Theory Compress

theory Compress

imports Kyber_spec
        Mod_Plus_Minus
        Abs_Qr
        "HOL-Analysis.Finite_Cartesian_Product"

begin

lemma prime_half:
  assumes "prime (p::int)"
          "p > 2"
  shows "⌈p / 2⌉ > ⌊p / 2⌋"
proof -
  have "odd p" using prime_odd_int[OF assms] .
  then have "⌈p / 2⌉ > p/2" 
  by (smt (verit, best) cos_npi_int cos_zero_iff_int 
    le_of_int_ceiling mult.commute times_divide_eq_right)
  then have "⌊p / 2⌋ < p/2" 
  by (meson floor_less_iff less_ceiling_iff)
  then show ?thesis using ‹⌈p / 2⌉ > p/2› by auto
qed

lemma ceiling_int: 
  "⌈of_int a + b⌉ = a + ⌈b⌉"
unfolding ceiling_def by (simp add: add.commute)

lemma deg_Poly':
  assumes "Poly xs ≠ 0" 
  shows "degree (Poly xs) ≤ length xs - 1"
proof (induct xs)
  case (Cons a xs)
  then show ?case
    by simp (metis Poly.simps(1) Suc_le_eq Suc_pred 
    le_imp_less_Suc length_greater_0_conv)
qed simp


context kyber_spec begin

section ‹Compress and Decompress Functions›


text ‹Properties of the ‹mod+-› function.›

lemma two_mid_lt_q:
  "2 * ⌊real_of_int q/2⌋ < q" 
using oddE[OF prime_odd_int[OF q_prime q_gt_two]] by fastforce

lemma mod_plus_minus_range_q: 
  assumes "y ∈ {-⌊q/2⌋..⌊q/2⌋}"
  shows "y mod+- q = y"
using assms mod_plus_minus_rangeE q_gt_zero q_odd by presburger

 

text ‹Compression only works for $x \in \mathbb{Z}_q$ and outputs an integer 
  in $\{0,\dots, 2^d-1\}$ , where $d$ is a positive integer with 
  $d < \rceil\log_2 (q)\lceil$. 
  For compression we omit the least important bits. 
  Decompression rescales to the modulus q.›


definition compress :: "nat ⇒ int ⇒ int" where 
  "compress d x = 
  round (real_of_int (2^d * x) / real_of_int q) mod (2^d)"

definition decompress :: "nat ⇒ int ⇒ int" where 
  "decompress d x = 
  round (real_of_int q * real_of_int x / real_of_int 2^d)"

  

lemma compress_zero: "compress d 0 = 0"
unfolding compress_def by auto

lemma compress_less:
  ‹compress d x < 2 ^ d›
  by (simp add: compress_def)

lemma decompress_zero: "decompress d 0 = 0" 
unfolding decompress_def by auto


text ‹Properties of the exponent $d$.›

lemma d_lt_logq: 
  assumes "of_nat d < ⌈(log 2 q)::real⌉"
  shows "d< log 2 q"
using assms by linarith

lemma twod_lt_q: 
  assumes "of_nat d < ⌈(log 2 q)::real⌉"
  shows "2 powr (real d) < of_int q"
using assms less_log_iff[of 2 q d] d_lt_logq q_gt_zero 
by auto




lemma break_point_gt_q_div_two:
  assumes "of_nat d < ⌈(log 2 q)::real⌉"
  shows "⌈q-(q/(2*2^d))⌉ > ⌊q/2⌋"
proof -
  have "1/((2::real)^d) ≤ (1::real)" 
    using one_le_power[of 2 d] by simp
  have "⌈q-(q/(2*2^d))⌉ ≥ q-(q/2)* (1/(2^d))" by simp
  moreover have "q-(q/2)* (1/(2^d)) ≥ q - q/2" 
    using ‹1/((2::real)^d) ≤ (1::real)›  
    by (smt (z3) calculation divide_le_eq divide_nonneg_nonneg 
      divide_self_if mult_left_mono of_int_nonneg 
      times_divide_eq_right q_gt_zero)
  ultimately have "⌈q-(q/(2*2^d))⌉ ≥ ⌈q/2⌉ " by linarith
  moreover have "⌈q/2⌉ > ⌊q/2⌋" 
    using prime_half[OF q_prime q_gt_two] .
  ultimately show ?thesis by auto
qed

lemma decompress_zero_unique: 
  assumes "decompress d s = 0"
          "s ∈ {0..2^d - 1}"
          "of_nat d < ⌈(log 2 q)::real⌉"
  shows "s = 0"
proof -
  let ?x = "real_of_int q * real_of_int s / 
    real_of_int 2^d + 1/2"
  have "0 ≤ ?x ∧ ?x < 1" using assms(1)
     unfolding decompress_def round_def 
    using floor_correct[of ?x] by auto
  then have "real_of_int q * real_of_int s / 
    real_of_int 2^d < 1/2" by linarith
  moreover have "real_of_int q / real_of_int 2^d > 1" 
    using twod_lt_q[OF assms(3)] 
    by (simp add: powr_realpow)
  ultimately have "real_of_int s < 1/2" 
  by (smt (verit, best) divide_less_eq_1_pos field_sum_of_halves 
    pos_divide_less_eq times_divide_eq_left)
  then show ?thesis using assms(2) by auto 
qed

text ‹Range of compress and decompress functions›


lemma range_compress: 
  assumes "x∈{0..q-1}" "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "compress d x ∈ {0..2^d - 1}" 
using compress_def by auto

lemma range_decompress:
  assumes "x∈{0..2^d - 1}" "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "decompress d x ∈ {0..q-1}" 
using decompress_def assms
proof (auto, goal_cases)
case 1
  then show ?case 
  by (smt (verit, best) divide_eq_0_iff divide_numeral_1 
    less_divide_eq_1_pos mult_of_int_commute 
    nonzero_mult_div_cancel_right of_int_eq_0_iff 
    of_int_less_1_iff powr_realpow q_gt_zero q_nonzero 
    round_0 round_mono twod_lt_q zero_less_power)
next
case 2
  have "real_of_int q/2^d > 1" using twod_lt_q[OF assms(2)]
    by (simp add: powr_realpow)
  then have log: "real_of_int q - real_of_int q/2^d ≤ q-1" by simp
  have "x ≤ 2^d-1" using assms(1) by simp
  then have "real_of_int x ≤ 2^d - 1" by (simp add: int_less_real_le)
  then have "real_of_int q * real_of_int x / 2^d ≤ 
    real_of_int q * (2^d-1) / 2^d" 
    by (smt (verit, best) divide_strict_right_mono 
      mult_less_cancel_left_pos of_int_pos q_gt_zero 
      zero_less_power)
  also have "… = real_of_int q * 2^d /2^d - real_of_int q/2^d"
    by (simp add: diff_divide_distrib right_diff_distrib)
  finally have "real_of_int q * real_of_int x / 2^d ≤ 
    real_of_int q - real_of_int q/2^d" by simp
  then have "round (real_of_int q * real_of_int x / 2^d) ≤ 
    round (real_of_int q - real_of_int q/2^d)" 
    using round_mono by blast
  also have "… ≤ q - 1" 
    using log by (metis round_mono round_of_int)
  finally show ?case by auto
qed

