Theory Reversed_Bit_Lists

(*
 * Copyright Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

(* Author: Jeremy Dawson, NICTA *)

section ‹Bit values as reversed lists of bools›

theory Reversed_Bit_Lists
  imports
    "HOL-Library.Word"
    Typedef_Morphisms
    Least_significant_bit
    Most_significant_bit
    Even_More_List
    "HOL-Library.Sublist"
    Aligned
    Singleton_Bit_Shifts
    Legacy_Aliases
begin

context
  includes bit_operations_syntax
begin

lemma horner_sum_of_bool_2_concat:
  ‹horner_sum of_bool 2 (concat (map (λx. map (bit x) [0..<LENGTH('a)]) ws)) = horner_sum uint (2 ^ LENGTH('a)) ws›
  for ws :: ‹'a::len word list›
proof (induction ws)
  case Nil
  then show ?case
    by simp
next
  case (Cons w ws)
  moreover have ‹horner_sum of_bool 2 (map (bit w) [0..<LENGTH('a)]) = uint w›
  proof transfer
    fix k :: int
    have ‹map (λn. n < LENGTH('a) ∧ bit k n) [0..<LENGTH('a)]
      = map (bit k) [0..<LENGTH('a)]›
      by simp
    then show ‹horner_sum of_bool 2 (map (λn. n < LENGTH('a) ∧ bit k n) [0..<LENGTH('a)])
      = take_bit LENGTH('a) k›
      by (simp only: horner_sum_bit_eq_take_bit)
  qed
  ultimately show ?case
    by (simp add: horner_sum_append)
qed


subsection ‹Implicit augmentation of list prefixes›

primrec takefill :: "'a ⇒ nat ⇒ 'a list ⇒ 'a list"
where
    Z: "takefill fill 0 xs = []"
  | Suc: "takefill fill (Suc n) xs =
      (case xs of
        [] ⇒ fill # takefill fill n xs
      | y # ys ⇒ y # takefill fill n ys)"

lemma nth_takefill: "m < n ⟹ takefill fill n l ! m = (if m < length l then l ! m else fill)"
  apply (induct n arbitrary: m l)
   apply clarsimp
  apply clarsimp
  apply (case_tac m)
   apply (simp split: list.split)
  apply (simp split: list.split)
  done

lemma takefill_alt: "takefill fill n l = take n l @ replicate (n - length l) fill"
  by (induct n arbitrary: l) (auto split: list.split)

lemma takefill_replicate [simp]: "takefill fill n (replicate m fill) = replicate n fill"
  by (simp add: takefill_alt replicate_add [symmetric])

lemma takefill_le': "n = m + k ⟹ takefill x m (takefill x n l) = takefill x m l"
  by (induct m arbitrary: l n) (auto split: list.split)

lemma length_takefill [simp]: "length (takefill fill n l) = n"
  by (simp add: takefill_alt)

lemma take_takefill': "n = k + m ⟹ take k (takefill fill n w) = takefill fill k w"
  by (induct k arbitrary: w n) (auto split: list.split)

lemma drop_takefill: "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
  by (induct k arbitrary: w) (auto split: list.split)

lemma takefill_le [simp]: "m ≤ n ⟹ takefill x m (takefill x n l) = takefill x m l"
  by (auto simp: le_iff_add takefill_le')

lemma take_takefill [simp]: "m ≤ n ⟹ take m (takefill fill n w) = takefill fill m w"
  by (auto simp: le_iff_add take_takefill')

lemma takefill_append: "takefill fill (m + length xs) (xs @ w) = xs @ (takefill fill m w)"
  by (induct xs) auto

lemma takefill_same': "l = length xs ⟹ takefill fill l xs = xs"
  by (induct xs arbitrary: l) auto

lemmas takefill_same [simp] = takefill_same' [OF refl]

lemma tf_rev:
  "n + k = m + length bl ⟹ takefill x m (rev (takefill y n bl)) =
    rev (takefill y m (rev (takefill x k (rev bl))))"
  apply (rule nth_equalityI)
   apply (auto simp add: nth_takefill rev_nth)
  apply (rule_tac f = "λn. bl ! n" in arg_cong)
  apply arith
  done

lemma takefill_minus: "0 < n ⟹ takefill fill (Suc (n - 1)) w = takefill fill n w"
  by auto

lemmas takefill_Suc_cases =
  list.cases [THEN takefill.Suc [THEN trans]]

lemmas takefill_Suc_Nil = takefill_Suc_cases (1)
lemmas takefill_Suc_Cons = takefill_Suc_cases (2)

lemmas takefill_minus_simps = takefill_Suc_cases [THEN [2]
  takefill_minus [symmetric, THEN trans]]

lemma takefill_numeral_Nil [simp]:
  "takefill fill (numeral k) [] = fill # takefill fill (pred_numeral k) []"
  by (simp add: numeral_eq_Suc)

lemma takefill_numeral_Cons [simp]:
  "takefill fill (numeral k) (x # xs) = x # takefill fill (pred_numeral k) xs"
  by (simp add: numeral_eq_Suc)


