Quelle  SVP_vec.thy

theory SVP_vec

imports 
  BHLE
begin

section ‹SVP in $\ell_\infty$›
text ‹The reduction of SVP to BHLE in $l_\infty$ norm›

text ‹The shortest vector problem.›
definition is_shortest_vec :: "int_lattice ==> int vec ==> bool" where
  "is_shortest_vec L v ≡ (is_lattice L) ∧ (∀x∈L. ∥x∥_\<infinity> ≥ ∥v∥_\<infinity> ∧ v∈L) "


text ‹The decision problem associated with solving SVP exactly.›
definition gap_svp :: "(int_lattice × int) set" where
  "gap_svp ≡ {(L, r). (is_lattice L) ∧ (dim_lattice L ≥ 2) ∧
      (∃v∈L. ∥v∥_\<infinity> ≤ r ∧ v ≠ 0_v (dim_vec v))}"

text ‹Generate a lattice to solve SVP for reduction.›

text ‹Here, the factor $K''$ from the paper \cite{EmBo81}
 was changed to be $2\cdot\mathbf{k}\cdot(k+1)\cdot \sum_i \mathbf{| a_i |}$
  order for the proofs to finish.›

definition gen_svp_basis :: "int vec ==> int ==> int mat" where
  "gen_svp_basis a k = mat (dim_vec a + 1) (dim_vec a + 1)
    (λ (i, j). if (i < dim_vec a) ∧ (j< dim_vec a) then (if i = j then 1 else 0)
      else (if (i < dim_vec a) ∧ (j ≥ dim_vec a) then 0
      else (if (i ≥ dim_vec a) ∧ (j < dim_vec a) then (k+1) * (a $ j)
      else 2*k*(k+1)* (∑ i ∈ {0 ..< dim_vec a}. ∣a $ i∣) +1 )))"


text ‹Reduction SVP to bounded homogeneous linear equation problem in $\ell_\infty$ norm.›
definition reduce_svp_bhle:: "(int vec × int) ==> (int_lattice × int)" where
  "reduce_svp_bhle ≡ (λ (a, k). (gen_lattice (gen_svp_basis a k), k))"



text ‹Lemmas for proof›

lemma gen_svp_basis_mult: 
  assumes "dim_vec z = dim_vec a + 1"
  shows "(gen_svp_basis a k) *_v z = vec (dim_vec a + 1)
         (λi. if i < dim_vec a then z$i else (k+1) * (∑ i ∈ {0 ..< dim_vec a}. z $ i * a $ i) +
              (2*k*(k+1)* (∑ i ∈ {0 ..< dim_vec a}. ∣a $ i∣) +1) * (z$(dim_vec a)))"
using assms proof (subst vec_eq_iff, safe, goal_cases)
case 1
  then show ?case using assms unfolding gen_svp_basis_def by auto
next
case (2 i)
  then show ?case proof (cases "i<dim_vec a")
  case True
    have "{0..<dim_vec a} = insert i {0..<dim_vec a}" using True by auto
    then have "(∑ia = 0..<dim_vec a. (if i = ia then 1 else 0) * z $ ia) =
      (∑ia ∈ insert i {0..<dim_vec a}. (if i = ia then 1 else 0) * z $ ia)" by auto
    also have "… = z$i" by (subst sum.insert_remove, auto) 
    finally have "(∑ia = 0..<dim_vec a. (if i = ia then 1 else 0) * z $ ia) = z $ i" 
      by auto
    then show ?thesis unfolding mult_mat_vec_def gen_svp_basis_def scalar_prod_def 
      using True assms by auto
  next
  case False
    then have "i = dim_vec a" using 2 by auto
    then show ?thesis unfolding gen_svp_basis_def using assms 
      by (auto simp add: scalar_prod_def sum_distrib_left mult.commute mult.left_commute)
  qed
qed

