Q by presburger
       hence "0\<^sub>m n n = mat_of_cols n fs * ?Q' * ?Q - mat_of_cols n fs" using length_fs by auto
       also have "... = mat_of_cols n fs * ?Q' * ?Q - mat_of_cols n fs * 1\<^sub>m n"
         using fs_carrier by auto
       also have "... = mat_of_cols n fs * (?Q' * ?Q) - mat_of_cols n fs * 1\<^sub>m n"
         using Q Q' fs_carrier by auto
       also have "... = mat_of_cols n fs * (?Q' * ?Q - 1\<^sub>m n)"
         by (rule mult_minus_distrib_mat[symmetric, OF fs_carrier Q'Q], auto)      
       finally have "mat_of_cols n fs * (?Q' * ?Q - 1\<^sub>m n) = 0\<^sub>m n n" ..
       have "det (?Q' * ?Q) = 1"
-        by (smt Determinant.det_mult Q Q' Q'Q fs_fs_Q'Q assoc_mult_mat det_fs_not_zero 
+        by (smt (verit) Determinant.det_mult Q Q' Q'Q fs_fs_Q'Q assoc_mult_mat det_fs_not_zero 
             fs_carrier mult_cancel_left2 of_int_code(2))
       hence det_Q'_Q_1: "det ?Q * det ?Q' = 1"
         by (metis (no_types, lifting) Determinant.det_mult Groups.mult_ac(2) Q Q')
       hence "det ?Q = 1 \<or> det ?Q = -1" by (rule pos_zmult_eq_1_iff_lemma)
       thus ?thesis using invertible_iff_is_unit_JNF[OF Q] by fastforce
     qed
   qed
 qed
 
 
 corollary eq_lattice_imp_mat_mult_invertible_rows:
   assumes fs: "set fs \<subseteq> carrier_vec n"
   and gs: "set gs \<subseteq> carrier_vec n"  and ind_fs: "lin_indep fs" (*fs is a basis*)
   and length_fs: "length fs = n" and length_gs: "length gs = n" (*For the moment, only valid for square matrices*)
   and l: "lattice_of fs = lattice_of gs" 
 shows "\<exists>P \<in> carrier_mat n n. invertible_mat P \<and> mat_of_rows n fs = P * mat_of_rows n gs"
 proof -
   obtain Q where Q: "Q \<in> carrier_mat n n" and inv_Q: "invertible_mat Q" 
     and fs_gs_Q: "mat_of_cols n fs = mat_of_cols n gs * Q" 
     using eq_lattice_imp_mat_mult_invertible_cols[OF assms] by auto
   have "invertible_mat Q\<^sup>T" by (simp add: inv_Q invertible_mat_transpose)
   moreover have "mat_of_rows n fs = Q\<^sup>T * mat_of_rows n gs" using fs_gs_Q
     by (metis Matrix.transpose_mult Q length_gs mat_of_cols_carrier(1) transpose_mat_of_cols)
   moreover have "Q\<^sup>T \<in> carrier_mat n n" using Q by auto
   ultimately show ?thesis by blast
 qed
 end
 
 subsubsection \<open>Missing results\<close>
 
 text \<open>This is a new definition for upper triangular matrix, valid for rectangular matrices. 
 This definition will allow us to prove that echelon form implies upper triangular for any matrix.\<close>
 
 definition "upper_triangular' A = (\<forall>i < dim_row A. \<forall> j<dim_col A. j < i \<longrightarrow> A $$ (i,j) = 0)"
 
 lemma upper_triangular'D[elim] :
   "upper_triangular' A \<Longrightarrow> j<dim_col A \<Longrightarrow> j < i \<Longrightarrow> i < dim_row A \<Longrightarrow> A $$ (i,j) = 0"
 unfolding upper_triangular'_def by auto
 
 lemma upper_triangular'I[intro] :
   "(\<And>i j. j<dim_col A \<Longrightarrow> j < i \<Longrightarrow> i < dim_row A \<Longrightarrow> A $$ (i,j) = 0) \<Longrightarrow> upper_triangular' A"
   unfolding upper_triangular'_def by auto
 
 lemma prod_list_abs(*[simp]?*):
   fixes xs:: "int list"
   shows "prod_list (map abs xs) = abs (prod_list xs)"
   by (induct xs, auto simp add: abs_mult)
 
 lemma euclid_ext2_works:
   assumes "euclid_ext2 a b = (p,q,u,v,d)"
   shows "p*a+q*b = d" and "d = gcd a b" and "gcd a b * u = -b" and "gcd a b * v = a"
   and "u = -b div gcd a b" and "v = a div gcd a b"
   using assms unfolding euclid_ext2_def
   by (auto simp add: bezout_coefficients_fst_snd)
 
 lemma res_function_euclidean2: 
   "res_function (\<lambda>b n::'a::{unique_euclidean_ring}. n mod b)"
 proof- 
   have "n mod b = n" if "b=0" for n b::"'a :: unique_euclidean_ring" using that by auto
   hence "res_function_euclidean = (\<lambda>b n::'a. n mod b)" 
     by (unfold fun_eq_iff res_function_euclidean_def, auto)
   thus ?thesis using res_function_euclidean by auto
 qed
 
 lemma mult_row_1_id:
   fixes A:: "'a::semiring_1^'n^'m"
   shows "mult_row A b 1 = A" unfolding mult_row_def by vector
 
 text \<open>Results about appending rows\<close>
 
 lemma row_append_rows1:
   assumes A: "A \<in> carrier_mat m n"
   and B: "B \<in> carrier_mat p n"
   assumes i: "i < dim_row A"
   shows "Matrix.row (A @\<^sub>r B) i = Matrix.row A i"  
 proof (rule eq_vecI)
   have AB_carrier[simp]: "(A @\<^sub>r B) \<in> carrier_mat (m+p) n" by (rule carrier_append_rows[OF A B])
   thus "dim_vec (Matrix.row (A @\<^sub>r B) i) = dim_vec (Matrix.row A i)"
     using A B by (auto, insert carrier_matD(2), blast)
   fix j assume j: "j < dim_vec (Matrix.row A i)" 
   have "Matrix.row (A @\<^sub>r B) i $v j = (A @\<^sub>r B) $$ (i, j)"
     by (metis AB_carrier Matrix.row_def j A carrier_matD(2) index_row(2) index_vec)
   also have "... = (if i < dim_row A then A $$ (i, j) else B $$ (i - m, j))"
     by (rule append_rows_nth, insert assms j, auto)
   also have "... = A$$ (i,j)" using i by simp
   finally show "Matrix.row (A @\<^sub>r B) i $v j = Matrix.row A i $v j" using i j by simp  
 qed
 
 lemma row_append_rows2:
   assumes A: "A \<in> carrier_mat m n"
   and B: "B \<in> carrier_mat p n"
   assumes i: "i \<in> {m..<m+p}"
   shows "Matrix.row (A @\<^sub>r B) i = Matrix.row B (i - m)"
 proof (rule eq_vecI)
   have AB_carrier[simp]: "(A @\<^sub>r B) \<in> carrier_mat (m+p) n" by (rule carrier_append_rows[OF A B])
   thus "dim_vec (Matrix.row (A @\<^sub>r B) i) = dim_vec (Matrix.row B (i-m))"
     using A B by (auto, insert carrier_matD(2), blast)
   fix j assume j: "j < dim_vec (Matrix.row B (i-m))" 
   have "Matrix.row (A @\<^sub>r B) i $v j = (A @\<^sub>r B) $$ (i, j)"
     by (metis AB_carrier Matrix.row_def j B carrier_matD(2) index_row(2) index_vec)
   also have "... = (if i < dim_row A then A $$ (i, j) else B $$ (i - m, j))"
     by (rule append_rows_nth, insert assms j, auto)
   also have "... = B $$ (i - m, j)" using i A by simp
   finally show "Matrix.row (A @\<^sub>r B) i $v j = Matrix.row B (i-m) $v j" using i j A B by auto  
 qed
 
 
 lemma rows_append_rows:
   assumes A: "A \<in> carrier_mat m n"
   and B: "B \<in> carrier_mat p n"
 shows "Matrix.rows (A @\<^sub>r B) = Matrix.rows A @ Matrix.rows B"
 proof -
   have AB_carrier: "(A @\<^sub>r B) \<in> carrier_mat (m+p) n" 
     by (rule carrier_append_rows, insert A B, auto)
   hence 1: "dim_row (A @\<^sub>r B) = dim_row A + dim_row B" using A B by blast
   moreover have "Matrix.row (A @\<^sub>r B) i = (Matrix.rows A @ Matrix.rows B) ! i"
     if i: "i < dim_row (A @\<^sub>r B)" for i
   proof (cases "i<dim_row A")
     case True
     have "Matrix.row (A @\<^sub>r B) i = Matrix.row A i" using A True B row_append_rows1 by blast
     also have "... = Matrix.rows A ! i" unfolding Matrix.rows_def using True by auto
     also have "... = (Matrix.rows A @ Matrix.rows B) ! i" using True by (simp add: nth_append)
     finally show ?thesis .
   next
     case False
     have i_mp: "i < m + p" using AB_carrier A B i by fastforce
     have "Matrix.row (A @\<^sub>r B) i = Matrix.row B (i-m)" using A False B i row_append_rows2 i_mp
-      by (smt AB_carrier atLeastLessThan_iff carrier_matD(1) le_add1
+      by (smt (verit) AB_carrier atLeastLessThan_iff carrier_matD(1) le_add1
           linordered_semidom_class.add_diff_inverse row_append_rows2)
     also have "... = Matrix.rows B ! (i-m)" unfolding Matrix.rows_def using False i A 1 by auto
     also have "... = (Matrix.rows A @ Matrix.rows B) ! (i-m+m)"
       by (metis add_diff_cancel_right' A carrier_matD(1) length_rows not_add_less2 nth_append)
     also have "... =  (Matrix.rows A @ Matrix.rows B) ! i" using False A by auto
     finally show ?thesis .
   qed  
   ultimately show ?thesis unfolding list_eq_iff_nth_eq by auto  
 qed
 
 
 
 lemma append_rows_nth2:
   assumes A': "A' \<in> carrier_mat m n"
   and B: "B \<in> carrier_mat p n"
   and A_def: "A = (A' @\<^sub>r  B)"
   and a: "a<m" and ap: "a < p" and j: "j<n"
   shows "A $$ (a + m, j) = B $$ (a,j)" 
 proof -
   have "A $$ (a + m, j) = (if a + m < dim_row A' then A' $$ (a + m, j) else B $$ (a + m - m, j))"
     unfolding A_def by (rule append_rows_nth[OF A' B _ j], insert ap a, auto)
   also have "... = B $$ (a,j)" using ap a A' by auto
   finally show ?thesis .
 qed
 
 
 lemma append_rows_nth3:
   assumes A': "A' \<in> carrier_mat m n"
   and B: "B \<in> carrier_mat p n"
   and A_def: "A = (A' @\<^sub>r  B)"
   and a: "a\<ge>m" and ap: "a < m + p" and j: "j<n"
   shows "A $$ (a, j) = B $$ (a-m,j)" 
 proof -
   have "A $$ (a, j) = (if a < dim_row A' then A' $$ (a, j) else B $$ (a - m, j))"
     unfolding A_def by (rule append_rows_nth[OF A' B _ j], insert ap a, auto)
   also have "... = B $$ (a-m,j)" using ap a A' by auto
   finally show ?thesis .
 qed
 
 
 text \<open>Results about submatrices\<close>
 
 lemma pick_first_id: assumes i: "i<n" shows "pick {0..<n} i = i"
 proof -
   have "i = (card {a \<in> {0..<n}. a < i})" using i
-    by (auto, smt Collect_cong card_Collect_less_nat nat_SN.gt_trans)
+    by (auto, smt (verit) Collect_cong card_Collect_less_nat nat_SN.gt_trans)
   thus ?thesis using pick_card_in_set i
     by (metis atLeastLessThan_iff zero_order(1))
 qed
 
 lemma submatrix_index_id:
   assumes H: "H \<in> carrier_mat m n" and i: "i<k1" and j: "j<k2"
   and k1: "k1\<le>m" and k2: "k2\<le>n"
   shows "(submatrix H {0..<k1} {0..<k2}) $$ (i,j) = H $$ (i,j)" 
 proof -
   let ?I = "{0..<k1}"
   let ?J = "{0..<k2}"
   let ?H = "submatrix H ?I ?J"  
   have km: "k1\<le>m" and kn: "k2\<le>n" using k1 k2 by simp+
-  have card_mk: "card {i. i < m \<and> i < k1} = k1" using km 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
-  have card_nk: "card {i. i < n \<and> i < k2} = k2" using kn 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
+  then have "{i. i < m \<and> i < k1} = {..<k1}" "{i. i < n \<and> i < k2} = {..<k2}" by auto
+  then have card_mk: "card {i. i < m \<and> i < k1} = k1" and card_nk: "card {i. i < n \<and> i < k2} = k2"
+    by auto
   show ?thesis
   proof- 
     have pick_j: "pick ?J j = j" by (rule pick_first_id[OF j])
     have pick_i: "pick ?I i = i" by (rule pick_first_id[OF i])
     have "submatrix H ?I ?J $$ (i, j) = H $$ (pick ?I i, pick ?J j)" 
       by (rule submatrix_index, insert H i j card_mk card_nk, auto)
     also have "... = H $$ (i,j)" using pick_i pick_j by simp
     finally show ?thesis .
   qed
 qed
 
 lemma submatrix_carrier_first:
   assumes H: "H \<in> carrier_mat m n"
   and k1: "k1 \<le> m" and k2: "k2 \<le> n"
   shows"submatrix H {0..<k1} {0..<k2} \<in> carrier_mat k1 k2"
 proof -  
   have km: "k1\<le>m" and kn: "k2\<le>n" using k1 k2 by simp+
-  have card_mk: "card {i. i < m \<and> i < k1} = k1" using km 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
-  have card_nk: "card {i. i < n \<and> i < k2} = k2" using kn 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
+  then have "{i. i < m \<and> i < k1} = {..<k1}" "{i. i < n \<and> i < k2} = {..<k2}" by auto
+  then have card_mk: "card {i. i < m \<and> i < k1} = k1" and card_nk: "card {i. i < n \<and> i < k2} = k2"
+    by auto
   show ?thesis
-    by (smt Collect_cong H atLeastLessThan_iff card_mk card_nk carrier_matD 
+    by (smt (verit) Collect_cong H atLeastLessThan_iff card_mk card_nk carrier_matD 
         carrier_matI dim_submatrix zero_order(1))
 qed
 
 
 
 lemma Units_eq_invertible_mat:
   assumes "A \<in> carrier_mat n n"
   shows "A \<in> Group.Units (ring_mat TYPE('a::comm_ring_1) n b) = invertible_mat A" (is "?lhs = ?rhs")
 proof -
   interpret m: ring "ring_mat TYPE('a) n b" by (rule ring_mat)
   show ?thesis
   proof
     assume "?lhs" thus "?rhs"
       unfolding Group.Units_def 
       by (insert assms, auto simp add: ring_mat_def invertible_mat_def inverts_mat_def)
   next
     assume "?rhs" 
     from this obtain B where AB: "A * B = 1\<^sub>m n" and BA: "B * A = 1\<^sub>m n" and B: "B \<in> carrier_mat n n"
       by (metis assms carrier_matD(1) inverts_mat_def obtain_inverse_matrix)
     hence "\<exists>x\<in>carrier (ring_mat TYPE('a) n b). x \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> A = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub> 
       \<and> A \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"
       unfolding ring_mat_def by auto
     thus "?lhs" unfolding Group.Units_def using assms unfolding ring_mat_def by auto
   qed
 qed
 
 lemma map_first_rows_index:
   assumes "A \<in> carrier_mat M n" and "m \<le> M" and "i<m" and "ja<n"
   shows "map (Matrix.row A) [0..<m] ! i $v ja = A $$ (i, ja)"
   using assms by auto
 
 lemma matrix_append_rows_eq_if_preserves:
   assumes A: "A \<in> carrier_mat (m+p) n" and B: "B \<in> carrier_mat p n"
     and eq: "\<forall>i\<in>{m..<m+p}.\<forall>j<n. A$$(i,j) = B $$ (i-m,j)"
   shows "A = mat_of_rows n [Matrix.row A i. i \<leftarrow> [0..<m]] @\<^sub>r B" (is "_ = ?A' @\<^sub>r _")
 proof (rule eq_matI)
   have A': "?A' \<in> carrier_mat m n" by (simp add: mat_of_rows_def)
   hence A'B: "?A' @\<^sub>r B \<in> carrier_mat (m+p) n" using B by blast
   show "dim_row A = dim_row (?A' @\<^sub>r B)" and "dim_col A = dim_col (?A' @\<^sub>r B)" using A'B A by auto
   fix i j assume i: "i < dim_row (?A' @\<^sub>r B)"
     and j: "j < dim_col (?A' @\<^sub>r B)" 
   have jn: "j<n" using A
     by (metis append_rows_def dim_col_mat(1) index_mat_four_block(3) index_zero_mat(3) 
         j mat_of_rows_def nat_arith.rule0)
   let ?xs = "(map (Matrix.row A) [0..<m])"
   show "A $$ (i, j) = (?A' @\<^sub>r B) $$ (i, j)"
   proof (cases "i<m")
     case True
     have "(?A' @\<^sub>r B) $$ (i, j) = ?A' $$ (i,j)"      
       by (metis (no_types, lifting) Nat.add_0_right True append_rows_def diff_zero i 
           index_mat_four_block index_zero_mat(3) j length_map length_upt mat_of_rows_carrier(2))
     also have "... = ?xs ! i $v j" 
       by (rule mat_of_rows_index, insert i True j, auto simp add: append_rows_def)
     also have "... = A $$ (i,j)"
       by (rule map_first_rows_index, insert assms A True i jn, auto)
     finally show ?thesis ..
   next
     case False
     have "(?A' @\<^sub>r B) $$ (i, j) = B $$ (i-m,j)"      
-      by (smt (z3) A' carrier_matD(1) False append_rows_def i index_mat_four_block j jn length_map
+      by (smt (verit) A' carrier_matD(1) False append_rows_def i index_mat_four_block j jn length_map
           length_upt mat_of_rows_carrier(2,3))
     also have "... = A $$ (i,j)"
       by (metis False append_rows_def B eq atLeastLessThan_iff carrier_matD(1) diff_zero i 
           index_mat_four_block(2) index_zero_mat(2) jn le_add1 length_map length_upt 
           linordered_semidom_class.add_diff_inverse mat_of_rows_carrier(2))
     finally show ?thesis ..
   qed
 qed
 
 lemma invertible_mat_first_column_not0:
   fixes A::"'a :: comm_ring_1 mat"
   assumes A: "A \<in> carrier_mat n n" and inv_A: "invertible_mat A" and n0: "0<n"
   shows "col A 0 \<noteq> (0\<^sub>v n)"
 proof (rule ccontr)
   assume " \<not> col A 0 \<noteq> 0\<^sub>v n" hence col_A0: "col A 0 = 0\<^sub>v n" by simp
   have "(det A dvd 1)" using inv_A invertible_iff_is_unit_JNF[OF A] by auto
   hence 1: "det A \<noteq> 0" by auto
   have "det A = (\<Sum>i<n. A $$ (i, 0) * Determinant.cofactor A i 0)" 
     by (rule laplace_expansion_column[OF A n0])
   also have "... = 0" 
     by (rule sum.neutral, insert col_A0 n0 A, auto simp add: col_def,
         metis Matrix.zero_vec_def index_vec mult_zero_left)
   finally show False using 1 by contradiction 
 qed
 
 lemma invertible_mat_mult_int:
   assumes "A = P * B" 
     and "P \<in> carrier_mat n n"
     and "B \<in> carrier_mat n n"
     and "invertible_mat P" 
     and "invertible_mat (map_mat rat_of_int B)"
   shows "invertible_mat (map_mat rat_of_int A)"
   by (metis (no_types, opaque_lifting) assms dvd_field_iff 
       invertible_iff_is_unit_JNF invertible_mult_JNF map_carrier_mat not_is_unit_0 
       of_int_hom.hom_0 of_int_hom.hom_det of_int_hom.mat_hom_mult)
 
 
 lemma echelon_form_JNF_intro: 
   assumes "(\<forall>i<dim_row A. is_zero_row_JNF i A \<longrightarrow> \<not> (\<exists>j. j < dim_row A \<and> j>i \<and> \<not> is_zero_row_JNF j A))"
   and "(\<forall>i j. i<j \<and> j<dim_row A \<and> \<not> (is_zero_row_JNF i A) \<and> \<not> (is_zero_row_JNF j A) 
          \<longrightarrow> ((LEAST n. A $$ (i, n) \<noteq> 0) < (LEAST n. A $$ (j, n) \<noteq> 0)))"
   shows "echelon_form_JNF A" using assms unfolding echelon_form_JNF_def by simp
 
 
 lemma echelon_form_submatrix:
   assumes ef_H: "echelon_form_JNF H" and H: "H \<in> carrier_mat m n"
   and k: "k \<le> min m n"
 shows "echelon_form_JNF (submatrix H {0..<k} {0..<k})" 
 proof -
   let ?I = "{0..<k}"
   let ?H = "submatrix H ?I ?I"  
   have km: "k\<le>m" and kn: "k\<le>n" using k by simp+
-  have card_mk: "card {i. i < m \<and> i < k} = k" using km 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
-  have card_nk: "card {i. i < n \<and> i < k} = k" using kn 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
+  then have "{i. i < m \<and> i < k} = {..<k}" "{i. i < n \<and> i < k} = {..<k}" by auto
+  then have card_mk: "card {i. i < m \<and> i < k} = k" and card_nk: "card {i. i < n \<and> i < k} = k"
+    by auto
   have H_ij: "H $$ (i,j) = (submatrix H ?I ?I) $$ (i,j)"  if i: "i<k" and j: "j<k" for i j
   proof- 
     have pick_j: "pick ?I j = j" by (rule pick_first_id[OF j])
     have pick_i: "pick ?I i = i" by (rule pick_first_id[OF i])
     have "submatrix H ?I ?I $$ (i, j) = H $$ (pick ?I i, pick ?I j)" 
       by (rule submatrix_index, insert H i j card_mk card_nk, auto)
     also have "... = H $$ (i,j)" using pick_i pick_j by simp
     finally show ?thesis ..
   qed
   have H'[simp]: "?H \<in> carrier_mat k k" 
     using H dim_submatrix[of H "{0..<k}" "{0..<k}"] card_mk card_nk by auto
   show ?thesis
   proof (rule echelon_form_JNF_intro, auto)   
     fix i j assume iH'_0: "is_zero_row_JNF i ?H" and ij: "i < j" and j: "j < dim_row ?H"  
     have jm: "j<m"
       by (metis H' carrier_matD(1) j km le_eq_less_or_eq nat_SN.gt_trans)
     show "is_zero_row_JNF j ?H"
     proof (rule ccontr)
       assume j_not0_H': "\<not> is_zero_row_JNF j ?H"
       define a where "a = (LEAST n. ?H $$ (j,n) \<noteq> 0)"
       have H'_ja: "?H $$ (j,a) \<noteq> 0" 
         by (metis (mono_tags) LeastI j_not0_H' a_def is_zero_row_JNF_def)
       have a: "a < dim_col ?H"
-        by (smt j_not0_H' a_def is_zero_row_JNF_def linorder_neqE_nat not_less_Least order_trans_rules(19))
+        by (smt (verit) j_not0_H' a_def is_zero_row_JNF_def linorder_neqE_nat not_less_Least order_trans_rules(19))
       have j_not0_H: "\<not> is_zero_row_JNF j H"
         by (metis H' H'_ja H_ij a assms(2) basic_trans_rules(19) carrier_matD is_zero_row_JNF_def j kn le_eq_less_or_eq)
       hence i_not0_H: "\<not> is_zero_row_JNF i H" using ef_H j ij unfolding echelon_form_JNF_def
         by (metis H' \<open>\<not> is_zero_row_JNF j H\<close> assms(2) carrier_matD(1) ij j km 
             not_less_iff_gr_or_eq order.strict_trans order_trans_rules(21))
       hence least_ab: "(LEAST n. H $$ (i, n) \<noteq> 0) < (LEAST n. H $$ (j, n) \<noteq> 0)" using jm
         using j_not0_H assms(2) echelon_form_JNF_def ef_H ij by blast
       define b where "b = (LEAST n. H $$ (i, n) \<noteq> 0)"
       have H_ib: "H $$ (i, b) \<noteq> 0"
         by (metis (mono_tags, lifting) LeastI b_def i_not0_H is_zero_row_JNF_def)
       have b: "b < dim_col ?H" using least_ab a unfolding a_def b_def
         by (metis (mono_tags, lifting) H' H'_ja H_ij a_def carrier_matD dual_order.strict_trans j nat_neq_iff not_less_Least)
       have H'_ib: "?H $$ (i,b) \<noteq> 0" using H_ib b H_ij H' ij j 
         by (metis H' carrier_matD dual_order.strict_trans ij j)
       hence "\<not> is_zero_row_JNF i ?H" using b is_zero_row_JNF_def by blast
       thus False using iH'_0 by contradiction
     qed  
   next
     fix i j assume ij: "i < j" and j: "j < dim_row ?H"
     have jm: "j<m"
       by (metis H' carrier_matD(1) j km le_eq_less_or_eq nat_SN.gt_trans)
     assume not0_iH': "\<not> is_zero_row_JNF i ?H"
       and not0_jH': "\<not> is_zero_row_JNF j ?H"
     define a where "a = (LEAST n. ?H $$ (i, n) \<noteq> 0)"
     define b where "b = (LEAST n. ?H $$ (j, n) \<noteq> 0)"
     have H'_ia: "?H $$ (i,a) \<noteq> 0"
       by (metis (mono_tags) LeastI_ex a_def is_zero_row_JNF_def not0_iH')
     have H'_jb: "?H $$ (j,b) \<noteq> 0"
       by (metis (mono_tags) LeastI_ex b_def is_zero_row_JNF_def not0_jH')
     have a: "a < dim_row ?H"
-      by (smt H' a_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_iH' not_less_Least)
+      by (smt (verit) H' a_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_iH' not_less_Least)
     have b: "b < dim_row ?H"
-      by (smt H' b_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_jH' not_less_Least)
+      by (smt (verit) H' b_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_jH' not_less_Least)
     have a_eq: "a = (LEAST n. H $$ (i, n) \<noteq> 0)"
-      by (smt H' H'_ia H_ij LeastI_ex a a_def carrier_matD(1) ij j linorder_neqE_nat not_less_Least order_trans_rules(19))
+      by (smt (verit) H' H'_ia H_ij LeastI_ex a a_def carrier_matD(1) ij j linorder_neqE_nat not_less_Least order_trans_rules(19))
     have b_eq: "b = (LEAST n. H $$ (j, n) \<noteq> 0)"
-      by (smt H' H'_jb H_ij LeastI_ex b b_def carrier_matD(1) ij j linorder_neqE_nat not_less_Least order_trans_rules(19)) 
+      by (smt (verit) H' H'_jb H_ij LeastI_ex b b_def carrier_matD(1) ij j linorder_neqE_nat not_less_Least order_trans_rules(19)) 
     have not0_iH: "\<not> is_zero_row_JNF i H" 
       by (metis H' H'_ia H_ij a H carrier_matD ij is_zero_row_JNF_def j kn le_eq_less_or_eq order.strict_trans)
     have not0_jH: "\<not> is_zero_row_JNF j H" 
       by (metis H' H'_jb H_ij b H carrier_matD is_zero_row_JNF_def j kn le_eq_less_or_eq order.strict_trans)
     show "(LEAST n. ?H $$ (i, n) \<noteq> 0) < (LEAST n. ?H $$ (j, n) \<noteq> 0)"
       unfolding a_def[symmetric] b_def[symmetric] a_eq b_eq using not0_iH not0_jH ef_H ij jm H 
       unfolding echelon_form_JNF_def by auto
   qed
 qed
 
 
 lemma HNF_submatrix:
   assumes HNF_H: "Hermite_JNF associates res H" and H: "H \<in> carrier_mat m n"
   and k: "k \<le> min m n"
   shows "Hermite_JNF associates res (submatrix H {0..<k} {0..<k})" 
 proof -
   let ?I = "{0..<k}"
   let ?H = "submatrix H ?I ?I"  
   have km: "k\<le>m" and kn: "k\<le>n" using k by simp+
-  have card_mk: "card {i. i < m \<and> i < k} = k" using km 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
-  have card_nk: "card {i. i < n \<and> i < k} = k" using kn 
-    by (smt Collect_cong card_Collect_less_nat le_eq_less_or_eq nat_less_induct nat_neq_iff)
+  then have "{i. i < m \<and> i < k} = {..<k}" "{i. i < n \<and> i < k} = {..<k}" by auto
+  then have card_mk: "card {i. i < m \<and> i < k} = k" and card_nk: "card {i. i < n \<and> i < k} = k"
+    by auto
   have H_ij: "H $$ (i,j) = (submatrix H ?I ?I) $$ (i,j)"  if i: "i<k" and j: "j<k" for i j
   proof- 
     have pick_j: "pick ?I j = j" by (rule pick_first_id[OF j])
     have pick_i: "pick ?I i = i" by (rule pick_first_id[OF i])
     have "submatrix H ?I ?I $$ (i, j) = H $$ (pick ?I i, pick ?I j)" 
       by (rule submatrix_index, insert H i j card_mk card_nk, auto)
     also have "... = H $$ (i,j)" using pick_i pick_j by simp
     finally show ?thesis ..
   qed
   have H'[simp]: "?H \<in> carrier_mat k k" 
     using H dim_submatrix[of H "{0..<k}" "{0..<k}"] card_mk card_nk by auto
   have CS_ass: "Complete_set_non_associates associates" using HNF_H unfolding Hermite_JNF_def by simp
   moreover have CS_res: "Complete_set_residues res"  using HNF_H unfolding Hermite_JNF_def by simp
   have ef_H: "echelon_form_JNF H" using HNF_H unfolding Hermite_JNF_def by auto
   have ef_H': "echelon_form_JNF ?H"
     by (rule echelon_form_submatrix[OF ef_H H k])
   have HNF1: "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> associates" 
     and HNF2: "(\<forall>j<i. ?H $$ (j, LEAST n. ?H $$ (i, n) \<noteq> 0)
                \<in> res (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0)))"
     if i: "i<dim_row ?H" and not0_iH': "\<not> is_zero_row_JNF i ?H" for i
   proof -
     define a where "a = (LEAST n. ?H $$ (i, n) \<noteq> 0)"
     have im: "i<m"
       by (metis H' carrier_matD(1) km order.strict_trans2 that(1))
     have H'_ia: "?H $$ (i,a) \<noteq> 0"
       by (metis (mono_tags) LeastI_ex a_def is_zero_row_JNF_def not0_iH')
     have a: "a < dim_row ?H"
-      by (smt H' a_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_iH' not_less_Least)
+      by (smt (verit) H' a_def carrier_matD is_zero_row_JNF_def less_trans linorder_neqE_nat not0_iH' not_less_Least)
     have a_eq: "a = (LEAST n. H $$ (i, n) \<noteq> 0)"
-      by (smt H' H'_ia H_ij LeastI_ex a a_def carrier_matD(1) i linorder_neqE_nat not_less_Least order_trans_rules(19))
+      by (smt (verit) H' H'_ia H_ij LeastI_ex a a_def carrier_matD(1) i linorder_neqE_nat not_less_Least order_trans_rules(19))
     have H'_ia_H_ia: "?H $$ (i, a) = H $$ (i, a)"  by (metis H' H_ij a carrier_matD(1) i)
     have not'_iH: "\<not> is_zero_row_JNF i H"
       by (metis H' H'_ia H'_ia_H_ia a assms(2) carrier_matD(1) carrier_matD(2) is_zero_row_JNF_def kn order.strict_trans2)
     thus "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> associates" using im
       by (metis H'_ia_H_ia Hermite_JNF_def a_def a_eq HNF_H H carrier_matD(1))
     show "(\<forall>j<i. ?H $$ (j, LEAST n. ?H $$ (i, n) \<noteq> 0)
                \<in> res (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0)))" 
     proof -
       { fix nn :: nat
     have ff1: "\<forall>n. ?H $$ (n, a) = H $$ (n, a) \<or> \<not> n < k"
       by (metis (no_types) H' H_ij a carrier_matD(1))
       have ff2: "i < k"
     by (metis H' carrier_matD(1) that(1))
     then have "H $$ (nn, a) \<in> res (H $$ (i, a)) \<longrightarrow> H $$ (nn, a) \<in> res (?H $$ (i, a))"
     using ff1 by (metis (no_types))
       moreover
       { assume "H $$ (nn, a) \<in> res (?H $$ (i, a))"
         then have "?H $$ (nn, a) = H $$ (nn, a) \<longrightarrow> ?H $$ (nn, a) \<in> res (?H $$ (i, a))"
             by presburger
           then have "\<not> nn < i \<or> ?H $$ (nn, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> res (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0))"
             using ff2 ff1 a_def order.strict_trans by blast }
         ultimately have "\<not> nn < i \<or> ?H $$ (nn, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> res (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0))"
           using Hermite_JNF_def a_eq assms(1) assms(2) im not'_iH by blast }
       then show ?thesis
         by meson
     qed
   qed
   show ?thesis using HNF1 HNF2 ef_H' CS_res CS_ass unfolding Hermite_JNF_def by blast
 qed
 
 lemma HNF_of_HNF_id:
   fixes H :: "int mat"
   assumes HNF_H: "Hermite_JNF associates res H"
   and H: "H \<in> carrier_mat n n"
   and H_P1_H1: "H = P1 * H1"
   and inv_P1: "invertible_mat P1"
   and H1: "H1 \<in> carrier_mat n n" 
   and P1: "P1 \<in> carrier_mat n n"
   and HNF_H1: "Hermite_JNF associates res H1"
   and inv_H: "invertible_mat (map_mat rat_of_int H)"
   shows "H1 = H" 
 proof (rule HNF_unique_generalized_JNF[OF H P1 H1 _ H H_P1_H1])   
   show "H = (1\<^sub>m n) * H" using H by auto  
 qed (insert assms, auto)
 
 
 (*Some of the following lemmas could be moved outside this context*)
 
 context
   fixes n :: nat
 begin
 
 interpretation vec_module "TYPE(int)" .        
 
 lemma lattice_is_monotone:
   fixes S T
   assumes S: "set S \<subseteq> carrier_vec n"
   assumes T: "set T \<subseteq> carrier_vec n"
   assumes subs: "set S \<subseteq> set T"
   shows "lattice_of S \<subseteq> lattice_of T"
 proof -
   have "\<exists>fa. lincomb fa (set T) = lincomb f (set S)" for f
   proof -
     let ?f = "\<lambda>i. if i \<in> set T - set S then 0 else f i"
     have set_T_eq: "set T = set S \<union> (set T - set S)" using subs by blast
     have l0: "lincomb ?f (set T - set S) = 0\<^sub>v n" by (rule lincomb_zero, insert T, auto)
     have "lincomb ?f (set T) = lincomb ?f (set S \<union> (set T - set S))" using set_T_eq by simp
     also have "... = lincomb ?f (set S) + lincomb ?f (set T - set S)"
       by (rule lincomb_union, insert S T subs, auto)
     also have "... = lincomb ?f (set S)" using l0 by (auto simp add: S)
     also have "... = lincomb f (set S)" using S by fastforce
     finally show ?thesis by blast    
   qed
   thus ?thesis unfolding lattice_of_altdef_lincomb[OF S] lattice_of_altdef_lincomb[OF T]
     by auto
 qed
 
 lemma lattice_of_append:
   assumes fs: "set fs \<subseteq> carrier_vec n"
   assumes gs: "set gs \<subseteq> carrier_vec n" 
   shows "lattice_of (fs @ gs) = lattice_of (gs @ fs)"
 proof -
   have fsgs: "set (fs @ gs) \<subseteq> carrier_vec n" using fs gs by auto
   have gsfs: "set (gs @ fs) \<subseteq> carrier_vec n" using fs gs by auto
   show ?thesis
     unfolding lattice_of_altdef_lincomb[OF fsgs] lattice_of_altdef_lincomb[OF gsfs] 
     by auto (metis Un_commute)+
 qed
 
 lemma lattice_of_append_cons:
   assumes fs: "set fs \<subseteq> carrier_vec n"   and v: "v \<in> carrier_vec n"
   shows "lattice_of (v # fs) = lattice_of (fs @ [v])"
 proof -
   have v_fs: "set (v # fs) \<subseteq> carrier_vec n" using fs v by auto
   hence fs_v: "set (fs @ [v]) \<subseteq> carrier_vec n" by simp
   show ?thesis
     unfolding lattice_of_altdef_lincomb[OF v_fs] lattice_of_altdef_lincomb[OF fs_v] by auto
 qed
 
 lemma already_in_lattice_subset:
   assumes fs: "set fs \<subseteq> carrier_vec n" and inlattice: "v \<in> lattice_of fs"
   and v: "v \<in> carrier_vec n"
   shows "lattice_of (v # fs) \<subseteq> lattice_of fs"
 proof (cases "v\<in>set fs")
   case True
   then show ?thesis
     by (metis fs lattice_is_monotone set_ConsD subset_code(1))
 next
   case False note v_notin_fs = False  
   obtain g where v_g: "lincomb g (set fs) = v"
     using lattice_of_altdef_lincomb[OF fs] inlattice by auto
   have v_fs: "set (v # fs) \<subseteq> carrier_vec n" using v fs by auto
   have "\<exists>fa. lincomb fa (set fs) = lincomb f (insert v (set fs))" for f
   proof -
     have smult_rw: "f v \<cdot>\<^sub>v (lincomb g (set fs)) = lincomb (\<lambda>w. f v * g w) (set fs)" 
       by (rule lincomb_smult[symmetric, OF fs])
     have "lincomb f (insert v (set fs)) =  f v \<cdot>\<^sub>v v + lincomb f (set fs)" 
       by (rule lincomb_insert2[OF _ fs _ v_notin_fs v], auto)
     also have "... = f v \<cdot>\<^sub>v (lincomb g (set fs)) + lincomb f (set fs)" using v_g by simp
     also have "... = lincomb (\<lambda>w. f v * g w) (set fs)  + lincomb f (set fs)"
       unfolding smult_rw by auto
     also have "... = lincomb (\<lambda>w. (\<lambda>w. f v * g w) w + f w) (set fs)"
       by (rule lincomb_sum[symmetric, OF _ fs], simp)
     finally show ?thesis by auto
   qed
   thus ?thesis unfolding lattice_of_altdef_lincomb[OF v_fs] lattice_of_altdef_lincomb[OF fs] by auto
 qed
 
 
 lemma already_in_lattice:
   assumes fs: "set fs \<subseteq> carrier_vec n" and inlattice: "v \<in> lattice_of fs"
   and v: "v \<in> carrier_vec n"
   shows "lattice_of fs = lattice_of (v # fs)"
 proof - 
   have dir1: "lattice_of fs \<subseteq> lattice_of (v # fs)"
     by (intro lattice_is_monotone, insert fs v, auto)
   moreover have dir2: "lattice_of (v # fs) \<subseteq> lattice_of fs"
     by (rule already_in_lattice_subset[OF assms])
   ultimately show ?thesis by auto
 qed
 
 
 lemma already_in_lattice_append:
   assumes fs: "set fs \<subseteq> carrier_vec n" and inlattice: "lattice_of gs \<subseteq> lattice_of fs"
   and gs: "set gs \<subseteq> carrier_vec n"
 shows "lattice_of fs = lattice_of (fs @ gs)"
   using assms
 proof (induct gs arbitrary: fs)
   case Nil
   then show ?case by auto
 next
   case (Cons a gs)
   note fs = Cons.prems(1)
   note inlattice = Cons.prems(2)
   note gs = Cons.prems(3)
   have gs_in_fs: "lattice_of gs \<subseteq> lattice_of fs"
     by (meson basic_trans_rules(23) gs lattice_is_monotone local.Cons(3) set_subset_Cons)
   have a: "a \<in> lattice_of (fs @ gs)"
     using basis_in_latticeI fs gs gs_in_fs local.Cons(1) local.Cons(3) by auto
   have "lattice_of (fs @ a # gs) = lattice_of ((a # gs) @ fs)"
     by (rule lattice_of_append, insert fs gs, auto) 
   also have "... = lattice_of (a # (gs @ fs))" by auto
   also have "... = lattice_of (a # (fs @ gs))"
     by (rule lattice_of_eq_set, insert gs fs, auto)
   also have "... = lattice_of (fs @ gs)"
     by (rule already_in_lattice[symmetric, OF _ a], insert fs gs, auto)
   also have "... = lattice_of fs" by (rule Cons.hyps[symmetric, OF fs gs_in_fs], insert gs, auto)     
   finally show ?case ..
 qed
 
 lemma zero_in_lattice:
   assumes fs_carrier: "set fs \<subseteq> carrier_vec n"
   shows "0\<^sub>v n \<in> lattice_of fs"
 proof - 
   have "\<forall>f. lincomb (\<lambda>v. 0 * f v) (set fs) = 0\<^sub>v n"
       using fs_carrier lincomb_closed lincomb_smult lmult_0 by presburger
   hence "lincomb (\<lambda>i. 0) (set fs) = 0\<^sub>v n" by fastforce 
   thus ?thesis unfolding lattice_of_altdef_lincomb[OF fs_carrier] by auto
 qed
 
 
 lemma lattice_zero_rows_subset:
   assumes H: "H \<in> carrier_mat a n"
   shows "lattice_of (Matrix.rows (0\<^sub>m m n)) \<subseteq> lattice_of (Matrix.rows H)"
 proof 
   let ?fs = "Matrix.rows (0\<^sub>m m n)"
   let ?gs = "Matrix.rows H"
   have fs_carrier: "set ?fs \<subseteq> carrier_vec n" unfolding Matrix.rows_def by auto
   have gs_carrier: "set ?gs \<subseteq> carrier_vec n" using H unfolding Matrix.rows_def by auto
   fix x assume x: "x \<in> lattice_of (Matrix.rows (0\<^sub>m m n))"
   obtain f where fx: "lincomb (of_int \<circ> f) (set (Matrix.rows (0\<^sub>m m n))) = x"
     using x lattice_of_altdef_lincomb[OF fs_carrier] by blast
   have "lincomb (of_int \<circ> f) (set (Matrix.rows (0\<^sub>m m n))) = 0\<^sub>v n"
     unfolding lincomb_def by (rule M.finsum_all0, unfold Matrix.rows_def, auto)
   hence "x = 0\<^sub>v n" using fx by auto
   thus "x \<in> lattice_of (Matrix.rows H)" using zero_in_lattice[OF gs_carrier] by auto 
 qed
 
 (*TODO: move outside this context (the previous lemmas too)*)
 lemma lattice_of_append_zero_rows:
   assumes H': "H' \<in> carrier_mat m n"
   and H: "H = H' @\<^sub>r (0\<^sub>m m n)"
 shows "lattice_of (Matrix.rows H) = lattice_of (Matrix.rows H')"
 proof -
   have "Matrix.rows H = Matrix.rows H' @ Matrix.rows (0\<^sub>m m n)"
     by (unfold H, rule rows_append_rows[OF H'], auto)
   also have "lattice_of ... = lattice_of (Matrix.rows H')"
   proof (rule already_in_lattice_append[symmetric])
     show "lattice_of (Matrix.rows (0\<^sub>m m n)) \<subseteq> lattice_of (Matrix.rows H')"
       by (rule lattice_zero_rows_subset[OF H'])
   qed (insert H', auto simp add: Matrix.rows_def)
   finally show ?thesis .
 qed
 end
 
 text \<open>Lemmas about echelon form\<close>
 
 lemma echelon_form_JNF_1xn:
   assumes "A\<in>carrier_mat m n" and "m<2"  
 shows "echelon_form_JNF A"
   using assms unfolding echelon_form_JNF_def is_zero_row_JNF_def by fastforce
 
 
 lemma echelon_form_JNF_mx1:
   assumes "A\<in>carrier_mat m n" and "n<2"
   and "\<forall>i \<in> {1..<m}. A$$(i,0) = 0"
 shows "echelon_form_JNF A"
   using assms unfolding echelon_form_JNF_def is_zero_row_JNF_def
     using atLeastLessThan_iff less_2_cases by fastforce
 
 
 lemma echelon_form_mx0:
   assumes "A \<in> carrier_mat m 0"
   shows "echelon_form_JNF A" using assms unfolding echelon_form_JNF_def is_zero_row_JNF_def by auto
 
 lemma echelon_form_JNF_first_column_0:
   assumes eA: "echelon_form_JNF A" and A: "A \<in> carrier_mat m n"
     and i0: "0<i" and im: "i<m" and n0: "0<n"
   shows "A $$ (i,0) =0"
 proof (rule ccontr)
   assume Ai0: "A $$ (i, 0) \<noteq> 0"
   hence nz_iA:  "\<not> is_zero_row_JNF i A" using n0 A unfolding is_zero_row_JNF_def by auto
   hence nz_0A: "\<not> is_zero_row_JNF 0 A" using eA A unfolding echelon_form_JNF_def using i0 im by auto
   have "(LEAST n. A $$ (0, n) \<noteq> 0) < (LEAST n. A $$ (i, n) \<noteq> 0)"
     using nz_iA nz_0A eA A unfolding echelon_form_JNF_def using i0 im by blast
   moreover have "(LEAST n. A $$ (i, n) \<noteq> 0) = 0" using Ai0 by simp
   ultimately show False by auto
 qed
 
 
 lemma is_zero_row_JNF_multrow[simp]: 
   fixes A::"'a::comm_ring_1 mat"
   assumes "i<dim_row A"
   shows "is_zero_row_JNF i (multrow j (- 1) A) = is_zero_row_JNF i A"
   using assms unfolding is_zero_row_JNF_def by auto
 
 lemma echelon_form_JNF_multrow:
   assumes "A : carrier_mat m n" and "i<m" and eA: "echelon_form_JNF A"
   shows "echelon_form_JNF (multrow i (- 1) A)"
 proof (rule echelon_form_JNF_intro)
   have "A $$ (j, ja) = 0" if  "\<forall>j'<dim_col A. A $$ (ia, j') = 0" 
     and iaj: "ia < j" and j: "j < dim_row A" and ja: "ja < dim_col A" for ia j ja
     using assms that unfolding echelon_form_JNF_def is_zero_row_JNF_def 
     by (meson order.strict_trans) 
   thus " \<forall>ia<dim_row (multrow i (- 1) A). is_zero_row_JNF ia (multrow i (- 1) A) 
       \<longrightarrow> \<not> (\<exists>j<dim_row (multrow i (- 1) A). ia < j \<and> \<not> is_zero_row_JNF j (multrow i (- 1) A))"
     unfolding is_zero_row_JNF_def by simp 
   have Least_eq: "(LEAST n. multrow i (- 1) A $$ (ia, n) \<noteq> 0) = (LEAST n. A $$ (ia, n) \<noteq> 0)"
     if ia: "ia < dim_row A" and nz_ia_mrA: "\<not> is_zero_row_JNF ia (multrow i (- 1) A)" for ia
   proof (rule Least_equality)
     have nz_ia_A: "\<not> is_zero_row_JNF ia A" using nz_ia_mrA ia by auto
     have Least_Aian_n: "(LEAST n. A $$ (ia, n) \<noteq> 0) < dim_col A"
-      by (smt dual_order.strict_trans is_zero_row_JNF_def not_less_Least not_less_iff_gr_or_eq nz_ia_A)
+      by (smt (verit) dual_order.strict_trans is_zero_row_JNF_def not_less_Least not_less_iff_gr_or_eq nz_ia_A)
     show "multrow i (- 1) A $$ (ia, LEAST n. A $$ (ia, n) \<noteq> 0) \<noteq> 0"
-      by (smt LeastI Least_Aian_n class_cring.cring_simprules(22) equation_minus_iff ia
+      by (smt (verit) LeastI Least_Aian_n class_cring.cring_simprules(22) equation_minus_iff ia
           index_mat_multrow(1) is_zero_row_JNF_def mult_minus1 nz_ia_A)
     show " \<And>y. multrow i (- 1) A $$ (ia, y) \<noteq> 0 \<Longrightarrow> (LEAST n. A $$ (ia, n) \<noteq> 0) \<le> y"
       by (metis (mono_tags, lifting) Least_Aian_n class_cring.cring_simprules(22) ia 
           index_mat_multrow(1) leI mult_minus1 order.strict_trans wellorder_Least_lemma(2))
   qed
   have "(LEAST n. multrow i (- 1) A $$ (ia, n) \<noteq> 0) < (LEAST n. multrow i (- 1) A $$ (j, n) \<noteq> 0)"
     if ia_j: "ia < j" and
       j: "j < dim_row A"
       and nz_ia_A: "\<not> is_zero_row_JNF ia A"
       and nz_j_A: "\<not> is_zero_row_JNF j A"
     for ia j
   proof -
     have ia: "ia < dim_row A" using ia_j j by auto
     show ?thesis using Least_eq[OF ia] Least_eq[OF j] nz_ia_A nz_j_A 
         is_zero_row_JNF_multrow[OF ia] is_zero_row_JNF_multrow[OF j] eA ia_j j
       unfolding echelon_form_JNF_def by simp
   qed 
   thus "\<forall>ia j.
        ia < j \<and> j < dim_row (multrow i (- 1) A) \<and> \<not> is_zero_row_JNF ia (multrow i (- 1) A)
         \<and> \<not> is_zero_row_JNF j (multrow i (- 1) A) \<longrightarrow>
        (LEAST n. multrow i (- 1) A $$ (ia, n) \<noteq> 0) < (LEAST n. multrow i (- 1) A $$ (j, n) \<noteq> 0)"
     by auto 
 qed
 
 
 (*The following lemma is already in HOL Analysis (thm echelon_form_imp_upper_triagular),
 but only for square matrices. We prove it here for rectangular matrices.*)
 thm echelon_form_imp_upper_triagular
 
 (*First we prove an auxiliary statement*)
 lemma echelon_form_JNF_least_position_ge_diagonal:
   assumes eA: "echelon_form_JNF A" 
   and A: "A: carrier_mat m n"
   and nz_iA: "\<not> is_zero_row_JNF i A"
   and im: "i<m"
 shows "i\<le>(LEAST n. A $$ (i,n) \<noteq> 0)"
   using nz_iA im
 proof (induct i rule: less_induct)
   case (less i)
   note nz_iA = less.prems(1) 
   note im = less.prems(2)
   show ?case
   proof (cases "i=0")
     case True show ?thesis using True by blast
   next
     case False
     show ?thesis
     proof (rule ccontr)
       assume " \<not> i \<le> (LEAST n. A $$ (i, n) \<noteq> 0)"
       hence i_least: "i > (LEAST n. A $$ (i, n) \<noteq> 0)" by auto
       have nz_i1A: "\<not> is_zero_row_JNF (i-1) A" 
         using nz_iA im False A eA unfolding echelon_form_JNF_def
         by (metis Num.numeral_nat(7) Suc_pred carrier_matD(1) gr_implies_not0 
             lessI linorder_neqE_nat order.strict_trans)  
       have "i-1\<le>(LEAST n. A $$ (i-1,n) \<noteq> 0)" by (rule less.hyps, insert im nz_i1A False, auto)
       moreover have "(LEAST n. A $$ (i,n) \<noteq> 0) > (LEAST n. A $$ (i-1,n) \<noteq> 0)"
         using nz_i1A nz_iA im False A eA unfolding echelon_form_JNF_def by auto   
       ultimately show False using i_least by auto
     qed
   qed
 qed
 
 
 lemma echelon_form_JNF_imp_upper_triangular:
   assumes eA: "echelon_form_JNF A" 
   shows "upper_triangular A"
 proof
   fix i j assume ji: "j<i" and i: "i<dim_row A"
   have A: "A \<in> carrier_mat (dim_row A) (dim_col A)" by auto
   show "A $$ (i,j) = 0"
   proof (cases "is_zero_row_JNF i A")
     case False
     have "i\<le> (LEAST n. A $$(i,n) \<noteq> 0)"
       by (rule echelon_form_JNF_least_position_ge_diagonal[OF eA A False i])
     then show ?thesis 
       using ji not_less_Least order.strict_trans2 by blast
   next
     case True
       (*
         Problem detected: at this point, we don't know if j < dim_col A.
         That is, upper_triangular definition only works for matrices \<in> carrier_mat m n with n\<ge>m.
         The definition is:
            - upper_triangular A \<equiv> \<forall>i < dim_row A. \<forall> j < i. A $$ (i,j) = 0
         But we need here:
            - upper_triangular A \<equiv> \<forall>i < dim_row A. \<forall> j < dim_col A. j < i  \<longrightarrow> A $$ (i,j) = 0
 
         Anyway, the existing definition makes sense since upper triangular is usually
         restricted to square matrices.
       *)
     then show ?thesis unfolding is_zero_row_JNF_def oops
 
 
 (*We do the same with the new definition upper_triangular'*)
 lemma echelon_form_JNF_imp_upper_triangular:
   assumes eA: "echelon_form_JNF A" 
   shows "upper_triangular' A"
 proof
   fix i j assume ji: "j<i" and i: "i<dim_row A" and j: "j<dim_col A"
   have A: "A \<in> carrier_mat (dim_row A) (dim_col A)" by auto
   show "A $$ (i,j) = 0"
   proof (cases "is_zero_row_JNF i A")
     case False
     have "i\<le> (LEAST n. A $$(i,n) \<noteq> 0)"
       by (rule echelon_form_JNF_least_position_ge_diagonal[OF eA A False i])
     then show ?thesis 
       using ji not_less_Least order.strict_trans2 by blast
   next
     case True     
     then show ?thesis unfolding is_zero_row_JNF_def using j by auto
   qed
 qed
 
 
 lemma upper_triangular_append_zero:
   assumes uH: "upper_triangular' H" 
   and H: "H \<in> carrier_mat (m+m) n" and mn: "n\<le>m"
   shows "H = mat_of_rows n (map (Matrix.row H) [0..<m]) @\<^sub>r 0\<^sub>m m n" (is "_ = ?H' @\<^sub>r 0\<^sub>m m n")
 proof 
   have H': "?H' \<in> carrier_mat m n" using H uH by auto
   have H'0: "(?H' @\<^sub>r 0\<^sub>m m n) \<in> carrier_mat (m+m) n" by (simp add: H')
   thus dr: "dim_row H = dim_row (?H' @\<^sub>r 0\<^sub>m m n)" using H H'  by (simp add: append_rows_def) 
   show dc: "dim_col H = dim_col (?H' @\<^sub>r 0\<^sub>m m n)" using H H'  by (simp add: append_rows_def) 
   fix i j assume i: "i < dim_row (?H' @\<^sub>r 0\<^sub>m m n)" and j: "j < dim_col (?H' @\<^sub>r 0\<^sub>m m n)"
   show "H $$ (i, j) = (?H' @\<^sub>r 0\<^sub>m m n) $$ (i, j)"
   proof (cases "i<m")
     case True
     have "H $$ (i, j) = ?H' $$ (i,j)"
       by (metis True H' append_rows_def H carrier_matD index_mat_four_block(3) index_zero_mat(3) j 
           le_add1 map_first_rows_index mat_of_rows_carrier(2) mat_of_rows_index nat_arith.rule0)
     then show ?thesis
       by (metis (mono_tags, lifting) H' True add.comm_neutral append_rows_def 
           carrier_matD(1) i index_mat_four_block index_zero_mat(3) j)
   next
     case False 
     have imn: "i<m+m" using i dr H by auto
     have jn: "j<n" using j dc H by auto
     have ji: "j<i" using j i False mn jn by linarith
     hence "H $$ (i, j) = 0" using uH unfolding upper_triangular'_def dr imn using i jn 
       by (simp add: dc j)
     also have "... = (?H' @\<^sub>r 0\<^sub>m m n) $$ (i, j)"
-      by (smt False H' append_rows_def assms(2) carrier_matD(1) carrier_matD(2) dc imn
+      by (smt (verit) False H' append_rows_def assms(2) carrier_matD(1) carrier_matD(2) dc imn
           index_mat_four_block(1,3) index_zero_mat j less_diff_conv2 linorder_not_less)
     finally show ?thesis .
   qed
 qed
 
 subsubsection \<open>The algorithm is sound\<close>
 
 lemma find_fst_non0_in_row: 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_fst_non0_in_row l A = Some j"
   shows "A $$ (l,j) \<noteq> 0" "l \<le> j" "j < dim_col A"
 proof -
   let ?xs = "filter (\<lambda>j. A $$ (l, j) \<noteq> 0) [l ..< dim_col A]"
   from res[unfolded find_fst_non0_in_row_def Let_def]
   have xs: "?xs \<noteq> []" by (cases ?xs, auto)
   have j_in_xs: "j \<in> set ?xs" using res unfolding find_fst_non0_in_row_def Let_def
     by (metis (no_types, lifting) length_greater_0_conv list.case(2) list.exhaust nth_mem option.simps(1) xs)
   show "A $$ (l,j) \<noteq> 0" "l \<le> j" "j < dim_col A" using j_in_xs by auto+  
 qed
 
 
 lemma find_fst_non0_in_row_zero_before: 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_fst_non0_in_row l A = Some j"
   shows "\<forall>j'\<in>{l..<j}. A $$ (l,j') = 0"
 proof -
   let ?xs = "filter (\<lambda>j. A $$ (l, j) \<noteq> 0) [l ..< dim_col A]"
   from res[unfolded find_fst_non0_in_row_def Let_def]
   have xs: "?xs \<noteq> []" by (cases ?xs, auto)
   have j_in_xs: "j \<in> set ?xs" using res unfolding find_fst_non0_in_row_def Let_def
     by (metis (no_types, lifting) length_greater_0_conv list.case(2) list.exhaust nth_mem option.simps(1) xs)
   have j_xs0: "j = ?xs ! 0"
-    by (smt res[unfolded find_fst_non0_in_row_def Let_def] list.case(2) list.exhaust option.inject xs)
+    by (smt (verit) res[unfolded find_fst_non0_in_row_def Let_def] list.case(2) list.exhaust option.inject xs)
   show "\<forall>j'\<in>{l..<j}. A $$ (l,j') = 0"
   proof (rule+, rule ccontr)
     fix j' assume j': "j' : {l..<j}" and Alj': "A $$ (l, j') \<noteq> 0"
     have j'j: "j'<j" using j' by auto
     have j'_in_xs: "j' \<in> set ?xs"
       by (metis (mono_tags, lifting) A Set.member_filter j' Alj' res atLeastLessThan_iff filter_set
           find_fst_non0_in_row(3) nat_SN.gt_trans set_upt)  
     have l_rw: "[l..<dim_col A] = [l ..<j] @[j..<dim_col A]"
       using assms(1) assms(2) find_fst_non0_in_row(3) j' upt_append by auto
     have xs_rw: "?xs = filter (\<lambda>j. A $$ (l, j) \<noteq> 0) ([l ..<j] @[j..<dim_col A])"
       using l_rw by auto
     hence "filter (\<lambda>j. A $$ (l, j) \<noteq> 0) [l ..<j] = []" using j_xs0 
       by (metis (no_types, lifting) Set.member_filter atLeastLessThan_iff filter_append filter_set
           length_greater_0_conv nth_append nth_mem order_less_irrefl set_upt)
     thus False using j_xs0 j' j_xs0 
       by (metis Set.member_filter filter_empty_conv filter_set j'_in_xs set_upt)
   qed
 qed
 
 
 corollary find_fst_non0_in_row_zero_before': 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_fst_non0_in_row l A = Some j"
   and "j' \<in> {l..<j}"
   shows "A $$ (l,j') = 0" using find_fst_non0_in_row_zero_before assms by auto
 
 lemma find_fst_non0_in_row_LEAST: 
   assumes A: "A \<in> carrier_mat m n"
   and ut_A: "upper_triangular' A"
   and res: "find_fst_non0_in_row l A = Some j"
   and lm: "l<m"
 shows "j = (LEAST n. A $$ (l,n) \<noteq> 0)"
 proof (rule Least_equality[symmetric])
   show " A $$ (l, j) \<noteq> 0" using res find_fst_non0_in_row(1) by blast
   show "\<And>y. A $$ (l, y) \<noteq> 0 \<Longrightarrow> j \<le> y"
   proof (rule ccontr)
     fix y assume Aly: "A $$ (l, y) \<noteq> 0" and jy: " \<not> j \<le> y "
     have yn: "y < n"
       by (metis A jy carrier_matD(2) find_fst_non0_in_row(3) leI less_imp_le_nat nat_SN.compat res)
     have "A $$(l,y) = 0"
     proof (cases "y\<in>{l..<j}")
       case True
       show ?thesis by (rule find_fst_non0_in_row_zero_before'[OF A res True])
     next
       case False hence "y<l" using jy by auto
       thus ?thesis using ut_A A lm unfolding upper_triangular'_def using yn by blast
     qed
     thus False using Aly by contradiction
   qed 
 qed
 
 
 
 lemma find_fst_non0_in_row_None': 
   assumes A: "A \<in> carrier_mat m n"  
   and lm: "l<m"
 shows "(find_fst_non0_in_row l A = None) = (\<forall>j\<in>{l..<dim_col A}. A $$ (l,j) = 0)" (is "?lhs = ?rhs")
 proof
   assume res: ?lhs
   let ?xs = "filter (\<lambda>j. A $$ (l, j) \<noteq> 0) [l ..< dim_col A]"
   from res[unfolded find_fst_non0_in_row_def Let_def]
   have xs: "?xs = []" by (cases ?xs, auto)
   have "A $$ (l, j) = 0" if j: "j\<in>{l..<dim_col A}" for j
     using xs by (metis (mono_tags, lifting) empty_filter_conv j set_upt)
   thus "?rhs" by blast
 next
   assume rhs: ?rhs
   show ?lhs
   proof (rule ccontr)
     assume "find_fst_non0_in_row l A \<noteq> None" 
     from this obtain j where r: "find_fst_non0_in_row l A = Some j" by blast
     hence "A $$ (l,j) \<noteq> 0" and  "l\<le>j" and "j<dim_col A" using find_fst_non0_in_row[OF A r] by blast+
     thus False using rhs by auto  
   qed
 qed
 
 
 lemma find_fst_non0_in_row_None: 
   assumes A: "A \<in> carrier_mat m n"
   and ut_A: "upper_triangular' A"
   and lm: "l<m"
 shows "(find_fst_non0_in_row l A = None) = (is_zero_row_JNF l A)" (is "?lhs = ?rhs")
 proof
   assume res: ?lhs
   let ?xs = "filter (\<lambda>j. A $$ (l, j) \<noteq> 0) [l ..< dim_col A]"
   from res[unfolded find_fst_non0_in_row_def Let_def]
   have xs: "?xs = []" by (cases ?xs, auto)
   have "A $$ (l, j) = 0" if j: "j < dim_col A" for j
   proof (cases "j<l")
     case True
     then show ?thesis using ut_A A lm j unfolding upper_triangular'_def by blast
   next
     case False
     hence j_ln: "j \<in> {l..<dim_col A}" using A lm j by simp
     then show ?thesis using xs by (metis (mono_tags, lifting) empty_filter_conv set_upt)
   qed
   thus "?rhs" unfolding is_zero_row_JNF_def by blast
 next
   assume rhs: ?rhs
   show ?lhs
   proof (rule ccontr)
     assume "find_fst_non0_in_row l A \<noteq> None" 
     from this obtain j where r: "find_fst_non0_in_row l A = Some j" by blast
     hence "A $$ (l,j) \<noteq> 0" and "j<dim_col A" using find_fst_non0_in_row[OF A r] by blast+
     hence "\<not> is_zero_row_JNF l A" unfolding is_zero_row_JNF_def using lm A by auto
     thus False using rhs by contradiction    
   qed
 qed
 
 lemma make_first_column_positive_preserves_dimensions:
   shows [simp]: "dim_row (make_first_column_positive A) = dim_row A" 
     and [simp]: "dim_col (make_first_column_positive A) = dim_col A"
   by (auto)
 
 
 lemma make_first_column_positive_works: 
   assumes "A\<in>carrier_mat m n" and i: "i<m" and "0<n"
   shows "make_first_column_positive A $$ (i,0) \<ge> 0"
   and "j<n \<Longrightarrow> A $$ (i,0) < 0 \<Longrightarrow> (make_first_column_positive A) $$ (i,j) = - A $$ (i,j)"
   and "j<n \<Longrightarrow> A $$ (i,0) \<ge> 0 \<Longrightarrow> (make_first_column_positive A) $$ (i,j) = A $$ (i,j)"
   using assms by auto 
 
 
 lemma make_first_column_positive_invertible: 
   shows  "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (dim_row A) (dim_row A) 
   \<and> make_first_column_positive A = P * A" 
 proof -
   let ?P = "Matrix.mat (dim_row A) (dim_row A)
           (\<lambda>(i,j). if i = j then if A $$(i,0) < 0 then - 1 else 1 else 0::int)"
   have "invertible_mat ?P"
   proof -
     have "(map abs (diag_mat ?P)) = replicate (length ((map abs (diag_mat ?P)))) 1" 
       by (rule replicate_length_same[symmetric], auto simp add: diag_mat_def)
     hence m_rw: "(map abs (diag_mat ?P)) = replicate (dim_row A) 1" by (auto simp add: diag_mat_def)
     have "Determinant.det ?P = prod_list (diag_mat ?P)" by (rule det_upper_triangular, auto)
     also have "abs ... = prod_list (map abs (diag_mat ?P))" unfolding prod_list_abs by blast
     also have " ... = prod_list (replicate (dim_row A) 1)" using m_rw by simp
     also have "... = 1" by auto
     finally have "\<bar>Determinant.det ?P\<bar> = 1" by blast
     hence "Determinant.det ?P dvd 1" by fastforce
     thus ?thesis using invertible_iff_is_unit_JNF mat_carrier by blast (*Thanks to the new bridge*)
   qed    
   moreover have "make_first_column_positive A = ?P * A" (is "?M = _")
   proof (rule eq_matI)
     show "dim_row ?M = dim_row (?P * A)" and "dim_col ?M = dim_col (?P * A)" by auto
     fix i j assume i: "i < dim_row (?P * A)" and j: "j < dim_col (?P * A)"
     have set_rw: "{0..<dim_row A} = insert i ({0..<dim_row A} - {i})" using i by auto
       have rw0: "(\<Sum>ia \<in> {0..<dim_row A } - {i}. Matrix.row ?P i $v ia * col A j $v ia) = 0"
         by (rule sum.neutral, insert i, auto)        
     have "(?P * A) $$ (i, j) = Matrix.row ?P i \<bullet> col A j" using i j by auto
     also have "... = (\<Sum>ia = 0..<dim_row A. Matrix.row ?P i $v ia * col A j $v ia)"
         unfolding scalar_prod_def by auto
       also have "... =  (\<Sum>ia \<in> insert i ({0..<dim_row A} - {i}). Matrix.row ?P i $v ia * col A j $v ia)"
         using set_rw by argo
       also have "... = Matrix.row ?P i $v i * col A j $v i 
         + (\<Sum>ia \<in> {0..<dim_row A } - {i}. Matrix.row ?P i $v ia * col A j $v ia)" 
         by (rule sum.insert, auto)
       also have "... = Matrix.row ?P i $v i * col A j $v i" unfolding rw0 by simp
       finally have *: "(?P * A) $$ (i, j) = Matrix.row ?P i $v i * col A j $v i" .
     also have "... = ?M $$ (i,j)" 
       by (cases " A $$ (i, 0) < 0", insert i j, auto simp add: col_def)
     finally show "?M $$ (i, j) = (?P * A) $$ (i, j)" ..
   qed
   moreover have "?P \<in> carrier_mat (dim_row A) (dim_row A)" by auto
   ultimately show ?thesis by blast
 qed
 
 locale proper_mod_operation = mod_operation +
   assumes dvd_gdiv_mult_right[simp]: "b > 0 \<Longrightarrow> b dvd a \<Longrightarrow> (a gdiv b) * b = a"
     and gmod_gdiv: "y > 0 \<Longrightarrow> x gmod y = x - x gdiv y * y"
     and dvd_imp_gmod_0: "0 < a \<Longrightarrow> a dvd b \<Longrightarrow> b gmod a = 0" 
     and gmod_0_imp_dvd: "a gmod b = 0 \<Longrightarrow> b dvd a" 
     and gmod_0[simp]: "n gmod 0 = n" "n > 0 \<Longrightarrow> 0 gmod n = 0"
 begin
 lemma reduce_alt_def_not0: 
   assumes "A $$ (a,0) \<noteq> 0" and pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A $$ (b,0))"
   shows "reduce a b D A =
        Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                               if k = 0 then if D dvd r then D else r else r gmod D
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                               if k = 0 then r else r gmod D
                    else A$$(i,k))" (is "_ = ?rhs")
   and 
    "reduce_abs a b D A =
        Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                               if abs r > D then if k = 0 \<and> D dvd r then D else r gmod D else r 
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                               if abs r > D then r gmod D else r
                    else A$$(i,k))" (is "_ = ?rhs_abs")
 proof -
   have "reduce a b D A =
        (case euclid_ext2 (A$$(a,0)) (A $$ (b,0)) of (p,q,u,v,d) \<Rightarrow>
        Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                                if k = 0 then if D dvd r then D else r else r gmod D
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                                 if k = 0 then r else r gmod D
                    else A$$(i,k)
             ))" using assms by auto
   also have "... = ?rhs" unfolding reduce.simps Let_def 
     by (rule eq_matI, insert pquvd) (metis (no_types, lifting) split_conv)+
   finally show "reduce a b D A = ?rhs" .
   have "reduce_abs a b D A =
        (case euclid_ext2 (A$$(a,0)) (A $$ (b,0)) of (p,q,u,v,d) \<Rightarrow>
        Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then let r = (p*A$$(a,k) + q*A$$(b,k)) in
                                if abs r > D then if k = 0 \<and> D dvd r then D else r gmod D else r 
                    else if i = b then let r = u * A$$(a,k) + v * A$$(b,k) in
                                if abs r > D then r gmod D else r
                    else A$$(i,k)
             ))" using assms by auto
   also have "... = ?rhs_abs" unfolding reduce.simps Let_def 
     by (rule eq_matI, insert pquvd) (metis (no_types, lifting) split_conv)+
   finally show "reduce_abs a b D A = ?rhs_abs" .
 qed
 
 lemma reduce_preserves_dimensions:
   shows [simp]: "dim_row (reduce a b D A) = dim_row A" 
     and [simp]: "dim_col (reduce a b D A) = dim_col A"
   and [simp]: "dim_row (reduce_abs a b D A) = dim_row A" 
     and [simp]: "dim_col (reduce_abs a b D A) = dim_col A"
   by (auto simp add: Let_def split_beta)
 
 lemma reduce_carrier:
   assumes "A \<in> carrier_mat m n"
   shows "(reduce a b D A) \<in> carrier_mat m n" 
     and "(reduce_abs a b D A) \<in> carrier_mat m n" 
   by (insert assms, auto simp add: Let_def split_beta)
 
 lemma reduce_gcd: 
   assumes A: "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n" 
   and Aaj: "A $$ (a,0) \<noteq> 0"
 shows "(reduce a b D A) $$ (a,0) = (let r = gcd (A$$(a,0)) (A$$(b,0)) in if D dvd r then D else r)" (is "?lhs = ?rhs")
   and "(reduce_abs a b D A) $$ (a,0) = (let r = gcd (A$$(a,0)) (A$$(b,0)) in if D < r then
                       if D dvd r then D else r gmod D else r)" (is "?lhs_abs = ?rhs_abs")
 proof -
   obtain p q u v d where pquvd: "euclid_ext2 (A$$(a,0)) (A$$(b,0)) = (p,q,u,v,d)"
     using prod_cases5 by blast
   have "p * A $$ (a, 0) + q * A $$ (b, 0) = d" 
     using Aaj pquvd is_bezout_ext_euclid_ext2 unfolding is_bezout_ext_def 
-    by (smt Pair_inject bezout_coefficients_fst_snd euclid_ext2_def)
+    by (smt (verit) Pair_inject bezout_coefficients_fst_snd euclid_ext2_def)
   also have " ... = gcd (A$$(a,0)) (A$$(b,0))" by (metis euclid_ext2_def pquvd prod.sel(2))
   finally have pAaj_qAbj_gcd: "p * A $$ (a, 0) + q * A $$ (b, 0) = gcd (A$$(a,0)) (A$$(b,0))" .
   let ?f = "(\<lambda>(i, k). if i = a then let r = p * A $$ (a, k) + q * A $$ (b, k) in if k = 0 then if D dvd r then D else r else r gmod D
               else if i = b then let r = u * A $$ (a, k) + v * A $$ (b, k) in if k = 0 then r else r gmod D else A $$ (i, k))"
   have "(reduce a b D A) $$ (a,0) = Matrix.mat (dim_row A) (dim_col A) ?f $$ (a, 0)"
     using Aaj pquvd by auto 
   also have "... = (let r = p * A $$ (a, 0) + q * A $$ (b, 0) in if (0::nat) = 0 then if D dvd r then D else r else r gmod D)"
     using A a j by auto
   also have "... = (if D dvd gcd (A$$(a,0)) (A$$(b,0)) then D else 
       gcd (A$$(a,0)) (A$$(b,0)))" 
     by (simp add: pAaj_qAbj_gcd)
   finally show "?lhs = ?rhs" by auto
   let ?g = "(\<lambda>(i, k). if i = a then let r = p * A $$ (a, k) + q * A $$ (b, k) in 
                 if D < \<bar>r\<bar> then if k = 0 \<and> D dvd r then D else r gmod D else r
               else if i = b then let r = u * A $$ (a, k) + v * A $$ (b, k) in 
                     if D < \<bar>r\<bar> then r gmod D else r else A $$ (i, k))"
   have "(reduce_abs a b D A) $$ (a,0) = Matrix.mat (dim_row A) (dim_col A) ?g $$ (a, 0)"
     using Aaj pquvd by auto 
   also have "... = (let r = p * A $$ (a, 0) + q * A $$ (b, 0) in if D < \<bar>r\<bar> then
             if (0::nat) = 0 \<and> D dvd r then D else r gmod D else r)"
     using A a j by auto
   also have "... = (if D < \<bar>gcd (A$$(a,0)) (A$$(b,0))\<bar> then if D dvd gcd (A$$(a,0)) (A$$(b,0)) then D else 
       gcd (A$$(a,0)) (A$$(b,0)) gmod D else gcd (A$$(a,0)) (A$$(b,0)))"
     by (simp add: pAaj_qAbj_gcd)
   finally show "?lhs_abs = ?rhs_abs" by auto
 qed
 