subsection ‹Range projection›

definition bl_of_nth :: "nat ⇒ (nat ⇒ 'a) ⇒ 'a list"
  where "bl_of_nth n f = map f (rev [0..<n])"

lemma bl_of_nth_simps [simp, code]:
  "bl_of_nth 0 f = []"
  "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
  by (simp_all add: bl_of_nth_def)

lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n"
  by (simp add: bl_of_nth_def)

lemma nth_bl_of_nth [simp]: "m < n ⟹ rev (bl_of_nth n f) ! m = f m"
  by (simp add: bl_of_nth_def rev_map)

lemma bl_of_nth_inj: "(⋀k. k < n ⟹ f k = g k) ⟹ bl_of_nth n f = bl_of_nth n g"
  by (simp add: bl_of_nth_def)

lemma bl_of_nth_nth_le: "n ≤ length xs ⟹ bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs"
  apply (induct n arbitrary: xs)
   apply clarsimp
  apply clarsimp
  apply (rule trans [OF _ hd_Cons_tl])
   apply (frule Suc_le_lessD)
   apply (simp add: rev_nth trans [OF drop_Suc drop_tl, symmetric])
   apply (subst hd_drop_conv_nth)
    apply force
   apply simp_all
  apply (rule_tac f = "λn. drop n xs" in arg_cong)
  apply simp
  done

lemma bl_of_nth_nth [simp]: "bl_of_nth (length xs) ((!) (rev xs)) = xs"
  by (simp add: bl_of_nth_nth_le)


subsection ‹More›

definition rotater1 :: "'a list ⇒ 'a list"
  where "rotater1 ys =
    (case ys of [] ⇒ [] | x # xs ⇒ last ys # butlast ys)"

definition rotater :: "nat ⇒ 'a list ⇒ 'a list"
  where "rotater n = rotater1 ^^ n"

lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified]

lemma rotate1_rl': "rotater1 (l @ [a]) = a # l"
  by (cases l) (auto simp: rotater1_def)

lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l"
  apply (unfold rotater1_def)
  apply (cases "l")
   apply (case_tac [2] "list")
    apply auto
  done

lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l"
  by (cases l) (auto simp: rotater1_def)

lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)"
  by (cases "xs") (simp add: rotater1_def, simp add: rotate1_rl')

lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)"
  by (induct n) (auto simp: rotater_def intro: rotater1_rev')

lemma rotater_rev: "rotater n ys = rev (rotate n (rev ys))"
  using rotater_rev' [where xs = "rev ys"] by simp

lemma rotater_drop_take:
  "rotater n xs =
    drop (length xs - n mod length xs) xs @
    take (length xs - n mod length xs) xs"
  by (auto simp: rotater_rev rotate_drop_take rev_take rev_drop)

lemma rotater_Suc [simp]: "rotater (Suc n) xs = rotater1 (rotater n xs)"
  unfolding rotater_def by auto

lemma nth_rotater:
  ‹rotater m xs ! n = xs ! ((n + (length xs - m mod length xs)) mod length xs)› if ‹n < length xs›
  using that by (simp add: rotater_drop_take nth_append not_less less_diff_conv ac_simps le_mod_geq)

lemma nth_rotater1:
  ‹rotater1 xs ! n = xs ! ((n + (length xs - 1)) mod length xs)› if ‹n < length xs›
  using that nth_rotater [of n xs 1] by simp

lemma rotate_inv_plus [rule_format]:
  "∀k. k = m + n ⟶ rotater k (rotate n xs) = rotater m xs ∧
    rotate k (rotater n xs) = rotate m xs ∧
    rotater n (rotate k xs) = rotate m xs ∧
    rotate n (rotater k xs) = rotater m xs"
  by (induct n) (auto simp: rotater_def rotate_def intro: funpow_swap1 [THEN trans])

lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus]

lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified]

lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1]
lemmas rotate_rl [simp] = rotate_inv_eq [THEN conjunct2, THEN conjunct1]

lemma rotate_gal: "rotater n xs = ys ⟷ rotate n ys = xs"
  by auto

lemma rotate_gal': "ys = rotater n xs ⟷ xs = rotate n ys"
  by auto

lemma length_rotater [simp]: "length (rotater n xs) = length xs"
  by (simp add : rotater_rev)

lemma rotate_eq_mod: "m mod length xs = n mod length xs ⟹ rotate m xs = rotate n xs"
  apply (rule box_equals)
    defer
    apply (rule rotate_conv_mod [symmetric])+
  apply simp
  done

lemma restrict_to_left: "x = y ⟹ x = z ⟷ y = z"
  by simp

lemmas rotate_eqs =
  trans [OF rotate0 [THEN fun_cong] id_apply]
  rotate_rotate [symmetric]
  rotate_id
  rotate_conv_mod
  rotate_eq_mod

lemmas rrs0 = rotate_eqs [THEN restrict_to_left,
  simplified rotate_gal [symmetric] rotate_gal' [symmetric]]
lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]]
lemmas rotater_eqs = rrs1 [simplified length_rotater]
lemmas rotater_0 = rotater_eqs (1)
lemmas rotater_add = rotater_eqs (2)