text ‹Compression is a function qrom $\mathbb{Z} / q\mathbb{Z}$ to 
  $\mathbb{Z} / (2^d)\mathbb{Z}$.›

lemma compress_in_range: 
  assumes "x∈{0..⌈q-(q/(2*2^d))⌉-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "round (real_of_int (2^d * x) / real_of_int q) < 2^d " 
proof - 
  have "(2::int)^d ≠ 0" by simp  
  have "real_of_int x < real_of_int q - real_of_int q / (2 * 2^d)" 
    using assms(1) less_ceiling_iff by auto 
  then have "2^d * real_of_int x / real_of_int q < 
    2^d * (real_of_int q - real_of_int q / (2 * 2^d)) / 
    real_of_int q" 
    by (simp add: divide_strict_right_mono q_gt_zero) 
  also have "… = 2^d * ((real_of_int q / real_of_int q) -
    (real_of_int q / real_of_int q) / (2 * 2^d))" 
  by (simp add:algebra_simps diff_divide_distrib) 
  also have "… = 2^d * (1 - 1/(2*2^d))" 
    using q_nonzero by simp 
  also have "… = 2^d - 1/2" 
    using ‹2^d ≠ 0› by (simp add: right_diff_distrib') 
  finally have "2^d * real_of_int x / real_of_int q < 
    2^d - (1::real)/(2::real)" 
    by auto 
  then show ?thesis unfolding round_def 
    using floor_less_iff by fastforce 
qed

text ‹When does the modulo operation in the compression function change the output? 
  Only when  ‹x ≥ ⌈q-(q / (2*2^d))⌉›. Then we can determine that the compress function 
  maps to zero. This is why we need the ‹mod+-› in the definition of Compression.
  Otherwise the error bound would not hold.›

lemma compress_no_mod: 
  assumes "x∈{0..⌈q-(q / (2*2^d))⌉-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "compress d x = 
    round (real_of_int (2^d * x) / real_of_int q)" 
unfolding compress_def 
using compress_in_range[OF assms] assms(1) q_gt_zero 
by (smt (z3) atLeastAtMost_iff divide_nonneg_nonneg 
  mod_pos_pos_trivial mult_less_cancel_left_pos 
  of_int_nonneg of_nat_0_less_iff right_diff_distrib'
  round_0 round_mono zero_less_power)

lemma compress_2d: 
  assumes "x∈{⌈q-(q/(2*2^d))⌉..q-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "round (real_of_int (2^d * x) / real_of_int q) = 2^d " 
using assms proof - 
  have "(2::int)^d ≠ 0" by simp
  have "round (real_of_int (2^d * x) / real_of_int q) ≥ 2^d"
  proof -
    have "real_of_int x ≥ real_of_int q - real_of_int q / (2 * 2^d)"
      using assms(1) ceiling_le_iff by auto 
    then have "2^d * real_of_int x / real_of_int q ≥ 
      2^d * (real_of_int q - real_of_int q / (2 * 2^d)) / 
      real_of_int q" 
      using q_gt_zero by (simp add: divide_right_mono) 
    also have "2^d * (real_of_int q - real_of_int q / 
      (2 * 2^d)) / real_of_int q
      = 2^d * ((real_of_int q / real_of_int q) -
      (real_of_int q / real_of_int q) / (2 * 2^d))" 
      by (simp add:algebra_simps diff_divide_distrib) 
    also have "… = 2^d * (1 - 1/(2*2^d))" 
      using q_nonzero by simp 
    also have "… = 2^d - 1/2" 
      using ‹2^d ≠ 0› by (simp add: right_diff_distrib') 
    finally have "2^d * real_of_int x / real_of_int q ≥ 
      2^d - (1::real)/(2::real)" 
      by auto 
    then show ?thesis unfolding round_def using le_floor_iff by force
  qed
  moreover have "round (real_of_int (2^d * x) / real_of_int q) ≤ 2^d"
  proof -
    have "d < log 2 q" using assms(2) by linarith
    then have "2 powr (real d) < of_int q" 
      using less_log_iff[of 2 q d] q_gt_zero by auto
    have "x < q" using assms(1) by auto
    then have "real_of_int x/ real_of_int q < 1"
      by (simp add: q_gt_zero)
    then have "real_of_int (2^d * x) / real_of_int q < 
      real_of_int (2^d)" 
      by (auto) (smt (verit, best) divide_less_eq_1_pos 
        nonzero_mult_div_cancel_left times_divide_eq_right 
        zero_less_power)
    then show ?thesis unfolding round_def by linarith
  qed 
  ultimately show ?thesis by auto
qed

lemma compress_mod: 
  assumes "x∈{⌈q-(q/(2*2^d))⌉..q-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉" 
  shows "compress d x = 0" 
unfolding compress_def using compress_2d[OF assms] by simp


text ‹Error after compression and decompression of data.
  To prove the error bound, we distinguish the cases where the ‹mod+-› is relevant or not.›