 
 
 
 lemma reduce_preserves: 
   assumes A: "A \<in> carrier_mat m n" and j: "j<n" 
   and Aaj: "A $$ (a,0) \<noteq> 0" and ib: "i\<noteq>b" and ia: "i\<noteq>a" and im: "i<m"
 shows "(reduce a b D A) $$ (i,j) = A $$ (i,j)"  (is "?thesis1")
 and "(reduce_abs a b D A) $$ (i,j) = A $$ (i,j)" (is "?thesis2") 
 proof -
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(b,0))"
     using prod_cases5 by metis
   show ?thesis1 unfolding reduce_alt_def_not0[OF Aaj pquvd] using ia im j A ib by auto
   show ?thesis2 unfolding reduce_alt_def_not0[OF Aaj pquvd] using ia im j A ib by auto
 qed
 
 
 lemma reduce_0: 
   assumes A: "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n" and b: "b<m" and ab: "a \<noteq> b"
   and Aaj: "A $$ (a,0) \<noteq> 0"
   and D: "D \<ge> 0" 
 shows "(reduce a b D A) $$ (b,0) = 0" (is "?thesis1")
 and "(reduce_abs a b D A) $$ (b,0) = 0" (is "?thesis2")
 proof -
   obtain p q u v d where pquvd: "euclid_ext2 (A$$(a,0)) (A$$(b,0)) = (p,q,u,v,d)"
     using prod_cases5 by blast
   hence u: "u = - (A$$(b,0)) div gcd (A$$(a,0)) (A$$(b,0))"
     using euclid_ext2_works[OF pquvd] by auto
   have v: "v = A$$(a,0) div gcd (A$$(a,0)) (A$$(b,0))" using euclid_ext2_works[OF pquvd] by auto
   have uv0: "u * A$$(a,0) + v * A$$(b,0) = 0" using u v
   proof -
     have "\<forall>i ia. gcd (ia::int) i * (ia div gcd ia i) = ia"
     by (meson dvd_mult_div_cancel gcd_dvd1)
     then have "v * - A $$ (b, 0) = u * A $$ (a, 0)"
       by (metis (no_types) dvd_minus_iff dvd_mult_div_cancel gcd_dvd2 minus_minus mult.assoc mult.commute u v)
     then show ?thesis
       by simp
   qed
   let ?f = "(\<lambda>(i, k). if i = a then let r = p * A $$ (a, k) + q * A $$ (b, k) in 
             if k = 0 then if D dvd r then D else r else r gmod D
               else if i = b then let r = u * A $$ (a, k) + v * A $$ (b, k) in 
                    if k = 0 then r else r gmod D else A $$ (i, k))" 
   have "(reduce a b D A) $$ (b,0) = Matrix.mat (dim_row A) (dim_col A) ?f $$ (b, 0)"
     using Aaj pquvd by auto 
   also have "... = (let r = u * A$$(a,0) + v * A$$(b,0) in r)"
     using A a j ab b by auto
   also have "... = 0" using uv0 D 
-    by (smt (z3) gmod_0(1) gmod_0(2)) 
+    by (smt (verit) gmod_0(1) gmod_0(2)) 
   finally show ?thesis1 .
   let ?g = "(\<lambda>(i, k). if i = a then let r = p * A $$ (a, k) + q * A $$ (b, k) in 
           if D < \<bar>r\<bar> then if k = 0 \<and> D dvd r then D else r gmod D else r
               else if i = b then let r = u * A $$ (a, k) + v * A $$ (b, k) in 
                   if D < \<bar>r\<bar> then r gmod D else r else A $$ (i, k))" 
   have "(reduce_abs a b D A) $$ (b,0) = Matrix.mat (dim_row A) (dim_col A) ?g $$ (b, 0)"
     using Aaj pquvd by auto 
   also have "... = (let r = u * A$$(a,0) + v * A$$(b,0) in if D < \<bar>r\<bar> then r gmod D else r)"
     using A a j ab b by auto
   also have "... = 0" using uv0 D by simp
   finally show ?thesis2 .
 qed
 end
 
 
 text \<open>Let us show the key lemma: operations modulo determinant don't modify the (integer) row span.\<close>
 
 context LLL_with_assms
 begin
 
 lemma lattice_of_kId_subset_fs_init: 
   assumes k_det: "k = Determinant.det (mat_of_rows n fs_init)"
   and mn: "m=n"
   shows "lattice_of (Matrix.rows (k \<cdot>\<^sub>m (1\<^sub>m m))) \<subseteq> lattice_of fs_init"
 proof -
   let ?Z = "(mat_of_rows n fs_init)"
   let ?RAT = "of_int_hom.mat_hom :: int mat \<Rightarrow> rat mat"
   have RAT_fs_init: "?RAT (mat_of_rows n fs_init) \<in> carrier_mat n n"
       using len map_carrier_mat mat_of_rows_carrier(1) mn by blast
   have det_RAT_fs_init: "Determinant.det (?RAT ?Z) \<noteq> 0"
   proof (rule gs.lin_indpt_rows_imp_det_not_0[OF RAT_fs_init])   
     have rw: "Matrix.rows (?RAT (mat_of_rows n fs_init)) = RAT fs_init"
       by (metis cof_vec_space.lin_indpt_list_def fs_init lin_dep mat_of_rows_map rows_mat_of_rows)
     thus "gs.lin_indpt (set (Matrix.rows (?RAT (mat_of_rows n fs_init))))" 
       by (insert lin_dep, simp add: cof_vec_space.lin_indpt_list_def)
     show "distinct (Matrix.rows (?RAT (mat_of_rows n fs_init)))"
       using rw cof_vec_space.lin_indpt_list_def lin_dep by auto
   qed
   obtain inv_Z where inverts_Z: "inverts_mat (?RAT ?Z) inv_Z" and inv_Z: "inv_Z \<in> carrier_mat m m"
     by (metis mn det_RAT_fs_init dvd_field_iff invertible_iff_is_unit_JNF
         len map_carrier_mat mat_of_rows_carrier(1) obtain_inverse_matrix)
   have det_rat_Z_k: "Determinant.det (?RAT ?Z) = rat_of_int k"
     using k_det of_int_hom.hom_det by blast
   have "?RAT ?Z *  adj_mat (?RAT ?Z) = Determinant.det (?RAT ?Z) \<cdot>\<^sub>m 1\<^sub>m n" 
     by (rule adj_mat[OF RAT_fs_init])
   hence "inv_Z * (?RAT ?Z *  adj_mat (?RAT ?Z)) = inv_Z * (Determinant.det (?RAT ?Z) \<cdot>\<^sub>m 1\<^sub>m n)" by simp
   hence k_inv_Z_eq_adj: "(rat_of_int k) \<cdot>\<^sub>m inv_Z = adj_mat (?RAT ?Z)"
-    by (smt Determinant.mat_mult_left_right_inverse RAT_fs_init adj_mat(1,3) mn 
+    by (smt (verit) Determinant.mat_mult_left_right_inverse RAT_fs_init adj_mat(1,3) mn 
         carrier_matD det_RAT_fs_init det_rat_Z_k gs.det_nonzero_congruence inv_Z inverts_Z 
         inverts_mat_def mult_smult_assoc_mat smult_carrier_mat)
   have adj_mat_Z: "adj_mat (?RAT ?Z) $$ (i,j) \<in> \<int>" if i: "i<m" and j: "j<n" for i j
   proof -
     have det_mat_delete_Z: "Determinant.det (mat_delete (?RAT ?Z) j i) \<in> \<int>"
     proof (rule Ints_det)
       fix ia ja
       assume ia: "ia < dim_row  (mat_delete (?RAT ?Z) j i)"
         and ja: "ja < dim_col  (mat_delete (?RAT ?Z) j i)"
       have "(mat_delete (?RAT ?Z) j i) $$ (ia, ja) =  (?RAT ?Z) $$ (insert_index j ia, insert_index i ja)"        
         by (rule mat_delete_index[symmetric], insert i j mn len ia ja RAT_fs_init, auto)
       also have "... = rat_of_int (?Z $$ (insert_index j ia, insert_index i ja))"
         by (rule index_map_mat, insert i j ia ja, auto simp add: insert_index_def)
       also have "... \<in> \<int>" using Ints_of_int by blast
       finally show "(mat_delete (?RAT ?Z) j i) $$ (ia, ja) \<in> \<int>" .
     qed
     have "adj_mat (?RAT ?Z) $$ (i,j) = Determinant.cofactor (?RAT ?Z) j i"
       unfolding adj_mat_def
       by (simp add: len i j)
     also have "... =  (- 1) ^ (j + i) * Determinant.det (mat_delete (?RAT ?Z) j i)"
       unfolding Determinant.cofactor_def by auto
     also have "... \<in> \<int>" using det_mat_delete_Z by auto
     finally show ?thesis .
   qed                
   have kinvZ_in_Z: "((rat_of_int k) \<cdot>\<^sub>m inv_Z) $$ (i,j) \<in> \<int>" if i: "i<m" and j: "j<n" for i j
     using k_inv_Z_eq_adj by (simp add: adj_mat_Z i j)
   have "?RAT (k \<cdot>\<^sub>m (1\<^sub>m m)) = Determinant.det (?RAT ?Z) \<cdot>\<^sub>m (inv_Z * ?RAT ?Z)" (is "?lhs = ?rhs")
   proof - 
     have "(inv_Z * ?RAT ?Z) = (1\<^sub>m m)"
       by (metis Determinant.mat_mult_left_right_inverse RAT_fs_init mn carrier_matD(1)
           inv_Z inverts_Z inverts_mat_def)
     from this have "?rhs = rat_of_int k \<cdot>\<^sub>m (1\<^sub>m m)" using det_rat_Z_k by auto
     also have "... = ?lhs" by auto
     finally show ?thesis ..
   qed
   also have "... = (Determinant.det (?RAT ?Z) \<cdot>\<^sub>m inv_Z) * ?RAT ?Z"
     by (metis RAT_fs_init mn inv_Z mult_smult_assoc_mat)
   also have "... = ((rat_of_int k) \<cdot>\<^sub>m inv_Z) * ?RAT ?Z" by (simp add: k_det)
   finally have r': "?RAT (k \<cdot>\<^sub>m (1\<^sub>m m)) = ((rat_of_int k) \<cdot>\<^sub>m inv_Z) * ?RAT ?Z" .
   have r: "(k \<cdot>\<^sub>m (1\<^sub>m m)) = ((map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z))) * ?Z"
   proof -
     have "?RAT ((map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z))) = ((rat_of_int k) \<cdot>\<^sub>m inv_Z)"
     proof (rule eq_matI, auto)
       fix i j assume i: "i < dim_row inv_Z" and j: "j < dim_col inv_Z"
       have "((rat_of_int k) \<cdot>\<^sub>m inv_Z) $$ (i,j) =  (rat_of_int k * inv_Z $$ (i, j))"
         using index_smult_mat i j by auto
       hence kinvZ_in_Z': "... \<in> \<int>" using kinvZ_in_Z i j inv_Z mn by simp
       show "rat_of_int (int_of_rat (rat_of_int k * inv_Z $$ (i, j))) = rat_of_int k * inv_Z $$ (i, j)" 
         by (rule int_of_rat, insert kinvZ_in_Z', auto)
     qed
     hence "?RAT (k \<cdot>\<^sub>m (1\<^sub>m m)) = ?RAT ((map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z))) * ?RAT ?Z"
       using r' by simp
     also have "... = ?RAT ((map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z)) * ?Z)"
       by (metis RAT_fs_init adj_mat(1) k_inv_Z_eq_adj map_carrier_mat of_int_hom.mat_hom_mult)
     finally show ?thesis by (rule of_int_hom.mat_hom_inj)
   qed
   show ?thesis
   proof (rule mat_mult_sub_lattice[OF _ fs_init])
     have rw: "of_int_hom.mat_hom (map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z)) 
       = map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z)" by auto
     have "mat_of_rows n (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m)) = (k \<cdot>\<^sub>m (1\<^sub>m m))" 
       by (metis mn index_one_mat(3) index_smult_mat(3) mat_of_rows_rows)
     also have "... = of_int_hom.mat_hom (map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z)) * mat_of_rows n fs_init" 
        using r rw by auto 
     finally show "mat_of_rows n (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m)) 
       = of_int_hom.mat_hom (map_mat int_of_rat ((rat_of_int k) \<cdot>\<^sub>m inv_Z)) * mat_of_rows n fs_init" .
    show "set (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m)) \<subseteq> carrier_vec n"using mn unfolding Matrix.rows_def by auto
    show "map_mat int_of_rat (rat_of_int k \<cdot>\<^sub>m inv_Z) \<in> carrier_mat (length (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m))) (length fs_init)"
      using len fs_init by (simp add: inv_Z)
   qed
 qed
 
 end
 
 context LLL_with_assms
 begin
 
 
 lemma lattice_of_append_det_preserves:  
   assumes k_det: "k = abs (Determinant.det (mat_of_rows n fs_init))"
   and mn: "m = n"
   and A: "A = (mat_of_rows n fs_init) @\<^sub>r (k \<cdot>\<^sub>m (1\<^sub>m m))"
 shows "lattice_of (Matrix.rows A) = lattice_of fs_init"
 proof -
   have "Matrix.rows (mat_of_rows n fs_init @\<^sub>r k \<cdot>\<^sub>m 1\<^sub>m m) = (Matrix.rows (mat_of_rows n fs_init) @ Matrix.rows (k \<cdot>\<^sub>m (1\<^sub>m m)))"
     by (rule rows_append_rows, insert fs_init len mn, auto)
   also have "... = (fs_init @ Matrix.rows (k \<cdot>\<^sub>m (1\<^sub>m m)))" by (simp add: fs_init)
   finally have rw: "Matrix.rows (mat_of_rows n fs_init @\<^sub>r k \<cdot>\<^sub>m 1\<^sub>m m) 
     = (fs_init @ Matrix.rows (k \<cdot>\<^sub>m (1\<^sub>m m)))" .
   have "lattice_of (Matrix.rows A) = lattice_of (fs_init @ Matrix.rows (k \<cdot>\<^sub>m (1\<^sub>m m)))"
     by (rule arg_cong[of _ _ lattice_of], auto simp add: A rw)
   also have "... = lattice_of fs_init" 
   proof (cases "k = Determinant.det (mat_of_rows n fs_init)")
     case True
     then show ?thesis 
     by (rule already_in_lattice_append[symmetric, OF fs_init 
              lattice_of_kId_subset_fs_init[OF _ mn]], insert mn, auto simp add: Matrix.rows_def)
   next
     case False
     hence k2: "k = -Determinant.det (mat_of_rows n fs_init)" using k_det by auto
     have l: "lattice_of (Matrix.rows (- k \<cdot>\<^sub>m 1\<^sub>m m)) \<subseteq> lattice_of fs_init"
       by (rule lattice_of_kId_subset_fs_init[OF _ mn], insert k2, auto)
     have l2: "lattice_of (Matrix.rows (- k \<cdot>\<^sub>m 1\<^sub>m m)) = lattice_of (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m))" 
     proof (rule mat_mult_invertible_lattice_eq)
       let ?P = "(- 1::int) \<cdot>\<^sub>m 1\<^sub>m m"
       show P: "?P \<in> carrier_mat m m" by simp
       have "det ?P = 1 \<or> det ?P = -1" unfolding det_smult by (auto simp add: minus_1_power_even)
-      hence "det ?P dvd 1" by (smt minus_dvd_iff one_dvd)
+      hence "det ?P dvd 1" by (smt (verit) minus_dvd_iff one_dvd)
       thus " invertible_mat ?P" unfolding invertible_iff_is_unit_JNF[OF P] .
       have "(- k \<cdot>\<^sub>m 1\<^sub>m m) = ?P * (k \<cdot>\<^sub>m 1\<^sub>m m)"
         unfolding mat_diag_smult[symmetric] unfolding mat_diag_diag by auto
       thus " mat_of_rows n (Matrix.rows (- k \<cdot>\<^sub>m 1\<^sub>m m)) = of_int_hom.mat_hom ?P * mat_of_rows n (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m))"
         by (metis mn index_one_mat(3) index_smult_mat(3) mat_of_rows_rows of_int_mat_hom_int_id)
       show " set (Matrix.rows (- k \<cdot>\<^sub>m 1\<^sub>m m)) \<subseteq> carrier_vec n"
         and "set (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m)) \<subseteq> carrier_vec n"
         using assms(2) one_carrier_mat set_rows_carrier smult_carrier_mat by blast+
     qed (insert mn, auto)
     hence l2: "lattice_of (Matrix.rows (k \<cdot>\<^sub>m 1\<^sub>m m)) \<subseteq> lattice_of fs_init" using l by auto
     show ?thesis by (rule already_in_lattice_append[symmetric, OF fs_init l2],
           insert mn one_carrier_mat set_rows_carrier smult_carrier_mat, blast)
   qed  
   finally show ?thesis .
 qed
 
 text \<open>This is another key lemma.
 Here, $A$ is the initial matrix @{text "(mat_of_rows n fs_init)"} augmented with $m$ rows 
 $(k,0,\dots,0),(0,k,0,\dots,0), \dots , (0,\dots,0,k)$ where $k$ is the determinant of 
 @{text "(mat_of_rows n fs_init)"}. 
 With the algorithm of the article, we obtain @{text "H = H' @\<^sub>r (0\<^sub>m m n)"} by means of an invertible 
 matrix $P$ (which is computable). Then, $H$ is the HNF of $A$.
 The lemma shows that $H'$ is the HNF of @{text "(mat_of_rows n fs_init)"}
 and that there exists an invertible matrix to carry out the transformation.\<close>
 
 lemma Hermite_append_det_id:
   assumes k_det: "k = abs (Determinant.det (mat_of_rows n fs_init))"
   and mn: "m = n"
   and A: "A = (mat_of_rows n fs_init) @\<^sub>r (k \<cdot>\<^sub>m (1\<^sub>m m))"
   and H': "H'\<in> carrier_mat m n"
   and H_append: "H = H' @\<^sub>r (0\<^sub>m m n)"
   and P: "P \<in> carrier_mat (m+m) (m+m)"
   and inv_P: "invertible_mat P"
   and A_PH: "A = P * H"
   and HNF_H: "Hermite_JNF associates res H"
 shows "Hermite_JNF associates res H'" 
   and "(\<exists>P'. invertible_mat P' \<and> P' \<in> carrier_mat m m \<and> (mat_of_rows n fs_init) = P' * H')"
 proof -
   have A_carrier: "A \<in> carrier_mat (m+m) n" using A mn len by auto
   let ?A' = "(mat_of_rows n fs_init)"
   let ?H' = "submatrix H {0..<m} {0..<n}"
   have nm: "n\<le>m" by (simp add: mn) 
   have H: "H \<in> carrier_mat (m + m) n" using H_append H' by auto
   have submatrix_carrier: "submatrix H {0..<m} {0..<n} \<in> carrier_mat m n"
     by (rule submatrix_carrier_first[OF H], auto)
   have H'_eq: "H' = ?H'"
   proof (rule eq_matI)
     fix i j assume i: "i < dim_row ?H'" and j: "j < dim_col ?H'"
     have im: "i<m" and jn: "j<n" using i j submatrix_carrier by auto
     have "?H' $$ (i,j) = H $$ (i,j)"
       by (rule submatrix_index_id[OF H], insert i j submatrix_carrier, auto)
     also have "... =  (if i < dim_row H' then H' $$ (i, j) else (0\<^sub>m m n) $$ (i - m, j))"
       unfolding H_append by (rule append_rows_nth[OF H'], insert im jn, auto)
     also have "... = H' $$ (i,j)" using H' im jn by simp
     finally show "H' $$ (i, j) = ?H' $$ (i, j)" ..
   qed (insert H' submatrix_carrier, auto)  
   show HNF_H': "Hermite_JNF associates res H'"
     unfolding H'_eq mn by (rule HNF_submatrix[OF HNF_H H], insert nm, simp)
   have L_fs_init_A: "lattice_of (fs_init) = lattice_of (Matrix.rows A)" 
     by (rule lattice_of_append_det_preserves[symmetric, OF k_det mn A])
   have L_H'_H: "lattice_of (Matrix.rows H') = lattice_of (Matrix.rows H)"
     using H_append H' lattice_of_append_zero_rows by blast
   have L_A_H: "lattice_of (Matrix.rows A) = lattice_of (Matrix.rows H)"
   proof (rule mat_mult_invertible_lattice_eq[OF _ _ P inv_P])
     show "set (Matrix.rows A) \<subseteq> carrier_vec n" using A_carrier set_rows_carrier by blast
     show "set (Matrix.rows H) \<subseteq> carrier_vec n" using H set_rows_carrier by blast
     show "length (Matrix.rows A) = m + m" using A_carrier by auto      
     show "length (Matrix.rows H) = m + m" using H by auto
     show "mat_of_rows n (Matrix.rows A) = of_int_hom.mat_hom P * mat_of_rows n (Matrix.rows H)"      
       by (metis A_carrier H A_PH carrier_matD(2) mat_of_rows_rows of_int_mat_hom_int_id)
   qed
   have L_fs_init_H': "lattice_of fs_init = lattice_of (Matrix.rows H')"
     using L_fs_init_A L_A_H L_H'_H by auto
   have exists_P2: 
       "\<exists>P2. P2 \<in> carrier_mat n n \<and> invertible_mat P2 \<and> mat_of_rows n  (Matrix.rows H') = P2 * H'"
     by (rule exI[of _ "1\<^sub>m n"], insert H' mn, auto)
   have exist_P': "\<exists>P'\<in>carrier_mat n n. invertible_mat P' 
       \<and> mat_of_rows n fs_init = P' * mat_of_rows n (Matrix.rows H')"
     by (rule eq_lattice_imp_mat_mult_invertible_rows[OF fs_init _ lin_dep len[unfolded mn] _ L_fs_init_H'],
         insert H' mn set_rows_carrier, auto)
   thus "\<exists>P'. invertible_mat P' \<and> P' \<in> carrier_mat m m \<and> (mat_of_rows n fs_init) = P' * H'"
     by (metis mn H' carrier_matD(2) mat_of_rows_rows)
 qed
 end
 
 
 
 context proper_mod_operation
 begin
 
 (* Perform the modulo D operation to reduce the element A$$(a,j), assuming A = A' @\<^sub>r  (D \<cdot>\<^sub>m (1\<^sub>m m))*)
 definition "reduce_element_mod_D (A::int mat) a j D m =  
   (if j = 0 then if D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A else A
   else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)"
 
 definition "reduce_element_mod_D_abs (A::int mat) a j D m =  
   (if j = 0 \<and> D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A 
   else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)"
 
 lemma reduce_element_mod_D_preserves_dimensions:
   shows [simp]: "dim_row (reduce_element_mod_D A a j D m) = dim_row A" 
     and [simp]: "dim_col (reduce_element_mod_D A a j D m) = dim_col A"
     and [simp]: "dim_row (reduce_element_mod_D_abs A a j D m) = dim_row A" 
     and [simp]: "dim_col (reduce_element_mod_D_abs A a j D m) = dim_col A"
   by (auto simp add: reduce_element_mod_D_def reduce_element_mod_D_abs_def Let_def split_beta)
 
 lemma reduce_element_mod_D_carrier:
   shows "reduce_element_mod_D A a j D m \<in> carrier_mat (dim_row A) (dim_col A)" 
     and "reduce_element_mod_D_abs A a j D m \<in> carrier_mat (dim_row A) (dim_col A)" by auto
 
 
 lemma reduce_element_mod_D_invertible_mat:
   assumes A_def: "A = A' @\<^sub>r  (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n" and mn: "m\<ge>n"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D A a j D m = P * A" (is ?thesis1)
     and "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D_abs A a j D m = P * A" (is ?thesis2)
   unfolding atomize_conj
 proof (rule conjI; cases "j = 0 \<and> D dvd A$$(a,j)")
   case True
   let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D) + 1) a (j + m)"
   have A: "A \<in> carrier_mat (m + n) n" using A_def A' mn by auto
   have "reduce_element_mod_D A a j D m = addrow (- (A $$ (a, j) gdiv D) + 1) a (j + m) A"
     unfolding reduce_element_mod_D_def using True by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have "reduce_element_mod_D A a j D m = ?P * A" .
   moreover have P: "?P \<in> carrier_mat (m+n) (m+n)" by simp
   moreover have inv_P: "invertible_mat ?P"
     by (metis addrow_mat_carrier a det_addrow_mat dvd_mult_right 
         invertible_iff_is_unit_JNF mult.right_neutral not_add_less2 semiring_gcd_class.gcd_dvd1)
   ultimately show ?thesis1 by blast
   have "reduce_element_mod_D_abs A a j D m = addrow (- (A $$ (a, j) gdiv D) + 1) a (j + m) A"
     unfolding reduce_element_mod_D_abs_def using True by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have "reduce_element_mod_D_abs A a j D m = ?P * A" .
   thus ?thesis2 using P inv_P by blast
 next
   case False note nc1 = False
   let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D)) a (j + m)"
   have A: "A \<in> carrier_mat (m + n) n" using A_def A' mn by auto
   have P: "?P \<in> carrier_mat (m+n) (m+n)" by simp
   have inv_P: "invertible_mat ?P"
     by (metis addrow_mat_carrier a det_addrow_mat dvd_mult_right 
         invertible_iff_is_unit_JNF mult.right_neutral not_add_less2 semiring_gcd_class.gcd_dvd1)
   show ?thesis1
   proof (cases "j = 0")
     case True
     have "reduce_element_mod_D A a j D m = A" 
       unfolding reduce_element_mod_D_def using True nc1 by auto
     thus ?thesis1
       by (metis A_def A' carrier_append_rows invertible_mat_one 
           left_mult_one_mat one_carrier_mat smult_carrier_mat)
   next
     case False   
     have "reduce_element_mod_D A a j D m =  addrow (- (A $$ (a, j) gdiv D)) a (j + m) A"
       unfolding reduce_element_mod_D_def using False by auto
     also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
     finally have "reduce_element_mod_D A a j D m = ?P * A" .    
     thus ?thesis using P inv_P by blast
   qed
   have "reduce_element_mod_D_abs A a j D m =  addrow (- (A $$ (a, j) gdiv D)) a (j + m) A"
     unfolding reduce_element_mod_D_abs_def using False by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have "reduce_element_mod_D_abs A a j D m = ?P * A" .    
   thus ?thesis2 using P inv_P by blast
 qed
 
 
 lemma reduce_element_mod_D_append:
   assumes A_def: "A = A' @\<^sub>r  (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n" and mn: "m\<ge>n"
 shows "reduce_element_mod_D A a j D m 
   = mat_of_rows n [Matrix.row (reduce_element_mod_D A a j D m) i. i \<leftarrow> [0..<m]] @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))" (is "?lhs = ?A' @\<^sub>r ?D")
 and "reduce_element_mod_D_abs A a j D m 
   = mat_of_rows n [Matrix.row (reduce_element_mod_D_abs A a j D m) i. i \<leftarrow> [0..<m]] @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))" (is "?lhs_abs = ?A'_abs @\<^sub>r ?D")
   unfolding atomize_conj
 proof (rule conjI; rule eq_matI)
   let ?xs = "(map (Matrix.row (reduce_element_mod_D A a j D m)) [0..<m])"
   let ?xs_abs = "(map (Matrix.row (reduce_element_mod_D_abs A a j D m)) [0..<m])"
   have lhs_carrier: "?lhs \<in> carrier_mat (m+n) n"
     and lhs_carrier_abs: "?lhs_abs \<in> carrier_mat (m+n) n"
     by (metis (no_types, lifting) add.comm_neutral append_rows_def A_def A' carrier_matD 
         carrier_mat_triv index_mat_four_block(2,3) index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) 
         reduce_element_mod_D_preserves_dimensions)+
   have map_A_carrier[simp]: "?A' \<in> carrier_mat m n" 
     and map_A_carrier_abs[simp]: "?A'_abs \<in> carrier_mat m n"
     by (simp add: mat_of_rows_def)+
   have AD_carrier[simp]: "?A' @\<^sub>r ?D \<in> carrier_mat (m+n) n" 
     and AD_carrier_abs[simp]: "?A'_abs @\<^sub>r ?D \<in> carrier_mat (m+n) n" 
     by (rule carrier_append_rows, insert lhs_carrier mn, auto)
   show "dim_row (?lhs) = dim_row (?A' @\<^sub>r ?D)"  and "dim_col (?lhs) = dim_col (?A' @\<^sub>r ?D)"
     "dim_row (?lhs_abs) = dim_row (?A'_abs @\<^sub>r ?D)"  and "dim_col (?lhs_abs) = dim_col (?A'_abs @\<^sub>r ?D)"
     using lhs_carrier lhs_carrier_abs AD_carrier AD_carrier_abs unfolding carrier_mat_def by simp+
   show "?lhs $$ (i, ja) = (?A' @\<^sub>r ?D) $$ (i, ja)" if i: "i < dim_row (?A' @\<^sub>r ?D)" and ja: "ja < dim_col (?A' @\<^sub>r ?D)" for i ja
   proof (cases "i<m")
     case True
     have ja_n: "ja < n"
       by (metis Nat.add_0_right append_rows_def index_mat_four_block(3) index_zero_mat(3) ja mat_of_rows_carrier(3))
     have "(?A' @\<^sub>r ?D) $$ (i, ja) = ?A' $$ (i,ja)"
       by (metis (no_types, lifting) Nat.add_0_right True append_rows_def diff_zero i 
           index_mat_four_block index_zero_mat(3) ja length_map length_upt mat_of_rows_carrier(2))
     also have "... = ?xs ! i $v ja" 
       by (rule mat_of_rows_index, insert i True ja , auto simp add: append_rows_def)
     also have "... = ?lhs $$ (i,ja)"
       by (rule map_first_rows_index, insert assms lhs_carrier True i ja_n, auto)
     finally show ?thesis ..
   next
     case False
     have ja_n: "ja < n"
       by (metis Nat.add_0_right append_rows_def index_mat_four_block(3) index_zero_mat(3) ja mat_of_rows_carrier(3))
     have "(?A' @\<^sub>r ?D) $$ (i, ja) =?D $$ (i-m,ja)"
-      by (smt False Nat.add_0_right map_A_carrier append_rows_def carrier_matD i 
+      by (smt (verit) False Nat.add_0_right map_A_carrier append_rows_def carrier_matD i 
           index_mat_four_block index_zero_mat(3) ja_n)
     also have "... = ?lhs $$ (i,ja)"
       by (metis (no_types, lifting) False Nat.add_0_right map_A_carrier append_rows_def A_def A' a 
           carrier_matD i index_mat_addrow(1) index_mat_four_block(1,2) index_zero_mat(3) ja_n 
           lhs_carrier reduce_element_mod_D_def reduce_element_mod_D_preserves_dimensions)
     finally show ?thesis ..
   qed
   fix i ja assume i: "i < dim_row (?A'_abs @\<^sub>r ?D)" and ja: "ja < dim_col (?A'_abs @\<^sub>r ?D)"
   have ja_n: "ja < n"
     by (metis Nat.add_0_right append_rows_def index_mat_four_block(3) index_zero_mat(3) ja mat_of_rows_carrier(3))
   show "?lhs_abs $$ (i, ja) = (?A'_abs @\<^sub>r ?D) $$ (i, ja)"
   proof (cases "i<m")
     case True
     have "(?A'_abs @\<^sub>r ?D) $$ (i, ja) = ?A'_abs $$ (i,ja)"
       by (metis (no_types, lifting) Nat.add_0_right True append_rows_def diff_zero i 
           index_mat_four_block index_zero_mat(3) ja length_map length_upt mat_of_rows_carrier(2))
     also have "... = ?xs_abs ! i $v ja" 
       by (rule mat_of_rows_index, insert i True ja , auto simp add: append_rows_def)
     also have "... = ?lhs_abs $$ (i,ja)"
       by (rule map_first_rows_index, insert assms lhs_carrier_abs True i ja_n, auto)
     finally show ?thesis ..
   next
     case False
     have "(?A'_abs @\<^sub>r ?D) $$ (i, ja) = ?D $$ (i-m,ja)"
-      by (smt False Nat.add_0_right map_A_carrier_abs append_rows_def carrier_matD i 
+      by (smt (verit) False Nat.add_0_right map_A_carrier_abs append_rows_def carrier_matD i 
           index_mat_four_block index_zero_mat(3) ja_n)
     also have "... = ?lhs_abs $$ (i,ja)"
       by (metis (no_types, lifting) False Nat.add_0_right map_A_carrier_abs append_rows_def A_def A' a 
           carrier_matD i index_mat_addrow(1) index_mat_four_block(1,2) index_zero_mat(3) ja_n 
           lhs_carrier_abs reduce_element_mod_D_abs_def reduce_element_mod_D_preserves_dimensions)
     finally show ?thesis ..
   qed
 qed
 
 
 lemma reduce_append_rows_eq:
   assumes A': "A' \<in> carrier_mat m n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))" and a: "a<m" and xm: "x<m" and "0<n"
   and Aaj: "A $$ (a,0) \<noteq> 0" 
   shows "reduce a x D A 
   = mat_of_rows n [Matrix.row ((reduce a x D A)) i. i \<leftarrow> [0..<m]] @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" (is ?thesis1)
   and "reduce_abs a x D A 
   = mat_of_rows n [Matrix.row ((reduce_abs a x D A)) i. i \<leftarrow> [0..<m]] @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" (is ?thesis2)
   unfolding atomize_conj
 proof (rule conjI; rule matrix_append_rows_eq_if_preserves)
   let ?reduce_ax = "reduce a x D A"
   let ?reduce_abs = "reduce_abs a x D A"
  obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
    by (metis prod_cases5)
   have A: "A: carrier_mat (m+n) n" by (simp add: A_def A')
   show D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" and "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp+
   show "?reduce_ax \<in> carrier_mat (m + n) n"  "?reduce_abs \<in> carrier_mat (m + n) n"
     by (metis Nat.add_0_right append_rows_def A' A_def carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2) index_zero_mat(3) reduce_preserves_dimensions)+
   show "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_ax $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" 
     and "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_abs $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)"
     unfolding atomize_conj
   proof (rule conjI; rule+)
     fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
     have ja_dc: "ja < dim_col A" and i_dr: "i < dim_row A" using i ja A by auto
     have i_not_a: "i \<noteq> a" using i a by auto
     have i_not_x: "i \<noteq> x" using i xm by auto
     have "?reduce_ax $$ (i,ja) = A $$ (i,ja)" 
       unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
     also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
       by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
     finally show "?reduce_ax $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
     have "?reduce_abs $$ (i,ja) = A $$ (i,ja)" 
       unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
     also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
       by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
     finally show "?reduce_abs $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
   qed
 qed
 
 fun reduce_row_mod_D
   where "reduce_row_mod_D A a [] D m = A" |
         "reduce_row_mod_D A a (x # xs) D m = reduce_row_mod_D (reduce_element_mod_D A a x D m) a xs D m"
 
 fun reduce_row_mod_D_abs
   where "reduce_row_mod_D_abs A a [] D m = A" |
         "reduce_row_mod_D_abs A a (x # xs) D m = reduce_row_mod_D_abs (reduce_element_mod_D_abs A a x D m) a xs D m"
 
 
 lemma reduce_row_mod_D_preserves_dimensions:
   shows [simp]: "dim_row (reduce_row_mod_D A a xs D m) = dim_row A" 
     and [simp]: "dim_col (reduce_row_mod_D A a xs D m) = dim_col A"
   by (induct A a xs D m rule: reduce_row_mod_D.induct, auto)
   
 lemma reduce_row_mod_D_preserves_dimensions_abs:
   shows [simp]: "dim_row (reduce_row_mod_D_abs A a xs D m) = dim_row A" 
     and [simp]: "dim_col (reduce_row_mod_D_abs A a xs D m) = dim_col A"
   by (induct A a xs D m rule: reduce_row_mod_D_abs.induct, auto)
 
 lemma reduce_row_mod_D_invertible_mat:
   assumes A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "\<forall>j\<in>set xs. j<n" and mn: "m\<ge>n"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D A a x D m = P * A" 
     by (rule reduce_element_mod_D_invertible_mat, insert "2.prems", auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> reduce_row_mod_D ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D A a x D m = ?A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       by (rule reduce_element_mod_D_append, insert "2.prems", auto)
   qed (insert "2.prems", auto)   
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions)
   ultimately show ?case by blast
 qed
 
 
 lemma reduce_row_mod_D_abs_invertible_mat:
   assumes A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "\<forall>j\<in>set xs. j<n" and mn: "m\<ge>n"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D_abs A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D_abs A a x D m = P * A" 
     by (rule reduce_element_mod_D_invertible_mat, insert "2.prems", auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D_abs A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> reduce_row_mod_D_abs ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D_abs A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D_abs A a x D m = ?A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       by (rule reduce_element_mod_D_append, insert "2.prems", auto)
   qed (insert "2.prems", auto)   
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D_abs ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D_abs A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions_abs)
   ultimately show ?case by blast
 qed
 end
 
 context proper_mod_operation
 begin
 lemma dvd_gdiv_mult_left[simp]: assumes "b > 0" "b dvd a" shows "b * (a gdiv b) = a"
   using dvd_gdiv_mult_right[OF assms] by (auto simp: ac_simps)
 
 
 lemma reduce_element_mod_D:
   assumes A_def: "A = A' @\<^sub>r  (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and A': "A' \<in> carrier_mat m n" and a: "a\<le>m" and j: "j<n" and mn: "m\<ge>n"
   and D: "D > 0" 
   shows "reduce_element_mod_D A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 then if D dvd A$$(i,k) 
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "_ = ?A")
     and "reduce_element_mod_D_abs A a j D m = Matrix.mat (dim_row A) (dim_col A)
       (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 \<and> D dvd A$$(i,k) then D else A$$(i,k) gmod D else A$$(i,k))" (is "_ = ?A_abs")
 unfolding atomize_conj
 proof (rule conjI; rule eq_matI)
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'  by simp
   have dr: "dim_row ?A = dim_row ?A_abs" and dc: "dim_col ?A = dim_col ?A_abs" by auto
   have 1: "reduce_element_mod_D A a j D m $$ (i, ja) = ?A $$ (i, ja)" (is ?thesis1)
     and 2: "reduce_element_mod_D_abs A a j D m $$ (i, ja) = ?A_abs $$ (i, ja)" (is ?thesis2)
     if i: "i < dim_row ?A" and ja: "ja < dim_col ?A" for i ja
     unfolding atomize_conj
   proof (rule conjI; cases "i=a")
     case False
     have "reduce_element_mod_D A a j D m = (if j = 0 then if D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A 
     else A
     else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)"
       unfolding reduce_element_mod_D_def by simp
     also have "... $$ (i,ja) = A $$ (i, ja)" unfolding mat_addrow_def using False ja i by auto     
     also have "... = ?A $$ (i,ja)" using False using i ja by auto
     finally show ?thesis1 .
     have "reduce_element_mod_D_abs A a j D m $$ (i,ja) = A $$ (i, ja)"
       unfolding reduce_element_mod_D_abs_def mat_addrow_def using False ja i by auto     
     also have "... = ?A_abs $$ (i,ja)" using False using i ja by auto
     finally show ?thesis2 .
   next
     case True note ia = True
     have "reduce_element_mod_D A a j D m 
       = (if j = 0 then if D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A else A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)" 
       unfolding reduce_element_mod_D_def by simp
     also have "... $$ (i,ja) = ?A $$ (i,ja)"
     proof (cases "ja = j")
       case True note ja_j = True
       have "A $$ (j + m, ja) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j ja A mn, auto)       
       also have "... = D * (1\<^sub>m n) $$ (j,ja)" by (rule index_smult_mat, insert ja j A mn, auto)
       also have "... = D" by (simp add: True j mn)
       finally have A_ja_jaD: "A $$ (j + m, ja) = D" .
       show ?thesis
       proof (cases "j=0 \<and> D dvd A$$(a,j)")
         case True         
         have 1: "reduce_element_mod_D A a j D m = addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A "
           using True ia ja_j unfolding reduce_element_mod_D_def by auto
         also have "... $$(i,ja) = (- (A $$ (a, j) gdiv D) + 1) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding mat_addrow_def using True ja_j ia
           using A i j by auto
         also have "... = D"
         proof -
           have "A $$ (i, ja) + D * - (A $$ (i, ja) gdiv D) = 0"
             using True ia ja_j D by force
           then show ?thesis
             by (metis A_ja_jaD ab_semigroup_add_class.add_ac(1) add.commute add_right_imp_eq ia int_distrib(2)
                 ja_j more_arith_simps(3) mult.commute mult_cancel_right1)
         qed   
         also have "... = ?A $$ (i,ja)" using True ia A i j ja_j by auto
         finally show ?thesis
           using True 1 by auto
       next
         case False
         show ?thesis
         proof (cases "ja=0")
           case True
           then show ?thesis
             using False i ja ja_j by force
         next
           case False
         have "?A $$ (i,ja) = A $$ (i, ja) gmod D" using True ia A i j False by auto
         also have "... = A $$ (i, ja) - ((A $$ (i, ja) gdiv D) * D)"
           by (subst gmod_gdiv[OF D], auto)
         also have "... =  - (A $$ (a, j) gdiv D) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding A_ja_jaD by (simp add: True ia)
         finally show ?thesis 
           using A False True i ia j by auto
       qed
     qed
   next
       case False
       have "A $$ (j + m, ja) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j mn ja A, auto)       
       also have "... = D * (1\<^sub>m n) $$ (j,ja)" by (rule index_smult_mat, insert ja j A mn, auto)
       also have "... = 0" using False using A a mn ja j by force        
       finally have A_am_ja0: "A $$ (j + m, ja) = 0" .
       then show ?thesis using False i ja by fastforce
     qed
     finally show ?thesis1 .
     have "reduce_element_mod_D_abs A a j D m 
       = (if j = 0 \<and> D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)" 
       unfolding reduce_element_mod_D_abs_def by simp
     also have "... $$ (i,ja) = ?A_abs $$ (i,ja)"
     proof (cases "ja = j")
       case True note ja_j = True
       have "A $$ (j + m, ja) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j ja A mn, auto)       
       also have "... = D * (1\<^sub>m n) $$ (j,ja)" by (rule index_smult_mat, insert ja j A mn, auto)
       also have "... = D" by (simp add: True j mn)
       finally have A_ja_jaD: "A $$ (j + m, ja) = D" .
       show ?thesis
       proof (cases "j=0 \<and> D dvd A$$(a,j)")
         case True         
         have 1: "reduce_element_mod_D_abs A a j D m = addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A "
           using True ia ja_j unfolding reduce_element_mod_D_abs_def by auto
         also have "... $$(i,ja) = (- (A $$ (a, j) gdiv D) + 1) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding mat_addrow_def using True ja_j ia
           using A i j by auto
         also have "... = D"
         proof -
           have "A $$ (i, ja) + D * - (A $$ (i, ja) gdiv D) = 0"
             using True ia ja_j D by force
           then show ?thesis
             by (metis A_ja_jaD ab_semigroup_add_class.add_ac(1) add.commute add_right_imp_eq ia int_distrib(2)
                 ja_j more_arith_simps(3) mult.commute mult_cancel_right1)
         qed   
         also have "... = ?A_abs $$ (i,ja)" using True ia A i j ja_j by auto
         finally show ?thesis
           using True 1 by auto
       next
         case False
         have i: "i<dim_row ?A_abs" and ja: "ja<dim_col ?A_abs" using i ja by auto
         have "?A_abs $$ (i,ja) = A $$ (i, ja) gmod D" using True ia A i j False by auto
         also have "... = A $$ (i, ja) - ((A $$ (i, ja) gdiv D) * D)"
           by (subst gmod_gdiv[OF D], auto)
         also have "... =  - (A $$ (a, j) gdiv D) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding A_ja_jaD by (simp add: True ia)
         finally show ?thesis 
           using A False True i ia j by auto
       qed    
   next
       case False
       have "A $$ (j + m, ja) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j mn ja A, auto)       
       also have "... = D * (1\<^sub>m n) $$ (j,ja)" by (rule index_smult_mat, insert ja j A mn, auto)
       also have "... = 0" using False using A a mn ja j by force        
       finally have A_am_ja0: "A $$ (j + m, ja) = 0" .
       then show ?thesis using False i ja by fastforce
     qed
     finally show ?thesis2 .  
   qed
   from this
   show "\<And>i ja. i<dim_row ?A \<Longrightarrow> ja < dim_col ?A \<Longrightarrow> reduce_element_mod_D A a j D m $$ (i, ja) = ?A $$ (i, ja)" 
     and "\<And>i ja. i<dim_row ?A_abs \<Longrightarrow> ja < dim_col ?A_abs \<Longrightarrow> reduce_element_mod_D_abs A a j D m $$ (i, ja) = ?A_abs $$ (i, ja)" 
     using dr dc by auto
 next
   show "dim_row (reduce_element_mod_D A a j D m) = dim_row ?A" 
     and "dim_col (reduce_element_mod_D A a j D m) = dim_col ?A"
     "dim_row (reduce_element_mod_D_abs A a j D m) = dim_row ?A_abs" 
     and "dim_col (reduce_element_mod_D_abs A a j D m) = dim_col ?A_abs"
     by auto
 qed
 
 
 lemma reduce_row_mod_D:
   assumes A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "\<forall>j\<in>set xs. j<n"
     and d: "distinct xs" and "m\<ge>n"
     and "D > 0" 
   shows "reduce_row_mod_D A a xs D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 then if D dvd A$$(i,k) 
            then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 then if D dvd A$$(i,k)
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" if "j<n" for j
     by (rule reduce_element_mod_D, insert "2.prems" that, auto)
   have "reduce_row_mod_D ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs then 
     if k=0 then if D dvd ?reduce_xs $$ (i, k) then D else ?reduce_xs $$ (i, k)
     else ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D A a x D m = ?A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       by (rule reduce_element_mod_D_append, insert "2.prems", auto)
   qed (insert "2.prems", auto)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 then if D dvd A$$(i,k)
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" by (simp add: "2.prems"(4))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x"
         using "2.prems"(5) True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x 
-          by (smt "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
+          by (smt (verit) "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
         finally show ?thesis .
       next
         case False note nc1 = False
         show ?thesis
         proof (cases "j=0")
           case True
           then show ?thesis
-            by (smt (z3) "2" False case_prod_conv dim_col_mat(1) dim_row_mat(1) i index_mat(1) j j_not_x xn)
+            by (smt (verit) "2" False case_prod_conv dim_col_mat(1) dim_row_mat(1) i index_mat(1) j j_not_x xn)
         next
           case False          
       have "?lhs $$ (i,j) = ?reduce_xs $$ (i, j) gmod D"
         using True False i j by auto
       also have "... = A $$ (i,j) gmod D" using 2[OF xn] j_not_x i j by auto
       also have "... = ?rhs $$ (i,j)" using i j j_not_x \<open>D > 0\<close>        
         using False True dim_col_mat(1) dim_row_mat(1) index_mat(1) list.set_intros(2) old.prod.case
         by auto
       finally show ?thesis .
     qed
   qed
   next
       case False
       show ?thesis using 2 i j xn 
-        by (smt False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
+        by (smt (verit) False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
     qed   
   qed  
   finally show ?case using 1 by simp
 qed
 
 
 
 
 lemma reduce_row_mod_D_abs:
   assumes A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "\<forall>j\<in>set xs. j<n"
     and d: "distinct xs" and "m\<ge>n"
     and "D > 0" 
   shows "reduce_row_mod_D_abs A a xs D m = Matrix.mat (dim_row A) (dim_col A)
              (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 \<and> D dvd A$$(i,k)
               then D else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D_abs A a j D m = Matrix.mat (dim_row A) (dim_col A)
   (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 \<and> D dvd A$$(i,k) then D 
     else A$$(i,k) gmod D else A$$(i,k))" if "j<n" for j
     by (rule reduce_element_mod_D, insert "2.prems" that, auto)
   have "reduce_row_mod_D_abs ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs then 
     if k=0 \<and> D dvd ?reduce_xs $$ (i, k) then D
     else ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D_abs A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D_abs A a x D m = ?A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       by (rule reduce_element_mod_D_append, insert "2.prems", auto)
   qed (insert "2.prems", auto)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
             (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 \<and> D dvd A$$(i,k)
             then D else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" by (simp add: "2.prems"(4))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x"
         using "2.prems"(5) True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x 
-          by (smt "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
+          by (smt (verit) "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
         finally show ?thesis .
       next
         case False 
       have "?lhs $$ (i,j) = ?reduce_xs $$ (i, j) gmod D"
         using True False i j by auto
       also have "... = A $$ (i,j) gmod D" using 2[OF xn] j_not_x i j by auto
       also have "... = ?rhs $$ (i,j)" using i j j_not_x \<open>D > 0\<close>  
         using "2" False True dim_col_mat(1) dim_row_mat(1) index_mat(1) list.set_intros(2) 
           old.prod.case xn by auto     
       finally show ?thesis .
     qed  
   next
       case False
       show ?thesis using 2 i j xn 
-        by (smt False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
+        by (smt (verit) False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
     qed   
   qed  
   finally show ?case using 1 by simp
 qed
 end
 
 
 text \<open>Now, we prove some transfer rules to connect B\'ezout matrices in HOL Analysis and JNF\<close>
 (*Connecting Bezout Matrix in HOL Analysis (thm bezout_matrix_def) and JNF (thm bezout_matrix_JNF_def)*)
 lemma HMA_bezout_matrix[transfer_rule]:
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> 'a :: {bezout_ring} ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) 
   ===> (Mod_Type_Connect.HMA_I :: _ \<Rightarrow> 'm \<Rightarrow> _) ===> (Mod_Type_Connect.HMA_I :: _ \<Rightarrow> 'm \<Rightarrow> _) 
   ===> (Mod_Type_Connect.HMA_I :: _ \<Rightarrow> 'n \<Rightarrow> _) ===> (=) ===> (Mod_Type_Connect.HMA_M)) 
   (bezout_matrix_JNF) (bezout_matrix)" 
 proof (intro rel_funI, goal_cases)
   case (1 A A' a a' b b' j j' bezout bezout')
   note HMA_AA'[transfer_rule] = "1"(1)
   note HMI_aa'[transfer_rule] = "1"(2)
   note HMI_bb'[transfer_rule] = "1"(3)
   note HMI_jj'[transfer_rule] = "1"(4)
   note eq_bezout'[transfer_rule] = "1"(5)
   show ?case unfolding Mod_Type_Connect.HMA_M_def Mod_Type_Connect.from_hma\<^sub>m_def 
   proof (rule eq_matI) 
     let ?A = "Matrix.mat CARD('m) CARD('m) (\<lambda>(i, j). bezout_matrix A' a' b' j' bezout' 
         $h mod_type_class.from_nat i $h mod_type_class.from_nat j)"
     show "dim_row (bezout_matrix_JNF A a b j bezout) = dim_row ?A"
       and "dim_col (bezout_matrix_JNF A a b j bezout) = dim_col ?A"
       using Mod_Type_Connect.dim_row_transfer_rule[OF HMA_AA']
       unfolding bezout_matrix_JNF_def by auto  
     fix i ja assume i: "i < dim_row ?A" and ja: "ja < dim_col ?A"
     let ?i = "mod_type_class.from_nat i :: 'm"
     let ?ja = "mod_type_class.from_nat ja :: 'm"    
     have i_A: "i < dim_row A"
       using HMA_AA' Mod_Type_Connect.dim_row_transfer_rule i by fastforce
     have ja_A: "ja < dim_row A"
       using Mod_Type_Connect.dim_row_transfer_rule[OF HMA_AA'] ja by fastforce
     have HMA_I_ii'[transfer_rule]: "Mod_Type_Connect.HMA_I i ?i"
       unfolding Mod_Type_Connect.HMA_I_def using from_nat_not_eq i by auto
     have HMA_I_ja'[transfer_rule]: "Mod_Type_Connect.HMA_I ja ?ja"
       unfolding Mod_Type_Connect.HMA_I_def using from_nat_not_eq ja by auto
     have Aaj: "A' $h a' $h j' = A $$ (a,j)" unfolding index_hma_def[symmetric] by (transfer, simp)
     have Abj: "A' $h b' $h j' = A $$ (b, j)" unfolding index_hma_def[symmetric] by (transfer, simp) 
     have "?A $$ (i, ja) = bezout_matrix A' a' b' j' bezout' $h ?i $h ?ja" using i ja by auto
     also have "... = (let (p, q, u, v, d) = bezout' (A' $h a' $h j') (A' $h b' $h j')
             in if ?i = a' \<and> ?ja = a' then p else if ?i = a' \<and> ?ja = b' then q else if ?i = b' \<and> ?ja = a' 
             then u else if ?i = b' \<and> ?ja = b' then v else if ?i = ?ja then 1 else 0)" 
       unfolding bezout_matrix_def by auto
     also have "... =  (let 
         (p, q, u, v, d) = bezout (A $$ (a, j)) (A $$ (b, j)) 
        in
          if i = a \<and> ja = a then p else
          if i = a \<and> ja = b then q else
          if i = b \<and> ja = a then u else
          if i = b \<and> ja = b then v else
          if i = ja then 1 else 0)" unfolding eq_bezout' Aaj Abj by (transfer, simp)
     also have "... = bezout_matrix_JNF A a b j bezout $$ (i,ja)"
       unfolding bezout_matrix_JNF_def using i_A ja_A by auto
     finally show "bezout_matrix_JNF A a b j bezout $$ (i, ja) = ?A $$ (i, ja)" ..
   qed
 qed
 
 (*thm invertible_bezout_matrix must be transferred from HOL Analysis to JNF*)
 
 context
 begin
 
 private lemma invertible_bezout_matrix_JNF_mod_type:
   fixes A::"'a::{bezout_ring_div} mat"
   assumes "A \<in> carrier_mat CARD('m::mod_type) CARD('n::mod_type)"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<CARD('m)" and j: "j<CARD('n)"
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
 proof -
   define A' where "A' = (Mod_Type_Connect.to_hma\<^sub>m A :: 'a ^'n :: mod_type ^'m :: mod_type)"
   define a' where "a' = (Mod_Type.from_nat a :: 'm)"
   define b' where "b' = (Mod_Type.from_nat b :: 'm)"
   define j' where "j' = (Mod_Type.from_nat j :: 'n)"
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'"
     unfolding Mod_Type_Connect.HMA_M_def using assms A'_def by auto
   have aa'[transfer_rule]: "Mod_Type_Connect.HMA_I a a'"
     unfolding Mod_Type_Connect.HMA_I_def a'_def using assms
     using from_nat_not_eq order.strict_trans by blast
   have bb'[transfer_rule]: "Mod_Type_Connect.HMA_I b b'"
     unfolding Mod_Type_Connect.HMA_I_def b'_def using assms
     using from_nat_not_eq order.strict_trans by blast
   have jj'[transfer_rule]: "Mod_Type_Connect.HMA_I j j'"
     unfolding Mod_Type_Connect.HMA_I_def j'_def using assms
     using from_nat_not_eq order.strict_trans by blast
   have [transfer_rule]: "bezout = bezout" ..
   have [transfer_rule]: "Mod_Type_Connect.HMA_M (bezout_matrix_JNF A a b j bezout) 
       (bezout_matrix A' a' b' j' bezout)"
     by transfer_prover
   have "invertible (bezout_matrix A' a' b' j' bezout)"
   proof (rule invertible_bezout_matrix[OF ib])
     show "a' < b'" using a_less_b by (simp add: a'_def b b'_def from_nat_mono)
     show "A' $h a' $h j' \<noteq> 0" unfolding index_hma_def[symmetric] using aj by (transfer, simp)
   qed
   thus ?thesis by (transfer, simp)
 qed 
 
 private lemma invertible_bezout_matrix_JNF_nontriv_mod_ring:
   fixes A::"'a::{bezout_ring_div} mat"
   assumes "A \<in> carrier_mat CARD('m::nontriv mod_ring) CARD('n::nontriv mod_ring)"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<CARD('m)" and j: "j<CARD('n)"
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
-  using assms invertible_bezout_matrix_JNF_mod_type by (smt CARD_mod_ring) 
+  using assms invertible_bezout_matrix_JNF_mod_type by (smt (verit) CARD_mod_ring) 
 
 
 (*We internalize both sort constraints in one step*)
 lemmas invertible_bezout_matrix_JNF_internalized = 
   invertible_bezout_matrix_JNF_nontriv_mod_ring[unfolded CARD_mod_ring, 
       internalize_sort "'m::nontriv", internalize_sort "'c::nontriv"]
 
 context
   fixes m::nat and n::nat
   assumes local_typedef1: "\<exists>(Rep :: ('b \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<m :: int}"
   assumes local_typedef2: "\<exists>(Rep :: ('c \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<n :: int}"
   and m: "m>1"
   and n: "n>1"
 begin
 
 lemma type_to_set1:
   shows "class.nontriv TYPE('b)" (is ?a) and "m=CARD('b)" (is ?b)
 proof -
   from local_typedef1 obtain Rep::"('b \<Rightarrow> int)" and Abs 
     where t: "type_definition Rep Abs {0..<m :: int}" by auto
   have "card (UNIV :: 'b set) = card {0..<m}" using t type_definition.card by fastforce
   also have "... = m" by auto
   finally show ?b ..
   then show ?a unfolding class.nontriv_def using m by auto
 qed
 
 lemma type_to_set2:
   shows "class.nontriv TYPE('c)" (is ?a) and "n=CARD('c)" (is ?b)
 proof -
   from local_typedef2 obtain Rep::"('c \<Rightarrow> int)" and Abs 
     where t: "type_definition Rep Abs {0..<n :: int}" by blast
   have "card (UNIV :: 'c set) = card {0..<n}" using t type_definition.card by force
   also have "... = n" by auto
   finally show ?b ..
   then show ?a unfolding class.nontriv_def using n by auto
 qed
 
 
 lemma invertible_bezout_matrix_JNF_nontriv_mod_ring_aux:
   fixes A::"'a::{bezout_ring_div} mat"
   assumes "A \<in> carrier_mat m n"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<m" and j: "j<n"
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
   using invertible_bezout_matrix_JNF_internalized[OF type_to_set2(1) type_to_set(1), where ?'aa = 'b]
   using assms 
   using type_to_set1(2) type_to_set2(2) local_typedef1 m by blast
 end
 
 
 (*Canceling the first local type definitions*)
 context
 begin
 
 (*Canceling the first*)
 private lemma invertible_bezout_matrix_JNF_cancelled_first:
 "\<exists>Rep Abs. type_definition Rep Abs {0..<int n} \<Longrightarrow> {0..<int m} \<noteq> {} \<Longrightarrow>
 1 < m \<Longrightarrow> 1 < n \<Longrightarrow>
 (A::'a::bezout_ring_div mat) \<in> carrier_mat m n \<Longrightarrow> is_bezout_ext bezout 
 \<Longrightarrow> a < b \<Longrightarrow> b < m \<Longrightarrow> j < n \<Longrightarrow> A $$ (a, j) \<noteq> 0 \<Longrightarrow> invertible_mat (bezout_matrix_JNF A a b j bezout)"
   using invertible_bezout_matrix_JNF_nontriv_mod_ring_aux[cancel_type_definition] by blast
 
 (*Canceling the second*)
 private lemma invertible_bezout_matrix_JNF_cancelled_both:
 "{0..<int n} \<noteq> {} \<Longrightarrow> {0..<int m} \<noteq> {} \<Longrightarrow> 1 < m \<Longrightarrow> 1 < n \<Longrightarrow>
 1 < m \<Longrightarrow> 1 < n \<Longrightarrow>
 (A::'a::bezout_ring_div mat) \<in> carrier_mat m n \<Longrightarrow> is_bezout_ext bezout 
 \<Longrightarrow> a < b \<Longrightarrow> b < m \<Longrightarrow> j < n \<Longrightarrow> A $$ (a, j) \<noteq> 0 \<Longrightarrow> invertible_mat (bezout_matrix_JNF A a b j bezout)"
   using invertible_bezout_matrix_JNF_cancelled_first[cancel_type_definition] by blast
 
 (*The final result in JNF*)
 lemma invertible_bezout_matrix_JNF':
   fixes A::"'a::{bezout_ring_div} mat"
   assumes "A \<in> carrier_mat m n"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<m" and j: "j<n" 
   and "n>1" (* Required from the mod_type restrictions*)
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
   using invertible_bezout_matrix_JNF_cancelled_both assms by auto
 
 (*Trick: we want to get rid out the "n>1" assumption, which has appeared since CARD('m::mod_type)>1.
 Given an mx1 matrix, we just append another column and the bezout_matrix is the same, so it will
 also be invertible by the previous transfered theorem
 *)
 lemma invertible_bezout_matrix_JNF_n1:
   fixes A::"'a::{bezout_ring_div} mat"
   assumes A: "A \<in> carrier_mat m n"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<m" and j: "j<n" 
   and n1: "n=1" (* Required from the mod_type restrictions*)
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
 proof -
   let ?A = "A @\<^sub>c (0\<^sub>m m n)"
   have "(A @\<^sub>c 0\<^sub>m m n) $$ (a, j) =  (if j < dim_col A then A $$ (a, j) else (0\<^sub>m m n) $$ (a, j - n))"     
     by (rule append_cols_nth[OF A], insert assms, auto)
   also have "... = A $$ (a,j)" using assms by auto
   finally have Aaj: "(A @\<^sub>c 0\<^sub>m m n) $$ (a, j) =  A $$ (a,j)" .
   have "(A @\<^sub>c 0\<^sub>m m n) $$ (b, j) =  (if j < dim_col A then A $$ (b, j) else (0\<^sub>m m n) $$ (b, j - n))"     
     by (rule append_cols_nth[OF A], insert assms, auto)
   also have "... = A $$ (b,j)" using assms by auto
   finally have Abj: "(A @\<^sub>c 0\<^sub>m m n) $$ (b, j) = A $$ (b, j)" .
   have dr: "dim_row A = dim_row ?A"  by (simp add: append_cols_def)
   have dc: "dim_col ?A = 2"
     by (metis Suc_1 append_cols_def A n1 carrier_matD(2) index_mat_four_block(3) 
         index_zero_mat(3) plus_1_eq_Suc)
   have bz_eq: "bezout_matrix_JNF A a b j bezout = bezout_matrix_JNF ?A a b j bezout"
     unfolding bezout_matrix_JNF_def Aaj Abj dr by auto
   have "invertible_mat (bezout_matrix_JNF ?A a b j bezout)"
     by (rule invertible_bezout_matrix_JNF', insert assms Aaj Abj dr dc, auto)
   thus ?thesis using bz_eq by simp
 qed
 
 (*The final result in JNF without requiring n>1*)
 corollary invertible_bezout_matrix_JNF:
   fixes A::"'a::{bezout_ring_div} mat"
   assumes "A \<in> carrier_mat m n"
   assumes ib: "is_bezout_ext bezout"
   and a_less_b: "a < b" and b: "b<m" and j: "j<n" 
   and aj: "A $$ (a, j) \<noteq> 0"
 shows "invertible_mat (bezout_matrix_JNF A a b j bezout)" 
   using invertible_bezout_matrix_JNF_n1 invertible_bezout_matrix_JNF' assms
   by (metis One_nat_def gr_implies_not0 less_Suc0 not_less_iff_gr_or_eq)
 
 end
 end
 
 text \<open>We continue with the soundness of the algorithm\<close>
 
 lemma bezout_matrix_JNF_mult_eq:
   assumes A': "A' \<in> carrier_mat m n" and a: "a\<le>m"  and b: "b\<le>m" and ab: "a \<noteq> b" 
   and A_def: "A = A' @\<^sub>r B" and B: "B \<in> carrier_mat n n"
   assumes pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,j)) (A$$(b,j))"
   shows "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             ) = (bezout_matrix_JNF A a b j euclid_ext2) * A" (is "?A = ?BM * A")
 proof (rule eq_matI) 
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' B by simp
   hence A_carrier: "?A \<in> carrier_mat (m+n) n" by auto  
   show dr: "dim_row ?A = dim_row (?BM * A)" and dc: "dim_col ?A = dim_col (?BM * A)"
     unfolding bezout_matrix_JNF_def by auto
   fix i ja assume i: "i < dim_row  (?BM * A)" and ja: "ja < dim_col (?BM * A)"
   let ?f = "\<lambda>ia. (bezout_matrix_JNF A a b j euclid_ext2) $$ (i,ia) * A $$ (ia,ja)"
   have dv: "dim_vec (col A ja) = m+n" using A by auto
   have i_dr: "i<dim_row A" using i A unfolding bezout_matrix_JNF_def by auto
   have a_dr: "a<dim_row A" using A a ja by auto
   have b_dr: "b<dim_row A" using A b ja by auto
   show "?A $$ (i,ja) = (?BM * A) $$ (i,ja)"
   proof -
     have "(?BM * A) $$ (i,ja) = Matrix.row ?BM i \<bullet> col A ja"
       by (rule index_mult_mat, insert i ja, auto)
     also have "... = (\<Sum>ia = 0..<dim_vec (col A ja). 
           Matrix.row (bezout_matrix_JNF A a b j euclid_ext2) i $v ia * col A ja $v ia)"
       by (simp add: scalar_prod_def)
     also have "... = (\<Sum>ia = 0..<m+n. ?f ia)"
-      by (rule sum.cong, insert A i dr dc, auto) (smt bezout_matrix_JNF_def carrier_matD(1) 
+      by (rule sum.cong, insert A i dr dc, auto) (smt (verit) bezout_matrix_JNF_def carrier_matD(1) 
           dim_col_mat(1) index_col index_mult_mat(3) index_row(1) ja)
     also have "... = (\<Sum>ia \<in> ({a,b} \<union> ({0..<m+n} - {a,b})). ?f ia)"
       by (rule sum.cong, insert a a_dr b A ja, auto)
     also have "... = sum ?f {a,b} + sum ?f ({0..<m+n} - {a,b})" 
       by (rule sum.union_disjoint, auto)
     finally have BM_A_ija_eq: "(?BM * A) $$ (i,ja) = sum ?f {a,b} + sum ?f ({0..<m+n} - {a,b})" by auto
     show ?thesis
     proof (cases "i = a")
       case True
       have sum0: "sum ?f ({0..<m+n} - {a,b}) = 0"
       proof (rule sum.neutral, rule)
         fix x assume x: "x \<in> {0..<m + n} - {a, b}"
         hence xm: "x < m+n" by auto
         have x_not_i: "x \<noteq> i" using True x by blast
         have x_dr: "x < dim_row A" using x A by auto
         have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
           unfolding bezout_matrix_JNF_def 
           unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
         thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
       qed
       have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = p" 
         unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using True pquvd 
         by (auto, metis split_conv)
       have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = q"
         unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using True pquvd ab
         by (auto, metis split_conv)
       have "sum ?f {a,b} + sum ?f ({0..<m+n} - {a,b}) = ?f a + ?f b" using sum0 by (simp add: ab)
       also have "... = p * A $$ (a, ja) + q * A $$ (b, ja)" unfolding fa fb by simp
       also have "... = ?A $$ (i,ja)" using A True dr i ja by auto
       finally show ?thesis using BM_A_ija_eq by simp
     next
       case False note i_not_a = False
       show ?thesis
       proof (cases "i=b")
         case True
         have sum0: "sum ?f ({0..<m+n} - {a,b}) = 0"
         proof (rule sum.neutral, rule)
           fix x assume x: "x \<in> {0..<m + n} - {a, b}"
           hence xm: "x < m+n" by auto
           have x_not_i: "x \<noteq> i" using True x by blast
           have x_dr: "x < dim_row A" using x A by auto
           have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
             unfolding bezout_matrix_JNF_def 
             unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
           thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
         qed
         have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = u" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using True i_not_a pquvd 
           by (auto, metis split_conv)
         have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = v"
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using True i_not_a pquvd ab
           by (auto, metis split_conv)
         have "sum ?f {a,b} + sum ?f ({0..<m+n} - {a,b}) = ?f a + ?f b" using sum0 by (simp add: ab)
         also have "... = u * A $$ (a, ja) + v * A $$ (b, ja)" unfolding fa fb by simp
         also have "... = ?A $$ (i,ja)" using A True i_not_a dr i ja by auto
         finally show ?thesis using BM_A_ija_eq by simp
       next
         case False note i_not_b = False
         have sum0: "sum ?f ({0..<m+n} - {a,b} - {i}) = 0" 
         proof (rule sum.neutral, rule)
           fix x assume x: "x \<in> {0..<m + n} - {a, b} - {i}"
           hence xm: "x < m+n" by auto
           have x_not_i: "x \<noteq> i" using x by blast
           have x_dr: "x < dim_row A" using x A by auto
           have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
             unfolding bezout_matrix_JNF_def 
             unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
           thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
         qed
         have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = 0" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using False i_not_a pquvd 
           by auto
         have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = 0" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using False i_not_a pquvd 
           by auto
         have "sum ?f ({0..<m+n} - {a,b}) = sum ?f (insert i ({0..<m+n} - {a,b} - {i}))"
           by (rule sum.cong, insert i_dr A i_not_a i_not_b, auto)
         also have "... = ?f i + sum ?f ({0..<m+n} - {a,b} - {i})" by (rule sum.insert, auto)
         also have "... = ?f i" using sum0 by simp
         also have "... = ?A $$ (i,ja)"
           unfolding bezout_matrix_JNF_def using i_not_a i_not_b  A dr i ja by fastforce
         finally show ?thesis unfolding BM_A_ija_eq by (simp add: ab fa fb)
       qed    
     qed
   qed
 qed
 
 
 
 
 context proper_mod_operation
 begin
 
 lemma reduce_invertible_mat: 
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n" and b: "b<m" and ab: "a \<noteq> b" 
   and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and Aaj: "A $$ (a,0) \<noteq> 0"
   and a_less_b: "a < b"
   and mn: "m\<ge>n"
   and D_ge0: "D > 0"
 shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> (reduce a b D A) = P * A" (is ?thesis1)
 proof -
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(b,0))"
     by (metis prod_cases5)
   let ?A = "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             )"
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' by simp
   hence A_carrier: "?A \<in> carrier_mat (m+n) n" by auto
 
   let ?BM = "bezout_matrix_JNF A a b 0 euclid_ext2" 
   have A'_BZ_A: "?A = ?BM * A"
     by (rule bezout_matrix_JNF_mult_eq[OF A' _ _ ab A_def D pquvd], insert a b, auto)  
   have invertible_bezout: "invertible_mat ?BM"
     by (rule invertible_bezout_matrix_JNF[OF A is_bezout_ext_euclid_ext2 a_less_b _ j Aaj],
         insert a_less_b b, auto)      
   have BM: "?BM \<in> carrier_mat (m+n) (m+n)" unfolding bezout_matrix_JNF_def using A by auto
 
   define xs where "xs = [0..<n]"
   let ?reduce_a = "reduce_row_mod_D ?A a xs D m"
   let ?A' = "mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [0..<m]]"
   have A_A'_D: "?A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   proof (rule matrix_append_rows_eq_if_preserves[OF A_carrier D], rule+)
     fix i j assume i: "i \<in> {m..<m + n}" and j: "j < n"
     have "?A $$ (i,j) = A $$ (i,j)" using i a b A j by auto
     also have "... = (if i < dim_row A' then A' $$(i,j) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,j))"
       by (unfold A_def, rule append_rows_nth[OF A' D _ j], insert i, auto)
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)" using i A' by auto
     finally show "?A $$ (i,j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)" .   
   qed
   have reduce_a_eq: "?reduce_a = Matrix.mat (dim_row ?A) (dim_col ?A) 
     (\<lambda>(i, k). if i = a \<and> k \<in> set xs then if k = 0 then if D dvd ?A$$(i,k) then D
               else ?A $$ (i, k) else ?A $$ (i, k) gmod D else ?A $$ (i, k))"
     by (rule reduce_row_mod_D[OF A_A'_D _ a _], insert xs_def mn D_ge0, auto)  
   have reduce_a: "?reduce_a \<in> carrier_mat (m+n) n"  using reduce_a_eq A by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_a = P * ?A"
     by (rule reduce_row_mod_D_invertible_mat[OF A_A'_D _ a], insert xs_def mn, auto)    
   from this obtain P where P: "P \<in> carrier_mat (m + n) (m + n)" and inv_P: "invertible_mat P" 
     and reduce_a_PA: "?reduce_a = P * ?A" by blast
   define ys where "ys = [1..<n]"
   let ?reduce_b = "reduce_row_mod_D ?reduce_a b ys D m"
   let ?B' = "mat_of_rows n [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   have reduce_a_B'_D: "?reduce_a = ?B' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   proof (rule matrix_append_rows_eq_if_preserves[OF reduce_a D], rule+)
     fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
     have i_not_a:"i\<noteq>a" and i_not_b: "i\<noteq>b" using i a b by auto
     have "?reduce_a $$ (i,ja) = ?A $$ (i, ja)"
       unfolding reduce_a_eq using i i_not_a i_not_b ja A by auto      
     also have "... = A $$ (i,ja)"  using i i_not_a i_not_b ja A by auto
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)"
-      by (smt D append_rows_nth A' A_def atLeastLessThan_iff 
+      by (smt (verit) D append_rows_nth A' A_def atLeastLessThan_iff 
           carrier_matD(1) i ja less_irrefl_nat nat_SN.compat)    
     finally show "?reduce_a $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .
   qed
   have reduce_b_eq: "?reduce_b = Matrix.mat (dim_row ?reduce_a) (dim_col ?reduce_a) 
     (\<lambda>(i, k). if i = b \<and> k \<in> set ys then if k = 0 then if D dvd ?reduce_a$$(i,k) then D else ?reduce_a $$ (i, k)
       else ?reduce_a $$ (i, k) gmod D else ?reduce_a $$ (i, k))"
     by (rule reduce_row_mod_D[OF reduce_a_B'_D _ b _ _ mn], unfold ys_def, insert D_ge0, auto)
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_b = P * ?reduce_a"
     by (rule reduce_row_mod_D_invertible_mat[OF reduce_a_B'_D _ b _ mn], insert ys_def, auto)    
   from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)" and inv_Q: "invertible_mat Q" 
     and reduce_b_Q_reduce: "?reduce_b = Q * ?reduce_a" by blast
   have reduce_b_eq_reduce: "?reduce_b = (reduce a b D A)"
   proof (rule eq_matI)
     show dr_eq: "dim_row ?reduce_b = dim_row (reduce a b D A)" 
       and dc_eq: "dim_col ?reduce_b = dim_col (reduce a b D A)"
       using reduce_preserves_dimensions by auto
     fix i ja assume i: "i<dim_row (reduce a b D A)" and ja: "ja< dim_col (reduce a b D A)"
     have im: "i<m+n" using A i reduce_preserves_dimensions(1) by auto
     have ja_n: "ja<n" using A ja reduce_preserves_dimensions(2) by auto
     show "?reduce_b $$ (i,ja) = (reduce a b D A) $$ (i,ja)"
     proof (cases "(i\<noteq>a \<and> i\<noteq>b)")
       case True
       have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq 
-        by (smt True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions)
+        by (smt (verit) True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions)
       also have "... = ?A $$ (i,ja)"
-        by (smt A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
+        by (smt (verit) A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
             reduce_a_eq reduce_preserves_dimensions(1) split_conv)
       also have "... = A $$ (i,ja)" using A True im ja_n by auto
       also have "... = (reduce a b D A) $$ (i,ja)" unfolding reduce_alt_def_not0[OF Aaj pquvd]
         using im ja_n A True by auto
       finally show ?thesis .      
     next
       case False note a_or_b = False
       show ?thesis
       proof (cases "i=a")
         case True note ia = True
         hence i_not_b: "i\<noteq>b" using ab by auto
         show ?thesis
         proof -
           have ja_in_xs: "ja \<in> set xs"
             unfolding xs_def using True ja_n im a A unfolding set_filter by auto
           have 1: "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions(2))
           show ?thesis 
           proof (cases "ja = 0 \<and> D dvd p*A$$(a,ja) + q*A$$(b,ja)")
             case True
             have "?reduce_a $$ (i,ja) = D"
               unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
             also have "... = (reduce a b D A) $$ (i,ja)"
               unfolding reduce_alt_def_not0[OF Aaj pquvd]
               using True a_or_b i_not_b ja_n im A False                
               by auto 
             finally show ?thesis using 1 by simp
           next
             case False note nc1 = False
             show ?thesis
             proof (cases "ja=0")
               case True
               then show ?thesis
-                by (smt (z3) "1" A assms(3) assms(7) dim_col_mat(1) dim_row_mat(1) euclid_ext2_works i ia im index_mat(1)
+                by (smt (verit) "1" A assms(3) assms(7) dim_col_mat(1) dim_row_mat(1) euclid_ext2_works i ia im index_mat(1)
                     ja ja_in_xs old.prod.case pquvd reduce_gcd reduce_preserves_dimensions reduce_a_eq)
             next
               case False
               have "?reduce_a $$ (i,ja) = ?A $$ (i, ja) gmod D"
                 unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
               also have "... = (reduce a b D A) $$ (i,ja)"
                 unfolding reduce_alt_def_not0[OF Aaj pquvd] using True a_or_b i_not_b ja_n im A False by auto
               finally show ?thesis using 1 by simp
           qed    
         qed        
       qed
       next
         case False note i_not_a = False
         have i_drb: "i<dim_row ?reduce_b"
           and i_dra: "i<dim_row ?reduce_a"
           and ja_drb: "ja < dim_col ?reduce_b"
           and ja_dra: "ja < dim_col ?reduce_a" using reduce_carrier[OF A] i ja A dr_eq dc_eq by auto
           have ib: "i=b" using False a_or_b by auto
         show ?thesis
         proof (cases "ja \<in> set  ys")
           case True note ja_in_ys = True     
           hence ja_not0: "ja \<noteq> 0" unfolding ys_def by auto
           have "?reduce_b $$ (i,ja) = (if ja = 0 then if D dvd ?reduce_a$$(i,ja) then D
                 else ?reduce_a $$ (i, ja) else ?reduce_a $$ (i, ja) gmod D)"
             unfolding reduce_b_eq using i_not_a True  ja ja_in_ys 
-            by (smt i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
+            by (smt (verit) i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
           also have "... = (if ja = 0 then if D dvd ?reduce_a$$(i,ja) then D else ?A $$ (i, ja) else ?A $$ (i, ja) gmod D)"
             unfolding reduce_a_eq using True ab a_or_b ib False ja_n im a A ja_in_ys by auto
           also have "... = (reduce a b D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using True ja_not0 False a_or_b ib ja_n im A 
             using i_not_a by auto                
           finally show ?thesis .
         next
           case False
           hence ja0:"ja = 0" using ja_n unfolding ys_def by auto
           have rw0: "u * A $$ (a, ja) + v * A $$ (b, ja) = 0"
             unfolding euclid_ext2_works[OF pquvd[symmetric]] ja0
-            by (smt euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
+            by (smt (verit) euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
           have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt False a_or_b dc_eq dim_row_mat(1) dr_eq i index_mat(1) ja 
+            by (smt (verit) False a_or_b dc_eq dim_row_mat(1) dr_eq i index_mat(1) ja 
                 prod.simps(2) reduce_b_eq reduce_row_mod_D_preserves_dimensions(2))
           also have "... = ?A $$ (i, ja)"
             unfolding reduce_a_eq using False ab a_or_b i_not_a ja_n im a A  by auto
           also have "... = u * A $$ (a, ja) + v * A $$ (b, ja)" 
             by (smt (verit, ccfv_SIG) A \<open>ja = 0\<close> assms(3) assms(5) carrier_matD(2) i ib index_mat(1)
                 old.prod.case reduce_preserves_dimensions(1))  
           also have "... = (reduce a b D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] 
             using False a_or_b i_not_a ja_n im A ja0 by auto
           finally show ?thesis .
         qed
       qed      
     qed    
   qed
   have inv_QPBM: "invertible_mat (Q * P * ?BM)"
     by (meson BM P Q inv_P inv_Q invertible_bezout invertible_mult_JNF mult_carrier_mat)
   moreover have "(Q*P*?BM) \<in> carrier_mat (m + n) (m + n)" using BM P Q by auto
   moreover have "(reduce a b D A) = (Q*P*?BM) * A"
   proof -
     have "?BM * A = ?A" using A'_BZ_A by auto
     hence "P * (?BM * A) = ?reduce_a" using reduce_a_PA by auto
     hence "Q * (P * (?BM * A)) = ?reduce_b" using reduce_b_Q_reduce by auto
     thus ?thesis using reduce_b_eq_reduce
-      by (smt A A'_BZ_A A_carrier BM P Q assoc_mult_mat mn mult_carrier_mat reduce_a_PA)  
+      by (smt (verit) A A'_BZ_A A_carrier BM P Q assoc_mult_mat mn mult_carrier_mat reduce_a_PA)  
   qed
   ultimately show ?thesis by blast
 qed
 
 
 lemma reduce_abs_invertible_mat: 
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n" and b: "b<m" and ab: "a \<noteq> b" 
   and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and Aaj: "A $$ (a,0) \<noteq> 0"
   and a_less_b: "a < b"
   and mn: "m\<ge>n"
   and D_ge0: "D > 0"
 shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> (reduce_abs a b D A) = P * A" (is ?thesis1)
 proof -
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(b,0))"
     by (metis prod_cases5)
   let ?A = "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             )"
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' by simp
   hence A_carrier: "?A \<in> carrier_mat (m+n) n" by auto
 
   let ?BM = "bezout_matrix_JNF A a b 0 euclid_ext2" 
   have A'_BZ_A: "?A = ?BM * A"
     by (rule bezout_matrix_JNF_mult_eq[OF A' _ _ ab A_def D pquvd], insert a b, auto)  
   have invertible_bezout: "invertible_mat ?BM"
     by (rule invertible_bezout_matrix_JNF[OF A is_bezout_ext_euclid_ext2 a_less_b _ j Aaj],
         insert a_less_b b, auto)      
   have BM: "?BM \<in> carrier_mat (m+n) (m+n)" unfolding bezout_matrix_JNF_def using A by auto
 
   define xs where "xs = filter (\<lambda>i. abs (?A $$ (a,i)) > D) [0..<n]"
   let ?reduce_a = "reduce_row_mod_D_abs ?A a xs D m"
   let ?A' = "mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [0..<m]]"
   have A_A'_D: "?A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   proof (rule matrix_append_rows_eq_if_preserves[OF A_carrier D], rule+)
     fix i j assume i: "i \<in> {m..<m + n}" and j: "j < n"
     have "?A $$ (i,j) = A $$ (i,j)" using i a b A j by auto
     also have "... = (if i < dim_row A' then A' $$(i,j) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,j))"
       by (unfold A_def, rule append_rows_nth[OF A' D _ j], insert i, auto)
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)" using i A' by auto
     finally show "?A $$ (i,j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)" .   
   qed
   have reduce_a_eq: "?reduce_a = Matrix.mat (dim_row ?A) (dim_col ?A) 
     (\<lambda>(i, k). if i = a \<and> k \<in> set xs then 
       if k = 0 \<and> D dvd ?A$$(i,k) then D else ?A $$ (i, k) gmod D else ?A $$ (i, k))"
     by (rule reduce_row_mod_D_abs[OF A_A'_D _ a _], insert xs_def mn D_ge0, auto)  
   have reduce_a: "?reduce_a \<in> carrier_mat (m+n) n"  using reduce_a_eq A by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_a = P * ?A"
     by (rule reduce_row_mod_D_abs_invertible_mat[OF A_A'_D _ a], insert xs_def mn, auto)    
   from this obtain P where P: "P \<in> carrier_mat (m + n) (m + n)" and inv_P: "invertible_mat P" 
     and reduce_a_PA: "?reduce_a = P * ?A" by blast
   define ys where "ys = filter (\<lambda>i. abs (?A $$ (b,i)) > D) [0..<n]"
   let ?reduce_b = "reduce_row_mod_D_abs ?reduce_a b ys D m"
   let ?B' = "mat_of_rows n [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   have reduce_a_B'_D: "?reduce_a = ?B' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   proof (rule matrix_append_rows_eq_if_preserves[OF reduce_a D], rule+)
     fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
     have i_not_a:"i\<noteq>a" and i_not_b: "i\<noteq>b" using i a b by auto
     have "?reduce_a $$ (i,ja) = ?A $$ (i, ja)"
       unfolding reduce_a_eq using i i_not_a i_not_b ja A by auto      
     also have "... = A $$ (i,ja)"  using i i_not_a i_not_b ja A by auto
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)"
-      by (smt D append_rows_nth A' A_def atLeastLessThan_iff 
+      by (smt (verit) D append_rows_nth A' A_def atLeastLessThan_iff 
           carrier_matD(1) i ja less_irrefl_nat nat_SN.compat)    
     finally show "?reduce_a $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .
   qed
   have reduce_b_eq: "?reduce_b = Matrix.mat (dim_row ?reduce_a) (dim_col ?reduce_a) 
     (\<lambda>(i, k). if i = b \<and> k \<in> set ys then if k = 0 \<and> D dvd ?reduce_a$$(i,k) then D 
       else ?reduce_a $$ (i, k) gmod D else ?reduce_a $$ (i, k))"
     by (rule reduce_row_mod_D_abs[OF reduce_a_B'_D _ b _ _ mn], unfold ys_def, insert D_ge0, auto)
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_b = P * ?reduce_a"
     by (rule reduce_row_mod_D_abs_invertible_mat[OF reduce_a_B'_D _ b _ mn], insert ys_def, auto)    
   from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)" and inv_Q: "invertible_mat Q" 
     and reduce_b_Q_reduce: "?reduce_b = Q * ?reduce_a" by blast
   have reduce_b_eq_reduce: "?reduce_b = (reduce_abs a b D A)"
   proof (rule eq_matI)
     show dr_eq: "dim_row ?reduce_b = dim_row (reduce_abs a b D A)" 
       and dc_eq: "dim_col ?reduce_b = dim_col (reduce_abs a b D A)"
       using reduce_preserves_dimensions by auto
     fix i ja assume i: "i<dim_row (reduce_abs a b D A)" and ja: "ja< dim_col (reduce_abs a b D A)"
     have im: "i<m+n" using A i reduce_preserves_dimensions(3) by auto
     have ja_n: "ja<n" using A ja reduce_preserves_dimensions(4) by auto
     show "?reduce_b $$ (i,ja) = (reduce_abs a b D A) $$ (i,ja)"
     proof (cases "(i\<noteq>a \<and> i\<noteq>b)")
       case True
       have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq 
-        by (smt True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions_abs)
+        by (smt (verit) True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions_abs)
       also have "... = ?A $$ (i,ja)"
-        by (smt A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
+        by (smt (verit) A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
             reduce_a_eq reduce_preserves_dimensions(3) split_conv)
       also have "... = A $$ (i,ja)" using A True im ja_n by auto
       also have "... = (reduce_abs a b D A) $$ (i,ja)" unfolding reduce_alt_def_not0[OF Aaj pquvd]
         using im ja_n A True by auto
       finally show ?thesis .      
     next
       case False note a_or_b = False
       show ?thesis
       proof (cases "i=a")
         case True note ia = True
         hence i_not_b: "i\<noteq>b" using ab by auto
         show ?thesis
         proof (cases "abs((p*A$$(a,ja) + q*A$$(b,ja))) > D")
           case True note ge_D = True
           have ja_in_xs: "ja \<in> set xs"
             unfolding xs_def using True ja_n im a A unfolding set_filter by auto
           have 1: "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions_abs(2))
           show ?thesis 
           proof (cases "ja = 0 \<and> D dvd p*A$$(a,ja) + q*A$$(b,ja)")
             case True
             have "?reduce_a $$ (i,ja) = D"
               unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
             also have "... = (reduce_abs a b D A) $$ (i,ja)"
               unfolding reduce_alt_def_not0[OF Aaj pquvd]
               using True a_or_b i_not_b ja_n im A False ge_D               
               by auto 
             finally show ?thesis using 1 by simp
           next
             case False
             have "?reduce_a $$ (i,ja) = ?A $$ (i, ja) gmod D"
               unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
             also have "... = (reduce_abs a b D A) $$ (i,ja)"
               unfolding reduce_alt_def_not0[OF Aaj pquvd] using True a_or_b i_not_b ja_n im A False by auto
             finally show ?thesis using 1 by simp
           qed        
         next
           case False
           have ja_in_xs: "ja \<notin> set xs"
             unfolding xs_def using False ja_n im a A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions_abs(2))
           also have "... = ?A $$ (i, ja)"
             unfolding reduce_a_eq using False ab a_or_b i_not_b ja_n im a A ja_in_xs by auto
           also have "... = (reduce_abs a b D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using False a_or_b i_not_b ja_n im A by auto
           finally show ?thesis .
         qed      
       next
         case False note i_not_a = False
         have i_drb: "i<dim_row ?reduce_b"
           and i_dra: "i<dim_row ?reduce_a"
           and ja_drb: "ja < dim_col ?reduce_b"
           and ja_dra: "ja < dim_col ?reduce_a" using reduce_carrier[OF A] i ja A dr_eq dc_eq by auto
           have ib: "i=b" using False a_or_b by auto
         show ?thesis
         proof (cases "abs((u*A$$(a,ja) + v * A$$(b,ja))) > D")
           case True note ge_D = True
           have ja_in_ys: "ja \<in> set ys"
             unfolding ys_def using True False ib ja_n im a b A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = (if ja = 0 \<and> D dvd ?reduce_a$$(i,ja) then D else ?reduce_a $$ (i, ja) gmod D)"          
             unfolding reduce_b_eq using i_not_a True  ja ja_in_ys 
-            by (smt i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
+            by (smt (verit) i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
           also have "... = (if ja = 0 \<and> D dvd ?reduce_a$$(i,ja) then D else ?A $$ (i, ja) gmod D)"   
             unfolding reduce_a_eq using True ab a_or_b ib False ja_n im a A ja_in_ys by auto
           also have "... = (reduce_abs a b D A) $$ (i,ja)"
           proof (cases "ja = 0 \<and> D dvd ?reduce_a$$(i,ja)")
             case True
             have ja0: "ja=0" using True by auto
             have "u * A $$ (a, ja) + v * A $$ (b, ja) = 0"
               unfolding euclid_ext2_works[OF pquvd[symmetric]] ja0
-              by (smt euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
+              by (smt (verit) euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
             hence abs_0: "abs((u*A$$(a,ja) + v * A$$(b,ja))) = 0" by auto
             show ?thesis using abs_0 D_ge0 ge_D by linarith           
           next
             case False
             then show ?thesis 
               unfolding reduce_alt_def_not0[OF Aaj pquvd] using True ge_D False a_or_b ib ja_n im A 
               using i_not_a by auto           
           qed              
           finally show ?thesis .
         next
           case False
           have ja_in_ys: "ja \<notin> set ys"
             unfolding ys_def using i_not_a False ib ja_n im a b A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq                         
             using i_dra ja_dra ja_in_ys by auto
           also have "... = ?A $$ (i, ja)"
             unfolding reduce_a_eq using False ab a_or_b i_not_a ja_n im a A  by auto
           also have "... = u * A $$ (a, ja) + v * A $$ (b, ja)" 
             unfolding reduce_a_eq using False ab a_or_b i_not_a ja_n im a A ja_in_ys by auto            
           also have "... = (reduce_abs a b D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] 
             using False a_or_b i_not_a ja_n im A by auto
           finally show ?thesis .
         qed
       qed      
     qed    
   qed
   have inv_QPBM: "invertible_mat (Q * P * ?BM)"
     by (meson BM P Q inv_P inv_Q invertible_bezout invertible_mult_JNF mult_carrier_mat)
   moreover have "(Q*P*?BM) \<in> carrier_mat (m + n) (m + n)" using BM P Q by auto
   moreover have "(reduce_abs a b D A) = (Q*P*?BM) * A"
   proof -
     have "?BM * A = ?A" using A'_BZ_A by auto
     hence "P * (?BM * A) = ?reduce_a" using reduce_a_PA by auto
     hence "Q * (P * (?BM * A)) = ?reduce_b" using reduce_b_Q_reduce by auto
     thus ?thesis using reduce_b_eq_reduce
-      by (smt A A'_BZ_A A_carrier BM P Q assoc_mult_mat mn mult_carrier_mat reduce_a_PA)  
+      by (smt (verit) A A'_BZ_A A_carrier BM P Q assoc_mult_mat mn mult_carrier_mat reduce_a_PA)  
   qed
   ultimately show ?thesis by blast
 qed
 