lemma butlast_map: "xs ≠ [] ⟹ butlast (map f xs) = map f (butlast xs)"
  by (induct xs) auto

lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)"
  by (cases xs) (auto simp: rotater1_def last_map butlast_map)

lemma rotater_map: "rotater n (map f xs) = map f (rotater n xs)"
  by (induct n) (auto simp: rotater_def rotater1_map)

lemma but_last_zip [rule_format] :
  "∀ys. length xs = length ys ⟶ xs ≠ [] ⟶
    last (zip xs ys) = (last xs, last ys) ∧
    butlast (zip xs ys) = zip (butlast xs) (butlast ys)"
  apply (induct xs)
   apply auto
     apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
  done

lemma but_last_map2 [rule_format] :
  "∀ys. length xs = length ys ⟶ xs ≠ [] ⟶
    last (map2 f xs ys) = f (last xs) (last ys) ∧
    butlast (map2 f xs ys) = map2 f (butlast xs) (butlast ys)"
  apply (induct xs)
   apply auto
     apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+
  done

lemma rotater1_zip:
  "length xs = length ys ⟹
    rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)"
  apply (unfold rotater1_def)
  apply (cases xs)
   apply auto
   apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+
  done

lemma rotater1_map2:
  "length xs = length ys ⟹
    rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)"
  by (simp add: rotater1_map rotater1_zip)

lemmas lrth =
  box_equals [OF asm_rl length_rotater [symmetric]
                 length_rotater [symmetric],
              THEN rotater1_map2]

lemma rotater_map2:
  "length xs = length ys ⟹
    rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)"
  by (induct n) (auto intro!: lrth)

lemma rotate1_map2:
  "length xs = length ys ⟹
    rotate1 (map2 f xs ys) = map2 f (rotate1 xs) (rotate1 ys)"
  by (cases xs; cases ys) auto

lemmas lth = box_equals [OF asm_rl length_rotate [symmetric]
  length_rotate [symmetric], THEN rotate1_map2]

lemma rotate_map2:
  "length xs = length ys ⟹
    rotate n (map2 f xs ys) = map2 f (rotate n xs) (rotate n ys)"
  by (induct n) (auto intro!: lth)


subsection ‹Explicit bit representation of \<^typ>‹int››

primrec bl_to_bin_aux :: "bool list ⇒ int ⇒ int"
  where
    Nil: "bl_to_bin_aux [] w = w"
  | Cons: "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (of_bool b + 2 * w)"

definition bl_to_bin :: "bool list ⇒ int"
  where "bl_to_bin bs = bl_to_bin_aux bs 0"

primrec bin_to_bl_aux :: "nat ⇒ int ⇒ bool list ⇒ bool list"
  where
    Z: "bin_to_bl_aux 0 w bl = bl"
  | Suc: "bin_to_bl_aux (Suc n) w bl = bin_to_bl_aux n (w div 2) (odd w # bl)"

definition bin_to_bl :: "nat ⇒ int ⇒ bool list"
  where "bin_to_bl n w = bin_to_bl_aux n w []"

lemma bin_to_bl_aux_zero_minus_simp [simp]:
  "0 < n ⟹ bin_to_bl_aux n 0 bl = bin_to_bl_aux (n - 1) 0 (False # bl)"
  by (cases n) auto

lemma bin_to_bl_aux_minus1_minus_simp [simp]:
  "0 < n ⟹ bin_to_bl_aux n (- 1) bl = bin_to_bl_aux (n - 1) (- 1) (True # bl)"
  by (cases n) auto

lemma bin_to_bl_aux_one_minus_simp [simp]:
  "0 < n ⟹ bin_to_bl_aux n 1 bl = bin_to_bl_aux (n - 1) 0 (True # bl)"
  by (cases n) auto

lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
  "0 < n ⟹
    bin_to_bl_aux n (numeral (Num.Bit0 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (False # bl)"
  by (cases n) simp_all

lemma bin_to_bl_aux_Bit1_minus_simp [simp]:
  "0 < n ⟹
    bin_to_bl_aux n (numeral (Num.Bit1 w)) bl = bin_to_bl_aux (n - 1) (numeral w) (True # bl)"
  by (cases n) simp_all

lemma bl_to_bin_aux_append: "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)"
  by (induct bs arbitrary: w) auto

lemma bin_to_bl_aux_append: "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)"
  by (induct n arbitrary: w bs) auto

lemma bl_to_bin_append: "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)"
  unfolding bl_to_bin_def by (rule bl_to_bin_aux_append)

lemma bin_to_bl_aux_alt: "bin_to_bl_aux n w bs = bin_to_bl n w @ bs"
  by (simp add: bin_to_bl_def bin_to_bl_aux_append)

lemma bin_to_bl_0 [simp]: "bin_to_bl 0 bs = []"
  by (auto simp: bin_to_bl_def)

lemma size_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
  by (induct n arbitrary: w bs) auto

lemma size_bin_to_bl [simp]: "length (bin_to_bl n w) = n"
  by (simp add: bin_to_bl_def size_bin_to_bl_aux)