text ‹First let us look at the error bound for no ‹mod+-› reduction.›

lemma decompress_compress_no_mod: 
  assumes "x∈{0..⌈q-(q/(2*2^d))⌉-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉"
  shows "abs (decompress d (compress d x) - x) ≤ 
    round ( real_of_int q / real_of_int (2^(d+1)))" 
proof - 
  have "abs (decompress d (compress d x) - x) = 
    abs (real_of_int (decompress d (compress d x)) - 
    real_of_int q / real_of_int (2^d) * 
    (real_of_int (2^d * x) / real_of_int q))"
    using q_gt_zero by force
  also have "… ≤ abs (real_of_int (decompress d (compress d x)) -
    real_of_int q / real_of_int (2^d) * real_of_int (compress d x)) +
    abs (real_of_int q / real_of_int (2^d) * 
    real_of_int (compress d x) - real_of_int q / real_of_int (2^d) *
    real_of_int (2^d) / real_of_int q * x)"
    using abs_triangle_ineq[of 
      "real_of_int (decompress d (compress d x)) -
       real_of_int q / real_of_int (2^d) * real_of_int (compress d x)"
      "real_of_int q / real_of_int (2^d) * real_of_int (compress d x)
       - real_of_int q / real_of_int (2^d) * real_of_int (2^d) / 
       real_of_int q * real_of_int x"] by auto
  also have "… ≤ 1/2 + abs (real_of_int q / real_of_int (2^d) *
    (real_of_int (compress d x) - 
     real_of_int (2^d) / real_of_int q * real_of_int x))"
    proof -
      have part_one: 
        "abs (real_of_int (decompress d (compress d x)) -
        real_of_int q / real_of_int (2^d) * real_of_int (compress d x))
        ≤ 1/2"
        unfolding decompress_def 
        using of_int_round_abs_le[of "real_of_int q * 
          real_of_int (compress d x) / real_of_int 2^d"] by simp
      have "real_of_int q * real_of_int (compress d x) / 2^d -
        real_of_int x =
        real_of_int q * (real_of_int (compress d x) - 
        2^d * real_of_int x / real_of_int q) / 2^d" 
        by (smt (verit, best) divide_cancel_right 
        nonzero_mult_div_cancel_left of_int_eq_0_iff 
        q_nonzero right_diff_distrib times_divide_eq_left 
        zero_less_power)
      then have part_two: 
        "abs (real_of_int q / real_of_int (2^d) * 
        real_of_int (compress d x) -
        real_of_int q / real_of_int (2^d) * real_of_int (2^d) /
        real_of_int q * x) =
        abs (real_of_int q / real_of_int (2^d) * 
        (real_of_int (compress d x) - real_of_int (2^d) /
        real_of_int q * x)) " by auto
      show ?thesis using part_one part_two by auto
   qed
  also have "… = 1/2 + (real_of_int q / real_of_int (2^d)) *
      abs (real_of_int (compress d x) - real_of_int (2^d) /
      real_of_int q * real_of_int x)"
    by (subst abs_mult) (smt (verit, best) assms(2) 
      less_divide_eq_1_pos of_int_add of_int_hom.hom_one 
      of_int_power powr_realpow twod_lt_q zero_less_power)
  also have "… ≤ 1/2 + (real_of_int q / real_of_int (2^d)) * (1/2) "
    using compress_no_mod[OF assms] 
    using of_int_round_abs_le[of "real_of_int (2^d) *
      real_of_int x / real_of_int q"]
    by (smt (verit, ccfv_SIG) divide_nonneg_nonneg left_diff_distrib 
      mult_less_cancel_left_pos of_int_mult of_int_nonneg q_gt_zero 
      times_divide_eq_left zero_le_power)
  finally have "real_of_int (abs (decompress d (compress d x) - x)) ≤
                real_of_int q / real_of_int (2*2^d) + 1/2" 
    by simp
  then show ?thesis unfolding round_def using le_floor_iff 
    by fastforce
qed


lemma no_mod_plus_minus: 
  assumes "abs y ≤ round ( real_of_int q / real_of_int (2^(d+1)))"
          "d>0"
  shows "abs y = abs (y mod+- q)"
proof -
  have "round (real_of_int q / real_of_int (2^(d+1))) ≤ ⌊q/2⌋" 
  unfolding round_def 
  proof -
    have "real_of_int q/real_of_int (2^d) ≤ real_of_int q/2" 
    using ‹d>0› 
    proof -
      have "1 / real_of_int (2^d) ≤ 1/2" 
        using ‹d>0› inverse_of_nat_le[of 2 "2^d"]
        by (simp add: self_le_power)
      then show ?thesis using q_gt_zero 
        by (smt (verit, best) frac_less2 of_int_le_0_iff zero_less_power)
    qed
    moreover have "real_of_int q/2 + 1 ≤ real_of_int q" 
      using q_gt_two by auto
    ultimately have "real_of_int q / real_of_int (2^d) + 1 ≤ 
      real_of_int q" by linarith
    then have fact: "real_of_int q / real_of_int (2 ^ (d + 1)) + 
      1/2 ≤ real_of_int q/2" 
      by auto
    then show "⌊real_of_int q / real_of_int (2 ^ (d + 1)) + 1/2⌋ ≤
      ⌊real_of_int q/2⌋" 
      using floor_mono[OF fact] by auto
  qed
  then have "abs y ≤ ⌊q/2⌋" using assms by auto
  then show ?thesis using mod_plus_minus_range_odd[OF q_gt_zero q_odd] 
  by (smt (verit, del_insts) mod_plus_minus_def mod_pos_pos_trivial neg_mod_plus_minus 
    q_odd two_mid_lt_q)
qed


lemma decompress_compress_no_mod_plus_minus: 
  assumes "x∈{0..⌈q-(q/(2*2^d))⌉-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉"
          "d>0"
  shows "abs ((decompress d (compress d x) - x) mod+- q) ≤ 
          round ( real_of_int q / real_of_int (2^(d+1)))"
proof -
  have "abs ((decompress d (compress d x) - x) mod+- q) =
        abs ((decompress d (compress d x) - x)) " 
    using no_mod_plus_minus[OF decompress_compress_no_mod
      [OF assms(1) assms(2)] assms(3)] by auto
  then show ?thesis using decompress_compress_no_mod
    [OF assms(1) assms(2)] by auto
qed

text ‹Now lets look at what happens when the ‹mod+-› reduction comes into action.›


lemma decompress_compress_mod: 
  assumes "x∈{⌈q-(q/(2*2^d))⌉..q-1}" 
          "of_nat d < ⌈(log 2 q)::real⌉"
  shows "abs ((decompress d (compress d x) - x) mod+- q) ≤ 
         round ( real_of_int q / real_of_int (2^(d+1)))"
proof -
  have "(decompress d (compress d x) - x) = - x" 
    using compress_mod[OF assms] unfolding decompress_def 
    by auto
  moreover have "-x mod+- q = -x+q"
  proof -
    have range_x: "x ∈ {⌊real_of_int q / 2⌋<..q - 1}" using assms(1) 
      break_point_gt_q_div_two[OF assms(2)] by auto
    then have *: "- x ∈ {-q + 1..< -⌊real_of_int q / 2⌋}" by auto
    have **: "-x + q ∈{0..<q-⌊real_of_int q / 2⌋}" using * by auto 
    have "-x + q ∈{0..<q}" 
    proof (subst atLeastLessThan_iff)
      have "q-⌊real_of_int q / 2⌋ ≤ q" using q_gt_zero by auto
      moreover have "0 ≤ - x + q ∧ - x + q < q-⌊real_of_int q / 2⌋" using ** by auto
      ultimately show "0 ≤ - x + q ∧ - x + q < q" by linarith
    qed
    then have rew: "-x mod q = -x + q" using mod_rangeE by fastforce
    have "-x mod q < q - ⌊real_of_int q / 2⌋" using ** by (subst rew)(auto simp add: * range_x)
    then have "⌊real_of_int q / 2⌋ ≥ - x mod q" by linarith
    then show ?thesis unfolding mod_plus_minus_def using rew by auto
  qed
  moreover have "abs (q - x) ≤ round ( real_of_int q / 
    real_of_int (2^(d+1)))" 
  proof -
    have "abs (q-x) = q-x" 
      using assms(1) by auto
    also have "… ≤ q - ⌈q - q/(2*2^d)⌉" 
      using assms(1) by simp
    also have "… = - ⌈- q/(2*2^d)⌉" 
      using ceiling_int[of q "- q/(2*2^d)"] by auto
    also have "… = ⌊q/(2*2^d)⌋" 
      by (simp add: ceiling_def)
    also have "… ≤ round (q/(2*2^d))" 
      using floor_le_round by blast
    finally have "abs (q-x) ≤ round (q/(2^(d+1)))" by auto
    then show ?thesis by auto
  qed
  ultimately show ?thesis by auto
qed