 
 
 
 lemma reduce_element_mod_D_case_m':
   assumes A_def: "A = A' @\<^sub>r  B" and B: "B\<in>carrier_mat n n"
   and A': "A' \<in> carrier_mat m n" and a: "a\<le>m" and j: "j<n" 
   and mn: "m>=n" and B1: "B $$ (j, j) = D" and B2: "(\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"
   and D0: "D > 0" 
   shows "reduce_element_mod_D A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 then if D dvd A$$(i,k) then D
                    else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "_ = ?A")
 proof (rule eq_matI)
   have jm: "j<m" using mn j by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' B mn by simp
   fix i ja assume i: "i < dim_row ?A" and ja: "ja < dim_col ?A"
   show "reduce_element_mod_D A a j D m $$ (i, ja) = ?A $$ (i, ja)"
  proof (cases "i=a")
     case False
     have "reduce_element_mod_D A a j D m = (if j = 0 then if D dvd A$$(a,j)
         then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A else A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)"
       unfolding reduce_element_mod_D_def by simp
     also have "... $$ (i,ja) = A $$ (i, ja)" unfolding mat_addrow_def using False ja i by auto     
     also have "... = ?A $$ (i,ja)" using False using i ja by auto
     finally show ?thesis .
   next
     case True note ia = True
     have "reduce_element_mod_D A a j D m 
       = (if j = 0 then if D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A else A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)" 
       unfolding reduce_element_mod_D_def by simp
     also have "... $$ (i,ja) = ?A $$ (i,ja)"
     proof (cases "ja = j")
       case True note ja_j = True
       have "A $$ (j + m, ja) = B $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j ja A B mn, auto)  
       also have "... = D" using True j mn B1 B2 B by auto      
       finally have A_ja_jaD: "A $$ (j + m, ja) = D" .
 
       show ?thesis
       proof (cases "j=0 \<and> D dvd A$$(a,j)")
         case True         
         have 1: "reduce_element_mod_D A a j D m = addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A "
           using True ia ja_j unfolding reduce_element_mod_D_def by auto
         also have "... $$(i,ja) = (- (A $$ (a, j) gdiv D) + 1) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding mat_addrow_def using True ja_j ia
           using A i j by auto
         also have "... = D"
         proof -
           have "A $$ (i, ja) + D * - (A $$ (i, ja) gdiv D) = 0"
             using True ia ja_j using D0 by force
           then show ?thesis
             by (metis A_ja_jaD ab_semigroup_add_class.add_ac(1) add.commute add_right_imp_eq ia int_distrib(2)
                 ja_j more_arith_simps(3) mult.commute mult_cancel_right1)
         qed   
         also have "... = ?A $$ (i,ja)" using True ia A i j ja_j by auto
         finally show ?thesis
           using True 1 by auto
       next
         case False
         show ?thesis
         proof (cases "j=0")
           case True
           then show ?thesis 
             using False i ja by auto
         next
           case False
           have "?A $$ (i,ja) = A $$ (i, ja) gmod D" using True ia A i j False by auto
           also have "... = A $$ (i, ja) - ((A $$ (i, ja) gdiv D) * D)"
             by (subst gmod_gdiv[OF D0], auto)
           also have "... =  - (A $$ (a, j) gdiv D) * A $$ (j + m, ja) + A $$ (i, ja)"
             unfolding A_ja_jaD by (simp add: True ia)
           finally show ?thesis 
             using A False True i ia j by auto
       qed
     qed
     next
       case False
       have "A $$ (j + m, ja) = B $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j mn ja A B, auto)
       also have "... = 0" using False using A a mn ja j B2 by force        
       finally have A_am_ja0: "A $$ (j + m, ja) = 0" .
       then show ?thesis using False i ja by fastforce
     qed
     finally show ?thesis .
   qed
 next
   show "dim_row (reduce_element_mod_D A a j D m) = dim_row ?A" 
     and "dim_col (reduce_element_mod_D A a j D m) = dim_col ?A"
     using reduce_element_mod_D_def by auto
 qed
 
 
 
 
 lemma reduce_element_mod_D_abs_case_m':
   assumes A_def: "A = A' @\<^sub>r  B" and B: "B\<in>carrier_mat n n"
   and A': "A' \<in> carrier_mat m n" and a: "a\<le>m" and j: "j<n" 
   and mn: "m>=n" and B1: "B $$ (j, j) = D" and B2: "(\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"
   and D0: "D > 0" 
   shows "reduce_element_mod_D_abs A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 \<and> D dvd A$$(i,k) then D else A$$(i,k) gmod D else A$$(i,k))" (is "_ = ?A")
 proof (rule eq_matI)
   have jm: "j<m" using mn j by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' B mn by simp
   fix i ja assume i: "i < dim_row ?A" and ja: "ja < dim_col ?A"
   show "reduce_element_mod_D_abs A a j D m $$ (i, ja) = ?A $$ (i, ja)"
  proof (cases "i=a")
     case False
     have "reduce_element_mod_D_abs A a j D m = (if j = 0 \<and> D dvd A$$(a,j)
         then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)"
       unfolding reduce_element_mod_D_abs_def by simp
     also have "... $$ (i,ja) = A $$ (i, ja)" unfolding mat_addrow_def using False ja i by auto     
     also have "... = ?A $$ (i,ja)" using False using i ja by auto
     finally show ?thesis .
   next
     case True note ia = True
     have "reduce_element_mod_D_abs A a j D m 
       = (if j = 0 \<and> D dvd A$$(a,j) then addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A
         else addrow (-((A$$(a,j) gdiv D))) a (j + m) A)" 
       unfolding reduce_element_mod_D_abs_def by simp
     also have "... $$ (i,ja) = ?A $$ (i,ja)"
     proof (cases "ja = j")
       case True note ja_j = True
       have "A $$ (j + m, ja) = B $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j ja A B mn, auto)  
       also have "... = D" using True j mn B1 B2 B by auto      
       finally have A_ja_jaD: "A $$ (j + m, ja) = D" .
 
       show ?thesis
       proof (cases "j=0 \<and> D dvd A$$(a,j)")
         case True         
         have 1: "reduce_element_mod_D_abs A a j D m = addrow (-((A$$(a,j) gdiv D)) + 1) a (j + m) A "
           using True ia ja_j unfolding reduce_element_mod_D_abs_def by auto
         also have "... $$(i,ja) = (- (A $$ (a, j) gdiv D) + 1) * A $$ (j + m, ja) + A $$ (i, ja)"
           unfolding mat_addrow_def using True ja_j ia
           using A i j by auto
         also have "... = D"
         proof -
           have "A $$ (i, ja) + D * - (A $$ (i, ja) gdiv D) = 0"
             using True ia ja_j using D0 by force
           then show ?thesis
             by (metis A_ja_jaD ab_semigroup_add_class.add_ac(1) add.commute add_right_imp_eq ia int_distrib(2)
                 ja_j more_arith_simps(3) mult.commute mult_cancel_right1)
         qed   
         also have "... = ?A $$ (i,ja)" using True ia A i j ja_j by auto
         finally show ?thesis
           using True 1 by auto
       next
         case False        
           have "?A $$ (i,ja) = A $$ (i, ja) gmod D" using True ia A i j False by auto
           also have "... = A $$ (i, ja) - ((A $$ (i, ja) gdiv D) * D)"
             by (subst gmod_gdiv[OF D0], auto)
           also have "... =  - (A $$ (a, j) gdiv D) * A $$ (j + m, ja) + A $$ (i, ja)"
             unfolding A_ja_jaD by (simp add: True ia)
           finally show ?thesis 
             using A False True i ia j by auto
         qed    
     next
       case False
       have "A $$ (j + m, ja) = B $$ (j,ja)"
         by (rule append_rows_nth2[OF A' _ A_def ], insert j mn ja A B, auto)
       also have "... = 0" using False using A a mn ja j B2 by force        
       finally have A_am_ja0: "A $$ (j + m, ja) = 0" .
       then show ?thesis using False i ja by fastforce
     qed
     finally show ?thesis .
   qed
 next
   show "dim_row (reduce_element_mod_D_abs A a j D m) = dim_row ?A" 
     and "dim_col (reduce_element_mod_D_abs A a j D m) = dim_col ?A"
     using reduce_element_mod_D_abs_def by auto
 qed
 
 
 lemma reduce_row_mod_D_case_m':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and "a < m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and "m\<ge>n"
     and D: "D > 0" 
   shows "reduce_row_mod_D A a xs D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 then if D dvd A$$(i,k) then D
                    else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 then if D dvd A$$(i,k) 
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" if "j\<in>set (x#xs)" for j
     by (rule reduce_element_mod_D_case_m'[OF A_A'B B A'], insert "2.prems" that, auto)
   have "reduce_row_mod_D ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs 
     then if k = 0 then if D dvd ?reduce_xs $$ (i, k) then D else ?reduce_xs $$ (i, k) else
     ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps"[OF _ B _ a _ _ mn])
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D A a x D m = ?A' @\<^sub>r B"
     proof (rule matrix_append_rows_eq_if_preserves[OF reduce_xs_carrier B])
       show " \<forall>i\<in>{m..<m + n}. \<forall>j<n. reduce_element_mod_D A a x D m $$ (i, j) = B $$ (i - m, j) "       
-        by (smt A_A'B A' B a Metric_Arith.nnf_simps(7) add_diff_cancel_left' atLeastLessThan_iff
+        by (smt (verit) A_A'B A' B a Metric_Arith.nnf_simps(7) add_diff_cancel_left' atLeastLessThan_iff
             carrier_matD index_mat_addrow(1) index_row(1) le_add_diff_inverse2 less_diff_conv
             reduce_element_mod_D_def reduce_element_mod_D_preserves_dimensions reduce_xs_carrier
             row_append_rows2)        
     qed
   qed (insert "2.prems", auto simp add: mat_of_rows_def)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 then if D dvd A$$(i,k)
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" 
       by (simp add: "2.prems"(5))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x" using d True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x 
-          by (smt "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
+          by (smt (verit) "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
         finally show ?thesis .
       next
         case False
         show ?thesis
         proof (cases "j=0")
           case True
           then show ?thesis
-            by (smt (z3) "2" dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j list.set(2) old.prod.case)
+            by (smt (verit) "2" dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j list.set(2) old.prod.case)
         next
           case False
           have "?lhs $$ (i,j) = ?reduce_xs $$ (i, j) gmod D"
             using True False i j by auto
           also have "... = A $$ (i,j) gmod D" using 2[OF ] j_not_x i j by auto
           also have "... = ?rhs $$ (i,j)" using i j j_not_x 
             using False True dim_col_mat(1) dim_row_mat(1) index_mat(1) 
               list.set_intros(2) old.prod.case by auto
           finally show ?thesis .
         qed
     qed
   next
       case False
       show ?thesis using 2 i j xn 
-        by (smt False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
+        by (smt (verit) False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
     qed   
   qed  
   finally show ?case using 1 by simp
 qed
 
 
 
 
 lemma reduce_row_mod_D_abs_case_m':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and "a < m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and "m\<ge>n"
     and D: "D > 0" 
   shows "reduce_row_mod_D_abs A a xs D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 \<and> D dvd A$$(i,k) then D
                    else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D_abs A a j D m = Matrix.mat (dim_row A) (dim_col A)
            (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 \<and> D dvd A$$(i,k) 
             then D else A$$(i,k) gmod D else A$$(i,k))" if "j\<in>set (x#xs)" for j
     by (rule reduce_element_mod_D_abs_case_m'[OF A_A'B B A'], insert "2.prems" that, auto)
   have "reduce_row_mod_D_abs ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs 
     then if k = 0 \<and> D dvd ?reduce_xs $$ (i, k) then D else
     ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps"[OF _ B _ a _ _ mn])
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D_abs A a x D m) i. i \<leftarrow> [0..<m]]"
     show "reduce_element_mod_D_abs A a x D m = ?A' @\<^sub>r B"
     proof (rule matrix_append_rows_eq_if_preserves[OF reduce_xs_carrier B])
       show " \<forall>i\<in>{m..<m + n}. \<forall>j<n. reduce_element_mod_D_abs A a x D m $$ (i, j) = B $$ (i - m, j) "       
-        by (smt A_A'B A' B a Metric_Arith.nnf_simps(7) add_diff_cancel_left' atLeastLessThan_iff
+        by (smt (verit) A_A'B A' B a Metric_Arith.nnf_simps(7) add_diff_cancel_left' atLeastLessThan_iff
             carrier_matD index_mat_addrow(1) index_row(1) le_add_diff_inverse2 less_diff_conv
             reduce_element_mod_D_abs_def reduce_element_mod_D_preserves_dimensions reduce_xs_carrier
             row_append_rows2)        
     qed
   qed (insert "2.prems", auto simp add: mat_of_rows_def)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 \<and> D dvd A$$(i,k)
           then D else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" 
       by (simp add: "2.prems"(5))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x" using d True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x 
-          by (smt "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
+          by (smt (verit) "2" calculation dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2) xn)
         finally show ?thesis .
       next
         case False        
         have "?lhs $$ (i,j) = ?reduce_xs $$ (i, j) gmod D"
          using True False i j by auto
        also have "... = A $$ (i,j) gmod D" using 2[OF ] j_not_x i j by auto
        also have "... = ?rhs $$ (i,j)" using i j j_not_x
-         by (smt False True \<open>Matrix.mat (dim_row ?reduce_xs) 
+         by (smt (verit) False True \<open>Matrix.mat (dim_row ?reduce_xs) 
            (dim_col ?reduce_xs) (\<lambda>(i, k). if i = a \<and> k \<in> set xs 
            then if k = 0 \<and> D dvd  ?reduce_xs $$ (i, k) 
            then D else  ?reduce_xs $$ (i, k) gmod D 
            else  ?reduce_xs $$ (i, k)) $$ (i, j) =  ?reduce_xs $$ (i, j) gmod D\<close> 
                calculation dim_col_mat(1) dim_row_mat(1) dvd_imp_gmod_0[OF \<open>D > 0\<close>] index_mat(1) 
                insert_iff list.set(2) gmod_0_imp_dvd prod.simps(2))
        finally show ?thesis .
      qed   
   next
       case False
       show ?thesis using 2 i j xn 
-        by (smt False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
+        by (smt (verit) False dim_col_mat(1) dim_row_mat(1) index_mat(1) insert_iff list.set(2) prod.simps(2))
     qed   
   qed  
   finally show ?case using 1 by simp
 qed
 
 
 
 lemma
   assumes A_def: "A = A' @\<^sub>r B" and B: "B \<in> carrier_mat n n"
   and A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n" and mn: "m\<ge>n"
 shows reduce_element_mod_D_invertible_mat_case_m: 
   "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> reduce_element_mod_D A a j D m = P * A" (is ?thesis1)
   and reduce_element_mod_D_abs_invertible_mat_case_m:
   "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
       reduce_element_mod_D_abs A a j D m = P * A" (is ?thesis2)
   unfolding atomize_conj
 proof (rule conjI; cases "j = 0 \<and> D dvd A$$(a,j)")
   case True
   let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D) + 1) a (j + m)"
   have A: "A \<in> carrier_mat (m + n) n" using A_def A' B mn by auto
   have "reduce_element_mod_D_abs A a j D m =  addrow (- (A $$ (a, j) gdiv D) + 1) a (j + m) A"
     unfolding reduce_element_mod_D_abs_def using True by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have rw: "reduce_element_mod_D_abs A a j D m = ?P * A" .
   have "reduce_element_mod_D A a j D m =  addrow (- (A $$ (a, j) gdiv D) + 1) a (j + m) A"
     unfolding reduce_element_mod_D_def using True by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have "reduce_element_mod_D A a j D m = ?P * A" .
   moreover have "?P \<in> carrier_mat (m+n) (m+n)" by simp
   moreover have "invertible_mat ?P"
     by (metis addrow_mat_carrier a det_addrow_mat dvd_mult_right 
         invertible_iff_is_unit_JNF mult.right_neutral not_add_less2 semiring_gcd_class.gcd_dvd1)
   ultimately show ?thesis1 and ?thesis2 using rw by blast+
 next
   case False
   show ?thesis1
   proof (cases "j=0")
     case True
     have "reduce_element_mod_D A a j D m = A" unfolding reduce_element_mod_D_def using False True by auto
     then show ?thesis
       by (metis A_def assms(2) assms(3) carrier_append_rows invertible_mat_one left_mult_one_mat one_carrier_mat)
   next
     case False
     let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D)) a (j + m)"
     have A: "A \<in> carrier_mat (m + n) n" using A_def B A' mn by auto
     have "reduce_element_mod_D A a j D m =  addrow (- (A $$ (a, j) gdiv D)) a (j + m) A"
       unfolding reduce_element_mod_D_def using False by auto
     also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
     finally have "reduce_element_mod_D A a j D m = ?P * A" .
     moreover have "?P \<in> carrier_mat (m+n) (m+n)" by simp
     moreover have "invertible_mat ?P"
       by (metis addrow_mat_carrier a det_addrow_mat dvd_mult_right 
           invertible_iff_is_unit_JNF mult.right_neutral not_add_less2 semiring_gcd_class.gcd_dvd1)
     ultimately show ?thesis by blast
   qed
   show ?thesis2
   proof -
     let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D)) a (j + m)"
     have A: "A \<in> carrier_mat (m + n) n" using A_def B A' mn by auto
     have "reduce_element_mod_D_abs A a j D m =  addrow (- (A $$ (a, j) gdiv D)) a (j + m) A"
       unfolding reduce_element_mod_D_abs_def using False by auto
     also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
     finally have "reduce_element_mod_D_abs A a j D m = ?P * A" .
     moreover have "?P \<in> carrier_mat (m+n) (m+n)" by simp
     moreover have "invertible_mat ?P"
       by (metis addrow_mat_carrier a det_addrow_mat dvd_mult_right 
           invertible_iff_is_unit_JNF mult.right_neutral not_add_less2 semiring_gcd_class.gcd_dvd1)
     ultimately show ?thesis by blast
   qed
 qed
 
 
 lemma reduce_row_mod_D_invertible_mat_case_m:
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and a: "a < m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and mn: "m\<ge>n"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   note A_def = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(6)  
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D A a x D m = P * A" 
     by (rule reduce_element_mod_D_invertible_mat_case_m, insert "2.prems", auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
       \<and> reduce_row_mod_D ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [0..<m]]"
     let ?B' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<m+n]]"
 
     show reduce_xs_A'B': "?reduce_xs = ?A' @\<^sub>r ?B'"
-      by (smt "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
+      by (smt (verit) "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
           index_mat_four_block(3) index_mult_mat(2) index_zero_mat(3) le_add1 reduce_element_mod_D_preserves_dimensions(2))
     show "\<forall>j\<in>set xs. j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
     proof
       fix j assume j_in_xs: "j \<in> set xs"
       have jn: "j<n" using j_in_xs j by auto
       have "?B' $$ (j, j) = ?reduce_xs $$ (m+j,j)"
-        by (smt "2"(7) Groups.add_ac(2) jn reduce_xs_A'B' add_diff_cancel_left' append_rows_nth2
-            diff_zero length_map length_upt mat_of_rows_carrier(1) nat_SN.compat)
+        using "2"(7)  add_diff_cancel_left'   jn length_map length_upt 
+        by (smt (verit, ccfv_SIG) "1" "2"(4) A_def B P inv_P R_P add_strict_left_mono carrier_append_rows carrier_matD(1) 
+            carrier_matD(2) index_mult_mat(2) index_row(1) mat_of_rows_index nth_map_upt reduce_row_mod_D_preserves_dimensions(2))
       also have "... = B $$ (j,j)" 
-        by (smt "2"(2) "2"(5) A' P R_P add_diff_cancel_left' append_rows_def carrier_matD
+        by (smt (verit) "2"(2) "2"(5) A' P R_P add_diff_cancel_left' append_rows_def carrier_matD
             group_cancel.rule0 index_mat_addrow(1) index_mat_four_block(1) index_mat_four_block(2,3)
             index_mult_mat(2) index_zero_mat(3) jn le_add1 linorder_not_less nat_SN.plus_gt_right_mono 
             reduce_element_mod_D_def reduce_element_mod_D_preserves_dimensions(1))
       also have "... = D" using j j_in_xs by auto
       finally have B'_jj: "?B' $$ (j, j) = D" by auto
       moreover have "\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0" 
       proof 
         fix j' assume j': "j' \<in>{0..<n} - {j}"
-        have "?B' $$ (j, j') = ?reduce_xs $$ (m+j,j')"
-          by (smt mn Diff_iff j' add.commute add_diff_cancel_left' 
+        then have "?B' $$ (j, j') = ?reduce_xs $$ (m+j,j')"
+          by (smt (z3) mn Diff_iff add.commute add_diff_cancel_left' 
               append_rows_nth2 atLeastLessThan_iff diff_zero jn length_map length_upt 
               mat_of_rows_carrier(1) nat_SN.compat reduce_xs_A'B')
         also have "... = B $$ (j,j')"
-          by (smt "2"(2) "2"(5) A' Diff_iff P R_P j' add.commute add_diff_cancel_left'  
+          by (smt (verit) "2"(2) "2"(5) A' Diff_iff P R_P j' add.commute add_diff_cancel_left'  
             append_rows_def atLeastLessThan_iff carrier_matD group_cancel.rule0 index_mat_addrow(1)
             index_mat_four_block index_mult_mat(2) index_zero_mat(3) jn nat_SN.plus_gt_right_mono 
             not_add_less2 reduce_element_mod_D_def reduce_element_mod_D_preserves_dimensions(1))
         also have "... = 0" using j j_in_xs j' by auto
         finally show "?B' $$ (j, j') = 0" .
       qed
       ultimately show "j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
         using jn by blast
     qed
     show "?A' : carrier_mat m n" by auto      
     show "?B' : carrier_mat n n" by auto
     show "a<m" using "2.prems" by auto
     show "n\<le>m" using "2.prems" by auto    
   qed
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions)
   ultimately show ?case by blast
 qed
 
 
 
 
 lemma reduce_row_mod_D_abs_invertible_mat_case_m:
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and a: "a < m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and mn: "m\<ge>n"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D_abs A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   note A_def = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(6)  
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D_abs A a x D m = P * A" 
     by (rule reduce_element_mod_D_abs_invertible_mat_case_m, insert "2.prems", auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D_abs A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
       \<and> reduce_row_mod_D_abs ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [0..<m]]"
     let ?B' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<m+n]]"
 
     show reduce_xs_A'B': "?reduce_xs = ?A' @\<^sub>r ?B'"
-      by (smt "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
+      by (smt (verit) "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
           index_mat_four_block(3) index_mult_mat(2) index_zero_mat(3) le_add1 reduce_element_mod_D_preserves_dimensions(4))
     show "\<forall>j\<in>set xs. j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
     proof
       fix j assume j_in_xs: "j \<in> set xs"
       have jn: "j<n" using j_in_xs j by auto
       have "?B' $$ (j, j) = ?reduce_xs $$ (m+j,j)"
-        by (smt "2"(7) Groups.add_ac(2) jn reduce_xs_A'B' add_diff_cancel_left' append_rows_nth2
+        by (smt (z3) "2"(7) Groups.add_ac(2) jn reduce_xs_A'B' add_diff_cancel_left' append_rows_nth2
             diff_zero length_map length_upt mat_of_rows_carrier(1) nat_SN.compat)
       also have "... = B $$ (j,j)" 
-        by (smt "2"(2) "2"(5) A' P R_P add_diff_cancel_left' append_rows_def carrier_matD
+        by (smt (verit) "2"(2) "2"(5) A' P R_P add_diff_cancel_left' append_rows_def carrier_matD
             group_cancel.rule0 index_mat_addrow(1) index_mat_four_block(1) index_mat_four_block(2,3)
             index_mult_mat(2) index_zero_mat(3) jn le_add1 linorder_not_less nat_SN.plus_gt_right_mono 
             reduce_element_mod_D_abs_def reduce_element_mod_D_preserves_dimensions(3))
       also have "... = D" using j j_in_xs by auto
       finally have B'_jj: "?B' $$ (j, j) = D" by auto
       moreover have "\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0" 
       proof 
         fix j' assume j': "j' \<in>{0..<n} - {j}"
+        then
         have "?B' $$ (j, j') = ?reduce_xs $$ (m+j,j')"
-          by (smt mn Diff_iff j' add.commute add_diff_cancel_left' 
-              append_rows_nth2 atLeastLessThan_iff diff_zero jn length_map length_upt 
-              mat_of_rows_carrier(1) nat_SN.compat reduce_xs_A'B')
+          apply simp
+          by (metis (mono_tags, lifting) "2"(4) A_def B P R_P add_less_cancel_left carrier_append_rows carrier_matD(1) carrier_matD(2) diff_add_inverse index_mult_mat(2) index_mult_mat(3) index_row(1) jn length_map length_upt mat_of_rows_index nth_map_upt)
         also have "... = B $$ (j,j')"
-          by (smt "2"(2) "2"(5) A' Diff_iff P R_P j' add.commute add_diff_cancel_left'  
+          by (smt (verit) "2"(2) "2"(5) A' Diff_iff P R_P j' add.commute add_diff_cancel_left'  
             append_rows_def atLeastLessThan_iff carrier_matD group_cancel.rule0 index_mat_addrow(1)
             index_mat_four_block index_mult_mat(2) index_zero_mat(3) jn nat_SN.plus_gt_right_mono 
             not_add_less2 reduce_element_mod_D_abs_def reduce_element_mod_D_preserves_dimensions(3))
         also have "... = 0" using j j_in_xs j' by auto
         finally show "?B' $$ (j, j') = 0" .
       qed
       ultimately show "j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
         using jn by blast
     qed
     show "?A' : carrier_mat m n" by auto      
     show "?B' : carrier_mat n n" by auto
     show "a<m" using "2.prems" by auto
     show "n\<le>m" using "2.prems" by auto    
   qed
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D_abs ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D_abs A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions_abs)
   ultimately show ?case by blast
 qed
 
 
 
 
 (*Similar to thm reduce_row_mod_D_case_m' but including the case a = m. 
 This could substitute the previous version.*)
 lemma reduce_row_mod_D_case_m'':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and "a \<le> m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and "m\<ge>n" and "0 \<notin> set xs"
     and "D > 0" 
   shows "reduce_row_mod_D A a xs D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 then if D dvd A$$(i,k) then D
                     else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   note zero_not_xs = "2.prems"(8)
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have A: "A:carrier_mat (m+n) n" using A' B A_A'B by blast
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 then if D dvd A$$(i,k)
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" if "j\<in>set (x#xs)" for j
     by (rule reduce_element_mod_D_case_m'[OF A_A'B B A'], insert "2.prems" that, auto)
   have "reduce_row_mod_D ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs 
     then if k=0 then if D dvd ?reduce_xs $$ (i, k) then D else ?reduce_xs $$ (i, k)
     else ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps"[OF _ _ _ a _ _ mn])
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D A a x D m) i. i \<leftarrow> [0..<m]]"
     define B' where "B' = mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<dim_row A]]"
     show A'': "?A' : carrier_mat m n" by auto
     show B': "B' : carrier_mat n n" unfolding B'_def using mn A by auto 
     show reduce_split: "?reduce_xs = ?A' @\<^sub>r B'" 
       by (metis B'_def append_rows_split carrier_matD
           reduce_element_mod_D_preserves_dimensions(1) reduce_xs_carrier le_add1)
     show "\<forall>j\<in>set xs. j<n \<and> (B' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B' $$ (j, j') = 0)"
     proof 
       fix j assume j_xs: "j\<in>set xs"
       have "B $$ (j,j') = B' $$ (j,j')" if j': "j'<n" for j'
       proof -
         have "B $$ (j,j') = A $$ (m+j,j')"
-          by (smt A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
+          by (smt (verit) A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
               index_mat_four_block(1) index_mat_four_block(2,3) insert_iff j less_diff_conv list.set(2) not_add_less1)
         also have "... = ?reduce_xs $$ (m+j,j')"
           by (smt (verit, ccfv_threshold) A'' diff_add_zero index_mat_addrow(3) neq0_conv
               a j zero_not_xs A add.commute add_diff_cancel_left' reduce_element_mod_D_def
               cancel_comm_monoid_add_class.diff_cancel carrier_matD index_mat_addrow(1) j'
               j_xs le_eq_less_or_eq less_diff_conv less_not_refl2 list.set_intros(2) nat_SN.compat)
         also have "... = B'$$ (j,j')"
-          by (smt B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
+          by (smt (verit) B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
               append_rows_def carrier_matD index_mat_four_block j j_xs less_diff_conv list.set_intros(2))
         finally show ?thesis .
       qed
       thus "j < n \<and> B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. B' $$ (j, j') = 0)" using j
         by (metis Diff_iff atLeastLessThan_iff insert_iff j_xs list.simps(15))
     qed          
   qed (insert "2.prems", auto simp add: mat_of_rows_def)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 then if D dvd A$$(i,k) 
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" 
       by (simp add: "2.prems"(5))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x" using d True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x
           by (metis "2.prems"(8) True ia_jxs list.set_intros(2))
         finally show ?thesis .
       next
         case False   
         show ?thesis
-          by (smt (z3) "2" "2.prems"(8) dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j j_not_x list.set(2) old.prod.case)     
+          by (smt (verit) "2" "2.prems"(8) dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j j_not_x list.set(2) old.prod.case)     
     qed
   next
       case False
       show ?thesis using 2 i j xn
-        by (smt (z3) "2.prems"(8) False carrier_matD(2) dim_row_mat(1) index_mat(1) 
+        by (smt (verit) "2.prems"(8) False carrier_matD(2) dim_row_mat(1) index_mat(1) 
             insert_iff jn list.set(2) old.prod.case reduce_element_mod_D_preserves_dimensions(2) reduce_xs_carrier)
     qed   
   qed  
   finally show ?case using 1 by simp
 qed
 
 
 
 
 (*Similar to thm reduce_row_mod_D_abs_case_m' but including the case a = m. 
 This could substitute the previous version.*)
 lemma reduce_row_mod_D_abs_case_m'':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and "a \<le> m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and "m\<ge>n" and "0 \<notin> set xs"
     and "D > 0" 
   shows "reduce_row_mod_D_abs A a xs D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set xs then if k = 0 \<and> D dvd A$$(i,k) then D
                    else A$$(i,k) gmod D else A$$(i,k))"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   then show ?case by force
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   note zero_not_xs = "2.prems"(8)
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have A: "A:carrier_mat (m+n) n" using A' B A_A'B by blast
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have 2: "reduce_element_mod_D_abs A a j D m = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k = j then if j = 0 \<and> D dvd A$$(i,k)
           then D else A$$(i,k) gmod D else A$$(i,k))" if "j\<in>set (x#xs)" for j
     by (rule reduce_element_mod_D_abs_case_m'[OF A_A'B B A'], insert "2.prems" that, auto)
   have "reduce_row_mod_D_abs ?reduce_xs a xs D m =
     Matrix.mat (dim_row ?reduce_xs) (dim_col ?reduce_xs) (\<lambda>(i,k). if i = a \<and> k \<in> set xs 
     then if k=0 \<and> D dvd ?reduce_xs $$ (i, k) then D 
     else ?reduce_xs $$ (i, k) gmod D else ?reduce_xs $$ (i, k))"
   proof (rule "2.hyps"[OF _ _ _ a _ _ mn])
     let ?A' = "mat_of_rows n [Matrix.row (reduce_element_mod_D_abs A a x D m) i. i \<leftarrow> [0..<m]]"
     define B' where "B' = mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<dim_row A]]"
     show A'': "?A' : carrier_mat m n" by auto
     show B': "B' : carrier_mat n n" unfolding B'_def using mn A by auto 
     show reduce_split: "?reduce_xs = ?A' @\<^sub>r B'" 
       by (metis B'_def append_rows_split carrier_matD
           reduce_element_mod_D_preserves_dimensions(3) reduce_xs_carrier le_add1)
     show "\<forall>j\<in>set xs. j<n \<and> (B' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B' $$ (j, j') = 0)"
     proof 
       fix j assume j_xs: "j\<in>set xs"
       have "B $$ (j,j') = B' $$ (j,j')" if j': "j'<n" for j'
       proof -
         have "B $$ (j,j') = A $$ (m+j,j')"
-          by (smt A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
+          by (smt (verit) A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
               index_mat_four_block(1) index_mat_four_block(2,3) insert_iff j less_diff_conv list.set(2) not_add_less1)
         also have "... = ?reduce_xs $$ (m+j,j')"
           by (smt (verit, ccfv_threshold) A'' diff_add_zero index_mat_addrow(3) neq0_conv
               a j zero_not_xs A add.commute add_diff_cancel_left' reduce_element_mod_D_abs_def
               cancel_comm_monoid_add_class.diff_cancel carrier_matD index_mat_addrow(1) j'
               j_xs le_eq_less_or_eq less_diff_conv less_not_refl2 list.set_intros(2) nat_SN.compat)
         also have "... = B'$$ (j,j')"
-          by (smt B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
+          by (smt (verit) B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
               append_rows_def carrier_matD index_mat_four_block j j_xs less_diff_conv list.set_intros(2))
         finally show ?thesis .
       qed
       thus "j < n \<and> B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. B' $$ (j, j') = 0)" using j
         by (metis Diff_iff atLeastLessThan_iff insert_iff j_xs list.simps(15))
     qed          
   qed (insert "2.prems", auto simp add: mat_of_rows_def)
   also have "... = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a \<and> k \<in> set (x # xs) then if k = 0 then if D dvd A$$(i,k) 
           then D else A$$(i,k) else A$$(i,k) gmod D else A$$(i,k))" (is "?lhs = ?rhs")
   proof (rule eq_matI) 
     show "dim_row ?lhs = dim_row ?rhs" and "dim_col ?lhs = dim_col ?rhs" by auto
     fix i j assume i: "i<dim_row ?rhs" and j: "j < dim_col ?rhs"
     have jn: "j<n" using j "2.prems" by (simp add: append_rows_def)
     have xn: "x < n" 
       by (simp add: "2.prems"(5))
     show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     proof (cases "i=a \<and> j \<in> set xs")
       case True note ia_jxs = True
       have j_not_x: "j\<noteq>x" using d True by auto
       show ?thesis
       proof (cases "j=0 \<and> D dvd ?reduce_xs $$(i,j)")
         case True
         have "?lhs $$ (i,j) = D"
           using True i j ia_jxs by auto
         also have "... = ?rhs $$ (i,j)" using i j j_not_x
           by (metis "2.prems"(8) True ia_jxs list.set_intros(2))
         finally show ?thesis .
       next
         case False   
         show ?thesis
-          by (smt (z3) "2" "2.prems"(8) dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j j_not_x list.set(2) old.prod.case)     
+          by (smt (verit) "2" "2.prems"(8) dim_col_mat(1) dim_row_mat(1) i index_mat(1) insert_iff j j_not_x list.set(2) old.prod.case)     
     qed
   next
       case False
       show ?thesis using 2 i j xn
-        by (smt (z3) "2.prems"(8) False carrier_matD(2) dim_row_mat(1) index_mat(1) 
+        by (smt (verit) "2.prems"(8) False carrier_matD(2) dim_row_mat(1) index_mat(1) 
             insert_iff jn list.set(2) old.prod.case reduce_element_mod_D_preserves_dimensions(4) reduce_xs_carrier)
     qed   
   qed  
   finally show ?case using 1
     by (smt (verit, ccfv_SIG) "2.prems"(8) cong_mat split_conv)
 qed
 
 
 
 lemma
   assumes A_def: "A = A' @\<^sub>r B" and B: "B \<in> carrier_mat n n"
   and A': "A' \<in> carrier_mat m n" and a: "a\<le>m" and j: "j<n" and mn: "m\<ge>n" and j0: "j\<noteq>0"
 shows reduce_element_mod_D_invertible_mat_case_m':
   "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> reduce_element_mod_D A a j D m = P * A" (is ?thesis1)
   and reduce_element_mod_D_abs_invertible_mat_case_m': 
   "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> reduce_element_mod_D_abs A a j D m = P * A" (is ?thesis2)
 proof -
   let ?P = "addrow_mat (m+n) (- (A $$ (a, j) gdiv D)) a (j + m)"
   have jm: "j+m \<noteq>a" using j0 a by auto
   have A: "A \<in> carrier_mat (m + n) n" using A_def A' B mn by auto
   have rw: "reduce_element_mod_D A a j D m = reduce_element_mod_D_abs A a j D m" 
     unfolding reduce_element_mod_D_def reduce_element_mod_D_abs_def using j0 by auto
   have "reduce_element_mod_D A a j D m =  addrow (- (A $$ (a, j) gdiv D)) a (j + m) A"
     unfolding reduce_element_mod_D_def using j0 by auto
   also have "... = ?P * A" by (rule addrow_mat[OF A], insert j mn, auto)
   finally have "reduce_element_mod_D A a j D m = ?P * A" .
   moreover have "?P \<in> carrier_mat (m+n) (m+n)" by simp
   moreover have "invertible_mat ?P"
     by (metis addrow_mat_carrier det_addrow_mat dvd_mult_right jm
         invertible_iff_is_unit_JNF mult.right_neutral semiring_gcd_class.gcd_dvd1)
   ultimately show ?thesis1 and ?thesis2 using rw by metis+
 qed
 
 (*Similar to reduce_row_mod_D_invertible_mat_case_m but including the case a = m, and then
 adding the assumption 0 not in set xs.*)
 lemma reduce_row_mod_D_invertible_mat_case_m':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and a: "a \<le> m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and mn: "m\<ge>n" and "0\<notin> set xs"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   note zero_not_xs = "2.prems"(8)
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have A: "A:carrier_mat (m+n) n" using A' B A_A'B by blast 
   let ?reduce_xs = "(reduce_element_mod_D A a x D m)"
   have 1: "reduce_row_mod_D A a (x # xs) D m 
     = reduce_row_mod_D ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D A a x D m = P * A"
     by (rule reduce_element_mod_D_invertible_mat_case_m'[OF A_A'B B A' a _ mn],
         insert zero_not_xs j, auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
       \<and> reduce_row_mod_D ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [0..<m]]"
     let ?B' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<m+n]]"
     show B': "?B' \<in> carrier_mat n n" by auto
     show A'': "?A' : carrier_mat m n" by auto
     show reduce_split: "?reduce_xs = ?A' @\<^sub>r ?B'"
-      by (smt "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
+      by (smt (verit) "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
           index_mat_four_block(3) index_mult_mat(2) index_zero_mat(3) le_add1 reduce_element_mod_D_preserves_dimensions(2))
     show "\<forall>j\<in>set xs. j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
     proof
       fix j assume j_xs: "j\<in>set xs"
       have "B $$ (j,j') = ?B' $$ (j,j')" if j': "j'<n" for j'
       proof -
         have "B $$ (j,j') = A $$ (m+j,j')"
-          by (smt A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
+          by (smt (verit) A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
               index_mat_four_block(1) index_mat_four_block(2,3) insert_iff j less_diff_conv list.set(2) not_add_less1)
         also have "... = ?reduce_xs $$ (m+j,j')"
           by (smt (verit, ccfv_SIG) not_add_less1
               a j zero_not_xs A add.commute add_diff_cancel_left' reduce_element_mod_D_def
               cancel_comm_monoid_add_class.diff_cancel carrier_matD index_mat_addrow(1) j'
               j_xs le_eq_less_or_eq less_diff_conv less_not_refl2 list.set_intros(2) nat_SN.compat)
         also have "... = ?B'$$ (j,j')"
-          by (smt B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
+          by (smt (verit) B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
               append_rows_def carrier_matD index_mat_four_block j j_xs less_diff_conv list.set_intros(2))
         finally show ?thesis .
       qed
       thus "j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)" using j
         by (metis Diff_iff atLeastLessThan_iff insert_iff j_xs list.simps(15))
     qed        
   qed (insert d zero_not_xs a mn, auto)
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions)
   ultimately show ?case by blast
 qed
 
 
 
 lemma reduce_row_mod_D_abs_invertible_mat_case_m':
   assumes A_def: "A = A' @\<^sub>r B" and "B \<in> carrier_mat n n"
     and A': "A' \<in> carrier_mat m n" and a: "a \<le> m" 
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)" 
     and d: "distinct xs" and mn: "m\<ge>n" and "0\<notin> set xs"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_row_mod_D_abs A a xs D m = P * A"
   using assms
 proof (induct A a xs D m arbitrary: A' B rule: reduce_row_mod_D_abs.induct)
   case (1 A a D m)
   show ?case by (rule exI[of _ "1\<^sub>m (m+n)"], insert "1.prems", auto simp add: append_rows_def)
 next
   case (2 A a x xs D m)
   note A_A'B = "2.prems"(1)
   note B = "2.prems"(2)
   note A' = "2.prems"(3)
   note a = "2.prems"(4)
   note j = "2.prems"(5)
   note mn = "2.prems"(7)
   note d = "2.prems"(6)
   note zero_not_xs = "2.prems"(8)
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have reduce_xs_carrier: "?reduce_xs \<in> carrier_mat (m + n) n"
     by (metis "2.prems"(1) "2.prems"(2) "2.prems"(3) add.right_neutral append_rows_def 
             carrier_matD carrier_mat_triv index_mat_four_block(2,3) index_zero_mat(2,3)
             reduce_element_mod_D_preserves_dimensions)
   have A: "A:carrier_mat (m+n) n" using A' B A_A'B by blast 
   let ?reduce_xs = "(reduce_element_mod_D_abs A a x D m)"
   have 1: "reduce_row_mod_D_abs A a (x # xs) D m 
     = reduce_row_mod_D_abs ?reduce_xs a xs D m" by simp
   have "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> 
     reduce_element_mod_D_abs A a x D m = P * A"
     by (rule reduce_element_mod_D_abs_invertible_mat_case_m'[OF A_A'B B A' a _ mn],
         insert zero_not_xs j, auto)
   from this obtain P where P: "P \<in> carrier_mat (m+n) (m+n)" and inv_P: "invertible_mat P"
     and R_P: "reduce_element_mod_D_abs A a x D m = P * A" by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
       \<and> reduce_row_mod_D_abs ?reduce_xs a xs D m = P * ?reduce_xs"
   proof (rule "2.hyps")
     let ?A' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [0..<m]]"
     let ?B' = "mat_of_rows n [Matrix.row ?reduce_xs i. i \<leftarrow> [m..<m+n]]"
     show B': "?B' \<in> carrier_mat n n" by auto
     show A'': "?A' : carrier_mat m n" by auto
     show reduce_split: "?reduce_xs = ?A' @\<^sub>r ?B'"
-      by (smt "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
+      by (smt (verit) "2"(2) "2"(4) P R_P add.comm_neutral append_rows_def append_rows_split carrier_matD
           index_mat_four_block(3) index_mult_mat(2) index_zero_mat(3) le_add1 reduce_element_mod_D_preserves_dimensions(4))
     show "\<forall>j\<in>set xs. j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)"
     proof
       fix j assume j_xs: "j\<in>set xs"
       have "B $$ (j,j') = ?B' $$ (j,j')" if j': "j'<n" for j'
       proof -
         have "B $$ (j,j') = A $$ (m+j,j')"
-          by (smt A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
+          by (smt (verit) A_A'B A A' Groups.add_ac(2) j_xs add_diff_cancel_left' append_rows_def carrier_matD j'
               index_mat_four_block(1) index_mat_four_block(2,3) insert_iff j less_diff_conv list.set(2) not_add_less1)
         also have "... = ?reduce_xs $$ (m+j,j')"
           by (smt (verit, ccfv_SIG) not_add_less1
               a j zero_not_xs A add.commute add_diff_cancel_left' reduce_element_mod_D_abs_def
               cancel_comm_monoid_add_class.diff_cancel carrier_matD index_mat_addrow(1) j'
               j_xs le_eq_less_or_eq less_diff_conv less_not_refl2 list.set_intros(2) nat_SN.compat)
         also have "... = ?B'$$ (j,j')"
-          by (smt B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
+          by (smt (verit) B A A' A_A'B B' A'' reduce_split add.commute add_diff_cancel_left' j' not_add_less1
               append_rows_def carrier_matD index_mat_four_block j j_xs less_diff_conv list.set_intros(2))
         finally show ?thesis .
       qed
       thus "j < n \<and> ?B' $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. ?B' $$ (j, j') = 0)" using j
         by (metis Diff_iff atLeastLessThan_iff insert_iff j_xs list.simps(15))
     qed        
   qed (insert d zero_not_xs a mn, auto)
   from this obtain P2 where P2: "P2 \<in> carrier_mat (m + n) (m + n)" and inv_P2: "invertible_mat P2"
     and R_P2: "reduce_row_mod_D_abs ?reduce_xs a xs D m = P2 * ?reduce_xs"
     by auto
   have "invertible_mat (P2 * P)" using P P2 inv_P inv_P2 invertible_mult_JNF by blast
   moreover have "(P2 * P) \<in> carrier_mat (m+n) (m+n)" using P2 P by auto
   moreover have "reduce_row_mod_D_abs A a (x # xs) D m = (P2 * P) * A" 
-    by (smt P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
+    by (smt (verit) P P2 R_P R_P2 1 assoc_mult_mat carrier_matD carrier_mat_triv
         index_mult_mat reduce_row_mod_D_preserves_dimensions_abs)
   ultimately show ?case by blast
 qed
 
 
 lemma reduce_invertible_mat_case_m: 
   assumes A': "A' \<in> carrier_mat m n" and B: "B \<in> carrier_mat n n"
     and a: "a<m" and ab: "a \<noteq> m" 
   and A_def: "A = A' @\<^sub>r B"
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"  
   and Aaj: "A $$ (a,0) \<noteq> 0"
   and mn: "m\<ge>n"
   and n0: "0<n"
   and pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(m,0))"
   and A2_def: "A2 = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(m,k))
                    else if i = m then u * A$$(a,k) + v * A$$(m,k)
                    else A$$(i,k)
             )"
   and xs_def: "xs = [1..<n]"
   and ys_def: "ys = [1..<n]"
     and j_ys: "\<forall>j\<in>set ys. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"
     and D0: "D > 0"
   and Am0_D: "A $$ (m, 0) \<in> {0,D}"
   and Am0_D2: "A $$ (m, 0) = 0 \<longrightarrow> A $$ (a, 0) = D"
 shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> (reduce a m D A) = P * A"
 proof -
   let ?A = "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(m,k))
                    else if i = m then u * A$$(a,k) + v * A$$(m,k)
                    else A$$(i,k)
             )"
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" using mn by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' B mn by simp
   hence A_carrier: "?A \<in> carrier_mat (m+n) n" by auto
 
   let ?BM = "bezout_matrix_JNF A a m 0 euclid_ext2"
   
   have A'_BZ_A: "?A = ?BM * A"
     by (rule bezout_matrix_JNF_mult_eq[OF A' _ _ ab A_def B pquvd], insert a, auto)  
   have invertible_bezout: "invertible_mat ?BM"
     by (rule invertible_bezout_matrix_JNF[OF A is_bezout_ext_euclid_ext2 a _ _ Aaj], insert a n0, auto)      
   have BM: "?BM \<in> carrier_mat (m+n) (m+n)" unfolding bezout_matrix_JNF_def using A by auto
   let ?reduce_a = "reduce_row_mod_D ?A a xs D m"
   define A'1 where "A'1 = mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [0..<m]]"
   define A'2 where "A'2 = mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]]"
   have A_A'_D: "?A = A'1 @\<^sub>r A'2" using append_rows_split A
     by (metis (no_types, lifting) A'1_def A'2_def A_carrier carrier_matD le_add1)
   have j_A'1_A'2: "\<forall>j\<in>set xs. j < n \<and> A'2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. A'2 $$ (j, j') = 0)"
     proof (rule ballI)
       fix ja assume ja: "ja\<in>set xs"
       have ja_n: "ja < n" using ja unfolding xs_def by auto
       have ja2: "ja < dim_row A - m" using A mn ja_n by auto
       have ja_m: "ja < m" using ja_n mn by auto
       have ja_not_0: "ja \<noteq> 0" using ja unfolding xs_def by auto
       show "ja < n \<and> A'2 $$ (ja, ja) = D \<and> (\<forall>j'\<in>{0..<n} - {ja}. A'2 $$ (ja, j') = 0)"
       proof -
         have "A'2 $$ (ja, ja) = [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]] ! ja $v ja"
           by (metis (no_types, lifting) A A'2_def add_diff_cancel_left' carrier_matD(1) 
               ja_n length_map length_upt mat_of_rows_index)
         also have "... = ?A $$ (m + ja, ja)" using A mn ja_n by auto
         also have "... = A $$ (m+ja, ja)" using A a mn ja_n ja_not_0 by auto
         also have "... =  (A' @\<^sub>r B) $$ (m + ja, ja)" unfolding A_def ..
         also have "... = B $$ (ja, ja)"
           by (metis B Groups.add_ac(2) append_rows_nth2 assms(1) ja_n mn nat_SN.compat)
         also have "... = D" using j ja by blast
         finally have A2_D: "A'2 $$ (ja, ja) = D" .
 
         moreover have "(\<forall>j'\<in>{0..<n} - {ja}. A'2 $$ (ja, j') = 0)"
         proof (rule ballI)
           fix j' assume j': "j': {0..<n} - {ja}"
           have "A'2 $$ (ja, j') = [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]] ! ja $v j'"
             unfolding A'2_def by (rule mat_of_rows_index, insert j' ja_n ja2, auto)
           also have "... = ?A $$ (m + ja, j')" using A mn ja_n j' by auto
           also have "... = A $$ (m+ja, j')" using A a mn ja_n ja_not_0 j' by auto
           also have "... =  (A' @\<^sub>r B) $$ (ja + m, j')" unfolding A_def
             by (simp add: add.commute)
           also have "... = B $$ (ja, j')"
             by (rule append_rows_nth2[OF A' B _ ja_m ja_n], insert j', auto)
           also have "... = 0" using mn j' ja_n j ja by auto
           finally show "A'2 $$ (ja, j') = 0" .
         qed
         ultimately show ?thesis using ja_n by simp
       qed
     qed
   have reduce_a_eq: "?reduce_a = Matrix.mat (dim_row ?A) (dim_col ?A) 
     (\<lambda>(i, k). if i = a \<and> k \<in> set xs then if k = 0 then if D dvd ?A $$ (i, k) then D
     else ?A $$ (i, k) else ?A $$ (i, k) gmod D else ?A $$ (i, k))"
   proof (rule reduce_row_mod_D_case_m'[OF A_A'_D _ _ a j_A'1_A'2 _ mn D0])    
     show "A'2 \<in> carrier_mat n n" using A A'2_def by auto
     show "A'1 \<in> carrier_mat m n" by (simp add: A'1_def mat_of_rows_def) 
     show "distinct xs" using distinct_filter distinct_upt xs_def by blast
   qed
   have reduce_a: "?reduce_a \<in> carrier_mat (m+n) n" using reduce_a_eq A by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_a = P * ?A"
     by (rule reduce_row_mod_D_invertible_mat_case_m[OF A_A'_D _ _ _ j_A'1_A'2 mn],
         insert a A A'2_def A'1_def, auto)
   from this obtain P where P: "P \<in> carrier_mat (m + n) (m + n)" and inv_P: "invertible_mat P" 
     and reduce_a_PA: "?reduce_a = P * ?A" by blast
   let ?reduce_b = "reduce_row_mod_D ?reduce_a m ys D m"
   let ?B' = "mat_of_rows n [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   define reduce_a1 where "reduce_a1 = mat_of_rows (dim_col ?reduce_a) [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   define reduce_a2 where "reduce_a2 = mat_of_rows (dim_col ?reduce_a) [Matrix.row ?reduce_a i. i \<leftarrow> [m..<dim_row ?reduce_a]]"
   have reduce_a_split: "?reduce_a = reduce_a1 @\<^sub>r reduce_a2"
     by (unfold reduce_a1_def reduce_a2_def, rule append_rows_split, insert mn A, auto)  
   have zero_notin_ys: "0 \<notin> set ys"
   proof -
     have m: "m<dim_row A" using A n0 by auto
     have "?A $$ (m,0) =  u * A $$ (a, 0) + v * A $$ (m, 0)" using m n0 a A by auto
     also have "... = 0" using pquvd
-      by (smt dvd_mult_div_cancel euclid_ext2_def euclid_ext2_works(3) more_arith_simps(11)
+      by (smt (verit) dvd_mult_div_cancel euclid_ext2_def euclid_ext2_works(3) more_arith_simps(11)
           mult.commute mult_minus_left prod.sel(1) prod.sel(2) semiring_gcd_class.gcd_dvd1)
     finally show ?thesis using D0 unfolding ys_def by auto
   qed
   have reduce_a2: "reduce_a2 \<in> carrier_mat n n" unfolding reduce_a2_def using A by auto
   have reduce_a1: "reduce_a1 \<in> carrier_mat m n" unfolding reduce_a1_def using A by auto
   have j2: "\<forall>j\<in>set ys. j < n \<and> reduce_a2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. reduce_a2 $$ (j, j') = 0)"
   proof
     fix j assume j_in_ys: "j \<in> set ys"
     have a_jm: "a \<noteq> j+m" using a by auto
     have m_not_jm: "m \<noteq> j + m" using zero_notin_ys j_in_ys by fastforce
     have jm: "j+m < dim_row ?A" using A_carrier j_in_ys unfolding ys_def by auto
     have jn: "j < dim_col ?A" using A_carrier j_in_ys unfolding ys_def by auto
     have jm': "j+m < dim_row A" using A_carrier j_in_ys unfolding ys_def by auto
     have jn': "j < dim_col A" using A_carrier j_in_ys unfolding ys_def by auto
     have "reduce_a2 $$ (j, j') = B $$ (j,j')" if j': "j'<n" for j'
     proof -
       have "reduce_a2 $$ (j, j') = ?reduce_a $$ (j+m,j')"
         by (rule append_rows_nth2[symmetric, OF reduce_a1 reduce_a2 reduce_a_split],
             insert j_in_ys mn j', auto simp add: ys_def)
       also have "... = ?A $$ (j+m, j')" using reduce_a_eq jm jn a_jm j' A_carrier by auto          
       also have "... = A $$ (j+m, j')" using a_jm m_not_jm jm' jn' j' A_carrier by auto
       also have "... = B $$ (j,j')"
-        by (smt A append_rows_nth2 A' B A_def mn carrier_matD(2) jn' le_Suc_ex that trans_less_add1)
+        using assms(1) assms(2) assms(5) assms(8,14) unfolding A_def
+        by (meson append_rows_nth2 less_le_trans j' j_in_ys)
       finally show ?thesis .
     qed
     thus "j < n \<and> reduce_a2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. reduce_a2 $$ (j, j') = 0)"
       using j_ys j_in_ys by auto
   qed
   have reduce_b_eq: "?reduce_b = Matrix.mat (dim_row ?reduce_a) (dim_col ?reduce_a) 
     (\<lambda>(i, k). if i = m \<and> k \<in> set ys then if k = 0 then if D dvd ?reduce_a $$ (i, k) then D
       else ?reduce_a $$ (i, k) else ?reduce_a $$ (i, k) gmod D else ?reduce_a $$ (i, k))"
     by (rule reduce_row_mod_D_case_m''[OF reduce_a_split reduce_a2 reduce_a1 _ j2 _ mn zero_notin_ys],
         insert D0, auto simp add: ys_def)    
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_b = P * ?reduce_a"
     by (rule reduce_row_mod_D_invertible_mat_case_m'[OF reduce_a_split reduce_a2 reduce_a1 _ j2 _ mn zero_notin_ys],
         auto simp add: ys_def) 
   from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)" and inv_Q: "invertible_mat Q" 
     and reduce_b_Q_reduce: "?reduce_b = Q * ?reduce_a" by blast
   have reduce_b_eq_reduce: "?reduce_b = (reduce a m D A)"
   proof (rule eq_matI)
     show dr_eq: "dim_row ?reduce_b = dim_row (reduce a m D A)" 
       and dc_eq: "dim_col ?reduce_b = dim_col (reduce a m D A)"
       using reduce_preserves_dimensions by auto
     fix i ja assume i: "i<dim_row (reduce a m D A)" and ja: "ja< dim_col (reduce a m D A)"
     have im: "i<m+n" using A i reduce_preserves_dimensions(1) by auto
     have ja_n: "ja<n" using A ja reduce_preserves_dimensions(2) by auto
     show "?reduce_b $$ (i,ja) = (reduce a m D A) $$ (i,ja)"    
     proof (cases "(i\<noteq>a \<and> i\<noteq>m)")
       case True
       have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq 
-        by (smt True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions)
+        by (smt (verit) True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions)
       also have "... = ?A $$ (i,ja)"
-        by (smt A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
+        by (smt (verit) A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
             reduce_a_eq reduce_preserves_dimensions(1) split_conv)
       also have "... = A $$ (i,ja)" using A True im ja_n by auto
       also have "... = (reduce a m D A) $$ (i,ja)" unfolding reduce_alt_def_not0[OF Aaj pquvd]
         using im ja_n A True by auto
       finally show ?thesis .      
     next
       case False note a_or_b = False
       have gcd_pq: "p * A $$ (a, 0) + q * A $$ (m, 0) = gcd (A $$ (a, 0)) (A $$ (m, 0))"
         by (metis assms(10) euclid_ext2_works(1) euclid_ext2_works(2))
       have gcd_le_D: "gcd (A $$ (a, 0)) (A $$ (m, 0)) \<le> D"        
         by (metis Am0_D D0 assms(17) empty_iff gcd_le1_int gcd_le2_int insert_iff)
       show ?thesis
       proof (cases "i=a")
         case True note ia = True
         hence i_not_b: "i\<noteq>m" using ab by auto
         have 1: "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions(2))
         show ?thesis
         proof (cases "ja=0")
           case True note ja0 = True           
           hence ja_notin_xs: "ja \<notin> set xs" unfolding xs_def by auto
           have "?reduce_a $$ (i,ja) = p * A $$ (a, 0) + q * A $$ (m, 0)" 
             unfolding reduce_a_eq using True ja0 ab a_or_b i_not_b ja_n im a A False ja_notin_xs
             by auto
           also have "... = (reduce a m D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd]
             using True a_or_b i_not_b ja_n im A False               
             using gcd_le_D gcd_pq Am0_D Am0_D2 by auto 
           finally show ?thesis using 1 by auto
         next
           case False
           hence ja_in_xs: "ja \<in> set xs"
             unfolding xs_def using True ja_n im a A unfolding set_filter by auto
           have "?reduce_a $$ (i,ja) = ?A $$ (i, ja) gmod D"
             unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
           also have "... = (reduce a m D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using True a_or_b i_not_b ja_n im A False by auto
           finally show ?thesis using 1 by simp
         qed
       next
         case False note i_not_a = False
         have i_drb: "i<dim_row ?reduce_b" 
           and i_dra: "i<dim_row ?reduce_a" 
           and ja_drb: "ja < dim_col ?reduce_b"
           and ja_dra: "ja < dim_col ?reduce_a" using i ja reduce_carrier[OF A] A ja_n im by auto
         have ib: "i=m" using False a_or_b by auto
         show ?thesis
         proof (cases "ja = 0")
           case True note ja0 = True
           have uv: "u * A $$ (a, ja) + v * A $$ (m, ja) = 0"
             unfolding euclid_ext2_works[OF pquvd[symmetric]] True
-            by (smt euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
+            by (smt (verit) euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
           have "?reduce_b $$ (i,ja) = u * A $$ (a, ja) + v * A $$ (m, ja)"
-            by (smt (z3) A A_carrier True assms(4) carrier_matD i ib index_mat(1) reduce_a_eq
+            by (smt (verit) A A_carrier True assms(4) carrier_matD i ib index_mat(1) reduce_a_eq
                 ja_dra old.prod.case reduce_preserves_dimensions(1) zero_notin_ys reduce_b_eq
                 reduce_row_mod_D_preserves_dimensions)
           also have "... = 0" using uv by blast
           also have "... = (reduce a m D A) $$ (i,ja)" 
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using True False a_or_b ib ja_n im A 
               using i_not_a uv by auto
           finally show ?thesis by auto
         next
           case False
           have ja_in_ys: "ja \<in> set ys"
             unfolding ys_def using False ib ja_n im a  A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = (if ja = 0 then if D dvd ?reduce_a$$(i,ja) then D
                                   else ?reduce_a $$ (i, ja) else ?reduce_a $$ (i, ja) gmod D)"
             unfolding reduce_b_eq using i_not_a  ja ja_in_ys 
-            by (smt i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
+            by (smt (verit) i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
           also have "... = (if ja = 0 then if D dvd ?reduce_a$$(i,ja) then D
                             else ?A $$ (i, ja) else ?A $$ (i, ja) gmod D)"
             unfolding reduce_a_eq using ab a_or_b ib False ja_n im a A ja_in_ys by auto
           also have "... = (reduce a m D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using  False a_or_b ib ja_n im A 
             using i_not_a  by auto         
           finally show ?thesis .
         qed
       qed
     qed
   qed
   have r: "?reduce_a = (P*?BM) * A" using A A'_BZ_A BM P reduce_a_PA by auto
   have "Q * P * ?BM : carrier_mat (m+n) (m+n)" using P BM Q by auto
   moreover have "invertible_mat (Q * P*?BM)"
     using inv_P invertible_bezout BM P invertible_mult_JNF inv_Q Q by (metis mult_carrier_mat)
   moreover have "(reduce a m D A) = (Q * P * ?BM) * A" using reduce_a_eq r reduce_b_eq_reduce
-    by (smt BM P Q assoc_mult_mat carrier_matD carrier_mat_triv 
+    by (smt (verit) BM P Q assoc_mult_mat carrier_matD carrier_mat_triv 
         dim_row_mat(1) index_mult_mat(2,3) reduce_b_Q_reduce)
   ultimately show ?thesis by auto
 qed
 
 
 
 lemma reduce_abs_invertible_mat_case_m: 
   assumes A': "A' \<in> carrier_mat m n" and B: "B \<in> carrier_mat n n"
     and a: "a<m" and ab: "a \<noteq> m" 
   and A_def: "A = A' @\<^sub>r B"
     and j: "\<forall>j\<in>set xs. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"  
   and Aaj: "A $$ (a,0) \<noteq> 0"
   and mn: "m\<ge>n"
   and n0: "0<n"
   and pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(m,0))"
   and A2_def: "A2 = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(m,k))
                    else if i = m then u * A$$(a,k) + v * A$$(m,k)
                    else A$$(i,k)
             )"
   and xs_def: "xs = filter (\<lambda>i. abs (A2 $$ (a,i)) > D) [0..<n]"
   and ys_def: "ys = filter (\<lambda>i. abs (A2 $$ (m,i)) > D) [0..<n]"
     and j_ys: "\<forall>j\<in>set ys. j<n \<and> (B $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. B $$ (j, j') = 0)"
     and D0: "D > 0"
 shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> (reduce_abs a m D A) = P * A"
 proof -
   let ?A = "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(m,k))
                    else if i = m then u * A$$(a,k) + v * A$$(m,k)
                    else A$$(i,k)
             )"
   note xs_def = xs_def[unfolded A2_def]
   note ys_def = ys_def[unfolded A2_def]
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" using mn by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A' B mn by simp
   hence A_carrier: "?A \<in> carrier_mat (m+n) n" by auto
 
   let ?BM = "bezout_matrix_JNF A a m 0 euclid_ext2"
   
   have A'_BZ_A: "?A = ?BM * A"
     by (rule bezout_matrix_JNF_mult_eq[OF A' _ _ ab A_def B pquvd], insert a, auto)  
   have invertible_bezout: "invertible_mat ?BM"
     by (rule invertible_bezout_matrix_JNF[OF A is_bezout_ext_euclid_ext2 a _ _ Aaj], insert a n0, auto)      
   have BM: "?BM \<in> carrier_mat (m+n) (m+n)" unfolding bezout_matrix_JNF_def using A by auto
   let ?reduce_a = "reduce_row_mod_D_abs ?A a xs D m"
   define A'1 where "A'1 = mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [0..<m]]"
   define A'2 where "A'2 = mat_of_rows n [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]]"
   have A_A'_D: "?A = A'1 @\<^sub>r A'2" using append_rows_split A
     by (metis (no_types, lifting) A'1_def A'2_def A_carrier carrier_matD le_add1)
   have j_A'1_A'2: "\<forall>j\<in>set xs. j < n \<and> A'2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. A'2 $$ (j, j') = 0)"
     proof (rule ballI)
       fix ja assume ja: "ja\<in>set xs"
       have ja_n: "ja < n" using ja unfolding xs_def by auto
       have ja2: "ja < dim_row A - m" using A mn ja_n by auto
       have ja_m: "ja < m" using ja_n mn by auto
       have abs_A_a_ja_D: "\<bar>(?A $$ (a,ja))\<bar> > D" using ja unfolding xs_def by auto
       have ja_not_0: "ja \<noteq> 0"
       proof (rule ccontr, simp)
         assume ja_a: "ja = 0" 
         have A_mja_D: "A$$(m,ja) = D"
         proof -
           have "A$$(m,ja) = (A' @\<^sub>r B) $$ (m, ja)" unfolding A_def ..
           also have "... = B $$ (m-m,ja)"                        
             by (metis B append_rows_nth A' assms(9) carrier_matD(1) ja_a less_add_same_cancel1 less_irrefl_nat)
           also have "... = B $$ (0,0)" unfolding ja_a by auto
           also have "... = D" using mn unfolding ja_a using ja_n ja j ja_a by auto 
           finally show ?thesis .
         qed
         have "?A $$ (a, ja) = p*A$$(a,ja) + q*A$$(m,ja)" using A_carrier ja_n a A by auto
         also have "... = d" using pquvd A assms(2) ja_n ja_a
           by (simp add: bezout_coefficients_fst_snd euclid_ext2_def)
         also have "... = gcd (A$$(a,ja)) (A$$(m,ja))"
           by (metis euclid_ext2_works(2) ja_a pquvd)
         also have "abs(...) \<le> D" using A_mja_D  by (simp add: D0)       
         finally have "abs (?A $$ (a, ja)) \<le> D" .
         thus False using abs_A_a_ja_D by auto
       qed      
       show "ja < n \<and> A'2 $$ (ja, ja) = D \<and> (\<forall>j'\<in>{0..<n} - {ja}. A'2 $$ (ja, j') = 0)"
       proof -
         have "A'2 $$ (ja, ja) = [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]] ! ja $v ja"
           by (metis (no_types, lifting) A A'2_def add_diff_cancel_left' carrier_matD(1) 
               ja_n length_map length_upt mat_of_rows_index)
         also have "... = ?A $$ (m + ja, ja)" using A mn ja_n by auto
         also have "... = A $$ (m+ja, ja)" using A a mn ja_n ja_not_0 by auto
         also have "... =  (A' @\<^sub>r B) $$ (m + ja, ja)" unfolding A_def ..
         also have "... = B $$ (ja, ja)"
           by (metis B Groups.add_ac(2) append_rows_nth2 assms(1) ja_n mn nat_SN.compat)
         also have "... = D" using j ja by blast
         finally have A2_D: "A'2 $$ (ja, ja) = D" .
 
