Theory Aligned

(*
 * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
Proofs tidied by LCP, 2024-09
 *)

section "Word Alignment"

theory Aligned
  imports
    "HOL-Library.Word"
    More_Word
    Bit_Shifts_Infix_Syntax

begin

context
  includes bit_operations_syntax
begin

lift_definition is_aligned :: ‹'a::len word ⇒ nat ⇒ bool›
  is ‹λk n. 2 ^ n dvd take_bit LENGTH('a) k›
  by simp

lemma is_aligned_iff_udvd:
  ‹is_aligned w n ⟷ 2 ^ n udvd w›
  by transfer (simp flip: take_bit_eq_0_iff add: min_def)

lemma is_aligned_iff_take_bit_eq_0:
  ‹is_aligned w n ⟷ take_bit n w = 0›
  by (simp add: is_aligned_iff_udvd take_bit_eq_0_iff exp_dvd_iff_exp_udvd)

lemma is_aligned_iff_dvd_int:
  ‹is_aligned ptr n ⟷ 2 ^ n dvd uint ptr›
  by transfer simp

lemma is_aligned_iff_dvd_nat:
  ‹is_aligned ptr n ⟷ 2 ^ n dvd unat ptr›
proof -
  have ‹unat ptr = nat ¦uint ptr¦›
    by transfer simp
  then have ‹2 ^ n dvd unat ptr ⟷ 2 ^ n dvd uint ptr›
    by (simp only: dvd_nat_abs_iff) simp
  then show ?thesis
    by (simp add: is_aligned_iff_dvd_int)
qed

lemma is_aligned_0 [simp]:
  ‹is_aligned 0 n›
  by transfer simp

lemma is_aligned_at_0 [simp]:
  ‹is_aligned w 0›
  by transfer simp

lemma is_aligned_beyond_length:
  ‹is_aligned w n ⟷ w = 0› if ‹LENGTH('a) ≤ n› for w :: ‹'a::len word›
  using that by (simp add: is_aligned_iff_take_bit_eq_0 take_bit_word_beyond_length_eq)

lemma is_alignedI [intro?]:
  ‹is_aligned x n› if ‹x = 2 ^ n * k› for x :: ‹'a::len word›
proof (unfold is_aligned_iff_udvd)
  from that show ‹2 ^ n udvd x›
    using dvd_triv_left exp_dvd_iff_exp_udvd by blast
qed

lemma is_alignedE:
  fixes w :: ‹'a::len word›
  assumes ‹is_aligned w n›
  obtains q where ‹w = 2 ^ n * word_of_nat q› ‹q < 2 ^ (LENGTH('a) - n)›
proof (cases ‹n < LENGTH('a)›)
  case False
  with assms have ‹w = 0›
    by (simp add: is_aligned_beyond_length)
  with that [of 0] show thesis
    by simp
next
  case True
  moreover define m where ‹m = LENGTH('a) - n›
  ultimately have l: ‹LENGTH('a) = n + m› and ‹m ≠ 0›
    by simp_all
  from ‹n < LENGTH('a)› have *: ‹unat (2 ^ n :: 'a word) = 2 ^ n›
    by transfer simp
  from assms have ‹2 ^ n udvd w›
    by (simp add: is_aligned_iff_udvd)
  then obtain v :: ‹'a word›
    where ‹unat w = unat (2 ^ n :: 'a word) * unat v› ..
  moreover define q where ‹q = unat v›
  ultimately have unat_w: ‹unat w = 2 ^ n * q›
    by (simp add: *)
  then have ‹word_of_nat (unat w) = (word_of_nat (2 ^ n * q) :: 'a word)›
    by simp
  then have w: ‹w = 2 ^ n * word_of_nat q›
    by simp
  moreover have ‹q < 2 ^ (LENGTH('a) - n)›
  proof (rule ccontr)
    assume ‹¬ q < 2 ^ (LENGTH('a) - n)›
    then have ‹2 ^ (LENGTH('a) - n) ≤ q›
      by simp
    then have ‹2 ^ LENGTH('a) ≤ 2 ^ n * q›
      by (simp add: l power_add)
    with unat_w [symmetric] show False
      by (metis le_antisym nat_less_le unsigned_less)
  qed
  ultimately show thesis
    using that by blast
qed

lemma is_alignedE' [elim?]:
  fixes w :: ‹'a::len word›
  assumes ‹is_aligned w n›
  obtains q where ‹w = push_bit n (word_of_nat q)› ‹q < 2 ^ (LENGTH('a) - n)›
proof -
  from assms
  obtain q where ‹w = 2 ^ n * word_of_nat q› ‹q < 2 ^ (LENGTH('a) - n)›
    by (rule is_alignedE)
  then have ‹w = push_bit n (word_of_nat q)›
    by (simp add: push_bit_eq_mult)
  with that show thesis
    using ‹q < 2 ^ (LENGTH('a) - n)› .
qed

lemma is_aligned_mask:
  ‹is_aligned w n ⟷ w AND mask n = 0›
  by (simp add: is_aligned_iff_take_bit_eq_0 take_bit_eq_mask)

lemma is_aligned_imp_not_bit:
  ‹¬ bit w m› if ‹is_aligned w n› and ‹m < n›
  for w :: ‹'a::len word›
proof -
  from ‹is_aligned w n›
  obtain q where ‹w = push_bit n (word_of_nat q)› ‹q < 2 ^ (LENGTH('a) - n)› ..
  moreover have ‹¬ bit (push_bit n (word_of_nat q :: 'a word)) m›
    using ‹m < n› by (simp add: bit_simps)
  ultimately show ?thesis
    by simp
qed

lemma is_aligned_weaken:
  "⟦ is_aligned w x; x ≥ y ⟧ ⟹ is_aligned w y"
  unfolding is_aligned_iff_dvd_nat
  by (erule dvd_trans [rotated]) (simp add: le_imp_power_dvd)

lemma is_alignedE_pre:
  fixes w::"'a::len word"
  assumes aligned: "is_aligned w n"
  shows        rl: "∃q. w = 2 ^ n * (of_nat q) ∧ q < 2 ^ (LENGTH('a) - n)"
  using aligned is_alignedE by blast

lemma aligned_add_aligned:
  fixes x::"'a::len word"
  assumes aligned1: "is_aligned x n"
  and     aligned2: "is_aligned y m"
  and           lt: "m ≤ n"
  shows   "is_aligned (x + y) m"
proof cases
  assume nlt: "n < LENGTH('a)"
  show ?thesis
    unfolding is_aligned_iff_dvd_nat dvd_def
  proof -
    from aligned2 obtain q2 where yv: "y = 2 ^ m * of_nat q2"
      and q2v: "q2 < 2 ^ (LENGTH('a) - m)"
      by (auto elim: is_alignedE)