text ‹Together, we can determine the general error bound on 
  decompression of compression of the data.
  This error needs to be small enough not to disturb the encryption and decryption process.›

lemma decompress_compress: 
  assumes "x∈{0..<q}"
          "of_nat d < ⌈(log 2 q)::real⌉"
          "d>0"
  shows "let x' = decompress d (compress d x) in 
         abs ((x' - x) mod+- q) ≤ 
          round ( real_of_int q / real_of_int (2^(d+1)) )" 
proof (cases "x<⌈q-(q/(2*2^d))⌉")
case True
  then have range_x: "x∈{0..⌈q-(q/(2*2^d))⌉-1}" 
    using assms(1) by auto
  show ?thesis unfolding Let_def 
    using decompress_compress_no_mod_plus_minus[OF 
    range_x assms(2) assms(3)] by auto
next
case False
  then have range_x: "x∈{⌈q-(q/(2*2^d))⌉..q-1}" 
    using assms(1) by auto
  show ?thesis unfolding Let_def 
    using decompress_compress_mod[OF range_x assms(2)] 
    by auto
qed

text ‹We have now defined compression only on integers (ie ‹{0..<q}›, corresponding to ‹ℤ_q›). 
  We need to extend this notion to the ring ‹ℤ_q[X]/(X^n+1)›. Here, a compressed polynomial 
  is the compression on every coefficient.›

text ‹
How to channel through the types
\begin{itemize}
\item ‹to_qr :: 'a mod_ring poly ⇒ 'a qr›
\item ‹Poly ::  'a mod_ring list ⇒ 'a mod_ring poly›
\item ‹map of_int_mod_ring :: int list ⇒ 'a mod_ring list›
\item ‹map compress :: int list ⇒ int list›
\item ‹map to_int_mod_ring :: 'a mod_ring list ⇒ int list›
\item ‹coeffs :: 'a mod_ring poly ⇒ 'a mod_ring list›
\item ‹of_qr :: 'a qr ⇒ 'a mod_ring poly›
\end{itemize}
›
definition compress_poly :: "nat ⇒ 'a qr ⇒ 'a qr" where
  "compress_poly d = 
        to_qr ∘
        Poly ∘
        (map of_int_mod_ring) ∘
        (map (compress d)) ∘
        (map to_int_mod_ring) ∘
        coeffs ∘
        of_qr"

definition decompress_poly :: "nat ⇒ 'a qr ⇒ 'a qr" where
  "decompress_poly d = 
        to_qr ∘
        Poly ∘
        (map of_int_mod_ring) ∘
        (map (decompress d)) ∘
        (map to_int_mod_ring) ∘
        coeffs ∘
        of_qr"


text ‹Lemmas for compression error for polynomials. Lemma telescope to go qrom module level 
    down to integer coefficients and back up again.›



lemma of_int_mod_ring_eq_0:
  "((of_int_mod_ring x :: 'a mod_ring) = 0) ⟷ 
    (x mod q = 0)"
by (metis CARD_a mod_0 of_int_code(2) 
  of_int_mod_ring.abs_eq of_int_mod_ring.rep_eq 
  of_int_of_int_mod_ring)

lemma dropWhile_mod_ring:
  "dropWhile ((=)0) (map of_int_mod_ring xs :: 'a mod_ring list) = 
   map of_int_mod_ring (dropWhile (λx. x mod q = 0) xs)"
proof (induct xs)
  case (Cons x xs)
  have "dropWhile ((=) 0) (map of_int_mod_ring (x # xs)) = 
        dropWhile ((=) 0) ((of_int_mod_ring x :: 'a mod_ring) # 
        (map of_int_mod_ring xs))"
    by auto
  also have "… = (if 0 = (of_int_mod_ring x :: 'a mod_ring) 
    then dropWhile ((=) 0) (map of_int_mod_ring xs) 
    else map of_int_mod_ring (x # xs))" 
    unfolding dropWhile.simps(2)[of "((=) 0)" 
      "of_int_mod_ring x :: 'a mod_ring" "map of_int_mod_ring xs"]
    by auto
  also have "… = (if x mod q = 0 
    then map of_int_mod_ring (dropWhile (λx. x mod q = 0) xs)
    else map of_int_mod_ring (x # xs))" 
    using of_int_mod_ring_eq_0 unfolding Cons.hyps by auto
  also have "… = map of_int_mod_ring (dropWhile (λx. x mod q = 0) 
    (x # xs))"
    unfolding dropWhile.simps(2) by auto
  finally show ?case by blast
qed simp

lemma strip_while_mod_ring:
"(strip_while ((=) 0) (map of_int_mod_ring xs :: 'a mod_ring list)) =
  map of_int_mod_ring (strip_while (λx. x mod q = 0)  xs)"
unfolding strip_while_def comp_def rev_map dropWhile_mod_ring by auto


lemma of_qr_to_qr_Poly: 
  assumes "length (xs :: int list) < Suc (nat n)"
          "xs ≠ []"
  shows "of_qr (to_qr 
    (Poly (map (of_int_mod_ring :: int ⇒ 'a mod_ring) xs))) =
     Poly (map (of_int_mod_ring :: int ⇒ 'a mod_ring) xs)"
    (is "_ = ?Poly")
proof -
  have deg: "degree (?Poly) < n"
    using deg_Poly'[of "map of_int_mod_ring xs"] assms 
    by (smt (verit, del_insts) One_nat_def Suc_pred degree_0 
      length_greater_0_conv length_map less_Suc_eq_le 
      order_less_le_trans zless_nat_eq_int_zless)
  then show ?thesis
    using of_qr_to_qr[of "?Poly"] deg_mod_qr_poly[of "?Poly"] 
      deg_qr_n by (smt (verit, best) of_nat_less_imp_less)
qed