         moreover have "(\<forall>j'\<in>{0..<n} - {ja}. A'2 $$ (ja, j') = 0)"
         proof (rule ballI)
           fix j' assume j': "j': {0..<n} - {ja}"
           have "A'2 $$ (ja, j') = [Matrix.row ?A i. i \<leftarrow> [m..<dim_row A]] ! ja $v j'"
             unfolding A'2_def by (rule mat_of_rows_index, insert j' ja_n ja2, auto)
           also have "... = ?A $$ (m + ja, j')" using A mn ja_n j' by auto
           also have "... = A $$ (m+ja, j')" using A a mn ja_n ja_not_0 j' by auto
           also have "... =  (A' @\<^sub>r B) $$ (ja + m, j')" unfolding A_def
             by (simp add: add.commute)
           also have "... = B $$ (ja, j')"
             by (rule append_rows_nth2[OF A' B _ ja_m ja_n], insert j', auto)
           also have "... = 0" using mn j' ja_n j ja by auto
           finally show "A'2 $$ (ja, j') = 0" .
         qed
         ultimately show ?thesis using ja_n by simp
       qed
     qed
   have reduce_a_eq: "?reduce_a = Matrix.mat (dim_row ?A) (dim_col ?A) 
     (\<lambda>(i, k). if i = a \<and> k \<in> set xs then if k = 0 \<and> D dvd ?A $$ (i, k) then D else ?A $$ (i, k) gmod D else ?A $$ (i, k))"
   proof (rule reduce_row_mod_D_abs_case_m'[OF A_A'_D _ _ a j_A'1_A'2 _ mn D0])    
     show "A'2 \<in> carrier_mat n n" using A A'2_def by auto
     show "A'1 \<in> carrier_mat m n" by (simp add: A'1_def mat_of_rows_def) 
     show "distinct xs" using distinct_filter distinct_upt xs_def by blast
   qed
   have reduce_a: "?reduce_a \<in> carrier_mat (m+n) n" using reduce_a_eq A by auto
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_a = P * ?A"
     by (rule reduce_row_mod_D_abs_invertible_mat_case_m[OF A_A'_D _ _ _ j_A'1_A'2 mn],
         insert a A A'2_def A'1_def, auto)
   from this obtain P where P: "P \<in> carrier_mat (m + n) (m + n)" and inv_P: "invertible_mat P" 
     and reduce_a_PA: "?reduce_a = P * ?A" by blast
   let ?reduce_b = "reduce_row_mod_D_abs ?reduce_a m ys D m"
   let ?B' = "mat_of_rows n [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   define reduce_a1 where "reduce_a1 = mat_of_rows (dim_col ?reduce_a) [Matrix.row ?reduce_a i. i \<leftarrow> [0..<m]]"
   define reduce_a2 where "reduce_a2 = mat_of_rows (dim_col ?reduce_a) [Matrix.row ?reduce_a i. i \<leftarrow> [m..<dim_row ?reduce_a]]"
   have reduce_a_split: "?reduce_a = reduce_a1 @\<^sub>r reduce_a2"
     by (unfold reduce_a1_def reduce_a2_def, rule append_rows_split, insert mn A, auto)  
   have zero_notin_ys: "0 \<notin> set ys"
   proof -
     have m: "m<dim_row A" using A n0 by auto
     have "?A $$ (m,0) =  u * A $$ (a, 0) + v * A $$ (m, 0)" using m n0 a A by auto
     also have "... = 0" using pquvd
-      by (smt dvd_mult_div_cancel euclid_ext2_def euclid_ext2_works(3) more_arith_simps(11)
+      by (smt (verit) dvd_mult_div_cancel euclid_ext2_def euclid_ext2_works(3) more_arith_simps(11)
           mult.commute mult_minus_left prod.sel(1) prod.sel(2) semiring_gcd_class.gcd_dvd1)
     finally show ?thesis using D0 unfolding ys_def by auto
   qed
   have reduce_a2: "reduce_a2 \<in> carrier_mat n n" unfolding reduce_a2_def using A by auto
   have reduce_a1: "reduce_a1 \<in> carrier_mat m n" unfolding reduce_a1_def using A by auto
   have j2: "\<forall>j\<in>set ys. j < n \<and> reduce_a2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. reduce_a2 $$ (j, j') = 0)"
   proof
     fix j assume j_in_ys: "j \<in> set ys"
     have a_jm: "a \<noteq> j+m" using a by auto
     have m_not_jm: "m \<noteq> j + m" using zero_notin_ys j_in_ys by fastforce
     have jm: "j+m < dim_row ?A" using A_carrier j_in_ys unfolding ys_def by auto
     have jn: "j < dim_col ?A" using A_carrier j_in_ys unfolding ys_def by auto
     have jm': "j+m < dim_row A" using A_carrier j_in_ys unfolding ys_def by auto
     have jn': "j < dim_col A" using A_carrier j_in_ys unfolding ys_def by auto
     have "reduce_a2 $$ (j, j') = B $$ (j,j')" if j': "j'<n" for j'
     proof -
       have "reduce_a2 $$ (j, j') = ?reduce_a $$ (j+m,j')"
         by (rule append_rows_nth2[symmetric, OF reduce_a1 reduce_a2 reduce_a_split],
             insert j_in_ys mn j', auto simp add: ys_def)
       also have "... = ?A $$ (j+m, j')" using reduce_a_eq jm jn a_jm j' A_carrier by auto          
       also have "... = A $$ (j+m, j')" using a_jm m_not_jm jm' jn' j' A_carrier by auto
       also have "... = B $$ (j,j')"
-        by (smt A append_rows_nth2 A' B A_def mn carrier_matD(2) jn' le_Suc_ex that trans_less_add1)
+        unfolding A_def
+        by (meson B append_rows_nth2 assms(1) j_in_ys j_ys mn nat_SN.compat that)
       finally show ?thesis .
     qed
     thus "j < n \<and> reduce_a2 $$ (j, j) = D \<and> (\<forall>j'\<in>{0..<n} - {j}. reduce_a2 $$ (j, j') = 0)"
       using j_ys j_in_ys by auto
   qed
   have reduce_b_eq: "?reduce_b = Matrix.mat (dim_row ?reduce_a) (dim_col ?reduce_a) 
     (\<lambda>(i, k). if i = m \<and> k \<in> set ys then if k = 0 \<and> D dvd ?reduce_a $$ (i, k) then D else ?reduce_a $$ (i, k) gmod D else ?reduce_a $$ (i, k))"
     by (rule reduce_row_mod_D_abs_case_m''[OF reduce_a_split reduce_a2 reduce_a1 _ j2 _ mn zero_notin_ys],
         insert D0, auto simp add: ys_def)    
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?reduce_b = P * ?reduce_a"
     by (rule reduce_row_mod_D_abs_invertible_mat_case_m'[OF reduce_a_split reduce_a2 reduce_a1 _ j2 _ mn zero_notin_ys],
         auto simp add: ys_def) 
   from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)" and inv_Q: "invertible_mat Q" 
     and reduce_b_Q_reduce: "?reduce_b = Q * ?reduce_a" by blast
   have reduce_b_eq_reduce: "?reduce_b = (reduce_abs a m D A)"
   proof (rule eq_matI)
     show dr_eq: "dim_row ?reduce_b = dim_row (reduce_abs a m D A)" 
       and dc_eq: "dim_col ?reduce_b = dim_col (reduce_abs a m D A)"
       using reduce_preserves_dimensions by auto
     fix i ja assume i: "i<dim_row (reduce_abs a m D A)" and ja: "ja< dim_col (reduce_abs a m D A)"
     have im: "i<m+n" using A i reduce_preserves_dimensions(3) by auto
     have ja_n: "ja<n" using A ja reduce_preserves_dimensions(4) by auto
     show "?reduce_b $$ (i,ja) = (reduce_abs a m D A) $$ (i,ja)"    
     proof (cases "(i\<noteq>a \<and> i\<noteq>m)")
       case True
       have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq 
-        by (smt True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions_abs)
+        by (smt (verit) True dr_eq dc_eq i index_mat(1) ja prod.simps(2) reduce_row_mod_D_preserves_dimensions_abs)
       also have "... = ?A $$ (i,ja)"
-        by (smt A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
+        by (smt (verit) A True carrier_matD(2) dim_col_mat(1) dim_row_mat(1) i index_mat(1) ja_n 
             reduce_a_eq reduce_preserves_dimensions(3) split_conv)
       also have "... = A $$ (i,ja)" using A True im ja_n by auto
       also have "... = (reduce_abs a m D A) $$ (i,ja)" unfolding reduce_alt_def_not0[OF Aaj pquvd]
         using im ja_n A True by auto
       finally show ?thesis .      
     next
       case False note a_or_b = False
       show ?thesis
       proof (cases "i=a")
         case True note ia = True
         hence i_not_b: "i\<noteq>m" using ab by auto
         show ?thesis
         proof (cases "abs((p*A$$(a,ja) + q*A$$(m,ja))) > D")
           case True note ge_D = True
           have ja_in_xs: "ja \<in> set xs"
             unfolding xs_def using True ja_n im a A unfolding set_filter by auto
           have 1: "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions_abs(2))
           show ?thesis 
           proof (cases "ja = 0 \<and> D dvd p*A$$(a,ja) + q*A$$(m,ja)")
             case True
             have "?reduce_a $$ (i,ja) = D"
               unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
             also have "... = (reduce_abs a m D A) $$ (i,ja)"
               unfolding reduce_alt_def_not0[OF Aaj pquvd]
               using True a_or_b i_not_b ja_n im A False ge_D               
               by auto 
             finally show ?thesis using 1 by simp
           next
             case False
             have "?reduce_a $$ (i,ja) = ?A $$ (i, ja) gmod D"
               unfolding reduce_a_eq using True ab a_or_b i_not_b ja_n im a A ja_in_xs False by auto
             also have "... = (reduce_abs a m D A) $$ (i,ja)"
               unfolding reduce_alt_def_not0[OF Aaj pquvd] using True a_or_b i_not_b ja_n im A False by auto
             finally show ?thesis using 1 by simp
           qed        
         next
           case False
           have ja_in_xs: "ja \<notin> set xs"
             unfolding xs_def using False ja_n im a A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
+            by (smt (verit) ab dc_eq dim_row_mat(1) dr_eq i ia index_mat(1) ja prod.simps(2)
                 reduce_b_eq reduce_row_mod_D_preserves_dimensions_abs(2))
           also have "... = ?A $$ (i, ja)"
             unfolding reduce_a_eq using False ab a_or_b i_not_b ja_n im a A ja_in_xs by auto
           also have "... = (reduce_abs a m D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using False a_or_b i_not_b ja_n im A by auto
           finally show ?thesis .
         qed
       next
         case False note i_not_a = False
         have i_drb: "i<dim_row ?reduce_b" 
           and i_dra: "i<dim_row ?reduce_a" 
           and ja_drb: "ja < dim_col ?reduce_b"
           and ja_dra: "ja < dim_col ?reduce_a" using i ja reduce_carrier[OF A] A ja_n im by auto
           have ib: "i=m" using False a_or_b by auto
         show ?thesis
         proof (cases "abs((u*A$$(a,ja) + v * A$$(m,ja))) > D")
           case True note ge_D = True
           have ja_in_ys: "ja \<in> set ys"
             unfolding ys_def using True False ib ja_n im a  A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = (if ja = 0 \<and> D dvd ?reduce_a$$(i,ja) then D else ?reduce_a $$ (i, ja) gmod D)"
             unfolding reduce_b_eq using i_not_a True  ja ja_in_ys 
-            by (smt i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
+            by (smt (verit) i_dra ja_dra a_or_b index_mat(1) prod.simps(2))
           also have "... = (if ja = 0 \<and> D dvd ?reduce_a$$(i,ja) then D else ?A $$ (i, ja) gmod D)"
             unfolding reduce_a_eq using True ab a_or_b ib False ja_n im a A ja_in_ys by auto
           also have "... = (reduce_abs a m D A) $$ (i,ja)"
           proof (cases "ja = 0 \<and> D dvd ?reduce_a$$(i,ja)")
             case True
             have ja0: "ja=0" using True by auto
             have "u * A $$ (a, ja) + v * A $$ (m, ja) = 0"
               unfolding euclid_ext2_works[OF pquvd[symmetric]] ja0
-              by (smt euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
+              by (smt (verit) euclid_ext2_works[OF pquvd[symmetric]] more_arith_simps(11) mult.commute mult_minus_left)
             hence abs_0: "abs((u*A$$(a,ja) + v * A$$(m,ja))) = 0" by auto
             show ?thesis using abs_0 D0 ge_D by linarith           
           next
             case False
             then show ?thesis 
               unfolding reduce_alt_def_not0[OF Aaj pquvd] using True ge_D False a_or_b ib ja_n im A 
               using i_not_a by auto           
           qed      
           finally show ?thesis .
         next
           case False
           have ja_in_ys: "ja \<notin> set ys"
             unfolding ys_def using i_not_a False ib ja_n im a  A unfolding set_filter by auto
           have "?reduce_b $$ (i,ja) = ?reduce_a $$ (i,ja)" unfolding reduce_b_eq             
-            by (smt False a_or_b dc_eq dim_row_mat(1) dr_eq i index_mat(1) ja ja_in_ys
+            by (smt (verit) False a_or_b dc_eq dim_row_mat(1) dr_eq i index_mat(1) ja ja_in_ys
                 prod.simps(2) reduce_b_eq reduce_row_mod_D_preserves_dimensions_abs(2))
           also have "... = ?A $$ (i, ja)"
             unfolding reduce_a_eq using False ab a_or_b i_not_a ja_n im a A ja_in_ys by auto
           also have "... = (reduce_abs a m D A) $$ (i,ja)"
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using False a_or_b i_not_a ja_n im A by auto
           finally show ?thesis .
         qed
       qed      
     qed    
   qed
   have r: "?reduce_a = (P*?BM) * A" using A A'_BZ_A BM P reduce_a_PA by auto
   have "Q * P * ?BM : carrier_mat (m+n) (m+n)" using P BM Q by auto
   moreover have "invertible_mat (Q * P*?BM)"
     using inv_P invertible_bezout BM P invertible_mult_JNF inv_Q Q by (metis mult_carrier_mat)
   moreover have "(reduce_abs a m D A) = (Q * P * ?BM) * A" using reduce_a_eq r reduce_b_eq_reduce
-    by (smt BM P Q assoc_mult_mat carrier_matD carrier_mat_triv 
+    by (smt (verit) BM P Q assoc_mult_mat carrier_matD carrier_mat_triv 
         dim_row_mat(1) index_mult_mat(2,3) reduce_b_Q_reduce)
   ultimately show ?thesis by auto
 qed
 
 
 
 
 lemma reduce_not0:
   assumes A: "A \<in> carrier_mat m n" and a: "a<m" and a_less_b: "a<b" and j: "0<n" and b: "b<m"
     and Aaj: "A $$ (a,0) \<noteq> 0" and D0: "D \<noteq> 0"
   shows "reduce a b D A $$ (a, 0) \<noteq> 0" (is "?reduce $$ (a,0) \<noteq> _")
   and "reduce_abs a b D A $$ (a, 0) \<noteq> 0" (is "?reduce_abs $$ (a,0) \<noteq> _")
 proof -
   have "?reduce $$ (a,0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if D dvd r then D else r)" 
     by (rule reduce_gcd[OF A _ j Aaj], insert a, simp)
   also have "... \<noteq> 0" unfolding Let_def using D0 
-    by (smt Aaj gcd_eq_0_iff gmod_0_imp_dvd)
+    by (smt (verit) Aaj gcd_eq_0_iff gmod_0_imp_dvd)
   finally show "reduce a b D A $$ (a, 0) \<noteq> 0" .
   have "?reduce_abs $$ (a,0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in 
         if D < r then if D dvd r then D else r gmod D else r)"
     by (rule reduce_gcd[OF A _ j Aaj], insert a, simp)
   also have "... \<noteq> 0" unfolding Let_def using D0 
-    by (smt Aaj gcd_eq_0_iff gmod_0_imp_dvd)
+    by (smt (verit) Aaj gcd_eq_0_iff gmod_0_imp_dvd)
   finally show "reduce_abs a b D A $$ (a, 0) \<noteq> 0" .
 qed
 
 lemma reduce_below_not0:
  assumes A: "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n" 
     and Aaj: "A $$ (a,0) \<noteq> 0" 
 and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D\<noteq> 0"
   shows "reduce_below a xs D A $$ (a, 0) \<noteq> 0" (is "?R $$ (a,0) \<noteq> _")
   using assms
 proof (induct a xs D A arbitrary: A rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note Aaj = "2.prems"(4)
   note d = "2.prems"(5)
   note D0 = "2.prems"(7)
   note x_less_xxs = "2.prems"(6)
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat m n"
     by (metis (no_types, lifting) A carrier_matD carrier_mat_triv reduce_preserves_dimensions)
   have h: "reduce_below a xs D (reduce a x D A) $$ (a,0) \<noteq> 0"
   proof (rule "2.hyps")
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A a _ j xm Aaj D0], insert x_less_xxs, simp)
   qed (insert A a j Aaj d x_less_xxs xm reduce_ax D0, auto)
   thus ?case by auto
 qed
 
 
 
 lemma reduce_below_abs_not0:
  assumes A: "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n" 
     and Aaj: "A $$ (a,0) \<noteq> 0" 
 and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D\<noteq> 0"
   shows "reduce_below_abs a xs D A $$ (a, 0) \<noteq> 0" (is "?R $$ (a,0) \<noteq> _")
   using assms
 proof (induct a xs D A arbitrary: A rule: reduce_below_abs.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note Aaj = "2.prems"(4)
   note d = "2.prems"(5)
   note D0 = "2.prems"(7)
   note x_less_xxs = "2.prems"(6)
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce_abs a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat m n"
     by (metis (no_types, lifting) A carrier_matD carrier_mat_triv reduce_preserves_dimensions)
   have h: "reduce_below_abs a xs D (reduce_abs a x D A) $$ (a,0) \<noteq> 0"
   proof (rule "2.hyps")
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A a _ j xm Aaj D0], insert x_less_xxs, simp)
   qed (insert A a j Aaj d x_less_xxs xm reduce_ax D0, auto)
   thus ?case by auto
 qed
 
 
 
 lemma reduce_below_not0_case_m:
  assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
     and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "D \<noteq> 0"
   shows "reduce_below a (xs@[m]) D A $$ (a, 0) \<noteq> 0" (is "?R $$ (a,0) \<noteq> _")
   using assms
 proof (induct a xs D A arbitrary: A A' rule: reduce_below.induct)
   case (1 a D A)
   note A' = "1.prems"(1)
   note a = "1.prems"(2)
   note n = "1.prems"(3)
   note A_def = "1.prems"(4)
   note Aaj = "1.prems"(5)
   note mn = "1.prems"(6)
   note all_less_xxs = "1.prems"(7)
   note D0 = "1.prems"(8)
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   have "reduce_below a ([] @ [m]) D A $$ (a, 0) = reduce_below a [m] D A $$ (a, 0)" by auto
   also have "... = reduce a m D A $$ (a, 0)" by auto
   also have "... \<noteq> 0"
     by (rule reduce_not0[OF A _ a n _ Aaj D0], insert a n, auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note n = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note x_less_xxs = "2.prems"(7)
   note D0= "2.prems"(8)
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m+n) n"
     by (metis (no_types, lifting) A carrier_matD carrier_mat_triv reduce_preserves_dimensions)
   have h: "reduce_below a (xs@[m]) D (reduce a x D A) $$ (a,0) \<noteq> 0"
   proof (rule "2.hyps")
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ _ D0], insert x_less_xxs j Aaj, auto)
     let ?reduce_ax' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "?reduce_ax = ?reduce_ax' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule reduce_append_rows_eq[OF A' A_def a xm n Aaj])
   qed (insert A a j Aaj x_less_xxs xm reduce_ax mn D0, auto)
   thus ?case by auto
 qed
 
 lemma reduce_below_abs_not0_case_m:
  assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
     and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "D \<noteq> 0"
   shows "reduce_below_abs a (xs@[m]) D A $$ (a, 0) \<noteq> 0" (is "?R $$ (a,0) \<noteq> _")
   using assms
 proof (induct a xs D A arbitrary: A A' rule: reduce_below_abs.induct)
   case (1 a D A)
   note A' = "1.prems"(1)
   note a = "1.prems"(2)
   note n = "1.prems"(3)
   note A_def = "1.prems"(4)
   note Aaj = "1.prems"(5)
   note mn = "1.prems"(6)
   note all_less_xxs = "1.prems"(7)
   note D0 = "1.prems"(8)
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   have "reduce_below_abs a ([] @ [m]) D A $$ (a, 0) = reduce_below_abs a [m] D A $$ (a, 0)" by auto
   also have "... = reduce_abs a m D A $$ (a, 0)" by auto
   also have "... \<noteq> 0"
     by (rule reduce_not0[OF A _ a n _ Aaj D0], insert a n, auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note n = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note x_less_xxs = "2.prems"(7)
   note D0= "2.prems"(8)
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce_abs a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m+n) n"
     by (metis (no_types, lifting) A carrier_matD carrier_mat_triv reduce_preserves_dimensions)
   have h: "reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) $$ (a,0) \<noteq> 0"
   proof (rule "2.hyps")
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ _ D0], insert x_less_xxs j Aaj, auto)
     let ?reduce_ax' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "?reduce_ax = ?reduce_ax' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule reduce_append_rows_eq[OF A' A_def a xm n Aaj])
   qed (insert A a j Aaj x_less_xxs xm reduce_ax mn D0, auto)
   thus ?case by auto
 qed
 
 
 
 
 
 lemma reduce_below_invertible_mat:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "m\<ge>n"
     and "D>0"
   shows "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> reduce_below a xs D A = P * A)"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below.induct)
   case (1 a D A)
   then show ?case
     by (metis append_rows_def carrier_matD(1) index_mat_four_block(2) reduce_below.simps(1)
         index_smult_mat(2) index_zero_mat(2) invertible_mat_one left_mult_one_mat' one_carrier_mat)
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note d = "2.prems"(6)
   note x_less_xxs = "2.prems"(7)
   note mn = "2.prems"(8)
   note D_ge0 = "2.prems"(9)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have h: "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below a xs D (reduce a x D A) = P * reduce a x D A)"
   proof (rule "2.hyps"[OF _ a j _ _ ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
   qed (insert mn d x_less_xxs D_ge0, auto)
   from this obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
     and rb_Pr: "reduce_below a xs D (reduce a x D A) = P * reduce a x D A" by blast
   have *: "reduce_below a (x # xs) D A = reduce_below a xs D (reduce a x D A)" by simp
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m+n) (m+n) \<and> (reduce a x D A) = Q * A"
     by (rule reduce_invertible_mat[OF A' a j xm _ A_def Aaj ], insert "2.prems", auto)
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and r_QA: "reduce a x D A = Q * A" by blast
   have "invertible_mat (P*Q)" using inv_P inv_Q P Q invertible_mult_JNF by blast
   moreover have "P * Q \<in> carrier_mat (m+n) (m+n)" using P Q by auto
   moreover have "reduce_below a (x # xs) D A = (P*Q) * A" 
-    by (smt P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
+    by (smt (verit) P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
         r_QA rb_Pr reduce_preserves_dimensions(1))
   ultimately show ?case by blast
 qed
 
 
 lemma reduce_below_abs_invertible_mat:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "m\<ge>n"
     and "D>0"
   shows "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> reduce_below_abs a xs D A = P * A)"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below_abs.induct)
   case (1 a D A)
   then show ?case
     by (metis carrier_append_rows invertible_mat_one left_mult_one_mat one_carrier_mat
         reduce_below_abs.simps(1) smult_carrier_mat)
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note d = "2.prems"(6)
   note x_less_xxs = "2.prems"(7)
   note mn = "2.prems"(8)
   note D_ge0 = "2.prems"(9)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   have xm: "x < m" using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by simp
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce_abs a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have h: "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below_abs a xs D (reduce_abs a x D A) = P * reduce_abs a x D A)"
   proof (rule "2.hyps"[OF _ a j _ _ ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
   qed (insert mn d x_less_xxs D_ge0, auto)
   from this obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
     and rb_Pr: "reduce_below_abs a xs D (reduce_abs a x D A) = P * reduce_abs a x D A" by blast
   have *: "reduce_below_abs a (x # xs) D A = reduce_below_abs a xs D (reduce_abs a x D A)" by simp
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m+n) (m+n) \<and> (reduce_abs a x D A) = Q * A"
     by (rule reduce_abs_invertible_mat[OF A' a j xm _ A_def Aaj ], insert "2.prems", auto)
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and r_QA: "reduce_abs a x D A = Q * A" by blast
   have "invertible_mat (P*Q)" using inv_P inv_Q P Q invertible_mult_JNF by blast
   moreover have "P * Q \<in> carrier_mat (m+n) (m+n)" using P Q by auto
   moreover have "reduce_below_abs a (x # xs) D A = (P*Q) * A" 
-    by (smt P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
+    by (smt (verit) P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
         r_QA rb_Pr reduce_preserves_dimensions(3))
   ultimately show ?case by blast
 qed
 
 
 
 lemma reduce_below_preserves:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<notin> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n"
   and "D>0"
   shows "reduce_below a xs D A $$ (i,j) = A $$ (i,j)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(12)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) 2 add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below a (x # xs) D A $$ (i, j) = reduce_below a xs D (reduce a x D A) $$ (i, j)"
     by auto
   also have "... = reduce a x D A $$ (i, j)"
   proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm D_ge0])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm _ Aaj], insert j, auto)  
     show "i \<notin> set xs" using i_set_xxs by auto
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
   qed  
   also have "... = A $$ (i,j)" by (rule reduce_preserves[OF A j Aaj], insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 
 
 lemma reduce_below_abs_preserves:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<notin> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n"
   and "D>0"
   shows "reduce_below_abs a xs D A $$ (i,j) = A $$ (i,j)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below_abs.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(12)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce_abs a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) 2 add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below_abs a (x # xs) D A $$ (i, j) = reduce_below_abs a xs D (reduce_abs a x D A) $$ (i, j)"
     by auto
   also have "... = reduce_abs a x D A $$ (i, j)"
   proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm D_ge0])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm _ Aaj], insert j, auto)  
     show "i \<notin> set xs" using i_set_xxs by auto
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
   qed  
   also have "... = A $$ (i,j)" by (rule reduce_preserves[OF A j Aaj], insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 
 lemma reduce_below_0:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D>0"
   shows "reduce_below a xs D A $$ (i,0) = 0"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note D_ge0 = "2.prems"(10)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   show ?case
   proof (cases "i=x")
     case True
     have "reduce_below a (x # xs) D A $$ (i, 0) = reduce_below a xs D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = (reduce a x D A) $$ (i, 0)"
     proof (rule reduce_below_preserves[OF _ a j _ _ mn ])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
         by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])      
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
       show "i \<notin> set xs" using True d by auto
       show "i \<noteq> a" using "2.prems" by blast
       show "i < m + n"
         by (simp add: True trans_less_add1 xm)
     qed (insert D_ge0)
     also have "... = 0" unfolding True by (rule reduce_0[OF A _ j _ _ Aaj], insert "2.prems", auto)
     finally show ?thesis .
   next
     case False note i_not_x = False    
     have h: "reduce_below a xs D (reduce a x D A) $$ (i, 0) = 0 "
     proof (rule "2.hyps"[OF _ a j _ _ mn])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       proof (rule matrix_append_rows_eq_if_preserves[OF reduce_ax D1])
         show "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_ax $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" 
         proof (rule+)
           fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
           have ja_dc: "ja < dim_col A" and i_dr: "i < dim_row A" using i ja A by auto
           have i_not_a: "i \<noteq> a" using i a by auto
           have i_not_x: "i \<noteq> x" using i xm by auto
           have "?reduce_ax $$ (i,ja) = A $$ (i,ja)" 
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
           also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
             by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
           finally show "?reduce_ax $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
         qed
       qed
       show "i \<in> set xs" using i_set_xxs i_not_x by auto
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
     qed (insert D_ge0)
     have "reduce_below a (x # xs) D A $$ (i, 0) = reduce_below a xs D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = 0" using h .
     finally show ?thesis .
   qed
 qed
 
 lemma reduce_below_abs_0:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D>0"
   shows "reduce_below_abs a xs D A $$ (i,0) = 0"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below_abs.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note D_ge0 = "2.prems"(10)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce_abs a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   show ?case
   proof (cases "i=x")
     case True
     have "reduce_below_abs a (x # xs) D A $$ (i, 0) = reduce_below_abs a xs D (reduce_abs a x D A) $$ (i, 0)"
       by auto
     also have "... = (reduce_abs a x D A) $$ (i, 0)"
     proof (rule reduce_below_abs_preserves[OF _ a j _ _ mn ])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
         by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])      
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
       show "i \<notin> set xs" using True d by auto
       show "i \<noteq> a" using "2.prems" by blast
       show "i < m + n"
         by (simp add: True trans_less_add1 xm)
     qed (insert D_ge0)
     also have "... = 0" unfolding True by (rule reduce_0[OF A _ j _ _ Aaj], insert "2.prems", auto)
     finally show ?thesis .
   next
     case False note i_not_x = False    
     have h: "reduce_below_abs a xs D (reduce_abs a x D A) $$ (i, 0) = 0 "
     proof (rule "2.hyps"[OF _ a j _ _ mn])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       proof (rule matrix_append_rows_eq_if_preserves[OF reduce_ax D1])
         show "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_ax $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" 
         proof (rule+)
           fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
           have ja_dc: "ja < dim_col A" and i_dr: "i < dim_row A" using i ja A by auto
           have i_not_a: "i \<noteq> a" using i a by auto
           have i_not_x: "i \<noteq> x" using i xm by auto
           have "?reduce_ax $$ (i,ja) = A $$ (i,ja)" 
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
           also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
             by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
           finally show "?reduce_ax $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
         qed
       qed
       show "i \<in> set xs" using i_set_xxs i_not_x by auto
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
     qed (insert D_ge0)
     have "reduce_below_abs a (x # xs) D A $$ (i, 0) = reduce_below_abs a xs D (reduce_abs a x D A) $$ (i, 0)"
       by auto
     also have "... = 0" using h .
     finally show ?thesis .
   qed
 qed
 
 
 
 
 lemma reduce_below_preserves_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<notin> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n" and "i\<noteq> m"
   and "D>0"
   shows "reduce_below a (xs @ [m]) D A $$ (i,j) = A $$ (i,j)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below.induct)
   case (1 a D A)
   have "reduce_below a ([] @ [m]) D A $$ (i, j) =  reduce_below a [m] D A $$ (i, j)" by auto
   also have "... = reduce a m D A $$ (i,j)" by auto
   also have "... = A $$ (i,j)"
     by (rule reduce_preserves, insert "1", auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(13)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) A' A_def add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below a ((x # xs) @ [m]) D A $$ (i, j) 
       = reduce_below a (xs@[m]) D (reduce a x D A) $$ (i, j)"
     by auto
   also have "... = reduce a x D A $$ (i, j)"
   proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm _ D_ge0])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm _ Aaj], insert j, auto)  
     show "i \<notin> set xs" using i_set_xxs by auto
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
     show "i\<noteq>m" using "2.prems" by auto
   qed  
   also have "... = A $$ (i,j)" by (rule reduce_preserves[OF A j Aaj], insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 lemma reduce_below_abs_preserves_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "j<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<notin> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n" and "i\<noteq> m"
   and "D>0"
   shows "reduce_below_abs a (xs @ [m]) D A $$ (i,j) = A $$ (i,j)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below_abs.induct)
   case (1 a D A)
   have "reduce_below_abs a ([] @ [m]) D A $$ (i, j) = reduce_below_abs a [m] D A $$ (i, j)" by auto
   also have "... = reduce_abs a m D A $$ (i,j)" by auto
   also have "... = A $$ (i,j)"
     by (rule reduce_preserves, insert "1", auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(13)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce_abs a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) A' A_def add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below_abs a ((x # xs) @ [m]) D A $$ (i, j) 
       = reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) $$ (i, j)"
     by auto
   also have "... = reduce_abs a x D A $$ (i, j)"
   proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm _ D_ge0])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm _ Aaj], insert j, auto)  
     show "i \<notin> set xs" using i_set_xxs by auto
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
     show "i\<noteq>m" using "2.prems" by auto
   qed  
   also have "... = A $$ (i,j)" by (rule reduce_preserves[OF A j Aaj], insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 
 lemma reduce_below_0_case_m1:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "m\<noteq>a"
   and "D>0"
   shows "reduce_below a (xs @ [m]) D A $$ (m,0) = 0"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below.induct)
   case (1 a D A)
   have A: "A \<in> carrier_mat (m+n) n" using "1" by auto
   have " reduce_below a ([] @ [m]) D A $$ (m, 0) =  reduce_below a [m] D A $$ (m, 0)" by auto
   also have "... = reduce a m D A $$ (m,0)" by auto
   also have "... = 0" by (rule reduce_0[OF A], insert "1.prems", auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note d = "2.prems"(7)
   note xxs_less_m = "2.prems"(8)
   note ma = "2.prems"(9)
   note D_ge0 = "2.prems"(10)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below a ((x # xs) @ [m]) D A $$ (m, 0) = reduce_below a (xs@[m]) D (reduce a x D A) $$ (m, 0)"
     by auto
   also have "... = 0"
   proof (rule "2.hyps"[OF ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
   qed (insert "2.prems", auto)
   finally show ?case .
 qed
 
 lemma reduce_below_abs_0_case_m1:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "m\<noteq>a"
   and "D>0"
   shows "reduce_below_abs a (xs @ [m]) D A $$ (m,0) = 0"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below_abs.induct)
   case (1 a D A)
   have A: "A \<in> carrier_mat (m+n) n" using "1" by auto
   have " reduce_below_abs a ([] @ [m]) D A $$ (m, 0) =  reduce_below_abs a [m] D A $$ (m, 0)" by auto
   also have "... = reduce_abs a m D A $$ (m,0)" by auto
   also have "... = 0" by (rule reduce_0[OF A], insert "1.prems", auto)
   finally show ?case .
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note d = "2.prems"(7)
   note xxs_less_m = "2.prems"(8)
   note ma = "2.prems"(9)
   note D_ge0 = "2.prems"(10)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce_abs a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below_abs a ((x # xs) @ [m]) D A $$ (m, 0) = reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) $$ (m, 0)"
     by auto
   also have "... = 0"
   proof (rule "2.hyps"[OF ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
     show "?A' \<in> carrier_mat m n" by auto    
   qed (insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 
 lemma reduce_below_preserves_case_m2:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n"
   and "D>0"
   shows "reduce_below a (xs @ [m]) D A $$ (i,0) = reduce_below a xs D A $$ (i,0)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(12)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) A_def A' add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   show ?case
   proof (cases "i=x")
     case True
     have "reduce_below a ((x # xs) @ [m]) D A $$ (i, 0) 
       = reduce_below a (xs @ [m]) D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = (reduce a x D A) $$ (i, 0)"
     proof (rule reduce_below_preserves_case_m[OF _ a j _ _ mn _ _ _ _ _ _ D_ge0])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       proof (rule matrix_append_rows_eq_if_preserves[OF reduce_ax D1])
         show "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_ax $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" 
         proof (rule+)
           fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
           have ja_dc: "ja < dim_col A" and i_dr: "i < dim_row A" using i ja A by auto
           have i_not_a: "i \<noteq> a" using i a by auto
           have i_not_x: "i \<noteq> x" using i xm by auto
           have "?reduce_ax $$ (i,ja) = A $$ (i,ja)" 
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
           also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
             by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
           finally show "?reduce_ax $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
         qed
       qed
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
       show "i \<notin> set xs" using True d by auto
       show "i \<noteq> a" using "2.prems" by blast
       show "i < m + n" 
         by (simp add: True trans_less_add1 xm)
       show "i \<noteq> m"  by (simp add: True less_not_refl3 xm)
     qed
     also have "... = 0" unfolding True by (rule reduce_0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     also have "... = reduce_below a (x # xs) D A $$ (i, 0) "
       unfolding True by (rule reduce_below_0[symmetric], insert "2.prems", auto)
     finally show ?thesis .
   next
     case False
     have "reduce_below a ((x # xs) @ [m]) D A $$ (i, 0) 
       = reduce_below a (xs@[m]) D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = reduce_below a xs D (reduce a x D A) $$ (i, 0)"
     proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm D_ge0])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
         by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
       show "i \<in> set xs" using i_set_xxs False by auto
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto    
     qed  
     also have "... = reduce_below a (x # xs) D A $$ (i, 0)" by auto
     finally show ?thesis .
   qed
 qed
 
 
 lemma reduce_below_abs_preserves_case_m2:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m+n"
   and "D>0"
   shows "reduce_below_abs a (xs @ [m]) D A $$ (i,0) = reduce_below_abs a xs D A $$ (i,0)"
   using assms
 proof (induct a xs D A arbitrary: A' i rule: reduce_below_abs.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note i_set_xxs = "2.prems"(7)
   note d = "2.prems"(8)
   note xxs_less_m = "2.prems"(9)
   note ia = "2.prems"(10)
   note imm = "2.prems"(11)
   note D_ge0 = "2.prems"(12)
   have D0: "D\<noteq>0" using D_ge0 by simp
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce_abs a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) A_def A' add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   show ?case
   proof (cases "i=x")
     case True
     have "reduce_below_abs a ((x # xs) @ [m]) D A $$ (i, 0) 
       = reduce_below_abs a (xs @ [m]) D (reduce_abs a x D A) $$ (i, 0)"
       by auto
     also have "... = (reduce_abs a x D A) $$ (i, 0)"
     proof (rule reduce_below_abs_preserves_case_m[OF _ a j _ _ mn _ _ _ _ _ _ D_ge0])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       proof (rule matrix_append_rows_eq_if_preserves[OF reduce_ax D1])
         show "\<forall>i\<in>{m..<m + n}. \<forall>ja<n. ?reduce_ax $$ (i, ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" 
         proof (rule+)
           fix i ja assume i: "i \<in> {m..<m + n}" and ja: "ja < n"
           have ja_dc: "ja < dim_col A" and i_dr: "i < dim_row A" using i ja A by auto
           have i_not_a: "i \<noteq> a" using i a by auto
           have i_not_x: "i \<noteq> x" using i xm by auto
           have "?reduce_ax $$ (i,ja) = A $$ (i,ja)" 
             unfolding reduce_alt_def_not0[OF Aaj pquvd] using ja_dc i_dr i_not_a i_not_x by auto 
           also have "... = (if i < dim_row A' then A' $$(i,ja) else (D \<cdot>\<^sub>m (1\<^sub>m n))$$(i-m,ja))"
             by (unfold A_def, rule append_rows_nth[OF A' D1 _ ja], insert A i_dr, simp)
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" using i A' by auto
           finally show "?reduce_ax $$ (i,ja) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, ja)" .   
         qed
       qed
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto
       show "i \<notin> set xs" using True d by auto
       show "i \<noteq> a" using "2.prems" by blast
       show "i < m + n" 
         by (simp add: True trans_less_add1 xm)
       show "i \<noteq> m"  by (simp add: True less_not_refl3 xm)
     qed
     also have "... = 0" unfolding True by (rule reduce_0[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     also have "... = reduce_below_abs a (x # xs) D A $$ (i, 0) "
       unfolding True by (rule reduce_below_abs_0[symmetric], insert "2.prems", auto)
     finally show ?thesis .
   next
     case False
     have "reduce_below_abs a ((x # xs) @ [m]) D A $$ (i, 0) 
       = reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) $$ (i, 0)"
       by auto
     also have "... = reduce_below_abs a xs D (reduce_abs a x D A) $$ (i, 0)"
     proof (rule "2.hyps"[OF _ a j _ _ mn _ _ _ ia imm D_ge0])
       let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
       show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
         by (rule reduce_append_rows_eq[OF A' A_def a xm j Aaj])  
       show "i \<in> set xs" using i_set_xxs False by auto
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
         by (rule reduce_not0[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "?A' \<in> carrier_mat m n" by auto    
     qed  
     also have "... = reduce_below_abs a (x # xs) D A $$ (i, 0)" by auto
     finally show ?thesis .
   qed
 qed
 
 
 lemma reduce_below_0_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set (xs @ [m])" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D>0"
   shows "reduce_below a (xs @ [m]) D A $$ (i,0) = 0"
 proof (cases "i=m")
   case True
   show ?thesis by (unfold True, rule reduce_below_0_case_m1, insert assms, auto)
 next
   case False
   have "reduce_below a (xs @ [m]) D A $$ (i,0) = reduce_below a (xs) D A $$ (i,0)"
     by (rule reduce_below_preserves_case_m2[OF A' a j A_def Aaj mn], insert assms False, auto) 
   also have "... = 0" by (rule reduce_below_0, insert assms False, auto)
   finally show ?thesis .
 qed
 
 
 lemma reduce_below_abs_0_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes "i \<in> set (xs @ [m])" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D>0"
   shows "reduce_below_abs a (xs @ [m]) D A $$ (i,0) = 0"
 proof (cases "i=m")
   case True
   show ?thesis by (unfold True, rule reduce_below_abs_0_case_m1, insert assms, auto)
 next
   case False
   have "reduce_below_abs a (xs @ [m]) D A $$ (i,0) = reduce_below_abs a (xs) D A $$ (i,0)"
     by (rule reduce_below_abs_preserves_case_m2[OF A' a j A_def Aaj mn], insert assms False, auto) 
   also have "... = 0" by (rule reduce_below_abs_0, insert assms False, auto)
   finally show ?thesis .
 qed
 
 
 lemma reduce_below_0_case_m_complete:
   assumes A': "A' \<in> carrier_mat m n" and a: "0<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (0,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes i_mn: "i < m+n" and d_xs: "distinct xs" and xs: "\<forall>x \<in> set xs. x < m \<and> 0 < x"
     and ia: "i\<noteq>0"
   and xs_def: "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and D: "D>0"
   shows "reduce_below 0 (xs @ [m]) D A $$ (i,0) = 0"
 proof (cases "i \<in> set (xs @ [m])")
   case True
   show ?thesis by (rule reduce_below_0_case_m[OF A' a j A_def Aaj mn True d_xs xs D])
 next
   case False
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by simp
   have "reduce_below 0 (xs @ [m]) D A $$ (i,0) = A $$ (i,0)"    
     by (rule reduce_below_preserves_case_m[OF A' a j A_def Aaj mn _ _ _ _ _ _ D],
         insert i_mn d_xs xs ia False, auto) 
   also have "... = 0"  using False ia i_mn A unfolding xs_def by auto    
   finally show ?thesis .
 qed
 
 
 
 lemma reduce_below_abs_0_case_m_complete:
   assumes A': "A' \<in> carrier_mat m n" and a: "0<m" and j: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (0,0) \<noteq> 0"
     and mn: "m\<ge>n"
   assumes i_mn: "i < m+n" and d_xs: "distinct xs" and xs: "\<forall>x \<in> set xs. x < m \<and> 0 < x"
     and ia: "i\<noteq>0"
   and xs_def: "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and D: "D>0"
   shows "reduce_below_abs 0 (xs @ [m]) D A $$ (i,0) = 0"
 proof (cases "i \<in> set (xs @ [m])")
   case True
   show ?thesis by (rule reduce_below_abs_0_case_m[OF A' a j A_def Aaj mn True d_xs xs D])
 next
   case False
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by simp
   have "reduce_below_abs 0 (xs @ [m]) D A $$ (i,0) = A $$ (i,0)"    
     by (rule reduce_below_abs_preserves_case_m[OF A' a j A_def Aaj mn _ _ _ _ _ _ D],
         insert i_mn d_xs xs ia False, auto) 
   also have "... = 0"  using False ia i_mn A unfolding xs_def by auto    
   finally show ?thesis .
 qed
 
 
 (*Now we take care of the mth row of A*)
 lemma reduce_below_invertible_mat_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and n0: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x" 
     and D0: "D>0"
   shows "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> reduce_below a (xs@[m]) D A = P * A)"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below.induct)
   case (1 a D A)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(m,0))"
     by (metis prod_cases5)
   have D: "D \<cdot>\<^sub>m (1\<^sub>m n) : carrier_mat n n" by auto
   note A' = "1.prems"(1)
   note a = "1.prems"(2)
   note j = "1.prems"(3)
   note A_def = "1.prems"(4)
   note Aaj = "1.prems"(5)
   note mn = "1.prems"(6)  
   note D0 = "1.prems"(9)
   have Am0_D: "A $$ (m, 0) = D"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (m-m,0)"
-      by (smt (z3) "1"(1) "1"(3) "1"(4) D append_rows_nth3 diff_is_0_eq diff_self_eq_0 less_add_same_cancel1)
+      unfolding "1"(4)
+      by (meson "1"(1) "1"(3) D append_rows_nth3 less_add_same_cancel1 order.refl)
     also have "... = D" by (simp add: n0)
     finally show ?thesis .
   qed
   have "reduce_below a ([]@[m]) D A = reduce a m D A" by auto
   let ?A = "Matrix.mat (dim_row A) (dim_col A) 
       (\<lambda>(i, k). if i = a then p * A $$ (a, k) + q * A $$ (m, k) else 
         if i = m then u * A $$ (a, k) + v * A $$ (m, k) else A $$ (i, k))"
   let ?xs = "[1..<n]"
   let ?ys = "[1..<n]"  
   have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and> reduce a m D A = P * A"
     by (rule reduce_invertible_mat_case_m[OF A' D a _ A_def _ Aaj mn n0 pquvd, of ?xs _ _ ?ys],
         insert a D0 Am0_D, auto)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note n0 = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note d = "2.prems"(7)
   note xxs_less_m = "2.prems"(8)
   note D0 = "2.prems"(9)
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   have Am0_D: "A $$ (m, 0) = D"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (m-m,0)"
-      by (smt (z3) "2"(2) "2"(4) "2"(5) D1 append_rows_nth3 
-          cancel_comm_monoid_add_class.diff_cancel diff_is_0_eq less_add_same_cancel1)
+      unfolding "2"(5)
+      by (meson "2"(2) "2"(4) D1 append_rows_nth3 less_add_same_cancel1 verit_comp_simplify(2))
     also have "... = D" by (simp add: n0)
     finally show ?thesis .
   qed
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have h: "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below a (xs@[m]) D (reduce a x D A) = P * reduce a x D A)"
   proof (rule "2.hyps"[OF _ a n0 _ _ ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm n0 Aaj])  
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ n0 _ Aaj], insert "2.prems", auto)
   qed (insert d xxs_less_m mn n0 D0, auto)
   from this obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
     and rb_Pr: "reduce_below a (xs@[m]) D (reduce a x D A) = P * reduce a x D A" by blast
   have *: "reduce_below a ((x # xs)@[m]) D A = reduce_below a (xs@[m]) D (reduce a x D A)" by simp
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m+n) (m+n) \<and> (reduce a x D A) = Q * A"
     by (rule reduce_invertible_mat[OF A' a n0 xm _ A_def Aaj _ mn D0], insert xxs_less_m, auto)
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and r_QA: "reduce a x D A = Q * A" by blast
   have "invertible_mat (P*Q)" using inv_P inv_Q P Q invertible_mult_JNF by blast
   moreover have "P * Q \<in> carrier_mat (m+n) (m+n)" using P Q by auto
   moreover have "reduce_below a ((x # xs)@[m]) D A = (P*Q) * A" 
-    by (smt P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
+    by (smt (verit) P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
         r_QA rb_Pr reduce_preserves_dimensions(1))
   ultimately show ?case by blast
 qed
 
 
 
 
 (*Now we take care of the mth row of A*)
 lemma reduce_below_abs_invertible_mat_case_m:
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m" and n0: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and Aaj: "A $$ (a,0) \<noteq> 0"
     and mn: "m\<ge>n" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x" 
     and D0: "D>0"
   shows "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m+n) (m+n) \<and> reduce_below_abs a (xs@[m]) D A = P * A)"
   using assms
 proof (induct a xs D A arbitrary: A' rule: reduce_below_abs.induct)
   case (1 a D A)
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(m,0))"
     by (metis prod_cases5)
   have D: "D \<cdot>\<^sub>m (1\<^sub>m n) : carrier_mat n n" by auto
   note A' = "1.prems"(1)
   note a = "1.prems"(2)
   note j = "1.prems"(3)
   note A_def = "1.prems"(4)
   note Aaj = "1.prems"(5)
   note mn = "1.prems"(6)  
   note D0 = "1.prems"(9)
   have Am0_D: "A $$ (m, 0) = D"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (m-m,0)"
-      by (smt (z3) "1"(1) "1"(3) "1"(4) D append_rows_nth3 diff_is_0_eq diff_self_eq_0 less_add_same_cancel1)
+      unfolding "1"(4)
+      by (meson "1"(1) "1"(3) D append_rows_nth3 le_refl less_add_same_cancel1)
     also have "... = D" by (simp add: n0)
     finally show ?thesis .
   qed
   have "reduce_below_abs a ([]@[m]) D A = reduce_abs a m D A" by auto
   let ?A = "Matrix.mat (dim_row A) (dim_col A) 
       (\<lambda>(i, k). if i = a then p * A $$ (a, k) + q * A $$ (m, k) else 
         if i = m then u * A $$ (a, k) + v * A $$ (m, k) else A $$ (i, k))"
   let ?xs = "filter (\<lambda>i. D < \<bar>?A $$ (a, i)\<bar>) [0..<n]"
   let ?ys = "filter (\<lambda>i. D < \<bar>?A $$ (m, i)\<bar>) [0..<n]"  
   have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and> reduce_abs a m D A = P * A"
     by (rule reduce_abs_invertible_mat_case_m[OF A' D a _ A_def _ Aaj mn n0 pquvd, of ?xs _ _ ?ys],
         insert a D0 Am0_D, auto)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A' = "2.prems"(1)
   note a = "2.prems"(2)
   note n0 = "2.prems"(3)
   note A_def = "2.prems"(4)
   note Aaj = "2.prems"(5)
   note mn = "2.prems"(6)
   note d = "2.prems"(7)
   note xxs_less_m = "2.prems"(8)
   note D0 = "2.prems"(9)
   have A: "A \<in> carrier_mat (m+n) n" using A' mn A_def by auto
   have xm: "x < m"  using "2.prems" by auto
   have D1: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" by (simp add: mn)
   have Am0_D: "A $$ (m, 0) = D"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (m-m,0)"
-      by (smt (z3) "2"(2) "2"(4) "2"(5) D1 append_rows_nth3 
-          cancel_comm_monoid_add_class.diff_cancel diff_is_0_eq less_add_same_cancel1)
+      unfolding "2"(5)
+      by (meson "2"(2) "2"(4) D1 append_rows_nth3 less_add_same_cancel1 order_refl)
     also have "... = D" by (simp add: n0)
     finally show ?thesis .
   qed
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce_abs a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat (m + n) n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have h: "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) = P * reduce_abs a x D A)"
   proof (rule "2.hyps"[OF _ a n0 _ _ ])
     let ?A' = "mat_of_rows n (map (Matrix.row ?reduce_ax) [0..<m])"
     show "reduce_abs a x D A = ?A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
       by (rule reduce_append_rows_eq[OF A' A_def a xm n0 Aaj])  
     show "reduce_abs a x D A $$ (a, 0) \<noteq> 0"
       by (rule reduce_not0[OF A _ _ n0 _ Aaj], insert "2.prems", auto)
   qed (insert d xxs_less_m mn n0 D0, auto)
   from this obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
     and rb_Pr: "reduce_below_abs a (xs@[m]) D (reduce_abs a x D A) = P * reduce_abs a x D A" by blast
   have *: "reduce_below_abs a ((x # xs)@[m]) D A = reduce_below_abs a (xs@[m]) D (reduce_abs a x D A)" by simp
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m+n) (m+n) \<and> (reduce_abs a x D A) = Q * A"
     by (rule reduce_abs_invertible_mat[OF A' a n0 xm _ A_def Aaj _ mn D0], insert xxs_less_m, auto)
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and r_QA: "reduce_abs a x D A = Q * A" by blast
   have "invertible_mat (P*Q)" using inv_P inv_Q P Q invertible_mult_JNF by blast
   moreover have "P * Q \<in> carrier_mat (m+n) (m+n)" using P Q by auto
   moreover have "reduce_below_abs a ((x # xs)@[m]) D A = (P*Q) * A" 
-    by (smt P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
+    by (smt (verit) P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
         r_QA rb_Pr reduce_preserves_dimensions(3))
   ultimately show ?case by blast
 qed
 
 end
 
 hide_const (open) C
 
 text \<open>This lemma will be very important, since it will allow us to prove that the output
 matrix is in echelon form.\<close>
 
 lemma echelon_form_four_block_mat:
   assumes A: "A \<in> carrier_mat 1 1"
   and B: "B \<in> carrier_mat 1 (n-1)"
   and D: "D \<in> carrier_mat (m-1) (n-1)"
   and H_def: "H = four_block_mat A B (0\<^sub>m (m-1) 1) D"
   and A00: "A $$ (0,0) \<noteq> 0"
   and e_D: "echelon_form_JNF D"
   and m: "m>0" and n: "n>0"
 shows "echelon_form_JNF H"
 proof (rule echelon_form_JNF_intro)
   have H: "H \<in> carrier_mat m n"
     by (metis H_def Num.numeral_nat(7) A D m n carrier_matD carrier_mat_triv
         index_mat_four_block(2,3) linordered_semidom_class.add_diff_inverse not_less_eq)
   have Hij_Dij: "H $$ (i+1,j+1) = D $$ (i,j)" if i: "i<m-1" and j: "j<n-1" for i j
   proof -
     have "H $$ (i+1,j+1) =  (if (i+1) < dim_row A then if (j+1) < dim_col A then A $$ ((i+1), (j+1)) 
     else B $$ ((i+1), (j+1) - dim_col A) else if (j+1) < dim_col A then 
     (0\<^sub>m (m-1) 1) $$ ((i+1) - dim_row A, (j+1)) else D $$ ((i+1) - dim_row A, (j+1) - dim_col A))"
       unfolding H_def by (rule index_mat_four_block, insert A D i j, auto)
     also have "... = D $$ ((i+1) - dim_row A, (j+1) - dim_col A)" using A D i j B m n by auto
     also have "... = D $$ (i,j)" using A by auto
     finally show ?thesis .
   qed
   have Hij_Dij':  "H $$ (i,j) = D $$ (i-1,j-1)" 
     if i: "i<m" and j: "j<n" and i0: "i>0" and j0: "j>0" for i j
     by (metis (no_types, lifting) H H_def Num.numeral_nat(7) A carrier_matD 
         index_mat_four_block less_Suc0 less_not_refl3 i j i0 j0)
   have Hi0: "H$$(i,0) = 0" if i: "i\<in>{1..<m}" for i
   proof -
     have "H $$ (i,0) =  (if i < dim_row A then if 0 < dim_col A then A $$ (i, 0) 
       else B $$ (i, 0 - dim_col A) else if 0 < dim_col A then 
       (0\<^sub>m (m-1) 1) $$ (i - dim_row A, 0) else D $$ (i - dim_row A, 0 - dim_col A))"
       unfolding H_def by (rule index_mat_four_block, insert A D i, auto)
     also have "... = (0\<^sub>m (m-1) 1) $$ (i - dim_row A, 0)" using A D i m n by auto
     also have "... = 0" using i A n by auto
     finally show ?thesis .
   qed
   have A00_H00: "A $$ (0,0) = H $$ (0,0)" unfolding H_def using A by auto
   have "is_zero_row_JNF j H" if zero_iH: "is_zero_row_JNF i H" and ij: "i < j" and j: "j < dim_row H"
     for i j 
   proof -
     have "\<not> is_zero_row_JNF 0 H" unfolding is_zero_row_JNF_def using m n H A00 A00_H00 by auto
     hence i_not0: "i\<noteq>0" using zero_iH by meson
     have "is_zero_row_JNF (i-1) D" using zero_iH i_not0 Hij_Dij m n D H unfolding is_zero_row_JNF_def
-      by (auto, smt (z3) Suc_leI carrier_matD(1) le_add_diff_inverse2 Hij_Dij One_nat_def Suc_pred carrier_matD(1) j le_add_diff_inverse2
+      by (auto, smt (verit) Suc_leI carrier_matD(1) le_add_diff_inverse2 Hij_Dij One_nat_def Suc_pred carrier_matD(1) j le_add_diff_inverse2
           less_diff_conv less_imp_add_positive plus_1_eq_Suc that(2) trans_less_add1)
     hence "is_zero_row_JNF (j-1) D" using ij e_D D j m i_not0 unfolding echelon_form_JNF_def
-      by (auto, smt H Nat.lessE Suc_pred carrier_matD(1) diff_Suc_1 diff_Suc_less order.strict_trans)
+      by (auto, smt (verit) H Nat.lessE Suc_pred carrier_matD(1) diff_Suc_1 diff_Suc_less order.strict_trans)
     thus ?thesis
-      by (smt A H H_def Hi0 D atLeastLessThan_iff carrier_matD index_mat_four_block(1)
+      by (smt (verit) A H H_def Hi0 D atLeastLessThan_iff carrier_matD index_mat_four_block(1)
           is_zero_row_JNF_def le_add1 less_one linordered_semidom_class.add_diff_inverse not_less_eq
           plus_1_eq_Suc ij j zero_order(3))
   qed
   thus "\<forall>i<dim_row H. is_zero_row_JNF i H \<longrightarrow> \<not> (\<exists>j<dim_row H. i < j \<and> \<not> is_zero_row_JNF j H)"
     by blast
   have "(LEAST n. H $$ (i, n) \<noteq> 0) < (LEAST n. H $$ (j, n) \<noteq> 0)"
     if ij: "i < j" and j: "j < dim_row H" and not_zero_iH: "\<not> is_zero_row_JNF i H" 
     and not_zero_jH: "\<not> is_zero_row_JNF j H" for i j
   proof (cases "i = 0")
     case True
     have "(LEAST n. H $$ (i, n) \<noteq> 0) = 0" unfolding True using A00_H00 A00 by auto
     then show ?thesis
       by (metis (mono_tags) H Hi0 LeastI True atLeastLessThan_iff carrier_matD(1) 
           is_zero_row_JNF_def leI less_one not_gr0 ij j not_zero_jH)
   next
     case False note i_not0 = False
     let ?least_H = "(LEAST n. H $$ (i, n) \<noteq> 0)"
     let ?least_Hj = "(LEAST n. H $$ (j, n) \<noteq> 0)"
 
     have least_not0: "(LEAST n. H $$ (i, n) \<noteq> 0) \<noteq> 0" 
     proof -
       have \<open>dim_row H = m\<close>
         using H by auto
       with \<open>i < j\<close> \<open>j < dim_row H\<close> have \<open>i < m\<close>
         by simp
       then have \<open>H $$ (i, 0) = 0\<close>
         using i_not0 by (auto simp add: Suc_le_eq intro: Hi0)
       moreover from is_zero_row_JNF_def [of i H] not_zero_iH
       obtain n where \<open>H $$ (i, n) \<noteq> 0\<close>
         by blast
       ultimately show ?thesis
         by (metis (mono_tags, lifting) LeastI)
     qed
     have least_not0j: "(LEAST n. H $$ (j, n) \<noteq> 0) \<noteq> 0"
     proof -
       have "\<exists>n. H $$ (j, 0) = 0 \<and> H $$ (j, n) \<noteq> 0"
         by (metis (no_types) H Hi0 LeastI_ex Num.numeral_nat(7) atLeastLessThan_iff carrier_matD(1)
             is_zero_row_JNF_def linorder_neqE_nat not_gr0 not_less_Least not_less_eq order_trans_rules(19)
             ij j not_zero_jH wellorder_Least_lemma(2))
       then show ?thesis
         by (metis (mono_tags, lifting) LeastI_ex)
     qed
     have least_n: "?least_H<n"
-      by (smt H carrier_matD(2) dual_order.strict_trans is_zero_row_JNF_def 
+      by (smt (verit) H carrier_matD(2) dual_order.strict_trans is_zero_row_JNF_def 
           not_less_Least not_less_iff_gr_or_eq not_zero_iH)
     have Hil: "H $$ (i,?least_H) \<noteq> 0" and ln':"(\<forall>n'. (H $$ (i, n') \<noteq> 0) \<longrightarrow> ?least_H \<le> n')" 
       by (metis (mono_tags, lifting) is_zero_row_JNF_def that(3) wellorder_Least_lemma)+
     have Hil_Dil: "H $$ (i,?least_H) = D $$ (i-1,?least_H - 1)"      
     proof -
       have "H $$ (i,?least_H) = (if i < dim_row A then if ?least_H < dim_col A then A $$ (i, ?least_H) 
       else B $$ (i, ?least_H - dim_col A) else if ?least_H < dim_col A then 
       (0\<^sub>m (m-1) 1) $$ (i - dim_row A, ?least_H) else D $$ (i - dim_row A, ?least_H - dim_col A))"
         unfolding H_def
         by (rule index_mat_four_block, insert False j ij H A D n least_n, auto simp add: H_def)
       also have "... = D $$ (i - 1, ?least_H - 1)"
         using False j ij H A D n least_n B Hi0 Hil by auto
       finally show ?thesis .
     qed
     have not_zero_iD: "\<not> is_zero_row_JNF (i-1) D" 
       by (metis (no_types, lifting) Hil Hil_Dil D carrier_matD(2) is_zero_row_JNF_def le_add1 
           le_add_diff_inverse2 least_n least_not0 less_diff_conv less_one
           linordered_semidom_class.add_diff_inverse)
     have not_zero_jD: "\<not> is_zero_row_JNF (j-1) D"
-      by (smt H Hij_Dij' One_nat_def Suc_pred D m carrier_matD diff_Suc_1 ij is_zero_row_JNF_def j
+      by (smt (verit) H Hij_Dij' One_nat_def Suc_pred D m carrier_matD diff_Suc_1 ij is_zero_row_JNF_def j
           least_not0j less_Suc0 less_Suc_eq_0_disj less_one neq0_conv not_less_Least not_less_eq
           plus_1_eq_Suc not_zero_jH zero_order(3))
     have "?least_H - 1 = (LEAST n. D $$ (i-1, n) \<noteq> 0 \<and> n<dim_col D)"
     proof (rule Least_equality[symmetric], rule)
       show "D $$ (i - 1, ?least_H - 1) \<noteq> 0" using Hil Hil_Dil by auto
       show "(LEAST n. H $$ (i, n) \<noteq> 0) - 1 < dim_col D" using least_n least_not0 H D n by auto
       fix n' assume "D $$ (i - 1, n') \<noteq> 0 \<and> n' < dim_col D" 
       hence Di1n'1: "D $$ (i - 1, n') \<noteq> 0" and n': "n' < dim_col D" by auto
       have "(LEAST n. H $$ (i, n) \<noteq> 0) \<le> n' + 1"
       proof (rule Least_le)
         have "H $$ (i, n'+1) = D $$ (i -1, (n'+1)-1)"
           by (rule Hij_Dij', insert i_not0 False H A ij j n' D, auto)
         thus Hin': "H $$ (i, n'+1) \<noteq> 0" using False Di1n'1 Hij_Dij' by auto
       qed
       thus  "(LEAST n. H $$ (i, n) \<noteq> 0) -1 \<le> n'" using least_not0 by auto
     qed
     also have "... = (LEAST n. D $$ (i-1, n) \<noteq> 0)"
     proof (rule Least_equality)
       have "D $$ (i - 1, LEAST n. D $$ (i - 1, n) \<noteq> 0) \<noteq> 0" 
         by (metis (mono_tags, lifting) Hil Hil_Dil LeastI_ex)
       moreover have leastD: "(LEAST n. D $$ (i - 1, n) \<noteq> 0) < dim_col D"
-        by (smt dual_order.strict_trans is_zero_row_JNF_def linorder_neqE_nat
+        by (smt (verit) dual_order.strict_trans is_zero_row_JNF_def linorder_neqE_nat
             not_less_Least not_zero_iD) 
       ultimately show "D $$ (i - 1, LEAST n. D $$ (i - 1, n) \<noteq> 0) \<noteq> 0 
         \<and> (LEAST n. D $$ (i - 1, n) \<noteq> 0) < dim_col D" by simp  
       fix y assume "D $$ (i - 1, y) \<noteq> 0 \<and> y < dim_col D" 
       thus "(LEAST n. D $$ (i - 1, n) \<noteq> 0) \<le> y" by (meson wellorder_Least_lemma(2))
     qed
     finally have leastHi_eq: "?least_H - 1 = (LEAST n. D $$ (i-1, n) \<noteq> 0)" .
     have least_nj: "?least_Hj<n"
-      by (smt H carrier_matD(2) dual_order.strict_trans is_zero_row_JNF_def 
+      by (smt (verit) H carrier_matD(2) dual_order.strict_trans is_zero_row_JNF_def 
           not_less_Least not_less_iff_gr_or_eq not_zero_jH)
     have Hjl: "H $$ (j,?least_Hj) \<noteq> 0" and ln':"(\<forall>n'. (H $$ (j, n') \<noteq> 0) \<longrightarrow> ?least_Hj \<le> n')" 
       by (metis (mono_tags, lifting) is_zero_row_JNF_def not_zero_jH wellorder_Least_lemma)+
     have Hjl_Djl: "H $$ (j,?least_Hj) = D $$ (j-1,?least_Hj - 1)"      
     proof -
       have "H $$ (j,?least_Hj) = (if j < dim_row A then if ?least_Hj < dim_col A then A $$ (j, ?least_Hj) 
       else B $$ (j, ?least_Hj - dim_col A) else if ?least_Hj < dim_col A then 
       (0\<^sub>m (m-1) 1) $$ (j - dim_row A, ?least_Hj) else D $$ (j - dim_row A, ?least_Hj - dim_col A))"
         unfolding H_def
         by (rule index_mat_four_block, insert False j ij H A D n least_nj, auto simp add: H_def)
       also have "... = D $$ (j - 1, ?least_Hj - 1)"
         using False j ij H A D n least_n B Hi0 Hjl by auto
       finally show ?thesis .
     qed
     have "(LEAST n. H $$ (j, n) \<noteq> 0) - 1 = (LEAST n. D $$ (j-1, n) \<noteq> 0 \<and> n<dim_col D)"
     proof (rule Least_equality[symmetric], rule)
       show "D $$ (j - 1, ?least_Hj - 1) \<noteq> 0" using Hil Hil_Dil
-        by (smt H Hij_Dij' LeastI_ex carrier_matD is_zero_row_JNF_def j least_not0j 
+        by (smt (verit) H Hij_Dij' LeastI_ex carrier_matD is_zero_row_JNF_def j least_not0j 
             linorder_neqE_nat not_gr0 not_less_Least order.strict_trans ij not_zero_jH)
       show "(LEAST n. H $$ (j, n) \<noteq> 0) - 1 < dim_col D" using least_nj least_not0j H D n by auto
       fix n' assume "D $$ (j - 1, n') \<noteq> 0 \<and> n' < dim_col D" 
       hence Di1n'1: "D $$ (j - 1, n') \<noteq> 0" and n': "n' < dim_col D" by auto
       have "(LEAST n. H $$ (j, n) \<noteq> 0) \<le> n' + 1"
       proof (rule Least_le)
         have "H $$ (j, n'+1) = D $$ (j -1, (n'+1)-1)"
           by (rule Hij_Dij', insert i_not0 False H A ij j n' D, auto)
         thus Hin': "H $$ (j, n'+1) \<noteq> 0" using False Di1n'1 Hij_Dij' by auto
       qed
       thus  "(LEAST n. H $$ (j, n) \<noteq> 0) -1 \<le> n'" using least_not0 by auto
     qed
     also have "... = (LEAST n. D $$ (j-1, n) \<noteq> 0)"
     proof (rule Least_equality)
       have "D $$ (j - 1, LEAST n. D $$ (j - 1, n) \<noteq> 0) \<noteq> 0" 
         by (metis (mono_tags, lifting) Hjl Hjl_Djl LeastI_ex)
       moreover have leastD: "(LEAST n. D $$ (j - 1, n) \<noteq> 0) < dim_col D"
-        by (smt dual_order.strict_trans is_zero_row_JNF_def linorder_neqE_nat
+        by (smt (verit) dual_order.strict_trans is_zero_row_JNF_def linorder_neqE_nat
             not_less_Least not_zero_jD) 
       ultimately show "D $$ (j - 1, LEAST n. D $$ (j - 1, n) \<noteq> 0) \<noteq> 0 
         \<and> (LEAST n. D $$ (j - 1, n) \<noteq> 0) < dim_col D" by simp  
       fix y assume "D $$ (j - 1, y) \<noteq> 0 \<and> y < dim_col D" 
       thus "(LEAST n. D $$ (j - 1, n) \<noteq> 0) \<le> y" by (meson wellorder_Least_lemma(2))
     qed
     finally have leastHj_eq: "(LEAST n. H $$ (j, n) \<noteq> 0) - 1 = (LEAST n. D $$ (j-1, n) \<noteq> 0)" .
     have ij': "i-1 < j-1" using ij False by auto
     have "j-1 < dim_row D "  using D H ij j by auto
     hence "(LEAST n. D $$ (i-1, n) \<noteq> 0) < (LEAST n. D $$ (j-1, n) \<noteq> 0)" 
       using e_D echelon_form_JNF_def ij' not_zero_jD order.strict_trans by blast    
     thus ?thesis using leastHj_eq leastHi_eq by auto
   qed
   thus "\<forall>i j. i < j \<and> j < dim_row H \<and> \<not> is_zero_row_JNF i H \<and> \<not> is_zero_row_JNF j H 
   \<longrightarrow> (LEAST n. H $$ (i, n) \<noteq> 0) < (LEAST n. H $$ (j, n) \<noteq> 0)" by blast    
 qed
 
 context mod_operation
 begin
 
 
 lemma reduce_below:
   assumes "A \<in> carrier_mat m n"
   shows "reduce_below a xs D A \<in> carrier_mat m n" 
   using assms 
   by (induct a xs D A rule: reduce_below.induct, auto simp add: Let_def euclid_ext2_def) 
 
 lemma reduce_below_preserves_dimensions:
  shows [simp]: "dim_row (reduce_below a xs D A) = dim_row A" 
     and [simp]: "dim_col (reduce_below a xs D A) = dim_col A"
   using reduce_below[of A "dim_row A" "dim_col A"] by auto
 
 
 lemma reduce_below_abs:
   assumes "A \<in> carrier_mat m n"
   shows "reduce_below_abs a xs D A \<in> carrier_mat m n" 
   using assms 
   by (induct a xs D A rule: reduce_below_abs.induct, auto simp add: Let_def euclid_ext2_def) 
 
 lemma reduce_below_abs_preserves_dimensions:
  shows [simp]: "dim_row (reduce_below_abs a xs D A) = dim_row A" 
     and [simp]: "dim_col (reduce_below_abs a xs D A) = dim_col A"
   using reduce_below_abs[of A "dim_row A" "dim_col A"] by auto
 
 
 lemma FindPreHNF_1xn:
  assumes A: "A \<in> carrier_mat m n" and "m<2 \<or> n = 0"
  shows "FindPreHNF abs_flag D A \<in> carrier_mat m n" using assms by auto
 
 lemma FindPreHNF_mx1:
  assumes A: "A \<in> carrier_mat m n" and "m\<ge>2" and "n \<noteq> 0" "n<2"
  shows "FindPreHNF abs_flag D A \<in> carrier_mat m n"
 proof (cases "abs_flag")
   case True
   let ?nz = "(filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<m])"
   have "FindPreHNF abs_flag D A =  (let non_zero_positions = filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [Suc 0..<m]
      in reduce_below_abs 0 non_zero_positions D (if A $$ (0, 0) \<noteq> 0 then A else 
   let i = non_zero_positions ! 0 in swaprows 0 i A))" 
     using assms True by auto
   also have "... =  reduce_below_abs 0 ?nz D (if A $$ (0, 0) \<noteq> 0 then A 
   else let i = ?nz ! 0 in swaprows 0 i A)" unfolding Let_def by auto
   also have "... \<in> carrier_mat m n" using A by auto
   finally show ?thesis .  
 next
   case False
   let ?nz = "(filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<m])"
   have "FindPreHNF abs_flag D A =  (let non_zero_positions = filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [Suc 0..<m]
      in reduce_below 0 non_zero_positions D (if A $$ (0, 0) \<noteq> 0 then A else 
   let i = non_zero_positions ! 0 in swaprows 0 i A))" 
     using assms False by auto
   also have "... =  reduce_below 0 ?nz D (if A $$ (0, 0) \<noteq> 0 then A 
   else let i = ?nz ! 0 in swaprows 0 i A)" unfolding Let_def by auto
   also have "... \<in> carrier_mat m n" using A by auto
   finally show ?thesis .
 qed
   
 
 lemma FindPreHNF_mxn2:
  assumes A: "A \<in> carrier_mat m n" and m: "m\<ge>2" and n: "n\<ge>2"
  shows "FindPreHNF abs_flag D A \<in> carrier_mat m n" 
 using assms
 proof (induct abs_flag D A arbitrary: m n rule: FindPreHNF.induct)
   case (1 abs_flag D A)
   note A = "1.prems"(1)
   note m = "1.prems"(2)
   note n = "1.prems"(3)
   define non_zero_positions where "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   define A' where "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
   define Reduce where [simp]: "Reduce = (if abs_flag then reduce_below_abs else reduce_below)"
   obtain A'_UL A'_UR A'_DL A'_DR where A'_split: "(A'_UL, A'_UR, A'_DL, A'_DR) 
     = split_block (Reduce 0 non_zero_positions D (make_first_column_positive A')) 1 1"     
     by (metis prod_cases4)
   define sub_PreHNF where "sub_PreHNF = FindPreHNF abs_flag D A'_DR"
   have A': "A' \<in> carrier_mat m n" unfolding A'_def using A by auto
   have A'_DR: "A'_DR \<in> carrier_mat (m -1) (n-1)"
     by (cases abs_flag; rule split_block(4)[OF A'_split[symmetric]], insert Reduce_def A A' m n, auto)
   have sub_PreHNF: "sub_PreHNF \<in> carrier_mat (m - 1) (n-1)"
   proof (cases "m-1<2")
     case True
     show ?thesis using A'_DR True unfolding sub_PreHNF_def by auto
   next
     case False note m' = False
     show ?thesis
     proof (cases "n-1<2")
       case True
       show ?thesis 
         unfolding sub_PreHNF_def by (rule FindPreHNF_mx1[OF A'_DR _ _ True], insert n m', auto)
     next
       case False
       show ?thesis
         by (unfold sub_PreHNF_def, rule "1.hyps"
             [of m n, OF _ _ _ non_zero_positions_def A'_def Reduce_def _ A'_split _ _ _ A'_DR],
             insert A False n m' Reduce_def, auto)
     qed  
   qed      
   have A'_UL: "A'_UL \<in> carrier_mat 1 1"
     by (cases abs_flag; rule split_block(1)[OF A'_split[symmetric], of "m-1" "n-1"], insert n m A', auto) 
   have A'_UR: "A'_UR \<in> carrier_mat 1 (n-1)"
     by (cases abs_flag; rule split_block(2)[OF A'_split[symmetric], of "m-1"], insert n m A', auto)
   have A'_DL: "A'_DL \<in> carrier_mat (m - 1) 1"
      by (cases abs_flag; rule split_block(3)[OF A'_split[symmetric], of _ "n-1"], insert n m A', auto)
   have *: "(dim_col A = 0) = False" using 1(2-) by auto
   have FindPreHNF_as_fbm: "FindPreHNF abs_flag D A = four_block_mat A'_UL A'_UR A'_DL sub_PreHNF" 
     unfolding FindPreHNF.simps[of abs_flag D A] using A'_split m n A
     unfolding Let_def sub_PreHNF_def  A'_def non_zero_positions_def * 
     apply (cases abs_flag)
-    by (smt (z3) Reduce_def carrier_matD(1) carrier_matD(2) linorder_not_less prod.simps(2))+
+    by (smt (verit) Reduce_def carrier_matD(1) carrier_matD(2) linorder_not_less prod.simps(2))+
   also have "... \<in> carrier_mat m n"
-    by (smt m A'_UL One_nat_def add.commute carrier_matD carrier_mat_triv index_mat_four_block(2,3) 
+    by (smt (verit) m A'_UL One_nat_def add.commute carrier_matD carrier_mat_triv index_mat_four_block(2,3) 
         le_add_diff_inverse2 le_eq_less_or_eq lessI n nat_SN.compat numerals(2) sub_PreHNF)  
   finally show ?case .
 qed
 
 
 lemma FindPreHNF:
  assumes A: "A \<in> carrier_mat m n"
  shows "FindPreHNF abs_flag D A \<in> carrier_mat m n" 
   using assms FindPreHNF_mxn2[OF A] FindPreHNF_mx1[OF A] FindPreHNF_1xn[OF A]
   using linorder_not_less by blast
 end
 
 lemma make_first_column_positive_append_id:
  assumes A': "A' \<in> carrier_mat m n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and D0: "D>0"
   and n0: "0<n"
   shows "make_first_column_positive A 
   = mat_of_rows n (map (Matrix.row (make_first_column_positive A)) [0..<m]) @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
 proof (rule matrix_append_rows_eq_if_preserves)
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   thus "make_first_column_positive A \<in> carrier_mat (m + n) n" by auto
   have "make_first_column_positive A $$ (i, j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)"
     if j: "j<n" and i: "i \<in> {m..<m + n}" for i j
   proof -
     have i_mn: "i<m+n" using i by auto
     have "A $$ (i,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, 0)" unfolding A_def
-      by (smt A append_rows_def assms(1) assms(2) atLeastLessThan_iff carrier_matD 
+      by (smt (verit) A append_rows_def assms(1) assms(2) atLeastLessThan_iff carrier_matD 
           index_mat_four_block less_irrefl_nat nat_SN.compat j i n0)
     also have "... \<ge> 0" using D0 mult_not_zero that(2) by auto
     finally have Ai0: "A$$(i,0)\<ge>0" .
     have "make_first_column_positive A $$ (i, j) = A$$(i,j)"
       using make_first_column_positive_works[OF A i_mn n0] j Ai0 by auto
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)" unfolding A_def
-      by (smt A append_rows_def A' A_def atLeastLessThan_iff carrier_matD 
+      by (smt (verit) A append_rows_def A' A_def atLeastLessThan_iff carrier_matD 
           index_mat_four_block less_irrefl_nat nat_SN.compat i j)
     finally show ?thesis .
   qed
   thus "\<forall>i\<in>{m..<m + n}. \<forall>j<n. make_first_column_positive A $$ (i, j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i - m, j)"
     by simp
 qed (auto)
 
 
 lemma A'_swaprows_invertible_mat:
   fixes A::"int mat"
   assumes A: "A\<in>carrier_mat m n"
   assumes A'_def: "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
   and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and nz_empty: "A$$(0,0) =0 \<Longrightarrow> non_zero_positions \<noteq> []"
   and m0: "0<m"
 shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> A' = P * A"
 proof (cases "A$$(0,0) \<noteq> 0")
   case True
   then show ?thesis
     by (metis A A'_def invertible_mat_one left_mult_one_mat one_carrier_mat)
 next
   case False
   have nz_empty: "non_zero_positions \<noteq> []" using nz_empty False by simp
   let ?i = "non_zero_positions ! 0"
   let ?M = "(swaprows_mat m 0 ?i) :: int mat"
   have i_set_nz: "?i \<in> set (non_zero_positions)" using nz_empty by auto
   have im: "?i < m" using A nz_def i_set_nz by auto
   have i_not0: "?i \<noteq> 0" using A nz_def i_set_nz by auto
   have "A' = swaprows 0 ?i A" using False A'_def by simp
   also have "... = ?M * A"
     by (rule swaprows_mat[OF A], insert nz_def nz_empty False A m0 im, auto) 
   finally have 1: "A' = ?M * A" .
   have 2: "?M \<in> carrier_mat m m" by auto
   have "Determinant.det ?M = - 1"
     by (rule det_swaprows_mat[OF m0 im i_not0[symmetric]])
   hence 3: "invertible_mat ?M" using invertible_iff_is_unit_JNF[OF 2] by auto  
   show ?thesis using 1 2 3 by blast
 qed
 
 lemma swaprows_append_id:
  assumes A': "A' \<in> carrier_mat m n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
   and i:"i<m"
   shows "swaprows 0 i A 
   = mat_of_rows n (map (Matrix.row (swaprows 0 i A)) [0..<m]) @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
 proof (rule matrix_append_rows_eq_if_preserves)
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   show swap: "swaprows 0 i A \<in> carrier_mat (m + n) n" by (simp add: A)
   have "swaprows 0 i A $$ (ia, j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (ia - m, j)"
     if ia: "ia \<in> {m..<m + n}" and j: "j<n" for ia j
   proof -
     have "swaprows 0 i A $$ (ia, j) = A $$ (ia,j)" using i ia j A by auto
     also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (ia - m, j)" 
-      by (smt A append_rows_def A' A_def atLeastLessThan_iff carrier_matD 
+      by (smt (verit) A append_rows_def A' A_def atLeastLessThan_iff carrier_matD 
           index_mat_four_block less_irrefl_nat nat_SN.compat ia j)
     finally show "swaprows 0 i A $$ (ia, j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (ia - m, j)" .
   qed
   thus "\<forall>ia\<in>{m..<m + n}. \<forall>j<n. swaprows 0 i A $$ (ia, j) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (ia - m, j)" by simp
 qed (simp)
 
 
 
 lemma non_zero_positions_xs_m:
   fixes A::"'a::comm_ring_1 mat"
   assumes A_def: "A = A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A': "A' \<in> carrier_mat m n" 
   and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and m0: "0<m" and n0: "0<n"
   and D0: "D \<noteq> 0"
 shows "\<exists>xs. non_zero_positions = xs @ [m] \<and> distinct xs \<and> (\<forall>x\<in>set xs. x < m \<and> 0 < x)"
 proof -  
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   let ?xs = "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
   have l_rw: "[1..<dim_row A] = [1..<m+1]@[m+1..<dim_row A]" using A m0 n0
     by (auto, metis Suc_leI less_add_same_cancel1 upt_add_eq_append upt_conv_Cons)
   have f0: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A]) = []"
   proof (rule filter_False) 
     have "A $$ (i,0) = 0" if i: "i\<in>set [m + 1..<dim_row A]" for i
     proof -
       have "A $$ (i,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i-m,0)"
         by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert i A, auto)
       also have "... = 0" using i A by auto
       finally show ?thesis .        
     qed
     thus "\<forall>x\<in>set [m + 1..<dim_row A]. \<not> A $$ (x, 0) \<noteq> 0" by blast
   qed
   have fm: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [m] = [m]"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m,0)"
       by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert n0, auto)
     also have "... = D" using m0 n0 by auto
     finally show ?thesis using D0 by auto
   qed
   have "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m+1]@[m+1..<dim_row A])"
     using nz_def l_rw by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1] @ filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A])"
     by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1]" using f0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m]@[m])" using m0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m] @ [m]" using fm by auto
   finally have "non_zero_positions = ?xs @ [m]" .
   moreover have "distinct ?xs" by auto
   moreover have "(\<forall>x\<in>set ?xs. x < m \<and> 0 < x)" by auto
   ultimately show ?thesis by blast
 qed
 
