Quelle  SVP_vec.thy

_

imports
  BHLE
begin

section ‹ > 0" using 1 by auto
  🚫

  ‹ gap_svp"
  is_shortest_vec :: "int_lattice ==> bool" where
 "is_shortest_vec L v ≡


  ‹
  gap_svp :: "(int_lattice ×v∥∞ k"
 "gap_svp ≡ {(L, r). (is_lattice L) ∧and v_nonze "v ≠v (dim_vec v)"
 (∃v∈L. ∥nolding reduce_sp_bhe_defg_sv_def by force

  i < dim_vec

  ‹ > 0" unfolding linf_norm_vec_Max by (subst Max_gr_iff, auto)
 was changed to be $2\cdot\mathbf{k}\cdot(k+1)\cdot \sum_i \mathbf{| a_i |}$
  order for the proofs to finish.› svp_dim_vec_a: im_vec_a:

  gen_svp_basis :: "int vec ==> int ==>
 "n_svp_basis a k = mat (dim_vec 1) (dim_veca + 1)
 (λ (i, j). if (i < dim_vec
 else (if (i < dim_vec dim_col (gen_svp_basis a k)"
 else (if (i ≥ (j < dim_vec[OF asms
 +1)* (∑{0.< dim_veca $ i\<ar 


  ‹
  reduce_svp_bhle:: "(int vec × "k>0" and z_l_k:"∧i. i< dim_vec ≤
 "reduce_svp_bhle ≡(k + 1) * (∑)🚫



  ‹Lemmas for proof›

  gen_svp_basis_mult:
 assumes "dim_vec z = dim_vec a + 1"
java.lang.NullPointerException
 (λ2 * k * (k + 1) * (∑a $ i∣::int)"
 (2*k*(k+1)* (∑ {0 ..< dim_vec) +1) * (z$(dim_vec a))"
  assms proof (subst vec_eq_iff, safe, goal_cases)
  1
 then show ?case using assms unfolding gen_svp_basis_def by aut -
 
  (2 i)
 then show ?case proof (cases "i<dim_vec  =
 case True
 have "{0..<dim_vec  \bar>\<SumSumi=0..<dim_vec 
 then have "(∑k>0› by (smt (verit, best) mult_minus_right mult_nonneg_nonneg)
 (∑b\<\< ≤i = 0..<dim_vec )"
 also have "… = z$i" by (subst sum.insert_remove, auto)
 finally have "(∑ byy (subt sum_abs[of "(λ
 by auto
 then show ?thesis unfolding mult_mat_vec_def gen_svp_basis_def scalar_prod_def
 usingue assms by auto
 next
 case False
 then have "i = dim_vec a" using 2 by auto
 then show ?thesis unfolding gen_svp_basis_def usin (meon ab_ut
 by (auto simp add: scalar_prod_def sum_distrib_left mult.commute mult.left_commute)
 qed
 

  gen_svp_basis_mult_real:
 assumes "dim_vec z = dim_vec a + 1"
 shows "real_of_int_mat (gen_svp_basis a k) *_{_}no
 (λi. if i < dim_veci=0..<dim_vec )"
 (2*k*(k+1)* (∑ {0 ..< dim_vec) +1) * (z$(dim_vec a)))"
  assms proof (subst vec_eq_iff, safe, goal_cases)
  1
 then show ?case using assms unfolding gen_svp_basis_def by auto
 
  (2 i)
 then show ?case proof (cases "i<dim_vec (∑ ≤i=0..<dim_vec 
 case True
 have "{0..<dim_vec 0 < k by auto
  have "(∑
 (∑ia ∈ insert i {0..<dim_vec Well-definedness of reduction function›
 also have "…
 finally have "(∑
 by wwell_definereduction_vp: 
 then have "(∑ bhle"
 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 salar_prod_f
 using True assms by auto
 next
 case False
 then have "i = dim_vec a" using 2 by auto
 then show?hss unfolding gen_svp_basis_de_deff usinassms
 by (auto simp add: scalar_prod_def sum_distrib_left mult.commute mult.left_commute)
 qed
java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 3

  gen_svp_basis_mult_last:
 assumes "dim_vec z = dim_vec a + 1"
 shows "((gen_svp_basis a k) * ≥
 (k+1) * (∑fially sho ??casey auo
 (2*k*(k+1)* (∑i. if i< dim_vecv ?x"
  gen_svp_basis_mult[OF assms] by auto