lemma telescope_stripped:
  assumes "length (xs :: int list) < Suc (nat n)"
    "strip_while (λx. x mod q = 0) xs = xs"
    "set xs ⊆ {0..<q}"
  shows "(map to_int_mod_ring) 
    (coeffs (of_qr (to_qr (Poly 
    (map (of_int_mod_ring :: int ⇒ 'a mod_ring) xs))))) = xs"
proof (cases "xs = []")
case False
  have ge_zero: "0≤x" and lt_q:"x < int CARD ('a)" 
    if "x∈set xs" for x 
    using assms(3) CARD_a atLeastLessThan_iff that by auto
  have to_int_of_int: "map (to_int_mod_ring ∘ 
    (of_int_mod_ring :: int ⇒ 'a mod_ring)) xs = xs"
    using to_int_mod_ring_of_int_mod_ring[OF ge_zero lt_q] 
    by (simp add: map_idI)
  show ?thesis using assms(2) 
    of_qr_to_qr_Poly[OF assms(1) False] 
    by (auto simp add: to_int_of_int strip_while_mod_ring) 
qed (simp)

lemma map_to_of_mod_ring:
  assumes "set xs ⊆ {0..<q}"
  shows "map (to_int_mod_ring ∘ 
    (of_int_mod_ring :: int ⇒ 'a mod_ring)) xs = xs"
using assms by (induct xs) (simp_all add: CARD_a)

lemma telescope:
  assumes "length (xs :: int list) < Suc (nat n)"
    "set xs ⊆ {0..<q}"
  shows "(map to_int_mod_ring) 
    (coeffs (of_qr (to_qr (Poly 
    (map (of_int_mod_ring :: int ⇒ 'a mod_ring) xs))))) =
    strip_while (λx. x mod q = 0) xs"
proof (cases "xs = strip_while (λx. x mod q = 0) xs")
case True
  then show ?thesis using telescope_stripped assms 
    by auto
next
case False
  let ?of_int = "(map (of_int_mod_ring :: 
    int ⇒ 'a mod_ring) xs)"
  have "xs ≠ []" using False by auto
  then have "(map to_int_mod_ring) 
    (coeffs (of_qr (to_qr (Poly ?of_int)))) = 
    (map to_int_mod_ring) (coeffs (Poly ?of_int))"
    using of_qr_to_qr_Poly[OF assms(1)] by auto
  also have "… = (map to_int_mod_ring) 
    (strip_while ((=) 0) ?of_int)" 
    by auto
  also have "… = map (to_int_mod_ring ∘ 
    (of_int_mod_ring :: int ⇒ 'a mod_ring))
    (strip_while (λx. x mod q = 0) xs)" 
    using strip_while_mod_ring by auto
  also have "… = strip_while (λx. x mod q = 0) xs"
  using assms(2) proof (induct xs rule: rev_induct)
  case (snoc y ys)
    let ?to_of_mod_ring = "to_int_mod_ring ∘ 
      (of_int_mod_ring :: int ⇒ 'a mod_ring)"
    have "map ?to_of_mod_ring 
      (strip_while (λx. x mod q = 0) (ys @ [y])) =
      (if y mod q = 0 
       then map ?to_of_mod_ring (strip_while (λx. x mod q = 0) ys)
       else map ?to_of_mod_ring ys @ [?to_of_mod_ring y])"
      by (subst strip_while_snoc) auto
    also have "… = (if y mod q = 0 
        then strip_while (λx. x mod q = 0) ys
        else map ?to_of_mod_ring ys @ [?to_of_mod_ring y])"
    using snoc by fastforce
    also have "… = (if y mod q = 0 
        then strip_while (λx. x mod q = 0) ys
        else ys @ [y])"
      using map_to_of_mod_ring[OF snoc(2)] by auto
    also have "… = strip_while (λx. x mod q = 0) (ys @ [y])" 
      by auto
    finally show ?case .
  qed simp
  finally show ?thesis by auto
qed

lemma length_coeffs_of_qr: 
  "length (coeffs (of_qr (x ::'a qr))) < Suc (nat n)"
proof (cases "x=0")
case False
  then have "of_qr x ≠ 0" by simp
  then show ?thesis 
    using length_coeffs_degree[of "of_qr x"] deg_of_qr[of x]
    using deg_qr_n by fastforce
qed  (auto simp add: n_gt_zero) 
end

lemma strip_while_change: 
  assumes "⋀x. P x ⟶ S x" "⋀x. (¬ P x) ⟶ (¬ S x)"
  shows "strip_while P xs = strip_while S xs"
proof (induct xs rule: rev_induct)
case (snoc x xs)
  have "P x = S x" using assms[of x] by blast
  then show ?case by (simp add: snoc.hyps)
qed simp

lemma strip_while_change_subset: 
  assumes "set xs ⊆ s" 
    "∀x∈s. P x ⟶ S x" 
    "∀x∈s. (¬ P x) ⟶ (¬ S x)"
  shows "strip_while P xs = strip_while S xs"
using assms(1) proof (induct xs rule: rev_induct)
case (snoc x xs)
  have "x∈s" using snoc(2) by simp
  then have "P x ⟶ S x" and "(¬ P x) ⟶ (¬ S x)" 
    using assms(2) assms(3) by auto
  then have "P x = S x" by blast
  then show ?case
  using snoc.hyps snoc.prems by auto
qed simp