 
 
 
 lemma non_zero_positions_xs_m':
   fixes A::"'a::comm_ring_1 mat"
   assumes A_def: "A = A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A': "A' \<in> carrier_mat m n" 
   and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and m0: "0<m" and n0: "0<n"
   and D0: "D \<noteq> 0"
 shows "non_zero_positions = (filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]) @ [m] 
   \<and> distinct (filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]) 
   \<and> (\<forall>x\<in>set (filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]). x < m \<and> 0 < x)"
 proof -  
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   let ?xs = "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
   have l_rw: "[1..<dim_row A] = [1..<m+1]@[m+1..<dim_row A]" using A m0 n0
     by (auto, metis Suc_leI less_add_same_cancel1 upt_add_eq_append upt_conv_Cons)
   have f0: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A]) = []"
   proof (rule filter_False) 
     have "A $$ (i,0) = 0" if i: "i\<in>set [m + 1..<dim_row A]" for i
     proof -
       have "A $$ (i,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i-m,0)"
         by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert i A, auto)
       also have "... = 0" using i A by auto
       finally show ?thesis .        
     qed
     thus "\<forall>x\<in>set [m + 1..<dim_row A]. \<not> A $$ (x, 0) \<noteq> 0" by blast
   qed
   have fm: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [m] = [m]"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m,0)"
       by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert n0, auto)
     also have "... = D" using m0 n0 by auto
     finally show ?thesis using D0 by auto
   qed
   have "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m+1]@[m+1..<dim_row A])"
     using nz_def l_rw by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1] @ filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A])"
     by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1]" using f0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m]@[m])" using m0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m] @ [m]" using fm by auto
   finally have "non_zero_positions = ?xs @ [m]" .
   moreover have "distinct ?xs" by auto
   moreover have "(\<forall>x\<in>set ?xs. x < m \<and> 0 < x)" by auto
   ultimately show ?thesis by blast
 qed
 
 lemma A_A'D_eq_first_n_rows:
  assumes A_def: "A = A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A': "A' \<in> carrier_mat m n" 
   and mn: "m\<ge>n"
 shows "(mat_of_rows n (map (Matrix.row A') [0..<n])) 
   = (mat_of_rows n (map (Matrix.row A) [0..<n]))" (is "?lhs = ?rhs")
 proof (rule eq_matI) 
   show dr: "dim_row ?lhs = dim_row ?rhs" and dc: "dim_col ?lhs = dim_col ?rhs" by auto
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n : carrier_mat n n" by simp
   fix i j assume i: "i<dim_row ?rhs" and j: "j<dim_col ?rhs"
   have "?lhs $$ (i,j) = A' $$ (i,j)" using i j dr dc A' mn by (simp add: mat_of_rows_def)
   also have "... = A $$ (i,j)" using append_rows_nth[OF A' D] i j dr dc A' mn A_def by auto
   also have "... = ?rhs $$ (i,j)" using i j dr dc A' A_def mn
     by (metis D calculation carrier_matD(1) diff_zero gr_implies_not0 length_map length_upt 
         linordered_semidom_class.add_diff_inverse mat_of_rows_carrier(2,3)
         mat_of_rows_index nat_SN.compat nth_map_upt row_append_rows1)
   finally show "?lhs $$ (i,j) = ?rhs $$ (i,j)" .
 qed
 
 lemma non_zero_positions_xs_m_invertible:  
   assumes A_def: "A = A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A': "A' \<in> carrier_mat m n" 
   and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and m0: "0<m" and n0: "0<n"
   and D0: "D \<noteq> 0"
   and inv_A'': "invertible_mat (map_mat rat_of_int (mat_of_rows n (map (Matrix.row A') [0..<n])))"
   and A'00: "A' $$ (0,0) = 0"
   and mn: "m\<ge>n"
 shows "length non_zero_positions > 1"
 proof -  
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   have D: "D \<cdot>\<^sub>m 1\<^sub>m n : carrier_mat n n" by auto
   let ?RAT = "map_mat rat_of_int"
   let ?A'' = "(mat_of_rows n (map (Matrix.row A') [0..<n]))"
   have A'': "?A'' \<in> carrier_mat n n" by auto
   have RAT_A'': "?RAT ?A'' \<in> carrier_mat n n" by auto
   let ?ys = "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
   let ?xs = "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<n]"
   have xs_not_empty:"?xs \<noteq> []"
   proof (rule ccontr)
     assume "\<not> ?xs \<noteq> []" hence xs0: "?xs = []" by simp
     have A00: "A $$ (0,0) = 0" 
     proof -
       have "A $$ (0,0) = A'$$(0,0)" unfolding A_def using append_rows_nth[OF A' D] m0 n0 A' by auto
       thus ?thesis using A'00 by simp
     qed
     hence "(\<forall>i\<in>set [1..<n]. A $$ (i,0) = 0)"
       by (metis (mono_tags, lifting) empty_filter_conv xs0)
     hence *: "(\<forall>i<n. A $$ (i,0) = 0)" using A00 n0 using linorder_not_less by force
     have "col ?A'' 0 = 0\<^sub>v n"
     proof (rule eq_vecI)
       show "dim_vec (col ?A'' 0) = dim_vec (0\<^sub>vn)" using A' by auto
       fix i assume i: "i < dim_vec (0\<^sub>v n)" 
       have "col ?A'' 0 $v i = ?A'' $$ (i,0)" by (rule index_col, insert i A' n0, auto)
       also have "... = A $$ (i,0)" 
         unfolding A_def using i A append_rows_nth[OF A' D _ n0] A' mn 
         by (metis A'' n0 carrier_matD(1) index_zero_vec(2) le_add2 map_first_rows_index
             mat_of_rows_carrier(2) mat_of_rows_index nat_SN.compat)
       also have "... = 0" using * i by auto
       finally show "col ?A'' 0 $v i = 0\<^sub>v n $v i" using i by auto
     qed
     hence "col (?RAT ?A'') 0 = 0\<^sub>v n" by auto
     hence "\<not> invertible_mat (?RAT ?A'')"
       using invertible_mat_first_column_not0[OF RAT_A'' _ n0] by auto
     thus False using inv_A'' by contradiction
   qed
   have l_rw: "[1..<dim_row A] = [1..<m+1]@[m+1..<dim_row A]" using A m0 n0
     by (auto, metis Suc_leI less_add_same_cancel1 upt_add_eq_append upt_conv_Cons)
   have f0: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A]) = []"
   proof (rule filter_False) 
     have "A $$ (i,0) = 0" if i: "i\<in>set [m + 1..<dim_row A]" for i
     proof -
       have "A $$ (i,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i-m,0)"
         by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert i A, auto)
       also have "... = 0" using i A by auto
       finally show ?thesis .        
     qed
     thus "\<forall>x\<in>set [m + 1..<dim_row A]. \<not> A $$ (x, 0) \<noteq> 0" by blast
   qed
   have fm: "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [m] = [m]"
   proof -
     have "A $$ (m, 0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m,0)"
       by (rule append_rows_nth3[OF A' _ A_def _ _ n0], insert n0, auto)
     also have "... = D" using m0 n0 by auto
     finally show ?thesis using D0 by auto
   qed
   have "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m+1]@[m+1..<dim_row A])"
     using nz_def l_rw by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1] @ filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([m+1..<dim_row A])"
     by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m+1]" using f0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) ([1..<m]@[m])" using m0 by auto
   also have "... = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m] @ [m]" using fm by auto
   finally have nz: "non_zero_positions = ?ys @ [m]" .
   moreover have ys_not_empty: "?ys \<noteq> []" using xs_not_empty mn
     by (metis (no_types, lifting) atLeastLessThan_iff empty_filter_conv nat_SN.compat set_upt)
   show ?thesis unfolding nz using ys_not_empty by auto  
 qed
 
 
 
 corollary non_zero_positions_length_xs:  
   assumes A_def: "A = A' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A': "A' \<in> carrier_mat m n" 
   and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   and m0: "0<m" and n0: "0<n"
   and D0: "D \<noteq> 0"
   and inv_A'': "invertible_mat (map_mat rat_of_int (mat_of_rows n (map (Matrix.row A') [0..<n])))"
   and A'00: "A' $$ (0,0) = 0"
   and mn: "m\<ge>n"
   and nz_xs_m: "non_zero_positions = xs @ [m]"
 shows "length xs > 0"
 proof -  
   have "length non_zero_positions > 1" 
     by (rule non_zero_positions_xs_m_invertible[OF A_def A' nz_def m0 n0 D0 inv_A'' A'00 mn])
   thus ?thesis using nz_xs_m by auto  
 qed
 
 
 
 lemma make_first_column_positive_nz_conv:
   assumes "i<dim_row A" and "j<dim_col A"
   shows "(make_first_column_positive A $$ (i, j) \<noteq> 0) = (A $$ (i, j) \<noteq> 0)"
   using assms unfolding make_first_column_positive.simps by auto
 
 
 
 lemma make_first_column_positive_00:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
     and A'': "A'' : carrier_mat m n"
   assumes nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
     and A'_def: "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
     and m0: "0<m" and n0: "0<n" and D0: "D \<noteq> 0" and mn: "m\<ge>n"
   shows "make_first_column_positive A' $$ (0, 0) \<noteq> 0"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   hence A': "A' \<in> carrier_mat (m+n) n" unfolding A'_def by auto 
   have "(make_first_column_positive A' $$ (0, 0) \<noteq> 0) = (A' $$ (0, 0) \<noteq> 0)"
     by (rule make_first_column_positive_nz_conv, insert m0 n0 A', auto)
   moreover have "A' $$ (0, 0) \<noteq> 0"
   proof (cases "A $$ (0, 0) \<noteq> 0")
     case True
     then show ?thesis unfolding A'_def by auto
   next
     case False
     have "A $$ (0, 0) = A'' $$ (0, 0)"
-      by (smt add_gr_0 append_rows_def A_def A'' carrier_matD index_mat_four_block(1) mn n0 nat_SN.compat)
+      by (smt (verit) add_gr_0 append_rows_def A_def A'' carrier_matD index_mat_four_block(1) mn n0 nat_SN.compat)
     hence A''00: "A''$$(0,0) = 0" using False by auto
     let ?i = "non_zero_positions ! 0"
     obtain xs where non_zero_positions_xs_m: "non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
       and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
       using non_zero_positions_xs_m[OF A_def A'' nz_def m0 n0] using D0 by fast    
-    have Ai0:"A $$ (?i,0) \<noteq> 0"
-      by (smt append.simps(1) append_Cons append_same_eq nz_def in_set_conv_nth length_greater_0_conv
-          list.simps(3) local.non_zero_positions_xs_m mem_Collect_eq set_filter)  
+    have Ai0: "A $$ (?i,0) \<noteq> 0"
+      by (smt (verit, ccfv_threshold) add_gr_0 length_append less_numeral_extra(1) list.size(4) local.non_zero_positions_xs_m mem_Collect_eq nth_mem nz_def plus_1_eq_Suc set_filter)
     have "A' $$ (0, 0) = swaprows 0 ?i A $$ (0,0)"  using False A'_def by auto
     also have "... \<noteq> 0" using A Ai0 n0 by auto  
     finally show ?thesis .
   qed
   ultimately show ?thesis by blast
 qed
 
 
 context proper_mod_operation
 begin
 lemma reduce_below_0_case_m_make_first_column_positive:
   assumes A': "A' \<in> carrier_mat m n" and m0: "0<m" and n0: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and mn: "m\<ge>n"
   assumes i_mn: "i < m+n" and d_xs: "distinct xs" and xs: "\<forall>x \<in> set xs. x < m \<and> 0 < x"
     and ia: "i\<noteq>0"
     and A''_def: "A'' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
     and D0: "D>0"
     and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   shows "reduce_below 0 non_zero_positions D (make_first_column_positive A'') $$ (i,0) = 0"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   define xs where "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
   have nz_xs_m: "non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
     and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
     using non_zero_positions_xs_m'[OF A_def A' nz_def m0 n0] using D0 A unfolding nz_def xs_def by auto
   have A'': "A'' \<in> carrier_mat (m+n) n" using A' A_def A''_def by auto
   have D_not0: "D\<noteq>0" using D0 by auto
   have Ai0: "A $$ (i, 0) = 0" if im: "i>m" and imn: "i<m+n" for i
   proof-
     have D: "(D \<cdot>\<^sub>m (1\<^sub>m n)) \<in> carrier_mat n n" by simp
     have "A $$ (i, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (i-m, 0)"
       unfolding A_def using append_rows_nth[OF A' D imn n0] im A' by auto 
     also have "... = 0" using im imn n0 by auto
     finally show ?thesis .
   qed
   let ?M' = "mat_of_rows n (map (Matrix.row (make_first_column_positive A'')) [0..<m])"
   have M': "?M' \<in> carrier_mat m n" using A'' by auto
   have mk0: "make_first_column_positive A'' $$ (0, 0) \<noteq> 0"
     by (rule make_first_column_positive_00[OF A_def A' nz_def A''_def m0 n0 D_not0 mn])
   have M_M'D: "make_first_column_positive A'' = ?M' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" if xs_empty: "xs \<noteq> []"
   proof (cases "A$$(0,0) \<noteq> 0")
     case True
     then have *: "make_first_column_positive A'' = make_first_column_positive A"
       unfolding A''_def by auto        
     show ?thesis 
       by (unfold *, rule make_first_column_positive_append_id[OF A' A_def D0 n0])
   next
     case False
     then have *: "make_first_column_positive A'' 
                   = make_first_column_positive (swaprows 0 (non_zero_positions ! 0) A)"
       unfolding A''_def by auto
     show ?thesis
     proof (unfold *, rule make_first_column_positive_append_id)
       let ?S = "mat_of_rows n (map (Matrix.row (swaprows 0 (non_zero_positions ! 0) A)) [0..<m])"
       show "swaprows 0 (non_zero_positions ! 0) A =  ?S @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       proof (rule swaprows_append_id[OF A' A_def])
         have A'00: "A' $$ (0, 0) = 0"          
           by (metis (no_types, lifting) A False add_pos_pos append_rows_def A' A_def 
               carrier_matD index_mat_four_block m0 n0)
         have length_xs: "length xs > 0" using xs_empty by auto
         have "non_zero_positions ! 0 = xs ! 0" unfolding nz_xs_m
           by (meson length_xs nth_append)
         thus "non_zero_positions ! 0 < m" using all_less_m length_xs by simp
       qed
     qed (insert n0 D0, auto)            
   qed
   show ?thesis
   proof (cases "xs = []")
     case True note xs_empty = True
     have "reduce_below 0 non_zero_positions D (make_first_column_positive A'') 
       = reduce 0 m D (make_first_column_positive A'')"
       unfolding nz_xs_m True by auto
 
     also have "...  $$ (i, 0) = 0"
     proof (cases "i=m")
       case True
       from D0 have "D \<ge> 1" "D \<ge> 0" by auto
       then show ?thesis using D0 True
         by (metis A add_sign_intros(2) A''_def carrier_matD(1) carrier_matD(2) carrier_matI 
             index_mat_swaprows(2) index_mat_swaprows(3) less_add_same_cancel1 m0 
             make_first_column_positive_preserves_dimensions mk0 n0 neq0_conv reduce_0)
     next
       case False note i_not_m = False
       have nz_m: "non_zero_positions ! 0 = m" unfolding nz_xs_m True by auto
       let ?M = "make_first_column_positive A''"
       have M: "?M \<in> carrier_mat (m+n) n" using A'' by auto
       show ?thesis
       proof (cases "A$$(0,0) = 0")
         case True
         have "reduce 0 m D ?M $$ (i, 0) = ?M $$ (i,0)" 
           by (rule reduce_preserves[OF M n0 mk0 False ia i_mn])
         also have Mi0: "... = abs (A'' $$ (i,0))"
-          by (smt M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
+          by (smt (verit) M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
               make_first_column_positive_preserves_dimensions n0 prod.simps(2))
         also have Mi02: "... = abs (A $$ (i,0)) " unfolding A''_def nz_m
           using True A False i_mn ia n0 by auto
         also have "... = 0"
         proof -
           have "filter (\<lambda>n. A $$ (n, 0) \<noteq> 0) [1..<m] = []"
             using xs_empty xs_def by presburger
           then have "\<forall>n. A $$ (n, 0) = 0 \<or> n \<notin> set [1..<m]" using filter_empty_conv by fast          
           then show ?thesis
             by (metis (no_types) Ai0 False arith_simps(43) assms(9) atLeastLessThan_iff i_mn
                 le_eq_less_or_eq less_one linorder_neqE_nat set_upt)
         qed
         finally show ?thesis .
       next
         case False hence A00: "A $$ (0,0) \<noteq> 0" by simp
         have "reduce 0 m D ?M $$ (i, 0) = ?M $$ (i,0)" 
           by (rule reduce_preserves[OF M n0 mk0 i_not_m ia i_mn])
         also have Mi0: "... = abs (A'' $$ (i,0))"
-          by (smt M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
+          by (smt (verit) M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
               make_first_column_positive_preserves_dimensions n0 prod.simps(2))
         also have Mi02: "... = abs (swaprows 0 m A $$ (i,0)) " unfolding A''_def nz_m
           using A00 A i_not_m i_mn ia n0 by auto
         also have "... = abs (A $$ (i,0))" using False ia A00 Mi0 A''_def calculation Mi02 by presburger  
         also have "... = 0"
         proof -
           have "filter (\<lambda>n. A $$ (n, 0) \<noteq> 0) [1..<m] = []"
             using True xs_def by presburger
           then have "\<forall>n. A $$ (n, 0) = 0 \<or> n \<notin> set [1..<m]" using filter_empty_conv by fast          
           then show ?thesis
             by (metis (no_types) Ai0 i_not_m arith_simps(43) ia atLeastLessThan_iff i_mn
                 le_eq_less_or_eq less_one linorder_neqE_nat set_upt)
         qed
         finally show ?thesis .
       qed
     qed
     finally show ?thesis .
   next
     case False note xs_not_empty = False
     note M_M'D = M_M'D[OF xs_not_empty]
     show ?thesis   
     proof (cases "i \<in> set (xs @ [m])")
       case True
       show ?thesis  
         by (unfold nz_xs_m, rule reduce_below_0_case_m[OF M' m0 n0 M_M'D mk0 mn True d_xs all_less_m D0])
     next
       case False note i_notin_xs_m = False
       have 1: "reduce_below 0 (xs @ [m]) D (make_first_column_positive A'') $$ (i,0) 
         = (make_first_column_positive A'') $$ (i,0)"    
         by (rule reduce_below_preserves_case_m[OF M' m0 n0 M_M'D mk0 mn _ d_xs all_less_m ia i_mn _ D0],
             insert False, auto) 
       have "((make_first_column_positive A'') $$ (i,0) \<noteq> 0) = (A'' $$ (i,0) \<noteq> 0)"
         by (rule make_first_column_positive_nz_conv, insert A'' i_mn n0, auto)
       hence 2: "((make_first_column_positive A'') $$ (i,0) = 0) = (A'' $$ (i,0) = 0)" by auto
       have 3: "(A'' $$ (i,0) = 0)"
       proof (cases "A$$(0,0) \<noteq> 0")
         case True
         then have "A'' $$ (i, 0) = A $$ (i, 0)" unfolding A''_def by auto
         also have "... = 0" using False ia i_mn A nz_xs_m Ai0 unfolding nz_def xs_def by auto
         finally show ?thesis by auto
       next
         case False hence A00: "A $$ (0,0) = 0" by simp
         let ?i = "non_zero_positions ! 0"
         have i_noti: "i\<noteq>?i" 
           using i_notin_xs_m unfolding nz_xs_m
           by (metis Nil_is_append_conv length_greater_0_conv list.distinct(2) nth_mem)
         have "A''$$(i,0) = (swaprows 0 ?i A) $$ (i,0)" using False unfolding A''_def by auto
         also have "... = A $$ (i,0)" using i_notin_xs_m ia i_mn A i_noti n0 unfolding xs_def by fastforce  
         also have "... = 0" using i_notin_xs_m ia i_mn A i_noti n0 unfolding xs_def 
-          by (smt nz_def atLeastLessThan_iff carrier_matD(1) less_one linorder_not_less
+          by (smt (verit) nz_def atLeastLessThan_iff carrier_matD(1) less_one linorder_not_less
               mem_Collect_eq nz_xs_m set_filter set_upt xs_def) 
         finally show ?thesis .
       qed
       show ?thesis using 1 2 3 nz_xs_m by argo
     qed
   qed
 qed
 
 
 lemma reduce_below_abs_0_case_m_make_first_column_positive:
   assumes A': "A' \<in> carrier_mat m n" and m0: "0<m" and n0: "0<n"
     and A_def: "A = A' @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
     and mn: "m\<ge>n"
   assumes i_mn: "i < m+n" and d_xs: "distinct xs" and xs: "\<forall>x \<in> set xs. x < m \<and> 0 < x"
     and ia: "i\<noteq>0"
     and A''_def: "A'' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
     and D0: "D>0"
     and nz_def: "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   shows "reduce_below_abs 0 non_zero_positions D (make_first_column_positive A'') $$ (i,0) = 0"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A' A_def by auto
   define xs where "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
   have nz_xs_m: "non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
     and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
     using non_zero_positions_xs_m'[OF A_def A' nz_def m0 n0] using D0 A unfolding nz_def xs_def by auto
   have A'': "A'' \<in> carrier_mat (m+n) n" using A' A_def A''_def by auto
   have D_not0: "D\<noteq>0" using D0 by auto
   have Ai0: "A $$ (i, 0) = 0" if im: "i>m" and imn: "i<m+n" for i
   proof-
     have D: "(D \<cdot>\<^sub>m (1\<^sub>m n)) \<in> carrier_mat n n" by simp
     have "A $$ (i, 0) = (D \<cdot>\<^sub>m (1\<^sub>m n)) $$ (i-m, 0)"
       unfolding A_def using append_rows_nth[OF A' D imn n0] im A' by auto 
     also have "... = 0" using im imn n0 by auto
     finally show ?thesis .
   qed
   let ?M' = "mat_of_rows n (map (Matrix.row (make_first_column_positive A'')) [0..<m])"
   have M': "?M' \<in> carrier_mat m n" using A'' by auto
   have mk0: "make_first_column_positive A'' $$ (0, 0) \<noteq> 0"
     by (rule make_first_column_positive_00[OF A_def A' nz_def A''_def m0 n0 D_not0 mn])
   have M_M'D: "make_first_column_positive A'' = ?M' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" if xs_empty: "xs \<noteq> []"
   proof (cases "A$$(0,0) \<noteq> 0")
     case True
     then have *: "make_first_column_positive A'' = make_first_column_positive A"
       unfolding A''_def by auto        
     show ?thesis 
       by (unfold *, rule make_first_column_positive_append_id[OF A' A_def D0 n0])
   next
     case False
     then have *: "make_first_column_positive A'' 
                   = make_first_column_positive (swaprows 0 (non_zero_positions ! 0) A)"
       unfolding A''_def by auto
     show ?thesis
     proof (unfold *, rule make_first_column_positive_append_id)
       let ?S = "mat_of_rows n (map (Matrix.row (swaprows 0 (non_zero_positions ! 0) A)) [0..<m])"
       show "swaprows 0 (non_zero_positions ! 0) A =  ?S @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       proof (rule swaprows_append_id[OF A' A_def])
         have A'00: "A' $$ (0, 0) = 0"          
           by (metis (no_types, lifting) A False add_pos_pos append_rows_def A' A_def 
               carrier_matD index_mat_four_block m0 n0)
         have length_xs: "length xs > 0" using xs_empty by auto
         have "non_zero_positions ! 0 = xs ! 0" unfolding nz_xs_m
           by (meson length_xs nth_append)
         thus "non_zero_positions ! 0 < m" using all_less_m length_xs by simp
       qed
     qed (insert n0 D0, auto)            
   qed
   show ?thesis
   proof (cases "xs = []")
     case True note xs_empty = True
     have "reduce_below_abs 0 non_zero_positions D (make_first_column_positive A'') 
       = reduce_abs 0 m D (make_first_column_positive A'')"
       unfolding nz_xs_m True by auto
 
     also have "...  $$ (i, 0) = 0"
     proof (cases "i=m")
       case True
       from D0 have "D \<ge> 1" "D \<ge> 0" by auto
       then show ?thesis using D0 True
         by (metis A add_sign_intros(2) A''_def carrier_matD(1) carrier_matD(2) carrier_matI 
             index_mat_swaprows(2) index_mat_swaprows(3) less_add_same_cancel1 m0 
             make_first_column_positive_preserves_dimensions mk0 n0 neq0_conv reduce_0)
     next
       case False note i_not_m = False
       have nz_m: "non_zero_positions ! 0 = m" unfolding nz_xs_m True by auto
       let ?M = "make_first_column_positive A''"
       have M: "?M \<in> carrier_mat (m+n) n" using A'' by auto
       show ?thesis
       proof (cases "A$$(0,0) = 0")
         case True
         have "reduce_abs 0 m D ?M $$ (i, 0) = ?M $$ (i,0)" 
           by (rule reduce_preserves[OF M n0 mk0 False ia i_mn])
         also have Mi0: "... = abs (A'' $$ (i,0))"
-          by (smt M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
+          by (smt (verit) M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
               make_first_column_positive_preserves_dimensions n0 prod.simps(2))
         also have Mi02: "... = abs (A $$ (i,0)) " unfolding A''_def nz_m
           using True A False i_mn ia n0 by auto
         also have "... = 0"
         proof -
           have "filter (\<lambda>n. A $$ (n, 0) \<noteq> 0) [1..<m] = []"
             using xs_empty xs_def by presburger
           then have "\<forall>n. A $$ (n, 0) = 0 \<or> n \<notin> set [1..<m]" using filter_empty_conv by fast          
           then show ?thesis
             by (metis (no_types) Ai0 False arith_simps(43) assms(9) atLeastLessThan_iff i_mn
                 le_eq_less_or_eq less_one linorder_neqE_nat set_upt)
         qed
         finally show ?thesis .
       next
         case False hence A00: "A $$ (0,0) \<noteq> 0" by simp
         have "reduce_abs 0 m D ?M $$ (i, 0) = ?M $$ (i,0)" 
           by (rule reduce_preserves[OF M n0 mk0 i_not_m ia i_mn])
         also have Mi0: "... = abs (A'' $$ (i,0))"
-          by (smt M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
+          by (smt (verit) M carrier_matD(1) carrier_matD(2) i_mn index_mat(1) make_first_column_positive.simps
               make_first_column_positive_preserves_dimensions n0 prod.simps(2))
         also have Mi02: "... = abs (swaprows 0 m A $$ (i,0)) " unfolding A''_def nz_m
           using A00 A i_not_m i_mn ia n0 by auto
         also have "... = abs (A $$ (i,0))" using False ia A00 Mi0 A''_def calculation Mi02 by presburger  
         also have "... = 0"
         proof -
           have "filter (\<lambda>n. A $$ (n, 0) \<noteq> 0) [1..<m] = []"
             using True xs_def by presburger
           then have "\<forall>n. A $$ (n, 0) = 0 \<or> n \<notin> set [1..<m]" using filter_empty_conv by fast          
           then show ?thesis
             by (metis (no_types) Ai0 i_not_m arith_simps(43) ia atLeastLessThan_iff i_mn
                 le_eq_less_or_eq less_one linorder_neqE_nat set_upt)
         qed
         finally show ?thesis .
       qed
     qed
     finally show ?thesis .
   next
     case False note xs_not_empty = False
     note M_M'D = M_M'D[OF xs_not_empty]
     show ?thesis   
     proof (cases "i \<in> set (xs @ [m])")
       case True
       show ?thesis  
         by (unfold nz_xs_m, rule reduce_below_abs_0_case_m[OF M' m0 n0 M_M'D mk0 mn True d_xs all_less_m D0])
     next
       case False note i_notin_xs_m = False
       have 1: "reduce_below_abs 0 (xs @ [m]) D (make_first_column_positive A'') $$ (i,0) 
         = (make_first_column_positive A'') $$ (i,0)"    
         by (rule reduce_below_abs_preserves_case_m[OF M' m0 n0 M_M'D mk0 mn _ d_xs all_less_m ia i_mn _ D0],
             insert False, auto) 
       have "((make_first_column_positive A'') $$ (i,0) \<noteq> 0) = (A'' $$ (i,0) \<noteq> 0)"
         by (rule make_first_column_positive_nz_conv, insert A'' i_mn n0, auto)
       hence 2: "((make_first_column_positive A'') $$ (i,0) = 0) = (A'' $$ (i,0) = 0)" by auto
       have 3: "(A'' $$ (i,0) = 0)"
       proof (cases "A$$(0,0) \<noteq> 0")
         case True
         then have "A'' $$ (i, 0) = A $$ (i, 0)" unfolding A''_def by auto
         also have "... = 0" using False ia i_mn A nz_xs_m Ai0 unfolding nz_def xs_def by auto
         finally show ?thesis by auto
       next
         case False hence A00: "A $$ (0,0) = 0" by simp
         let ?i = "non_zero_positions ! 0"
         have i_noti: "i\<noteq>?i" 
           using i_notin_xs_m unfolding nz_xs_m
           by (metis Nil_is_append_conv length_greater_0_conv list.distinct(2) nth_mem)
         have "A''$$(i,0) = (swaprows 0 ?i A) $$ (i,0)" using False unfolding A''_def by auto
         also have "... = A $$ (i,0)" using i_notin_xs_m ia i_mn A i_noti n0 unfolding xs_def by fastforce  
         also have "... = 0" using i_notin_xs_m ia i_mn A i_noti n0 unfolding xs_def 
-          by (smt nz_def atLeastLessThan_iff carrier_matD(1) less_one linorder_not_less
+          by (smt (verit) nz_def atLeastLessThan_iff carrier_matD(1) less_one linorder_not_less
               mem_Collect_eq nz_xs_m set_filter set_upt xs_def) 
         finally show ?thesis .
       qed
       show ?thesis using 1 2 3 nz_xs_m by argo
     qed
   qed
 qed
 
 
 lemma FindPreHNF_invertible_mat_2xn:
   assumes A: "A \<in> carrier_mat m n" and "m<2"
   shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> FindPreHNF abs_flag D A = P * A"
   using assms
   by (auto, metis invertible_mat_one left_mult_one_mat one_carrier_mat)
 
 
 lemma FindPreHNF_invertible_mat_mx2:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A'': "A'' \<in> carrier_mat m n" and n2: "n<2" and n0: "0<n" and D_g0: "D>0" and mn: "m\<ge>n"
 shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> FindPreHNF abs_flag D A = P * A"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   have m0: "m>0" using mn n2 n0 by auto
   have D0: "D\<noteq>0" using D_g0 by auto
   show ?thesis
   proof (cases "m+n<2")
     case True
     show ?thesis by (rule FindPreHNF_invertible_mat_2xn[OF A True])
   next
     case False note mn_le_2 = False
     have dr_A: "dim_row A \<ge>2" using False n2 A by auto
     have dc_A: "dim_col A < 2" using n2 A by auto
     let ?non_zero_positions = "filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [Suc 0..<dim_row A]" 
     let ?A' = "(if A $$ (0, 0) \<noteq> 0 then A else let i = ?non_zero_positions ! 0 in swaprows 0 i A)"
     define xs where "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
     let ?Reduce = "(if abs_flag then reduce_below_abs else reduce_below)"
     have nz_xs_m: "?non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
       and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
       using non_zero_positions_xs_m'[OF A_def A'' _ m0 n0 D0] using D0 A unfolding xs_def by auto  
     have *: "FindPreHNF abs_flag D A = (if abs_flag then reduce_below_abs 0 ?non_zero_positions D ?A' 
         else reduce_below 0 ?non_zero_positions D ?A')"
       using dr_A dc_A by (auto simp add: Let_def) 
     have l: "length ?non_zero_positions > 1" if "xs\<noteq>[]" using that unfolding nz_xs_m by auto
     have inv: "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
       \<and> reduce_below 0 ?non_zero_positions D ?A' = P * ?A'"
     proof (cases "A $$ (0,0) \<noteq>0")
       case True
       show ?thesis
         by (unfold nz_xs_m, rule reduce_below_invertible_mat_case_m
             [OF A'' m0 n0 _ _ mn d_xs all_less_m], insert A_def True D_g0, auto)
     next
       case False hence A00: "A $$ (0,0) = 0" by auto
       let ?S = "swaprows 0 (?non_zero_positions ! 0) A"
       have rw: "(if A $$ (0, 0) \<noteq> 0 then A else let i = ?non_zero_positions ! 0 in swaprows 0 i A)
           = ?S" using False by auto
       show ?thesis
       proof (cases "xs = []")
         case True
         have nz_m: "?non_zero_positions = [m]" using True nz_xs_m by simp
         obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
           by (metis prod_cases5)        
         have Am0: "A $$ (m,0) = D"
         proof -
           have "A $$ (m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m, 0)"
-            by (smt (z3) A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
-                less_add_same_cancel1 less_diff_conv diff_add nat_less_le)
+            unfolding A_def
+            by (meson append_rows_nth3 assms(2) assms(4) less_add_same_cancel1 one_carrier_mat order_refl smult_carrier_mat)
           also have "... = D" by (simp add: n0)
           finally show ?thesis .
         qed
         have Sm0: "(swaprows 0 m A) $$ (m,0) = 0" using A False n0 by auto
         have S00: "(swaprows 0 m A) $$ (0,0) = D" using A Am0 n0 by auto
         have pquvd2: "(p,q,u,v,d) = euclid_ext2 (A $$ (m, 0)) (A $$ (0, 0))"
           using pquvd Sm0 S00 Am0 A00 by auto
         have "reduce_below 0 ?non_zero_positions D ?A' = reduce 0 m D ?A'" unfolding nz_m by auto
         also have "... = reduce 0 m D (swaprows 0 m A)" using True False rw nz_m by auto
         have " \<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and>
            reduce 0 m D (swaprows 0 m A) = P * (swaprows 0 m A)"
         proof (rule reduce_invertible_mat_case_m[OF _ _ m0 _ _ _ _ mn n0])
           show "swaprows 0 m A $$ (0, 0) \<noteq> 0" using S00 D0 by auto
           define S' where  "S' = mat_of_rows n (map (Matrix.row ?S) [0..<m])"
           define S'' where "S'' = mat_of_rows n (map (Matrix.row ?S) [m..<m+n])"
           define A2 where "A2 = Matrix.mat (dim_row (swaprows 0 m A)) (dim_col (swaprows 0 m A))
              (\<lambda>(i, k). if i = 0 then p * A $$ (m, k) + q * A $$ (0, k)
              else if i = m then u * A $$ (m, k) + v * A $$ (0, k) else A $$ (i, k))"
           show S_S'_S'': "swaprows 0 m A = S' @\<^sub>r S''" unfolding S'_def S''_def
             by (metis A append_rows_split carrier_matD index_mat_swaprows(2,3) le_add1 nth_Cons_0 nz_m)
           show S': "S' \<in> carrier_mat m n" unfolding S'_def by fastforce
           show S'': "S'' \<in> carrier_mat n n" unfolding S''_def by fastforce
           show "0 \<noteq> m" using m0 by simp
           show "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
             using pquvd by simp
           show "A2 = Matrix.mat (dim_row (swaprows 0 m A)) (dim_col (swaprows 0 m A))
          (\<lambda>(i, k). if i = 0 then p * swaprows 0 m A $$ (0, k) + q * swaprows 0 m A $$ (m, k)
          else if i = m then u * swaprows 0 m A $$ (0, k) + v * swaprows 0 m A $$ (m, k) else swaprows 0 m A $$ (i, k))"
             (is "_ = ?rhs") using A A2_def by auto
           define xs' where "xs' = [1..<n]"
           define ys' where  "ys' = [1..<n]"
           show "xs' = [1..<n]" unfolding xs'_def by auto
           show "ys' = [1..<n]" unfolding ys'_def by auto
           have S''D: "(S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"
             if jn: "j<n" and j0: "j>0" for j
           proof -
             have "S'' $$ (j, i) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)" if i_n: "i<n" for i
             proof -
               have "S'' $$ (j, i) = swaprows 0 m A $$ (j+m,i)"
                 by (metis S' S'' S_S'_S'' append_rows_nth2 mn nat_SN.compat i_n jn)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .
             qed
             thus ?thesis using jn j0 by auto
           qed
           have "0 \<notin> set xs'"
           proof -
             have "A2 $$ (0,0) = p * A $$ (m, 0) + q * A $$ (0, 0)" 
               using A A2_def n0 by auto
             also have "... = gcd (A $$ (m, 0)) (A $$ (0, 0))"
               by (metis euclid_ext2_works(1) euclid_ext2_works(2) pquvd2)
             also have "... = D" using Am0 A00 D_g0 by auto             
             finally have "A2 $$ (0,0) = D" .
             thus ?thesis unfolding xs'_def using D_g0 by auto
           qed
           thus "\<forall>j\<in>set xs'. j<n \<and> (S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"  
             using S''D xs'_def by auto
           have "0 \<notin> set ys'"
           proof -
             have "A2 $$ (m,0) = u * A $$ (m, 0) + v * A $$ (0, 0)"
               using A A2_def n0 m0 by auto
             also have "... = - A $$ (0, 0) div gcd (A $$ (m, 0)) (A $$ (0, 0)) * A $$ (m, 0) 
               + A $$ (m, 0) div gcd (A $$ (m, 0)) (A $$ (0, 0)) * A $$ (0, 0)"
               by (simp add: euclid_ext2_works[OF pquvd2[symmetric]])
             also have "... = 0" using A00 Am0 by auto
             finally have "A2 $$ (m,0) = 0" .
             thus ?thesis unfolding ys'_def using D_g0 by auto
           qed
           thus "\<forall>j\<in>set ys'. j<n \<and> (S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"
             using S''D ys'_def by auto    
           show "swaprows 0 m A $$ (m, 0) \<in> {0, D}" using Sm0 by blast
           thus "swaprows 0 m A $$ (m, 0) = 0 \<longrightarrow> swaprows 0 m A $$ (0, 0) = D"
             using S00 by linarith                  
         qed (insert D_g0)      
         then show ?thesis by (simp add: False nz_m)
       next
         case False note xs_not_empty = False
         show ?thesis       
       proof (unfold nz_xs_m, rule reduce_below_invertible_mat_case_m[OF _ m0 n0 _ _ mn d_xs all_less_m D_g0])
         let ?S' = "mat_of_rows n (map (Matrix.row ?S) [0..<m])"
         show "?S' \<in> carrier_mat m n" by auto
         have l: "length ?non_zero_positions > 1" using l False by blast
         hence nz0_less_m: "?non_zero_positions ! 0 < m"
           by (metis One_nat_def add.commute add.left_neutral all_less_m append_Cons_nth_left 
               length_append less_add_same_cancel1 list.size(3,4) nth_mem nz_xs_m)    
         have "?S = ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule swaprows_append_id[OF A'' A_def nz0_less_m])
         thus "(if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A)= ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" 
           using rw nz_xs_m by argo
-        have "?S $$ (0, 0) \<noteq> 0"
-          by (smt A l add_pos_pos carrier_matD index_mat_swaprows(1) le_eq_less_or_eq length_greater_0_conv
-              less_one linorder_not_less list.size(3) m0 mem_Collect_eq n0 nth_mem set_filter)
+        have "A $$ (filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [Suc 0..<dim_row A] ! 0, 0) \<noteq> 0"
+          by (metis (mono_tags, lifting) Cons_eq_filterD l length_nth_simps(1) length_nth_simps(3) list.exhaust not_one_less_zero)
+        then have "?S $$ (0, 0) \<noteq> 0"
+          by (metis A add_sign_intros(2) carrier_matD(1) carrier_matD(2) index_mat_swaprows(1) m0 n0)
         thus "(if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A) $$ (0, 0) \<noteq> 0"
           using rw nz_xs_m by algebra
       qed
     qed
   qed
     have inv2: "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) 
       \<and> reduce_below_abs 0 ?non_zero_positions D ?A' = P * ?A'"
     proof (cases "A $$ (0,0) \<noteq>0")
       case True
       show ?thesis
         by (unfold nz_xs_m, rule reduce_below_abs_invertible_mat_case_m
             [OF A'' m0 n0 _ _ mn d_xs all_less_m], insert A_def True D_g0, auto)
     next
       case False hence A00: "A $$ (0,0) = 0" by auto
       let ?S = "swaprows 0 (?non_zero_positions ! 0) A"
       have rw: "(if A $$ (0, 0) \<noteq> 0 then A else let i = ?non_zero_positions ! 0 in swaprows 0 i A)
           = ?S" using False by auto
       show ?thesis
       proof (cases "xs = []")
         case True
         have nz_m: "?non_zero_positions = [m]" using True nz_xs_m by simp
         obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
           by (metis prod_cases5)        
         have Am0: "A $$ (m,0) = D"
         proof -
           have "A $$ (m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m, 0)"
-            by (smt (z3) A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
-                less_add_same_cancel1 less_diff_conv diff_add nat_less_le)
+            unfolding A_def
+            by (meson append_rows_nth3 assms(2) assms(4) less_add_same_cancel1 one_carrier_mat order.refl smult_carrier_mat)
           also have "... = D" by (simp add: n0)
           finally show ?thesis .
         qed
         have Sm0: "(swaprows 0 m A) $$ (m,0) = 0" using A False n0 by auto
         have S00: "(swaprows 0 m A) $$ (0,0) = D" using A Am0 n0 by auto
         have pquvd2: "(p,q,u,v,d) = euclid_ext2 (A $$ (m, 0)) (A $$ (0, 0))"
           using pquvd Sm0 S00 Am0 A00 by auto
         have "reduce_below 0 ?non_zero_positions D ?A' = reduce 0 m D ?A'" unfolding nz_m by auto
         also have "... = reduce 0 m D (swaprows 0 m A)" using True False rw nz_m by auto
         have " \<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and>
            reduce_abs 0 m D (swaprows 0 m A) = P * (swaprows 0 m A)"
         proof (rule reduce_abs_invertible_mat_case_m[OF _ _ m0 _ _ _ _ mn n0])
           show "swaprows 0 m A $$ (0, 0) \<noteq> 0" using S00 D0 by auto
           define S' where  "S' = mat_of_rows n (map (Matrix.row ?S) [0..<m])"
           define S'' where "S'' = mat_of_rows n (map (Matrix.row ?S) [m..<m+n])"
           define A2 where "A2 = Matrix.mat (dim_row (swaprows 0 m A)) (dim_col (swaprows 0 m A))
              (\<lambda>(i, k). if i = 0 then p * A $$ (m, k) + q * A $$ (0, k)
              else if i = m then u * A $$ (m, k) + v * A $$ (0, k) else A $$ (i, k))"
           show S_S'_S'': "swaprows 0 m A = S' @\<^sub>r S''" unfolding S'_def S''_def
             by (metis A append_rows_split carrier_matD index_mat_swaprows(2,3) le_add1 nth_Cons_0 nz_m)
           show S': "S' \<in> carrier_mat m n" unfolding S'_def by fastforce
           show S'': "S'' \<in> carrier_mat n n" unfolding S''_def by fastforce
           show "0 \<noteq> m" using m0 by simp
           show "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
             using pquvd by simp
           show "A2 = Matrix.mat (dim_row (swaprows 0 m A)) (dim_col (swaprows 0 m A))
          (\<lambda>(i, k). if i = 0 then p * swaprows 0 m A $$ (0, k) + q * swaprows 0 m A $$ (m, k)
          else if i = m then u * swaprows 0 m A $$ (0, k) + v * swaprows 0 m A $$ (m, k) else swaprows 0 m A $$ (i, k))"
             (is "_ = ?rhs") using A A2_def by auto
           define xs' where "xs' = filter (\<lambda>i. abs (A2 $$ (0,i)) > D) [0..<n]"
           define ys' where  "ys' = filter (\<lambda>i. abs (A2 $$ (m,i)) > D) [0..<n]"
           show "xs' = filter (\<lambda>i. abs (A2 $$ (0,i)) > D) [0..<n]" unfolding xs'_def by auto
           show "ys' = filter (\<lambda>i. abs (A2 $$ (m,i)) > D) [0..<n]" unfolding ys'_def by auto
           have S''D: "(S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"
             if jn: "j<n" and j0: "j>0" for j
           proof -
             have "S'' $$ (j, i) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)" if i_n: "i<n" for i
             proof -
               have "S'' $$ (j, i) = swaprows 0 m A $$ (j+m,i)"
                 by (metis S' S'' S_S'_S'' append_rows_nth2 mn nat_SN.compat i_n jn)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .
             qed
             thus ?thesis using jn j0 by auto
           qed
           have "0 \<notin> set xs'"
           proof -
             have "A2 $$ (0,0) = p * A $$ (m, 0) + q * A $$ (0, 0)" 
               using A A2_def n0 by auto
             also have "... = gcd (A $$ (m, 0)) (A $$ (0, 0))"
               by (metis euclid_ext2_works(1) euclid_ext2_works(2) pquvd2)
             also have "... = D" using Am0 A00 D_g0 by auto             
             finally have "A2 $$ (0,0) = D" .
             thus ?thesis unfolding xs'_def using D_g0 by auto
           qed
           thus "\<forall>j\<in>set xs'. j<n \<and> (S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"  
             using S''D xs'_def by auto
           have "0 \<notin> set ys'"
           proof -
             have "A2 $$ (m,0) = u * A $$ (m, 0) + v * A $$ (0, 0)"
               using A A2_def n0 m0 by auto
             also have "... = - A $$ (0, 0) div gcd (A $$ (m, 0)) (A $$ (0, 0)) * A $$ (m, 0) 
               + A $$ (m, 0) div gcd (A $$ (m, 0)) (A $$ (0, 0)) * A $$ (0, 0)"
               by (simp add: euclid_ext2_works[OF pquvd2[symmetric]])
             also have "... = 0" using A00 Am0 by auto
             finally have "A2 $$ (m,0) = 0" .
             thus ?thesis unfolding ys'_def using D_g0 by auto
           qed
           thus "\<forall>j\<in>set ys'. j<n \<and> (S'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. S'' $$ (j, j') = 0)"
             using S''D ys'_def by auto    
         qed (insert D_g0)      
         then show ?thesis by (simp add: False nz_m)
       next
         case False note xs_not_empty = False
         show ?thesis       
       proof (unfold nz_xs_m, rule reduce_below_abs_invertible_mat_case_m[OF _ m0 n0 _ _ mn d_xs all_less_m D_g0])
         let ?S' = "mat_of_rows n (map (Matrix.row ?S) [0..<m])"
         show "?S' \<in> carrier_mat m n" by auto
         have l: "length ?non_zero_positions > 1" using l False by blast
         hence nz0_less_m: "?non_zero_positions ! 0 < m"
           by (metis One_nat_def add.commute add.left_neutral all_less_m append_Cons_nth_left 
               length_append less_add_same_cancel1 list.size(3,4) nth_mem nz_xs_m)    
         have "?S = ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule swaprows_append_id[OF A'' A_def nz0_less_m])
         thus "(if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A)= ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" 
           using rw nz_xs_m by argo
         have "?S $$ (0, 0) \<noteq> 0"
-          by (smt A l add_pos_pos carrier_matD index_mat_swaprows(1) le_eq_less_or_eq length_greater_0_conv
+          by (smt (verit) A l add_pos_pos carrier_matD index_mat_swaprows(1) le_eq_less_or_eq length_greater_0_conv
               less_one linorder_not_less list.size(3) m0 mem_Collect_eq n0 nth_mem set_filter)
         thus "(if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A) $$ (0, 0) \<noteq> 0"
           using rw nz_xs_m by algebra
       qed
     qed
   qed
   show ?thesis
   proof (cases abs_flag)
     case False
     from inv obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
       and r_PA': "reduce_below 0 ?non_zero_positions D ?A' = P * ?A'" by blast
     have Find_rw: "FindPreHNF abs_flag D A = reduce_below 0 ?non_zero_positions D ?A'"
       using n0 A dr_A dc_A False * by (auto simp add: Let_def)
     have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?A' = P * A" 
       by (rule A'_swaprows_invertible_mat[OF A], insert non_zero_positions_xs_m n0 m0 l nz_xs_m, auto) 
     from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)"
       and inv_Q: "invertible_mat Q" and A'_QA: "?A' = Q * A" by blast
     have "reduce_below 0 ?non_zero_positions D ?A' = (P * Q) * A" using Q A'_QA P r_PA' A by auto
     moreover have "invertible_mat (P*Q)" using P Q inv_P inv_Q invertible_mult_JNF by blast
     moreover have "(P*Q) \<in> carrier_mat (m + n) (m + n)" using P Q by auto
     ultimately show ?thesis using Find_rw by metis
   next
     case True
     from inv2 obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat (m + n) (m + n)"
       and r_PA': "reduce_below_abs 0 ?non_zero_positions D ?A' = P * ?A'" by blast
     have Find_rw: "FindPreHNF abs_flag D A = reduce_below_abs 0 ?non_zero_positions D ?A'"
       using n0 A dr_A dc_A True * by (auto simp add: Let_def)
     have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> ?A' = P * A" 
       by (rule A'_swaprows_invertible_mat[OF A], insert non_zero_positions_xs_m n0 m0 l nz_xs_m, auto) 
     from this obtain Q where Q: "Q \<in> carrier_mat (m + n) (m + n)"
       and inv_Q: "invertible_mat Q" and A'_QA: "?A' = Q * A" by blast
     have "reduce_below_abs 0 ?non_zero_positions D ?A' = (P * Q) * A" using Q A'_QA P r_PA' A by auto
     moreover have "invertible_mat (P*Q)" using P Q inv_P inv_Q invertible_mult_JNF by blast
     moreover have "(P*Q) \<in> carrier_mat (m + n) (m + n)" using P Q by auto
     ultimately show ?thesis using Find_rw by metis
   qed     
   qed
 qed
 
 
 corollary FindPreHNF_echelon_form_mx0:
   assumes "A \<in> carrier_mat m 0"
   shows "echelon_form_JNF (FindPreHNF abs_flag D A)"
   by (rule echelon_form_mx0, rule FindPreHNF[OF assms])
                
 
 lemma FindPreHNF_echelon_form_mx1:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A'': "A'' \<in> carrier_mat m n" and n2: "n<2" and D_g0: "D>0" and mn: "m\<ge>n"
 shows "echelon_form_JNF (FindPreHNF abs_flag D A)"
 proof (cases "n=0")
   case True
   have A: "A \<in> carrier_mat m 0" using A_def A'' True 
     by (metis add.comm_neutral append_rows_def carrier_matD carrier_matI index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3)) 
   show ?thesis unfolding True by (rule FindPreHNF_echelon_form_mx0, insert A, auto)
 next
   case False hence n0: "0<n" by auto
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   have m0: "m>0" using mn n2 n0 by auto
   have D0: "D\<noteq>0" using D_g0 by auto
   show ?thesis
   proof (cases "m+n<2")
     case True
     show ?thesis by (rule echelon_form_JNF_1xn[OF _ True], rule FindPreHNF[OF A])
   next
     case False note mn_le_2 = False
     have dr_A: "dim_row A \<ge>2" using False n2 A by auto
     have dc_A: "dim_col A < 2" using n2 A by auto
     let ?non_zero_positions = "filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [Suc 0..<dim_row A]" 
     let ?A' = "(if A $$ (0, 0) \<noteq> 0 then A else let i = ?non_zero_positions ! 0 in swaprows 0 i A)" 
     define xs where "xs = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<m]"
     let ?Reduce = "(if abs_flag then reduce_below_abs else reduce_below)"
     have nz_xs_m: "?non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
       and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
       using non_zero_positions_xs_m'[OF A_def A'' _ m0 n0 D0] using D0 A unfolding xs_def by auto  
     have *: "FindPreHNF abs_flag D A = (if abs_flag then reduce_below_abs 0 ?non_zero_positions D ?A' 
         else reduce_below 0 ?non_zero_positions D ?A')"
       using dr_A dc_A by (auto simp add: Let_def) 
     have l: "length ?non_zero_positions > 1" if "xs\<noteq>[]" using that unfolding nz_xs_m by auto
     have e: "echelon_form_JNF (reduce_below 0 ?non_zero_positions D ?A')"
     proof (cases "A $$ (0,0) \<noteq>0")
       case True note A00 = True
       have 1: "reduce_below 0 ?non_zero_positions D ?A' = reduce_below 0 ?non_zero_positions D A"
         using True by auto
       have "echelon_form_JNF (reduce_below 0 ?non_zero_positions D A)"
       proof (rule echelon_form_JNF_mx1[OF _ n2]) 
         show "reduce_below 0 ?non_zero_positions D A \<in> carrier_mat (m+n) n" using A by auto
         show "\<forall>i\<in>{1..<m + n}. reduce_below 0 ?non_zero_positions D A $$ (i, 0) = 0"
         proof 
           fix i assume i: "i \<in> {1..<m + n}"
           show "reduce_below 0 ?non_zero_positions D A $$ (i, 0) =0"
           proof (cases "i\<in>set ?non_zero_positions")
             case True
             show ?thesis unfolding nz_xs_m 
               by (rule reduce_below_0_case_m[OF A'' m0 n0 A_def A00 mn _ d_xs all_less_m D_g0],
                   insert nz_xs_m True, auto)
           next
             case False note i_notin_set = False
             have "reduce_below 0 ?non_zero_positions D A $$ (i, 0) = A $$ (i, 0)" unfolding nz_xs_m 
               by (rule reduce_below_preserves_case_m[OF A'' m0 n0 A_def A00 mn _ d_xs all_less_m _ _ _ D_g0],
                   insert i nz_xs_m i_notin_set, auto)
             also have "... = 0" using i_notin_set i A unfolding set_filter by auto
             finally show ?thesis .        
           qed
         qed
       qed
       thus ?thesis using 1 by argo
     next
       case False hence A00: "A $$ (0,0) = 0" by simp
       let ?i = "((xs @ [m]) ! 0)"
       let ?S = "swaprows 0 ?i A"
       let ?S' = "mat_of_rows n (map (Matrix.row (swaprows 0 ?i A)) [0..<m])"
       have rw: "(if A$$(0, 0) \<noteq> 0 then A else let i = ?non_zero_positions!0 in swaprows 0 i A) = ?S"
         using A00 nz_xs_m by auto
       have S: "?S \<in> carrier_mat (m+n) n" using A by auto
       have A00_eq_A'00: "A $$ (0, 0) = A'' $$ (0, 0)" 
         by (metis A'' A_def add_gr_0 append_rows_def n0 carrier_matD index_mat_four_block(1) m0)
       show ?thesis
       proof (cases "xs=[]")
         case True
         have nz_m: "?non_zero_positions = [m]" using True nz_xs_m by simp
         obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
           by (metis prod_cases5)        
         have Am0: "A $$ (m,0) = D"
         proof -
           have "A $$ (m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m, 0)"
-            by (smt A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
+            by (smt (verit) A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
                 less_add_same_cancel1 less_diff_conv ordered_cancel_comm_monoid_diff_class.diff_add
                 nat_less_le)
           also have "... = D" by (simp add: n0)
           finally show ?thesis .
         qed
         have Sm0: "(swaprows 0 m A) $$ (m,0) = 0" using A False n0 by auto
         have S00: "(swaprows 0 m A) $$ (0,0) = D" using A Am0 n0 by auto
         have pquvd2: "(p,q,u,v,d) = euclid_ext2 (A $$ (m, 0)) (A $$ (0, 0))"
           using pquvd Sm0 S00 Am0 A00 by auto
         have "reduce_below 0 ?non_zero_positions D ?A' = reduce 0 m D ?A'" unfolding nz_m by auto
         also have "... = reduce 0 m D (swaprows 0 m A)" using True False rw nz_m by auto
         finally have *: "reduce_below 0 ?non_zero_positions D ?A' = reduce 0 m D (swaprows 0 m A)" .
         have "echelon_form_JNF (reduce 0 m D (swaprows 0 m A))"
         proof (rule echelon_form_JNF_mx1[OF _ n2])
           show "reduce 0 m D (swaprows 0 m A) \<in> carrier_mat (m+n) n"
             using A n2 reduce_carrier by (auto simp add: Let_def) 
           show "\<forall>i\<in>{1..<m+n}. reduce 0 m D (swaprows 0 m A) $$ (i, 0) = 0"
           proof
             fix i assume i: "i \<in> {1..<m+n}"
             show "reduce 0 m D (swaprows 0 m A) $$ (i, 0) = 0"
             proof (cases "i=m")
               case True
               show ?thesis
               proof (unfold True, rule reduce_0[OF _ _ n0])
                 show "swaprows 0 m A \<in> carrier_mat (m+n) n" using A by auto
               qed (insert m0 n0 S00 D_g0, auto)
             next
               case False
               have "reduce 0 m D (swaprows 0 m A) $$ (i, 0) = (swaprows 0 m A) $$ (i, 0)"
               proof (rule reduce_preserves[OF _ n0])
                 show "swaprows 0 m A \<in> carrier_mat (m+n) n" using A by auto  
               qed (insert m0 n0 S00 D_g0 False i, auto)
               also have "... = A $$ (i, 0)" using i False A n0 by auto
               also have "... = 0"
               proof (rule ccontr)
                 assume "A $$ (i, 0) \<noteq> 0" hence "i \<in> set ?non_zero_positions" using i A by auto 
                 hence "i=m" using nz_xs_m True by auto
                 thus False using False by contradiction
               qed
               finally show ?thesis .
             qed 
           qed
         qed       
         then show ?thesis using * by presburger
       next
         case False        
       have l: "length ?non_zero_positions > 1"    using False nz_xs_m by auto   
       hence l_xs: "length xs > 0" using nz_xs_m by auto
       hence xs_m_less_m: "(xs@[m]) ! 0 < m" by (simp add: all_less_m nth_append)
-      have S00: "?S $$ (0,0) \<noteq> 0"
-        by (smt A add_pos_pos append_Cons_nth_left n0 carrier_matD index_mat_swaprows(1)
-            l_xs m0 mem_Collect_eq nth_mem set_filter xs_def)
+      have "A $$ ((xs @ [m]) ! 0, 0) \<noteq> 0"
+        by (metis (mono_tags, lifting) Cons_eq_filterD List.min_list.cases append_is_Nil_conv nth_Cons_0 nz_xs_m)
+      then have S00: "?S $$ (0,0) \<noteq> 0"
+        using A n0 by auto
       have S': "?S' \<in> carrier_mat m n" using A by auto
       have S_S'D: "?S = ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule swaprows_append_id[OF A'' A_def xs_m_less_m]) 
       have 2: "reduce_below 0 ?non_zero_positions D ?A' = reduce_below 0 ?non_zero_positions D ?S"
         using A00 nz_xs_m by algebra
       have "echelon_form_JNF (reduce_below 0 ?non_zero_positions D ?S)"
       proof (rule echelon_form_JNF_mx1[OF _ n2])
         show "reduce_below 0 ?non_zero_positions D ?S \<in> carrier_mat (m+n) n" using A by auto
         show "\<forall>i\<in>{1..<m + n}. reduce_below 0 ?non_zero_positions D ?S $$ (i, 0) = 0"
         proof 
           fix i assume i: "i \<in> {1..<m + n}"
           show "reduce_below 0 ?non_zero_positions D ?S $$ (i, 0) =0"
           proof (cases "i\<in>set ?non_zero_positions")
             case True
             show ?thesis unfolding nz_xs_m 
               by (rule reduce_below_0_case_m[OF S' m0 n0 S_S'D S00 mn _ d_xs all_less_m D_g0],
                   insert True nz_xs_m, auto)
           next
             case False note i_notin_set = False
             have "reduce_below 0 ?non_zero_positions D ?S $$ (i, 0) = ?S $$ (i, 0)" unfolding nz_xs_m 
               by (rule reduce_below_preserves_case_m[OF S' m0 n0 S_S'D S00 mn _ d_xs all_less_m _ _ _ D_g0],
                   insert i nz_xs_m i_notin_set, auto)
             also have "... = 0" using i_notin_set i A S00 n0 unfolding set_filter by auto              
             finally show ?thesis .        
           qed
         qed
       qed
       thus ?thesis using 2 by argo
     qed
   qed
     have e2: "echelon_form_JNF (reduce_below_abs 0 ?non_zero_positions D ?A')"
     proof (cases "A $$ (0,0) \<noteq>0")
       case True note A00 = True
       have 1: "reduce_below_abs 0 ?non_zero_positions D ?A' = reduce_below_abs 0 ?non_zero_positions D A"
         using True by auto
       have "echelon_form_JNF (reduce_below_abs 0 ?non_zero_positions D A)"
       proof (rule echelon_form_JNF_mx1[OF _ n2]) 
         show "reduce_below_abs 0 ?non_zero_positions D A \<in> carrier_mat (m+n) n" using A by auto
         show "\<forall>i\<in>{1..<m + n}. reduce_below_abs 0 ?non_zero_positions D A $$ (i, 0) = 0"
         proof 
           fix i assume i: "i \<in> {1..<m + n}"
           show "reduce_below_abs 0 ?non_zero_positions D A $$ (i, 0) =0"
           proof (cases "i\<in>set ?non_zero_positions")
             case True
             show ?thesis unfolding nz_xs_m 
               by (rule reduce_below_abs_0_case_m[OF A'' m0 n0 A_def A00 mn _ d_xs all_less_m D_g0],
                   insert nz_xs_m True, auto)
           next
             case False note i_notin_set = False
             have "reduce_below_abs 0 ?non_zero_positions D A $$ (i, 0) = A $$ (i, 0)" unfolding nz_xs_m 
               by (rule reduce_below_abs_preserves_case_m[OF A'' m0 n0 A_def A00 mn _ d_xs all_less_m _ _ _ D_g0],
                   insert i nz_xs_m i_notin_set, auto)
             also have "... = 0" using i_notin_set i A unfolding set_filter by auto
             finally show ?thesis .        
           qed
         qed
       qed
       thus ?thesis using 1 by argo
     next
       case False hence A00: "A $$ (0,0) = 0" by simp
       let ?i = "((xs @ [m]) ! 0)"
       let ?S = "swaprows 0 ?i A"
       let ?S' = "mat_of_rows n (map (Matrix.row (swaprows 0 ?i A)) [0..<m])"
       have rw: "(if A$$(0, 0) \<noteq> 0 then A else let i = ?non_zero_positions!0 in swaprows 0 i A) = ?S"
         using A00 nz_xs_m by auto
       have S: "?S \<in> carrier_mat (m+n) n" using A by auto
       have A00_eq_A'00: "A $$ (0, 0) = A'' $$ (0, 0)" 
         by (metis A'' A_def add_gr_0 append_rows_def n0 carrier_matD index_mat_four_block(1) m0)
       show ?thesis
       proof (cases "xs=[]")
         case True
         have nz_m: "?non_zero_positions = [m]" using True nz_xs_m by simp
         obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (swaprows 0 m A $$ (0, 0)) (swaprows 0 m A $$ (m, 0))"
           by (metis prod_cases5)        
         have Am0: "A $$ (m,0) = D"
         proof -
           have "A $$ (m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m, 0)"
-            by (smt A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
+            by (smt (verit) A append_rows_def A_def A'' n0 carrier_matD diff_self_eq_0 index_mat_four_block
                 less_add_same_cancel1 less_diff_conv ordered_cancel_comm_monoid_diff_class.diff_add
                 nat_less_le)
           also have "... = D" by (simp add: n0)
           finally show ?thesis .
         qed
         have Sm0: "(swaprows 0 m A) $$ (m,0) = 0" using A False n0 by auto
         have S00: "(swaprows 0 m A) $$ (0,0) = D" using A Am0 n0 by auto
         have pquvd2: "(p,q,u,v,d) = euclid_ext2 (A $$ (m, 0)) (A $$ (0, 0))"
           using pquvd Sm0 S00 Am0 A00 by auto
         have "reduce_below_abs 0 ?non_zero_positions D ?A' = reduce_abs 0 m D ?A'" unfolding nz_m by auto
         also have "... = reduce_abs 0 m D (swaprows 0 m A)" using True False rw nz_m by auto
         finally have *: "reduce_below_abs 0 ?non_zero_positions D ?A' = reduce_abs 0 m D (swaprows 0 m A)" .
         have "echelon_form_JNF (reduce_abs 0 m D (swaprows 0 m A))"
         proof (rule echelon_form_JNF_mx1[OF _ n2])
           show "reduce_abs 0 m D (swaprows 0 m A) \<in> carrier_mat (m+n) n"
             using A n2 reduce_carrier by (auto simp add: Let_def) 
           show "\<forall>i\<in>{1..<m+n}. reduce_abs 0 m D (swaprows 0 m A) $$ (i, 0) = 0"
           proof
             fix i assume i: "i \<in> {1..<m+n}"
             show "reduce_abs 0 m D (swaprows 0 m A) $$ (i, 0) = 0"
             proof (cases "i=m")
               case True
               show ?thesis
               proof (unfold True, rule reduce_0[OF _ _ n0])
                 show "swaprows 0 m A \<in> carrier_mat (m+n) n" using A by auto
               qed (insert m0 n0 S00 D_g0, auto)
             next
               case False
               have "reduce_abs 0 m D (swaprows 0 m A) $$ (i, 0) = (swaprows 0 m A) $$ (i, 0)"
               proof (rule reduce_preserves[OF _ n0])
                 show "swaprows 0 m A \<in> carrier_mat (m+n) n" using A by auto  
               qed (insert m0 n0 S00 D_g0 False i, auto)
               also have "... = A $$ (i, 0)" using i False A n0 by auto
               also have "... = 0"
               proof (rule ccontr)
                 assume "A $$ (i, 0) \<noteq> 0" hence "i \<in> set ?non_zero_positions" using i A by auto 
                 hence "i=m" using nz_xs_m True by auto
                 thus False using False by contradiction
               qed
               finally show ?thesis .
             qed 
           qed
         qed       
         then show ?thesis using * by presburger
       next
         case False        
       have l: "length ?non_zero_positions > 1"    using False nz_xs_m by auto   
       hence l_xs: "length xs > 0" using nz_xs_m by auto
       hence xs_m_less_m: "(xs@[m]) ! 0 < m" by (simp add: all_less_m nth_append)
-      have S00: "?S $$ (0,0) \<noteq> 0"
-        by (smt A add_pos_pos append_Cons_nth_left n0 carrier_matD index_mat_swaprows(1)
-            l_xs m0 mem_Collect_eq nth_mem set_filter xs_def)
+      have "A $$ ((xs @ [m]) ! 0, 0) \<noteq> 0"
+        by (smt (verit) append_Cons_nth_left l_xs mem_Collect_eq nth_mem set_filter xs_def)
+      then have S00: "?S $$ (0,0) \<noteq> 0"
+        using A n0 by auto
       have S': "?S' \<in> carrier_mat m n" using A by auto
       have S_S'D: "?S = ?S' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" by (rule swaprows_append_id[OF A'' A_def xs_m_less_m]) 
       have 2: "reduce_below_abs 0 ?non_zero_positions D ?A' = reduce_below_abs 0 ?non_zero_positions D ?S"
         using A00 nz_xs_m by algebra
       have "echelon_form_JNF (reduce_below_abs 0 ?non_zero_positions D ?S)"
       proof (rule echelon_form_JNF_mx1[OF _ n2])
         show "reduce_below_abs 0 ?non_zero_positions D ?S \<in> carrier_mat (m+n) n" using A by auto
         show "\<forall>i\<in>{1..<m + n}. reduce_below_abs 0 ?non_zero_positions D ?S $$ (i, 0) = 0"
         proof 
           fix i assume i: "i \<in> {1..<m + n}"
           show "reduce_below_abs 0 ?non_zero_positions D ?S $$ (i, 0) =0"
           proof (cases "i\<in>set ?non_zero_positions")
             case True
             show ?thesis unfolding nz_xs_m 
               by (rule reduce_below_abs_0_case_m[OF S' m0 n0 S_S'D S00 mn _ d_xs all_less_m D_g0],
                   insert True nz_xs_m, auto)
           next
             case False note i_notin_set = False
             have "reduce_below_abs 0 ?non_zero_positions D ?S $$ (i, 0) = ?S $$ (i, 0)" unfolding nz_xs_m 
               by (rule reduce_below_abs_preserves_case_m[OF S' m0 n0 S_S'D S00 mn _ d_xs all_less_m _ _ _ D_g0],
                   insert i nz_xs_m i_notin_set, auto)
             also have "... = 0" using i_notin_set i A S00 n0 unfolding set_filter by auto              
             finally show ?thesis .        
           qed
         qed
       qed
       thus ?thesis using 2 by argo
     qed
   qed
     thus ?thesis using * e by presburger
   qed
 qed
 
 
 lemma FindPreHNF_works_n_ge2:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A'': "A'' \<in> carrier_mat m n" and "n\<ge>2" and m_le_n: "m\<ge>n" and "D>0"
 shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> FindPreHNF abs_flag D A = P * A \<and> echelon_form_JNF (FindPreHNF abs_flag D A)"
   using assms
 proof (induct abs_flag D A arbitrary: A'' m n rule: FindPreHNF.induct)
   case (1 abs_flag D A)  
   note A_def = "1.prems"(1)
   note A'' = "1.prems"(2)
   note n = "1.prems"(3)
   note m_le_n = "1.prems"(4)
   note D0 = "1.prems"(5)
   let ?RAT = "map_mat rat_of_int"
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   have mn: "2\<le>m+n" using n by auto
   have m0: "0<m" using n m_le_n by auto
   have n0: "0<n" using n by simp
   have D_not0: "D\<noteq>0" using D0 by auto
   define non_zero_positions where "non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]"
   define A' where "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)"
   let ?Reduce = "(if abs_flag then reduce_below_abs else reduce_below)"
   obtain A'_UL A'_UR A'_DL A'_DR where A'_split: "(A'_UL, A'_UR, A'_DL, A'_DR) 
     = split_block (?Reduce 0 non_zero_positions D (make_first_column_positive A')) 1 1"     
     by (metis prod_cases4)
   define sub_PreHNF where "sub_PreHNF = FindPreHNF abs_flag D A'_DR"
   obtain xs where non_zero_positions_xs_m: "non_zero_positions = xs @ [m]" and d_xs: "distinct xs"
     and all_less_m: "\<forall>x\<in>set xs. x < m \<and> 0 < x" 
     using non_zero_positions_xs_m[OF A_def A'' non_zero_positions_def m0 n0] using D0 by fast
   define M where "M = (make_first_column_positive A')"  
   have A': "A' \<in> carrier_mat (m+n) n" unfolding A'_def using A by auto 
   have mk_A'_not0:"make_first_column_positive A' $$ (0,0) \<noteq> 0"
     by (rule make_first_column_positive_00[OF A_def A'' non_zero_positions_def
           A'_def m0 n0 D_not0 m_le_n])  
   have M: "M \<in> carrier_mat (m+n) n" using A' M_def by auto
   let ?M' = "mat_of_rows n (map (Matrix.row (make_first_column_positive A')) [0..<m])"
   have M': "?M' \<in> carrier_mat m n" by auto            
   have M_M'D: "make_first_column_positive A' = ?M' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" if xs_empty: "xs \<noteq> []"
   proof (cases "A$$(0,0) \<noteq> 0")
     case True
     then have *: "make_first_column_positive A' = make_first_column_positive A"
       unfolding A'_def by auto        
     show ?thesis 
       by (unfold *, rule make_first_column_positive_append_id[OF A'' A_def D0 n0])
   next
     case False
     then have *: "make_first_column_positive A' 
                   = make_first_column_positive (swaprows 0 (non_zero_positions ! 0) A)"
       unfolding A'_def by auto
     show ?thesis
     proof (unfold *, rule make_first_column_positive_append_id)
       let ?S = "mat_of_rows n (map (Matrix.row (swaprows 0 (non_zero_positions ! 0) A)) [0..<m])"
       show "swaprows 0 (non_zero_positions ! 0) A =  ?S @\<^sub>r (D \<cdot>\<^sub>m (1\<^sub>m n))"
       proof (rule swaprows_append_id[OF A'' A_def])
         have A''00: "A'' $$ (0, 0) = 0"
           by (metis (no_types, lifting) A A'' A_def False add_sign_intros(2) append_rows_def
               carrier_matD index_mat_four_block m0 n0)
         have length_xs: "length xs > 0" using xs_empty by auto
         have "non_zero_positions ! 0 = xs ! 0" unfolding non_zero_positions_xs_m
           by (meson length_xs nth_append)
         thus "non_zero_positions ! 0 < m" using all_less_m length_xs by simp
       qed
     qed (insert n0 D0, auto)            
   qed
   have A'_DR: "A'_DR \<in> carrier_mat (m + (n-1)) (n-1)"
     by (rule split_block(4)[OF A'_split[symmetric]], insert n M M_def, auto)  
   have sub_PreHNF: "sub_PreHNF \<in> carrier_mat (m + (n -1)) (n-1)"
     unfolding sub_PreHNF_def by (rule FindPreHNF[OF A'_DR])
   hence sub_PreHNF': "sub_PreHNF \<in> carrier_mat (m+n - 1) (n-1)" using n by auto
   have A'_UL: "A'_UL \<in> carrier_mat 1 1"
     by (rule split_block(1)[OF A'_split[symmetric], of "m+n-1" "n-1"], insert n A', auto) 
   have A'_UR: "A'_UR \<in> carrier_mat 1 (n-1)"
     by (rule split_block(2)[OF A'_split[symmetric], of "m+n-1"], insert n A', auto)
   have A'_DL: "A'_DL \<in> carrier_mat (m + (n - 1)) 1"
     by (rule split_block(3)[OF A'_split[symmetric], of _ "n-1"], insert n A', auto)
 
   show ?case
   proof (cases abs_flag)
     case True note abs_flag = True
       hence A'_split: "(A'_UL, A'_UR, A'_DL, A'_DR) 
     = split_block (reduce_below_abs 0 non_zero_positions D (make_first_column_positive A')) 1 1"   using A'_split by auto
     let ?R = "reduce_below_abs 0 non_zero_positions D (make_first_column_positive A')"
    have fbm_R: "four_block_mat A'_UL A'_UR A'_DL A'_DR 
      = reduce_below_abs 0 non_zero_positions D (make_first_column_positive A')"
     by (rule split_block(5)[symmetric, OF A'_split[symmetric], of "m+n-1" "n-1"], insert A' n, auto)
   have A'_DL0: "A'_DL = (0\<^sub>m (m + (n - 1)) 1)"   
   proof (rule eq_matI)
     show "dim_row A'_DL = dim_row (0\<^sub>m (m + (n - 1)) 1)"
       and "dim_col A'_DL = dim_col (0\<^sub>m (m + (n - 1)) 1)" using A'_DL by auto    
     fix i j assume i: "i < dim_row (0\<^sub>m (m + (n - 1)) 1)" and j: "j < dim_col (0\<^sub>m (m + (n - 1)) 1)"
     have j0: "j=0" using j by auto
     have "0 = ?R $$ (i+1,j)"
     proof (unfold M_def non_zero_positions_xs_m j0, 
         rule reduce_below_abs_0_case_m_make_first_column_positive[symmetric,
           OF A'' m0 n0 A_def m_le_n _  d_xs all_less_m _ _ D0 _ ])
       show "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A)"
         using A'_def non_zero_positions_def non_zero_positions_xs_m by presburger
       show "xs @ [m] = filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A]"
         using A'_def non_zero_positions_def non_zero_positions_xs_m by presburger
     qed (insert i n0, auto)
     also have "... = four_block_mat A'_UL A'_UR A'_DL A'_DR $$ (i+1,j)" unfolding fbm_R ..
     also have "... = (if i+1 < dim_row A'_UL then if j < dim_col A'_UL 
             then A'_UL $$ (i+1, j) else A'_UR $$ (i+1, j - dim_col A'_UL)
             else if j < dim_col A'_UL then A'_DL $$ (i+1 - dim_row A'_UL, j)
             else A'_DR $$ (i+1 - dim_row A'_UL, j - dim_col A'_UL))"
       by (rule index_mat_four_block, insert A'_UL A'_DR i j, auto)
     also have "... = A'_DL $$ (i, j)" using A'_UL A'_UR i j by auto
     finally show "A'_DL $$ (i, j) = 0\<^sub>m (m + (n - 1)) 1 $$ (i, j)" using i j by auto
   qed
 