    from lt obtain k where kv: "m + k = n" by (auto simp: le_iff_add)
    with aligned1 obtain q1 where xv: "x = 2 ^ (m + k) * of_nat q1"
      and q1v: "q1 < 2 ^ (LENGTH('a) - (m + k))"
      by (auto elim: is_alignedE)

    have l1: "2 ^ (m + k) * q1 < 2 ^ LENGTH('a)"
      by (rule nat_less_power_trans [OF q1v])
         (subst kv, rule order_less_imp_le [OF nlt])

    have l2: "2 ^ m * q2 < 2 ^ LENGTH('a)"
      by (rule nat_less_power_trans [OF q2v],
          rule order_less_imp_le [OF order_le_less_trans])
         fact+

    have "x = of_nat (2 ^ (m + k) *  q1)" using xv
      by simp

    moreover have "y = of_nat (2 ^ m * q2)" using yv
      by simp

    ultimately have upls: "unat x + unat y = 2 ^ m * (2 ^ k * q1 + q2)"
    proof -
      have f1: "unat x = 2 ^ (m + k) * q1"
        using q1v unat_mult_power_lem xv by blast
      have "unat y = 2 ^ m * q2"
        using q2v unat_mult_power_lem yv by blast
      then show ?thesis
        using f1 by (simp add: power_add semiring_normalization_rules(34))
    qed

    (* (2 ^ k * q1 + q2) *)
    show "∃d. unat (x + y) = 2 ^ m * d"
    proof (cases "unat x + unat y < 2 ^ LENGTH('a)")
      case True

      have "unat (x + y) = unat x + unat y"
        by (subst unat_plus_if', rule if_P) fact

      also have "… = 2 ^ m * (2 ^ k * q1 + q2)" by (rule upls)
      finally show ?thesis ..
    next
      case False
      then have "unat (x + y) = (unat x + unat y) mod 2 ^ LENGTH('a)"
        by (subst unat_word_ariths(1)) simp

      also have "… = (2 ^ m * (2 ^ k * q1 + q2)) mod 2 ^ LENGTH('a)"
        by (subst upls, rule refl)

      also
      have "… = 2 ^ m * ((2 ^ k * q1 +  q2) mod 2 ^ (LENGTH('a) - m))"
      proof -
        have "m ≤ len_of (TYPE('a))"
          by (meson le_trans less_imp_le_nat lt nlt)
        then show ?thesis
          by (metis mult_mod_right ordered_cancel_comm_monoid_diff_class.add_diff_inverse power_add)
      qed

      finally show ?thesis ..
    qed
  qed
next
  assume "¬ n < LENGTH('a)"
  with assms
  show ?thesis
    by (simp add: is_aligned_mask not_less take_bit_eq_mod power_overflow word_arith_nat_defs(7) flip: take_bit_eq_mask)
qed

corollary aligned_sub_aligned:
  "⟦is_aligned (x::'a::len word) n; is_aligned y m; m ≤ n⟧
   ⟹ is_aligned (x - y) m"
  by (metis (no_types, lifting) diff_zero is_aligned_mask is_aligned_weaken mask_eqs(4))

lemma is_aligned_shift:
  fixes k::"'a::len word"
  shows "is_aligned (k << m) m"
proof cases
  assume mv: "m < LENGTH('a)"
  from mv obtain q where mq: "m + q = LENGTH('a)" and "0 < q"
    by (auto dest: less_imp_add_positive)

  have "(2::nat) ^ m dvd unat (push_bit m k)"
  proof
    have kv: "(unat k div 2 ^ q) * 2 ^ q + unat k mod 2 ^ q = unat k"
      by (rule div_mult_mod_eq)

    have "unat (push_bit m k) = unat (2 ^ m * k)"
      by (simp add: push_bit_eq_mult ac_simps)
    also have "… = (2 ^ m * unat k) mod (2 ^ LENGTH('a))" using mv
      by (simp add: unat_word_ariths(2))
    also have "… = 2 ^ m * (unat k mod 2 ^ q)"
      by (subst mq [symmetric], subst power_add, subst mod_mult2_eq) simp
    finally show "unat (push_bit m k) = 2 ^ m * (unat k mod 2 ^ q)" .
  qed
  then show ?thesis by (unfold is_aligned_iff_dvd_nat shiftl_def)
next
  assume "¬ m < LENGTH('a)"
  then show ?thesis
    by (simp add: not_less power_overflow is_aligned_mask word_size shiftl_def)
qed

lemma aligned_mod_eq_0:
  fixes p::"'a::len word"
  assumes al: "is_aligned p sz"
  shows   "p mod 2 ^ sz = 0"
proof cases
  assume szv: "sz < LENGTH('a)"
  with al
  show ?thesis
    unfolding is_aligned_iff_dvd_nat
    by (simp add: and_mask_dvd_nat p2_gt_0 word_mod_2p_is_mask)
next
  assume "¬ sz < LENGTH('a)"
  with al show ?thesis
    by (simp add: is_aligned_mask flip: take_bit_eq_mask take_bit_eq_mod)
qed

lemma is_aligned_triv: "is_aligned (2 ^ n ::'a::len word) n"
  by (rule is_alignedI [where k = 1], simp)

lemma is_aligned_mult_triv1: "is_aligned (2 ^ n * x  ::'a::len word) n"
  by (rule is_alignedI [OF refl])

lemma is_aligned_mult_triv2: "is_aligned (x * 2 ^ n ::'a::len word) n"
  by (subst mult.commute, simp add: is_aligned_mult_triv1)

lemma word_power_less_0_is_0:
  fixes x :: "'a::len word"
  shows "x < a ^ 0 ⟹ x = 0" by simp

lemma is_aligned_no_wrap:
  fixes off :: "'a::len word"
  fixes ptr :: "'a::len word"
  assumes al: "is_aligned ptr sz"
  and    off: "off < 2 ^ sz"
  shows  "unat ptr + unat off < 2 ^ LENGTH('a)"
proof -
  have szv: "sz < LENGTH('a)"
  using off p2_gt_0 word_neq_0_conv by fastforce

  from al obtain q where ptrq: "ptr = 2 ^ sz * of_nat q" and
    qv: "q < 2 ^ (LENGTH('a) - sz)" by (auto elim: is_alignedE)