lemma bl_bin_bl': "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = bin_to_bl_aux n w bs"
  apply (induct bs arbitrary: w n)
   apply auto
    apply (simp_all only: add_Suc [symmetric])
    apply (auto simp add: bin_to_bl_def)
  done

lemma bl_bin_bl [simp]: "bin_to_bl (length bs) (bl_to_bin bs) = bs"
  unfolding bl_to_bin_def
  apply (rule box_equals)
    apply (rule bl_bin_bl')
   prefer 2
   apply (rule bin_to_bl_aux.Z)
  apply simp
  done

lemma bl_to_bin_inj: "bl_to_bin bs = bl_to_bin cs ⟹ length bs = length cs ⟹ bs = cs"
  apply (rule_tac box_equals)
    defer
    apply (rule bl_bin_bl)
   apply (rule bl_bin_bl)
  apply simp
  done

lemma bl_to_bin_False [simp]: "bl_to_bin (False # bl) = bl_to_bin bl"
  by (auto simp: bl_to_bin_def)

lemma bl_to_bin_Nil [simp]: "bl_to_bin [] = 0"
  by (auto simp: bl_to_bin_def)

lemma bin_to_bl_zero_aux: "bin_to_bl_aux n 0 bl = replicate n False @ bl"
  by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)

lemma bin_to_bl_zero: "bin_to_bl n 0 = replicate n False"
  by (simp add: bin_to_bl_def bin_to_bl_zero_aux)

lemma bin_to_bl_minus1_aux: "bin_to_bl_aux n (- 1) bl = replicate n True @ bl"
  by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)

lemma bin_to_bl_minus1: "bin_to_bl n (- 1) = replicate n True"
  by (simp add: bin_to_bl_def bin_to_bl_minus1_aux)


subsection ‹Semantic interpretation of \<^typ>‹bool list› as \<^typ>‹int››

lemma bin_bl_bin': "bl_to_bin (bin_to_bl_aux n w bs) = bl_to_bin_aux bs (take_bit n w)"
  by (induct n arbitrary: w bs) (auto simp: bl_to_bin_def take_bit_Suc ac_simps mod_2_eq_odd)

lemma bin_bl_bin [simp]: "bl_to_bin (bin_to_bl n w) = take_bit n w"
  by (auto simp: bin_to_bl_def bin_bl_bin')

lemma bl_to_bin_rep_F: "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
  by (simp add: bin_to_bl_zero_aux [symmetric] bin_bl_bin') (simp add: bl_to_bin_def)

lemma bin_to_bl_trunc [simp]: "n ≤ m ⟹ bin_to_bl n (take_bit m w) = bin_to_bl n w"
  by (auto intro: bl_to_bin_inj)

lemma bin_to_bl_aux_bintr:
  "bin_to_bl_aux n (take_bit m bin) bl =
    replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl"
  apply (induct n arbitrary: m bin bl)
   apply clarsimp
  apply clarsimp
  apply (case_tac "m")
   apply (clarsimp simp: bin_to_bl_zero_aux)
   apply (erule thin_rl)
   apply (induct_tac n)
    apply (auto simp add: take_bit_Suc)
  done

lemma bin_to_bl_bintr:
  "bin_to_bl n (take_bit m bin) = replicate (n - m) False @ bin_to_bl (min n m) bin"
  unfolding bin_to_bl_def by (rule bin_to_bl_aux_bintr)

lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = 0"
  by (induct n) auto

lemma len_bin_to_bl_aux: "length (bin_to_bl_aux n w bs) = n + length bs"
  by (fact size_bin_to_bl_aux)

lemma len_bin_to_bl: "length (bin_to_bl n w) = n"
  by (fact size_bin_to_bl) (* FIXME: duplicate *)

lemma sign_bl_bin': "bin_sign (bl_to_bin_aux bs w) = bin_sign w"
  by (induction bs arbitrary: w) (simp_all add: bin_sign_def)

lemma sign_bl_bin: "bin_sign (bl_to_bin bs) = 0"
  by (simp add: bl_to_bin_def sign_bl_bin')

lemma bl_sbin_sign_aux: "hd (bin_to_bl_aux (Suc n) w bs) = (bin_sign (signed_take_bit n w) = -1)"
  by (induction n arbitrary: w bs) (auto simp add: bin_sign_def even_iff_mod_2_eq_zero bit_Suc)

lemma bl_sbin_sign: "hd (bin_to_bl (Suc n) w) = (bin_sign (signed_take_bit n w) = -1)"
  unfolding bin_to_bl_def by (rule bl_sbin_sign_aux)

lemma bin_nth_of_bl_aux:
  "bit (bl_to_bin_aux bl w) n =
    (n < size bl ∧ rev bl ! n ∨ n ≥ length bl ∧ bit w (n - size bl))"
  apply (induction bl arbitrary: w)
   apply simp_all
  apply safe
                      apply (simp_all add: not_le nth_append bit_double_iff even_bit_succ_iff split: if_splits)
  done