   let ?A'_DR_m = "mat_of_rows (n-1) [Matrix.row A'_DR i. i \<leftarrow> [0..<m]]"
   have A'_DR_m: "?A'_DR_m \<in> carrier_mat m (n-1)" by auto
   have A'DR_A'DR_m_D: "A'_DR = ?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)"
   proof (rule eq_matI)
     show dr: "dim_row A'_DR = dim_row (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))" 
       by (metis A'_DR A'_DR_m append_rows_def carrier_matD(1) index_mat_four_block(2) 
           index_one_mat(2) index_smult_mat(2) index_zero_mat(2))
     show dc: "dim_col A'_DR = dim_col (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
       by (metis A'_DR A'_DR_m add.comm_neutral append_rows_def
           carrier_matD(2) index_mat_four_block(3) index_zero_mat(3))  
     fix i j assume i: "i < dim_row(?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
       and j: "j<dim_col (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
     have jn1: "j<n-1" using dc j A'_DR by auto
     show "A'_DR $$ (i,j) = (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j)"
     proof (cases "i<m")
       case True
       have "A'_DR $$ (i,j) = ?A'_DR_m $$ (i,j)"
         by (metis A'_DR A'_DR_m True dc carrier_matD(1) carrier_matD(2) j le_add1 
             map_first_rows_index mat_of_rows_carrier(2) mat_of_rows_index)
       also have "... = (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j)"
         by (metis (mono_tags, lifting) A'_DR A'_DR_m True append_rows_def 
             carrier_matD dc i index_mat_four_block j)
       finally show ?thesis .
     next
       case False note i_ge_m = False
       let ?reduce_below = "reduce_below_abs 0 non_zero_positions D (make_first_column_positive A')"
       have 1: "(?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j) = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
-        by (smt A'_DR A'_DR_m False append_rows_nth carrier_matD carrier_mat_triv dc dr i
+        by (smt (verit) A'_DR A'_DR_m False append_rows_nth carrier_matD carrier_mat_triv dc dr i
             index_one_mat(2) index_one_mat(3) index_smult_mat(2,3) j)
       have "?reduce_below = four_block_mat A'_UL A'_UR A'_DL A'_DR" using fbm_R ..
       also have "... $$ (i+1,j+1) = (if i+1 < dim_row A'_UL then if j+1 < dim_col A'_UL 
               then A'_UL $$ (i+1, j+1) else A'_UR $$ (i+1, j+1 - dim_col A'_UL)
               else if j+1 < dim_col A'_UL then A'_DL $$ (i+1 - dim_row A'_UL, j+1)
               else A'_DR $$ (i+1 - dim_row A'_UL, j+1 - dim_col A'_UL))"
         by (rule index_mat_four_block, insert i j A'_UL A'_DR dr dc, auto)
       also have "... = A'_DR $$ (i,j)" using A'_UL by auto
       finally have 2: "?reduce_below $$ (i+1,j+1) = A'_DR $$ (i,j)" .
       show ?thesis 
       proof (cases "xs = []")
         case True note xs_empty = True
         have i1_m: "i + 1 \<noteq> m" 
           using False less_add_one by blast
         have j1n: "j+1<n"
           using jn1 less_diff_conv by blast
         have i1_mn: "i+1<m + n"
           using i i_ge_m
           by (metis A'_DR carrier_matD(1) dr less_diff_conv sub_PreHNF sub_PreHNF')
         have "?reduce_below = reduce_abs 0 m D M"
           unfolding non_zero_positions_xs_m xs_empty M_def by auto
         also have "... $$ (i+1,j+1) = M $$ (i+1, j+1)"
           by (rule reduce_preserves[OF M j1n _ i1_m _ i1_mn], insert M_def mk_A'_not0, auto)         
         also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ ((i+1)-m, j+1)"
         proof (cases "A $$ (0,0) = 0")
           case True
           let ?S = "(swaprows 0 m A)"
           have S: "?S \<in> carrier_mat (m+n) n" using A by auto
           have Si10: "?S $$ (i+1,0) = 0"
           proof -
             have "?S $$ (i+1,0) = A $$ (i+1,0)" using i1_m n0 i1_mn S by auto
             also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,0)"
-              by (smt A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
+              by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
                   index_mat_four_block less_imp_diff_less n0)
             also have "... = 0" using i_ge_m n0 i1_mn by auto
             finally show ?thesis .
           qed
           have "M $$ (i+1, j+1) = (make_first_column_positive ?S) $$ (i+1,j+1)"
             by (simp add: A'_def M_def True non_zero_positions_xs_m xs_empty)
           also have "... = (if ?S $$ (i+1,0) < 0 then - ?S $$ (i+1,j+1) else ?S $$ (i+1,j+1))" 
             unfolding make_first_column_positive.simps using S i1_mn j1n by auto
           also have "... = ?S $$ (i+1,j+1)" using Si10 by auto
           also have "... = A $$ (i+1,j+1)" using i1_m n0 i1_mn S jn1 by auto
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,j+1)"
-            by (smt A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
+            by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
                 index_one_mat(2) index_smult_mat(2) index_zero_mat(2) j1n less_imp_diff_less add_diff_cancel_right')
           finally show ?thesis .
         next
           case False         
           have Ai10: "A $$ (i+1,0) = 0"
           proof -
             have "A $$ (i+1,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,0)"
-              by (smt A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
+              by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
                   index_mat_four_block less_imp_diff_less n0)
             also have "... = 0" using i_ge_m n0 i1_mn by auto
             finally show ?thesis .
           qed          
           have "M $$ (i+1, j+1) = (make_first_column_positive A) $$ (i+1,j+1)"
             by (simp add: A'_def M_def False True non_zero_positions_xs_m)
           also have "... = (if A $$ (i+1,0) < 0 then - A $$ (i+1,j+1) else A $$ (i+1,j+1))" 
             unfolding make_first_column_positive.simps using A i1_mn j1n by auto
           also have "... = A $$ (i+1,j+1)" using Ai10 by auto
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,j+1)"
-            by (smt A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
+            by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
                 index_one_mat(2) index_smult_mat(2) index_zero_mat(2) j1n less_imp_diff_less add_diff_cancel_right')
           finally show ?thesis .
         qed
         also have "... = D * (1\<^sub>m n) $$ ((i+1)-m, j+1)"
           by (rule index_smult_mat, insert i jn1 A'_DR False dr, auto)            
         also have "... = D *(1\<^sub>m (n - 1)) $$ (i-m,j)" using dc dr i j A'_DR i_ge_m
-          by (smt Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
+          by (smt (verit) Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
               linorder_not_less add_diff_cancel_right' add_diff_cancel_right' add_diff_cancel_left')
         also have "... = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
           by (rule index_smult_mat[symmetric], insert i jn1 A'_DR False dr, auto)
         finally show ?thesis using 1 2 by auto
       next
         case False     
       have "?reduce_below $$ (i+1, j+1) = M $$ (i+1, j+1)"
       proof (unfold non_zero_positions_xs_m M_def,
           rule reduce_below_abs_preserves_case_m[OF M' m0 _ M_M'D mk_A'_not0 m_le_n _ d_xs all_less_m _ _ _ D0])            
         show "j + 1 < n" using jn1 by auto
         show "i + 1 \<notin> set xs" using all_less_m i_ge_m non_zero_positions_xs_m by auto
         show "i + 1 \<noteq> 0" by auto
         show " i + 1 < m + n" using i_ge_m i dr A'_DR by auto
         show " i + 1 \<noteq> m" using i_ge_m by auto
       qed (insert False)
       also have "... = (?M' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1, j+1)" unfolding M_def using False M_M'D by argo
       also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ ((i+1)-m, j+1)"
       proof -
         have f1: "1 + j < n"
           by (metis Groups.add_ac(2) jn1 less_diff_conv)
         have f2: "\<forall>n. \<not> n + i < m"
           by (meson i_ge_m linorder_not_less nat_SN.compat not_add_less2)
         have "i < m + (n - 1)"
           by (metis (no_types) A'_DR carrier_matD(1) dr i)
         then have "1 + i < m + n"
           using f1 by linarith
         then show ?thesis
           using f2 f1 by (metis (no_types) Groups.add_ac(2) M' append_rows_def carrier_matD(1) 
               dim_col_mat(1) index_mat_four_block(1) index_one_mat(2) index_smult_mat(2) 
               index_zero_mat(2,3) mat_of_rows_def nat_arith.rule0)
       qed
       also have "... = D * (1\<^sub>m n) $$ ((i+1)-m, j+1)"
         by (rule index_smult_mat, insert i jn1 A'_DR False dr, auto)            
       also have "... = D *(1\<^sub>m (n - 1)) $$ (i-m,j)" using dc dr i j A'_DR i_ge_m
-        by (smt Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
+        by (smt (verit) Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
             linorder_not_less add_diff_cancel_right' add_diff_cancel_left')
       also have "... = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
         by (rule index_smult_mat[symmetric], insert i jn1 A'_DR False dr, auto)
       finally have 3: "?reduce_below $$ (i+1,j+1) = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)" .            
       show ?thesis using 1 2 3 by presburger
     qed              
   qed
 qed
   let ?A'_DR_n = "mat_of_rows (n - 1) (map (Matrix.row A'_DR) [0..<n - 1])"
   have hyp: "\<exists>P. P\<in>carrier_mat (m + (n-1)) (m + (n-1)) \<and> invertible_mat P \<and> sub_PreHNF = P * A'_DR 
   \<and> echelon_form_JNF sub_PreHNF" 
   proof (cases "2 \<le> n - 1")
     case True
     show ?thesis
       by (unfold sub_PreHNF_def, rule "1.hyps"[OF _ _ _ non_zero_positions_def A'_def _ _ _ _ _])
          (insert A n D0 m_le_n True A'DR_A'DR_m_D A A'_split abs_flag, auto)
   next
     case False
     have "\<exists>P. P\<in>carrier_mat (m + (n-1)) (m + (n-1)) \<and> invertible_mat P \<and> sub_PreHNF = P * A'_DR"
       by (unfold sub_PreHNF_def, rule FindPreHNF_invertible_mat_mx2
           [OF A'DR_A'DR_m_D A'_DR_m _ _ D0 _])
           (insert False m_le_n n0 m0 "1"(4), auto)
     moreover have "echelon_form_JNF sub_PreHNF" unfolding sub_PreHNF_def 
       by (rule FindPreHNF_echelon_form_mx1[OF A'DR_A'DR_m_D A'_DR_m _ D0 _],
           insert False n0 m_le_n, auto) 
     ultimately show ?thesis by simp
   qed  
   from this obtain P where P: "P \<in> carrier_mat (m + (n - 1)) (m + (n - 1))"
     and inv_P: "invertible_mat P" and sub_PreHNF_P_A'_DR: "sub_PreHNF = P * A'_DR" by blast
   define P' where "P' = (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m+(n-1))) (0\<^sub>m (m+(n-1)) 1) P)"
   have P': "P' \<in> carrier_mat (m+n) (m+n)" 
   proof -
     have "P' \<in> carrier_mat (1 + (m+(n-1))) (1 + (m+(n-1))) "
       unfolding P'_def by (rule four_block_carrier_mat[OF _ P], simp) 
     thus ?thesis using n by auto
   qed
   have inv_P': "invertible_mat P'" 
     unfolding P'_def by (rule invertible_mat_four_block_mat_lower_right[OF P inv_P])
   have dr_A2: "dim_row A \<ge> 2" using A m0 n by auto
   have dc_A2: "dim_col A \<ge> 2" using n A by blast
   have *: "(dim_col A = 0) = False" using dc_A2 by auto
   have FindPreHNF_as_fbm: "FindPreHNF abs_flag D A = four_block_mat A'_UL A'_UR A'_DL sub_PreHNF" 
     unfolding FindPreHNF.simps[of abs_flag D A] using A'_split mn n A dr_A2 dc_A2 abs_flag
     unfolding Let_def sub_PreHNF_def M_def A'_def non_zero_positions_def *    
-    by (smt (z3) linorder_not_less split_conv)
+    by (smt (verit) linorder_not_less split_conv)
   also have "... = P' * (reduce_below_abs 0 non_zero_positions D M)"
   proof -
     have "P' * (reduce_below_abs 0 non_zero_positions D M) 
       = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m + (n - 1))) (0\<^sub>m (m + (n - 1)) 1) P 
       * four_block_mat A'_UL A'_UR A'_DL A'_DR"
       unfolding P'_def fbm_R[unfolded M_def[symmetric], symmetric] ..
     also have "... = four_block_mat 
             ((1\<^sub>m 1) * A'_UL + (0\<^sub>m 1 (m + (n - 1)) * A'_DL)) 
             ((1\<^sub>m 1) * A'_UR + (0\<^sub>m 1 (m + (n - 1))) * A'_DR) 
             ((0\<^sub>m (m + (n - 1)) 1) * A'_UL + P * A'_DL) 
             ((0\<^sub>m (m + (n - 1)) 1) * A'_UR + P * A'_DR)" 
       by (rule mult_four_block_mat[OF _ _ _ P A'_UL A'_UR A'_DL A'_DR], auto)
     also have "... = four_block_mat A'_UL A'_UR (P * A'_DL) (P * A'_DR)"
       by (rule cong_four_block_mat, insert A'_UL A'_UR A'_DL A'_DR P, auto)
     also have "... = four_block_mat A'_UL A'_UR (0\<^sub>m (m + (n - 1)) 1) sub_PreHNF"
       unfolding A'_DL0 sub_PreHNF_P_A'_DR using P by simp
     also have "... = four_block_mat A'_UL A'_UR A'_DL sub_PreHNF"
       unfolding A'_DL0 by simp
     finally show ?thesis ..
   qed
   finally have Find_P'_reduceM: "FindPreHNF abs_flag D A =  P' * (reduce_below_abs 0 non_zero_positions D M)" .
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below_abs 0 (xs @ [m]) D M = Q * M"
   proof (cases "xs = []")
     case True note xs_empty = True
     have rw: "reduce_below_abs 0 (xs @ [m]) D M = reduce_abs 0 m D M" using True by auto
     obtain p q u v d where pquvd: "(p, q, u, v, d) = euclid_ext2 (M $$ (0, 0)) (M $$ (m, 0))"
       by (simp add: euclid_ext2_def)
     have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and> reduce_abs 0 m D M = P * M"  
     proof (rule reduce_abs_invertible_mat_case_m[OF _ _ m0 _ _ _ _ m_le_n n0 pquvd])
           show "M $$ (0, 0) \<noteq> 0" 
             using M_def mk_A'_not0 by blast
           define M' where  "M' = mat_of_rows n (map (Matrix.row M) [0..<m])"
           define M'' where "M'' = mat_of_rows n (map (Matrix.row M) [m..<m+n])"
           define A2 where "A2 = Matrix.mat (dim_row M) (dim_col M)
           (\<lambda>(i, k). if i = 0 then p * M $$ (0, k) + q * M $$ (m, k) 
                     else if i = m then u * M $$ (0, k) + v * M $$ (m, k)
                     else M $$ (i, k))"
           show M_M'_M'': "M = M' @\<^sub>r M''" unfolding M'_def M''_def            
             by (metis M append_rows_split carrier_matD le_add1)
           show M': "M' \<in> carrier_mat m n" unfolding M'_def by fastforce
           show M'': "M'' \<in> carrier_mat n n" unfolding M''_def by fastforce
           show "0 \<noteq> m" using m0 by simp
           show "A2 = Matrix.mat (dim_row M) (dim_col M)
           (\<lambda>(i, k). if i = 0 then p * M $$ (0, k) + q * M $$ (m, k) 
                     else if i = m then u * M $$ (0, k) + v * M $$ (m, k)
                     else M $$ (i, k))"
             (is "_ = ?rhs") using A A2_def by auto
           define xs' where "xs' =  filter (\<lambda>i. abs (A2 $$ (0,i)) > D) [0..<n]"
           define ys' where  "ys' = filter (\<lambda>i. abs (A2 $$ (m,i)) > D) [0..<n]"
           show "xs' = filter (\<lambda>i. abs (A2 $$ (0,i)) > D) [0..<n]" unfolding xs'_def by auto
           show "ys' = filter (\<lambda>i. abs (A2 $$ (m,i)) > D) [0..<n]" unfolding ys'_def by auto
           have M''D: "(M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"
             if jn: "j<n" and j0: "j>0" for j
           proof -
             have Ajm0: "A $$ (j+m,0) = 0"
             proof -
               have "A $$ (j+m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j+m-m,0)"
-                by (smt "1"(2) "1"(3) M M' M'' M_M'_M'' add.commute append_rows_def carrier_matD
+                by (smt (verit) "1"(2) "1"(3) M M' M'' M_M'_M'' add.commute append_rows_def carrier_matD
                     diff_add_inverse2 index_mat_four_block index_one_mat(2) index_smult_mat(2)
                     le_add2 less_diff_conv2 n0 not_add_less2 that(1))
               also have "... = 0" using jn j0 by auto
               finally show ?thesis .
             qed
             have "M'' $$ (j, i) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)" if i_n: "i<n" for i
             proof (cases "A$$(0,0) = 0")
               case True 
-              have "M'' $$ (j, i) = make_first_column_positive (swaprows 0 m A) $$ (j+m,i)"                
-                by (smt A'_def Groups.add_ac(2) M' M'' M_M'_M'' M_def True append.simps(1) 
-                    append_rows_nth3 diff_add_inverse2 jn le_add2 local.non_zero_positions_xs_m
-                    nat_add_left_cancel_less nth_Cons_0 that xs_empty)
+              have "M'' $$ (j, i) = make_first_column_positive (swaprows 0 m A) $$ (j+m,i)"  
+                using M' M'' M_M'_M'' unfolding M_def A_def 
+                by (metis (no_types, lifting) "1"(2) "1"(5) A'_def True append.simps(1) append_rows_nth2 i_n jn 
+                    non_zero_positions_xs_m nat_SN.compat nth_Cons_0 xs_empty)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n Ajm0 by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .            
             next
               case False
               have "A' = A" unfolding A'_def non_zero_positions_xs_m using False True by auto
-              hence "M'' $$ (j, i) = make_first_column_positive A $$ (j+m,i)"                
-                by (smt m_le_n M' M'' M_M'_M'' M_def append_rows_nth2 jn nat_SN.compat that)                
+              hence "M'' $$ (j, i) = make_first_column_positive A $$ (j+m,i)"     
+                using M' M'' M_M'_M'' unfolding M_def
+                by (metis (no_types, opaque_lifting) "1"(5) append_rows_nth2 jn nat_SN.compat that)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n Ajm0 by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .
             qed                           
             thus ?thesis using jn j0 by auto
           qed
           have Am0D: "A$$(m,0) = D"
           proof -
             have "A$$(m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m,0)"
-              by (smt "1"(2) "1"(3) M M' M'' M_M'_M'' append_rows_def carrier_matD
+              by (smt (verit) "1"(2) "1"(3) M M' M'' M_M'_M'' append_rows_def carrier_matD
                   diff_less_mono2 diff_self_eq_0 index_mat_four_block index_one_mat(2) 
                   index_smult_mat(2) less_add_same_cancel1 n0 semiring_norm(137))
             also have "... = D" using m0 n0 by auto
             finally show ?thesis .
           qed
           hence S00D: "(swaprows 0 m A) $$ (0,0) = D" using n0 m0 A by auto
           have Sm00: "(swaprows 0 m A) $$ (m,0) = A$$(0,0)" using n0 m0 A by auto
           have M00D: "M $$ (0, 0) = D" if A00: "A$$(0,0) = 0"
           proof -
             have "M $$ (0,0) = (make_first_column_positive (swaprows 0 m A)) $$ (0,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if (swaprows 0 m A) $$ (0,0) < 0 then - (swaprows 0 m A) $$(0,0)
                               else (swaprows 0 m A) $$(0,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = (swaprows 0 m A) $$(0,0)" using S00D D0 by auto
             also have "... = D" using S00D by auto
             finally show ?thesis .
           qed
           have Mm00: "M $$ (m, 0) = 0" if A00: "A$$(0,0) = 0"
           proof -
             have "M $$ (m,0) = (make_first_column_positive (swaprows 0 m A)) $$ (m,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if (swaprows 0 m A) $$ (m,0) < 0 then - (swaprows 0 m A) $$(m,0)
                               else (swaprows 0 m A) $$(m,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = (swaprows 0 m A) $$(m,0)" using Sm00 A00 D0 by auto
             also have "... = 0" using Sm00 A00 by auto
             finally show ?thesis .
           qed
           have M000: "M $$ (0, 0) = abs (A$$(0,0))" if A00: "A$$(0,0) \<noteq> 0"
           proof -
             have "M $$ (0,0) = (make_first_column_positive A) $$ (0,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if A $$ (0,0) < 0 then - A $$(0,0)
                               else A $$(0,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = abs (A$$(0,0))" using Sm00 A00 by auto
             finally show ?thesis .
           qed
           have Mm0D: "M $$ (m, 0) = D" if A00: "A $$ (0,0) \<noteq> 0"
           proof -
             have "M $$ (m,0) = (make_first_column_positive A) $$ (m,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if A $$ (m,0) < 0 then - A $$(m,0)
                               else A $$(m,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = A $$(m,0)" using S00D D0 Am0D by auto
             also have "... = D" using Am0D D0 by auto
             finally show ?thesis .
           qed
           have "0 \<notin> set xs'"
           proof -
             have "A2 $$ (0,0) = p * M $$ (0, 0) + q * M $$ (m, 0)" 
               using A A2_def n0 M by auto
             also have "... = gcd (M $$ (0, 0)) (M $$ (m, 0))"
               by (metis euclid_ext2_works(1,2) pquvd)
             also have "abs ... \<le> D" using M00D Mm00 M000 Mm0D using gcd_0_int D0 by fastforce
             finally have "abs (A2 $$ (0,0)) \<le> D" .
             thus ?thesis unfolding xs'_def using D0 by auto
           qed
           thus "\<forall>j\<in>set xs'. j<n \<and> (M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"  
             using M''D xs'_def by auto
           have "0 \<notin> set ys'"
           proof -
             have "A2 $$ (m,0) = u * M $$ (0, 0) + v * M $$ (m, 0)"
               using A A2_def n0 m0 M by auto
             also have "... = - M $$ (m, 0) div gcd (M $$ (0, 0)) (M $$ (m, 0)) * M $$ (0, 0) 
                 + M $$ (0, 0) div gcd (M $$ (0, 0)) (M $$ (m, 0)) * M $$ (m, 0) "
               by (simp add: euclid_ext2_works[OF pquvd[symmetric]])
             also have "... = 0" using M00D Mm00 M000 Mm0D
-              by (smt dvd_div_mult_self euclid_ext2_works(3) euclid_ext2_works(5)
+              by (smt (verit) dvd_div_mult_self euclid_ext2_works(3) euclid_ext2_works(5)
                   more_arith_simps(11) mult.commute mult_minus_left pquvd semiring_gcd_class.gcd_dvd1)
             finally have "A2 $$ (m,0) = 0" .
             thus ?thesis unfolding ys'_def using D0 by auto
           qed
           thus "\<forall>j\<in>set ys'. j<n \<and> (M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"
             using M''D ys'_def by auto   
         qed (insert D0)
         then show ?thesis using rw by auto
   next
     case False
     show ?thesis
       by (unfold M_def, rule reduce_below_abs_invertible_mat_case_m[OF M' m0 n0 M_M'D[OF False] 
           mk_A'_not0 m_le_n d_xs all_less_m D0])
   qed
    
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and reduce_QM: "reduce_below_abs 0 (xs @ [m]) D M = Q * M" by blast
   have "\<exists>R. invertible_mat R 
     \<and> R \<in> carrier_mat (dim_row A') (dim_row A') \<and> M = R * A'"
     by (unfold M_def, rule make_first_column_positive_invertible)
   from this obtain R where inv_R: "invertible_mat R"
     and R: "R \<in> carrier_mat (dim_row A') (dim_row A')" and M_RA': "M = R * A'" by blast
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> A' = P * A" 
     by (rule A'_swaprows_invertible_mat[OF A A'_def non_zero_positions_def],
         insert non_zero_positions_xs_m n m0, auto)
   from this obtain S where inv_S: "invertible_mat S"
    and S: "S \<in> carrier_mat (dim_row A) (dim_row A)" and A'_SA: "A' = S * A" 
     using A by auto
   have "(P'*Q*R*S) \<in> carrier_mat (m+n) (m+n)" using P' Q R S A' A by auto
   moreover have "FindPreHNF abs_flag D A = (P'*Q*R*S) * A" using Find_P'_reduceM reduce_QM 
     unfolding  M_RA' A'_SA M_def
-    by (smt A' A'_SA P' Q R S assoc_mult_mat carrier_matD carrier_mat_triv index_mult_mat(2,3) 
+    by (smt (verit) A' A'_SA P' Q R S assoc_mult_mat carrier_matD carrier_mat_triv index_mult_mat(2,3) 
         non_zero_positions_xs_m)
   moreover have "invertible_mat (P'*Q*R*S)" using inv_P' inv_Q inv_R inv_S using P' Q R S A' A 
     by (metis carrier_matD carrier_mat_triv index_mult_mat(2,3) invertible_mult_JNF)
   ultimately have exists_inv: "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
     \<and> FindPreHNF abs_flag D A = P * A" by blast
   moreover have "echelon_form_JNF (FindPreHNF abs_flag D A)" 
   proof (rule echelon_form_four_block_mat[OF A'_UL A'_UR sub_PreHNF'   ])
     show "FindPreHNF abs_flag D A = four_block_mat A'_UL A'_UR (0\<^sub>m (m + n - 1) 1) sub_PreHNF" 
       using A'_DL0 FindPreHNF_as_fbm sub_PreHNF sub_PreHNF' by auto
     have "A'_UL $$ (0, 0) = ?R $$ (0,0)"
       by (metis (mono_tags, lifting) A A'_DR A'_UL Find_P'_reduceM M_def 
           \<open>FindPreHNF abs_flag D A = P' * Q * R * S * A\<close> add_Suc_right add_sign_intros(2) carrier_matD fbm_R
           index_mat_four_block(1,3) index_mult_mat(3) m0 n0 plus_1_eq_Suc
           zero_less_one_class.zero_less_one)
     also have "... \<noteq> 0"
     proof (cases "xs=[]")
       case True
       have "?R $$ (0,0) = reduce_abs 0 m D M $$ (0,0)"
         unfolding non_zero_positions_xs_m True M_def by simp
       also have "... \<noteq> 0"
         by (metis D_not0 M M_def add_pos_pos less_add_same_cancel1 m0 mk_A'_not0 n0 reduce_not0)
       finally show ?thesis .
     next
       case False
       show ?thesis
         by (unfold non_zero_positions_xs_m,
           rule reduce_below_abs_not0_case_m[OF M' m0 n0 M_M'D[OF False] mk_A'_not0 m_le_n all_less_m D_not0])     
     qed       
     finally show "A'_UL $$ (0, 0) \<noteq> 0" .
   qed (insert mn n hyp, auto)
   ultimately show ?thesis by blast
   next
     case False
       hence A'_split: "(A'_UL, A'_UR, A'_DL, A'_DR) 
     = split_block (reduce_below 0 non_zero_positions D (make_first_column_positive A')) 1 1"   using A'_split by auto
     let ?R = "reduce_below 0 non_zero_positions D (make_first_column_positive A')"
    have fbm_R: "four_block_mat A'_UL A'_UR A'_DL A'_DR 
      = reduce_below 0 non_zero_positions D (make_first_column_positive A')"
     by (rule split_block(5)[symmetric, OF A'_split[symmetric], of "m+n-1" "n-1"], insert A' n, auto)
   have A'_DL0: "A'_DL = (0\<^sub>m (m + (n - 1)) 1)"   
   proof (rule eq_matI)
     show "dim_row A'_DL = dim_row (0\<^sub>m (m + (n - 1)) 1)"
       and "dim_col A'_DL = dim_col (0\<^sub>m (m + (n - 1)) 1)" using A'_DL by auto    
     fix i j assume i: "i < dim_row (0\<^sub>m (m + (n - 1)) 1)" and j: "j < dim_col (0\<^sub>m (m + (n - 1)) 1)"
     have j0: "j=0" using j by auto
     have "0 = ?R $$ (i+1,j)"
     proof (unfold M_def non_zero_positions_xs_m j0, 
         rule reduce_below_0_case_m_make_first_column_positive[symmetric,
           OF A'' m0 n0 A_def m_le_n _  d_xs all_less_m _ _ D0 _ ])
       show "A' = (if A $$ (0, 0) \<noteq> 0 then A else let i = (xs @ [m]) ! 0 in swaprows 0 i A)"
         using A'_def non_zero_positions_def non_zero_positions_xs_m by presburger
       show "xs @ [m] = filter (\<lambda>i. A $$ (i, 0) \<noteq> 0) [1..<dim_row A]"
         using A'_def non_zero_positions_def non_zero_positions_xs_m by presburger
     qed (insert i n0, auto)
     also have "... = four_block_mat A'_UL A'_UR A'_DL A'_DR $$ (i+1,j)" unfolding fbm_R ..
     also have "... = (if i+1 < dim_row A'_UL then if j < dim_col A'_UL 
             then A'_UL $$ (i+1, j) else A'_UR $$ (i+1, j - dim_col A'_UL)
             else if j < dim_col A'_UL then A'_DL $$ (i+1 - dim_row A'_UL, j)
             else A'_DR $$ (i+1 - dim_row A'_UL, j - dim_col A'_UL))"
       by (rule index_mat_four_block, insert A'_UL A'_DR i j, auto)
     also have "... = A'_DL $$ (i, j)" using A'_UL A'_UR i j by auto
     finally show "A'_DL $$ (i, j) = 0\<^sub>m (m + (n - 1)) 1 $$ (i, j)" using i j by auto
   qed
 
   let ?A'_DR_m = "mat_of_rows (n-1) [Matrix.row A'_DR i. i \<leftarrow> [0..<m]]"
   have A'_DR_m: "?A'_DR_m \<in> carrier_mat m (n-1)" by auto
   have A'DR_A'DR_m_D: "A'_DR = ?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)"
   proof (rule eq_matI)
     show dr: "dim_row A'_DR = dim_row (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))" 
       by (metis A'_DR A'_DR_m append_rows_def carrier_matD(1) index_mat_four_block(2) 
           index_one_mat(2) index_smult_mat(2) index_zero_mat(2))
     show dc: "dim_col A'_DR = dim_col (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
       by (metis A'_DR A'_DR_m add.comm_neutral append_rows_def
           carrier_matD(2) index_mat_four_block(3) index_zero_mat(3))  
     fix i j assume i: "i < dim_row(?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
       and j: "j<dim_col (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1))"
     have jn1: "j<n-1" using dc j A'_DR by auto
     show "A'_DR $$ (i,j) = (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j)"
     proof (cases "i<m")
       case True
       have "A'_DR $$ (i,j) = ?A'_DR_m $$ (i,j)"
         by (metis A'_DR A'_DR_m True dc carrier_matD(1) carrier_matD(2) j le_add1 
             map_first_rows_index mat_of_rows_carrier(2) mat_of_rows_index)
       also have "... = (?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j)"
         by (metis (mono_tags, lifting) A'_DR A'_DR_m True append_rows_def 
             carrier_matD dc i index_mat_four_block j)
       finally show ?thesis .
     next
       case False note i_ge_m = False
       let ?reduce_below = "reduce_below 0 non_zero_positions D (make_first_column_positive A')"
       have 1: "(?A'_DR_m @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i,j) = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
-        by (smt A'_DR A'_DR_m False append_rows_nth carrier_matD carrier_mat_triv dc dr i
+        by (smt (verit) A'_DR A'_DR_m False append_rows_nth carrier_matD carrier_mat_triv dc dr i
             index_one_mat(2) index_one_mat(3) index_smult_mat(2,3) j)
       have "?reduce_below = four_block_mat A'_UL A'_UR A'_DL A'_DR" using fbm_R ..
       also have "... $$ (i+1,j+1) = (if i+1 < dim_row A'_UL then if j+1 < dim_col A'_UL 
               then A'_UL $$ (i+1, j+1) else A'_UR $$ (i+1, j+1 - dim_col A'_UL)
               else if j+1 < dim_col A'_UL then A'_DL $$ (i+1 - dim_row A'_UL, j+1)
               else A'_DR $$ (i+1 - dim_row A'_UL, j+1 - dim_col A'_UL))"
         by (rule index_mat_four_block, insert i j A'_UL A'_DR dr dc, auto)
       also have "... = A'_DR $$ (i,j)" using A'_UL by auto
       finally have 2: "?reduce_below $$ (i+1,j+1) = A'_DR $$ (i,j)" .
       show ?thesis 
       proof (cases "xs = []")
         case True note xs_empty = True
         have i1_m: "i + 1 \<noteq> m" 
           using False less_add_one by blast
         have j1n: "j+1<n"
           using jn1 less_diff_conv by blast
         have i1_mn: "i+1<m + n"
           using i i_ge_m
           by (metis A'_DR carrier_matD(1) dr less_diff_conv sub_PreHNF sub_PreHNF')
         have "?reduce_below = reduce 0 m D M"
           unfolding non_zero_positions_xs_m xs_empty M_def by auto
         also have "... $$ (i+1,j+1) = M $$ (i+1, j+1)"
           by (rule reduce_preserves[OF M j1n _ i1_m _ i1_mn], insert M_def mk_A'_not0, auto)         
         also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ ((i+1)-m, j+1)"
         proof (cases "A $$ (0,0) = 0")
           case True
           let ?S = "(swaprows 0 m A)"
           have S: "?S \<in> carrier_mat (m+n) n" using A by auto
           have Si10: "?S $$ (i+1,0) = 0"
           proof -
             have "?S $$ (i+1,0) = A $$ (i+1,0)" using i1_m n0 i1_mn S by auto
             also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,0)"
-              by (smt A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
+              by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
                   index_mat_four_block less_imp_diff_less n0)
             also have "... = 0" using i_ge_m n0 i1_mn by auto
             finally show ?thesis .
           qed
           have "M $$ (i+1, j+1) = (make_first_column_positive ?S) $$ (i+1,j+1)"
             by (simp add: A'_def M_def True non_zero_positions_xs_m xs_empty)
           also have "... = (if ?S $$ (i+1,0) < 0 then - ?S $$ (i+1,j+1) else ?S $$ (i+1,j+1))" 
             unfolding make_first_column_positive.simps using S i1_mn j1n by auto
           also have "... = ?S $$ (i+1,j+1)" using Si10 by auto
           also have "... = A $$ (i+1,j+1)" using i1_m n0 i1_mn S jn1 by auto
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,j+1)"
-            by (smt A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
+            by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
                 index_one_mat(2) index_smult_mat(2) index_zero_mat(2) j1n less_imp_diff_less add_diff_cancel_right')
           finally show ?thesis .
         next
           case False         
           have Ai10: "A $$ (i+1,0) = 0"
           proof -
             have "A $$ (i+1,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,0)"
-              by (smt A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
+              by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD diff_add_inverse2 i1_mn 
                   index_mat_four_block less_imp_diff_less n0)
             also have "... = 0" using i_ge_m n0 i1_mn by auto
             finally show ?thesis .
           qed          
           have "M $$ (i+1, j+1) = (make_first_column_positive A) $$ (i+1,j+1)"
             by (simp add: A'_def M_def False True non_zero_positions_xs_m)
           also have "... = (if A $$ (i+1,0) < 0 then - A $$ (i+1,j+1) else A $$ (i+1,j+1))" 
             unfolding make_first_column_positive.simps using A i1_mn j1n by auto
           also have "... = A $$ (i+1,j+1)" using Ai10 by auto
           also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1 - m,j+1)"
-            by (smt A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
+            by (smt (verit) A_def A'' A i_ge_m append_rows_def carrier_matD i1_mn index_mat_four_block(1,3)
                 index_one_mat(2) index_smult_mat(2) index_zero_mat(2) j1n less_imp_diff_less add_diff_cancel_right')
           finally show ?thesis .
         qed
         also have "... = D * (1\<^sub>m n) $$ ((i+1)-m, j+1)"
           by (rule index_smult_mat, insert i jn1 A'_DR False dr, auto)            
         also have "... = D *(1\<^sub>m (n - 1)) $$ (i-m,j)" using dc dr i j A'_DR i_ge_m
-          by (smt Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
+          by (smt (verit) Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
               linorder_not_less add_diff_cancel_right' add_diff_cancel_right' add_diff_cancel_left')
         also have "... = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
           by (rule index_smult_mat[symmetric], insert i jn1 A'_DR False dr, auto)
         finally show ?thesis using 1 2 by auto
       next
         case False     
       have "?reduce_below $$ (i+1, j+1) = M $$ (i+1, j+1)"
       proof (unfold non_zero_positions_xs_m M_def,
           rule reduce_below_preserves_case_m[OF M' m0 _ M_M'D mk_A'_not0 m_le_n _ d_xs all_less_m _ _ _ D0])            
         show "j + 1 < n" using jn1 by auto
         show "i + 1 \<notin> set xs" using all_less_m i_ge_m non_zero_positions_xs_m by auto
         show "i + 1 \<noteq> 0" by auto
         show " i + 1 < m + n" using i_ge_m i dr A'_DR by auto
         show " i + 1 \<noteq> m" using i_ge_m by auto
       qed (insert False)
       also have "... = (?M' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n) $$ (i+1, j+1)" unfolding M_def using False M_M'D by argo
       also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ ((i+1)-m, j+1)"
       proof -
         have f1: "1 + j < n"
           by (metis Groups.add_ac(2) jn1 less_diff_conv)
         have f2: "\<forall>n. \<not> n + i < m"
           by (meson i_ge_m linorder_not_less nat_SN.compat not_add_less2)
         have "i < m + (n - 1)"
           by (metis (no_types) A'_DR carrier_matD(1) dr i)
         then have "1 + i < m + n"
           using f1 by linarith
         then show ?thesis
           using f2 f1 by (metis (no_types) Groups.add_ac(2) M' append_rows_def carrier_matD(1) 
               dim_col_mat(1) index_mat_four_block(1) index_one_mat(2) index_smult_mat(2) 
               index_zero_mat(2,3) mat_of_rows_def nat_arith.rule0)
       qed
       also have "... = D * (1\<^sub>m n) $$ ((i+1)-m, j+1)"
         by (rule index_smult_mat, insert i jn1 A'_DR False dr, auto)            
       also have "... = D *(1\<^sub>m (n - 1)) $$ (i-m,j)" using dc dr i j A'_DR i_ge_m
-        by (smt Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
+        by (smt (verit) Nat.add_diff_assoc2 carrier_matD(1) index_one_mat(1) jn1 less_diff_conv 
             linorder_not_less add_diff_cancel_right' add_diff_cancel_left')
       also have "... = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)"
         by (rule index_smult_mat[symmetric], insert i jn1 A'_DR False dr, auto)
       finally have 3: "?reduce_below $$ (i+1,j+1) = (D \<cdot>\<^sub>m 1\<^sub>m (n - 1)) $$ (i-m,j)" .            
       show ?thesis using 1 2 3 by presburger
     qed              
   qed
 qed
   let ?A'_DR_n = "mat_of_rows (n - 1) (map (Matrix.row A'_DR) [0..<n - 1])"
   have hyp: "\<exists>P. P\<in>carrier_mat (m + (n-1)) (m + (n-1)) \<and> invertible_mat P \<and> sub_PreHNF = P * A'_DR 
   \<and> echelon_form_JNF sub_PreHNF" 
   proof (cases "2 \<le> n - 1")
     case True
     show ?thesis
       by (unfold sub_PreHNF_def, rule "1.hyps"[OF _ _ _ non_zero_positions_def A'_def _ _ _ _ _])
          (insert A n D0 m_le_n True A'DR_A'DR_m_D A A'_split False, auto)
   next
     case False
     have "\<exists>P. P\<in>carrier_mat (m + (n-1)) (m + (n-1)) \<and> invertible_mat P \<and> sub_PreHNF = P * A'_DR"
       by (unfold sub_PreHNF_def, rule FindPreHNF_invertible_mat_mx2
           [OF A'DR_A'DR_m_D A'_DR_m _ _ D0 _])
           (insert False m_le_n n0 m0 "1"(4), auto)
     moreover have "echelon_form_JNF sub_PreHNF" unfolding sub_PreHNF_def 
       by (rule FindPreHNF_echelon_form_mx1[OF A'DR_A'DR_m_D A'_DR_m _ D0 _],
           insert False n0 m_le_n, auto) 
     ultimately show ?thesis by simp
   qed  
   from this obtain P where P: "P \<in> carrier_mat (m + (n - 1)) (m + (n - 1))"
     and inv_P: "invertible_mat P" and sub_PreHNF_P_A'_DR: "sub_PreHNF = P * A'_DR" by blast
   define P' where "P' = (four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m+(n-1))) (0\<^sub>m (m+(n-1)) 1) P)"
   have P': "P' \<in> carrier_mat (m+n) (m+n)" 
   proof -
     have "P' \<in> carrier_mat (1 + (m+(n-1))) (1 + (m+(n-1))) "
       unfolding P'_def by (rule four_block_carrier_mat[OF _ P], simp) 
     thus ?thesis using n by auto
   qed
   have inv_P': "invertible_mat P'" 
     unfolding P'_def by (rule invertible_mat_four_block_mat_lower_right[OF P inv_P])
   have dr_A2: "dim_row A \<ge> 2" using A m0 n by auto
   have dc_A2: "dim_col A \<ge> 2" using n A by blast
   have *: "(dim_col A = 0) = False" using dc_A2 by auto
   have FindPreHNF_as_fbm: "FindPreHNF abs_flag D A = four_block_mat A'_UL A'_UR A'_DL sub_PreHNF" 
     unfolding FindPreHNF.simps[of abs_flag D A] using A'_split mn n A dr_A2 dc_A2 False
     unfolding Let_def sub_PreHNF_def M_def A'_def non_zero_positions_def *    
-    by (smt (z3) linorder_not_less split_conv)
+    by (smt (verit) linorder_not_less split_conv)
   also have "... = P' * (reduce_below 0 non_zero_positions D M)"
   proof -
     have "P' * (reduce_below 0 non_zero_positions D M) 
       = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m + (n - 1))) (0\<^sub>m (m + (n - 1)) 1) P 
       * four_block_mat A'_UL A'_UR A'_DL A'_DR"
       unfolding P'_def fbm_R[unfolded M_def[symmetric], symmetric] ..
     also have "... = four_block_mat 
             ((1\<^sub>m 1) * A'_UL + (0\<^sub>m 1 (m + (n - 1)) * A'_DL)) 
             ((1\<^sub>m 1) * A'_UR + (0\<^sub>m 1 (m + (n - 1))) * A'_DR) 
             ((0\<^sub>m (m + (n - 1)) 1) * A'_UL + P * A'_DL) 
             ((0\<^sub>m (m + (n - 1)) 1) * A'_UR + P * A'_DR)" 
       by (rule mult_four_block_mat[OF _ _ _ P A'_UL A'_UR A'_DL A'_DR], auto)
     also have "... = four_block_mat A'_UL A'_UR (P * A'_DL) (P * A'_DR)"
       by (rule cong_four_block_mat, insert A'_UL A'_UR A'_DL A'_DR P, auto)
     also have "... = four_block_mat A'_UL A'_UR (0\<^sub>m (m + (n - 1)) 1) sub_PreHNF"
       unfolding A'_DL0 sub_PreHNF_P_A'_DR using P by simp
     also have "... = four_block_mat A'_UL A'_UR A'_DL sub_PreHNF"
       unfolding A'_DL0 by simp
     finally show ?thesis ..
   qed
   finally have Find_P'_reduceM: "FindPreHNF abs_flag D A =  P' * (reduce_below 0 non_zero_positions D M)" .
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat (m + n) (m + n) 
     \<and> reduce_below 0 (xs @ [m]) D M = Q * M"
   proof (cases "xs = []")
     case True note xs_empty = True
     have rw: "reduce_below 0 (xs @ [m]) D M = reduce 0 m D M" using True by auto
     obtain p q u v d where pquvd: "(p, q, u, v, d) = euclid_ext2 (M $$ (0, 0)) (M $$ (m, 0))"
       by (simp add: euclid_ext2_def)
     have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (m + n) (m + n) \<and> reduce 0 m D M = P * M"  
     proof (rule reduce_invertible_mat_case_m[OF _ _ m0 _ _ _ _ m_le_n n0 pquvd])
           show "M $$ (0, 0) \<noteq> 0" 
             using M_def mk_A'_not0 by blast
           define M' where  "M' = mat_of_rows n (map (Matrix.row M) [0..<m])"
           define M'' where "M'' = mat_of_rows n (map (Matrix.row M) [m..<m+n])"
           define A2 where "A2 = Matrix.mat (dim_row M) (dim_col M)
           (\<lambda>(i, k). if i = 0 then p * M $$ (0, k) + q * M $$ (m, k) 
                     else if i = m then u * M $$ (0, k) + v * M $$ (m, k)
                     else M $$ (i, k))"
           show M_M'_M'': "M = M' @\<^sub>r M''" unfolding M'_def M''_def            
             by (metis M append_rows_split carrier_matD le_add1)
           show M': "M' \<in> carrier_mat m n" unfolding M'_def by fastforce
           show M'': "M'' \<in> carrier_mat n n" unfolding M''_def by fastforce
           show "0 \<noteq> m" using m0 by simp
           show "A2 = Matrix.mat (dim_row M) (dim_col M)
           (\<lambda>(i, k). if i = 0 then p * M $$ (0, k) + q * M $$ (m, k) 
                     else if i = m then u * M $$ (0, k) + v * M $$ (m, k)
                     else M $$ (i, k))"
             (is "_ = ?rhs") using A A2_def by auto
           define xs' where "xs' = [1..<n]"
           define ys' where  "ys' = [1..<n]"
           show "xs' = [1..<n]" unfolding xs'_def by auto
           show "ys' = [1..<n]" unfolding ys'_def by auto
           have M''D: "(M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"
             if jn: "j<n" and j0: "j>0" for j
           proof -
             have Ajm0: "A $$ (j+m,0) = 0"
             proof -
               have "A $$ (j+m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j+m-m,0)"
-                by (smt "1"(2) "1"(3) M M' M'' M_M'_M'' add.commute append_rows_def carrier_matD
+                by (smt (verit) "1"(2) "1"(3) M M' M'' M_M'_M'' add.commute append_rows_def carrier_matD
                     diff_add_inverse2 index_mat_four_block index_one_mat(2) index_smult_mat(2)
                     le_add2 less_diff_conv2 n0 not_add_less2 that(1))
               also have "... = 0" using jn j0 by auto
               finally show ?thesis .
             qed
             have "M'' $$ (j, i) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)" if i_n: "i<n" for i
             proof (cases "A$$(0,0) = 0")
               case True 
-              have "M'' $$ (j, i) = make_first_column_positive (swaprows 0 m A) $$ (j+m,i)"                
-                by (smt A'_def Groups.add_ac(2) M' M'' M_M'_M'' M_def True append.simps(1) 
-                    append_rows_nth3 diff_add_inverse2 jn le_add2 local.non_zero_positions_xs_m
-                    nat_add_left_cancel_less nth_Cons_0 that xs_empty)
+              have "M'' $$ (j, i) = make_first_column_positive (swaprows 0 m A) $$ (j+m,i)"    
+                using M' M'' M_M'_M''  
+                by (simp add: M_def A'_def True append_rows_nth3 i_n jn local.non_zero_positions_xs_m xs_empty)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n Ajm0 by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .            
             next
               case False
               have "A' = A" unfolding A'_def non_zero_positions_xs_m using False True by auto
               hence "M'' $$ (j, i) = make_first_column_positive A $$ (j+m,i)"                
-                by (smt m_le_n M' M'' M_M'_M'' M_def append_rows_nth2 jn nat_SN.compat that)                
+                using M' M'' M_M'_M'' unfolding M_def
+                by (metis (no_types, opaque_lifting)  "1"(5) append_rows_nth2 jn nat_SN.compat that)
               also have "... = A $$ (j+m,i)" using A jn j0 i_n Ajm0 by auto
               also have "... = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (j,i)"
-                by (smt A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
+                by (smt (verit) A Groups.add_ac(2) add_mono_thms_linordered_field(1) append_rows_def A_def A'' i_n
                     carrier_matD index_mat_four_block(1,2) add_diff_cancel_right' not_add_less2 jn trans_less_add1)
               finally show ?thesis .
             qed                           
             thus ?thesis using jn j0 by auto
           qed
           have Am0D: "A$$(m,0) = D"
           proof -
             have "A$$(m,0) = (D \<cdot>\<^sub>m 1\<^sub>m n) $$ (m-m,0)"
-              by (smt "1"(2) "1"(3) M M' M'' M_M'_M'' append_rows_def carrier_matD
+              by (smt (verit) "1"(2) "1"(3) M M' M'' M_M'_M'' append_rows_def carrier_matD
                   diff_less_mono2 diff_self_eq_0 index_mat_four_block index_one_mat(2) 
                   index_smult_mat(2) less_add_same_cancel1 n0 semiring_norm(137))
             also have "... = D" using m0 n0 by auto
             finally show ?thesis .
           qed
           hence S00D: "(swaprows 0 m A) $$ (0,0) = D" using n0 m0 A by auto
           have Sm00: "(swaprows 0 m A) $$ (m,0) = A$$(0,0)" using n0 m0 A by auto
           have M00D: "M $$ (0, 0) = D" if A00: "A$$(0,0) = 0"
           proof -
             have "M $$ (0,0) = (make_first_column_positive (swaprows 0 m A)) $$ (0,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if (swaprows 0 m A) $$ (0,0) < 0 then - (swaprows 0 m A) $$(0,0)
                               else (swaprows 0 m A) $$(0,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = (swaprows 0 m A) $$(0,0)" using S00D D0 by auto
             also have "... = D" using S00D by auto
             finally show ?thesis .
           qed
           have Mm00: "M $$ (m, 0) = 0" if A00: "A$$(0,0) = 0"
           proof -
             have "M $$ (m,0) = (make_first_column_positive (swaprows 0 m A)) $$ (m,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if (swaprows 0 m A) $$ (m,0) < 0 then - (swaprows 0 m A) $$(m,0)
                               else (swaprows 0 m A) $$(m,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = (swaprows 0 m A) $$(m,0)" using Sm00 A00 D0 by auto
             also have "... = 0" using Sm00 A00 by auto
             finally show ?thesis .
           qed
           have M000: "M $$ (0, 0) = abs (A$$(0,0))" if A00: "A$$(0,0) \<noteq> 0"
           proof -
             have "M $$ (0,0) = (make_first_column_positive A) $$ (0,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if A $$ (0,0) < 0 then - A $$(0,0)
                               else A $$(0,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = abs (A$$(0,0))" using Sm00 A00 by auto
             finally show ?thesis .
           qed
           have Mm0D: "M $$ (m, 0) = D" if A00: "A $$ (0,0) \<noteq> 0"
           proof -
             have "M $$ (m,0) = (make_first_column_positive A) $$ (m,0)"
               unfolding M_def A'_def using A00
               by (simp add: True non_zero_positions_xs_m)
             also have "... = (if A $$ (m,0) < 0 then - A $$(m,0)
                               else A $$(m,0))"
               unfolding make_first_column_positive.simps using m0 n0 A by auto
             also have "... = A $$(m,0)" using S00D D0 Am0D by auto
             also have "... = D" using Am0D D0 by auto
             finally show ?thesis .
           qed
           have "0 \<notin> set xs'"
           proof -
             have "A2 $$ (0,0) = p * M $$ (0, 0) + q * M $$ (m, 0)" 
               using A A2_def n0 M by auto
             also have "... = gcd (M $$ (0, 0)) (M $$ (m, 0))"
               by (metis euclid_ext2_works(1,2) pquvd)
             also have "abs ... \<le> D" using M00D Mm00 M000 Mm0D using gcd_0_int D0 by fastforce
             finally have "abs (A2 $$ (0,0)) \<le> D" .
             thus ?thesis unfolding xs'_def using D0 by auto
           qed
           thus "\<forall>j\<in>set xs'. j<n \<and> (M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"  
             using M''D xs'_def by auto
           have "0 \<notin> set ys'"
           proof -
             have "A2 $$ (m,0) = u * M $$ (0, 0) + v * M $$ (m, 0)"
               using A A2_def n0 m0 M by auto
             also have "... = - M $$ (m, 0) div gcd (M $$ (0, 0)) (M $$ (m, 0)) * M $$ (0, 0) 
                 + M $$ (0, 0) div gcd (M $$ (0, 0)) (M $$ (m, 0)) * M $$ (m, 0) "
               by (simp add: euclid_ext2_works[OF pquvd[symmetric]])
             also have "... = 0" using M00D Mm00 M000 Mm0D
-              by (smt dvd_div_mult_self euclid_ext2_works(3) euclid_ext2_works(5)
+              by (smt (verit) dvd_div_mult_self euclid_ext2_works(3) euclid_ext2_works(5)
                   more_arith_simps(11) mult.commute mult_minus_left pquvd semiring_gcd_class.gcd_dvd1)
             finally have "A2 $$ (m,0) = 0" .
             thus ?thesis unfolding ys'_def using D0 by auto
           qed
           thus "\<forall>j\<in>set ys'. j<n \<and> (M'' $$ (j, j) = D) \<and> (\<forall>j'\<in>{0..<n}-{j}. M'' $$ (j, j') = 0)"
             using M''D ys'_def by auto   
           show "M $$ (m, 0) \<in> {0,D}" using Mm00 Mm0D by blast
           show " M $$ (m, 0) = 0 \<longrightarrow> M $$ (0, 0) = D"  using Mm00 Mm0D D_not0 M00D by blast          
         qed (insert D0)
         then show ?thesis using rw by auto
   next
     case False
     show ?thesis
       by (unfold M_def, rule reduce_below_invertible_mat_case_m[OF M' m0 n0 M_M'D[OF False] 
           mk_A'_not0 m_le_n d_xs all_less_m D0])
   qed
    
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat (m + n) (m + n)"
     and reduce_QM: "reduce_below 0 (xs @ [m]) D M = Q * M" by blast
   have "\<exists>R. invertible_mat R 
     \<and> R \<in> carrier_mat (dim_row A') (dim_row A') \<and> M = R * A'"
     by (unfold M_def, rule make_first_column_positive_invertible)
   from this obtain R where inv_R: "invertible_mat R"
     and R: "R \<in> carrier_mat (dim_row A') (dim_row A')" and M_RA': "M = R * A'" by blast
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> A' = P * A" 
     by (rule A'_swaprows_invertible_mat[OF A A'_def non_zero_positions_def],
         insert non_zero_positions_xs_m n m0, auto)
   from this obtain S where inv_S: "invertible_mat S"
    and S: "S \<in> carrier_mat (dim_row A) (dim_row A)" and A'_SA: "A' = S * A" 
     using A by auto
   have "(P'*Q*R*S) \<in> carrier_mat (m+n) (m+n)" using P' Q R S A' A by auto
   moreover have "FindPreHNF abs_flag D A = (P'*Q*R*S) * A" using Find_P'_reduceM reduce_QM 
     unfolding  M_RA' A'_SA M_def
-    by (smt A' A'_SA P' Q R S assoc_mult_mat carrier_matD carrier_mat_triv index_mult_mat(2,3) 
+    by (smt (verit) A' A'_SA P' Q R S assoc_mult_mat carrier_matD carrier_mat_triv index_mult_mat(2,3) 
         non_zero_positions_xs_m)
   moreover have "invertible_mat (P'*Q*R*S)" using inv_P' inv_Q inv_R inv_S using P' Q R S A' A 
     by (metis carrier_matD carrier_mat_triv index_mult_mat(2,3) invertible_mult_JNF)
   ultimately have exists_inv: "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P 
     \<and> FindPreHNF abs_flag D A = P * A" by blast
   moreover have "echelon_form_JNF (FindPreHNF abs_flag D A)" 
   proof (rule echelon_form_four_block_mat[OF A'_UL A'_UR sub_PreHNF'   ])
     show "FindPreHNF abs_flag D A = four_block_mat A'_UL A'_UR (0\<^sub>m (m + n - 1) 1) sub_PreHNF" 
       using A'_DL0 FindPreHNF_as_fbm sub_PreHNF sub_PreHNF' by auto
     have "A'_UL $$ (0, 0) = ?R $$ (0,0)"
       by (metis (mono_tags, lifting) A A'_DR A'_UL Find_P'_reduceM M_def 
           \<open>FindPreHNF abs_flag D A = P' * Q * R * S * A\<close> add_Suc_right add_sign_intros(2) carrier_matD fbm_R
           index_mat_four_block(1,3) index_mult_mat(3) m0 n0 plus_1_eq_Suc
           zero_less_one_class.zero_less_one)
     also have "... \<noteq> 0"
     proof (cases "xs=[]")
       case True
       have "?R $$ (0,0) = reduce 0 m D M $$ (0,0)"
         unfolding non_zero_positions_xs_m True M_def by simp
       also have "... \<noteq> 0"
         by (metis D_not0 M M_def add_pos_pos less_add_same_cancel1 m0 mk_A'_not0 n0 reduce_not0)
       finally show ?thesis .
     next
       case False
       show ?thesis
         by (unfold non_zero_positions_xs_m,
           rule reduce_below_not0_case_m[OF M' m0 n0 M_M'D[OF False] mk_A'_not0 m_le_n all_less_m D_not0])     
     qed       
     finally show "A'_UL $$ (0, 0) \<noteq> 0" .
   qed (insert mn n hyp, auto)
   ultimately show ?thesis by blast
 qed
 qed
 
 lemma
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
   and A'': "A'' \<in> carrier_mat m n" and "n\<ge>2" and m_le_n: "m\<ge>n" and "D>0"
 shows FindPreHNF_invertible_mat_n_ge2: "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> FindPreHNF abs_flag D A = P * A"
 and FindPreHNF_echelon_form_n_ge2: "echelon_form_JNF (FindPreHNF abs_flag D A)"
   using FindPreHNF_works_n_ge2[OF assms] by blast+
 
 lemma FindPreHNF_invertible_mat:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
     and A'': "A'' \<in> carrier_mat m n" and n0: "0<n" and mn: "m\<ge>n" and D: "D>0"
   shows "\<exists>P. P \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat P \<and> FindPreHNF abs_flag D A = P * A"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   show ?thesis
   proof (cases "m+n<2")
     case True
     show ?thesis by (rule FindPreHNF_invertible_mat_2xn[OF A True])
   next
     case False note m_ge2 = False
     show ?thesis
     proof (cases "n<2")
       case True
       show ?thesis by (rule FindPreHNF_invertible_mat_mx2[OF A_def A'' True n0 D mn])
     next
       case False
       show ?thesis 
         by (rule FindPreHNF_invertible_mat_n_ge2[OF A_def A'' _ mn D], insert False, auto)
     qed  
   qed
 qed
 
 
 lemma FindPreHNF_echelon_form:
   assumes A_def: "A = A'' @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n"
     and A'': "A'' \<in> carrier_mat m n" and mn: "m\<ge>n" and D: "D>0"
   shows "echelon_form_JNF (FindPreHNF abs_flag D A)"
 proof -
   have A: "A \<in> carrier_mat (m+n) n" using A_def A'' by auto
   have FindPreHNF: "(FindPreHNF abs_flag D A) \<in> carrier_mat (m+n) n" by (rule FindPreHNF[OF A])
   show ?thesis
   proof (cases "m+n<2")
     case True
     show ?thesis by (rule echelon_form_JNF_1xn[OF FindPreHNF True])
   next
     case False note m_ge2 = False
     show ?thesis
     proof (cases "n<2")
       case True
       show ?thesis by (rule FindPreHNF_echelon_form_mx1[OF A_def A'' True D mn])
     next
       case False
       show ?thesis 
         by (rule FindPreHNF_echelon_form_n_ge2[OF A_def A'' _ mn D], insert False, auto)
     qed  
   qed
 qed
 end
 
 text \<open>We connect the algorithm developed in the Hermite AFP entry with ours. This would permit
 to reuse many existing results and prove easily the soundness.\<close>
 
 (*In HOL Analysis*)
 thm Hermite.Hermite_reduce_above.simps
 thm Hermite.Hermite_of_row_i_def
 thm Hermite.Hermite_of_upt_row_i_def
 thm Hermite.Hermite_of_def
 
 (*In JNF*)
 thm Hermite_reduce_above.simps
 thm Hermite_of_row_i_def
 thm Hermite_of_list_of_rows.simps
 thm mod_operation.Hermite_mod_det_def
 
 (*Connecting Hermite.Hermite_reduce_above and Hermite_reduce_above*)
 thm Hermite.Hermite_reduce_above.simps Hermite_reduce_above.simps
 
 context includes lifting_syntax
 begin
 
 definition "res_int = (\<lambda>b n::int. n mod b)"
 
 lemma res_function_res_int: 
   "res_function res_int"
   using res_function_euclidean2 unfolding res_int_def by auto
 
 lemma HMA_Hermite_reduce_above[transfer_rule]: 
   assumes "n<CARD('m)"
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) 
     ===> (Mod_Type_Connect.HMA_I) ===> (Mod_Type_Connect.HMA_I) ===> (Mod_Type_Connect.HMA_M)) 
   (\<lambda>A i j. Hermite_reduce_above A n i j)
   (\<lambda>A i j. Hermite.Hermite_reduce_above A n i j res_int)"
 proof (intro rel_funI, goal_cases)
   case (1 A A' i i' j j')
   then show ?case using assms
   proof (induct n arbitrary: A A')
     case 0
     then show ?case by auto
   next
     case (Suc n)
     note AA'[transfer_rule] = "Suc.prems"(1)
     note ii'[transfer_rule] = "Suc.prems"(2)
     note jj'[transfer_rule] = "Suc.prems"(3)
     note Suc_n_less_m = "Suc.prems"(4)
 
     let ?H_JNF = "HNF_Mod_Det_Algorithm.Hermite_reduce_above"
     let ?H_HMA = "Hermite.Hermite_reduce_above"
     let ?from_nat_rows = "mod_type_class.from_nat :: _ \<Rightarrow> 'm"
     have nn[transfer_rule]: "Mod_Type_Connect.HMA_I n (?from_nat_rows n)" 
       unfolding Mod_Type_Connect.HMA_I_def
       by (simp add: Suc_lessD Suc_n_less_m mod_type_class.from_nat_to_nat)
 
     have Anj: "A' $h (?from_nat_rows n) $h j' = A $$ (n,j)" 
       by (unfold index_hma_def[symmetric], transfer, simp)
     have Aij: "A' $h i' $h j' = A $$ (i,j)" by (unfold index_hma_def[symmetric], transfer, simp)
     let ?s = "(- (A $$ (n, j) div A $$ (i, j)))"
     let ?s' = "((res_int (A' $h i' $h j') (A' $h ?from_nat_rows n $h j') 
       - A' $h ?from_nat_rows n $h j') div A' $h i' $h j')"
     have ss'[transfer_rule]: "?s = ?s'" unfolding res_int_def Anj Aij
       by (metis (no_types, opaque_lifting) Groups.add_ac(2) add_diff_cancel_left' div_by_0 
           minus_div_mult_eq_mod more_arith_simps(7) nat_arith.rule0 nonzero_mult_div_cancel_right
           uminus_add_conv_diff)
     have H_JNF_eq: "?H_JNF A (Suc n) i j = ?H_JNF (addrow (- (A $$ (n, j) div A $$ (i, j))) n i A) n i j"
       by auto
     have H_HMA_eq: "?H_HMA A' (Suc n) i' j' res_int = ?H_HMA (row_add A' (?from_nat_rows n) i' ?s') n i' j' res_int"
       by (auto simp add: Let_def)
     have "Mod_Type_Connect.HMA_M (?H_JNF (addrow ?s n i A) n i j) 
       (?H_HMA (row_add A' (?from_nat_rows n) i' ?s') n i' j' res_int)" 
       by (rule "Suc.hyps"[OF _ ii' jj'], transfer_prover, insert Suc_n_less_m, simp)
     thus ?case using H_JNF_eq H_HMA_eq by auto
   qed
 qed
 
 
 corollary HMA_Hermite_reduce_above': 
   assumes "n<CARD('m)"
   and "Mod_Type_Connect.HMA_M A (A':: int ^ 'n :: mod_type ^ 'm :: mod_type)"
   and "Mod_Type_Connect.HMA_I i i'" and "Mod_Type_Connect.HMA_I j j'"
   shows"Mod_Type_Connect.HMA_M (Hermite_reduce_above A n i j) (Hermite.Hermite_reduce_above A' n i' j' res_int)"
   using HMA_Hermite_reduce_above assms unfolding rel_fun_def by metis
 
 
 lemma HMA_Hermite_of_row_i[transfer_rule]: 
   assumes upt_A: "upper_triangular' A"
   and AA': "Mod_Type_Connect.HMA_M A (A':: int ^ 'n :: mod_type ^ 'm :: mod_type)"
   and ii': "Mod_Type_Connect.HMA_I i i'"
   shows "Mod_Type_Connect.HMA_M (Hermite_of_row_i A i)
   (Hermite.Hermite_of_row_i ass_function_euclidean res_int A' i')"
 proof -
   note AA'[transfer_rule]
   note ii'[transfer_rule]
   have i: "i<dim_row A"
     by (metis (full_types) AA' ii' Mod_Type_Connect.HMA_I_def 
         Mod_Type_Connect.dim_row_transfer_rule mod_type_class.to_nat_less_card)  
   show ?thesis
   proof (cases "is_zero_row i' A'")
     case True
     hence "is_zero_row_JNF i A" by (transfer, simp)
     hence "find_fst_non0_in_row i A = None" using find_fst_non0_in_row_None[OF _ upt_A i] by auto
     thus ?thesis using True AA' unfolding Hermite.Hermite_of_row_i_def Hermite_of_row_i_def by auto
   next
     case False
     have nz_iA: "\<not> is_zero_row_JNF i A" using False by transfer
     hence "find_fst_non0_in_row i A \<noteq> None" using find_fst_non0_in_row_None[OF _ upt_A i] by auto
     from this obtain j where j: "find_fst_non0_in_row i A = Some j" by blast
     have j_eq: "j = (LEAST n. A $$ (i,n) \<noteq> 0)"
       by (rule find_fst_non0_in_row_LEAST[OF _ upt_A j i], auto)
     have H_JNF_rw: "(Hermite_of_row_i A i) = 
     (if A $$ (i, j) < 0 then Hermite_reduce_above (multrow i (- 1) A) i i j
      else Hermite_reduce_above A i i j)" unfolding Hermite_of_row_i_def using j by auto
     let ?H_HMA = "Hermite.Hermite_of_row_i"
     let ?j' = "(LEAST n. A' $h i' $h n \<noteq> 0)"
     have ii'2: "(mod_type_class.to_nat i') = i" using ii'
       by (simp add: Mod_Type_Connect.HMA_I_def)                                            
     have jj'[transfer_rule]: "Mod_Type_Connect.HMA_I j ?j'"
       unfolding j_eq index_hma_def[symmetric] by (rule HMA_LEAST[OF AA' ii' nz_iA])
     have Aij: "A $$ (i, j) = A' $h i' $h (LEAST n. A' $h i' $h n \<noteq> 0)"
       by (subst index_hma_def[symmetric], transfer', simp)
     have H_HMA_rw: "?H_HMA ass_function_euclidean res_int A' i' = 
     Hermite.Hermite_reduce_above (mult_row A' i' (\<bar>A' $h i' $h ?j'\<bar> 
       div A' $h i' $h ?j'))
      (mod_type_class.to_nat i') i' ?j' res_int" 
       unfolding Hermite.Hermite_of_row_i_def Let_def ass_function_euclidean_def
       by (auto simp add: False)
     have im: "i < CARD('m)" using ii' unfolding Mod_Type_Connect.HMA_I_def
       using mod_type_class.to_nat_less_card by blast
     show ?thesis
     proof (cases "A $$ (i, j) < 0")
       case True
       have A'i'j'_le_0: "A' $h i' $h ?j' < 0" using Aij True by auto
       hence 1: "(\<bar>A' $h i' $h ?j'\<bar> div A' $h i' $h ?j') 
           = -1" using div_pos_neg_trivial by auto
       have [transfer_rule]: "Mod_Type_Connect.HMA_M (multrow i (- 1) A) 
       (mult_row A' i' (\<bar>A' $h i' $h ?j'\<bar> 
       div A' $h i' $h ?j'))" unfolding 1 by transfer_prover      
       have H_HMA_rw2: "Hermite_of_row_i A i = Hermite_reduce_above (multrow i (- 1) A) i i j"
         using True H_JNF_rw by auto
       have *: "Mod_Type_Connect.HMA_M (Hermite_reduce_above (multrow i (- 1) A) i i j) 
         (Hermite.Hermite_reduce_above (mult_row A' i' (\<bar>A' $h i' $h ?j'\<bar> 
         div A' $h i' $h ?j'))
         (mod_type_class.to_nat i') i' ?j' res_int) "
         unfolding 1 ii'2
         by (rule HMA_Hermite_reduce_above'[OF im _ ii' jj'], transfer_prover)
        show ?thesis unfolding H_JNF_rw H_HMA_rw unfolding H_HMA_rw2 using True * by auto
     next
       case False
       have Aij_not0: "A $$ (i, j) \<noteq> 0" using j_eq nz_iA
         by (metis (mono_tags) LeastI is_zero_row_JNF_def)
       have A'i'j'_le_0: "A' $h i' $h ?j' > 0" using False Aij_not0 Aij by auto
       hence 1: "(\<bar>A' $h i' $h ?j'\<bar> div A' $h i' $h ?j') = 1" by auto
       have H_HMA_rw2: "Hermite_of_row_i A i = Hermite_reduce_above A i i j"
         using False H_JNF_rw by auto
       have *: "?H_HMA ass_function_euclidean res_int A' i' = 
       (Hermite.Hermite_reduce_above A' (mod_type_class.to_nat i') i' ?j' res_int)"
         using H_HMA_rw unfolding 1 unfolding mult_row_1_id by simp
       have "Mod_Type_Connect.HMA_M (Hermite_reduce_above A i i j) 
         (Hermite.Hermite_reduce_above A' (mod_type_class.to_nat i') i' ?j' res_int)"
         unfolding 1 ii'2
         by (rule HMA_Hermite_reduce_above'[OF im AA' ii' jj'])
       then show ?thesis using H_HMA_rw * H_HMA_rw2 by presburger
     qed        
   qed
 qed
 
 
 lemma Hermite_of_list_of_rows_append:
 "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i (Hermite_of_list_of_rows A xs) x"
   by (induct xs arbitrary: A, auto)
 
 
 lemma Hermite_reduce_above[simp]: "Hermite_reduce_above A n i j \<in> carrier_mat (dim_row A) (dim_col A)"  
 proof (induct n arbitrary: A)
   case 0
   then show ?case by auto
 next
   case (Suc n)    
   let ?A = "(addrow (- (A $$ (n, j) div A $$ (i, j))) n i A)"  
   have "Hermite_reduce_above A (Suc n) i j = Hermite_reduce_above ?A n i j"
     by (auto simp add: Let_def)
   also have "... \<in> carrier_mat (dim_row ?A) (dim_col ?A)" by(rule Suc.hyps)
   finally show ?case by auto  
 qed                           
 
 
 lemma Hermite_of_row_i: "Hermite_of_row_i A i \<in> carrier_mat (dim_row A) (dim_col A)" 
 proof -
   have "Hermite_reduce_above (multrow i (- 1) A) i i a 
     \<in> carrier_mat (dim_row (multrow i (- 1) A)) (dim_col (multrow i (- 1) A))" 
     for a by (rule Hermite_reduce_above)
   thus ?thesis
     unfolding Hermite_of_row_i_def using Hermite_reduce_above 
     by (cases "find_fst_non0_in_row i A", auto)
 qed
 
 end
 
 
 text \<open>We now move more lemmas from HOL Analysis (with mod-type restrictions) to the JNF matrix
 representation.\<close>
 
 (*thm echelon_form_Hermite_of_row will be transferred from HOL Analysis to JNF*)
 
 context
 begin
 
 private lemma echelon_form_Hermite_of_row_mod_type:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::mod_type) CARD('n::mod_type)"
   assumes eA: "echelon_form_JNF A"
   and i: "i<CARD('m)"
   shows "echelon_form_JNF (Hermite_of_row_i A i)"
 proof -
   have uA: "upper_triangular' A" by (rule echelon_form_JNF_imp_upper_triangular[OF eA])
   define A' where "A' = (Mod_Type_Connect.to_hma\<^sub>m A :: int ^'n :: mod_type ^'m :: mod_type)"
   define i' where "i' = (Mod_Type.from_nat i :: 'm)"
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'"
     unfolding Mod_Type_Connect.HMA_M_def using assms A'_def by auto
   have ii'[transfer_rule]: "Mod_Type_Connect.HMA_I i i'"
     unfolding Mod_Type_Connect.HMA_I_def i'_def using assms
     using from_nat_not_eq order.strict_trans by blast 
   have eA'[transfer_rule]: "echelon_form A'" using eA by transfer
   have [transfer_rule]: "Mod_Type_Connect.HMA_M 
     (HNF_Mod_Det_Algorithm.Hermite_of_row_i A i) 
     (Hermite.Hermite_of_row_i ass_function_euclidean res_int A' i')"
     by (rule HMA_Hermite_of_row_i[OF uA AA' ii'])
   have "echelon_form (Hermite.Hermite_of_row_i ass_function_euclidean res_int A' i')"
     by (rule echelon_form_Hermite_of_row[OF ass_function_euclidean res_function_res_int eA'])    
   thus ?thesis by (transfer, simp)
 qed 
 
 
 private lemma echelon_form_Hermite_of_row_nontriv_mod_ring:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::nontriv mod_ring) CARD('n::nontriv mod_ring)"
   assumes eA: "echelon_form_JNF A"
   and "i<CARD('m)"
   shows "echelon_form_JNF (Hermite_of_row_i A i)"
-using assms echelon_form_Hermite_of_row_mod_type by (smt CARD_mod_ring) 
+using assms echelon_form_Hermite_of_row_mod_type by (smt (verit) CARD_mod_ring) 
 
 (*We internalize both sort constraints in one step*)
 lemmas echelon_form_Hermite_of_row_nontriv_mod_ring_internalized = 
   echelon_form_Hermite_of_row_nontriv_mod_ring[unfolded CARD_mod_ring, 
       internalize_sort "'m::nontriv", internalize_sort "'b::nontriv"]
 
 context
   fixes m::nat and n::nat
   assumes local_typedef1: "\<exists>(Rep :: ('b \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<m :: int}"
   assumes local_typedef2: "\<exists>(Rep :: ('c \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<n :: int}"
   and m: "m>1"
   and n: "n>1"
 begin
 
 lemma echelon_form_Hermite_of_row_nontriv_mod_ring_aux:
  fixes A::"int mat"
   assumes "A \<in> carrier_mat m n"
   assumes eA: "echelon_form_JNF A"
   and "i<m"
 shows "echelon_form_JNF (Hermite_of_row_i A i)"  
   using echelon_form_Hermite_of_row_nontriv_mod_ring_internalized
     [OF type_to_set2(1)[OF local_typedef1 local_typedef2] 
         type_to_set1(1)[OF local_typedef1 local_typedef2]]
   using assms 
   using type_to_set1(2) local_typedef1 local_typedef2 n m by metis 
 
 end
 
 (*Canceling the first local type definitions*)
 context
 begin
 
 (*Canceling the first*)
 
 private lemma echelon_form_Hermite_of_row_i_cancelled_first:
 "\<exists>Rep Abs. type_definition Rep Abs {0..<int n} \<Longrightarrow> 1 < m \<Longrightarrow> 1 < n 
   \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> echelon_form_JNF A \<Longrightarrow> i < m 
   \<Longrightarrow> echelon_form_JNF (HNF_Mod_Det_Algorithm.Hermite_of_row_i A i)"
   using echelon_form_Hermite_of_row_nontriv_mod_ring_aux[cancel_type_definition, of m n A i]
   by auto  
 
 (*Canceling the second*)
 private lemma echelon_form_Hermite_of_row_i_cancelled_both:
 "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> echelon_form_JNF A \<Longrightarrow> i < m 
   \<Longrightarrow> echelon_form_JNF (HNF_Mod_Det_Algorithm.Hermite_of_row_i A i)"
   using echelon_form_Hermite_of_row_i_cancelled_first[cancel_type_definition, of n m A i] by simp
 
 (*The final results in JNF*)
 
 lemma echelon_form_JNF_Hermite_of_row_i':
  fixes A::"int mat"
   assumes "A \<in> carrier_mat m n"
   assumes eA: "echelon_form_JNF A"
   and "i<m"
   and "1 < m" and "1 < n" (*Required from the mod_type restrictions*)
 shows "echelon_form_JNF (Hermite_of_row_i A i)"  
   using echelon_form_Hermite_of_row_i_cancelled_both assms by auto
 
 
 corollary echelon_form_JNF_Hermite_of_row_i:
   fixes A::"int mat"
   assumes eA: "echelon_form_JNF A"
     and i: "i<dim_row A"
   shows "echelon_form_JNF (Hermite_of_row_i A i)"  
 proof (cases "dim_row A < 2")
   case True
   show ?thesis 
     by (rule echelon_form_JNF_1xn[OF Hermite_of_row_i True])
 next
   case False note m_ge2 = False
   show ?thesis
   proof (cases "1<dim_col A")
     case True
     show ?thesis by (rule echelon_form_JNF_Hermite_of_row_i'[OF _ eA i _ True], insert m_ge2, auto)
   next
     case False
     hence dc_01: "dim_col A \<in> {0,1}" by auto
     show ?thesis
     proof (cases "dim_col A = 0")
       case True
       have H: "Hermite_of_row_i A i \<in> carrier_mat (dim_row A) (dim_col A)"
         using Hermite_of_row_i by blast
       show ?thesis by (rule echelon_form_mx0, insert True H, auto)
     next
       case False
       hence dc_1: "dim_col A = 1" using dc_01 by simp
       then show ?thesis     
       proof (cases "i=0")
         case True      
         have eA': "echelon_form_JNF (multrow 0 (- 1) A)"
           by (rule echelon_form_JNF_multrow[OF _ _ eA], insert m_ge2, auto)
         show ?thesis using True unfolding Hermite_of_row_i_def
           by (cases "find_fst_non0_in_row 0 A", insert eA eA', auto)
       next
         case False  
         have all_zero: "(\<forall>j\<in>{i..<dim_col A}. A $$ (i, j) = 0)" unfolding dc_1 using False by auto
         hence "find_fst_non0_in_row i A = None" using find_fst_non0_in_row_None'[OF _ i] by blast
         hence "Hermite_of_row_i A i = A" unfolding Hermite_of_row_i_def by auto
         then show ?thesis using eA by auto
       qed  
     qed  
   qed
 qed
 
 
 
 lemma Hermite_of_list_of_rows:
  "(Hermite_of_list_of_rows A xs) \<in> carrier_mat (dim_row A) (dim_col A)"
 proof (induct xs arbitrary: A rule: rev_induct)
   case Nil
   then show ?case by auto
 next
   case (snoc x xs)
   let ?A = "(Hermite_of_list_of_rows A xs)"
   have hyp: "(Hermite_of_list_of_rows A xs) \<in> carrier_mat (dim_row A) (dim_col A)" by (rule snoc.hyps)
   have "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i ?A x"
     using Hermite_of_list_of_rows_append by auto
   also have "... \<in> carrier_mat (dim_row ?A) (dim_col ?A)" using Hermite_of_row_i by auto
   finally show ?case using hyp by auto  
 qed
 
 lemma echelon_form_JNF_Hermite_of_list_of_rows:
   assumes "A\<in>carrier_mat m n"
  and "\<forall>x\<in>set xs. x < m"
   and "echelon_form_JNF A"
 shows "echelon_form_JNF (Hermite_of_list_of_rows A xs)"
   using assms
 proof (induct xs arbitrary: A rule: rev_induct)
   case Nil
   then show ?case by auto
 next
   case (snoc x xs)
   have hyp: "echelon_form_JNF (Hermite_of_list_of_rows A xs)"
     by (rule snoc.hyps, insert snoc.prems, auto)
   have H_Axs: "(Hermite_of_list_of_rows A xs) \<in> carrier_mat (dim_row A) (dim_col A)" 
     by (rule Hermite_of_list_of_rows)
   have "(Hermite_of_list_of_rows A (xs @ [x])) = Hermite_of_row_i (Hermite_of_list_of_rows A xs) x"
     using Hermite_of_list_of_rows_append by simp
   also have "echelon_form_JNF ..."
   proof (rule echelon_form_JNF_Hermite_of_row_i[OF hyp])
     show "x < dim_row (Hermite_of_list_of_rows A xs)" using snoc.prems H_Axs by auto
   qed
   finally show ?case .
 qed
 
 
 
 
 lemma HMA_Hermite_of_upt_row_i[transfer_rule]: 
   assumes "xs = [0..<i]"
     and "\<forall>x\<in>set xs. x < CARD('m)"
   assumes "Mod_Type_Connect.HMA_M A (A':: int ^ 'n :: mod_type ^ 'm :: mod_type)" 
     and "echelon_form_JNF A"
   shows "Mod_Type_Connect.HMA_M (Hermite_of_list_of_rows A xs)
   (Hermite.Hermite_of_upt_row_i A' i ass_function_euclidean res_int)"
   using assms
 proof (induct xs arbitrary: A A' i rule: rev_induct)
   case Nil
   have "i=0" using Nil by (metis le_0_eq upt_eq_Nil_conv)
   then show ?case using Nil unfolding Hermite_of_upt_row_i_def by auto
 next
   case (snoc x xs)
   note xs_x_eq = snoc.prems(1)
   note all_xm = snoc.prems(2)
   note AA' = snoc.prems(3)
   note upt_A = snoc.prems(4)
   let ?x' = "(mod_type_class.from_nat x::'m)"
   have xm: "x < CARD('m)" using all_xm by auto
   have xx'[transfer_rule]: "Mod_Type_Connect.HMA_I x ?x'"
     unfolding Mod_Type_Connect.HMA_I_def using from_nat_not_eq xm by blast
   have last_i1: "last [0..<i] = i-1"
     by (metis append_is_Nil_conv last_upt list.simps(3) neq0_conv xs_x_eq upt.simps(1))
   have "last (xs @ [x]) = i-1" using xs_x_eq last_i1 by auto
   hence x_i1: "x = i-1" by auto
   have xs_eq: "xs = [0..<x]" using xs_x_eq x_i1
     by (metis add_diff_inverse_nat append_is_Nil_conv append_same_eq less_one list.simps(3)
         plus_1_eq_Suc upt_Suc upt_eq_Nil_conv)
   have list_rw: "[0..<i] = 0 #[1..<i]" 
     by (auto, metis append_is_Nil_conv list.distinct(2) upt_rec xs_x_eq)
   have 1: "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i (Hermite_of_list_of_rows A xs) x"
     unfolding Hermite_of_list_of_rows_append by auto
   let ?H_upt_HA = "Hermite.Hermite_of_upt_row_i"
   let ?H_HA = "Hermite.Hermite_of_row_i ass_function_euclidean res_int"
   have "(Hermite_of_upt_row_i A' i ass_function_euclidean res_int) = 
     foldl ?H_HA A' (map mod_type_class.from_nat [0..<i])"
     unfolding Hermite_of_upt_row_i_def by auto
   also have "... = foldl ?H_HA A' ((map mod_type_class.from_nat [0..<i-1])@[?x'])"
     by (metis list.simps(8) list.simps(9) map_append x_i1 xs_eq xs_x_eq)
   also have "... = foldl ?H_HA (?H_upt_HA A' (i - 1) ass_function_euclidean res_int) [?x']"
     unfolding foldl_append unfolding Hermite_of_upt_row_i_def[symmetric] by auto
   also have "... = ?H_HA (Hermite_of_upt_row_i A' (i - 1) ass_function_euclidean res_int) ?x'" by auto
   finally have 2: "?H_upt_HA A' i ass_function_euclidean res_int =
     ?H_HA (Hermite_of_upt_row_i A' (i - 1) ass_function_euclidean res_int) ?x'" .
 
   have hyp[transfer_rule]: "Mod_Type_Connect.HMA_M (Hermite_of_list_of_rows A xs)
               (Hermite_of_upt_row_i A' (i - 1) ass_function_euclidean res_int)"
     by (rule snoc.hyps[OF _ _ AA' upt_A], insert xs_eq x_i1 xm, auto)
 
   have upt_H_Axs:"upper_triangular' (Hermite_of_list_of_rows A xs)"
   proof (rule echelon_form_JNF_imp_upper_triangular, 
       rule echelon_form_JNF_Hermite_of_list_of_rows[OF _ _ upt_A])
     show "A\<in>carrier_mat (CARD('m)) (CARD('n))"
       using Mod_Type_Connect.dim_col_transfer_rule
         Mod_Type_Connect.dim_row_transfer_rule snoc(4) by blast
     show "\<forall>x\<in>set xs. x < CARD('m)" using all_xm by auto
   qed
   show ?case unfolding 1 2 
   by (rule HMA_Hermite_of_row_i[OF upt_H_Axs hyp xx'])
 qed
 