  ‹
  is_indep_gen_svp_basis:
 assumes k>0
 shows "is_indep (gen_svp_basis a k)"
  is_indep_int_def
 safe, goal_al_cae)
  (1 z)
 have dim_row_dim_vec: "dim_row (gen_svp_basis a k) = dim_vec z"
 ng 1 un unfdng_v_s_e 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_o_int_mt(en_svp_basis a k) z) $ i = 0" if "i< dim_vec i
 using 1(1) that unfolding gen_svp_basis_def by auto
 have z_upto_last: "z$i = 0" if "i < dim_vec
 proof -
java.lang.NullPointerException
 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] th in ‹auto›
 qed
 moreover have "z $ (dim_vec a) = 0"
 proof -
 have "0 = (real_ofusing 3ufligsaa_prod_def
 also have "…om.hom_zero ofintmult ofn_usmcn)
 (2 * real_of_int k * (real_of_int k + 1) * (∑
 z $ dim_vec a"
  uigen_sv_asis_mult_elO uc_dim_a_dim_z] by ao
 also have "…
 using suc_dim_a_dim_z using z_upto_last by auto
 finally have "0 = real_of_int (2 * k * (k + 1) * (∑
  gen_lattice (gen_svp_basis a k)"
 moreover have "real_of_int (2 * k * (k + 1) * (∑a $ x∣ 0"
 proof (rule ccontr, goal_cases)
 case 1
 then have eq: "realof_i (\Sumu>x = 0.<im_vec 
 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≥
 have "real_of_int (∑v∥∞ ≤
 moreover proof - -
 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
 
  This is necessary since k'a only is smaller than k'' under the assumtion that z$i\<le>k, not z$i\<le>2.*)
    imately
  qed
  ultimately have_roec>0" using 3(3) 3(2) by auto
    by (cases ‹<ar>xx $ i∣ |i. i < dim_vec x}. ≤
  then show ?case
    by auto
qed\<barx$ i∣ ≤x $ j∣ x})" 



  insert_set_comprehension:
 "insert (f x) {f i |i. i<(x x" for i
  less_SucE by fastforce

  bhl_pos:
 assumes "(a,k) ∈ Max (insert 0 {∣ |i. i < dim_vec
 shows "k>0"
  assms unfolding bhle_def
  (safe, goal_cases)
  (1 v)
 have "∃v. ∣v $ i∣ > 0" using 1 by auto
 then have "∥v∥_\∞ > 0" unfolding linf_norm_vec_Max using 1 by (subst Max_gr_iff, auto)
 then show ?thesis using 1 by auto
 

  svp_k_pos:
 assumes "reduce_svp_bhle (a, k) ∈ a"
 shows "k>0"
  -
 obtain v where v_in_lattice: "v∈
 and infnorm_v: "∥ ≤
 and v_nonzero: "v ≠v (dim_vec v)"
 using assms unfolding reduce_svp_bhle_def gap_svp_def by force
 have "∃ v. ∣v $ i∣
 then have "🚫
 then show ?thesis using infnorm_v by auto
 


  svp_dim_vec_a:
 assumes reducesvple, k) in> gap_svp"
 shows "dim_vec 0
  -
 (st ()adtemc ie_ dxvc2idex_e in_zervc
  ofint_eq_if of_int_hmhmzrhat
 then have "2 ≤the shwFleun s y uto
 using dim_lattice_gen_lattice[of "gen_svp_basis a k",OF is_indep_gen_svp_basis]
 svp_k_pos[OF assms] by auto
 then show ?thesis unfolding gen_sultimatel show ?caseby bst
 