  show ?thesis
  proof (cases "sz = 0")
    case True
    then show ?thesis using off ptrq qv
      by simp
  next
    case False
    then have sne: "0 < sz" ..
    show ?thesis
    proof -
      have uq: "unat (of_nat q ::'a::len word) = q"
        by (metis diff_less le_unat_uoi len_gt_0 order_less_imp_le qv sne unat_power_lower)
      have uptr: "unat ptr = 2 ^ sz * q"
        by (simp add: ptrq qv unat_mult_power_lem)
      show "unat ptr + unat off < 2 ^ LENGTH('a)" using szv
        by (metis le_add_diff_inverse2 less_imp_le mult.commute nat_add_offset_less off qv unat_less_power uptr)
    qed
  qed
qed

lemma is_aligned_no_wrap':
  fixes ptr :: "'a::len word"
  assumes al: "is_aligned ptr sz"
  and    off: "off < 2 ^ sz"
  shows  "ptr ≤ ptr + off"
  by (subst no_plus_overflow_unat_size, subst word_size, rule is_aligned_no_wrap) fact+

lemma is_aligned_no_overflow':
  fixes p :: "'a::len word"
  assumes al: "is_aligned p n"
  shows "p ≤ p + (2 ^ n - 1)"
proof cases
  assume "n<LENGTH('a)"
  with al
  have "2^n - (1::'a::len word) < 2^n"
    by (simp add: word_less_nat_alt unat_sub_if_size)
  with al
  show ?thesis by (rule is_aligned_no_wrap')
next
  assume "¬ n<LENGTH('a)"
  with al
  show ?thesis
  by (simp add: not_less power_overflow is_aligned_mask mask_2pm1)
qed

lemma is_aligned_no_overflow:
  "is_aligned ptr sz ⟹ ptr ≤ ptr + 2^sz - 1"
  by (drule is_aligned_no_overflow') (simp add: field_simps)

lemma replicate_not_True:
  "⋀n. xs = replicate n False ⟹ True ∉ set xs"
  by (induct xs) auto

lemma map_zip_replicate_False_xor:
  "n = length xs ⟹ map (λ(x, y). x = (¬ y)) (zip xs (replicate n False)) = xs"
  by (induct xs arbitrary: n, auto)

lemma drop_minus_lem:
  "⟦ n ≤ length xs; 0 < n; n' = length xs ⟧ ⟹ drop (n' - n) xs = rev xs ! (n - 1)  # drop (Suc (n' - n)) xs"
proof (induct xs)
  case Nil then show ?case by simp
next
  case (Cons y ys)
  show ?case
  proof (cases "n = Suc (length ys)")
    case True
    then show ?thesis
      using Cons.prems(3) rev_nth by fastforce
  next
    case False
    with Cons show ?thesis
      apply (simp add: rev_nth nth_append)
      by (simp add: Cons_nth_drop_Suc Suc_diff_le)
  qed
qed

lemma drop_minus:
  "⟦ n < length xs; n' = length xs ⟧ ⟹ drop (n' - Suc n) xs = rev xs ! n  # drop (n' - n) xs"
  by (simp add: Suc_diff_Suc drop_minus_lem)

lemma aligned_add_xor:
  ‹(x + 2 ^ n) XOR 2 ^ n = x›
  if al: ‹is_aligned (x::'a::len word) n'› and le: ‹n < n'›
proof -
  have ‹¬ bit x n›
    using that by (rule is_aligned_imp_not_bit)
  then have ‹x + 2 ^ n = x OR 2 ^ n›
    by (subst disjunctive_add) (auto simp add: bit_simps disjunctive_add)
  moreover have ‹(x OR 2 ^ n) XOR 2 ^ n = x›
    by (rule bit_word_eqI) (auto simp add: bit_simps ‹¬ bit x n›)
  ultimately show ?thesis
    by simp
qed

lemma is_aligned_add_mult_multI:
  fixes p :: "'a::len word"
  shows "⟦is_aligned p m; n ≤ m; n' = n⟧ ⟹ is_aligned (p + x * 2 ^ n * z) n'"
  by (metis aligned_add_aligned is_alignedI mult.assoc mult.commute)

lemma is_aligned_add_multI:
  fixes p :: "'a::len word"
  shows "⟦is_aligned p m; n ≤ m; n' = n⟧ ⟹ is_aligned (p + x * 2 ^ n) n'"
  by (simp add: aligned_add_aligned is_aligned_mult_triv2)

lemma is_aligned_no_wrap''':
  fixes ptr :: "'a::len word"
  shows "⟦ is_aligned ptr sz; sz < LENGTH('a); off < 2 ^ sz ⟧
         ⟹ unat ptr + off < 2 ^ LENGTH('a)"
  by (metis is_alignedE le_add_diff_inverse2 less_imp_le mult.commute nat_add_offset_less unat_mult_power_lem)

lemma is_aligned_get_word_bits:
  fixes p :: "'a::len word"
  assumes "is_aligned p n"
  obtains "n < LENGTH('a)" | "p = 0" "n ≥ LENGTH('a)"
  by (meson assms is_aligned_beyond_length linorder_not_le)

lemma aligned_small_is_0:
  "⟦ is_aligned x n; x < 2 ^ n ⟧ ⟹ x = 0"
  by (simp add: is_aligned_mask less_mask_eq)

corollary is_aligned_less_sz:
  "⟦is_aligned a sz; a ≠ 0⟧ ⟹ ¬ a < 2 ^ sz"
  by (rule notI, drule(1) aligned_small_is_0, erule(1) notE)

lemma aligned_at_least_t2n_diff:
  assumes x: "is_aligned x n"
      and y: "is_aligned y n"
      and "x < y"
    shows "x ≤ y - 2 ^ n"
  by (meson assms is_aligned_iff_udvd udvd_minus_le')

lemma is_aligned_no_overflow'':
  "⟦is_aligned x n; x + 2 ^ n ≠ 0⟧ ⟹ x ≤ x + 2 ^ n"
  using is_aligned_no_overflow order_trans word_sub_1_le by blast

lemma is_aligned_bitI:
  ‹is_aligned p m› if ‹⋀n. n < m ⟹ ¬ bit p n›
  by (simp add: bit_take_bit_iff bit_word_eqI is_aligned_iff_take_bit_eq_0 that)

lemma is_aligned_nth:
  "is_aligned p m = (∀n < m. ¬ bit p n)"
  using is_aligned_bitI is_aligned_imp_not_bit by blast