lemma bin_nth_of_bl: "bit (bl_to_bin bl) n = (n < length bl ∧ rev bl ! n)"
  by (simp add: bl_to_bin_def bin_nth_of_bl_aux)

lemma bin_nth_bl: "n < m ⟹ bit w n = nth (rev (bin_to_bl m w)) n"
  by (metis bin_bl_bin bin_nth_of_bl nth_bintr size_bin_to_bl)

lemma nth_bin_to_bl_aux:
  "n < m + length bl ⟹ (bin_to_bl_aux m w bl) ! n =
    (if n < m then bit w (m - 1 - n) else bl ! (n - m))"
  apply (induction bl arbitrary: w)
   apply simp_all
   apply (simp add: bin_nth_bl [of ‹m - Suc n› m] rev_nth flip: bin_to_bl_def)
   apply (metis One_nat_def Suc_pred add_diff_cancel_left'
     add_diff_cancel_right' bin_to_bl_aux_alt bin_to_bl_def
     diff_Suc_Suc diff_is_0_eq diff_zero less_Suc_eq_0_disj
     less_antisym less_imp_Suc_add list.size(3) nat_less_le nth_append size_bin_to_bl_aux)
  done

lemma nth_bin_to_bl: "n < m ⟹ (bin_to_bl m w) ! n = bit w (m - Suc n)"
  by (simp add: bin_to_bl_def nth_bin_to_bl_aux)

lemma takefill_bintrunc: "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))"
  apply (rule nth_equalityI)
   apply simp
  apply (clarsimp simp: nth_takefill rev_nth nth_bin_to_bl bin_nth_of_bl)
  done

lemma bl_bin_bl_rtf: "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))"
  by (simp add: takefill_bintrunc)

lemma bl_to_bin_lt2p_aux: "bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
proof (induction bs arbitrary: w)
  case Nil
  then show ?case
    by simp
next
  case (Cons b bs)
  from Cons.IH [of ‹1 + 2 * w›] Cons.IH [of ‹2 * w›]
  show ?case
    apply (auto simp add: algebra_simps)
    apply (subst mult_2 [of ‹2 ^ length bs›])
    apply (simp only: add.assoc)
    apply (rule pos_add_strict)
     apply simp_all
    done
qed

lemma bl_to_bin_lt2p_drop: "bl_to_bin bs < 2 ^ length (dropWhile Not bs)"
proof (induct bs)
  case Nil
  then show ?case by simp
next
  case (Cons b bs)
  with bl_to_bin_lt2p_aux[where w=1] show ?case
    by (simp add: bl_to_bin_def)
qed

lemma bl_to_bin_lt2p: "bl_to_bin bs < 2 ^ length bs"
  by (metis bin_bl_bin bintr_lt2p bl_bin_bl)

lemma bl_to_bin_ge2p_aux: "bl_to_bin_aux bs w ≥ w * (2 ^ length bs)"
proof (induction bs arbitrary: w)
  case Nil
  then show ?case
    by simp
next
  case (Cons b bs)
  from Cons.IH [of ‹1 + 2 * w›] Cons.IH [of ‹2 * w›]
  show ?case
    apply (auto simp add: algebra_simps)
    apply (rule add_le_imp_le_left [of ‹2 ^ length bs›])
    apply (rule add_increasing)
    apply simp_all
    done
qed

lemma bl_to_bin_ge0: "bl_to_bin bs ≥ 0"
  apply (unfold bl_to_bin_def)
  apply (rule xtrans(4))
   apply (rule bl_to_bin_ge2p_aux)
  apply simp
  done

lemma butlast_rest_bin: "butlast (bin_to_bl n w) = bin_to_bl (n - 1) (w div 2)"
  apply (unfold bin_to_bl_def)
  apply (cases n, clarsimp)
  apply clarsimp
  apply (auto simp add: bin_to_bl_aux_alt)
  done

lemma butlast_bin_rest: "butlast bl = bin_to_bl (length bl - Suc 0) (bl_to_bin bl div 2)"
  using butlast_rest_bin [where w="bl_to_bin bl" and n="length bl"] by simp

lemma butlast_rest_bl2bin_aux:
  "bl ≠ [] ⟹ bl_to_bin_aux (butlast bl) w = bl_to_bin_aux bl w div 2"
  by (induct bl arbitrary: w) auto

lemma butlast_rest_bl2bin: "bl_to_bin (butlast bl) = bl_to_bin bl div 2"
  by (cases bl) (auto simp: bl_to_bin_def butlast_rest_bl2bin_aux)

lemma trunc_bl2bin_aux:
  "take_bit m (bl_to_bin_aux bl w) =
    bl_to_bin_aux (drop (length bl - m) bl) (take_bit (m - length bl) w)"
proof (induct bl arbitrary: w)
  case Nil
  show ?case by simp
next
  case (Cons b bl)
  show ?case
  proof (cases "m - length bl")
    case 0
    then have "Suc (length bl) - m = Suc (length bl - m)" by simp
    with Cons show ?thesis by simp
  next
    case (Suc n)
    then have "m - Suc (length bl) = n" by simp
    with Cons Suc show ?thesis by (simp add: take_bit_Suc ac_simps)
  qed
qed