lemma gen_svp_basis_mult_real: 
  assumes "dim_vec z = dim_vec a + 1"
  shows "real_of_int_mat (gen_svp_basis a k) *_v z = vec (dim_vec a + 1)
         (λi. if i < dim_vec a then z$i else (k+1) * (∑ i ∈ {0 ..< dim_vec a}. z $ i * a $ i) +
              (2*k*(k+1)* (∑ i ∈ {0 ..< dim_vec a}. ∣a $ i∣) +1) * (z$(dim_vec a)))"
using assms proof (subst vec_eq_iff, safe, goal_cases)
case 1
  then show ?case using assms unfolding gen_svp_basis_def by auto
next
case (2 i)
  then show ?case proof (cases "i<dim_vec a")
  case True
    have "{0..<dim_vec a} = insert i {0..<dim_vec a}" using True by auto
    then have "(∑ia = 0..<dim_vec a. (if i = ia then 1 else 0) * z $ ia) =
      (∑ia ∈ insert i {0..<dim_vec a}. (if i = ia then 1 else 0) * z $ ia)" by auto
    also have "… = z$i" by (subst sum.insert_remove, auto) 
    finally have "(∑ia = 0..<dim_vec a. (if i = ia then 1 else 0) * z $ ia) = z $ i" 
      by auto
    then have "(∑ia = 0..<dim_vec a. real_of_int (if i = ia then 1 else 0) * z $ ia) = z $ i" 
      by (smt (verit, best) of_int_hom.hom_one of_int_hom.hom_zero sum.cong)
    then show ?thesis unfolding mult_mat_vec_def gen_svp_basis_def scalar_prod_def 
      using True assms by auto
  next
  case False
    then have "i = dim_vec a" using 2 by auto
    then show ?thesis unfolding gen_svp_basis_def using assms 
      by (auto simp add: scalar_prod_def sum_distrib_left mult.commute mult.left_commute)
  qed
qed

lemma gen_svp_basis_mult_last: 
  assumes "dim_vec z = dim_vec a + 1"
  shows "((gen_svp_basis a k) *_v z) $ (dim_vec a) =
         (k+1) * (∑ i ∈ {0 ..< dim_vec a}. z $ i * a $ i) +
              (2*k*(k+1)* (∑ i ∈ {0 ..< dim_vec a}. ∣a $ i∣) +1) * (z$(dim_vec a))"
using gen_svp_basis_mult[OF assms] by auto

text ‹The set generated by ‹gen_svp_basis› is indeed linearly independent.›
lemma is_indep_gen_svp_basis: 
  assumes "k>0"
  shows "is_indep (gen_svp_basis a k)"
unfolding is_indep_int_def
proof (safe, goal_cases)
case (1 z)
  have dim_row_dim_vec: "dim_row (gen_svp_basis a k) = dim_vec z" 
    using 1 unfolding gen_svp_basis_def by auto
  then have suc_dim_a_dim_z: "dim_vec z = dim_vec a + 1" unfolding gen_svp_basis_def by auto
  have alt1: "(real_of_int_mat (gen_svp_basis a k) *_v z) $ i = 0" if "i< dim_vec a +1"for i 
    using 1(1) that unfolding gen_svp_basis_def by auto
  have z_upto_last: "z$i = 0" if "i < dim_vec a" for i 
  proof -
    have elem: "(real_of_int_mat (gen_svp_basis a k) *_v z) $ i = z $ i" 
      using gen_svp_basis_mult_real[OF suc_dim_a_dim_z] that by auto
    show ?thesis by (subst elem[symmetric]) (use alt1[of i] that in ‹auto›) 
  qed
  moreover have "z $ (dim_vec a) = 0"
  proof -
    have "0 = (real_of_int_mat (gen_svp_basis a k) *_v z) $ (dim_vec a)" using alt1 by auto
    also have "… = (real_of_int k + 1) * (∑i = 0..<dim_vec a. z $ i * real_of_int (a $ i)) +
      (2 * real_of_int k * (real_of_int k + 1) * (∑x = 0..<dim_vec a. real_of_int (∣a $ x∣)) + 1) *
      z $ dim_vec a" 
      using gen_svp_basis_mult_real[OF suc_dim_a_dim_z] by auto
    also have "… = real_of_int (2 * k * (k + 1) * (∑x = 0..<dim_vec a. ∣a $ x∣) + 1 ) * (z$dim_vec a)"
      using suc_dim_a_dim_z using z_upto_last by auto
    finally have "0 = real_of_int (2 * k * (k + 1) * (∑x = 0..<dim_vec a. ∣a $ x∣) + 1 ) *
                      (z$dim_vec a)" by blast
    moreover have "real_of_int (2 * k * (k + 1) * (∑x = 0..<dim_vec a. ∣a $ x∣) + 1 ) ≠ 0"
    proof (rule ccontr, goal_cases)
    case 1
      then have eq: "real_of_int (∑x = 0..<dim_vec a. ∣a $ x∣) = - 1 / real_of_int (k * (k + 1))"
         using assms
         by (smt (verit, best) mult_minus_right of_int_hom.hom_0 pos_zmult_eq_1_iff 
          pos_zmult_eq_1_iff_lemma)
      have "k≥1" using assms by auto
      have "real_of_int (∑x = 0..<dim_vec a. ∣a $ x∣) ∈ ℤ" by auto
      moreover have "- 1 / real_of_int (k * (k + 1)) ∉ ℤ" using ‹k≥1›
      by (smt (verit, del_insts) "1" mult_pos_pos
        mult_minus_right of_int_hom.hom_0 pos_zmult_eq_1_iff)
      ultimately show False using eq by auto
    qed
(*Problems here when changing 2*(k+1) to k*(k+1). 
  This is necessary since k'a only is smaller than k'' under the assumtion that z$i≤k, not z$i≤2.*)
    ultimately show ?thesis by auto
  qed
  ultimately have "z$i = 0" if "i < dim_vec z" for i using that suc_dim_a_dim_z
    by (cases ‹dim_vec a = i›) simp_all
  then show ?case
    by auto
qed