 (*This is the lemma that I will transfer to JNF to get the soundness*)
 lemma Hermite_Hermite_of_upt_row_i:
   assumes a: "ass_function ass"
     and r: "res_function res"
     and eA: "echelon_form A"
   shows "Hermite (range ass) (\<lambda>c. range (res c)) (Hermite_of_upt_row_i A (nrows A) ass res)" 
 proof -
   let ?H = "(Hermite_of_upt_row_i A (nrows A) ass res)"  
   show ?thesis
   proof (rule Hermite_intro, auto)
     show "Complete_set_non_associates (range ass)"
       by (simp add: ass_function_Complete_set_non_associates a)
     show "Complete_set_residues (\<lambda>c. range (res c))"
       by (simp add: r res_function_Complete_set_residues)
     show "echelon_form ?H"
       by (rule echelon_form_Hermite_of_upt_row_i[OF eA a r])  
     fix i
     assume i: "\<not> is_zero_row i ?H" 
     show "?H $ i $ (LEAST n. ?H $ i $ n \<noteq> 0) \<in> range ass"
     proof -
       have non_zero_i_eA: "\<not> is_zero_row i A"
         using Hermite_of_upt_row_preserves_zero_rows[OF _ _ a r] i eA by blast   
       have least: "(LEAST n. ?H $h i $h n \<noteq> 0) = (LEAST n. A $h i $h n \<noteq> 0)"
         by (rule Hermite_of_upt_row_i_Least[OF non_zero_i_eA eA a r], simp)
       have "?H $ i $ (LEAST n. A $ i $ n \<noteq> 0) \<in> range ass"
         by (rule Hermite_of_upt_row_i_in_range[OF non_zero_i_eA eA a r], auto)
       thus ?thesis unfolding least by auto
     qed
   next
     fix i j assume i: "\<not> is_zero_row i ?H" and j: "j < i"
     show "?H $ j $ (LEAST n. ?H $ i $ n \<noteq> 0)
     \<in> range (res (?H $ i $ (LEAST n. ?H $ i $ n \<noteq> 0)))"
     proof -
       have non_zero_i_eA: "\<not> is_zero_row i A"
         using Hermite_of_upt_row_preserves_zero_rows[OF _ _ a r] i eA by blast   
       have least: "(LEAST n. ?H $h i $h n \<noteq> 0) = (LEAST n. A $h i $h n \<noteq> 0)"
         by (rule Hermite_of_upt_row_i_Least[OF non_zero_i_eA eA a r], simp)
       have "?H $ j $ (LEAST n. A $ i $ n \<noteq> 0) \<in> range (res (?H $ i $ (LEAST n. A $ i $ n \<noteq> 0)))"
         by (rule Hermite_of_upt_row_i_in_range_res[OF non_zero_i_eA eA a r _ _ j], auto)
       thus ?thesis unfolding least by auto
     qed
   qed
 qed
 
 
 lemma Hermite_of_row_i_0:
   "Hermite_of_row_i A 0 = A \<or> Hermite_of_row_i A 0 = multrow 0 (- 1) A"
     by (cases "find_fst_non0_in_row 0 A", unfold Hermite_of_row_i_def, auto)
 
 lemma Hermite_JNF_intro:
 assumes 
 "Complete_set_non_associates associates" "(Complete_set_residues res)" "echelon_form_JNF A"
  "(\<forall>i<dim_row A. \<not> is_zero_row_JNF i A \<longrightarrow> A $$ (i, LEAST n. A $$ (i, n) \<noteq> 0) \<in> associates)"
  "(\<forall>i<dim_row A. \<not> is_zero_row_JNF i A \<longrightarrow> (\<forall>j. j<i \<longrightarrow> A $$ (j, (LEAST n. A $$ (i, n) \<noteq> 0)) 
      \<in> res (A $$ (i,(LEAST n. A $$ (i,n) \<noteq> 0)))))"
 shows "Hermite_JNF associates res A"
   using assms unfolding Hermite_JNF_def by auto
 
 lemma least_multrow:
   assumes "A \<in> carrier_mat m n" and "i<m" and eA: "echelon_form_JNF A"
   assumes ia: "ia < dim_row A" and nz_ia_mrA: "\<not> is_zero_row_JNF ia (multrow i (- 1) A)"
   shows "(LEAST n. multrow i (- 1) A $$ (ia, n) \<noteq> 0) = (LEAST n. A $$ (ia, n) \<noteq> 0)"
 proof (rule Least_equality)
   have nz_ia_A: "\<not> is_zero_row_JNF ia A" using nz_ia_mrA ia by auto
   have Least_Aian_n: "(LEAST n. A $$ (ia, n) \<noteq> 0) < dim_col A"
-    by (smt dual_order.strict_trans is_zero_row_JNF_def not_less_Least not_less_iff_gr_or_eq nz_ia_A)
+    by (smt (verit) dual_order.strict_trans is_zero_row_JNF_def not_less_Least not_less_iff_gr_or_eq nz_ia_A)
   show "multrow i (- 1) A $$ (ia, LEAST n. A $$ (ia, n) \<noteq> 0) \<noteq> 0"
-    by (smt LeastI Least_Aian_n class_cring.cring_simprules(22) equation_minus_iff ia
+    by (smt (verit) LeastI Least_Aian_n class_cring.cring_simprules(22) equation_minus_iff ia
         index_mat_multrow(1) is_zero_row_JNF_def mult_minus1 nz_ia_A)
   show " \<And>y. multrow i (- 1) A $$ (ia, y) \<noteq> 0 \<Longrightarrow> (LEAST n. A $$ (ia, n) \<noteq> 0) \<le> y"
     by (metis (mono_tags, lifting) Least_Aian_n class_cring.cring_simprules(22) ia 
         index_mat_multrow(1) leI mult_minus1 order.strict_trans wellorder_Least_lemma(2))
 qed
 
 
 lemma Hermite_Hermite_of_row_i:
   assumes A: "A \<in> carrier_mat 1 n"
   shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_of_row_i A 0)"
 proof (rule Hermite_JNF_intro)
   show "Complete_set_non_associates (range ass_function_euclidean)"   
     using ass_function_Complete_set_non_associates ass_function_euclidean by blast
   show "Complete_set_residues (\<lambda>c. range (res_int c))"
     using res_function_Complete_set_residues res_function_res_int by blast
   show "echelon_form_JNF (HNF_Mod_Det_Algorithm.Hermite_of_row_i A 0)"
     by (metis (full_types) assms carrier_matD(1) echelon_form_JNF_Hermite_of_row_i
         echelon_form_JNF_def less_one not_less_zero)
   let ?H = "Hermite_of_row_i A 0"
   show "\<forall>i<dim_row ?H. \<not> is_zero_row_JNF i ?H 
       \<longrightarrow> ?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> range ass_function_euclidean"
   proof (auto)
     fix i assume i: "i<dim_row ?H" and nz_iH: "\<not> is_zero_row_JNF i ?H"
     have nz_iA: "\<not> is_zero_row_JNF i A"
       by (metis (full_types) Hermite_of_row_i Hermite_of_row_i_0 carrier_matD(1)
           i is_zero_row_JNF_multrow nz_iH)
     have "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<ge> 0"
     proof (cases "find_fst_non0_in_row 0 A")
       case None
        then show ?thesis using nz_iH unfolding Hermite_of_row_i_def
-         by (smt HNF_Mod_Det_Algorithm.Hermite_of_row_i_def upper_triangular'_def assms 
+         by (smt (verit) HNF_Mod_Det_Algorithm.Hermite_of_row_i_def upper_triangular'_def assms 
              carrier_matD(1) find_fst_non0_in_row_None i less_one not_less_zero option.simps(4))
     next
       case (Some a)
       have upA: "upper_triangular' A" using A unfolding upper_triangular'_def by auto
       have eA: "echelon_form_JNF A" by (metis A Suc_1 echelon_form_JNF_1xn lessI)
       have i0: "i=0" using Hermite_of_row_i[of A 0] A i by auto        
       have Aia: "A $$ (i,a) \<noteq> 0" and a0: "0 \<le> a" and an: "a<n"
         using i0 Some assms find_fst_non0_in_row by auto+
       have l: "(LEAST n. A $$ (i, n) \<noteq> 0) = (LEAST n.  multrow 0 (- 1) A $$ (i, n) \<noteq> 0)"
         by (rule least_multrow[symmetric, OF A _ eA _], insert nz_iA i A i0, auto)
       have a1: "a = (LEAST n. A $$ (i, n) \<noteq> 0)"
         by (rule find_fst_non0_in_row_LEAST[OF A upA], insert Some i0, auto)
       hence a2: "a = (LEAST n.  multrow 0 (- 1) A $$ (i, n) \<noteq> 0)" unfolding l by simp
       have m1: "multrow 0 (- 1) A $$ (i, LEAST n. multrow 0 (- 1) A $$ (i, n) \<noteq> 0) 
           = (- 1) * A $$ (i, LEAST n. A $$ (i, n) \<noteq> 0)"
         by (metis Hermite_of_row_i_0 a1 a2 an assms carrier_matD(2) i i0 index_mat_multrow(1,4))
       then show ?thesis using nz_iH Some a1 Aia a2 i0 unfolding Hermite_of_row_i_def by auto 
     qed
     thus "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> range ass_function_euclidean"
       using ass_function_int ass_function_int_UNIV by auto
     qed
     show "\<forall>i<dim_row ?H. \<not> is_zero_row_JNF i ?H \<longrightarrow> (\<forall>j<i. ?H $$ (j, LEAST n. ?H $$ (i, n) \<noteq> 0)
     \<in> range (res_int (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0))))"
       using Hermite_of_row_i[of A 0] A by auto
   qed
   
 lemma Hermite_of_row_i_0_eq_0:
   assumes A: "A\<in>carrier_mat m n" and i: "i>0" and eA: "echelon_form_JNF A" and im: "i<m"
     and n0: "0<n"
   shows "Hermite_of_row_i A 0 $$ (i, 0) = 0"
 proof -
   have Ai0: "A $$ (i, 0) = 0" by (rule echelon_form_JNF_first_column_0[OF eA A i im n0]) 
   show ?thesis
   proof (cases "find_fst_non0_in_row 0 A")
     case None  
     thus ?thesis using Ai0 unfolding Hermite_of_row_i_def by auto 
   next
     case (Some a)
     have "A $$ (0, a) \<noteq> 0"  and a0: "0 \<le> a" and an: "a<n"
       using find_fst_non0_in_row[OF A Some] A by auto
     then show ?thesis using Some Ai0 A an a0 im unfolding Hermite_of_row_i_def mat_multrow_def by auto
   qed
 qed
 
   
 
 lemma Hermite_Hermite_of_row_i_mx1:
   assumes A: "A \<in> carrier_mat m 1" and eA: "echelon_form_JNF A"
   shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_of_row_i A 0)"
 proof (rule Hermite_JNF_intro)
   show "Complete_set_non_associates (range ass_function_euclidean)"   
     using ass_function_Complete_set_non_associates ass_function_euclidean by blast
   show "Complete_set_residues (\<lambda>c. range (res_int c))"
     using res_function_Complete_set_residues res_function_res_int by blast
   have H: "Hermite_of_row_i A 0 : carrier_mat m 1" using A Hermite_of_row_i[of A] by auto
   have upA: "upper_triangular' A"
     by (simp add: eA echelon_form_JNF_imp_upper_triangular)
   show eH: "echelon_form_JNF (Hermite_of_row_i A 0)"
   proof (rule echelon_form_JNF_mx1[OF H])
     show "\<forall>i\<in>{1..<m}. HNF_Mod_Det_Algorithm.Hermite_of_row_i A 0 $$ (i, 0) = 0"
       using Hermite_of_row_i_0_eq_0 assms by auto
   qed (simp)
   let ?H = "Hermite_of_row_i A 0"
   show "\<forall>i<dim_row ?H. \<not> is_zero_row_JNF i ?H 
       \<longrightarrow> ?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> range ass_function_euclidean"
   proof (auto)
     fix i assume i: "i<dim_row ?H" and nz_iH: "\<not> is_zero_row_JNF i ?H"
     have nz_iA: "\<not> is_zero_row_JNF i A"
       by (metis (full_types) Hermite_of_row_i Hermite_of_row_i_0 carrier_matD(1)
           i is_zero_row_JNF_multrow nz_iH)    
     have "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<ge> 0"
     proof (cases "find_fst_non0_in_row 0 A")
       case None
       have "is_zero_row_JNF i A"
         by (metis H upper_triangular'_def None assms(1) carrier_matD find_fst_non0_in_row_None
             i is_zero_row_JNF_def less_one linorder_neqE_nat not_less0 upA)
        then show ?thesis using nz_iH None unfolding Hermite_of_row_i_def by auto
      next
       case (Some a)
       have Aia: "A $$ (0,a) \<noteq> 0" and a0: "0 \<le> a" and an: "a<1"
         using find_fst_non0_in_row[OF A Some] A by auto  
       have nz_j_mA: "is_zero_row_JNF j (multrow 0 (- 1) A)" if j0: "j>0" and jm: "j<m" for j 
         unfolding is_zero_row_JNF_def using A j0 jm upA by auto
       show ?thesis
       proof (cases "i=0")
         case True
         then show ?thesis
           using nz_iH Some nz_j_mA A H i Aia an unfolding Hermite_of_row_i_def by auto
       next
         case False
         have nz_iA: "is_zero_row_JNF i A"
           by (metis False H Hermite_of_row_i_0 carrier_matD(1) i is_zero_row_JNF_multrow not_gr0 nz_iH nz_j_mA)
         hence "is_zero_row_JNF i (multrow 0 (- 1) A)" using A H i by auto
         then show ?thesis using nz_iH Some nz_j_mA False nz_iA 
           unfolding Hermite_of_row_i_def by fastforce
       qed
     qed
     thus "?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0) \<in> range ass_function_euclidean"
       using ass_function_int ass_function_int_UNIV by auto
     qed
     show "\<forall>i<dim_row ?H. \<not> is_zero_row_JNF i ?H \<longrightarrow> (\<forall>j<i. ?H $$ (j, LEAST n. ?H $$ (i, n) \<noteq> 0)
     \<in> range (res_int (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0))))"
     proof auto
       fix i j assume i: "i<dim_row ?H" and nz_iH: "\<not> is_zero_row_JNF i ?H" and ji: "j<i"
       have "i=0"
         by (metis H upper_triangular'_def One_nat_def nz_iH eH i carrier_matD(2) nat_neq_iff
             echelon_form_JNF_imp_upper_triangular is_zero_row_JNF_def less_Suc0 not_less_zero)
       thus "?H $$ (j, LEAST n. ?H $$ (i, n) \<noteq> 0)
            \<in> range (res_int (?H $$ (i, LEAST n. ?H $$ (i, n) \<noteq> 0)))" using ji by auto
     qed
 qed
 
 
 lemma Hermite_of_list_of_rows_1xn:
   assumes A: "A \<in> carrier_mat 1 n"
     and eA: "echelon_form_JNF A" 
     and x: "\<forall>x \<in> set xs. x < 1" and xs: "xs\<noteq>[]"
   shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A xs)"
   using x xs
 proof (induct xs rule: rev_induct)
   case Nil
   then show ?case by auto
 next
   case (snoc x xs)
   have x0: "x=0" using snoc.prems by auto
   show ?case 
   proof (cases "xs = []")
     case True    
     have "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i A 0"
       unfolding Hermite_of_list_of_rows_append x0 using True by auto
     then show ?thesis using Hermite_Hermite_of_row_i[OF A] by auto
   next
     case False
     have x0: "x=0" using snoc.prems by auto
     have hyp: "Hermite_JNF (range ass_function_euclidean) 
       (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A xs)"
         by (rule snoc.hyps, insert snoc.prems False, auto)
     have "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i (Hermite_of_list_of_rows A xs) 0"
       unfolding Hermite_of_list_of_rows_append hyp x0 ..
     thus ?thesis
       by (metis A Hermite_Hermite_of_row_i Hermite_of_list_of_rows carrier_matD(1))
   qed
 qed
 
 
 lemma Hermite_of_row_i_id_mx1:
   assumes H': "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) A"
     and x: "x<dim_row A" and A: "A\<in>carrier_mat m 1"
 shows "Hermite_of_row_i A x = A"  
 proof (cases "find_fst_non0_in_row x A")
   case None
   then show ?thesis unfolding Hermite_of_row_i_def by auto
 next
   case (Some a)
   have eH: "echelon_form_JNF A" using H' unfolding Hermite_JNF_def by simp
   have ut_A: "upper_triangular' A" by (simp add: eH echelon_form_JNF_imp_upper_triangular)
   have a_least: "a = (LEAST n. A $$ (x,n) \<noteq> 0)" 
     by (rule find_fst_non0_in_row_LEAST[OF _ ut_A Some], insert x, auto)
   have Axa: "A $$ (x, a) \<noteq> 0" and xa: "x\<le>a" and a: "a<dim_col A"
     using find_fst_non0_in_row[OF A Some] unfolding a_least by auto
   have nz_xA: "\<not> is_zero_row_JNF x A" using Axa xa x a unfolding is_zero_row_JNF_def by blast
   have a0: "a = 0" using a A by auto
   have x0: "x=0" using echelon_form_JNF_first_column_0[OF eH A] Axa a0 xa by blast  
   have "A $$ (x, a) \<in> (range ass_function_euclidean)" 
     using nz_xA H' x unfolding a_least unfolding Hermite_JNF_def by auto
   hence "A $$ (x, a) > 0" using Axa unfolding image_def ass_function_euclidean_def by auto
   then show ?thesis unfolding Hermite_of_row_i_def using Some x0 by auto
 qed
 
 lemma Hermite_of_row_i_id_mx1':
   assumes eA: "echelon_form_JNF A"
     and x: "x<dim_row A" and A: "A\<in>carrier_mat m 1"
 shows "Hermite_of_row_i A x = A \<or> Hermite_of_row_i A x = multrow 0 (- 1) A" 
 proof (cases "find_fst_non0_in_row x A")
   case None
   then show ?thesis unfolding Hermite_of_row_i_def by auto
 next
   case (Some a)
   have ut_A: "upper_triangular' A" by (simp add: eA echelon_form_JNF_imp_upper_triangular)
   have a_least: "a = (LEAST n. A $$ (x,n) \<noteq> 0)" 
     by (rule find_fst_non0_in_row_LEAST[OF _ ut_A Some], insert x, auto)
   have Axa: "A $$ (x, a) \<noteq> 0" and xa: "x\<le>a" and a: "a<dim_col A"
     using find_fst_non0_in_row[OF A Some] unfolding a_least by auto
   have nz_xA: "\<not> is_zero_row_JNF x A" using Axa xa x a unfolding is_zero_row_JNF_def by blast
   have a0: "a = 0" using a A by auto
   have x0: "x=0" using echelon_form_JNF_first_column_0[OF eA A] Axa a0 xa by blast
   show ?thesis by (cases "A $$(x,a)>0", unfold Hermite_of_row_i_def, insert Some x0, auto)
 qed
 
 
 lemma Hermite_of_list_of_rows_mx1:
   assumes A: "A \<in> carrier_mat m 1"
     and eA: "echelon_form_JNF A" 
     and x: "\<forall>x \<in> set xs. x < m" and xs: "xs=[0..<i]" and i: "i>0"
   shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A xs)"
   using x xs i
 proof (induct xs arbitrary: i rule: rev_induct)
   case Nil
   then show ?case by (metis neq0_conv not_less upt_eq_Nil_conv)
 next
   case (snoc x xs)
   note all_n_xs_x = snoc.prems(1)
   note xs_x = snoc.prems(2)
   note i0 = snoc.prems(3)
   have i_list_rw:"[0..<i] = [0..<i-1] @ [i-1]" using i0 less_imp_Suc_add by fastforce
   hence xs: "xs = [0..<i-1]" using xs_x by force
   hence x: "x=i-1" using i_list_rw xs_x by auto
   have H: "Hermite_of_list_of_rows A xs \<in> carrier_mat m 1"
     using A Hermite_of_list_of_rows[of A xs] by auto
   show ?case
   proof (cases "i-1=0")
     case True
     hence xs_empty: "xs = []" using xs by auto
     have *: "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i A 0"
       unfolding Hermite_of_list_of_rows_append xs_empty x True by simp    
     show ?thesis unfolding * by (rule Hermite_Hermite_of_row_i_mx1[OF A eA])
   next
     case False          
     have hyp: "Hermite_JNF (range ass_function_euclidean) 
       (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A xs)"
       by (rule snoc.hyps[OF _ xs], insert False all_n_xs_x, auto)
     have "Hermite_of_list_of_rows A (xs @ [x]) 
         = Hermite_of_row_i (Hermite_of_list_of_rows A xs) x" 
       unfolding Hermite_of_list_of_rows_append ..
     also have "... = (Hermite_of_list_of_rows A xs)"
       by (rule Hermite_of_row_i_id_mx1[OF hyp _ H], insert snoc.prems H x, auto)
     finally show ?thesis using hyp by auto
   qed
 qed
 
 
 
 lemma invertible_Hermite_of_list_of_rows_1xn:
   assumes "A \<in> carrier_mat 1 n"
   shows "\<exists>P. P \<in> carrier_mat 1 1 \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<1] = P * A"
 proof -
   let ?H = "Hermite_of_list_of_rows A [0..<1]"
   have "?H = Hermite_of_row_i A 0" by auto 
   hence H_or: "?H = A \<or> ?H = multrow 0 (- 1) A"
     using Hermite_of_row_i_0 by simp
   show ?thesis
   proof (cases "?H = A")
     case True
     then show ?thesis
       by (metis assms invertible_mat_one left_mult_one_mat one_carrier_mat)
   next
     case False
     hence H_mr: "?H = multrow 0 (- 1) A" using H_or by simp
     let ?M = "multrow_mat 1 0 (-1)::int mat"
     show ?thesis
     proof (rule exI[of _ "?M"])
       have "?M \<in> carrier_mat 1 1" by auto
       moreover have "invertible_mat ?M"
         by (metis calculation det_multrow_mat det_one dvd_mult_right invertible_iff_is_unit_JNF
             invertible_mat_one one_carrier_mat square_eq_1_iff zero_less_one_class.zero_less_one)
       moreover have "?H= ?M * A"
         by (metis H_mr assms multrow_mat)      
       ultimately show "?M \<in> carrier_mat 1 1 \<and> invertible_mat (?M) 
   \<and> Hermite_of_list_of_rows A [0..<1] = ?M * A" by blast
     qed
   qed
 qed
 
 
 
 lemma invertible_Hermite_of_list_of_rows_mx1':
   assumes A: "A \<in> carrier_mat m 1" and eA: "echelon_form_JNF A"
     and xs_i: "xs = [0..<i]" and xs_m: "\<forall>x\<in>set xs. x < m" and i: "i>0"
   shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A xs = P * A"
   using xs_i xs_m i
 proof (induct xs arbitrary: i rule: rev_induct)
   case Nil
   then show ?case by (metis diff_zero length_upt list.size(3) zero_order(3))
 next
   case (snoc x xs)
   note all_n_xs_x = snoc.prems(2)
   note xs_x = snoc.prems(1)
   note i0 = snoc.prems(3)
   have i_list_rw:"[0..<i] = [0..<i-1] @ [i-1]" using i0 less_imp_Suc_add by fastforce
   hence xs: "xs = [0..<i-1]" using xs_x by force
   hence x: "x=i-1" using i_list_rw xs_x by auto
   have H: "Hermite_of_list_of_rows A xs \<in> carrier_mat m 1"
     using A Hermite_of_list_of_rows[of A xs] by auto
   show ?case
   proof (cases "i-1=0")
     case True
     hence xs_empty: "xs = []" using xs by auto
     let ?H = "Hermite_of_list_of_rows A (xs @ [x])"
     have *: "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i A 0"
       unfolding Hermite_of_list_of_rows_append xs_empty x True by simp  
     hence H_or: "?H = A \<or> ?H = multrow 0 (- 1) A" using Hermite_of_row_i_0 by simp
     thus ?thesis
     proof (cases "?H=A")
       case True
       then show ?thesis unfolding * 
         by (metis A invertible_mat_one left_mult_one_mat one_carrier_mat)
     next
       case False
       hence H_mr: "?H = multrow 0 (- 1) A" using H_or by simp    
       let ?M = "multrow_mat m 0 (-1)::int mat"
     show ?thesis 
     proof (rule exI[of _ "?M"])
       have "?M \<in> carrier_mat m m" by auto
       moreover have "invertible_mat ?M"
         by (metis (full_types) det_multrow_mat dvd_mult_right invertible_iff_is_unit_JNF
             invertible_mat_zero more_arith_simps(10) mult_minus1_right multrow_mat_carrier neq0_conv)
       moreover have "?H = ?M * A" unfolding H_mr using A multrow_mat by blast            
       ultimately show "?M \<in> carrier_mat m m \<and> invertible_mat ?M \<and> ?H = ?M * A" by blast
     qed
   qed
   next
     case False
     let ?A = "(Hermite_of_list_of_rows A xs)"
     have A': "?A \<in> carrier_mat m 1" using A Hermite_of_list_of_rows[of A xs] by simp
     have hyp: "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A xs = P * A"
       by (rule snoc.hyps[OF xs], insert False all_n_xs_x, auto)
     have rw: "Hermite_of_list_of_rows A (xs @ [x]) 
         = Hermite_of_row_i (Hermite_of_list_of_rows A xs) x" 
       unfolding Hermite_of_list_of_rows_append ..
     have *: "Hermite_of_row_i ?A x = ?A \<or> Hermite_of_row_i ?A x = multrow 0 (- 1) ?A"
     proof (rule Hermite_of_row_i_id_mx1'[OF _ _ A'])
       show "echelon_form_JNF ?A"
         using A eA echelon_form_JNF_Hermite_of_list_of_rows snoc(3) by auto
       show "x < dim_row ?A" using A' x i A by (simp add: snoc(3))
     qed
     show ?thesis
     proof (cases "Hermite_of_row_i ?A x = ?A")
     case True
       then show ?thesis 
         by (simp add: hyp rw)
     next
       case False
       let ?M = "multrow_mat m 0 (-1)::int mat"
       obtain P where P: "P \<in> carrier_mat m m" 
         and inv_P: "invertible_mat P" and H_PA: "Hermite_of_list_of_rows A xs = P * A"
         using hyp by auto
       have M: "?M \<in> carrier_mat m m" by auto
       have inv_M: "invertible_mat ?M"
         by (metis (full_types) det_multrow_mat dvd_mult_right invertible_iff_is_unit_JNF
             invertible_mat_zero more_arith_simps(10) mult_minus1_right multrow_mat_carrier neq0_conv)
       have H_MA': "Hermite_of_row_i ?A x = ?M * ?A" using False * H multrow_mat by metis
       have inv_MP: "invertible_mat (?M*P)" using M inv_M P inv_P invertible_mult_JNF by blast
       moreover have MP: "(?M*P) \<in> carrier_mat m m" using M P by fastforce
       moreover have "Hermite_of_list_of_rows A (xs @ [x]) = (?M*P) * A"
         by (metis A H_MA' H_PA M P assoc_mult_mat rw)  
       ultimately show ?thesis by blast
     qed  
   qed
 qed
 
 
 corollary invertible_Hermite_of_list_of_rows_mx1:
   assumes "A \<in> carrier_mat m 1" and eA: "echelon_form_JNF A"
   shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<m] = P * A"
 proof (cases "m=0")
   case True
   then show ?thesis 
     by (auto, metis assms(1) invertible_mat_one left_mult_one_mat one_carrier_mat)
 next
   case False
   then show ?thesis using invertible_Hermite_of_list_of_rows_mx1' assms by simp
 qed
   
 
 lemma Hermite_of_list_of_rows_mx0:
   assumes A: "A \<in> carrier_mat m 0"
   and xs: "xs = [0..<i]" and x: "\<forall>x\<in> set xs. x < m"
 shows "Hermite_of_list_of_rows A xs = A"
   using xs x
 proof (induct xs arbitrary: i rule: rev_induct)
   case Nil
   then show ?case by auto
 next
   case (snoc x xs)    
   note all_n_xs_x = snoc.prems(2)
   note xs_x = snoc.prems(1)
   have i0: "i>0" using neq0_conv snoc(2) by fastforce
   have i_list_rw:"[0..<i] = [0..<i-1] @ [i-1]" using i0 less_imp_Suc_add by fastforce
   hence xs: "xs = [0..<i-1]" using xs_x by force
   hence x: "x=i-1" using i_list_rw xs_x by auto
   have H: "Hermite_of_list_of_rows A xs \<in> carrier_mat m 0"
     using A Hermite_of_list_of_rows[of A xs] by auto
   define A' where "A' = (Hermite_of_list_of_rows A xs)"
   have A'A: "A' = A" by (unfold A'_def, rule snoc.hyps, insert snoc.prems xs, auto)
   have "Hermite_of_list_of_rows A (xs @ [x]) = Hermite_of_row_i A' x"
     using Hermite_of_list_of_rows_append A'_def by auto
   also have "... = A"
   proof (cases "find_fst_non0_in_row x A'")
     case None
     then show ?thesis unfolding Hermite_of_row_i_def using A'A by auto
     next
       case (Some a)
     then show ?thesis
       by (metis (full_types) A'A A carrier_matD(2) find_fst_non0_in_row(3) zero_order(3))
   qed
   finally show ?case .
 qed
 
 
 text \<open>Again, we move more lemmas from HOL Analysis (with mod-type restrictions) to the JNF matrix
 representation.\<close>
 
 (*
 The following lemmas will be transferred from HOL Analysis to JNF:
 thm Hermite_Hermite_of_upt_row_i
 thm invertible_Hermite_of_upt_row_i
 *)
 
 context
 begin
 
 private lemma Hermite_Hermite_of_list_of_rows_mod_type:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::mod_type) CARD('n::mod_type)"
   assumes eA: "echelon_form_JNF A"
 shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<CARD('m)])"
 proof -
   define A' where "A' = (Mod_Type_Connect.to_hma\<^sub>m A :: int ^'n :: mod_type ^'m :: mod_type)"
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'"
     unfolding Mod_Type_Connect.HMA_M_def using assms A'_def by auto 
   have eA'[transfer_rule]: "echelon_form A'" using eA by transfer
   have [transfer_rule]: "Mod_Type_Connect.HMA_M (Hermite_of_list_of_rows A [0..<CARD('m)]) 
   (Hermite_of_upt_row_i A' (CARD('m)) ass_function_euclidean res_int)"
     by (rule HMA_Hermite_of_upt_row_i[OF _ _ AA' eA], auto)
   have [transfer_rule]: " (range ass_function_euclidean) =  (range ass_function_euclidean)" ..
   have [transfer_rule]: "(\<lambda>c. range (res_int c)) = (\<lambda>c. range (res_int c))" ..
   have n: "CARD('m) = nrows A'" using AA' unfolding nrows_def by auto
   have "Hermite (range ass_function_euclidean) (\<lambda>c. range (res_int c)) 
   (Hermite_of_upt_row_i A' (CARD('m)) ass_function_euclidean res_int)"
     by (unfold n, rule Hermite_Hermite_of_upt_row_i[OF ass_function_euclidean res_function_res_int eA'])    
   thus ?thesis by transfer
 qed 
 
 private lemma invertible_Hermite_of_list_of_rows_mod_type:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::mod_type) CARD('n::mod_type)"
   assumes eA: "echelon_form_JNF A"
   shows "\<exists>P. P \<in> carrier_mat CARD('m) CARD('m) \<and> 
     invertible_mat P \<and> Hermite_of_list_of_rows A [0..<CARD('m)] = P * A"
 proof -
   define A' where "A' = (Mod_Type_Connect.to_hma\<^sub>m A :: int ^'n :: mod_type ^'m :: mod_type)"
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'"
     unfolding Mod_Type_Connect.HMA_M_def using assms A'_def by auto 
   have eA'[transfer_rule]: "echelon_form A'" using eA by transfer
   have [transfer_rule]: "Mod_Type_Connect.HMA_M (Hermite_of_list_of_rows A [0..<CARD('m)]) 
   (Hermite_of_upt_row_i A' (CARD('m)) ass_function_euclidean res_int)"
     by (rule HMA_Hermite_of_upt_row_i[OF _ _ AA' eA], auto)
   have [transfer_rule]: " (range ass_function_euclidean) =  (range ass_function_euclidean)" ..
   have [transfer_rule]: "(\<lambda>c. range (res_int c)) = (\<lambda>c. range (res_int c))" ..
   have n: "CARD('m) = nrows A'" using AA' unfolding nrows_def by auto
   have "\<exists>P. invertible P \<and> Hermite_of_upt_row_i A' (CARD('m)) ass_function_euclidean res_int 
       = P ** A'" by (rule invertible_Hermite_of_upt_row_i[OF ass_function_euclidean])
   thus ?thesis by (transfer, auto)
 qed
 
 
 private lemma Hermite_Hermite_of_list_of_rows_nontriv_mod_ring:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::nontriv mod_ring) CARD('n::nontriv mod_ring)"
   assumes eA: "echelon_form_JNF A"
 shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<CARD('m)])"
-using assms Hermite_Hermite_of_list_of_rows_mod_type by (smt CARD_mod_ring) 
+using assms Hermite_Hermite_of_list_of_rows_mod_type by (smt (verit) CARD_mod_ring) 
 
 private lemma invertible_Hermite_of_list_of_rows_nontriv_mod_ring:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::nontriv mod_ring) CARD('n::nontriv mod_ring)"
   assumes eA: "echelon_form_JNF A"
   shows "\<exists>P. P \<in> carrier_mat CARD('m) CARD('m) \<and> 
     invertible_mat P \<and> Hermite_of_list_of_rows A [0..<CARD('m)] = P * A"
-using assms invertible_Hermite_of_list_of_rows_mod_type by (smt CARD_mod_ring) 
+using assms invertible_Hermite_of_list_of_rows_mod_type by (smt (verit) CARD_mod_ring) 
 
 
 (*We internalize both sort constraints in one step*)
 lemmas Hermite_Hermite_of_list_of_rows_nontriv_mod_ring_internalized = 
   Hermite_Hermite_of_list_of_rows_nontriv_mod_ring[unfolded CARD_mod_ring, 
       internalize_sort "'m::nontriv", internalize_sort "'b::nontriv"]
 
 lemmas invertible_Hermite_of_list_of_rows_nontriv_mod_ring_internalized = 
   invertible_Hermite_of_list_of_rows_nontriv_mod_ring[unfolded CARD_mod_ring, 
       internalize_sort "'m::nontriv", internalize_sort "'b::nontriv"]
 
 
 context
   fixes m::nat and n::nat
   assumes local_typedef1: "\<exists>(Rep :: ('b \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<m :: int}"
   assumes local_typedef2: "\<exists>(Rep :: ('c \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<n :: int}"
   and m: "m>1"
   and n: "n>1"
 begin
 
 lemma Hermite_Hermite_of_list_of_rows_nontriv_mod_ring_aux:
  fixes A::"int mat"
    assumes "A \<in> carrier_mat m n"
   assumes eA: "echelon_form_JNF A"
 shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<m])"
   using Hermite_Hermite_of_list_of_rows_nontriv_mod_ring_internalized
     [OF type_to_set2(1)[OF local_typedef1 local_typedef2] 
         type_to_set1(1)[OF local_typedef1 local_typedef2]]
   using assms 
   using type_to_set1(2) local_typedef1 local_typedef2 n m by metis 
 
 
 
 lemma invertible_Hermite_of_list_of_rows_nontriv_mod_ring_aux:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat m n"
   assumes eA: "echelon_form_JNF A"
   shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<m] = P * A"
 using invertible_Hermite_of_list_of_rows_nontriv_mod_ring_internalized
     [OF type_to_set2(1)[OF local_typedef1 local_typedef2] 
         type_to_set1(1)[OF local_typedef1 local_typedef2]]
   using assms 
   using type_to_set1(2) local_typedef1 local_typedef2 n m by metis 
 end
 
 
 (*Canceling the first local type definitions*)
 context
 begin
 
 (*Canceling the first*)
 private lemma invertible_Hermite_of_list_of_rows_cancelled_first:
   "\<exists>Rep Abs. type_definition Rep Abs {0..<int n} 
   \<Longrightarrow> 1 < m \<Longrightarrow> 1 < n \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> echelon_form_JNF A 
   \<Longrightarrow> \<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<m] = P * A"
   using invertible_Hermite_of_list_of_rows_nontriv_mod_ring_aux[cancel_type_definition, of m n A]
   by auto  
 
 (*Canceling the second*)
 private lemma invertible_Hermite_of_list_of_rows_cancelled_both:
   "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> echelon_form_JNF A 
   \<Longrightarrow> \<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<m] = P * A"
   using invertible_Hermite_of_list_of_rows_cancelled_first[cancel_type_definition, of n m A] by simp
 
 
 (*Canceling the first*)
 
 private lemma Hermite_Hermite_of_list_of_rows_cancelled_first:
 "\<exists>Rep Abs. type_definition Rep Abs {0..<int n} \<Longrightarrow>
   1 < m \<Longrightarrow>
   1 < n \<Longrightarrow>
   A \<in> carrier_mat m n \<Longrightarrow>
   echelon_form_JNF A 
   \<Longrightarrow> Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<m])"
   using Hermite_Hermite_of_list_of_rows_nontriv_mod_ring_aux[cancel_type_definition, of m n A]
   by auto  
 
 (*Canceling the second*)
 private lemma Hermite_Hermite_of_list_of_rows_cancelled_both:
 "1 < m \<Longrightarrow>
   1 < n \<Longrightarrow>
   A \<in> carrier_mat m n \<Longrightarrow>
   echelon_form_JNF A 
   \<Longrightarrow> Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<m])"
   using Hermite_Hermite_of_list_of_rows_cancelled_first[cancel_type_definition, of n m A] by simp
 
 
 (*The final results in JNF*)
 
 lemma Hermite_Hermite_of_list_of_rows':
  fixes A::"int mat"
   assumes "A \<in> carrier_mat m n"
     and "echelon_form_JNF A"  
   and "1 < m" and "1 < n" (*Required from the mod_type restrictions*)
 shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<m])"  
   using Hermite_Hermite_of_list_of_rows_cancelled_both assms by auto
 
 corollary Hermite_Hermite_of_list_of_rows:
  fixes A::"int mat"
   assumes A: "A \<in> carrier_mat m n"
     and eA: "echelon_form_JNF A"  
 shows "Hermite_JNF (range ass_function_euclidean) 
   (\<lambda>c. range (res_int c)) (Hermite_of_list_of_rows A [0..<m])"
 proof (cases "m=0 \<or> n=0")
   case True
   then show ?thesis 
     by (auto, metis Hermite_Hermite_of_row_i Hermite_JNF_def A eA carrier_matD(1) one_carrier_mat zero_order(3))   
      (metis Hermite_Hermite_of_row_i Hermite_JNF_def Hermite_of_list_of_rows A carrier_matD(2) 
        echelon_form_mx0 is_zero_row_JNF_def mat_carrier zero_order(3))
 next
   case False note not_m0_or_n0 = False
   show ?thesis
   proof (cases "m=1 \<or> n=1")
     case True
     then show ?thesis
       by (metis False Hermite_of_list_of_rows_1xn Hermite_of_list_of_rows_mx1 A eA 
           atLeastLessThan_iff linorder_not_less neq0_conv set_upt upt_eq_Nil_conv)
   next
     case False
     show ?thesis
       by (rule Hermite_Hermite_of_list_of_rows'[OF A eA], insert not_m0_or_n0 False, auto)
   qed
 qed
 
 lemma invertible_Hermite_of_list_of_rows:
   assumes A: "A \<in> carrier_mat m n"
   and eA: "echelon_form_JNF A"
 shows "\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> Hermite_of_list_of_rows A [0..<m] = P * A"
 proof (cases "m=0 \<or> n=0")
   case True
   have *: "Hermite_of_list_of_rows A [0..<m] = A" if n: "n=0"
     by (rule Hermite_of_list_of_rows_mx0, insert A n, auto)
   show ?thesis using True
     by (auto, metis assms(1) invertible_mat_one left_mult_one_mat one_carrier_mat)
        (metis (full_types) "*" assms(1) invertible_mat_one left_mult_one_mat one_carrier_mat)
 next
   case False note mn = False
   show ?thesis
   proof (cases "m=1 \<or> n=1")
     case True
     then show ?thesis 
       using A eA invertible_Hermite_of_list_of_rows_1xn invertible_Hermite_of_list_of_rows_mx1 by blast
   next
     case False
     then show ?thesis 
       using invertible_Hermite_of_list_of_rows_cancelled_both[OF _ _ A eA] False mn by auto
   qed  
 qed
 end
 end
 end
 end
 
 
 text \<open>Now we have all the required stuff to prove the soundness of the algorithm.\<close>
 
 context proper_mod_operation
 begin
 
 (*
 thm invertible_Hermite_of_list_of_rows
 thm Hermite_Hermite_of_list_of_rows
 thm LLL_with_assms.Hermite_append_det_id
 thm FindPreHNF_invertible_mat
 thm FindPreHNF_echelon_form
 *)
 
 lemma Hermite_mod_det_mx0:
   assumes "A \<in> carrier_mat m 0"
   shows "Hermite_mod_det abs_flag A = A"
   unfolding Hermite_mod_det_def Let_def using assms by auto
 
 lemma Hermite_JNF_mx0:
   assumes A: "A \<in> carrier_mat m 0"
   shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) A"
   unfolding Hermite_JNF_def using A echelon_form_mx0 unfolding is_zero_row_JNF_def 
   using ass_function_Complete_set_non_associates[OF ass_function_euclidean]
   using res_function_Complete_set_residues[OF res_function_res_int] by auto
   
 
 lemma Hermite_mod_det_soundness_mx0:
   assumes  A: "A \<in> carrier_mat m n"
   and n0: "n=0"
 shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_mod_det abs_flag A)" 
 and "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> (Hermite_mod_det abs_flag A) = P * A)"
 proof -
   have A: "A \<in> carrier_mat m 0" using A n0 by blast
   then show "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_mod_det abs_flag A)"
     using Hermite_JNF_mx0[OF A] Hermite_mod_det_mx0[OF A] by auto
   show "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> (Hermite_mod_det abs_flag A) = P * A)"
     by (metis A Hermite_mod_det_mx0 invertible_mat_one left_mult_one_mat one_carrier_mat)
 qed
 
 
 lemma Hermite_mod_det_soundness_mxn:
   assumes mn: "m = n"
   and A: "A \<in> carrier_mat m n"
   and n0: "0<n"
   and inv_RAT_A: "invertible_mat (map_mat rat_of_int A)"
 shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_mod_det abs_flag A)" 
   and "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> (Hermite_mod_det abs_flag A) = P * A)"
 proof -
   define D A' E H H' where D_def: "D = \<bar>Determinant.det A\<bar>"
   and A'_def: "A' = A @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n" and E_def: "E = FindPreHNF abs_flag D A'"
   and H_def: "H = Hermite_of_list_of_rows E [0..<m+n]"
   and H'_def: "H' = mat_of_rows n (map (Matrix.row H) [0..<m])"
   have A': "A' \<in> carrier_mat (m+n) n" using A A A'_def by auto
   let ?RAT = "of_int_hom.mat_hom :: int mat \<Rightarrow> rat mat"
   have RAT_A: "?RAT A \<in> carrier_mat n n"
     using A map_carrier_mat mat_of_rows_carrier(1) mn by auto 
   have det_RAT_fs_init: "det (?RAT A) \<noteq> 0"
     using inv_RAT_A unfolding invertible_iff_is_unit_JNF[OF RAT_A] by auto
   moreover have "mat_of_rows n (map (Matrix.row A') [0..<n]) = A"
   proof
     let ?A' = "mat_of_rows n (map (Matrix.row A') [0..<n])"    
     show dr: "dim_row ?A' = dim_row A" and dc: "dim_col ?A' = dim_col A" using A mn by auto
     fix i j assume i: "i < dim_row A" and j: "j < dim_col A"
     have D: "D \<cdot>\<^sub>m 1\<^sub>m n \<in> carrier_mat n n" using mn by auto
     have "?A' $$ (i,j) =  (map (Matrix.row A') [0..<n]) ! i $v j"
       by (rule mat_of_rows_index, insert i j dr dc A, auto) 
     also have "... = A' $$ (i,j)" using A' mn i j A by auto
     also have "... = A $$ (i,j)" unfolding A'_def using i append_rows_nth[OF A D] mn j A by auto
     finally show "?A' $$ (i, j) = A $$ (i, j)" .
   qed
   ultimately have inv_RAT_A'n: 
     "invertible_mat (map_mat rat_of_int (mat_of_rows n (map (Matrix.row A') [0..<n])))" 
     using inv_RAT_A by auto
   have eE: "echelon_form_JNF E"
     by (unfold E_def, rule FindPreHNF_echelon_form[OF A'_def A _ _],
         insert mn D_def det_RAT_fs_init, auto)
   have E: "E \<in> carrier_mat (m+n) n" unfolding E_def by (rule FindPreHNF[OF A'])
   have "\<exists>P. P \<in> carrier_mat (m + n) (m + n) \<and> invertible_mat P \<and> E = P * A'"
   by (unfold E_def, rule FindPreHNF_invertible_mat[OF A'_def A n0 _ _],
       insert mn D_def det_RAT_fs_init, auto)
   from this obtain P where P: "P \<in> carrier_mat (m + n) (m + n)"
     and inv_P: "invertible_mat P" and E_PA': "E = P * A'"
     by blast
   have "\<exists>Q. Q \<in> carrier_mat (m+n) (m+n) \<and> invertible_mat Q \<and> H = Q * E"
     by (unfold H_def, rule invertible_Hermite_of_list_of_rows[OF E eE])  
   from this obtain Q where Q: "Q \<in> carrier_mat (m+n) (m+n)"
     and inv_Q: "invertible_mat Q" and H_QE: "H = Q * E" by blast
   let ?ass ="(range ass_function_euclidean)"
   let ?res = "(\<lambda>c. range (res_int c))"
   have Hermite_H: "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) H"
     by (unfold H_def, rule Hermite_Hermite_of_list_of_rows[OF E eE])
   hence eH: "echelon_form_JNF H" unfolding Hermite_JNF_def by auto
   have H': "H' \<in> carrier_mat m n" using H'_def by auto
   have H_H'0: "H = H' @\<^sub>r 0\<^sub>m m n"
   proof (unfold H'_def, rule upper_triangular_append_zero)
     show "upper_triangular' H" using eH by (rule echelon_form_JNF_imp_upper_triangular)
     show "H \<in> carrier_mat (m + m) n"
       unfolding H_def using Hermite_of_list_of_rows[of E] E mn by auto
   qed (insert mn, simp)
   obtain P' where PP': "inverts_mat P P'"
     and P'P: "inverts_mat P' P" and P': "P' \<in> carrier_mat (m+n) (m+n)"         
     using P inv_P obtain_inverse_matrix by blast
   obtain Q' where QQ': "inverts_mat Q Q'"
     and Q'Q: "inverts_mat Q' Q" and Q': "Q' \<in> carrier_mat (m+n) (m+n)"         
     using Q inv_Q obtain_inverse_matrix by blast
   have P'Q': "(P'*Q') \<in> carrier_mat (m + m) (m + m)" using P' Q' mn by simp
   have A'_P'Q'H: "A' = P' * Q' * H"
   proof -
     have QP: "Q * P \<in> carrier_mat (m + m) (m + m)" using Q P mn by auto
     have "H = Q * (P * A')" using H_QE E_PA' by auto
     also have "... = (Q * P) * A'" using A' P Q by auto
     also have "(P' * Q') * ... = ((P' * Q') * (Q * P)) * A'" using A' P'Q' QP mn by auto
     also have "... =  (P' * (Q' * Q) * P) * A'"
-      by (smt P P' P'Q' Q Q' assms(1) assoc_mult_mat)
+      by (smt (verit) P P' P'Q' Q Q' assms(1) assoc_mult_mat)
     also have "... = (P'*P) * A'" 
       by (metis P' Q' Q'Q carrier_matD(1) inverts_mat_def right_mult_one_mat)
     also have "... = A'"
       by (metis A' P' P'P carrier_matD(1) inverts_mat_def left_mult_one_mat)
     finally show "A' = P' * Q' * H" ..
   qed
   have inv_P'Q': "invertible_mat (P' * Q')"
     by (metis P' P'P PP' Q' Q'Q QQ' carrier_matD(1) carrier_matD(2) invertible_mat_def
         invertible_mult_JNF square_mat.simps)
   interpret vec_module "TYPE(int)" .
   interpret B: cof_vec_space n "TYPE(rat)" .
   interpret A: LLL_with_assms n m "(Matrix.rows A)" "4/3"
   proof       
     show "length (rows A) = m " using A unfolding Matrix.rows_def by simp
     have s: "set (map of_int_hom.vec_hom (rows A)) \<subseteq> carrier_vec n"
       using A unfolding Matrix.rows_def by auto
     have rw: "(map of_int_hom.vec_hom (rows A)) = (rows (?RAT A))" 
       by (metis A s carrier_matD(2) mat_of_rows_map mat_of_rows_rows rows_mat_of_rows set_rows_carrier subsetI)
     have "B.lin_indpt (set (map of_int_hom.vec_hom (rows A)))"
       unfolding rw by (rule B.det_not_0_imp_lin_indpt_rows[OF RAT_A det_RAT_fs_init])
     moreover have "distinct (map of_int_hom.vec_hom (rows A)::rat Matrix.vec list)"
     proof (rule ccontr)
       assume " \<not> distinct (map of_int_hom.vec_hom (rows A)::rat Matrix.vec list)"
       from this obtain i j where "row (?RAT A) i = row (?RAT A) j" and "i \<noteq> j" and "i < n" and "j < n"
         unfolding rw
         by (metis Determinant.det_transpose RAT_A add_0 cols_transpose det_RAT_fs_init 
             not_add_less2 transpose_carrier_mat vec_space.det_rank_iff vec_space.non_distinct_low_rank)
       thus False using Determinant.det_identical_rows[OF RAT_A] using det_RAT_fs_init RAT_A by auto      
     qed      
     ultimately show "B.lin_indpt_list (map of_int_hom.vec_hom (rows A))"
       using s unfolding B.lin_indpt_list_def by auto
   qed (simp)
   have A_eq: "mat_of_rows n (Matrix.rows A) = A" using A mat_of_rows_rows by blast
   have D_A: "D = \<bar>det (mat_of_rows n (rows A))\<bar>" using D_def A_eq by auto
   have Hermite_H': "Hermite_JNF ?ass ?res H'"
     by (rule A.Hermite_append_det_id(1)[OF _ mn _ H' H_H'0 P'Q' inv_P'Q' A'_P'Q'H Hermite_H],
          insert D_def A'_def mn A inv_RAT_A D_A A_eq, auto) 
   have dc: "dim_row A = m" and dr: "dim_col A = n" using A by auto
   have Hermite_mod_det_H': "Hermite_mod_det abs_flag A = H'" 
     unfolding Hermite_mod_det_def Let_def H'_def H_def E_def A'_def D_def dc dr det_int by blast
   show "Hermite_JNF ?ass ?res (Hermite_mod_det abs_flag A)" using Hermite_mod_det_H' Hermite_H' by simp
   have "\<exists>R. invertible_mat R \<and> R \<in> carrier_mat m m \<and> A = R * H'"
     by (subst A_eq[symmetric], 
         rule A.Hermite_append_det_id(2)[OF _ mn _ H' H_H'0 P'Q' inv_P'Q' A'_P'Q'H Hermite_H],
         insert D_def A'_def mn A inv_RAT_A D_A A_eq, auto)
   from this obtain R where inv_R: "invertible_mat R" 
     and R: "R \<in> carrier_mat m m" and A_RH': "A = R * H'"
     by blast
   obtain R' where inverts_R: "inverts_mat R R'" and R': "R' \<in> carrier_mat m m"    
     by (meson R inv_R obtain_inverse_matrix)
   have inv_R': "invertible_mat R'" using inverts_R unfolding invertible_mat_def inverts_mat_def
     using R R' mat_mult_left_right_inverse by auto
   moreover have "H' = R' * A"
   proof -
     have "R' * A = R' * (R * H')" using A_RH' by auto
     also have "... = (R'*R) * H'" using H' R R' by auto
     also have "... = H'"
       by (metis H' R R' mat_mult_left_right_inverse carrier_matD(1) 
           inverts_R inverts_mat_def left_mult_one_mat)
     finally show ?thesis ..
   qed
   ultimately show "\<exists>S. invertible_mat S \<and> S \<in> carrier_mat m m \<and> Hermite_mod_det abs_flag A = S * A" 
     using R' Hermite_mod_det_H' by blast
 qed
 
 
 lemma Hermite_mod_det_soundness:
   assumes mn: "m = n"
   and A_def: "A \<in> carrier_mat m n"
   and i: "invertible_mat (map_mat rat_of_int A)"
 shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (Hermite_mod_det abs_flag A)" 
   and "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> (Hermite_mod_det abs_flag A) = P * A)" 
   using A_def Hermite_mod_det_soundness_mx0(1) Hermite_mod_det_soundness_mxn(1) mn i 
   by blast (insert Hermite_mod_det_soundness_mx0(2) Hermite_mod_det_soundness_mxn(2) assms, blast)
 
 
 text \<open>We can even move the whole echelon form algorithm @{text "echelon_form_of"} from HOL Analysis 
 to JNF and then we can combine it with @{text "Hermite_of_list_of_rows"} to have another 
 HNF algorithm which is not efficient, but valid for arbitrary matrices.\<close>
 
 lemma reduce_D0:
 "reduce a b 0 A = (let Aaj = A$$(a,0); Abj = A $$ (b,0)     
   in
   if Aaj = 0 then A else
   case euclid_ext2 Aaj Abj of (p,q,u,v,d) \<Rightarrow> 
        Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then  (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then  u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             )
   )" (is "?lhs = ?rhs")
 proof 
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A $$ (a, 0)) (A $$ (b, 0))"
     by (simp add: euclid_ext2_def) 
   have *:" Matrix.mat (dim_row A) (dim_col A)
         (\<lambda>(i, k).
             if i = a then let r = p * A $$ (a, k) + q * A $$ (b, k) in if 0 < \<bar>r\<bar> then 
                                   if k = 0 \<and> 0 dvd r then 0 else r mod 0 else r
             else if i = b then let r = u * A $$ (a, k) + v * A $$ (b, k) in 
                               if 0 < \<bar>r\<bar> then r mod 0 else r else A $$ (i, k)) 
         = Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then  (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then  u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k) 
             )" 
     by (rule eq_matI, auto simp add: Let_def)
   show "dim_row ?lhs = dim_row ?rhs" 
-    unfolding reduce.simps Let_def by (smt dim_row_mat(1) pquvd prod.simps(2))
+    unfolding reduce.simps Let_def by (smt (verit) dim_row_mat(1) pquvd prod.simps(2))
   show "dim_col ?lhs = dim_col ?rhs"
-    unfolding reduce.simps Let_def by (smt dim_col_mat(1) pquvd prod.simps(2))
+    unfolding reduce.simps Let_def by (smt (verit) dim_col_mat(1) pquvd prod.simps(2))
   fix i j assume i: "i<dim_row ?rhs" and j: "j<dim_col ?rhs"
   show "?lhs $$ (i,j) = ?rhs $$ (i,j)"
     by (cases " A $$ (a, 0) = 0", insert * pquvd i j, auto simp add: case_prod_beta Let_def)
 qed
 
 
 
 lemma bezout_matrix_JNF_mult_eq':
   assumes A': "A' \<in> carrier_mat m n" and a: "a<m"  and b: "b<m" and ab: "a \<noteq> b" 
   and A_def: "A = A' @\<^sub>r B" and B: "B \<in> carrier_mat t n"
   assumes pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,j)) (A$$(b,j))"
   shows "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             ) = (bezout_matrix_JNF A a b j euclid_ext2) * A" (is "?A = ?BM * A")
 proof (rule eq_matI) 
   have A: "A \<in> carrier_mat (m+t) n" using A_def A' B by simp
   hence A_carrier: "?A \<in> carrier_mat (m+t) n" by auto  
   show dr: "dim_row ?A = dim_row (?BM * A)" and dc: "dim_col ?A = dim_col (?BM * A)"
     unfolding bezout_matrix_JNF_def by auto
   fix i ja assume i: "i < dim_row  (?BM * A)" and ja: "ja < dim_col (?BM * A)"
   let ?f = "\<lambda>ia. (bezout_matrix_JNF A a b j euclid_ext2) $$ (i,ia) * A $$ (ia,ja)"
   have dv: "dim_vec (col A ja) = m+t" using A by auto
   have i_dr: "i<dim_row A" using i A unfolding bezout_matrix_JNF_def by auto
   have a_dr: "a<dim_row A" using A a ja by auto
   have b_dr: "b<dim_row A" using A b ja by auto
   show "?A $$ (i,ja) = (?BM * A) $$ (i,ja)"
   proof -
     have "(?BM * A) $$ (i,ja) = Matrix.row ?BM i \<bullet> col A ja"
       by (rule index_mult_mat, insert i ja, auto)
     also have "... = (\<Sum>ia = 0..<dim_vec (col A ja). 
           Matrix.row (bezout_matrix_JNF A a b j euclid_ext2) i $v ia * col A ja $v ia)"
       by (simp add: scalar_prod_def)
     also have "... = (\<Sum>ia = 0..<m+t. ?f ia)"
-      by (rule sum.cong, insert A i dr dc, auto) (smt bezout_matrix_JNF_def carrier_matD(1) 
+      by (rule sum.cong, insert A i dr dc, auto) (smt (verit) bezout_matrix_JNF_def carrier_matD(1) 
           dim_col_mat(1) index_col index_mult_mat(3) index_row(1) ja)
     also have "... = (\<Sum>ia \<in> ({a,b} \<union> ({0..<m+t} - {a,b})). ?f ia)"
       by (rule sum.cong, insert a a_dr b A ja, auto)
     also have "... = sum ?f {a,b} + sum ?f ({0..<m+t} - {a,b})" 
       by (rule sum.union_disjoint, auto)
     finally have BM_A_ija_eq: "(?BM * A) $$ (i,ja) = sum ?f {a,b} + sum ?f ({0..<m+t} - {a,b})" by auto
     show ?thesis
     proof (cases "i = a")
       case True
       have sum0: "sum ?f ({0..<m+t} - {a,b}) = 0"
       proof (rule sum.neutral, rule)
         fix x assume x: "x \<in> {0..<m + t} - {a, b}"
         hence xm: "x < m+t" by auto
         have x_not_i: "x \<noteq> i" using True x by blast
         have x_dr: "x < dim_row A" using x A by auto
         have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
           unfolding bezout_matrix_JNF_def 
           unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
         thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
       qed
       have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = p" 
         unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using True pquvd 
         by (auto, metis split_conv)
       have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = q"
         unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using True pquvd ab
         by (auto, metis split_conv)
       have "sum ?f {a,b} + sum ?f ({0..<m+t} - {a,b}) = ?f a + ?f b" using sum0 by (simp add: ab)
       also have "... = p * A $$ (a, ja) + q * A $$ (b, ja)" unfolding fa fb by simp
       also have "... = ?A $$ (i,ja)" using A True dr i ja by auto
       finally show ?thesis using BM_A_ija_eq by simp
     next
       case False note i_not_a = False
       show ?thesis
       proof (cases "i=b")
         case True
         have sum0: "sum ?f ({0..<m+t} - {a,b}) = 0"
         proof (rule sum.neutral, rule)
           fix x assume x: "x \<in> {0..<m + t} - {a, b}"
           hence xm: "x < m+t" by auto
           have x_not_i: "x \<noteq> i" using True x by blast
           have x_dr: "x < dim_row A" using x A by auto
           have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
             unfolding bezout_matrix_JNF_def 
             unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
           thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
         qed
         have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = u" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using True i_not_a pquvd 
           by (auto, metis split_conv)
         have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = v"
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using True i_not_a pquvd ab
           by (auto, metis split_conv)
         have "sum ?f {a,b} + sum ?f ({0..<m+t} - {a,b}) = ?f a + ?f b" using sum0 by (simp add: ab)
         also have "... = u * A $$ (a, ja) + v * A $$ (b, ja)" unfolding fa fb by simp
         also have "... = ?A $$ (i,ja)" using A True i_not_a dr i ja by auto
         finally show ?thesis using BM_A_ija_eq by simp
       next
         case False note i_not_b = False
         have sum0: "sum ?f ({0..<m+t} - {a,b} - {i}) = 0" 
         proof (rule sum.neutral, rule)
           fix x assume x: "x \<in> {0..<m + t} - {a, b} - {i}"
           hence xm: "x < m+t" by auto
           have x_not_i: "x \<noteq> i" using x by blast
           have x_dr: "x < dim_row A" using x A by auto
           have "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) = 0"
             unfolding bezout_matrix_JNF_def 
             unfolding index_mat(1)[OF i_dr x_dr] using x_not_i x by auto
           thus "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, x) * A $$ (x, ja) = 0" by auto
         qed
         have fa: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, a) = 0" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr a_dr] using False i_not_a pquvd 
           by auto
         have fb: "bezout_matrix_JNF A a b j euclid_ext2 $$ (i, b) = 0" 
           unfolding bezout_matrix_JNF_def index_mat(1)[OF i_dr b_dr] using False i_not_a pquvd 
           by auto
         have "sum ?f ({0..<m+t} - {a,b}) = sum ?f (insert i ({0..<m+t} - {a,b} - {i}))"
           by (rule sum.cong, insert i_dr A i_not_a i_not_b, auto)
         also have "... = ?f i + sum ?f ({0..<m+t} - {a,b} - {i})" by (rule sum.insert, auto)
         also have "... = ?f i" using sum0 by simp
         also have "... = ?A $$ (i,ja)"
           unfolding bezout_matrix_JNF_def using i_not_a i_not_b  A dr i ja by fastforce
         finally show ?thesis unfolding BM_A_ija_eq by (simp add: ab fa fb)
       qed    
     qed
   qed
 qed
 
 
 
 lemma bezout_matrix_JNF_mult_eq2:
   assumes A: "A \<in> carrier_mat m n" and a: "a<m"  and b: "b<m" and ab: "a \<noteq> b" 
   assumes pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,j)) (A$$(b,j))"
   shows "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k)
             ) = (bezout_matrix_JNF A a b j euclid_ext2) * A" (is "?A = ?BM * A")
 proof (rule bezout_matrix_JNF_mult_eq'[OF A a b ab _ _ pquvd])
   show "A = A @\<^sub>r (0\<^sub>m 0 n)" by (rule eq_matI, unfold append_rows_def, auto)
   show "(0\<^sub>m 0 n) \<in> carrier_mat 0 n" by auto
 qed
 
 
 lemma reduce_invertible_mat_D0_BM:
   assumes A: "A \<in> carrier_mat m n"
   and a: "a < m"
   and b: "b < m"
   and ab: "a \<noteq> b"
   and Aa0: "A$$(a,0) \<noteq> 0"
   shows "reduce a b 0 A = (bezout_matrix_JNF A a b 0 euclid_ext2) * A"
 proof -
  obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(b,0))"
     by (simp add: euclid_ext2_def)
   let ?BM = "bezout_matrix_JNF A a b 0 euclid_ext2"
   let ?A = "Matrix.mat (dim_row A) (dim_col A)
           (\<lambda>(i,k). if i = a then  (p*A$$(a,k) + q*A$$(b,k))
                    else if i = b then  u * A$$(a,k) + v * A$$(b,k)
                    else A$$(i,k))"
   have A'_BZ_A: "?A = ?BM * A"
     by (rule bezout_matrix_JNF_mult_eq2[OF A _ _ ab pquvd], insert a b, auto)  
   moreover have "?A = reduce a b 0 A" using pquvd Aa0 unfolding reduce_D0 Let_def
     by (metis (no_types, lifting) split_conv)
   ultimately show ?thesis by simp
 qed
 
 
 lemma reduce_invertible_mat_D0:
   assumes A: "A \<in> carrier_mat m n"
   and a: "a < m"
   and b: "b < m"
   and n0: "0<n"
   and ab: "a \<noteq> b"
   and a_less_b: "a<b"
   shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> reduce a b 0 A = P * A"
 proof (cases "A$$(a,0) = 0")
   case True
   then show ?thesis
-    by (smt A invertible_mat_one left_mult_one_mat one_carrier_mat reduce.simps)
+    by (smt (verit) A invertible_mat_one left_mult_one_mat one_carrier_mat reduce.simps)
 next
   case False
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(b,0))"
     by (simp add: euclid_ext2_def)
   let ?BM = "bezout_matrix_JNF A a b 0 euclid_ext2"
   have "reduce a b 0 A = ?BM * A"  by (rule reduce_invertible_mat_D0_BM[OF A a b ab False])
   moreover have invertible_bezout: "invertible_mat ?BM"
     by (rule invertible_bezout_matrix_JNF[OF A is_bezout_ext_euclid_ext2 a_less_b _ n0 False],
         insert a_less_b b, auto)      
   moreover have BM: "?BM \<in> carrier_mat m m" unfolding bezout_matrix_JNF_def using A by auto
   ultimately show ?thesis by blast
 qed
 
 lemma reduce_below_invertible_mat_D0:
   assumes A': "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D=0"
 shows "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> reduce_below a xs D A = P * A)"
   using assms
 proof (induct a xs D A arbitrary: A rule: reduce_below.induct)
   case (1 a D A)
   then show ?case
     by (auto, metis invertible_mat_one left_mult_one_mat one_carrier_mat)
 next
   case (2 a x xs D A)
   note A = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note d = "2.prems"(4)
   note x_xs = "2.prems"(5)
   note D0 = "2.prems"(6)
   have xm: "x < m" using "2.prems" by auto
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat m n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have h: "(\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m
     \<and> reduce_below a xs D (reduce a x D A) = P * reduce a x D A)"
     by (rule "2.hyps"[OF _ a j _ _ ],insert  d x_xs  D0 reduce_ax, auto)   
   from this obtain P where inv_P: "invertible_mat P" and P: "P \<in> carrier_mat m m"
     and rb_Pr: "reduce_below a xs D (reduce a x D A) = P * reduce a x D A" by blast
   have *: "reduce_below a (x # xs) D A = reduce_below a xs D (reduce a x D A)" by simp
   have "\<exists>Q. invertible_mat Q \<and> Q \<in> carrier_mat m m \<and> (reduce a x D A) = Q * A"
     by (unfold D0, rule reduce_invertible_mat_D0[OF A a xm j], insert "2.prems", auto)
   from this obtain Q where inv_Q: "invertible_mat Q" and Q: "Q \<in> carrier_mat m m"
     and r_QA: "reduce a x D A = Q * A" by blast
   have "invertible_mat (P*Q)" using inv_P inv_Q P Q invertible_mult_JNF by blast
   moreover have "P * Q \<in> carrier_mat m m" using P Q by auto
   moreover have "reduce_below a (x # xs) D A = (P*Q) * A" 
-    by (smt P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
+    by (smt (verit) P Q * assoc_mult_mat carrier_matD(1) carrier_mat_triv index_mult_mat(2) 
         r_QA rb_Pr reduce_preserves_dimensions(1))
   ultimately show ?case by blast
 qed
 
 
 (*This lemma permits to get rid of one assumption in reduce_not0*)
 lemma reduce_not0':
   assumes A: "A \<in> carrier_mat m n" and a: "a<m" and a_less_b: "a<b" and j: "0<n" and b: "b<m"
     and Aaj: "A $$ (a,0) \<noteq> 0"
   shows "reduce a b 0 A $$ (a, 0) \<noteq> 0" (is "?reduce_ab $$ (a,0) \<noteq> _")
 proof -
   have "?reduce_ab $$ (a,0) = (let r = gcd (A $$ (a, 0)) (A $$ (b, 0)) in if 0 dvd r then 0 else r)" 
     by (rule reduce_gcd[OF A _ j Aaj], insert a, simp)
   also have "... \<noteq> 0" unfolding Let_def
     by (simp add: assms(6))
   finally show ?thesis .
 qed
 
 
 lemma reduce_below_preserves_D0:
   assumes A': "A \<in> carrier_mat m n" and a: "a<m" and j: "j<n"
     and Aaj: "A $$ (a,0) \<noteq> 0"
   assumes "i \<notin> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
     and "i\<noteq>a" and "i<m"
   and "D=0"
   shows "reduce_below a xs D A $$ (i,j) = A $$ (i,j)"
   using assms
 proof (induct a xs D A arbitrary: A i rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto
 next
   case (2 a x xs D A)
   note A = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note Aaj = "2.prems"(4)
   note i_set_xxs = "2.prems"(5)
   note d = "2.prems"(6)
   note xxs_less_m = "2.prems"(7)
   note ia = "2.prems"(8)
   note im = "2.prems"(9)
   note D0 = "2.prems"(10)
   have xm: "x < m"  using "2.prems" by auto
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "(reduce a x D A)"
   have reduce_ax: "?reduce_ax \<in> carrier_mat m n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   have "reduce_below a (x # xs) D A $$ (i, j) = reduce_below a xs D (reduce a x D A) $$ (i, j)"
     by auto
   also have "... = reduce a x D A $$ (i, j)"
   proof (rule "2.hyps"[OF _ a j _ _ ])   
     show "i \<notin> set xs" using i_set_xxs by auto
     show "distinct xs" using d by auto
     show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
     show "reduce a x D A $$ (a, 0) \<noteq> 0"
       by (unfold D0, rule reduce_not0'[OF A _ _ _ _ Aaj], insert "2.prems", auto)
     show "reduce a x D A \<in> carrier_mat m n" using reduce_ax by linarith
   qed (insert "2.prems", auto)
   also have "... = A $$ (i,j)" by (rule reduce_preserves[OF A j Aaj], insert "2.prems", auto)
   finally show ?case .
 qed
 
 
 
 lemma reduce_below_0_D0:
   assumes A: "A \<in> carrier_mat m n" and a: "a<m" and j: "0<n"
     and Aaj: "A $$ (a,0) \<noteq> 0"
   assumes "i \<in> set xs" and "distinct xs" and "\<forall>x \<in> set xs. x < m \<and> a < x"
   and "D=0"
   shows "reduce_below a xs D A $$ (i,0) = 0"
   using assms
 proof (induct a xs D A arbitrary: A i rule: reduce_below.induct)
   case (1 a D A)
   then show ?case by auto 
 next
   case (2 a x xs D A)
   note A = "2.prems"(1)
   note a = "2.prems"(2)
   note j = "2.prems"(3)
   note Aaj = "2.prems"(4)
   note i_set_xxs = "2.prems"(5)
   note d = "2.prems"(6)
   note xxs_less_m = "2.prems"(7)
   note D0 = "2.prems"(8)
   have xm: "x < m"  using "2.prems" by auto
   obtain p q u v d where pquvd: "(p,q,u,v,d) = euclid_ext2 (A$$(a,0)) (A$$(x,0))"
     by (metis prod_cases5)
   let ?reduce_ax = "reduce a x D A"
   have reduce_ax: "?reduce_ax \<in> carrier_mat m n"
     by (metis (no_types, lifting) "2" add.comm_neutral append_rows_def 
         carrier_matD carrier_mat_triv index_mat_four_block(2,3)
         index_one_mat(2) index_smult_mat(2) index_zero_mat(2,3) reduce_preserves_dimensions)
   show ?case
   proof (cases "i=x")
     case True
     have "reduce_below a (x # xs) D A $$ (i, 0) = reduce_below a xs D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = (reduce a x D A) $$ (i, 0)"
     proof (rule reduce_below_preserves_D0[OF _ a j _ _ ])
       show "reduce a x D A \<in> carrier_mat m n" using reduce_ax by linarith
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (unfold D0, rule reduce_not0'[OF A _ _ j _ Aaj], insert "2.prems", auto)
       show "i \<notin> set xs" using True d by auto
       show "i \<noteq> a" using "2.prems" by blast
       show "i < m" by (simp add: True trans_less_add1 xm)
     qed (insert D0)
     also have "... = 0" unfolding True by (rule reduce_0[OF A _ j _ _ Aaj], insert "2.prems", auto)
     finally show ?thesis .
   next
     case False note i_not_x = False    
     have h: "reduce_below a xs D (reduce a x D A) $$ (i, 0) = 0 "
     proof (rule "2.hyps"[OF _ a j _ _ ])
       show "reduce a x D A \<in> carrier_mat m n" using reduce_ax by linarith
       show "i \<in> set xs" using i_set_xxs i_not_x by auto
       show "distinct xs" using d by auto
       show "\<forall>x\<in>set xs. x < m \<and> a < x" using xxs_less_m by auto
       show "reduce a x D A $$ (a, 0) \<noteq> 0"
         by (unfold D0, rule reduce_not0'[OF A _ _ j _ Aaj], insert "2.prems", auto)
     qed (insert D0)
     have "reduce_below a (x # xs) D A $$ (i, 0) = reduce_below a xs D (reduce a x D A) $$ (i, 0)"
       by auto
     also have "... = 0" using h .
     finally show ?thesis .
   qed
 qed
 
 end
 
 text \<open>Definition of the echelon form algorithm in JNF\<close>
 
 primrec bezout_iterate_JNF
 where "bezout_iterate_JNF A 0 i j bezout = A"
     | "bezout_iterate_JNF A (Suc n) i j bezout = 
         (if (Suc n) \<le> i then A else 
               bezout_iterate_JNF (bezout_matrix_JNF A i ((Suc n)) j bezout * A) n i j bezout)"
 
 definition 
   "echelon_form_of_column_k_JNF bezout A' k = 
     (let (A, i) = A' 
      in if (i = dim_row A) \<or> (\<forall>m \<in> {i..<dim_row A}. A $$ (m, k) = 0) then (A, i) else 
         if (\<forall>m\<in>{i+1..<dim_row A}. A $$ (m,k) = 0) then (A, i + 1) else
             let n = (LEAST n. A $$ (n,k) \<noteq> 0 \<and> i \<le> n); 
                 interchange_A = swaprows i n A
            in
             (bezout_iterate_JNF (interchange_A) (dim_row A - 1) i k bezout, i + 1) )"
 
 
 definition "echelon_form_of_upt_k_JNF A k bezout = (fst (foldl (echelon_form_of_column_k_JNF bezout) (A,0) [0..<Suc k]))"
 definition "echelon_form_of_JNF A bezout = echelon_form_of_upt_k_JNF A (dim_col A - 1) bezout"
 
 
 context includes lifting_syntax
 begin
 
 lemma HMA_bezout_iterate[transfer_rule]: 
   assumes "n<CARD('m)"
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) 
      ===> (Mod_Type_Connect.HMA_I) ===> (Mod_Type_Connect.HMA_I) ===> (=) ===> (Mod_Type_Connect.HMA_M)) 
   (\<lambda>A i j bezout. bezout_iterate_JNF A n i j bezout)  
   (\<lambda>A i j bezout. bezout_iterate A n i j bezout)
   "
 proof (intro rel_funI, goal_cases)
   case (1 A A' i i' j j' bezout bezout')
   then show ?case using assms
   proof (induct n arbitrary: A A')
     case 0
     then show ?case by auto
   next
     case (Suc n)
     note AA'[transfer_rule] = "Suc.prems"(1)
     note ii'[transfer_rule] = "Suc.prems"(2)
     note jj'[transfer_rule] = "Suc.prems"(3)
     note bb'[transfer_rule] = "Suc.prems"(4)
     note Suc_n_less_m = "Suc.prems"(5)
     let ?BI_JNF = "bezout_iterate_JNF"
     let ?BI_HMA = "bezout_iterate"
     let ?from_nat_rows = "mod_type_class.from_nat :: _ \<Rightarrow> 'm"
     have Sucn[transfer_rule]: "Mod_Type_Connect.HMA_I (Suc n) (?from_nat_rows (Suc n))"
       unfolding Mod_Type_Connect.HMA_I_def
       by (simp add: Suc_lessD Suc_n_less_m mod_type_class.from_nat_to_nat)
     have n: " n < CARD('m)" using Suc_n_less_m by simp
     have [transfer_rule]: 
       "Mod_Type_Connect.HMA_M (?BI_JNF (bezout_matrix_JNF A i (Suc n) j bezout * A) n i j bezout)
      (?BI_HMA (bezout_matrix A' i' (?from_nat_rows (Suc n)) j' bezout' ** A') n i' j' bezout')"    
       by (rule Suc.hyps[OF _ ii' jj' bb' n], transfer_prover)
     moreover have "Suc n \<le> i \<Longrightarrow>  Suc n \<le> mod_type_class.to_nat i'"
       and "Suc n > i \<Longrightarrow> Suc n > mod_type_class.to_nat i'" 
       by (metis "1"(2) Mod_Type_Connect.HMA_I_def)+
     ultimately show ?case using AA' by auto
   qed
 qed
 
 
 corollary HMA_bezout_iterate'[transfer_rule]: 
   fixes A'::"int ^ 'n :: mod_type ^ 'm :: mod_type"
   assumes n: "n<CARD('m)"
   and "Mod_Type_Connect.HMA_M A A'"
    and "Mod_Type_Connect.HMA_I i i'" and "Mod_Type_Connect.HMA_I j j'" 
  shows "Mod_Type_Connect.HMA_M (bezout_iterate_JNF A n i j bezout) (bezout_iterate A' n i' j' bezout)"
   using assms HMA_bezout_iterate[OF n] unfolding rel_fun_def by force
 
 
 
 lemma snd_echelon_form_of_column_k_JNF_le_dim_row:
   assumes "i<dim_row A"
   shows "snd (echelon_form_of_column_k_JNF bezout (A,i) k ) \<le> dim_row A"   
   using assms unfolding echelon_form_of_column_k_JNF_def by auto
 