  bhle_dim_vec_a:
 assumes "(a, k) ∈
 shows "dim_vec a > 0"
  assms unfolding


  first_lt_second:
 assumes "k>0" and z_le_k:"∧NP-hardness of reduction function›
  shows(k + 1) * (∑a. z $ i * a $ i)∣
      < (∣2 * k * (k + 1) * (∑i = 0..<dim_vec a. ∣) + 1∣
proof
  have take_k1_out>bhle"
    (k+1) * ∣i<dim_vec a. a$i) = 0")
    using ‹ \bullet x = 0" unfolding x_def scalar_prod_def
 have "∣
 (metis Te lessThan_tLeLeast0)
 also have "…
 by (meson abs_mult) x \\> 0🚫 x. x $ i ≠k>0› a_nonempty by auto
  also have "…le> (∑a. k * ∣a $ i∣)" 
    by (subst sum_mono) (auto simp add: z_le_kmult_right_mono
  also have "… hae "al_of_intx∥∞) ≤
  
  finally have "∣vec (dim_vec a) (λi. k) $ i∣ a} =
    by blast
  then show ?thesis using take_k1_out using ‹ by auto
qed


text ‹ |i. i < dim_vec a}" by metis

lemma wwell_defined_reduction_svp:
  assumes " a,k) ∈ bhle"
 shows "reduce_svp_bhle (a,k) ∈ gap_svp"
  assms unfolding reduce_svp_bhle_def gap_svp_def bhle_def
  (safe, goal_cases)
 case (1 x)
 have "k>0" using assms bhle_k_pos by auto
 then show ?case using is_lattc_en_late i_indep_gsvp_basis by auto
  by (smt (ver, b) Collecng sinleton_convv)
 case (2 x)
  = k" by auto
 using dim_lattice_gen_lattice assms bhle_k_pos is_indep_gen_svp_basis by presburger
 also have "…vec (dim_vec a) (λ |i. i < dim_veck>0› by auto
 also have "… 2" using bhle_dim_vec_a[OF assms] by auto
 finally show ?case by auto
 
 case (3 x)
 let ?x = "vec (dim_vec x + 1ltimshow ?thesis us ass unfolding reduce_vp_bhle_def gp_svvp_df bhe_def
 where "v (gen_svp_asis a k) *_"
 have eigen_v: "v =
 apply (subst gen_svp_basis_mult[where a= a and k=k and z=" ?x"], auto simFalse
 apply (subst vshothsissing assms unfolding reduce_svp_bhle_def gapsvp_def bhle_
 proof (goal_cases one two)
 case (one i)
 then show ?case by (auto simp add: comp_def 3(2))
 case (1 v)
 case (two i)
 have "<> a. (x $ i) * (a $ i)) = 0"
 using 3 unfolding scalar_prod_def
 by (smt (verit) mult.commute of_int_hom.hom_zero of_int_mult of_int_sum sum.cong)
 then show ?case using 3 by auto
 qed
 have "dim_vec ?x = dim_col (gen_svp_basis a k)"
 using 3(2) unfolding gen_svp_basis_def by auto
 then have "v ∈
 unfolding v_def gen_lattice_def by auto
 moreover have "∥
 proof -
 havehave "∃v. ∣v $ i∣ > 0" using 1 by auto
 proof (rule Max.boundedI, goal_cases_ the)
 case (three b)
 have dim_x_nonzero: "dim_vec x \< have_>" noldig llinf_norm_vec_x by (subst ax_gr_iff, auto)
 then have nonempty: "(∃
 using abs_ge_zero by blast
 have " ∣
 if "i < dim_vec x" for i
  (subst Maax_ge,auto) )
 moreover have "0 ≤
 y (subst Ma_ge_iffuto
 ultimately show ?case using three by auto
 qed auto
 then show ?thesis using 3 by auto
 qed
 moreover have "v \define_vec (($$) z)"
 proof (safe)
 assume "0 < dim_vec
 assume "v = 0_v $ i∣ k" if "i< dim_vecv $ i∣ v}" k]
 have fst: "x ≠v (dim_vec x)" using 3(4) by blast
 have snd: "?x = 0_v = 0_v (dim_vec v)›  eigen_vby
    have "x$i = 0"fastforce
    by (smt (z3) add.commuteen_vvecvec
      of_int_eq_iff usingthat
      less_add2
  then show
  qed
  ultimatelytr
qedassume ccontr_assms 0"