lemma insert_set_comprehension:
  "insert (f x) {f i |i. i<(x::nat)} = {f i | i. i<x+1}"
using less_SucE by fastforce

lemma bhle_k_pos:
  assumes "(a,k) ∈ bhle"
  shows "k>0"
using assms unfolding bhle_def
proof (safe, goal_cases)
case (1 v)
  have "∃i<dim_vec v. ∣v $ i∣openThe reduction of SVP to BHLE in $l_\infty$ norm›
  then have "∥v∥_\<infinity> > 0" unfolding linf_norm_vec_Max using 1 by (subst Max_gr_iff, auto)
  then show ?thesis using 1 by auto
qed

lemma svp_k_pos:
  assumes "reduce_svp_bhle (a, k) ∈ int vec ==>
  shows "k>0"
proof -
  obtain v where v_in_lattice: "v∈gen_lattice (gen_svp_basis a k)"
    and infnorm_v: "∥ ≤
java.lang.NullPointerException
    using assms ufldrduce_l ap_dfby force
  have "∃dim_vec v. ∣v $ i∣ > 0" using v_nonzero by auto
  then have "∥v∥∞
  then show ?thesis using infnorm_v by auto
qed


lemma svp_dim_ve
  assumes "reduce_svp_bhle (a, k) ∈ int mat" where
  "en_svp_basis a k = mat (dim_vec a +1) im a +)
proof -
  have "2 ≤ dim_lattice (gen_lattice (gen_svp_basis a k))"
    using assms unfolding reduce_svp_bhle_def gap_svp_def by auto
  then have "2 ≤ dim_vec a) ∧ a) then (k+1) * (a $ j)
    using dim_lattice_gen_lattice[of "gen_svp_basis a k",OF is_indep_gen_svp_basis]
      svp_k_posOF assssms] by auto
  then show ?thesis unfoldin else 2*k*(k+1)* (\Sum> i ∈ .< dim_vec a}. ∣a $ i∣ a}. ∣<>) +1+1 )))"
qed

lemma bhle_dim_vec_a:
  assumes
  shows "dim_vec a > 0"
using assms unfolding bhle_def by auto


lemma first_lt_second:
  assumes "k>0 and zze_k:"∧ a ==> ∣z $ i∣ k"
  shows "2 * ∣i = 0..<dim_vec a. z $ i * a $ i)🚫v z = vec (dim_vec a + 1)
      < (∣i = 0..<dim_vec a. ∣) + 1∣ i ∈ a}. ∣a $ i∣))"
proof
  have take_k1_out: "∣(k + 1) * (∑i = 0..<dim_vec a. z $ i * a $ i)∣a} = insert i {0..<dim_vec a}" using True by auto
    (k+1) *<bar∑a. z $ i * a $ i∣"
    using ‹
  have "∣Sum>i = 0..<dim_vec a. z $ i * a $ i∣ (∑a. ∣z $ i * a $ i∣
    substs sum_abss[of " "(\\lambda>i. z$i * a$i)" "{0..<dim_vec a}"],simp)
  also have " Tru assms by autoto
    by(meson sml)
  also have "…v z = vec (dim_vec a + 1)
    by (subst sum_mono) (auto simp add: z_le_k mult_right_moo)
  also have "… = k * (∑a. ∣a $ i∣ i ∈ ddim_vec a}. ∣a $ i∣
  by (metis mult_hom.hom_sum)
  finally have "∣i=0..<dim_vec a. z $ i * a $ i)∣ k * (∑a. ∣a $ i∣)"
    by blast
  then show ?thesis using take_k1_out using ‹›thenSumia = 0..<dim_vec a. (if i = ia then 1 else 0) * z $ ia) =
qed