lemma trunc_bl2bin: "take_bit m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
  by (simp add: bl_to_bin_def trunc_bl2bin_aux)

lemma trunc_bl2bin_len [simp]: "take_bit (length bl) (bl_to_bin bl) = bl_to_bin bl"
  by (simp add: trunc_bl2bin)

lemma bl2bin_drop: "bl_to_bin (drop k bl) = take_bit (length bl - k) (bl_to_bin bl)"
  apply (rule trans)
   prefer 2
   apply (rule trunc_bl2bin [symmetric])
  apply (cases "k ≤ length bl")
   apply auto
  done

lemma take_rest_power_bin: "m ≤ n ⟹ take m (bin_to_bl n w) = bin_to_bl m (((λw. w div 2) ^^ (n - m)) w)"
  apply (rule nth_equalityI)
   apply simp
  apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
  done

lemma last_bin_last': "size xs > 0 ⟹ last xs ⟷ odd (bl_to_bin_aux xs w)"
  by (induct xs arbitrary: w) auto

lemma last_bin_last: "size xs > 0 ⟹ last xs ⟷ odd (bl_to_bin xs)"
  unfolding bl_to_bin_def by (erule last_bin_last')

lemma bin_last_last: "odd w ⟷ last (bin_to_bl (Suc n) w)"
  by (simp add: bin_to_bl_def) (auto simp: bin_to_bl_aux_alt)

lemma drop_bin2bl_aux:
  "drop m (bin_to_bl_aux n bin bs) =
    bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
  apply (induction n arbitrary: m bin bs)
   apply auto
  apply (case_tac "m ≤ n")
   apply (auto simp add: not_le Suc_diff_le)
  apply (case_tac "m - n")
   apply auto
  apply (use Suc_diff_Suc in fastforce)
  done

lemma drop_bin2bl: "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin"
  by (simp add: bin_to_bl_def drop_bin2bl_aux)

lemma take_bin2bl_lem1: "take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
  apply (induct m arbitrary: w bs)
   apply clarsimp
  apply clarsimp
  apply (simp add: bin_to_bl_aux_alt)
  apply (simp add: bin_to_bl_def)
  apply (simp add: bin_to_bl_aux_alt)
  done

lemma take_bin2bl_lem: "take m (bin_to_bl_aux (m + n) w bs) = take m (bin_to_bl (m + n) w)"
  by (induct n arbitrary: w bs) (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1, simp)

lemma bin_split_take: "bin_split n c = (a, b) ⟹ bin_to_bl m a = take m (bin_to_bl (m + n) c)"
  apply (induct n arbitrary: b c)
   apply clarsimp
  apply (clarsimp simp: Let_def split: prod.split_asm)
  apply (simp add: bin_to_bl_def)
  apply (simp add: take_bin2bl_lem drop_bit_Suc)
  done

lemma bin_to_bl_drop_bit:
  "k = m + n ⟹ bin_to_bl m (drop_bit n c) = take m (bin_to_bl k c)"
  using bin_split_take by simp

lemma bin_split_take1:
  "k = m + n ⟹ bin_split n c = (a, b) ⟹ bin_to_bl m a = take m (bin_to_bl k c)"
  using bin_split_take by simp

lemma bl_bin_bl_rep_drop:
  "bin_to_bl n (bl_to_bin bl) =
    replicate (n - length bl) False @ drop (length bl - n) bl"
  by (simp add: bl_to_bin_inj bl_to_bin_rep_F trunc_bl2bin)

lemma bl_to_bin_aux_cat:
  "bl_to_bin_aux bs (concat_bit nv v w) =
    concat_bit (nv + length bs) (bl_to_bin_aux bs v) w"
  by (rule bit_eqI)
    (auto simp add: bin_nth_of_bl_aux bin_nth_cat algebra_simps)

lemma bin_to_bl_aux_cat:
  "bin_to_bl_aux (nv + nw) (concat_bit nw w v) bs =
    bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
  by (induction nw arbitrary: w bs) (simp_all add: concat_bit_Suc)

lemma bl_to_bin_aux_alt: "bl_to_bin_aux bs w = concat_bit (length bs) (bl_to_bin bs) w"
  using bl_to_bin_aux_cat [where nv = "0" and v = "0"]
  by (simp add: bl_to_bin_def [symmetric])

lemma bin_to_bl_cat:
  "bin_to_bl (nv + nw) (concat_bit nw w v) =
    bin_to_bl_aux nv v (bin_to_bl nw w)"
  by (simp add: bin_to_bl_def bin_to_bl_aux_cat)

lemmas bl_to_bin_aux_app_cat =
  trans [OF bl_to_bin_aux_append bl_to_bin_aux_alt]

lemmas bin_to_bl_aux_cat_app =
  trans [OF bin_to_bl_aux_cat bin_to_bl_aux_alt]

lemma bl_to_bin_app_cat:
  "bl_to_bin (bsa @ bs) = concat_bit (length bs) (bl_to_bin bs) (bl_to_bin bsa)"
  by (simp only: bl_to_bin_aux_app_cat bl_to_bin_def)