lemma range_inter:
  "({a..b} ∩ {c..d} = {}) = (∀x. ¬(a ≤ x ∧ x ≤ b ∧ c ≤ x ∧ x ≤ d))"
  by auto

lemma aligned_inter_non_empty:
  "⟦ {p..p + (2 ^ n - 1)} ∩ {p..p + 2 ^ m - 1} = {};
     is_aligned p n; is_aligned p m⟧ ⟹ False"
  by (simp add: is_aligned_no_overflow is_aligned_no_overflow')

lemma not_aligned_mod_nz:
  assumes al: "¬ is_aligned a n"
  shows "a mod 2 ^ n ≠ 0"
  by (meson assms is_aligned_iff_udvd mod_eq_0_imp_udvd)

lemma nat_add_offset_le:
  fixes x :: nat
  assumes yv: "y ≤ 2 ^ n"
  and     xv: "x < 2 ^ m"
  and     mn: "sz = m + n"
  shows   "x * 2 ^ n + y ≤ 2 ^ sz"
proof (subst mn)
  from yv obtain qy where "y + qy = 2 ^ n"
    by (auto simp: le_iff_add)

  have "x * 2 ^ n + y ≤ x * 2 ^ n + 2 ^ n"
    using yv xv by simp
  also have "… = (x + 1) * 2 ^ n" by simp
  also have "… ≤ 2 ^ (m + n)" using xv
    by (subst power_add) (rule mult_le_mono1, simp)
  finally show "x * 2 ^ n + y ≤ 2 ^ (m + n)" .
qed

lemma is_aligned_no_wrap_le:
  fixes ptr::"'a::len word"
  assumes al: "is_aligned ptr sz"
  and    szv: "sz < LENGTH('a)"
  and    off: "off ≤ 2 ^ sz"
  shows  "unat ptr + off ≤ 2 ^ LENGTH('a)"
proof -
  from al obtain q where ptrq: "ptr = 2 ^ sz * of_nat q" and
    qv: "q < 2 ^ (LENGTH('a) - sz)" by (auto elim: is_alignedE)

  show ?thesis
  proof (cases "sz = 0")
    case True
    then show ?thesis using off ptrq qv
      by (auto simp add: le_Suc_eq Suc_le_eq) (simp add: le_less)
  next
    case False
    then have sne: "0 < sz" ..

    show ?thesis
    proof -
      have uq: "unat (of_nat q :: 'a word) = q"
        by (metis diff_less le_unat_uoi len_gt_0 order_le_less qv sne unat_power_lower)
      have uptr: "unat ptr = 2 ^ sz * q"
        by (simp add: ptrq qv unat_mult_power_lem)
      show "unat ptr + off ≤ 2 ^ LENGTH('a)" using szv
        by (metis le_add_diff_inverse2 less_imp_le mult.commute nat_add_offset_le off qv uptr)
    qed
  qed
qed

lemma is_aligned_neg_mask:
  "m ≤ n ⟹ is_aligned (x AND NOT (mask n)) m"
  by (rule is_aligned_bitI) (simp add: bit_simps)

lemma unat_minus:
  "unat (- (x :: 'a :: len word)) = (if x = 0 then 0 else 2 ^ size x - unat x)"
  using unat_sub_if_size[where x="2 ^ size x" and y=x]
  by (simp add: unat_eq_0 word_size)

lemma is_aligned_minus:
  ‹is_aligned (- p) n› if ‹is_aligned p n› for p :: ‹'a::len word›
  by (metis is_alignedE is_alignedI mult_minus_right that)

lemma add_mask_lower_bits:
  fixes x :: "'a :: len word"
  assumes "is_aligned x n"
    and "∀n'≥n. n' < LENGTH('a) ⟶ ¬ bit p n'"
  shows "x + p AND NOT (mask n) = x"
proof -
  have "x AND p = 0"
    by (metis assms bit_and_iff bit_imp_le_length is_aligned_nth not_le_imp_less word_exists_nth)
  moreover 
  have "(x OR p) AND NOT (mask n) = x"
  proof (rule bit_word_eqI)
    fix k
    assume "k < LENGTH('a)"
    show "bit ((x OR p) AND NOT (mask n)) k = bit x k"
      by (metis assms is_aligned_mask mask_eq_0_eq_x neg_mask_test_bit word_ao_nth)
  qed
  ultimately show ?thesis
    by (simp add: word_plus_and_or_coroll)
qed

lemma is_aligned_andI1:
  "is_aligned x n ⟹ is_aligned (x AND y) n"
  by (simp add: is_aligned_nth bit_simps)

lemma is_aligned_andI2:
  "is_aligned y n ⟹ is_aligned (x AND y) n"
  by (simp add: is_aligned_nth bit_simps)

lemma is_aligned_shiftl:
  "is_aligned w (n - m) ⟹ is_aligned (w << m) n"
  by (simp add: is_aligned_nth bit_simps)

lemma is_aligned_shiftr:
  "is_aligned w (n + m) ⟹ is_aligned (w >> m) n"
  by (simp add: is_aligned_nth bit_simps)

lemma is_aligned_shiftl_self:
  "is_aligned (p << n) n"
  by (rule is_aligned_shift)

lemma is_aligned_neg_mask_eq:
  "is_aligned p n ⟹ p AND NOT (mask n) = p"
  by (simp add: is_aligned_mask mask_eq_x_eq_0)

lemma is_aligned_shiftr_shiftl:
  "is_aligned w n ⟹ w >> n << n = w"
  apply (rule bit_word_eqI)
  by (metis bit_shiftl_iff bit_shiftr_eq is_aligned_nth leI le_add_diff_inverse o_apply possible_bit_word)

lemma aligned_shiftr_mask_shiftl:
  assumes "is_aligned x n"
  shows "((x >> n) AND mask v) << n = x AND mask (v + n)"
proof -
  have "bit x m ⟹ m ≥ n" for m
    using assms is_aligned_imp_not_bit leI by blast
  with assms show ?thesis
    apply (intro word_eqI)
    by (metis bit_and_iff is_aligned_neg_mask_eq is_aligned_shiftr_shiftl push_bit_and push_bit_mask_eq shiftl_def)
qed

lemma mask_zero:
  "is_aligned x a ⟹ x AND mask a = 0"
  by (metis is_aligned_mask)

lemma is_aligned_neg_mask_eq_concrete:
  "⟦ is_aligned p n; msk AND NOT (mask n) = NOT (mask n) ⟧
   ⟹ p AND msk = p"
  by (metis word_bw_assocs(1) word_bw_comms(1) is_aligned_neg_mask_eq)