text ‹Estimate for decompress compress for polynomials. Using the inequality for integers,
  chain it up to the level of polynomials.›
context kyber_spec
begin
lemma decompress_compress_poly:
  assumes "of_nat d < ⌈(log 2 q)::real⌉"
          "d>0"
  shows "let x' = decompress_poly d (compress_poly d x) in 
    abs_infty_poly (x - x') ≤ 
    round ( real_of_int q / real_of_int (2^(d+1)) )" 
proof -
  let ?x' = "decompress_poly d (compress_poly d x)"
  have "abs_infty_q (poly.coeff (of_qr (x - ?x')) xa)
       ≤ round (real_of_int q / real_of_int (2 ^ (d + 1)))" 
  for xa
  proof -
    let ?telescope = "(λxs. (map to_int_mod_ring) 
      (coeffs (of_qr (to_qr (Poly 
      (map (of_int_mod_ring :: int ⇒ 'a mod_ring) xs))))))"
    define compress_x where 
      "compress_x =  map (compress d ∘ to_int_mod_ring) 
      (coeffs (of_qr x))"
    let ?to_Poly = "(λa. Poly (map ((of_int_mod_ring :: 
      int ⇒ 'a mod_ring) ∘ decompress d) a))"
    have "abs_infty_q (poly.coeff (of_qr x) xa -
      poly.coeff (of_qr (to_qr (?to_Poly 
      (?telescope compress_x)))) xa ) = 
      abs_infty_q (poly.coeff (of_qr x) xa - 
      poly.coeff (of_qr (to_qr (?to_Poly 
      (strip_while (λx. x = 0) compress_x)))) xa )" 
    proof (cases "x = 0")
    case True
      then have "compress_x = []" 
        unfolding compress_x_def by auto
      then show ?thesis by simp
    next
    case False
      then have nonempty:"compress_x ≠ []" 
        unfolding compress_x_def by simp
      have "length compress_x < Suc (nat n)" 
         by (auto simp add: compress_x_def length_coeffs_of_qr)
      moreover have "set compress_x ⊆ {0..<q}" 
      proof -
        have to: "to_int_mod_ring (s::'a mod_ring) ∈ 
            {0..q - 1}" for s
          using to_int_mod_ring_range by auto
        have "compress d (to_int_mod_ring (s::'a mod_ring)) ∈ 
          {0..<q}" for s
          using range_compress[OF to assms(1), of s] 
          twod_lt_q[OF assms(1)]
          by (simp add: powr_realpow)
        then show ?thesis unfolding compress_x_def by auto
      qed
      ultimately have "?telescope compress_x = 
        strip_while (λx. x mod q = 0) compress_x"
        by (intro telescope[of "compress_x"]) simp
      moreover have "strip_while (λx. x mod q = 0) compress_x = 
        strip_while (λx. x = 0) compress_x"
      proof -
        have ‹compress d s = 0› if ‹compress d s mod q = 0› for s
        proof -
          from ‹int d < ⌈log 2 (real_of_int q)⌉› twod_lt_q [of d]
          have ‹2 ^ d < q›
            by (simp add: powr_realpow)
          with compress_less [of d s] have ‹compress d s < q›
            by simp
          then have ‹compress d s = compress d s mod q›
            by (simp add: compress_def)
          with that show ?thesis
            by simp
        qed
        then have right: "⋀s. compress d s mod q = 0 ⟶ 
          compress d s = 0" by simp
        have  "¬ (compress d s = 0)" 
          if "¬ (compress d s mod q = 0)" for s
          using twod_lt_q compress_def that by force
        then have left: "⋀s. ¬ (compress d s mod q = 0) ⟶ 
          ¬ (compress d s = 0)" by simp
        have "strip_while (λx. x mod q = 0) compress_x = 
          strip_while (λx. x mod q = 0) (map (compress d) 
          (map to_int_mod_ring (coeffs (of_qr x))))"
          (is "_ = strip_while (λx. x mod q = 0) 
            (map (compress d) ?rest)")
          unfolding compress_x_def by simp
        also have "… = map (compress d) 
          (strip_while ((λy. y mod q = 0) ∘ compress d)
          (map to_int_mod_ring (coeffs (of_qr x))))"
          using strip_while_map[of "λy. y mod q = 0" "compress d"] 
          by blast
        also have "… = map (compress d) 
          (strip_while ((λy. y = 0) ∘ compress d)
          (map to_int_mod_ring (coeffs (of_qr x))))"
          by (smt (verit, best) comp_eq_dest_lhs left right 
            strip_while_change)
        also have "… = strip_while (λx. x = 0) 
          (map (compress d) ?rest)"
          using strip_while_map[of "λy. y = 0" 
            "compress d", symmetric] by blast
        finally show ?thesis 
          unfolding compress_x_def by auto 
      qed
      ultimately show ?thesis by auto
    qed
    also have "… = abs_infty_q (poly.coeff (of_qr x) xa -
      poly.coeff (?to_Poly (strip_while (λx. x = 0) compress_x)) xa)" 
    proof (cases "?to_Poly (strip_while (λx. x = 0) compress_x) = 0")
    case False
      have "degree (?to_Poly (strip_while (λx. x = 0) compress_x)) ≤
        length (map ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ 
        decompress d) (strip_while (λx. x = 0) compress_x)) - 1" 
        using deg_Poly'[OF False] .
      moreover have "length (map (of_int_mod_ring ∘ decompress d) 
          (strip_while (λx. x = 0) compress_x)) ≤ 
          length (coeffs (of_qr x))"
        unfolding compress_x_def 
        by (metis length_map length_strip_while_le)
      moreover have "length (coeffs (of_qr x)) - 1 < deg_qr TYPE('a)"
        using deg_of_qr degree_eq_length_coeffs by metis
      ultimately have deg: 
        "degree (?to_Poly (strip_while (λx. x = 0) compress_x)) < 
        deg_qr TYPE('a)" by auto
      show ?thesis using of_qr_to_qr'
        by (simp add: of_qr_to_qr'[OF deg])
    qed simp
    also have "… = abs_infty_q (poly.coeff (of_qr x) xa -
      poly.coeff (Poly (map of_int_mod_ring (strip_while (λx. x = 0) 
        (map (decompress d) compress_x)))) xa )" 
    proof -
      have  "s = 0" if "decompress d s = 0" "s ∈ {0..2^d - 1}" for s 
        using decompress_zero_unique[OF that assms(1)] .
      then have right: "∀s ∈ {0..2^d-1}. decompress d s = 0 ⟶ 
        s = 0" by simp
      have left: "∀ s ∈ {0..2^d-1}. decompress d s ≠ 0 ⟶ s ≠ 0" 
        using decompress_zero[of d] by auto
      have compress_x_range: "set compress_x ⊆ {0..2^d - 1}" 
        unfolding compress_x_def compress_def by auto
      have "map (decompress d) (strip_while (λx. x = 0) compress_x) =
        map (decompress d) (strip_while (λx. decompress d x = 0) 
          compress_x)"
      using strip_while_change_subset[OF compress_x_range right left] 
        by auto
      also have "… = strip_while (λx. x = 0) 
        (map (decompress d) compress_x)"
      proof -
        have "(λx. x = 0) ∘ decompress d = (λx. decompress d x = 0)" 
          using comp_def by blast
        then show ?thesis 
          using strip_while_map[symmetric, of "decompress d" 
            "λx. x=0" compress_x] by auto
      qed
      finally have "map (decompress d) (strip_while (λx. x = 0) 
        compress_x) =  strip_while (λx. x = 0) (map (decompress d)
        compress_x)" by auto
      then show ?thesis by (metis map_map)
    qed
    also have "… = abs_infty_q (poly.coeff (of_qr x) xa -
      poly.coeff (Poly (map of_int_mod_ring (strip_while 
      (λx. x mod q = 0) (map (decompress d) compress_x)))) xa )" 
    proof -
      have range: "set (map (decompress d) compress_x) ⊆ {0..<q}" 
        unfolding compress_x_def compress_def
        using range_decompress[OF _ assms(1)] by auto
      have right: " ∀x∈{0..<q}. x = 0 ⟶ x mod q = 0" by auto
      have left: "∀x∈{0..<q}. ¬ x = 0 ⟶ ¬ x mod q = 0" by auto
      have "strip_while (λx. x = 0) (map (decompress d) compress_x) =
        strip_while (λx. x mod q = 0) (map (decompress d) compress_x)"
      using strip_while_change_subset[OF range right left] by auto
      then show ?thesis by auto
    qed
    also have "… = abs_infty_q (poly.coeff (of_qr x) xa -
      poly.coeff (Poly (map of_int_mod_ring 
      (map (decompress d) compress_x))) xa )" 
      by (metis Poly_coeffs coeffs_Poly strip_while_mod_ring)
    also have "… = abs_infty_q (poly.coeff (of_qr x) xa -
      ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ decompress d ∘ 
        compress d ∘ to_int_mod_ring) (poly.coeff (of_qr x) xa))" 
    using coeffs_Poly 
    proof (cases "xa < length (coeffs (?to_Poly  compress_x))")
    case True
      have "poly.coeff (?to_Poly compress_x) xa =
            coeffs (?to_Poly compress_x) ! xa"
      using nth_coeffs_coeff[OF True] by simp
      also have "… = strip_while ((=) 0) (map (
        (of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ decompress d) 
        compress_x) ! xa"
        using coeffs_Poly by auto
      also have "… = (map ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ 
        decompress d) compress_x) ! xa" 
        using True by (metis coeffs_Poly nth_strip_while)
      also have "… = ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ 
        decompress d ∘ compress d ∘ to_int_mod_ring) 
        (coeffs (of_qr x) ! xa)" 
        unfolding compress_x_def 
        by (smt (z3) True coeffs_Poly compress_x_def length_map 
          length_strip_while_le map_map not_less nth_map order_trans)
      also have "… = ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ 
        decompress d ∘ compress d ∘ to_int_mod_ring) 
        (poly.coeff (of_qr x) xa)" 
        by (metis (no_types, lifting) True coeffs_Poly compress_x_def 
          length_map length_strip_while_le not_less nth_coeffs_coeff 
          order.trans)
      finally have no_coeff: "poly.coeff (?to_Poly compress_x) xa = 
          ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ decompress d ∘ 
          compress d ∘ to_int_mod_ring) (poly.coeff (of_qr x) xa)" 
        by auto
      show ?thesis unfolding compress_x_def 
      by (metis compress_x_def map_map no_coeff)
    next
    case False
      then have "poly.coeff (?to_Poly compress_x) xa = 0"
        by (metis Poly_coeffs coeff_Poly_eq nth_default_def)
      moreover have "((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ 
        decompress d ∘ compress d ∘ to_int_mod_ring) 
        (poly.coeff (of_qr x) xa) = 0" 
      proof (cases "poly.coeff (of_qr x) xa = 0")
      case True
        then show ?thesis using compress_zero decompress_zero 
          by auto
      next
      case False
        then show ?thesis 
        proof (subst ccontr, goal_cases)
        case 1
          then have "poly.coeff (?to_Poly compress_x) xa ≠ 0" 
            by (subst coeff_Poly) (metis (no_types, lifting) comp_apply 
              compress_x_def compress_zero decompress_zero map_map 
              nth_default_coeffs_eq nth_default_map_eq 
              of_int_mod_ring_hom.hom_zero to_int_mod_ring_hom.hom_zero)
          then show ?case using ‹poly.coeff (?to_Poly compress_x) xa = 0› 
            by auto
        qed auto
      qed
      ultimately show ?thesis by auto
    qed
    also have "… = abs_infty_q (
      ((of_int_mod_ring :: int ⇒ 'a mod_ring) ∘ decompress d ∘ 
        compress d ∘ to_int_mod_ring) (poly.coeff (of_qr x) xa) -
        poly.coeff (of_qr x) xa)" 
      using abs_infty_q_minus by (metis minus_diff_eq)
    also have "… = ¦((decompress d ∘ compress d ∘ to_int_mod_ring) 
      (poly.coeff (of_qr x) xa) -
       to_int_mod_ring (poly.coeff (of_qr x) xa)) mod+- q¦"
      unfolding abs_infty_q_def 
      using to_int_mod_ring_of_int_mod_ring 
      by (smt (verit, best) CARD_a comp_apply mod_plus_minus_def 
        of_int_diff of_int_mod_ring.rep_eq 
        of_int_mod_ring_to_int_mod_ring of_int_of_int_mod_ring)
    also have "… ≤ round (real_of_int q / real_of_int (2 ^ (d + 1)))" 
    proof -
      have range_to_int_mod_ring: 
        "to_int_mod_ring (poly.coeff (of_qr x) xa) ∈ {0..<q}"
        using to_int_mod_ring_range by auto
      then show ?thesis 
        unfolding abs_infty_q_def Let_def
        using decompress_compress[OF range_to_int_mod_ring assms] 
        by simp
    qed
    finally have "abs_infty_q (poly.coeff (of_qr x) xa - poly.coeff 
      (of_qr (to_qr (?to_Poly (?telescope compress_x)))) xa ) 
      ≤ round (real_of_int q / real_of_int (2 ^ (d + 1)))" by auto
    then show ?thesis unfolding compress_x_def decompress_poly_def 
      compress_poly_def by (auto simp add: o_assoc)
  qed
  moreover have finite: 
    "finite (range (abs_infty_q ∘ poly.coeff (of_qr (x - ?x'))))" 
    by (metis finite_Max image_comp image_image)
  ultimately show ?thesis unfolding abs_infty_poly_def 
    using Max_le_iff[OF finite] by auto
qed