lemma bin_to_bl_cat_app:
  "bin_to_bl (n + nw) (concat_bit nw wa w) = bin_to_bl n w @ bin_to_bl nw wa"
  by (simp only: bin_to_bl_def bin_to_bl_aux_cat_app)

text ‹‹bl_to_bin_app_cat_alt› and ‹bl_to_bin_app_cat› are easily interderivable.›
lemma bl_to_bin_app_cat_alt: "concat_bit n w (bl_to_bin cs) = bl_to_bin (cs @ bin_to_bl n w)"
  by (simp add: bl_to_bin_app_cat)

lemma mask_lem: "(bl_to_bin (True # replicate n False)) = bl_to_bin (replicate n True) + 1"
  apply (unfold bl_to_bin_def)
  apply (induct n)
   apply simp
  apply (simp only: Suc_eq_plus1 replicate_add append_Cons [symmetric] bl_to_bin_aux_append)
  apply simp
  done

lemma bin_exhaust:
  "(⋀x b. bin = of_bool b + 2 * x ⟹ Q) ⟹ Q" for bin :: int
  apply (cases ‹even bin›)
   apply (auto elim!: evenE oddE)
   apply fastforce
  apply fastforce
  done

primrec rbl_succ :: "bool list ⇒ bool list"
  where
    Nil: "rbl_succ Nil = Nil"
  | Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"

primrec rbl_pred :: "bool list ⇒ bool list"
  where
    Nil: "rbl_pred Nil = Nil"
  | Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)"

primrec rbl_add :: "bool list ⇒ bool list ⇒ bool list"
  where ― ‹result is length of first arg, second arg may be longer›
    Nil: "rbl_add Nil x = Nil"
  | Cons: "rbl_add (y # ys) x =
      (let ws = rbl_add ys (tl x)
       in (y ≠ hd x) # (if hd x ∧ y then rbl_succ ws else ws))"

primrec rbl_mult :: "bool list ⇒ bool list ⇒ bool list"
  where ― ‹result is length of first arg, second arg may be longer›
    Nil: "rbl_mult Nil x = Nil"
  | Cons: "rbl_mult (y # ys) x =
      (let ws = False # rbl_mult ys x
       in if y then rbl_add ws x else ws)"

lemma size_rbl_pred: "length (rbl_pred bl) = length bl"
  by (induct bl) auto

lemma size_rbl_succ: "length (rbl_succ bl) = length bl"
  by (induct bl) auto

lemma size_rbl_add: "length (rbl_add bl cl) = length bl"
  by (induct bl arbitrary: cl) (auto simp: Let_def size_rbl_succ)

lemma size_rbl_mult: "length (rbl_mult bl cl) = length bl"
  by (induct bl arbitrary: cl) (auto simp add: Let_def size_rbl_add)

lemmas rbl_sizes [simp] =
  size_rbl_pred size_rbl_succ size_rbl_add size_rbl_mult

lemmas rbl_Nils =
  rbl_pred.Nil rbl_succ.Nil rbl_add.Nil rbl_mult.Nil

lemma rbl_add_app2: "length blb ≥ length bla ⟹ rbl_add bla (blb @ blc) = rbl_add bla blb"
  apply (induct bla arbitrary: blb)
   apply simp
  apply clarsimp
  apply (case_tac blb, clarsimp)
  apply (clarsimp simp: Let_def)
  done

lemma rbl_add_take2:
  "length blb ≥ length bla ⟹ rbl_add bla (take (length bla) blb) = rbl_add bla blb"
  apply (induct bla arbitrary: blb)
   apply simp
  apply clarsimp
  apply (case_tac blb, clarsimp)
  apply (clarsimp simp: Let_def)
  done

lemma rbl_mult_app2: "length blb ≥ length bla ⟹ rbl_mult bla (blb @ blc) = rbl_mult bla blb"
  apply (induct bla arbitrary: blb)
   apply simp
  apply clarsimp
  apply (case_tac blb, clarsimp)
  apply (clarsimp simp: Let_def rbl_add_app2)
  done

lemma rbl_mult_take2:
  "length blb ≥ length bla ⟹ rbl_mult bla (take (length bla) blb) = rbl_mult bla blb"
  apply (rule trans)
   apply (rule rbl_mult_app2 [symmetric])
   apply simp
  apply (rule_tac f = "rbl_mult bla" in arg_cong)
  apply (rule append_take_drop_id)
  done

lemma rbl_add_split:
  "P (rbl_add (y # ys) (x # xs)) =
    (∀ws. length ws = length ys ⟶ ws = rbl_add ys xs ⟶
      (y ⟶ ((x ⟶ P (False # rbl_succ ws)) ∧ (¬ x ⟶ P (True # ws)))) ∧
      (¬ y ⟶ P (x # ws)))"
  by (cases y) (auto simp: Let_def)

lemma rbl_mult_split:
  "P (rbl_mult (y # ys) xs) =
    (∀ws. length ws = Suc (length ys) ⟶ ws = False # rbl_mult ys xs ⟶
      (y ⟶ P (rbl_add ws xs)) ∧ (¬ y ⟶ P ws))"
  by (auto simp: Let_def)