text ‹ = z$i" by (subst sum.insert_remove, auto)

lemma efieed_rduction_svp:
  assumes "(a,k) ∈calarrdef
  shows " teis unfolding gen_svp_basis_de us ms
using
proof (safe, goal_cases)
  case (1 x)
  have "k>0" using assms bhle_k_pos by auto
  then show ?case using is_lattice_gen_lattice is_indep_gen_svp_basis by auto
next
  case (2 x)
  have "dim_lattice (gen_lattice (gen_svp_basis a k)) = dim_col (gen_svp_basis a k)"
    using dim_lattice_gen_lattice assms bhle_k_pos is_indep_gen_svp_basis by presburger
  also have "… = dim_vec a + 1" unfolding gen_svp_basis_def by auto
  also have "… 2" using bhle_dim_vec_a[OF assms] by auto
  finally nall shw cb ato
next
  case (3 x)
  let ?x = "vec (dim_vec x + 1) (λdim_vec x then x$i else 0)"
java.lang.NullPointerException
  have eigen_v: "v = ?x" unfolding v_def
  apply (subassk>0"
  applysafe, goss
  proofng 1flig enspbaisdf
    have 1rof_ina gn_svp_basis a k) *\<^>v a +1"for
    then show ?case by (auto simp add: comp_def 3(2))
  next
    case (two i)
    have "(∑show ?thesis by [symmetric]) (use altt1[ihain ‹)
      ng nodn clrprod_def
      by (smt (verit) mult.commute of_int_hm.hom_zero of_ullt fitsm u.cg
    then show ?case using 3 by auto
  qedsn gn_spbsiss_mura[Fsu_dim_a_at
  have "dim_vec ?x = dim_col (gen_svp_basis a k)"
    using 3(2) unfolding gen_svp_basis_def by auto
  then have "v ∈x = 0..<dim_vec a. ∣) + 1 ) ≠lofnt<Sumx= .dim_vec a. ∣a $ x∣) = - 1 / real_of_int (k * (k + 1))"
    unfolding v_def gen_lattice_def by auto
  moreover have "∥ k"
  proof
    haveqed
    proof (rule Max
    case ultimashow ?thesis by auto
      have dim_x_nonzero: "dim_vec x ≠
      then have nonempty: " ∃a∈{r0le> a)"
 using abs_ge_zero by blast
 have " <>  Max (insert 0 {∣ |j. j < dim_vec::nat)} = {f i | i. i<x+1}"
 if "i < dim_vec
 using that bhle_k_pos:
 moreover have "0 ≤x $ i∣ x})" using 3 nonempty
 by (subst Max_ge_iff, auto)
 ultimately show ?case using three by auto
 qed auto
 then show ?thesis using 3 by auto
 qed
 moreover have "v ≠exists>i<dim_vec 
 p_k_pos: pos:
 assume "0 < dim_vecgen_lattice (gen_svp_basis a k)"
 \parallelv∥∞ k"
 have fst: "x ≠ 0 i < dim_vec > 0" using v_nonzero by auto
 have snd: "?x = 0\thenparall>v∥_\∞ > 0" unfolding linf_norm_vec_Max by (subst Max_gr_iff, auto)
 have "x$i = 0" if "i< dim_vecassume"reducesvp_bhl(a k) na > 0"
 by (smt (z3 add.commute d_vec egnvine_map_e() index_vec index_zero_ero_vc(1)
 of_int_eq_iff of_int_eqif of_int_o.o_eo t
 trans_less_add2)
 theno as sigftbauto 
 qed
 imatelyy show ??case by blalast
 bhle_dim_vec_a:





  ‹tshows "2 * ∣i = 0..<dim_vec a $ i∣::int)"

  NP_hardness_reduction_svp:
 assumes "reduce_svp_bhle ( -
 shows "(a,k) \<in 
  (cases "(∑
 case True
 have a_nonempty: "dim_vec a > 0" using svp_dim_vec_a[OF assms] by auto
 have "k>0" using svp_k_pos[OF assms] by auto
 define x where "x = vec (dim_vec a) (λi. k)"
 have "a∙
 by (auto simp add: True sum_distrib_right[symmetric])
 (metis Ts True lessThan_atLeast00)
 moreover have "dim_vec x = dim_vec a" unfolding x_def by auto
 moreover have "x <noteq\v (dim_vec x)"
 proof -
 have "∃i< dim_vec 0" unfolding x_def using ‹
 then show ?thesis using vec_eq_iff[of "x" "0 ≤i=0..<dim_vec  mu)
 qed
 moreoverv "real_of_(∥ k"
 proof -
 have "vec (dim_vec a) (λi. k
 then have "Max {∣ |i. i < dim_vec0 < k›
 Max {∣k∣
 also have "\<lemma iegn_lattice is_indep_gen_svgen_svp_basis by auto
 by (smt (veritit, besestColct_ong sngletconv
 also have "…
 finally have "Max {∣lambda>i. k) $ i∣ a} = k" by blast
 then show ?thesis unfolding x_def linf_norm_vec_Max using ‹ ≥
 qed
 ltimately how ?thesis ing assmfolding reducsvp_bhle_dp_sp_def bhle_def
 by fastw= (gen_svp_basis a k) *_v ?x"
 
 case alse
 show ?esis using assms unfonfoldinreduce_svpce_svp_bhle_def gapgap__def bhle__def
 proof (safe, goal_cases)
  (1 v)
 have a_nonempty: "dim_vec a > 0have "(∑= 0..<dim_vec  gen_lattice (gen_svp_basis a k)"
 have "k>0" unfolding linf_norm_vec_Max
 proof -
 "\existsi<dim_vec s _hre
 hen have "∥v∥∞ 0"ufolding linf_norm_vec_Max _Max by (sstMax_r_iff, auto)
 then show ?thesis using 1 by auto
 qed
 then have "k≥1" by auto
 obtain z where z_def:
 "v = (gen_svy (subst ge, auto)
 "dim_vec z = dim_vec a + 1"
 using 1 unfolding gen_lattice_def gen_svp_basis_def y (subst Max_ge_ auto))
 have dim_v_dim_a:"dim_vec v = dim_vec a + 1"
 using 1 unfolding gen_lattice_def gen_svp_basis_def by auto
  x where "x = vec (dim_ec a) (($) z)"
 have v_eq_z_upto_last: "v $ i = z $ i" if "i< dim_vec a"
 by (substv (dim_vec v)"
 have v_le_k: "∣ ≤ a + 1" for i
 using 1 dim_v_dim_a that unfolding linf_norm_vec_Max
 using Max_le_iff[of "{∣ |i. i < dim_vec 0_v (dim_vec v)" using ‹ by auto
 by fastforce
 then have z_le_k: "∣ dim_vec eigen_v index_map_vec(2) index_vec index_zero_vec(1)
  v_eq_z_upto_last[OF that] that
 by (metis less_add_ontrans_less_add2)

 have v_last_zero: "v$(dim_vec qed
 proof (rule ccontr) 
 assume ccontr_assms:"v $ dim_vec a ≠
 have v_la
 (k+1) * (∑NP-hardness of reduction function›
 (2*k*(k+1) * (∑reduce_svp_bhle k) n gap_svp"
 (is "v$(dim_vec a) = ?first + ?second")
 by (auto simp: z_def gen_svp_basis_mult_last)
 then have "?first ≠ ?second ≠
 using ccontr_assms by authave"a \bullet x = 0" unfolding x_def scalar_prod_def
 