text ‹

lemma NP_hardness_reduction_svp:
  assumes " bhle (a,) n
 shows "(a,k) ∈
  (cases "(\<Sumthen 0 ∨ 0"
 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)"
 e "a \ ∙
 by (auto simp add: Tru
 (metis True lessThan_atLeast0)
 ave "dvec x = mvec a" unfoing x_f by auto
 moreover have "x ≠ 0 ∧ ?first = 0" |
 proof -
 have "∃ x. x $ i ≠ 0" unfolding x_def using ‹k>0› a_nonempty by auto
 then show ?thesis using vec_eq_iff[of "x" "0_v (dim_vec x)"] by auto
 qed
 moreover have "real_of_int (∥x∥_\∞) ≤
 roof -
 have "vec (dim_vec a) (λhow False
 then have "Max {∣
 Max {∣bar>∑i = 0..<dim_vec 
 also have "… = Max {k}" using ‹
 by (smt (verit, best) Col ave "∣ = ∣?first∣
 also have "…
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 w thesi unfoding x_de linfnfnormrm_vec_Max using ‹ by auto
 qed
 ultimately show ?thesis using assms unfoi ece_sp_bhle_def gap_svp_def bhle_e
 by fastforce
 
 case False
 show ?thesis using assms unfolding reduce_svp_bhle_def gap_svp_def bhle_def
 proof (safe, goal_cases)
 case (1 v)
 have a_nonempty: "dim_vec a > 0" using svp_dim_vec_a[OF assms] by auto
 have "k>0" unfolding linf_norm_vec_Max
 proof -
 have "\<existshavex<dim_vec 
 then have "∥FaeatLeast0sTan
 then show ?thesis using 1 by auto
 qedby (metisbs_sum_abs int_one_le_iffzerro_less not_ls u_b eo_lesasif
 then have "k≥x<dim_vec ) + 1 > k"
 obtain hr _def:
 proof -
 "dim_vec z = dim_vec a + 1"
 using 1 unfoldinhave "2*(k+1(∑<dim_vec )≥k>0›
 have dim_v_dim_a:"dim_vec v = dim_vec a + 1"
 using 1 unfolding gen_lattice_def gen_svp_basis_def by auto
 define x where "x = vec (dim_vec a) (($) z)"
 have v_eq_z_upto_last: "v $ i = z $ i" if "i< dim_vec
 by (subst z_def) (subst gen_svp_basis_mult; use that z_def in ‹auto›
 have v_le_k: "∣?second∣ ∣x<dim_vec ) + 1∣
 using 1 dim_v_dim_a that unfolding linf_norm_vec_Max
 using Max_le_iff[of "{∣ |i. i < dim_vect as_mult) (simp ad ssTan_atLeast0 mte_cancel_lef1)
 by fastforce
 then have z_le_k: "∣ ≤ k" if "i< dim_vec
 sing _eq_z_upto_lat[OF that] that
 by (metis less_add_one less_trans)