text ‹More properties of compress and decompress, used for returning message at the end.›

lemma compress_1:
  shows "compress 1 x ∈ {0,1}"
unfolding compress_def by auto

lemma compress_poly_1:
  shows "∀i. poly.coeff (of_qr (compress_poly 1 x)) i ∈ {0,1}"
proof -
  have "poly.coeff (of_qr (compress_poly 1 x)) i ∈ {0,1}" 
    for i
  proof -
    have "set (map (compress 1) 
      ((map to_int_mod_ring ∘ coeffs ∘ of_qr) x)) ⊆ {0,1}"
      using compress_1 by auto
    then have "set ((map (compress 1) ∘ map to_int_mod_ring ∘ 
      coeffs ∘ of_qr) x) ⊆ {0,1}"
      (is "set (?compressed_1) ⊆ _")
      by auto
    then have "set (map (of_int_mod_ring :: int ⇒ 'a mod_ring) 
      ?compressed_1) ⊆ {0,1}"
      (is "set (?of_int_compressed_1)⊆_")
      by (smt (verit, best) imageE insert_iff of_int_mod_ring_hom.hom_zero 
        of_int_mod_ring_to_int_mod_ring set_map singletonD subsetD subsetI 
        to_int_mod_ring_hom.hom_one)
    then have "nth_default 0 (?of_int_compressed_1) i
       ∈ {0,1}"
       by (smt (verit, best) comp_apply compress_1 compress_zero  
        insert_iff nth_default_map_eq of_int_mod_ring_hom.hom_zero 
        of_int_mod_ring_to_int_mod_ring singleton_iff 
        to_int_mod_ring_hom.hom_one)
    moreover have "Poly (?of_int_compressed_1)
      = Poly (?of_int_compressed_1) mod qr_poly" 
    proof -
      have "degree (Poly (?of_int_compressed_1)) < deg_qr TYPE('a)"
      proof (cases "Poly ?of_int_compressed_1 ≠ 0")
      case True
        have "degree (Poly ?of_int_compressed_1) ≤ 
          length (map (of_int_mod_ring :: int ⇒ 'a mod_ring) 
          ?compressed_1) - 1"
          using deg_Poly'[OF True] by simp
        also have "… = length ((coeffs ∘ of_qr) x) - 1" 
          by simp 
        also have "… < n" unfolding comp_def 
          using length_coeffs_of_qr
          by (metis deg_qr_n deg_of_qr degree_eq_length_coeffs 
          nat_int zless_nat_conj)
        finally have "degree (Poly ?of_int_compressed_1) < n"
        using True ‹int (length ((coeffs ∘ of_qr) x) - 1) < n› 
          deg_Poly' by fastforce
        then show ?thesis using deg_qr_n by simp
      next
      case False
        then show ?thesis 
        using deg_qr_pos by auto
      qed
      then show ?thesis
        using deg_mod_qr_poly[of "Poly (?of_int_compressed_1)", 
          symmetric] by auto
    qed
    ultimately show ?thesis unfolding compress_poly_def comp_def
      using of_qr_to_qr[of "Poly (?of_int_compressed_1)"]  
      by auto
  qed
  then show ?thesis by auto
qed
end