lemma is_aligned_and_not_zero:
  "⟦ is_aligned n k; n ≠ 0 ⟧ ⟹ 2 ^ k ≤ n"
  using is_aligned_less_sz leI by blast

lemma is_aligned_and_2_to_k:
  "(n AND 2 ^ k - 1) = 0 ⟹ is_aligned (n :: 'a :: len word) k"
  by (simp add: is_aligned_mask mask_eq_decr_exp)

lemma is_aligned_power2:
  "b ≤ a ⟹ is_aligned (2 ^ a) b"
  by (metis is_aligned_triv is_aligned_weaken)

lemma aligned_sub_aligned':
  "⟦ is_aligned (a :: 'a :: len word) n; is_aligned b n; n < LENGTH('a) ⟧
   ⟹ is_aligned (a - b) n"
  by (simp add: aligned_sub_aligned)

lemma is_aligned_neg_mask_weaken:
  "⟦ is_aligned p n; m ≤ n ⟧ ⟹ p AND NOT (mask m) = p"
   using is_aligned_neg_mask_eq is_aligned_weaken by blast

lemma is_aligned_neg_mask2 [simp]:
  "is_aligned (a AND NOT (mask n)) n"
  by (rule is_aligned_bitI) (simp add: bit_simps)

lemma is_aligned_0':
  "is_aligned 0 n"
  by (fact is_aligned_0)

lemma aligned_add_offset_no_wrap:
  fixes off :: "('a::len) word"
  and     x :: "'a word"
  assumes al: "is_aligned x sz"
  and   offv: "off < 2 ^ sz"
  shows  "unat x + unat off < 2 ^ LENGTH('a)"
proof cases
  assume szv: "sz < LENGTH('a)"
  from al obtain k where xv: "x = 2 ^ sz * (of_nat k)"
    and kl: "k < 2 ^ (LENGTH('a) - sz)"
    by (auto elim: is_alignedE)

  show ?thesis using szv
    using al is_aligned_no_wrap offv by blast
next
  assume "¬ sz < LENGTH('a)"
  with offv show ?thesis by (simp add: not_less power_overflow )
qed

lemma aligned_add_offset_mod:
  fixes x :: "('a::len) word"
  assumes al: "is_aligned x sz"
  and     kv: "k < 2 ^ sz"
  shows   "(x + k) mod 2 ^ sz = k"
proof cases
  assume szv: "sz < LENGTH('a)"

  have ux: "unat x + unat k < 2 ^ LENGTH('a)"
    by (rule aligned_add_offset_no_wrap) fact+

  show ?thesis using al szv
    by (metis add_0 assms(2) less_mask_eq mask_eqs(1) mask_zero word_gt_a_gt_0 word_mod_2p_is_mask)
next
  assume "¬ sz < LENGTH('a)"
  with al show ?thesis
    by (simp add: not_less power_overflow is_aligned_mask mask_eq_decr_exp
                  word_mod_by_0)
qed

lemma aligned_neq_into_no_overlap:
  fixes x :: "'a::len word"
  assumes neq: "x ≠ y"
  and     alx: "is_aligned x sz"
  and     aly: "is_aligned y sz"
  shows  "{x .. x + (2 ^ sz - 1)} ∩ {y .. y + (2 ^ sz - 1)} = {}"
proof cases
  assume szv: "sz < LENGTH('a)"
  show ?thesis
  proof (rule equals0I, clarsimp)
    fix z
    assume xb: "x ≤ z" and xt: "z ≤ x + (2 ^ sz - 1)"
      and yb: "y ≤ z" and yt: "z ≤ y + (2 ^ sz - 1)"

    have rl: "⋀(p::'a word) k w. ⟦uint p + uint k < 2 ^ LENGTH('a); w = p + k; w ≤ p + (2 ^ sz - 1) ⟧
      ⟹ k < 2 ^ sz"
      by (smt (verit, best) no_olen_add szv uint_plus_simple_iff word_le_def word_less_sub_le)
    from xb obtain kx where
      kx: "z = x + kx" and
      kxl: "uint x + uint kx < 2 ^ LENGTH('a)"
      by (clarsimp dest!: word_le_exists')

    from yb obtain ky where
      ky: "z = y + ky" and
      kyl: "uint y + uint ky < 2 ^ LENGTH('a)"
      by (clarsimp dest!: word_le_exists')

    have "x = y"
    proof -
      have "kx = z mod 2 ^ sz"
      proof (subst kx, rule sym, rule aligned_add_offset_mod)
        show "kx < 2 ^ sz" by (rule rl) fact+
      qed fact+

      also have "… = ky"
      proof (subst ky, rule aligned_add_offset_mod)
        show "ky < 2 ^ sz"
          using kyl ky yt by (rule rl)
      qed fact+

      finally have kxky: "kx = ky" .
      moreover have "x + kx = y + ky" by (simp add: kx [symmetric] ky [symmetric])
      ultimately show ?thesis by simp
    qed
    then show False using neq by simp
  qed
next
  assume "¬ sz < LENGTH('a)"
  with neq alx aly
  have False by (simp add: is_aligned_mask mask_eq_decr_exp power_overflow)
  then show ?thesis ..
qed

lemma is_aligned_add_helper:
  "⟦ is_aligned p n; d < 2 ^ n ⟧
     ⟹ (p + d AND mask n = d) ∧ (p + d AND (NOT (mask n)) = p)"
  by (metis add_diff_cancel_left' add_mask_lower_bits less_2p_is_upper_bits_unset subtract_mask(2))

lemmas mask_inner_mask = mask_eqs(1)

lemma mask_add_aligned:
  "is_aligned p n ⟹ (p + q) AND mask n = q AND mask n"
  by (metis add_0 mask_inner_mask mask_zero)

lemma mask_out_add_aligned:
  assumes al: "is_aligned p n"
  shows "p + (q AND NOT (mask n)) = (p + q) AND NOT (mask n)"
  using mask_add_aligned [OF al]
  by (simp add: mask_out_sub_mask)

lemma is_aligned_add_or:
  "⟦is_aligned p n; d < 2 ^ n⟧ ⟹ p + d = p OR d"
  by (metis and_zero_eq less_mask_eq mask_zero word_bw_assocs(1) word_bw_comms(1) word_plus_and_or_coroll)