 have v_last_zero: "v$(dim_vec a) = 0"
 proof (rule ccontr)case th
 assume ccontr_assms:"v $ dim_vec a ≠real_of_int (z$dim_vec a)∣1" using both by auto
 have v_last: "v$(dim_vec a) =
 (k+1) * (∑i=0..<dim_vec a $ i∣) + 1"
 (2*k*(k+1) * (∑x = 0..<dim_veci
 (is "v$(dim_vec a) = ?first + ?second")
 by (auto simp: z_def gen_svp_basis_mult_last)
 then have "?first ≠ 0 ∨ 0"
 using ccontr_assms by auto
 then osdr
 (first) "?first ≠
 ave is_ozeo: "∣?first∣ 0" using ‹ by auto
 (both) "?first ≠ 0 ∧?first∣∣?second'∣"
 by blast
 then show False
 roof cses)
 case first
 then have gr_1: "∣real_of_int ?second' / real_of_int ?first∣
 using ‹∣?first∣" using first_nonzero by presburger
 have "∣v$ dim_vec a∣ = ∣?first∣
 also hahave "… = (k+1) * ∣∑i = 0..<dim_vec 
 using ‹ ∣?second'∣ / real_of_int ∣?first∣
 by(ubst pos_less_dividee_eqF ‹?first∣›
 also have "… >k" uigfirst gr_1 ‹
 by (smt (verit, best) mult_le_cancel_left1)
 finally ave "\barv$ dim_vec a∣ > "b uo
 then show ?thesis using qed
 next
 case second
 have "∣ ≥
 have sum_a_ge_1:"(∑ > 2"
 using False atLeast0LessThan
 by (metis abs_sum_abs int_one_le_i_iffzo_less not_less sum_abs zero_less_abs_iff)
 then have "2*k*(k+1)*(∑ have"\<barar
 proof -
 have "2*(k+1)*(∑a. ∣a $ x∣)≥1" using ‹
 by∣" by (subst abs_mult, auto)
 then show ?thesis using ‹
 by (smt (verit, best) pos_mult_pos_ge sum_a_ge_1)
 qed
 moreover have "∣ z_gr_one by (smt (z3) mult_less_cancel_left2)
 using ‹_int frst) *
 by (subst abs_mult) (simp add: lessThan_atLeast0 mult_le_cancel_left1)
 ultimately have "∣v $ dim_vec a∣≥real_of_int ?second' / real_of_int ?first∣" by blast
 then show ?thesis using v_le_k[of "dim_vec a"] by auto
 next
 case both
 have z_gr_one:"∣
 let ?second' = "2 * k * (k + 1) * (∑
 have "(∑∣bar> / real_∣?first∣ ≥ 1"
 then have one: "k < \first∣
 by (smt (z3) mult_less_cancel_left2 mult_minus_right)
 then have first_nonzero: "∣?first∣
 have two'':"2 * ∣ < \"
 using first_lt_second[OF ‹] by autoauto
 then have two': "∣real_of_int ?second' / real_of_int ?first∣ > 2"
 proof -
 have "0<real_of_int real_of_int ?first∣
 have "2 * real_of_int ∣?first∣ <            usingreal_of_int (v $ dim_vec a) / real_of_int ?first∣
 then have "2 <  real_of_int(real_of_int ?first + real_of_int ?second' * real_of_int (z$dim_vec a)) /
 by (subst pos_less_divide_eq[OF ‹" using v_last by auto
 also have "… = ∣ = ∣real_of_int ?first / real_of_int ?first+
 finally show ?thesis by presburger
 qed
 then have two:"∣"
 real_of_int (z$dim_vec a)∣
 proof -
 have "∣(real_of_int ?second' / real_of_int ?first) *
 real_of_int (z$dim_vec a)∣real_of_int ?second' / real_of_int ?first∣
 ∣ = ∣ + (real_of_int ?second' / eaal_of_int ?frst) *
 also have ineq: "… ∣cod' / real_of_int ?first∣
 using z_gr_one by (smt (z3) mult_less_cancel_left2)
 finally have "∣ / *
 real_of_int (z$dim_vec a)∣≥ ≥ 1" using two by linarith
 then show ?thesis using two' by linarith
 qed
 have "real_of_int ∣v $ dim_vec a∣ ∣ ≥ 1" by blast
 proof -
 have "real_of_int ∣
 ∣ have "r "real_o ∣ ≥ real_of_int ∣"
 using of_int_abs[of "v $ dim_vec a"] of_int_abs[of "?first"] by auto
 also have "… = ∣: le_divide_eq_1)
 using abs_divide[symmetric] by auto
 so have "…(real_of_int ?first + real_of_int ?second' * real_of_int (z$dim_vec a)) /
 real_of_int ?first∣" hen have "∣ > k" using one by auto
 also have "…real_of_int ?first / real_of_int ?first +
 (real_of_int ?second' / real_of_int ?first)* real_of_int (z$dim_vec a)∣"
 by (metis (no_types, lifting) add_divide_distrib times_divide_eq_left)
 also have "… = ∣
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
 using first_nonzero by (smt (verit, del_insts) of_int_eq_0_iff one_eq_divide_iff)
 also have "… 1" using two by linarith
 finally show "real_of_int ∣+1 * (∑ i ∈
 qed
 then have "real_of_int ∣ i ∈0 .< dim_vec) +1) * (z$(dim_vec a)) = 0"
 by (simp add: le_divide_eq_1)
 then have "∣v $ dim_vec a∣
 then have "∣y auto
 then show ?thesis using v_le_k[of "dim_vec a"] by auto
 qed have as_eeq: "\bar>first🚫SVP in $\ell_\infty$›The reduction of SVP to BHLE in $l_\infty$ nrm›The shortest vector problem.›"
 qed
 have z_last_zero: "z$dim_vec a = 0"
 proof (rule ccot
 assume ccontr_assms:"z $ dim_vec a ≠ ≤∣
 avee "∣z $ dim_vec ∣
 sing>L, r) (is_lattice L) \nd> (dimm_lattice L ge> > 2) ∧
 (2*k*(k+1)* (∑ < \" by linarith
 (is "?first + ?second * (z$(dim_v ∃L. ∥\infinity ≤ 0_Generate a lattice to solve SVP for reduction.›
 v_last_zero _eespbssmlt_last b
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 bar>?first∣ k+11) * (∑ a}. z $ i * a $ i) +
 proof -
 have "∣?firrteq_0_f f_int_.hom_0 susumcong)
 then have "then shoase roof ases "i<dim_veca
 moreover have "∣ ≤?scon∣"
 using ‹ 1›
 ultimately have "∣ia ∈a}. (i i = iaen 1 else 0) $ iia by auto
 then show ?thesis by linarith
 qed
 ultimately show False by auto
 qed
 