lemma of_int_mod_ring_mult:
  "of_int_mod_ring (a*b) = of_int_mod_ring a * of_int_mod_ring b"
unfolding of_int_mod_ring_def
by (metis (mono_tags, lifting) Rep_mod_ring_inverse mod_mult_eq 
  of_int_mod_ring.rep_eq of_int_mod_ring_def times_mod_ring.rep_eq)
  
context kyber_spec
begin
lemma decompress_1: 
  assumes "a∈{0,1}"
  shows "decompress 1 a = round(real_of_int q/2) * a" 
unfolding decompress_def using assms by auto 

lemma decompress_poly_1: 
  assumes "∀i. poly.coeff (of_qr x) i ∈ {0,1}"
  shows "decompress_poly 1 x = 
    to_module (round((real_of_int q)/2)) * x"
proof -
  have "poly.coeff (of_qr (decompress_poly 1 x)) i = 
   poly.coeff (of_qr (to_module (round((real_of_int q)/2)) * x)) i"
  for i 
  proof -
    have "set (map to_int_mod_ring (coeffs (of_qr x))) ⊆ {0,1}" 
    (is "set (?int_coeffs) ⊆ _")
    proof -
      have "set (coeffs (of_qr x)) ⊆ {0,1}" using assms 
      by (meson forall_coeffs_conv insert_iff subset_code(1))
      then show ?thesis by auto
    qed
    then have "map (decompress 1) (?int_coeffs) = 
      map ((*) (round (real_of_int q/2))) (?int_coeffs)"
    proof (induct "?int_coeffs")
    case (Cons a xa)
      then show ?case using decompress_1
      by (meson map_eq_conv subsetD)
    qed simp
    then have "poly.coeff (of_qr (decompress_poly 1 x)) i = 
      poly.coeff (of_qr (to_qr (Poly (map of_int_mod_ring
        (map (λa. round(real_of_int q/2) * a) 
        (?int_coeffs)))))) i"
      unfolding decompress_poly_def comp_def by presburger
    also have "… = poly.coeff (of_qr (to_qr (Poly 
        (map (λa. of_int_mod_ring ((round(real_of_int q/2)) * a)) 
        (?int_coeffs))))) i"
      using map_map[of of_int_mod_ring "((*) (round (real_of_int q/2)))"]
      by (smt (z3) map_eq_conv o_apply)
    also have "… = poly.coeff (of_qr (to_qr (Poly 
        (map (λa. of_int_mod_ring (round(real_of_int q/2)) * 
        of_int_mod_ring a) (?int_coeffs))))) i"
      by (simp add: of_int_mod_ring_mult[of "(round(real_of_int q/2))"])
    also have "… = poly.coeff (of_qr (to_qr (Poly 
        (map (λa. of_int_mod_ring (round(real_of_int q/2)) * a) 
        (map of_int_mod_ring (?int_coeffs)))))) i"
      using map_map[symmetric, of 
        "(λa. of_int_mod_ring (round (real_of_int q/2)) * a ::'a mod_ring)"
        "of_int_mod_ring"] unfolding comp_def by presburger
    also have "… = poly.coeff (of_qr (to_qr 
        (Polynomial.smult (of_int_mod_ring (round(real_of_int q/2))) 
        (Poly (map of_int_mod_ring (?int_coeffs)))))) i"
      using smult_Poly[symmetric, of 
        "(of_int_mod_ring (round (real_of_int q/2)))"] 
      by metis
    also have "… = poly.coeff (of_qr ((to_module 
      (round (real_of_int q/2)) * 
      to_qr (Poly (map of_int_mod_ring (?int_coeffs)))))) i"
      unfolding to_module_def 
      using to_qr_smult_to_module
        [of "of_int_mod_ring (round (real_of_int q/2))"]
      by metis
    also have "… = poly.coeff (of_qr
       (to_module (round (real_of_int q/2)) *
        to_qr (Poly (coeffs (of_qr x)))))i"
      by (subst map_map[of of_int_mod_ring to_int_mod_ring], 
        unfold comp_def)(subst of_int_mod_ring_to_int_mod_ring, auto)
    also have "… = poly.coeff (of_qr
       (to_module (round (real_of_int q/2)) * x))i"
      by (subst Poly_coeffs) (subst to_qr_of_qr, simp)
    finally show ?thesis by auto
  qed
  then have eq: "of_qr (decompress_poly 1 x) = 
    of_qr (to_module (round((real_of_int q)/2)) * x)"
    by (simp add: poly_eq_iff)
  show ?thesis using arg_cong[OF eq, of "to_qr"] 
    to_qr_of_qr[of "decompress_poly 1 x"] 
    to_qr_of_qr[of "to_module (round (real_of_int q/2)) * x"] 
    by auto
qed
end
text ‹Compression and decompression for vectors.›

definition map_vector :: 
  "('b ⇒ 'c) ⇒ ('b, 'n) vec ⇒ ('c, 'n::finite) vec" where
  "map_vector f v = (χ i. f (vec_nth v i))"

context kyber_spec 
begin
text ‹Compression and decompression of vectors in ‹ℤ_q[X]/(X^n+1)›.›

definition compress_vec :: 
  "nat ⇒ ('a qr, 'k) vec ⇒ ('a qr, 'k) vec" where
  "compress_vec d = map_vector (compress_poly d)"

definition decompress_vec :: 
  "nat ⇒ ('a qr, 'k) vec ⇒ ('a qr, 'k) vec" where
  "decompress_vec d = map_vector (decompress_poly d)"

end

end