lemma not_greatest_aligned:
  "⟦ x < y; is_aligned x n; is_aligned y n ⟧ ⟹ x + 2 ^ n ≠ 0"
  by (metis NOT_mask add_diff_cancel_right' diff_0 is_aligned_neg_mask_eq not_le word_and_le1)

lemma neg_mask_mono_le:
  "x ≤ y ⟹ x AND NOT(mask n) ≤ y AND NOT(mask n)" for x :: "'a :: len word"
proof (rule ccontr, simp add: linorder_not_le, cases "n < LENGTH('a)")
  case False
  then show "y AND NOT(mask n) < x AND NOT(mask n) ⟹ False"
    by (simp add: mask_eq_decr_exp linorder_not_less power_overflow)
next
  case True
  assume a: "x ≤ y" and b: "y AND NOT(mask n) < x AND NOT(mask n)"
  have word_bits: "n < LENGTH('a)" by fact
  have "y ≤ (y AND NOT(mask n)) + (y AND mask n)"
    by (simp add: word_plus_and_or_coroll2 add.commute)
  also have "… ≤ (y AND NOT(mask n)) + 2 ^ n"
  proof (rule word_plus_mono_right)
    show "y AND mask n ≤ 2 ^ n"
      by (metis and_mask_less' b linorder_not_le mask_exceed order_less_imp_le)
  next
    show "y AND NOT (mask n) ≤ (y AND NOT (mask n)) + 2 ^ n"
      using b is_aligned_neg_mask2 is_aligned_no_overflow'' not_greatest_aligned by blast
  qed
  also have "… ≤ x AND NOT(mask n)"
  proof -
    have "y AND NOT (mask n) ≤ (x AND NOT (mask n)) - 2 ^ n"
      by (simp add: aligned_at_least_t2n_diff b)
    moreover have "2 ^ n ≤ x AND NOT (mask n)"
      by (metis b is_aligned_mask is_aligned_neg_mask2 less_mask_eq linorder_not_le word_and_le2)
    ultimately show ?thesis
      by (metis add.commute le_plus)
  qed
  also have "… ≤ x" by (rule word_and_le2)
  also have "x ≤ y" by fact
  finally
  show "False" using b by simp
qed

lemma and_neg_mask_eq_iff_not_mask_le:
  "w AND NOT(mask n) = NOT(mask n) ⟷ NOT(mask n) ≤ w"
  for w :: ‹'a::len word›
  by (metis eq_iff neg_mask_mono_le word_and_le1 word_and_le2 word_bw_same(1))

lemma neg_mask_le_high_bits:
  ‹NOT (mask n) ≤ w ⟷ (∀i ∈ {n ..< size w}. bit w i)› (is ‹?P ⟷ ?Q›)
  for w :: ‹'a::len word›
proof
  assume ?Q
  then have ‹w AND NOT (mask n) = NOT (mask n)›
    by (auto simp add: bit_simps word_size intro: bit_word_eqI)
  then show ?P
    by (simp add: and_neg_mask_eq_iff_not_mask_le)
next
  assume ?P
  then have *: ‹w AND NOT (mask n) = NOT (mask n)›
    by (simp add: and_neg_mask_eq_iff_not_mask_le)
  show ‹?Q›
  proof (rule ccontr)
    assume ‹¬ (∀i∈{n..<size w}. bit w i)›
    then obtain m where m: ‹¬ bit w m› ‹n ≤ m› ‹m < LENGTH('a)›
      by (auto simp add: word_size)
    from * have ‹bit (w AND NOT (mask n)) m ⟷ bit (NOT (mask n :: 'a word)) m›
      by auto
    with m show False by (auto simp add: bit_simps)
  qed
qed

lemma is_aligned_add_less_t2n:
  fixes p :: "'a::len word"
  assumes "is_aligned p n"
      and "d < 2 ^ n"
      and "n ≤ m"
      and "p < 2 ^ m"
    shows "p + d < 2^m"
proof (cases "m < LENGTH('a)")
  case True
  have "m < size (p + d)"
    by (simp add: True word_size)
  moreover
  have "p + d AND mask m = p + d"
    using assms
    by (smt (verit, ccfv_SIG) is_aligned_add_helper less_mask_eq mask_eq_x_eq_0 mask_lower_twice)
  ultimately show ?thesis
    using True assms by (metis and_mask_less')
next
  case False
  then show ?thesis
    using ‹p < 2 ^ m› less_2p_is_upper_bits_unset by blast
qed

lemma aligned_offset_non_zero:
  "⟦ is_aligned x n; y < 2 ^ n; x ≠ 0 ⟧ ⟹ x + y ≠ 0"
  by (simp add: is_aligned_no_wrap' neq_0_no_wrap)

lemma is_aligned_over_length:
  "⟦ is_aligned p n; LENGTH('a) ≤ n ⟧ ⟹ (p::'a::len word) = 0"
  by (simp add: is_aligned_mask mask_over_length)

lemma is_aligned_no_overflow_mask:
  "is_aligned x n ⟹ x ≤ x + mask n"
  by (simp add: mask_eq_decr_exp) (erule is_aligned_no_overflow')

lemma aligned_mask_step:
  fixes p' :: "'a::len word"
  assumes "n' ≤ n"
      and "p' ≤ p + mask n"
      and "is_aligned p n"
      and "is_aligned p' n'"
    shows "p' + mask n' ≤ p + mask n"
proof (cases "LENGTH('a) ≤ n")
  case True
  then show ?thesis
    using assms(3) is_aligned_over_length mask_over_length by fastforce
next
  case False
  obtain k k' where kk: "2 ^ n' * k' ≤ 2 ^ n * k + unat (mask n::'a word)"
    "unat p = 2 ^ n * k" "unat p' = 2 ^ n' * k'"
    using assms
    by (metis is_alignedE is_aligned_no_overflow_mask unat_mult_power_lem unat_plus_simple word_le_nat_alt)
  then have "2 ^ n' * (k' + 1) ≤ 2 ^ n * (k + 1)"
    by (metis False Suc_2p_unat_mask assms(1) leI le_imp_less_Suc power_2_mult_step_le)
  then have "2 ^ n' * k' + unat (mask n'::'a word) ≤ 2 ^ n * k + unat (mask n::'a word)"
    by (smt (verit, best) False Suc_2p_unat_mask assms(1) leI not_less_eq_eq order_le_less_trans)
  with assms kk show ?thesis
    by (metis is_aligned_no_overflow_mask unat_plus_simple word_le_nat_alt)
qed