 
 
 lemma HMA_echelon_form_of_column_k[transfer_rule]: 
   assumes k: "k<CARD('n)"
   shows "((=) ===> rel_prod (Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) (\<lambda>a b. a=b \<and> a\<le>CARD('m))
     ===> (rel_prod (Mod_Type_Connect.HMA_M) (\<lambda>a b. a=b \<and> a\<le>CARD('m)))) 
   (\<lambda>bezout A. echelon_form_of_column_k_JNF bezout A k)  
   (\<lambda>bezout A. echelon_form_of_column_k bezout A k)
   "
 proof (intro rel_funI, goal_cases)
   case (1 bezout bezout' xa ya )
   obtain A i where xa: "xa = (A,i)" using surjective_pairing by blast
   obtain A' i' where ya: "ya = (A',i')" using surjective_pairing by blast
   have ii'[transfer_rule]: "i=i'" using "1"(2) xa ya by auto
   have i_le_m: "i\<le>CARD('m)"  using "1"(2) xa ya by auto
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'" using "1"(2) xa ya by auto
   have bb'[transfer_rule]: "bezout=bezout'" using "1" by auto
   let ?from_nat_rows = "mod_type_class.from_nat :: _ \<Rightarrow> 'm"
   let ?from_nat_cols = "mod_type_class.from_nat :: _ \<Rightarrow> 'n"
   have kk'[transfer_rule]: "Mod_Type_Connect.HMA_I k (?from_nat_cols k)"
     by (simp add: Mod_Type_Connect.HMA_I_def assms mod_type_class.to_nat_from_nat_id)
   have c1_eq: "(i = dim_row A) = (i = nrows A')"
     by (metis AA' Mod_Type_Connect.dim_row_transfer_rule nrows_def)
   have c2_eq: "(\<forall>m \<in> {i..<dim_row A}. A $$ (m, k) = 0) 
       = (\<forall>m\<ge>?from_nat_rows i. A' $ m $ ?from_nat_cols k = 0)" (is "?lhs = ?rhs") if i_not: "i\<noteq>dim_row A" 
   proof 
     assume lhs: "?lhs"
     show "?rhs"
     proof (rule+)    
       fix m 
       assume im: "?from_nat_rows i \<le> m"
       have im': "i<CARD('m)" using i_le_m i_not
         by (simp add: c1_eq dual_order.order_iff_strict nrows_def)
       let ?m' = "mod_type_class.to_nat m"
       have mm'[transfer_rule]: "Mod_Type_Connect.HMA_I ?m' m" 
         by (simp add: Mod_Type_Connect.HMA_I_def)
       from im have "mod_type_class.to_nat (?from_nat_rows i) \<le> ?m'"
         by (simp add: to_nat_mono')
       hence "?m' >= i" using im im' by (simp add: mod_type_class.to_nat_from_nat_id)
       hence "?m' \<in> {i..<dim_row A}"
         using AA' Mod_Type_Connect.dim_row_transfer_rule mod_type_class.to_nat_less_card by fastforce
       hence "A $$ (?m', k) = 0" using lhs by auto
       moreover have "A $$ (?m', k) = A' $h m $h ?from_nat_cols k" unfolding index_hma_def[symmetric] by transfer_prover
       ultimately show "A' $h m $h ?from_nat_cols k = 0" by simp
     qed
   next
     assume rhs: "?rhs"
     show "?lhs"
     proof (rule)
       fix m assume m: "m \<in> {i..<dim_row A}"
       let ?m = "?from_nat_rows m"
       have mm'[transfer_rule]: "Mod_Type_Connect.HMA_I m ?m" 
         by (metis AA' Mod_Type_Connect.HMA_I_def Mod_Type_Connect.dim_row_transfer_rule
             atLeastLessThan_iff m mod_type_class.from_nat_to_nat)
       have m_ge_i: "?m\<ge>?from_nat_rows i"
         using AA' Mod_Type_Connect.dim_row_transfer_rule from_nat_mono' m by fastforce
       hence "A' $h ?m $h ?from_nat_cols k = 0" using rhs by auto
       moreover have "A $$ (m, k) = A' $h ?m $h ?from_nat_cols k"
         unfolding index_hma_def[symmetric] by transfer_prover
       ultimately show "A $$ (m, k) = 0" by simp
     qed
   qed
   show ?case 
   proof (cases "(i = dim_row A) \<or> (\<forall>m \<in> {i..<dim_row A}. A $$ (m, k) = 0)")
     case True   
     hence *: "(\<forall>m\<ge>?from_nat_rows i. A' $ m $ ?from_nat_cols k = 0) \<or> (i = nrows A')"
       using c1_eq c2_eq by auto
     have "echelon_form_of_column_k_JNF bezout xa k = (A,i)" 
       unfolding echelon_form_of_column_k_JNF_def using True xa by auto
     moreover have "echelon_form_of_column_k bezout ya k = (A',i')"
       unfolding echelon_form_of_column_k_def Let_def using * ya ii' by simp
     ultimately show ?thesis unfolding xa ya rel_prod.simps using AA' ii' bb' i_le_m by blast    
   next
     case False note not_c1 = False
     hence im': "i<CARD('m)"
       by (metis c1_eq dual_order.order_iff_strict i_le_m nrows_def)
     have *: "(\<forall>m\<in>{i+1..<dim_row A}. A $$ (m,k) = 0) 
       = (\<forall>m>?from_nat_rows i. A' $ m $ ?from_nat_cols k = 0)" (is "?lhs = ?rhs")
     proof 
       assume lhs: "?lhs"
       show "?rhs"
       proof (rule+)    
         fix m 
         assume im: "?from_nat_rows i < m"     
         let ?m' = "mod_type_class.to_nat m"
         have mm'[transfer_rule]: "Mod_Type_Connect.HMA_I ?m' m" 
           by (simp add: Mod_Type_Connect.HMA_I_def)
         from im have "mod_type_class.to_nat (?from_nat_rows i) < ?m'"
           by (simp add: to_nat_mono)
         hence "?m' > i" using im im' by (simp add: mod_type_class.to_nat_from_nat_id)
         hence "?m' \<in> {i+1..<dim_row A}"
           using AA' Mod_Type_Connect.dim_row_transfer_rule mod_type_class.to_nat_less_card by fastforce
         hence "A $$ (?m', k) = 0" using lhs by auto
         moreover have "A $$ (?m', k) = A' $h m $h ?from_nat_cols k" unfolding index_hma_def[symmetric] by transfer_prover
         ultimately show "A' $h m $h ?from_nat_cols k = 0" by simp
       qed
     next
       assume rhs: "?rhs"
       show "?lhs"
       proof (rule)
         fix m assume m: "m \<in> {i+1..<dim_row A}"
         let ?m = "?from_nat_rows m"
         have mm'[transfer_rule]: "Mod_Type_Connect.HMA_I m ?m" 
           by (metis AA' Mod_Type_Connect.HMA_I_def Mod_Type_Connect.dim_row_transfer_rule
               atLeastLessThan_iff m mod_type_class.from_nat_to_nat)
         have m_ge_i: "?m>?from_nat_rows i"
           by (metis Mod_Type_Connect.HMA_I_def One_nat_def add_Suc_right atLeastLessThan_iff from_nat_mono
               le_simps(3) m mm' mod_type_class.to_nat_less_card nat_arith.rule0)
         hence "A' $h ?m $h ?from_nat_cols k = 0" using rhs by auto
         moreover have "A $$ (m, k) = A' $h ?m $h ?from_nat_cols k"
           unfolding index_hma_def[symmetric] by transfer_prover
         ultimately show "A $$ (m, k) = 0" by simp
       qed
     qed
     show ?thesis
     proof (cases "(\<forall>m\<in>{i+1..<dim_row A}. A $$ (m,k) = 0)")
       case True
       have "echelon_form_of_column_k_JNF bezout xa k = (A,i+1)" 
         unfolding echelon_form_of_column_k_JNF_def using True xa not_c1 by auto
       moreover have "echelon_form_of_column_k bezout ya k = (A',i'+1)"
         unfolding echelon_form_of_column_k_def Let_def using ya ii' * True c1_eq c2_eq not_c1 by auto
       ultimately show ?thesis unfolding xa ya rel_prod.simps using AA' ii' bb' i_le_m im'
         by auto
     next
       case False
       hence *: "\<not> (\<forall>m>?from_nat_rows i. A' $ m $ ?from_nat_cols k = 0)" using * by auto
       have **: "\<not> ((\<forall>m\<ge>?from_nat_rows i. A' $h m $h ?from_nat_cols k = 0) \<or> i = nrows A')"
         using c1_eq c2_eq not_c1 by auto
       define n where "n=(LEAST n. A $$ (n,k) \<noteq> 0 \<and> i \<le> n)"
       define n' where "n'=(LEAST n. A' $ n $ ?from_nat_cols k \<noteq> 0 \<and> ?from_nat_rows i \<le> n)"
       let ?interchange_A = "swaprows i n A"
       let ?interchange_A' = "interchange_rows A' (?from_nat_rows i') n'"      
       have nn'[transfer_rule]: "Mod_Type_Connect.HMA_I n n'"
       proof -
         let ?n' = "mod_type_class.to_nat n'"
         have exist: "\<exists>n. A' $ n $ ?from_nat_cols k \<noteq> 0 \<and> ?from_nat_rows i \<le> n"
           using * by auto
         from this obtain a where c: "A' $ a $ ?from_nat_cols k \<noteq> 0 \<and> ?from_nat_rows i \<le> a" by blast        
         have "n = ?n'"
         proof (unfold n_def, rule Least_equality)
           have n'n'[transfer_rule]: "Mod_Type_Connect.HMA_I ?n' n'"
             by (simp add: Mod_Type_Connect.HMA_I_def)
           have e: "(A' $ n' $ ?from_nat_cols k \<noteq> 0 \<and> ?from_nat_rows i \<le> n')"
             by (metis (mono_tags, lifting) LeastI c2_eq n'_def not_c1) 
           hence  "i \<le> mod_type_class.to_nat n'" 
             using im' mod_type_class.from_nat_to_nat to_nat_mono' by fastforce       
           moreover have "A' $ n' $ ?from_nat_cols k = A $$ (?n', k)" 
             unfolding index_hma_def[symmetric] by (transfer', auto)
           ultimately show "A $$ (?n', k) \<noteq> 0 \<and> i \<le> ?n'"
             using e by auto
           show " \<And>y. A $$ (y, k) \<noteq> 0 \<and> i \<le> y \<Longrightarrow> mod_type_class.to_nat n' \<le> y"
-            by (smt AA' Mod_Type_Connect.HMA_M_def Mod_Type_Connect.from_hma\<^sub>m_def assms from_nat_mono
-                from_nat_mono' index_mat(1) linorder_not_less mod_type_class.from_nat_to_nat_id
-                mod_type_class.to_nat_less_card n'_def order.strict_trans prod.simps(2) wellorder_Least_lemma(2))
+            unfolding n'_def 
+            by (smt (verit, del_insts) AA' Mod_Type_Connect.HMA_M_def Mod_Type_Connect.from_hma\<^sub>m_def assms from_nat_mono 
+                from_nat_mono' index_mat(1) less_trans mod_type_class.from_nat_to_nat_id mod_type_class.to_nat_less_card 
+                not_le prod.simps(2) wellorder_Least_lemma(2))
         qed
         thus ?thesis unfolding Mod_Type_Connect.HMA_I_def by auto
       qed
       have dr1[transfer_rule]: "(nrows A' - 1) = (dim_row A - 1)" unfolding nrows_def
         using AA' Mod_Type_Connect.dim_row_transfer_rule by force
       have ii'2[transfer_rule]: "Mod_Type_Connect.HMA_I i (?from_nat_rows i')"
         by (metis "**" Mod_Type_Connect.HMA_I_def i_le_m ii' le_neq_implies_less
             mod_type_class.to_nat_from_nat_id nrows_def)
       have ii'3[transfer_rule]: "Mod_Type_Connect.HMA_I i' (?from_nat_rows i')" 
         using ii' ii'2 by blast
       let ?BI_JNF = "(bezout_iterate_JNF (?interchange_A) (dim_row A - 1) i k bezout)"
       let ?BI_HA = "(bezout_iterate (?interchange_A') (nrows A' - 1) (?from_nat_rows i) (?from_nat_cols k) bezout)"
       have e_rw: "echelon_form_of_column_k_JNF bezout xa k = (?BI_JNF,i+1)"
         unfolding echelon_form_of_column_k_JNF_def n_def using False xa not_c1 by auto
       have e_rw2: "echelon_form_of_column_k bezout ya k = (?BI_HA,i+1)"
         unfolding echelon_form_of_column_k_def Let_def n'_def using * ya ** ii' by auto
       have s[transfer_rule]: "Mod_Type_Connect.HMA_M (swaprows i' n A) (interchange_rows A' (?from_nat_rows i') n')" 
         by transfer_prover
       have n_CARD: "(nrows A' - 1) < CARD('m)" unfolding nrows_def by auto
       note a[transfer_rule] = HMA_bezout_iterate[OF n_CARD]
       have BI[transfer_rule]:"Mod_Type_Connect.HMA_M ?BI_JNF ?BI_HA" unfolding ii' dr1 
         by (rule HMA_bezout_iterate'[OF _ s ii'3 kk'], insert n_CARD, transfer', simp)
       thus ?thesis using e_rw e_rw2 bb'
         by (metis (mono_tags, lifting) AA' False Mod_Type_Connect.dim_row_transfer_rule 
             atLeastLessThan_iff dual_order.trans order_less_imp_le rel_prod_inject)
     qed
   qed
 qed
 
 corollary HMA_echelon_form_of_column_k'[transfer_rule]: 
   assumes k: "k<CARD('n)" and "i\<le>CARD('m)"
   and "(Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) A A'"
   shows "(rel_prod (Mod_Type_Connect.HMA_M) (\<lambda>a b. a=b \<and> a\<le>CARD('m))) 
   (echelon_form_of_column_k_JNF bezout (A,i) k)  
   (echelon_form_of_column_k bezout (A',i) k)"
   using assms HMA_echelon_form_of_column_k[OF k] unfolding rel_fun_def by force
 
 lemma HMA_foldl_echelon_form_of_column_k:
   assumes k: "k\<le>CARD('n)"
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) ===> (=)
     ===> (rel_prod (Mod_Type_Connect.HMA_M) (\<lambda>a b. a=b \<and> a\<le>CARD('m)))) 
   (\<lambda>A bezout. (foldl (echelon_form_of_column_k_JNF bezout) (A,0) [0..<k]))  
   (\<lambda>A bezout. (foldl (echelon_form_of_column_k bezout) (A,0) [0..<k]))"
 proof (intro rel_funI, goal_cases)
   case (1 A A' bezout bezout')
   then show ?case using assms
   proof (induct k arbitrary: A A' )
     case 0
     then show ?case by auto
   next
      case (Suc k)
     note AA'[transfer_rule] = "Suc.prems"(1)
     note bb'[transfer_rule] = "Suc.prems"(2)
     note Suc_k_less_m = "Suc.prems"(3)
     let ?foldl_JNF = "foldl (echelon_form_of_column_k_JNF bezout) (A,0)"
     let ?foldl_HA = "foldl (echelon_form_of_column_k bezout') (A',0)"
     have set_rw: "[0..<Suc k] = [0..<k] @ [k]" by auto
     have f_JNF: "?foldl_JNF [0..<Suc k] = echelon_form_of_column_k_JNF bezout (?foldl_JNF [0..<k]) k" 
       by auto
     have f_HA: "?foldl_HA [0..<Suc k] = echelon_form_of_column_k bezout' (?foldl_HA [0..<k]) k" 
       by auto
     have hyp[transfer_rule]: "rel_prod Mod_Type_Connect.HMA_M (\<lambda>a b. a=b \<and> a\<le>CARD('m)) (?foldl_JNF [0..<k]) (?foldl_HA [0..<k])"
       by (rule Suc.hyps[OF AA'], insert Suc.prems, auto)
     show ?case unfolding f_JNF unfolding f_HA bb' using HMA_echelon_form_of_column_k'
-      by (smt "1"(2) Suc_k_less_m Suc_le_lessD hyp rel_prod.cases)
+      by (smt (verit) "1"(2) Suc_k_less_m Suc_le_lessD hyp rel_prod.cases)
   qed
 qed
 
 
 
 lemma HMA_echelon_form_of_upt_k[transfer_rule]:
   assumes k: "k<CARD('n)"
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) ===> (=)
     ===> (Mod_Type_Connect.HMA_M)) 
   (\<lambda>A bezout. echelon_form_of_upt_k_JNF A k bezout)  
   (\<lambda>A bezout. echelon_form_of_upt_k A k bezout)
   "
 proof (intro rel_funI, goal_cases)
   case (1 A A' bezout bezout')
   have k': "Suc k \<le> CARD('n)" using k by auto
   have rel_foldl: "(rel_prod (Mod_Type_Connect.HMA_M) (\<lambda>a b. a=b \<and> a\<le>CARD('m))) 
   (foldl (echelon_form_of_column_k_JNF bezout) (A,0) [0..<Suc k])  
   (foldl (echelon_form_of_column_k bezout) (A',0) [0..<Suc k])"
-    using HMA_foldl_echelon_form_of_column_k[OF k'] by (smt "1"(1) rel_fun_def)
+    using HMA_foldl_echelon_form_of_column_k[OF k'] by (smt (verit) "1"(1) rel_fun_def)
   then show ?case using assms unfolding echelon_form_of_upt_k_JNF_def echelon_form_of_upt_k_def    
     by (metis (no_types, lifting) "1"(2) prod.collapse rel_prod_inject)
 qed
  
 
 lemma HMA_echelon_form_of[transfer_rule]:
   shows "((Mod_Type_Connect.HMA_M :: _ \<Rightarrow> int ^ 'n :: mod_type ^ 'm :: mod_type \<Rightarrow> _) ===> (=)
     ===> (Mod_Type_Connect.HMA_M)) 
   (\<lambda>A bezout. echelon_form_of_JNF A bezout)  
   (\<lambda>A bezout. echelon_form_of A bezout)
   "
 proof (intro rel_funI, goal_cases)
   case (1 A A' bezout bezout')
   note AA'[transfer_rule] = 1(1)
   note bb'[transfer_rule] = 1(2)
   have *: "(dim_col A - 1) < CARD('n)" using 1
     using Mod_Type_Connect.dim_col_transfer_rule by force
   note **[transfer_rule] = HMA_echelon_form_of_upt_k[OF *]
   have [transfer_rule]: "(ncols A' - 1) = (dim_col A - 1)"
     by (metis "1"(1) Mod_Type_Connect.dim_col_transfer_rule ncols_def)
   have [transfer_rule]: "(dim_col A - 1) = (dim_col A - 1)" ..
   show ?case unfolding echelon_form_of_def echelon_form_of_JNF_def bb'
     by (metis (mono_tags) "**" "1"(1) \<open>ncols A' - 1 = dim_col A - 1\<close> rel_fun_def)
 qed
 end
 
 
 context
 begin
 
 private lemma echelon_form_of_euclidean_invertible_mod_type:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::mod_type) CARD('n::mod_type)"
   shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (CARD('m::mod_type)) (CARD('m::mod_type)) 
     \<and> P * A = echelon_form_of_JNF A euclid_ext2 
     \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"
 proof -
   define A' where "A' = (Mod_Type_Connect.to_hma\<^sub>m A :: int ^'n :: mod_type ^'m :: mod_type)"
   have AA'[transfer_rule]: "Mod_Type_Connect.HMA_M A A'"
     unfolding Mod_Type_Connect.HMA_M_def using assms A'_def by auto   
   have [transfer_rule]: "Mod_Type_Connect.HMA_M 
     (echelon_form_of_JNF A euclid_ext2) (echelon_form_of A' euclid_ext2)" 
     by transfer_prover  
   have "\<exists>P. invertible P \<and> P**A' = (echelon_form_of A' euclid_ext2) 
          \<and> echelon_form (echelon_form_of A' euclid_ext2)"
     by (rule echelon_form_of_euclidean_invertible)    
   thus ?thesis by (transfer, auto)
 qed 
 
 
 private lemma echelon_form_of_euclidean_invertible_nontriv_mod_ring:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat CARD('m::nontriv mod_ring) CARD('n::nontriv mod_ring)"
   shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat (CARD('m)) (CARD('m)) 
     \<and> P * A = echelon_form_of_JNF A euclid_ext2 
     \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"
-  using assms echelon_form_of_euclidean_invertible_mod_type by (smt CARD_mod_ring)
+  using assms echelon_form_of_euclidean_invertible_mod_type by (smt (verit) CARD_mod_ring)
 
 (*We internalize both sort constraints in one step*)
 lemmas echelon_form_of_euclidean_invertible_nontriv_mod_ring_internalized = 
   echelon_form_of_euclidean_invertible_nontriv_mod_ring[unfolded CARD_mod_ring, 
       internalize_sort "'m::nontriv", internalize_sort "'b::nontriv"]
 
 context
   fixes m::nat and n::nat
   assumes local_typedef1: "\<exists>(Rep :: ('b \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<m :: int}"
   assumes local_typedef2: "\<exists>(Rep :: ('c \<Rightarrow> int)) Abs. type_definition Rep Abs {0..<n :: int}"
   and m: "m>1"
   and n: "n>1"
 begin
 
 lemma echelon_form_of_euclidean_invertible_nontriv_mod_ring_aux:
   fixes A::"int mat"
   assumes "A \<in> carrier_mat m n"
   shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m
     \<and> P * A = echelon_form_of_JNF A euclid_ext2 
     \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"
  using echelon_form_of_euclidean_invertible_nontriv_mod_ring_internalized
     [OF type_to_set2(1)[OF local_typedef1 local_typedef2] 
         type_to_set1(1)[OF local_typedef1 local_typedef2]]
   using assms 
   using type_to_set1(2) local_typedef1 local_typedef2 n m by metis 
 
 end
 
 
 (*Canceling the first local type definitions*)
 context
 begin
 
 (*Canceling the first*)
 
 private lemma echelon_form_of_euclidean_invertible_cancelled_first:
 "\<exists>Rep Abs. type_definition Rep Abs {0..<int n} \<Longrightarrow> 1 < m \<Longrightarrow> 1 < n \<Longrightarrow>
   A \<in> carrier_mat m n \<Longrightarrow> \<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m
   \<and> P * (A::int mat) = echelon_form_of_JNF A euclid_ext2 \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"
   using echelon_form_of_euclidean_invertible_nontriv_mod_ring_aux[cancel_type_definition, of m n A]
   by force  
 
 (*Canceling the second*)
 private lemma echelon_form_of_euclidean_invertible_cancelled_both:
 "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> A \<in> carrier_mat m n \<Longrightarrow> \<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m 
   \<and> P * (A::int mat) = echelon_form_of_JNF A euclid_ext2 \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"
   using echelon_form_of_euclidean_invertible_cancelled_first[cancel_type_definition, of n m A]
   by force
 
 (*The final result in JNF*)
 
 lemma echelon_form_of_euclidean_invertible':
  fixes A::"int mat"
   assumes "A \<in> carrier_mat m n" 
   and "1 < m" and "1 < n" (*Required from the mod_type restrictions*)
   shows "\<exists>P. invertible_mat P \<and>
         P \<in> carrier_mat m m \<and> P * A = echelon_form_of_JNF A euclid_ext2 
         \<and> echelon_form_JNF (echelon_form_of_JNF A euclid_ext2)"  
   using echelon_form_of_euclidean_invertible_cancelled_both assms by auto
 end
 end
 
 context mod_operation
 begin
 
 definition "FindPreHNF_rectangular A 
   =   (let m = dim_row A; n = dim_col A in 
   if m < 2 \<or> n = 0 then A else \<comment> \<open> No operations are carried out if m = 1 \<close>
   if n = 1 then 
         let non_zero_positions = filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A] in
         if non_zero_positions = [] then A
         else let A' = (if A$$(0,0) \<noteq> 0 then A else let i = non_zero_positions ! 0 in swaprows 0 i A)
             in reduce_below_impl 0 non_zero_positions 0 A'
   else (echelon_form_of_JNF A euclid_ext2))"
 
 text \<open>This is the (non-efficient) HNF algorithm obtained from the echelon form and Hermite normal 
 form AFP entries\<close>
 
 definition "HNF_algorithm_from_HA A 
             = Hermite_of_list_of_rows (FindPreHNF_rectangular A) [0..<(dim_row A)]"
 
 (*
   Now we can combine FindPreHNF_rectangular, FindPreHNF and Hermite_of_list_of_rows to get
   an algorithm to compute the HNF of any matrix (if it is square and invertible, then the HNF is
   computed reducing entries modulo D)
 *)
 
 text \<open>Now we can combine @{text"FindPreHNF_rectangular"}, @{text"FindPreHNF"}
   and @{text"Hermite_of_list_of_rows"} to get an algorithm to compute the HNF of any matrix
   (if it is square and invertible, then the HNF is computed reducing entries modulo D)\<close>
 
 definition "HNF_algorithm abs_flag A = 
   (let m = dim_row A; n = dim_col A in
   if m \<noteq> n then Hermite_of_list_of_rows (FindPreHNF_rectangular A) [0..<m]
   else
     let D = abs (det_int A) in
     if D = 0 then Hermite_of_list_of_rows (FindPreHNF_rectangular A) [0..<m]
     else 
       let A' = A @\<^sub>r D \<cdot>\<^sub>m 1\<^sub>m n;
           E = FindPreHNF abs_flag D A';
           H = Hermite_of_list_of_rows E [0..<m+n]
       in mat_of_rows n (map (Matrix.row H) [0..<m]))"
 
 end
 
 declare mod_operation.FindPreHNF_rectangular_def[code]
 declare mod_operation.HNF_algorithm_from_HA_def[code]
 declare mod_operation.HNF_algorithm_def[code]
 
 context proper_mod_operation
 begin
 
 (*With some work, we could get this lemma for matrices whose elements belong to a Bézout domain*)
 lemma FindPreHNF_rectangular_soundness:
   fixes A::"int mat"
   assumes A: "A \<in> carrier_mat m n" 
   shows "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> P * A = FindPreHNF_rectangular A 
     \<and> echelon_form_JNF (FindPreHNF_rectangular A)"  
 proof (cases "m < 2 \<or> n = 0")
   case True
   then show ?thesis
-    by (smt A FindPreHNF_rectangular_def carrier_matD echelon_form_JNF_1xn echelon_form_mx0
+    by (smt (verit) A FindPreHNF_rectangular_def carrier_matD echelon_form_JNF_1xn echelon_form_mx0
         invertible_mat_one left_mult_one_mat one_carrier_mat)
 next
   case False 
   have m1: "m>1" using False by auto
   have n0: "n>0" using False by auto
   show ?thesis
   proof (cases "n=1")
     case True note n1 = True
     let ?nz = "filter (\<lambda>i. A $$ (i,0) \<noteq> 0) [1..<dim_row A]" 
     let ?A' = "(if A$$(0,0) \<noteq> 0 then A else let i = ?nz ! 0 in swaprows 0 i A)"
     have A': "?A' \<in> carrier_mat m n" using A by auto
     have A'00: "?A' $$ (0,0) \<noteq> 0" if "?nz \<noteq> []" 
-      by (smt True assms carrier_matD index_mat_swaprows(1) length_greater_0_conv m1
+      by (smt (verit) True assms carrier_matD index_mat_swaprows(1) length_greater_0_conv m1
           mem_Collect_eq nat_SN.gt_trans nth_mem set_filter that zero_less_one_class.zero_less_one)
     have e_r: "echelon_form_JNF (reduce_below 0 ?nz 0 ?A')" if nz_not_empty: "?nz \<noteq> []" 
     proof (rule echelon_form_JNF_mx1)
       show "(reduce_below 0 ?nz 0 ?A') \<in> carrier_mat m n"  using A reduce_below by auto
       have "(reduce_below 0 ?nz 0 ?A') $$ (i,0) = 0" if i: "i \<in> {1..<m}" for i
       proof (cases "i \<in> set ?nz")
         case True
         show ?thesis
           by (rule reduce_below_0_D0[OF A' _ _ A'00 True], insert m1 n0 True A nz_not_empty, auto)
       next
         case False
         have "(reduce_below 0 ?nz 0 ?A') $$ (i,0) = ?A' $$ (i,0)" 
           by (rule reduce_below_preserves_D0[OF A' _ _ A'00 False], insert m1 n0 True A i nz_not_empty, auto)
         also have "... = 0" using False n1 assms that by auto
         finally show ?thesis .
       qed
       thus "\<forall>i \<in> {1..<m}. (reduce_below 0 ?nz 0 ?A') $$ (i,0) = 0"
         by simp
     qed (insert True, simp)
     have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> reduce_below 0 ?nz 0 ?A' = P * ?A'" 
       by (rule reduce_below_invertible_mat_D0[OF A'], insert m1 n0 True A, auto)
     moreover have "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> ?A' = P * A"   if "?nz \<noteq> []"
       using A A'_swaprows_invertible_mat m1 that by blast 
     ultimately have e_inv: "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> reduce_below 0 ?nz 0 ?A' = P * A"
       if "?nz \<noteq> []"
-      by (smt that A assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
+      by (smt (verit) that A assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
     have e_r1: "echelon_form_JNF A" if nz_empty: "?nz = []" 
     proof (rule echelon_form_JNF_mx1[OF A])
       show "\<forall>i\<in>{1..<m}. A $$ (i, 0) = 0 " using nz_empty
         by (metis (mono_tags, lifting) A carrier_matD(1) empty_filter_conv set_upt)
     qed (insert n1, simp)
     have e_inv1: "\<exists>P. invertible_mat P \<and> P \<in> carrier_mat m m \<and> A = P * A"
       by (metis A invertible_mat_one left_mult_one_mat one_carrier_mat)
     have "FindPreHNF_rectangular A = (if ?nz = [] then A else reduce_below_impl 0 ?nz 0 ?A')"
       unfolding FindPreHNF_rectangular_def Let_def using m1 n1 A True by auto
     also have "reduce_below_impl 0 ?nz 0 ?A' = reduce_below 0 ?nz 0 ?A'"
       by (rule reduce_below_impl[OF _ _ _ _ A'], insert m1 n0 A, auto)
     finally show ?thesis using e_inv e_r e_r1 e_inv1 by metis
   next
     case False
     have f_rw: "FindPreHNF_rectangular A = echelon_form_of_JNF A euclid_ext2"
       unfolding FindPreHNF_rectangular_def Let_def using m1 n0 A False by auto
     show ?thesis unfolding f_rw 
       by (rule echelon_form_of_euclidean_invertible'[OF A], insert False n0 m1, auto)
   qed
 qed
 
 lemma HNF_algorithm_from_HA_soundness:
   assumes A: "A \<in> carrier_mat m n"
   shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm_from_HA A)
     \<and> (\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm_from_HA A) = P * A)"
 proof -
   have m: "dim_row A = m" using A by auto
   have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm_from_HA A) = P * (FindPreHNF_rectangular A))"
     unfolding HNF_algorithm_from_HA_def m
   proof (rule invertible_Hermite_of_list_of_rows)
     show "FindPreHNF_rectangular A \<in> carrier_mat m n"
-      by (smt A FindPreHNF_rectangular_soundness mult_carrier_mat)
+      by (smt (verit) A FindPreHNF_rectangular_soundness mult_carrier_mat)
     show "echelon_form_JNF (FindPreHNF_rectangular A)" 
       using FindPreHNF_rectangular_soundness by blast
   qed
   moreover have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (FindPreHNF_rectangular A) = P * A)"    
     by (metis A FindPreHNF_rectangular_soundness)
   ultimately have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm_from_HA A) = P * A)"
-    by (smt assms assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
+    by (smt (verit) assms assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
   moreover have "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm_from_HA A)"
     by (metis A FindPreHNF_rectangular_soundness HNF_algorithm_from_HA_def m 
         Hermite_Hermite_of_list_of_rows mult_carrier_mat)
   ultimately show ?thesis by simp
 qed
 
 
 text \<open>Soundness theorem for any matrix\<close>
 lemma HNF_algorithm_soundness:
   assumes A: "A \<in> carrier_mat m n"
   shows "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm abs_flag A)
     \<and> (\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm abs_flag A) = P * A)"
 proof (cases "m\<noteq>n \<or> Determinant.det A = 0")
   case True
   have H_rw: "HNF_algorithm abs_flag A = Hermite_of_list_of_rows (FindPreHNF_rectangular A) [0..<m]"
     using True A unfolding HNF_algorithm_def Let_def by auto
   have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm abs_flag A) = P * (FindPreHNF_rectangular A))"
     unfolding H_rw
   proof (rule invertible_Hermite_of_list_of_rows)
     show "FindPreHNF_rectangular A \<in> carrier_mat m n"
-      by (smt A FindPreHNF_rectangular_soundness mult_carrier_mat)
+      by (smt (verit) A FindPreHNF_rectangular_soundness mult_carrier_mat)
     show "echelon_form_JNF (FindPreHNF_rectangular A)" 
       using FindPreHNF_rectangular_soundness by blast
   qed
   moreover have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (FindPreHNF_rectangular A) = P * A)"    
     by (metis A FindPreHNF_rectangular_soundness)
   ultimately have "(\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> (HNF_algorithm abs_flag A) = P * A)"
-    by (smt assms assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
+    by (smt (verit) assms assoc_mult_mat invertible_mult_JNF mult_carrier_mat)
   moreover have "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm abs_flag A)"
     by (metis A FindPreHNF_rectangular_soundness H_rw Hermite_Hermite_of_list_of_rows mult_carrier_mat)
   ultimately show ?thesis by simp
 next
   case False
   hence mn: "m=n" and det_A_not0:"(Determinant.det A) \<noteq> 0" by auto
   have inv_RAT_A: "invertible_mat (map_mat rat_of_int A)"
   proof -
     have "det (map_mat rat_of_int A) \<noteq> 0" using det_A_not0 by auto
     thus ?thesis 
       by (metis False assms dvd_field_iff invertible_iff_is_unit_JNF map_carrier_mat)
   qed
   have "HNF_algorithm abs_flag A = Hermite_mod_det abs_flag A"
     unfolding HNF_algorithm_def Hermite_mod_det_def Let_def using False A by simp    
   then show ?thesis using Hermite_mod_det_soundness[OF mn A inv_RAT_A] by auto
 qed
 end
 
 
 text \<open>New predicate of soundness of a HNF algorithm, without providing explicitly the transformation matrix.\<close>
 
 definition "is_sound_HNF' algorithm associates res 
     = (\<forall>A. let H = algorithm A; m = dim_row A; n = dim_col A in Hermite_JNF associates res H 
         \<and> H \<in> carrier_mat m n \<and> (\<exists>P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> A = P * H))"
 
 lemma is_sound_HNF_conv:
   assumes s: "is_sound_HNF' algorithm associates res"
   shows "is_sound_HNF (\<lambda>A. let H = algorithm A in (SOME P. P \<in> carrier_mat (dim_row A) (dim_row A)
     \<and> invertible_mat P \<and> A = P * H, H)) associates res"
 proof (unfold is_sound_HNF_def Let_def prod.case, rule allI)
   fix A::"'a mat"
   define m where "m = dim_row A"
   obtain P where P: "P \<in> carrier_mat m m \<and> invertible_mat P \<and> A = P * (algorithm A)" 
     using s unfolding is_sound_HNF'_def Let_def m_def by auto
   let ?some_P = "(SOME P. P \<in> carrier_mat m m \<and> invertible_mat P \<and> A = P * algorithm A)"
   have some_P: "?some_P \<in> carrier_mat m m \<and> invertible_mat ?some_P \<and> A = ?some_P * algorithm A"
-    by (smt P verit_sko_ex_indirect)
+    by (metis (mono_tags, lifting) P someI_ex)
   moreover have "algorithm A \<in> carrier_mat (dim_row A) (dim_col A)"
     and  "Hermite_JNF associates res (algorithm A)" using s unfolding is_sound_HNF'_def Let_def by auto    
   ultimately show "?some_P \<in> carrier_mat m m \<and> algorithm A \<in> carrier_mat m (dim_col A) 
     \<and> invertible_mat ?some_P \<and> A = ?some_P * algorithm A \<and> Hermite_JNF associates res (algorithm A)"
     unfolding is_sound_HNF_def Let_def m_def by (auto split: prod.split)
 qed
 
 context proper_mod_operation
 begin
 corollary is_sound_HNF'_HNF_algorithm:
    "is_sound_HNF' (HNF_algorithm abs_flag) (range ass_function_euclidean) (\<lambda>c. range (res_int c))"  
 proof -
   have "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm abs_flag A)" for A
     using HNF_algorithm_soundness by blast
   moreover have "HNF_algorithm abs_flag A \<in> carrier_mat (dim_row A) (dim_col A)" for A
     by (metis HNF_algorithm_soundness carrier_matI mult_carrier_mat)
   moreover have "\<exists>P. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> A = P * HNF_algorithm abs_flag A" for A 
   proof -
     have "\<exists>P. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> HNF_algorithm abs_flag A = P *  A"
       using HNF_algorithm_soundness by blast
     from this obtain P where P: "P \<in> carrier_mat (dim_row A) (dim_row A)" and inv_P: "invertible_mat P"
       and H_PA: "HNF_algorithm abs_flag A = P *  A" 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
     have P': "P' \<in> carrier_mat (dim_row A) (dim_row A) "
       by (metis P PP' P'P carrier_matD carrier_mat_triv index_mult_mat(3) index_one_mat(3) inverts_mat_def)
     moreover have inv_P': "invertible_mat P'"
       by (metis P' P'P PP' carrier_matD(1) carrier_matD(2) invertible_mat_def square_mat.simps)
     moreover have "A = P' * HNF_algorithm abs_flag A"
-      by (smt H_PA P P'P assoc_mult_mat calculation(1) carrier_matD(1) carrier_matI inverts_mat_def left_mult_one_mat')
+      by (smt (verit) H_PA P P'P assoc_mult_mat calculation(1) carrier_matD(1) carrier_matI inverts_mat_def left_mult_one_mat')
     ultimately show ?thesis by auto
   qed
   ultimately show ?thesis
     unfolding is_sound_HNF'_def Let_def by auto
 qed
 
 
 corollary is_sound_HNF'_HNF_algorithm_from_HA:
    "is_sound_HNF' (HNF_algorithm_from_HA) (range ass_function_euclidean) (\<lambda>c. range (res_int c))"  
 proof -
   have "Hermite_JNF (range ass_function_euclidean) (\<lambda>c. range (res_int c)) (HNF_algorithm_from_HA A)" for A
     using HNF_algorithm_from_HA_soundness by blast
   moreover have "HNF_algorithm_from_HA A \<in> carrier_mat (dim_row A) (dim_col A)" for A
     by (metis HNF_algorithm_from_HA_soundness carrier_matI mult_carrier_mat)
   moreover have "\<exists>P. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> A = P * HNF_algorithm_from_HA A" for A 
   proof -
     have "\<exists>P. P \<in> carrier_mat (dim_row A) (dim_row A) \<and> invertible_mat P \<and> HNF_algorithm_from_HA A = P *  A"
       using HNF_algorithm_from_HA_soundness by blast
     from this obtain P where P: "P \<in> carrier_mat (dim_row A) (dim_row A)" and inv_P: "invertible_mat P"
       and H_PA: "HNF_algorithm_from_HA A = P *  A" 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
     have P': "P' \<in> carrier_mat (dim_row A) (dim_row A) "
       by (metis P PP' P'P carrier_matD carrier_mat_triv index_mult_mat(3) index_one_mat(3) inverts_mat_def)
     moreover have inv_P': "invertible_mat P'"
       by (metis P' P'P PP' carrier_matD(1) carrier_matD(2) invertible_mat_def square_mat.simps)
     moreover have "A = P' * HNF_algorithm_from_HA A"
-      by (smt H_PA P P'P assoc_mult_mat calculation(1) carrier_matD(1) carrier_matI inverts_mat_def left_mult_one_mat')
+      by (smt (verit) H_PA P P'P assoc_mult_mat calculation(1) carrier_matD(1) carrier_matI inverts_mat_def left_mult_one_mat')
     ultimately show ?thesis by auto
   qed
   ultimately show ?thesis
     unfolding is_sound_HNF'_def Let_def by auto
 qed
 end
 
 text \<open>Some work to make the algorithm executable\<close>
 
 definition find_non0' :: "nat \<Rightarrow> nat \<Rightarrow> 'a::comm_ring_1 mat \<Rightarrow> nat option" where
   "find_non0' i k A = (let is = [i ..< dim_row A];
     Ais = filter (\<lambda>j. A $$ (j, k) \<noteq> 0) is
     in case Ais of [] \<Rightarrow> None | _ \<Rightarrow> Some (Ais!0))"
 
 lemma find_non0': 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_non0' i k A = Some j"
   shows "A $$ (j,k) \<noteq> 0" "i \<le> j" "j < dim_row A"
 proof -
   let ?xs = "filter (\<lambda>j. A $$ (j,k) \<noteq> 0) [i ..< dim_row A]"
   from res[unfolded find_non0'_def Let_def]
   have xs: "?xs \<noteq> []" by (cases ?xs, auto)
   have j_in_xs: "j \<in> set ?xs" using res unfolding find_non0'_def Let_def
     by (metis (no_types, lifting) length_greater_0_conv list.case(2) list.exhaust nth_mem option.simps(1) xs)
   show "A $$ (j,k) \<noteq> 0" "i \<le> j" "j < dim_row A" using j_in_xs by auto+  
 qed
 
 
 lemma find_non0'_w_zero_before: 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_non0' i k A = Some j"
   shows "\<forall>j'\<in>{i..<j}. A $$ (j',k) = 0"
 proof -
   let ?xs = "filter (\<lambda>j. A $$ (j, k) \<noteq> 0) [i ..< dim_row A]"
   from res[unfolded find_non0'_def Let_def]
   have xs: "?xs \<noteq> []" by (cases ?xs, auto)
   have j_in_xs: "j \<in> set ?xs" using res unfolding find_non0'_def Let_def
     by (metis (no_types, lifting) length_greater_0_conv list.case(2) list.exhaust nth_mem option.simps(1) xs)
   have j_xs0: "j = ?xs ! 0"
-    by (smt res[unfolded find_non0'_def Let_def] list.case(2) list.exhaust option.inject xs)
+    by (smt (verit) res[unfolded find_non0'_def Let_def] list.case(2) list.exhaust option.inject xs)
   show "\<forall>j'\<in>{i..<j}. A $$ (j',k) = 0"
   proof (rule+, rule ccontr)
     fix j' assume j': "j' : {i..<j}" and Alj': "A $$ (j',k) \<noteq> 0"
     have j'j: "j'<j" using j' by auto
     have j'_in_xs: "j' \<in> set ?xs" 
       by (metis (mono_tags, lifting) A Alj' Set.member_filter atLeastLessThan_iff filter_set
           find_non0'(3) j' nat_SN.gt_trans res set_upt)
     have l_rw: "[i..<dim_row A] = [i ..<j] @[j..<dim_row A]"
       using assms(1) assms(2) find_non0'(3) j' upt_append 
       by (metis atLeastLessThan_iff le_trans linorder_not_le) 
     have xs_rw: "?xs = filter (\<lambda>j. A $$ (j,k) \<noteq> 0) ([i ..<j] @[j..<dim_row A])"
       using l_rw by auto
     hence "filter (\<lambda>j. A $$ (j,k) \<noteq> 0) [i ..<j] = []" using j_xs0 
       by (metis (no_types, lifting) Set.member_filter atLeastLessThan_iff filter_append filter_set
           length_greater_0_conv nth_append nth_mem order_less_irrefl set_upt)
     thus False using j_xs0 j' j_xs0 
       by (metis Set.member_filter filter_empty_conv filter_set j'_in_xs set_upt)
   qed
 qed
 
 
 lemma find_non0'_LEAST: 
   assumes A: "A \<in> carrier_mat m n"
   and res: "find_non0' i k A = Some j"
 shows "j = (LEAST n. A $$ (n,k) \<noteq> 0 \<and> i\<le>n)"
 proof (rule Least_equality[symmetric])
   show " A $$ (j, k) \<noteq> 0 \<and> i \<le> j" 
     using A res find_non0'[OF A] by auto
   show " \<And>y. A $$ (y, k) \<noteq> 0 \<and> i \<le> y \<Longrightarrow> j \<le> y"
     by (meson A res atLeastLessThan_iff find_non0'_w_zero_before linorder_not_le)
 qed
 
 lemma echelon_form_of_column_k_JNF_code[code]:
   "echelon_form_of_column_k_JNF bezout (A,i) k = 
     (if (i = dim_row A) \<or> (\<forall>m \<in> {i..<dim_row A}. A $$ (m, k) = 0) then (A, i) else 
         if (\<forall>m\<in>{i+1..<dim_row A}. A $$ (m,k) = 0) then (A, i + 1) else
             let n = the (find_non0' i k A); 
                 interchange_A = swaprows i n A
            in
             (bezout_iterate_JNF (interchange_A) (dim_row A - 1) i k bezout, i + 1))"
 proof (cases "\<not> ((i = dim_row A) \<or> (\<forall>m \<in> {i..<dim_row A}. A $$ (m, k) = 0)) 
             \<and> \<not> (\<forall>m\<in>{i+1..<dim_row A}. A $$ (m,k) = 0)")
   case True
   let ?n = "the (find_non0' i k A)"
   let ?interchange_A = "swaprows i ?n A"
   have f_rw: "(the (find_non0' i k A)) = (LEAST n. A $$ (n, k) \<noteq> 0 \<and> i \<le> n)"
   proof (rule find_non0'_LEAST)
     have "find_non0' i k A \<noteq> None" using True unfolding find_non0'_def Let_def
       by (auto split: list.split)
          (metis (mono_tags, lifting) atLeastLessThan_iff atLeastLessThan_upt empty_filter_conv)
     thus "find_non0' i k A = Some (the (find_non0' i k A))" by auto
   qed (auto)
   show ?thesis unfolding echelon_form_of_column_k_JNF_def Let_def f_rw using True by auto
 next
   case False
   then show ?thesis unfolding echelon_form_of_column_k_JNF_def by auto
 qed
 
 
 subsection \<open>Instantiation of the HNF-algorithm with modulo-operation\<close>
 
 text \<open>We currently use a Boolean flag to indicate whether standard-mod or symmetric modulo
   should be used.\<close>
 
 lemma sym_mod: "proper_mod_operation sym_mod sym_div" 
   by (unfold_locales, auto simp: sym_mod_sym_div)
 
 lemma standard_mod: "proper_mod_operation (mod) (div)" 
   by (unfold_locales, auto, intro HOL.nitpick_unfold(7))
 
 definition HNF_algorithm :: "bool \<Rightarrow> int mat \<Rightarrow> int mat" where
   "HNF_algorithm use_sym_mod = (if use_sym_mod 
     then mod_operation.HNF_algorithm sym_mod False else mod_operation.HNF_algorithm (mod) True)" 
 
 definition HNF_algorithm_from_HA :: "bool \<Rightarrow> int mat \<Rightarrow> int mat" where
   "HNF_algorithm_from_HA use_sym_mod = (if use_sym_mod 
     then mod_operation.HNF_algorithm_from_HA sym_mod else mod_operation.HNF_algorithm_from_HA (mod))"
 
 
 corollary is_sound_HNF'_HNF_algorithm:
    "is_sound_HNF' (HNF_algorithm use_sym_mod) (range ass_function_euclidean) (\<lambda>c. range (res_int c))"  
   using proper_mod_operation.is_sound_HNF'_HNF_algorithm[OF sym_mod]
     proper_mod_operation.is_sound_HNF'_HNF_algorithm[OF standard_mod]
   unfolding HNF_algorithm_def by (cases use_sym_mod, auto)
 
 corollary is_sound_HNF'_HNF_algorithm_from_HA:
    "is_sound_HNF' (HNF_algorithm_from_HA use_sym_mod) (range ass_function_euclidean) (\<lambda>c. range (res_int c))"  
   using proper_mod_operation.is_sound_HNF'_HNF_algorithm_from_HA[OF sym_mod]
     proper_mod_operation.is_sound_HNF'_HNF_algorithm_from_HA[OF standard_mod]
   unfolding HNF_algorithm_from_HA_def by (cases use_sym_mod, auto)
 
 
 (*Examples:*)
 (*Rectangular matrix (6x4)*)
 value [code]"let A = mat_of_rows_list 4 (
   [[0,3,1,4],
    [7,1,0,0],
    [8,0,19,16],
    [2,0,0,3::int],
    [9,-3,2,5],
    [6,3,2,4]]) in
   show (HNF_algorithm True A)"
 
 (*Rectangular matrix (4x6)*)
 
 value [code]"let A = mat_of_rows_list 6 (
   [[0,3,1,4,8,7],
    [7,1,0,0,4,1],
    [8,0,19,16,33,5],
    [2,0,0,3::int,-5,8]]) in
   show (HNF_algorithm False A)"
 
 (*Singular matrix*)
 value [code]"let A = mat_of_rows_list 6 (
   [[0,3,1,4,8,7],
    [7,1,0,0,4,1],
    [8,0,19,16,33,5],
    [0,3,1,4,8,7],
    [2,0,0,3::int,-5,8],
    [2,4,6,8,10,12]]) in
   show (Determinant.det A, HNF_algorithm True A)"
 
 (*Invertible matrix*)
 value [code]"let A = mat_of_rows_list 6 (
   [[0,3,1,4,8,7],
    [7,1,0,0,4,1],
    [8,0,19,16,33,5],
    [5,6,1,2,8,7],
    [2,0,0,3::int,-5,8],
    [2,4,6,8,10,12]]) in
   show (Determinant.det A, HNF_algorithm True A)"
 
 end
\ No newline at end of file
diff --git a/thys/Source_Coding_Theorem/Source_Coding_Theorem.thy b/thys/Source_Coding_Theorem/Source_Coding_Theorem.thy
--- a/thys/Source_Coding_Theorem/Source_Coding_Theorem.thy
+++ b/thys/Source_Coding_Theorem/Source_Coding_Theorem.thy
@@ -1,603 +1,603 @@
 (* Title:       One Part of Shannon's Source Coding Theorem
    Author:      Quentin Hibon <qh225@cl.cam.ac.uk>, Lawrence Paulson <lp15@cam.ac.uk>, 2014
    Maintainer:  Quentin Hibon <qh225@cl.cam.ac.uk>
 *)
 
 theory Source_Coding_Theorem
 imports "HOL-Probability.Information"
 begin
 section\<open>Basic types\<close>
 
 type_synonym bit = bool
 type_synonym bword = "bit list"
 type_synonym letter = nat
 type_synonym 'b word = "'b list"
 
 type_synonym 'b encoder = "'b word \<Rightarrow> bword"
 type_synonym 'b decoder = "bword \<Rightarrow> 'b word option"
 
 section\<open>Locale for the source coding theorem\<close>
 locale source_code = information_space +
   fixes fi :: "'b \<Rightarrow> real"
   fixes X :: "'a \<Rightarrow> 'b"
 
   assumes distr_i: "simple_distributed M X fi"
   assumes b_val: "b = 2"
 
   fixes enc::"'b encoder"
   fixes dec::"'b decoder"
   assumes real_code:
   "dec (enc x) = Some x"
   "enc w = [] \<longleftrightarrow> w = []"
   "x \<noteq> [] \<longrightarrow> enc x = enc [hd x] @ enc (tl x)"
 
 section\<open>Source coding theorem, direct: the entropy is a lower bound of the code rate\<close>
 context source_code
 begin
 subsection\<open>The letter set\<close>
 
 definition L :: "'b set" where
   "L \<equiv> X ` space M"
 
 lemma fin_L: "finite L"
     using L_def distr_i
     by auto
 
 lemma emp_L: "L \<noteq> {}"
     using L_def subprob_not_empty
     by auto
 
 subsection\<open>Codes and words\<close>
 
 abbreviation real_word :: "'b word \<Rightarrow> bool" where
   "real_word w \<equiv> (set w \<subseteq> L)"
 
 abbreviation k_words :: "nat \<Rightarrow> ('b word) set" where
   "k_words k \<equiv> {w. length w = k \<and> real_word w}"
 
 lemma rw_tail:
   assumes "real_word w"
 shows "w = [] \<or> real_word (tl w)"
     by (meson assms list.set_sel(2) subset_code(1))
 
 definition code_word_length :: "'e encoder \<Rightarrow> 'e \<Rightarrow> nat" where
   "code_word_length e l = length (e [l])"
 
 abbreviation cw_len :: "'b \<Rightarrow> nat" where
   "cw_len l \<equiv> code_word_length enc l"
 
 definition code_rate :: "'e encoder \<Rightarrow> ('a \<Rightarrow> 'e) \<Rightarrow> real" where
   "code_rate e Xo = expectation (\<lambda>a. (code_word_length e ((Xo) a)))"
 
 lemma fi_pos: "i\<in> L \<Longrightarrow> 0 \<le> fi i"
     using simple_distributed_nonneg[OF distr_i] L_def by auto
 
 lemma (in prob_space) simp_exp_composed:
   assumes X: "simple_distributed M X Px"
 shows "expectation (\<lambda>a. f (X a)) = (\<Sum>x \<in> X`space M. f x * Px x)"
   using distributed_integral[OF simple_distributed[OF X], of f]
     simple_distributed_nonneg[OF X]
     lebesgue_integral_count_space_finite[OF simple_distributed_finite[OF X], of "\<lambda>x. f x * Px x"]
   by (simp add:  ac_simps)
 
 lemma cr_rw:
   "code_rate enc X = (\<Sum>i \<in> X ` space M. fi i * cw_len i)"
     using simp_exp_composed[OF distr_i, of "cw_len"]
     by (simp add: mult.commute code_rate_def)
 
 abbreviation cw_len_concat :: "'b word \<Rightarrow> nat" where
   "cw_len_concat w \<equiv> foldr (\<lambda>x s. (cw_len x) + s) w 0"
 
 lemma cw_len_length: "cw_len_concat w = length (enc w)"
 proof (induction w)
     case Nil
     show ?case using real_code by simp
     case (Cons a w)
     have "cw_len_concat (a # w) = cw_len a + cw_len_concat w" by simp
     thus ?case using code_word_length_def real_code Cons
       by (metis length_append list.distinct(1) list.sel(1) list.sel(3))
 qed
 
 lemma maj_fold:
   assumes "\<And>l. l\<in>L \<Longrightarrow> f l \<le> bound"
   assumes "real_word w"
 shows "foldr (\<lambda>x s. f x + s) w 0 \<le> length w * bound"
     using assms
     by(induction w) (simp,fastforce)
 
 definition max_len :: "nat" where
   "max_len = Max ((\<lambda>x. cw_len x) ` L)"
 
 lemma max_cw:
   "l \<in> L \<Longrightarrow> cw_len l \<le> max_len"
     by (simp add: max_len_def fin_L)
 
 subsection\<open>Related to the Kraft theorem\<close>
 definition \<K> :: "real" where
   "\<K> = (\<Sum>i\<in>L. 1 / b ^ (cw_len i))"
 
 lemma pos_cw_len: "0 < 1 / b ^ cw_len i" using b_gt_1 by simp
 
 lemma \<K>_pos: "0 < \<K>"
     using emp_L fin_L pos_cw_len sum_pos \<K>_def
     by metis
 
 lemma \<K>_pow: "\<K> = (\<Sum>i\<in>L. 1 / b powr cw_len i)"
     using powr_realpow b_gt_1
     by (simp add: \<K>_def)
 
 lemma k_words_rel:
   "k_words (Suc k) = {w. (hd w \<in> L \<and> tl w \<in> k_words k \<and> w \<noteq> [])}"
 proof
     fix k
     show "k_words (Suc k) \<subseteq> {w. (hd w \<in> L \<and> tl w \<in> k_words k \<and> w \<noteq> [] )}" (is "?l \<subseteq> ?r")
   proof
       fix w
       assume w_kw: "w \<in> k_words (Suc k)"
       hence "real_word w" by simp
       hence "hd w \<in> L"
         by (metis (mono_tags) w_kw hd_in_set list.size(3) mem_Collect_eq nat.distinct(1) subset_code(1))
       moreover have "length w = Suc k" using w_kw by simp
       moreover hence "w \<noteq> []" by auto
       moreover have "real_word (tl w)" using \<open>real_word w\<close> calculation(3) rw_tail by auto
       ultimately show "w \<in> ?r" using w_kw by simp
   qed
 next
     fix k
     show "k_words (Suc k) \<supseteq> {w. (hd w \<in> L \<and> tl w \<in> k_words k \<and> w \<noteq> [])}"
   proof
       fix w
       assume asm: "w \<in> {w. hd w \<in> L \<and> tl w \<in> {w. length w = k \<and> real_word w} \<and> w \<noteq> []}"
       hence "hd w \<in> L \<and> length (tl w) = k \<and> real_word (tl w)" by simp
       hence "real_word w"
         by (metis empty_iff insert_subset list.collapse list.set(1) set_simps(2) subsetI)
       moreover hence "length w = Suc k" using asm by auto
       ultimately show "w \<in> k_words (Suc k)" by simp
   qed
 qed
 
 lemma bij_k_words:
 shows "bij_betw (\<lambda>wi. Cons (fst wi) (snd wi)) (L \<times> k_words k) (k_words (Suc k))"
     unfolding bij_betw_def
 proof
     fix k
     let ?f = "(\<lambda>wi. Cons (fst wi) (snd wi))"
     let ?S = "L \<times> (k_words k)"
     let ?T = "k_words (Suc k)"
     show "inj_on ?f ?S" by (simp add: inj_on_def)
     show "?f`?S = ?T"
   proof (rule ccontr)
       assume "?f ` ?S \<noteq> ?T"
       hence "\<exists>w. w\<in> ?T \<and> w \<notin> ?f`?S" by auto
       then obtain w where asm: "w\<in> ?T \<and> w \<notin> ?f`?S" by blast
       hence "w = ?f (hd w,tl w)" using k_words_rel by simp
       moreover have "(hd w,tl w) \<in> ?S" using k_words_rel asm by simp
       ultimately have "w \<in> ?f`?S" by blast
       thus "False" using asm by simp
   qed
 qed
 
 lemma finite_k_words: "finite (k_words k)"
 proof (induct k)
     case 0
     show ?case by simp
     case (Suc n)
     thus ?case using bij_k_words bij_betw_finite fin_L by blast
 qed
 
 lemma cartesian_product:
   fixes f::"('c \<Rightarrow> real)"
   fixes g::"('d \<Rightarrow> real)"
   assumes "finite A"
   assumes "finite B"
 shows "(\<Sum>b\<in>B. g b) * (\<Sum>a\<in>A. f a) = (\<Sum>ab\<in>A\<times>B. f (fst ab) * g (snd ab))"
     using bilinear_times bilinear_sum[where h="(\<lambda>x y. x * y)" and f="f" and g="g"] assms
     by (metis (erased, lifting) sum.cong split_beta' Groups.ab_semigroup_mult_class.mult.commute)
 
 lemma \<K>_power:
 shows "\<K>^k = (\<Sum>w \<in> (k_words k). 1 / b^(cw_len_concat w))"
 proof (induct k)
     case 0
     have "k_words 0 = {[]}" by auto
     thus ?case by simp
 next
     case (Suc n)
     have " \<K> ^Suc n = \<K> ^n * \<K> " by simp
     also have "\<dots> = (\<Sum>w \<in> k_words n. 1 / b^cw_len_concat w) * (\<Sum>i\<in>L. 1 / b^cw_len i)"
       using Suc.hyps \<K>_def by auto
     also have "\<dots> = (\<Sum>wi \<in> L \<times> k_words n. 1/b^cw_len (fst wi) * (1 / b^cw_len_concat (snd wi)))"
       using fin_L finite_k_words cartesian_product
       by blast
     also have "\<dots> = (\<Sum>wi \<in> L \<times> k_words n. 1 / b^(cw_len_concat (snd wi) + cw_len (fst wi)))"
       by (metis (no_types, lifting) power_add add.commute power_one_over)
     also have "\<dots> = (\<Sum>wi \<in> L \<times> k_words n. 1 / b^cw_len_concat (fst wi # snd wi))"
       by (metis (erased, lifting) add.commute comp_apply foldr.simps(2))
     also have "\<dots> = (\<Sum>w \<in> (k_words (Suc n)). 1 / b^(cw_len_concat w))"
       using sum.reindex_bij_betw [OF bij_k_words] by fastforce
     finally show ?case by simp
 qed
 
 lemma bound_len_concat:
 shows "w \<in> k_words k \<Longrightarrow> cw_len_concat w \<le> k * max_len"
     using max_cw maj_fold by blast
 
 subsection\<open>Inequality of the kraft sum (source coding theorem, direct)\<close>
 subsubsection\<open>Sum manipulation lemmas and McMillan theorem\<close>
 
 lemma sum_vimage_proof:
   fixes g::"nat \<Rightarrow> real"
   assumes "\<And>w. f w < bd"
 shows "finite S \<Longrightarrow> (\<Sum>w\<in>S. g (f w)) = (\<Sum> m=0..<bd. (card ((f-`{m}) \<inter> S) )* g m)"
 (is "_ \<Longrightarrow> _ = (\<Sum> m=0..<bd. ?ff m S)")
 proof (induct S rule: finite_induct)
     case empty
     show ?case by simp
 next
     case (insert x F)
     let ?rr = "(\<Sum>m = 0..<bd. ?ff m (insert x F))"
     have "(f x) \<in> {0..<bd}" using assms by simp
     hence "\<And>h::(nat \<Rightarrow> real). (\<Sum>m=0..<bd. h m) = (\<Sum>y\<in>({0..<bd} - {f x}).h y) + h (f x)"
       by (metis diff_add_cancel finite_atLeastLessThan sum_diff1)
     moreover hence
     "(\<Sum>m = 0..<bd. ?ff m (insert x F))
     = (\<Sum>m\<in>{0..<bd} - {f x}. ?ff m (insert x F)) + card (f -` {f x} \<inter> F) * g (f x) + g (f x)"
       by (simp add: semiring_normalization_rules(2), simp add: insert)
     ultimately have "(\<Sum>m = 0..<bd. ?ff m (insert x F)) = (\<Sum>m\<in>{0..<bd}. ?ff m F) + g (f x)"
       by fastforce
     thus ?case using insert by simp
 qed
 
 lemma sum_vimage:
   fixes g::"nat \<Rightarrow> real"
   assumes bounded: "\<And>w. w \<in> S \<Longrightarrow> f w < bd" and "0 < bd"
   assumes finite: "finite S"
 shows "(\<Sum>w\<in>S. g (f w)) = (\<Sum> m=0..<bd. (card ((f-`{m}) \<inter> S) ) * g m)"
 (is "?s1 = ?s2")
 proof -
     let ?ff = "(\<lambda>x. if x\<in>S then f x else 0)"
     let ?ss1 = "(\<Sum>w\<in>S. g (?ff w))"
     let ?ss2 = "(\<Sum> m=0..<bd. (card ((?ff-`{m}) \<inter> S) ) * g m)"
     have "?s1 =?ss1" by simp
     moreover have"\<And>m. ?ff -`{m} \<inter> S = f-`{m} \<inter> S" by auto
     moreover hence "?s2 = ?ss2" by simp
     moreover have "\<And>w . ?ff w < bd" using assms by simp
     moreover hence "?ss1 = ?ss2" using sum_vimage_proof[of "?ff"] finite by blast
     ultimately show "?s1 = ?s2" by metis
 qed
 
 lemma \<K>_rw:
   "(\<Sum>w \<in> (k_words k). 1 / b^(cw_len_concat w)) = (\<Sum>m=0..<Suc (k*max_len). card (k_words k \<inter>
 ((cw_len_concat) -` {m})) * (1 / b^m))" (is "?L = ?R")
 proof -
     have "\<And>w. w \<in> k_words k \<Longrightarrow> cw_len_concat w < Suc ( k * max_len)"
       by (simp add: bound_len_concat le_imp_less_Suc)
     moreover have
     "?R = (\<Sum>m = 0..<Suc (k * max_len).
   (card (cw_len_concat -` {m} \<inter> k_words k)) * (1 / b ^ m))"
       by (metis Int_commute)
     moreover have "0 < Suc (k*max_len)" by simp
     ultimately show ?thesis
       using finite_k_words
     sum_vimage[where f="cw_len_concat" and g = "\<lambda>i. 1/ (b^i)"]
       by fastforce
 qed
 
 definition set_of_k_words_length_m :: "nat \<Rightarrow> nat \<Rightarrow> 'b word set" where
   "set_of_k_words_length_m k m = {xk. xk \<in> k_words k} \<inter> (cw_len_concat)-`{m}"
 
 lemma am_inj_code: "inj_on enc ((cw_len_concat)-`{m})" (is "inj_on _ ?s")
   using inj_on_def[of enc "?s"] real_code
   by (metis option.inject)
 
 lemma img_inc: "enc`cw_len_concat-`{m} \<subseteq> {bl. length bl = m}" using cw_len_length by auto
 
 lemma bool_lists_card: "card {bl::bool list. length bl = m} = b^m"
  using card_lists_length_eq[of "UNIV::bool set"]
   by (simp add: b_val)
 
 lemma bool_list_fin: "finite {bl::bool list. length bl = m}"
   using finite_lists_length_eq[of "UNIV::bool set"]
   by (simp add: b_val)
 
 lemma set_of_k_words_bound:
 shows "card (set_of_k_words_length_m k m) \<le> b^m" (is "?c \<le> ?b")
 proof -
     have card_w_len_m_bound: "card (cw_len_concat-`{m}) \<le> b^m"
       by (metis (no_types, lifting) am_inj_code bool_list_fin bool_lists_card card_image card_mono
     img_inc of_nat_le_iff)
     have "set_of_k_words_length_m k m \<subseteq> (cw_len_concat)-`{m}"
       by (simp add: set_of_k_words_length_m_def)
     hence "card (set_of_k_words_length_m k m) \<le> card ((cw_len_concat)-`{m})"
       by (metis (no_types, lifting) am_inj_code bool_list_fin card.infinite card_0_eq
     card_image card_mono empty_iff finite_subset img_inc inf_img_fin_dom)
     thus ?thesis using card_w_len_m_bound by simp
 qed
 
 lemma empty_set_k_words:
   assumes "0 < k"
 shows "set_of_k_words_length_m k 0 = {}"
 proof(rule ccontr)
     assume "\<not> set_of_k_words_length_m k 0 = {}"
     hence "\<exists>x. x \<in> set_of_k_words_length_m k 0" by auto
     then obtain x where x_def: "x \<in> set_of_k_words_length_m k 0" by auto
     hence "x \<noteq> []" unfolding set_of_k_words_length_m_def using assms by auto
     moreover have "cw_len_concat (hd x#tl x) = cw_len_concat (tl x) + cw_len (hd x)"
       by (metis add.commute comp_apply foldr.simps(2))
     moreover have "enc [(hd x)] \<noteq> []" using assms real_code by blast
     moreover hence "0 < cw_len (hd x)" unfolding code_word_length_def by simp
     ultimately have "x \<notin> set_of_k_words_length_m k 0" by (simp add:set_of_k_words_length_m_def)
     thus "False" using x_def by simp
 qed
 
 lemma \<K>_rw2:
   assumes "0 < k"
 shows "(\<Sum>m=0..<Suc (k * max_len). card (set_of_k_words_length_m k m)/ b^m) \<le> (k * max_len)"
 proof -
     have
     "(\<Sum>m=1..<Suc (k * max_len). card (set_of_k_words_length_m k m) / b^m)
     \<le> (\<Sum>m=1..<Suc(k * max_len). b^m / b^m)"
       using set_of_k_words_bound b_val
     Groups_Big.sum_mono[of "{1..<Suc(k * max_len)}"
     "(\<lambda>m. (card (set_of_k_words_length_m k m))/b^m)" "\<lambda>m. b^m /b^m"]
       by simp
     moreover have"(\<Sum>m=1..<Suc(k * max_len). b^m / b^m) = (\<Sum>m=1..<Suc(k *max_len). 1)"
       using b_gt_1 by simp
     moreover have "(\<Sum>m=1..<Suc(k * max_len). 1) = (k * max_len)"
       by simp
     ultimately have
     "(\<Sum>m = 1..<Suc (k * max_len). card (set_of_k_words_length_m k m) / b ^ m) \<le> k * max_len"
       by (metis One_nat_def card_atLeastLessThan card_eq_sum diff_Suc_Suc real_of_card)
     thus ?thesis using empty_set_k_words assms
       by (simp add: sum_shift_lb_Suc0_0_upt split: if_split_asm)
 qed
 
 lemma \<K>_power_bound :
   assumes "0 < k"
 shows " \<K>^k \<le> k * max_len"
     using assms \<K>_power \<K>_rw \<K>_rw2
     by (simp add: set_of_k_words_length_m_def)
 
 theorem McMillan :
 shows "\<K> \<le> 1"
 proof -
     have ineq: "\<And>k. 0 < k \<Longrightarrow> \<K> \<le> root k k * root k max_len"
       using \<K>_pos \<K>_power_bound
       by (metis (no_types, opaque_lifting) not_less of_nat_0_le_iff of_nat_mult power_strict_mono real_root_mult real_root_pos_pos_le real_root_pos_unique real_root_power)
     hence "0 < max_len \<Longrightarrow> (\<lambda>k. root k k * root k max_len) \<longlonglongrightarrow> 1"
       by (auto intro!: tendsto_eq_intros LIMSEQ_root LIMSEQ_root_const)
     moreover have "\<forall>n\<ge>1. \<K> \<le> root n n * root n max_len"
       using ineq by simp
     moreover have "max_len = 0 \<Longrightarrow> \<K> \<le> 1" using ineq by fastforce
     ultimately show " \<K> \<le> 1" using LIMSEQ_le_const by blast
 qed
 
 lemma entropy_rw: "\<H>(X) = -(\<Sum>i \<in> L. fi i * log b (fi i))"
     using entropy_simple_distributed[OF distr_i]
     by (simp add: L_def)
 
 subsubsection\<open>Technical lemmas about the logarithm\<close>
 lemma log_mult_ext3:
   "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> 0 < z \<Longrightarrow> x * log b (x*y*z) = x * log b (x*y) + x * log b z"
     by(cases "x=0")(simp add: log_mult_eq abs_of_pos distrib_left less_eq_real_def)+
 
 lemma log_mult_ext2:
   "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> x * log b (x*y) = x * log b x + x * log b y"
     using log_mult_ext3[where y=1] by simp
 
 subsubsection \<open>KL divergence and properties\<close>
 definition KL_div ::"'b set \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> real" where
   "KL_div S a d = (\<Sum> i \<in> S. a i * log b (a i / d i))"
 
 lemma KL_div_mul:
   assumes "0 < d" "d \<le> 1"
   assumes "\<And>i. i\<in>S \<Longrightarrow> 0 \<le> a i"
   assumes "\<And>i. i\<in>S \<Longrightarrow> 0 < e i"
 shows "KL_div S a e \<ge> KL_div S a (\<lambda>i. e i / d)"
     unfolding KL_div_def
 proof -
     {
     fix i
     assume "i\<in>S"
     hence "a i / (e i / d) \<le> a i / e i" using assms
       by (metis (no_types) div_by_1 frac_le less_imp_triv not_less)
     hence "log b (a i / (e i / d)) \<le> log b (a i / e i)" using assms(1)
       by (metis (full_types) b_gt_1 divide_divide_eq_left inverse_divide le_less_linear log_le
     log_neg_const order_refl times_divide_eq_right zero_less_mult_iff)
     }
     thus "(\<Sum>i\<in>S. a i * log b (a i / (e i / d))) \<le> (\<Sum>i\<in>S. a i * log b (a i / e i))"
       by (meson mult_left_mono assms sum_mono)
 qed
 
 lemma KL_div_pos:
   fixes a e::"'b \<Rightarrow> real"
   assumes fin: "finite S"
   assumes nemp: "S \<noteq> {}"
   assumes non_null: "\<And>i. i\<in>S \<Longrightarrow> 0 < a i" "\<And>i. i\<in> S \<Longrightarrow> 0 < e i"
   assumes sum_a_one: "(\<Sum> i \<in> S. a i) = 1"
   assumes sum_c_one: "(\<Sum> i \<in> S. e i) = 1"
 shows "0 \<le> KL_div S a e"
     unfolding KL_div_def
 proof -
     let ?f = "\<lambda>i. e i / a i"
     have f_pos: "\<And>i. i\<in>S \<Longrightarrow> 0 < ?f i"
       using non_null
       by simp
     have a_pos: "\<And>i. i\<in> S \<Longrightarrow> 0 \<le> a i"
       using non_null
       by (simp add: order.strict_implies_order)
     have "- log b (\<Sum>i\<in>S. a i * e i / a i) \<le> (\<Sum>i\<in>S. a i * - log b (e i / a i))"
-      using convex_on_sum[OF fin nemp  minus_log_convex[OF b_gt_1] convex_real_interval(3)
+      using convex_on_sum[OF fin nemp  minus_log_convex[OF b_gt_1] 
                              sum_a_one a_pos, of "\<lambda>i. e i / a i"] f_pos by simp
     also have "-log b (\<Sum>i\<in>S. a i * e i / a i) = -log b (\<Sum>i\<in>S. e i)"
   proof -
       from non_null(1) have "\<And>i. i \<in> S \<Longrightarrow> a i * e i / a i = e i" by force
       thus ?thesis by simp
   qed
     finally have "0 \<le> (\<Sum>i\<in>S. a i * - log b (e i / a i))"
       by (simp add: sum_c_one)
     thus "0 \<le> (\<Sum>i\<in>S. a i * log b (a i / e i))"
       using b_gt_1 log_divide non_null
       by simp
 qed
 
 lemma KL_div_pos_emp:
   "0 \<le> KL_div {} a e" by (simp add: KL_div_def)
 
 lemma KL_div_pos_gen:
   fixes a d::"'b \<Rightarrow> real"
   assumes fin: "finite S"
   assumes non_null: "\<And>i. i\<in>S \<Longrightarrow> 0 < a i" "\<And>i. i\<in> S \<Longrightarrow> 0 < d i"
   assumes sum_a_one: "(\<Sum> i \<in> S. a i) = 1"
   assumes sum_d_one: "(\<Sum> i \<in> S. d i) = 1"
 shows "0 \<le> KL_div S a d"
     using KL_div_pos KL_div_pos_emp assms by metis
 
 theorem KL_div_pos2:
   fixes a d::"'b \<Rightarrow> real"
   assumes fin: "finite S"
   assumes non_null: "\<And>i. i\<in>S \<Longrightarrow> 0 \<le> a i" "\<And>i. i\<in> S \<Longrightarrow> 0 < d i"
   assumes sum_a_one: "(\<Sum> i \<in> S. a i) = 1"
   assumes sum_c_one: "(\<Sum> i \<in> S. d i) = 1"
 shows "0 \<le> KL_div S a d"
 proof -
     have "S = (S \<inter> {i. 0 < a i}) \<union> (S \<inter> {i. 0 = a i})" using non_null(1) by fastforce
     moreover have "(S \<inter> {i. 0 < a i}) \<inter> (S \<inter> {i. 0 = a i}) = {}" by auto
     ultimately have
     eq: "KL_div S a d = KL_div (S \<inter> {i. 0 < a i}) a d + KL_div (S \<inter> {i. 0 = a i}) a d"
       unfolding KL_div_def
       by (metis (mono_tags, lifting) fin finite_Un sum.union_disjoint)
     have "KL_div (S \<inter> {i. 0 = a i}) a d = 0" unfolding KL_div_def by simp
     hence "KL_div S a d = KL_div (S \<inter> {i. 0 < a i}) a d" using eq by simp
     moreover have "0 \<le> KL_div (S \<inter> {i. 0 < a i}) a d"
   proof(cases "(S \<inter> {i. 0 < a i}) = {}")
       case True
       thus ?thesis unfolding KL_div_def by simp
   next
       case False
       let ?c = "\<lambda>i. d i / (\<Sum>j \<in>(S \<inter> {i. 0 < a i}). d j)"
       have 1: "(\<And>i. i \<in> S \<inter> {i. 0 < a i} \<Longrightarrow> 0 < a i)" by simp
       have 2: "(\<And>i. i \<in> S \<inter> {i. 0 < a i} \<Longrightarrow> 0 < ?c i)"
         by (metis False IntD1 divide_pos_pos fin finite_Int non_null(2) sum_pos)
       have 3: "(\<Sum>i\<in> (S \<inter> {i. 0 < a i}). a i) = 1"
         using sum.cong[of S, of S, of "(\<lambda>x. if x \<in> {i. 0 < a i} then a x else 0)", of a]
       sum.inter_restrict[OF fin, of a] non_null(1) sum_a_one
         by fastforce
       have "(\<Sum>i\<in>S \<inter> {j. 0 < a j}. ?c i) = (\<Sum>i\<in>S \<inter> {j. 0 < a j}. d i) / (\<Sum>i\<in>S \<inter> {j. 0 < a j}. d i)"
         by (metis sum_divide_distrib)
       hence 5: "(\<Sum>i\<in>S \<inter> {j. 0 < a j}. ?c i) = 1" using 2 False by force
       hence "0 \<le> KL_div (S \<inter> {j. 0 < a j}) a ?c"
         using KL_div_pos_gen[
       OF finite_Int[OF disjI1, of S, of "{j. 0 < a j}"], of a, of ?c
       ] 1 2 3
         by (metis fin)
       have fstdb: "0 < (\<Sum>i\<in>S \<inter> {i. 0 < a i}. d i)" using non_null(2) False
         by (metis Int_Collect fin finite_Int sum_pos)
       have 6: "0 \<le> KL_div (S \<inter> {i. 0 < a i}) a (\<lambda>i. d i / (\<Sum>i\<in>(S \<inter> {i. 0 < a i}). d i))"
         using 2 3 5
         KL_div_pos_gen[
       OF finite_Int[OF disjI1, OF fin], of "{i. 0 < a i}", of "a", of "?c"
       ]
         by simp
       hence
       "KL_div (S \<inter> {j. 0 < a j}) a (\<lambda>i. d i / (\<Sum>i\<in>(S \<inter> {i. 0 < a i}). d i)) \<le> KL_div (S \<inter> {j. 0 < a j}) a d"
         using non_null sum.inter_restrict[OF fin, of d, of "{i. 0 < a i}"]
         sum_mono[of S, of "(\<lambda>x. if x \<in> {i. 0 < a i} then d x else 0)", of d] non_null(2) sum_c_one
         non_null(2) fstdb KL_div_mul
         by force
       moreover have "0 \<le> KL_div (S \<inter> {j. 0 < a j}) a (\<lambda>i. d i / (\<Sum>i\<in>(S \<inter> {i. 0 < a i}). d i))"
         using KL_div_pos_gen[ OF finite_Int[OF disjI1, OF fin]] using 2 3 5 by fastforce
       ultimately show "0 \<le> KL_div (S \<inter> {j. 0 < a j}) a d" by simp
   qed
     ultimately show ?thesis by simp
 qed
 
 lemma sum_div_1:
   fixes f::"'b \<Rightarrow> 'c::field"
   assumes "(\<Sum>i\<in>A. f i) \<noteq> 0"
 shows "(\<Sum>i\<in>A. f i / (\<Sum>j\<in>A. f j)) = 1"
     by (metis (no_types) assms right_inverse_eq sum_divide_distrib)
 
 theorem rate_lower_bound:
 shows "\<H>(X) \<le> code_rate enc X"
 proof -
     let ?cr = "code_rate enc X"
     let ?r = "(\<lambda>i. 1 / ((b powr cw_len i) * \<K>))"
     have pos_pi: "\<And>i. i \<in> L \<Longrightarrow> 0 \<le> fi i" using fi_pos by simp
     {
     fix i
     assume "i \<in> L"
     hence
     "fi i * (log b (1 / (1 / b powr (cw_len i))) + log b (fi i))
     = fi i * log b (fi i / (1 / b powr (cw_len i)))"
       using log_mult_ext2 [OF pos_pi, of i] b_gt_1
       by simp (simp add: algebra_simps)
     }
     hence eqpi:
     "\<And>i. i\<in> L \<Longrightarrow> fi i * (log b (1 / (1 / b powr (cw_len i))) + log b (fi i))
     = fi i * log b (fi i / (1 / b powr (cw_len i)))"
       by simp
     have sum_one_L: "(\<Sum> i \<in> L. fi i) = 1"
       using simple_distributed_sum_space[OF distr_i] by (simp add: L_def)
     {
     fix i
     assume "i \<in> L"
     hence h1: "0 \<le> fi i" using pos_pi by blast
     have h2: "0 < \<K> / (1/b powr cw_len i)" using b_gt_1 \<K>_pos by auto
     have h3: "0 < 1 / \<K>" using \<K>_pos by simp
     have
     "fi i * log b (fi i * \<K> / (1/b powr cw_len i) * (1/ \<K>)) =
     fi i * log b (fi i * \<K> / (1/b powr cw_len i)) + fi i * log b (1/ \<K>)"
       using log_mult_ext3[OF h1 h2 h3]
       by (metis times_divide_eq_right)
     } hence big_eq:
     "\<And>i. i \<in> L \<Longrightarrow> fi i * log b (fi i * \<K> / (1/b powr cw_len i) * (1 / \<K>)) =
     fi i * log b (fi i * \<K> / (1/b powr cw_len i)) + fi i * log b (1 / \<K>)"
       by (simp add: inverse_eq_divide)
     have 1: "?cr - \<H>(X) = (\<Sum>i \<in> L. fi i * cw_len i) + (\<Sum>i \<in> L. fi i * log b (fi i))"
       using \<K>_def entropy_rw cr_rw L_def by simp
     also have 2: "(\<Sum>i\<in>L. fi i * cw_len i) = (\<Sum>i \<in> L. fi i * (-log b (1/(b powr (cw_len i)))))"
       using b_gt_1 log_divide by simp
     also have "\<dots> = -1 * (\<Sum>i \<in> L. fi i * (log b (1/(b powr (cw_len i)))))"
       using sum_distrib_left[of "-1" "(\<lambda>i. fi i * (- 1 * log b (1 / b powr (cw_len i))))" L]
       by simp
     finally have
     "?cr - \<H>(X) = -(\<Sum>i \<in> L. fi i * log b (1/b powr cw_len i)) + (\<Sum>i \<in> L. fi i * log b (fi i))"
       by simp
     have "?cr - \<H>(X) = (\<Sum>i \<in> L. fi i * ((log b (1/ (1/(b powr (cw_len i))))) + log b (fi i)))"
       using b_gt_1 1
       by (simp add: distrib_left sum.distrib)
     also have "\<dots> = (\<Sum>i \<in> L. fi i *((log b (fi i / (1/(b powr (cw_len i)))))))"
       using Finite_Cartesian_Product.sum_cong_aux[OF eqpi] by simp
     also from big_eq have
     "\<dots> = (\<Sum>i\<in>L. fi i * (log b (fi i * \<K> / (1 / b powr (cw_len i))))) + (\<Sum>i \<in> L. fi i) * log b (1/ \<K>)"
       using \<K>_pos
       by (simp add: sum_distrib_right sum.distrib)
     also have "\<dots> = (\<Sum>i\<in>L. fi i * (log b (fi i * \<K> / (1 / b powr (cw_len i))))) - log b (\<K>)"
       using \<K>_pos
       by (simp add: log_inverse_eq divide_inverse sum_one_L)
     also have "\<dots> = (\<Sum> i \<in> L. fi i * log b (fi i / ?r i)) - log b (\<K>)"
       by (metis (mono_tags, opaque_lifting) divide_divide_eq_left divide_divide_eq_right)
     also have "\<dots> = KL_div L fi ?r - log b ( \<K>)"
       using b_gt_1 \<K>_pos log_inverse KL_div_def
       by simp
     also have "\<dots> = KL_div L fi ?r + log b (1 / \<K>)"
       using log_inverse b_val \<K>_pos
       by (simp add: inverse_eq_divide)
     finally have code_ent_kl_log: "?cr - \<H>(X) = KL_div L fi ?r + log b (1 / \<K>)" by simp
     have "(\<Sum>i\<in>L. ?r i) = 1"
       using sum_div_1[of "\<lambda>i. 1 / (b powr (cw_len i))"] \<K>_pos \<K>_pow
       by simp
     moreover have "\<And>i. 0 < ?r i" using b_gt_1 \<K>_pos by simp
     moreover have "(\<Sum>i\<in>L. fi i) = 1" using sum_one_L by simp
     ultimately have "0 \<le> KL_div L fi ?r"
       using KL_div_pos2[OF fin_L fi_pos] by simp
     hence "log b (1 / \<K>) \<le> ?cr - \<H>(X)" using code_ent_kl_log by simp
     moreover from McMillan have "0 \<le> log b (1 / \<K>)"
       using \<K>_pos
       by (simp add: b_gt_1)
     ultimately show ?thesis by simp
 qed
 
 end
 
 end