 (firave m_vec x = di_ec a" uunflding ef by aut
 (second) "?second ≠
 i< dim_vec k"
 by blaroof -
 then showlse
 proof (cases)
 case first
 then have gr_1: "\<\<a. z $ i * a $ i∣ ≥ 1"
 using \<openhavek>0› a_nonempty
 bar>v$ dim_vec a∣" using first v_last by auto
 also have "… = k" by auto
java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
 by (smt (z3) mult_le_0_iff mult_minus_right)
 also have "… > k" using first gr_1 ‹k>0›
 by (smt (verit, best) mult_le_cancel_left1)
 finally have "∣v$ dim_vec a∣then show esisunfldinng x_df li_nom_ve_Maax usin\openk>0›is usingassms unnfdngrdue_spbhlle_def gap_svp_def bhledf
 then show ?thesis using v_le_k[of "dim_vec a"] by auto
 next
 case second
 have "∣
 have sum_a_ge_1:"(∑a. ∣a $ x∣) ≥1"
 using as tLeat0eshn
 tis asum_abs int_one_l_if_z_less notessmaszro_ls_b_f)
 then have "2*k*(k+1)*(∑a. ∣a $ x∣zweezf:
 proof -
 *(k)∑a. ∣a $ x∣1" using ‹
 by (smt (verit, ccfv_SIG) int_distrib(2) sum_a_ge_1 zmult_zless_mono2)
 then show ?thesis using ‹)
 by (smt (verit, best) pos_mult_pos_ge sum_a_ge_1)
 qed
 moreover have "∣ ≥2*k*(k+1)*(∑a. ∣a $ x∣"
 v $ i∣ v}" k]
 by (subsbs_ult) (simp d:lesThantLeast ml_l_cancel_lt1
 ultimately have "∣z $ i∣ a" for i
 then show ?thesis using v_le_k[of "dim_vec a"] by au v_z_uptost[OF th]at
 next
 case bo
 have z_gr_one:"∣≥
 let ?second' = "2 * k * (k + 1) * (∑a. ∣
  * a $ i) ≠ 0" using both by auto
 then have one: "k < \ ?second ≠cnie
 by (smt (z3) mult_less_cancel_left2 mult_minus_right)
 then have fit_nnro: " ≠k>0›
 have two'':"2 * ∣ < \
 using first_lt_second[OF (aes)
 then have two': "∣ > 2"
 proof -
 have "0<real_of_int " using first v_last by auto
 have "2 * real_of_int \<also a. z $ i * a $ i∣"
 then have "2 <  real_of_int"
  sust poos_lless_divide_e[OF \<><real_of_int ∣\close], auto)
 also have "… sn irst gr_1 \>k>0›
 finally have"∣k yat
 
 then have two:"∣z $ dim_vec a∣ 1" using second by auto
 real_of_int (z$dim_vec a)∣_l_er_less not_lesssuum_abs zeo_less_abs_iff)
 proof -
 have "bar>(real_of_int ?second' / real_of_int ?first) *
 real_of_int (z$dim_vec a)∣x<dim_vec k>0›
 >real_of_int (z$dim_vec a)∣k>0›
          also
            using
          finally have "∣(real_of_int ?second' / real_of_n?irs) *
            real_of_int (z$dim_vec a)∣ ∣
          then show ?thesis using two' by linarith
        qed
        have "real_of_int <v $ dim_vec a<real_of_int <bar>?first"
        proof -
          have "real_of_int?first∣<bar>?second'∣k>0› z_le_k to
            ∣real_of_int (v $ dim_vec a)∣ / ∣"
             of_int_abs[of "v $ dim_vec a"] of_int_abs[of "?first"] by auto
          also have "… = ∣"
            using abs_divide[symmetric] by auto
          also have "… = ∣
            real_of_int ?first∣
          also have "… r
            (real_of_int ?second' / real_of_int ?first)* real_of_int (z$dim_vec a)∣ > 2"
            by (metis real_of_int = ∣ * 
          also have "…1(real_of_i_int ?second' / re_of_inist *
            real_of_int (z$dim_vec a)∣ ≥real_of_int ?seod elof_int ?firt\bar"
            using(real_of_int ?second'/ real_of_int ?first)*
          also have "…
          finally show "real_of_int ∣ / real_of_int?first∣
        qed
        theneeal_of_intv $ dim_vec a∣?first∣
          by (simp add
        then = ∣
        bar>v $ dim_vec a∣ = ∣
        then show ?thesiside_eq_left
      qed
    qed
    have z_last_zero: "z$dim_vec a = 0" 
    proof (rule ccontr)
      assume ccontr_assms:"z $ dim_vec a ≠ 0"
      then> ≥
      have "(k) * (∑ {0 ..< dim_vec a}. z $ i * a $ i) +
        (2*k*(k+1)* (∑ { < dim_vec a}. ∣a $ i∣
        (is "?first + ?second * (z$(dim_vec a)) = 0")
        using v_last_zero z_def gen_svp_basis_mult_last b
      n havebs_: \bar>first🚫_\inftyo\close
      moreoverhv \bar>fit< < ∣
      oof
        have "∣ 2" by auto
        then have "\<bar    otr)
        moreover have "∣\barsecond∣z$dim_vec a∣
           us> {. is_lattc L) \L<nd> (di_attice L\ge>) \and>
        ultimately have "∣?second∣z$dim_vec a∣ (\exists∈ r ∧v (dim_vec v))}"
        thenso eisb nrit
      qed
      ultimately show False by auto
    ed
 