lemma is_aligned_mask_offset_unat:
  fixes off :: "('a::len) word"
  and     x :: "'a word"
  assumes al: "is_aligned x sz"
  and   offv: "off ≤ mask sz"
  shows  "unat x + unat off < 2 ^ LENGTH('a)"
  using al is_aligned_no_overflow_mask no_olen_add_nat offv word_random by blast

lemma aligned_less_plus_1:
  assumes "is_aligned x n" and "0 < n"
  shows "x < x + 1"
proof (rule plus_one_helper2)
  show "x + 1 ≠ 0"
    using assms is_aligned_nth overflow_imp_lsb by blast
qed auto

lemma aligned_add_offset_less:
  assumes x: "is_aligned x n"
      and y: "is_aligned y n"
      and "x < y"
      and z: "z < 2 ^ n"
    shows "x + z < y"
proof (cases "y = 0 ∨ z = 0")
  case True
  then show ?thesis
    using ‹x<y› by auto 
next
  case False
  with y is_aligned_get_word_bits have §: "n < LENGTH('a)" "z ≠ 0"
    by auto
  then have "x ≤ y - 2 ^ n"
    by (simp add: aligned_at_least_t2n_diff assms(3) x y)
  with assms show ?thesis
    by (smt (verit, ccfv_SIG) False diff_add_cancel is_aligned_and_not_zero is_aligned_no_wrap' not_less olen_add_eqv word_sub_mono2)
qed

lemma gap_between_aligned:
  "⟦a < (b :: 'a ::len word); is_aligned a n; is_aligned b n; n < LENGTH('a) ⟧
  ⟹ a + (2^n - 1) < b"
  by (simp add: aligned_add_offset_less)

lemma is_aligned_add_step_le:
  "⟦ is_aligned (a::'a::len word) n; is_aligned b n; a < b; b ≤ a + mask n ⟧ ⟹ False"
  by (metis gap_between_aligned is_aligned_get_word_bits leD linorder_neq_iff mask_eq_decr_exp)

lemma aligned_add_mask_lessD:
  "⟦ x + mask n < y; is_aligned x n ⟧ ⟹ x < y" for y::"'a::len word"
  by (metis is_aligned_no_overflow' mask_2pm1 order_le_less_trans)

lemma aligned_add_mask_less_eq:
  "⟦ is_aligned x n; is_aligned y n;  n < LENGTH('a) ⟧ ⟹ (x + mask n < y) = (x < y)"
  for y::"'a::len word"
  using aligned_add_mask_lessD is_aligned_add_step_le word_le_not_less by blast

lemma is_aligned_diff:
  fixes m :: "'a::len word"
  assumes alm: "is_aligned m s1"
  and     aln: "is_aligned n s2"
  and    s2wb: "s2 < LENGTH('a)"
  and      nm: "m ∈ {n .. n + (2 ^ s2 - 1)}"
  and    s1s2: "s1 ≤ s2"
shows  "∃q. m - n = of_nat q * 2 ^ s1 ∧ q < 2 ^ (s2 - s1)"
proof (cases "s1=0")
  case True
  with nm have "unat(m-n) < 2 ^ s2"
    by simp (metis add.commute s2wb unat_less_power word_diff_ls'(4) word_less_sub_le)
  with True nm show ?thesis
    using unat_eq_of_nat by force
next
  case False
  have rl: "⋀m s. ⟦ m < 2 ^ (LENGTH('a) - s); s < LENGTH('a) ⟧ ⟹ unat ((2::'a word) ^ s * of_nat m) = 2 ^ s * m"
  proof -
    fix m :: nat and  s
    assume m: "m < 2 ^ (LENGTH('a) - s)" and s: "s < LENGTH('a)"
    then have "unat ((of_nat m) :: 'a word) = m"
      by (metis diff_le_self le_unat_uoi nat_less_le unat_of_nat_len unat_power_lower)
    then show "?thesis m s" using s m
      using unat_mult_power_lem by blast
  qed
  have s1wb: "s1 < LENGTH('a)" using s2wb s1s2 by simp
  from alm obtain mq where mmq: "m = 2 ^ s1 * of_nat mq" and mq: "mq < 2 ^ (LENGTH('a) - s1)"
    by (auto elim: is_alignedE simp: field_simps)
  from aln obtain nq where nnq: "n = 2 ^ s2 * of_nat nq" and nq: "nq < 2 ^ (LENGTH('a) - s2)"
    by (auto elim: is_alignedE simp: field_simps)
  from s1s2 obtain sq where sq: "s2 = s1 + sq" by (auto simp: le_iff_add)

  note us1 = rl [OF mq s1wb]
  note us2 = rl [OF nq s2wb]

  from nm have "n ≤ m" by clarsimp
  then have "(2::'a word) ^ s2 * of_nat nq ≤ 2 ^ s1 * of_nat mq" using nnq mmq by simp
  then have "2 ^ s2 * nq ≤ 2 ^ s1 * mq" using s1wb s2wb
    by (simp add: word_le_nat_alt us1 us2)
  then have nqmq: "2 ^ sq * nq ≤ mq" using sq by (simp add: power_add)

  have "m - n = 2 ^ s1 * of_nat mq - 2 ^ s2 * of_nat nq" using mmq nnq by simp
  also have "… = 2 ^ s1 * of_nat mq - 2 ^ s1 * 2 ^ sq * of_nat nq" using sq by (simp add: power_add)
  also have "… = 2 ^ s1 * (of_nat mq - 2 ^ sq * of_nat nq)" by (simp add: field_simps)
  also have "… = 2 ^ s1 * of_nat (mq - 2 ^ sq * nq)" using s1wb s2wb us1 us2 nqmq
    by simp
  finally have mn: "m - n = of_nat (mq - 2 ^ sq * nq) * 2 ^ s1" by simp
  moreover
  from nm have "m - n ≤ 2 ^ s2 - 1"
    by - (rule word_diff_ls', (simp add: field_simps)+)
  then have §: "(2::'a word) ^ s1 * of_nat (mq - 2 ^ sq * nq) < 2 ^ s2" using mn s2wb by (simp add: field_simps)
  then have "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (s2 - s1)"
  proof (rule word_power_less_diff)
    have mm: "mq - 2 ^ sq * nq < 2 ^ (LENGTH('a) - s1)" using mq by simp
    moreover have "LENGTH('a) - s1 < LENGTH('a)"
      using False diff_less by blast
    ultimately show "of_nat (mq - 2 ^ sq * nq) < (2::'a word) ^ (LENGTH('a) - s1)"
      using of_nat_power by blast
  qed
  then have "mq - 2 ^ sq * nq < 2 ^ (s2 - s1)" using mq s2wb
    by (smt (verit, best) § diff_le_self nat_power_less_diff order_le_less_trans unat_less_power unat_mult_power_lem)
  ultimately show ?thesis
    by auto
qed