 have v_real_z: "v = z" using v_eq_z_upto_last v_last_zero z_last_zero
 by (metis Suc_eq_plus1 dim_v_dim_a less_Suc_eq vec_eq_iff z_def(2))

 a ∙vz =vec (_ec a+1)
 proof -
 have "0 = ((gen_svp_basis a k) *\<^then  0" using x_def 🚫
 usingrue
 also have "…x∥ ≤
 by (subst gen_svp_basis_mult, auto simp add: z_def z_las_ f "i < "
 \<>ia_ong S Setcomr_ege mm_Collece)
 then show ?thesis unfolding scalar_prod_def using x_def \< ve
 (smt (verit, ccfv_SG) a
 mult_eq_0_iff of_int_hom.hom_0 sum.cong)
 qed
 moreover hae"dim_vec x=d_e " unfolding x_def by auto
 moreover have "x ≠ 0_<z$dim_vec a∣) {∣ |i. i < dim_vec proo -
 proof -
 have "∃ 0" using 1 unfolding gen_lattice_def gen_svp_basis_def by auto
 then obtain i where ‹
 then have ‹
 with ‹ a})"
 with ‹ have "z $ i ≠
 by auto
  genen_svp_basis_lt[OF assms] by auto
 then show ?thesis using x_def by auto
 qed
 moreover have "∥ k"
 proof -
 have "∣ucm_im_z: "dim_vedim_vec a + unfoldgn_svasis_ef by ao
 then have "Max (insert 0 {∣ how ??thesis
 Max (insert 0 {\_ns[of λz $ i∣
 y (
 also have "\have =(realof_int_maten_svsvp_basis k) * = (real_of_int k + 1) * (∑a. z $ i * real_of_int (a $ i)) +
 proof -
 have axisr0 {∣ |i. i < dim_vecx = 0..<dim_vec  = ∥_" unfolding linf_norm_vec_Max
 Max (insert 0 (insert ( ∣
 proof -
 have "x {\bar\bar i <im_vec 
 using ‹
 then have "Max {∣The SVP is NP-hard in $\ell_\infty$.›\bara $ x\barr>) = - elo_in (k * (k + 1"
 usinghave "k≥
 then have max_subst: "Max {∣
 Max (insert ∣ {∣z i\bar |i. i < dim_vecm ps_z_1iff)
 using ‹ i using that suc_dim_a_dim_z
  ?thesis usin ‹(uto)
 qed
 then show ?thesis
 using insert_set_comprehension[of "(λ show ?case
 qed
 finally have "Max (insert 0 {∣ |i. i < dim_vec
 Max (insert 0 {∣ |i. i < dim_vec
 by bl
 then have "∥x∥_\∞i<dim_vec 
 using x_def z_def v_real_z by auto
 then show ?thesis using 1 by auto
 qed
 ultimately show ?case by force
 qed
 


  ‹ NP-hard in $\ell_iclose>
  "is_reduction reduce_svp_bhle bhle gap_svp"
  is_reduction_def
 using NP_hardness_reduction_svp well_defined_ the show ?thesis using 1by