    have v_real_z: "v = z" using_qzut_lsvltzro lteo
      by using v_laszdef gn_s_ai_ultlastt by auto

    have "a ∙ (j < dim_vec a) then (k+1) * (a $ j)
    proof -
      have "0 = ((gen_svp_basis a k) *_v z) $ (dim_vec a)"
        using v_last_zerozd yu
      also have "…>{0 ..< dim_vec a}. z $ i * a $ i)"
        by (subst gen_svp_basis_mult, auto simp add: z_def z_last_zero)
llyk\Sum i ∈ a}. z $ i * a $ i) = 0" by auto
      en_svp_basisk*<> z = vec (dim_vec a + 1)
      by (smt (verit, ccfv_SIG) atLeastLessTha moreover have "∣ a then z$i else(k+) * (\Sum i ∈
         eq_0_ifffo_ntoo.hom_0 su su.ogng)) po (stvcq_fse o_ss
    qed
    moreover have "dim_vec x = dim_vec a" unfolding x_def by auto
    moreover have "x ≠v (dim_vec x)"
    proof -
      have "∃?econ\barz$dim_vec a∣
      then obtain i where ‹a. (if i = ia then 1 else 0) * z $ ia) =
      then have ‹ ≥ by (simp add: mult_le_cancel_left1)
      with ‹a}f i = ithn 1 lsse 00) * z * z $ ia)"y auuto
      with ‹
        bya ∙v z = vec (dimva + )
      then have "∃ a. x$i ≠open>ii<dim_vec a›lngnsv_bsdbyuto
      then show ?thesis using x_def by auto
    u Tue
    moreover have "∥∞ k"
    proof -
      r>" if " < < da" for i using that by auto
      then have "Max (insert 0 {∣ |i. i < dim_vec a}) =
            Max (insert 0 {∣ |i. i < dim_vec a})"
        by (smt (z3) Collectcng Setcompr_q_image mem_Collect_q
      lso have "… |i. i < dim_vec a + 1})"
      proofby mt (eritt, ccfv_IG)
        have v "c imvca" ung x_d
          Max (insert 0 (insert ( ∣z $ i∣ a}))"
        roof -
ave\existsii≤
            >im_vec a >0› by (subst Max_ge_iff, auto)
          then have gen_s:
            using z_last_zero by simp
          then have max_subst: "Max {∣ |i. i < dim_vec a} =
              Max (insert ∣ {∣autoto
            using ‹ ≤
then_ai_z:z: "dvec z = dvecaa +1unfolding gen_s_asisdef by auto
        qed
        hen how ?tess
          using insert_setcompreheion[of "(λ)" "dim_vec a"] by auto
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
      finally "Max (net 0 \barz $ i∣ a}) =
                    nsertr {{<>z \bar |i. i < dim_vec a +1})"
        by blastusingvp_basis_mult_realOF sucda__] byuo
      nve <>x\<arallel\
        using x_def z_def v_real_z by auto
      then show ?thesis using 1 by auto
    have "Max <bar>z $ i<bar> |i.i < dim_vec a} ≥ 0"
    ultimately show ?case by force
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
qed


text ‹a. \>x<bar>) 1/ra_fnt ( (k + 1))
lemmaby t(vrt bet)mult_iu_ightf_itho.ho_ poszult_eq1f
unfolding is_reduction_def
  using NP_hardness_reduction_svp well_defined_reduction_svp by fastforce

end