lemma is_aligned_addD1:
  assumes al1: "is_aligned (x + y) n"
  and     al2: "is_aligned (x::'a::len word) n"
  shows "is_aligned y n"
  using al2
proof (rule is_aligned_get_word_bits)
  assume "x = 0" then show ?thesis using al1 by simp
next
  assume nv: "n < LENGTH('a)"
  from al1 obtain q1
    where xy: "x + y = 2 ^ n * of_nat q1" and "q1 < 2 ^ (LENGTH('a) - n)"
    by (rule is_alignedE)
  moreover from al2 obtain q2
    where x: "x = 2 ^ n * of_nat q2" and "q2 < 2 ^ (LENGTH('a) - n)"
    by (rule is_alignedE)
  ultimately have "y = 2 ^ n * (of_nat q1 - of_nat q2)"
    by (simp add: field_simps)
  then show ?thesis using nv by (simp add: is_aligned_mult_triv1)
qed

lemmas is_aligned_addD2 =
       is_aligned_addD1[OF subst[OF add.commute,
                                 of "%x. is_aligned x n" for n]]

lemma is_aligned_add:
  "⟦is_aligned p n; is_aligned q n⟧ ⟹ is_aligned (p + q) n"
  by (simp add: is_aligned_mask mask_add_aligned)

lemma aligned_shift:
  fixes x :: "'a::len word"
  assumes x: "x < 2 ^ n"
      and "is_aligned y n"
      and "n ≤ LENGTH('a::len)"
    shows "(x + y) >> n = y >> n"
proof (subst word_plus_and_or_coroll)
  show "x AND y = 0"
    using assms 
    by (meson le_less_trans is_aligned_andI2 is_aligned_less_sz word_and_le2)
next
  show "x OR y >> n = y >> n"
    unfolding shiftr_def
    by (metis x drop_bit_or less_mask_eq or.left_neutral take_bit_eq_mask take_bit_eq_self_iff_drop_bit_eq_0)
qed

lemma aligned_shift':
  "⟦x < 2 ^ n; is_aligned (y :: 'a :: len word) n;n ≤ LENGTH('a)⟧
   ⟹ (y + x) >> n = y >> n"
  by (simp add: add.commute aligned_shift)

lemma and_neg_mask_plus_mask_mono: "(p AND NOT (mask n)) + mask n ≥ p"
  for p :: ‹'a::len word›
  by (metis le_word_or2 or_eq_and_not_plus)

lemma word_neg_and_le:
  "ptr ≤ (ptr AND NOT (mask n)) + (2 ^ n - 1)"
  for ptr :: ‹'a::len word›
  by (simp add: and_neg_mask_plus_mask_mono mask_2pm1[symmetric])

lemma is_aligned_sub_helper:
  "⟦ is_aligned (p - d) n; d < 2 ^ n ⟧
     ⟹ (p AND mask n = d) ∧ (p AND (NOT (mask n)) = p - d)"
  by (drule(1) is_aligned_add_helper, simp)

lemma is_aligned_after_mask:
  "⟦is_aligned k m;m≤ n⟧ ⟹ is_aligned (k AND mask n) m"
  by (rule is_aligned_andI1)

lemma and_mask_plus:
  assumes "is_aligned ptr m"
    and "m ≤ n"
    and "a < 2 ^ m"
  shows "ptr + a AND mask n = (ptr AND mask n) + a"
proof (rule mask_eqI)
  show "(ptr + a AND mask n) AND mask m = (ptr AND mask n) + a AND mask m"
    by (metis assms(2) mask_inner_mask mask_twice2)
next
  show "(ptr + a AND mask n) AND NOT (mask m) = (ptr AND mask n) + a AND NOT (mask m)"
    by (metis assms(1,3) is_aligned_add_helper is_aligned_neg_mask2 word_bw_assocs(1) word_bw_comms(1))
qed

lemma is_aligned_add_not_aligned:
  "⟦is_aligned (p::'a::len word) n; ¬ is_aligned (q::'a::len word) n⟧ ⟹ ¬ is_aligned (p + q) n"
  by (metis is_aligned_addD1)

lemma neg_mask_add_aligned:
  "⟦ is_aligned p n; q < 2 ^ n ⟧ ⟹ (p + q) AND NOT (mask n) = p AND NOT (mask n)"
  by (metis is_aligned_add_helper is_aligned_neg_mask_eq)

lemma word_add_power_off:
  fixes a :: "'a :: len word"
  assumes ak: "a < k"
  and kw: "k < 2 ^ (LENGTH('a) - m)"
  and mw: "m < LENGTH('a)"
  and off: "off < 2 ^ m"
shows "(a * 2 ^ m) + off < k * 2 ^ m"
proof (cases "m = 0")
  case True
  then show ?thesis using off ak by simp
next
  case False

  from ak have ak1: "a + 1 ≤ k" by (rule inc_le)
  then have "(a + 1) * 2 ^ m ≠ 0"
    by (meson ak kw less_is_non_zero_p1 mw order_le_less_trans word_power_nonzero)
  then have "(a * 2 ^ m) + off < ((a + 1) * 2 ^ m)" using kw mw
    by (simp add: distrib_right is_aligned_mult_triv2 is_aligned_no_overflow'' off word_plus_strict_mono_right)
  also have "… ≤ k * 2 ^ m" using ak1 mw kw False
    by (simp add: less_2p_is_upper_bits_unset nat_mult_power_less_eq unat_less_power word_mult_le_mono1)
  finally show ?thesis .
qed

lemma offset_not_aligned:
  "⟦ is_aligned (p::'a::len word) n; i > 0; i < 2 ^ n; n < LENGTH('a)⟧ ⟹
   ¬ is_aligned (p + of_nat i) n"
  by (metis of_nat_power Word.of_nat_neq_0 add.commute add.right_neutral is_aligned_addD1 is_aligned_less_sz 
is_aligned_no_wrap''' unat_0)

lemma le_or_mask:
  "w ≤ w' ⟹ w OR mask x ≤ w' OR mask x"
  for w w' :: ‹'a::len word›
  by (metis neg_mask_add_mask add.commute le_word_or1 mask_2pm1 neg_mask_mono_le word_plus_mono_left)

end

end