lemma rbl_pred: "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))"
proof (unfold bin_to_bl_def, induction n arbitrary: bin)
  case 0
  then show ?case
    by simp
next
  case (Suc n)
  obtain b k where ‹bin = of_bool b + 2 * k›
    using bin_exhaust by blast
  moreover have ‹(2 * k - 1) div 2 = k - 1›
    using even_succ_div_2 [of ‹2 * (k - 1)›]
    by simp
  ultimately show ?case
    using Suc [of ‹bin div 2›]
    by simp (auto simp add: bin_to_bl_aux_alt)
qed

lemma rbl_succ: "rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin + 1))"
  apply (unfold bin_to_bl_def)
  apply (induction n arbitrary: bin)
   apply simp_all
  apply (case_tac bin rule: bin_exhaust)
   apply (simp_all add: bin_to_bl_aux_alt ac_simps)
  done

lemma rbl_add:
  "⋀bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
    rev (bin_to_bl n (bina + binb))"
  apply (unfold bin_to_bl_def)
  apply (induct n)
   apply simp
  apply clarsimp
  apply (case_tac bina rule: bin_exhaust)
  apply (case_tac binb rule: bin_exhaust)
  apply (case_tac b)
   apply (case_tac [!] "ba")
     apply (auto simp: rbl_succ bin_to_bl_aux_alt Let_def ac_simps)
  done

lemma rbl_add_long:
  "m ≥ n ⟹ rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
    rev (bin_to_bl n (bina + binb))"
  apply (rule box_equals [OF _ rbl_add_take2 rbl_add])
   apply (rule_tac f = "rbl_add (rev (bin_to_bl n bina))" in arg_cong)
   apply (rule rev_swap [THEN iffD1])
   apply (simp add: rev_take drop_bin2bl)
  apply simp
  done

lemma rbl_mult_gt1:
  "m ≥ length bl ⟹
    rbl_mult bl (rev (bin_to_bl m binb)) =
    rbl_mult bl (rev (bin_to_bl (length bl) binb))"
  apply (rule trans)
   apply (rule rbl_mult_take2 [symmetric])
   apply simp_all
  apply (rule_tac f = "rbl_mult bl" in arg_cong)
  apply (rule rev_swap [THEN iffD1])
  apply (simp add: rev_take drop_bin2bl)
  done

lemma rbl_mult_gt:
  "m > n ⟹
    rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) =
    rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))"
  by (auto intro: trans [OF rbl_mult_gt1])

lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt]

lemma rbbl_Cons: "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (of_bool b + 2 * x))"
  by (simp add: bin_to_bl_def) (simp add: bin_to_bl_aux_alt)

lemma rbl_mult:
  "rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) =
    rev (bin_to_bl n (bina * binb))"
  apply (induct n arbitrary: bina binb)
   apply simp_all
  apply (unfold bin_to_bl_def)
  apply clarsimp
  apply (case_tac bina rule: bin_exhaust)
  apply (case_tac binb rule: bin_exhaust)
  apply (simp_all add: bin_to_bl_aux_alt)
  apply (simp_all add: rbbl_Cons rbl_mult_Suc rbl_add algebra_simps)
  done

lemma sclem: "size (concat (map (bin_to_bl n) xs)) = length xs * n"
  by (simp add: length_concat comp_def sum_list_triv)

lemma bin_cat_foldl_lem:
  "foldl (λu k. concat_bit n k u) x xs =
    concat_bit (size xs * n) (foldl (λu k. concat_bit n k u) y xs) x"
  apply (induct xs arbitrary: x)
   apply simp
  apply (simp (no_asm))
  apply (frule asm_rl)
  apply (drule meta_spec)
  apply (erule trans)
  apply (drule_tac x = "concat_bit n a y" in meta_spec)
  apply (simp add: bin_cat_assoc_sym)
  done

lemma bin_rcat_bl: "bin_rcat n wl = bl_to_bin (concat (map (bin_to_bl n) wl))"
  apply (unfold bin_rcat_eq_foldl)
  apply (rule sym)
  apply (induct wl)
   apply (auto simp add: bl_to_bin_append)
  apply (simp add: bl_to_bin_aux_alt sclem)
  apply (simp add: bin_cat_foldl_lem [symmetric])
  done

lemma bin_last_bl_to_bin: "odd (bl_to_bin bs) ⟷ bs ≠ [] ∧ last bs"
by(cases "bs = []")(auto simp add: bl_to_bin_def last_bin_last'[where w=0])

lemma bin_rest_bl_to_bin: "bl_to_bin bs div 2 = bl_to_bin (butlast bs)"
by(cases "bs = []")(simp_all add: bl_to_bin_def butlast_rest_bl2bin_aux)

lemma bl_xor_aux_bin:
  "map2 (λx y. x ≠ y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v XOR w) (map2 (λx y. x ≠ y) bs cs)"
  apply (induction n arbitrary: v w bs cs)
   apply auto
  apply (case_tac v rule: bin_exhaust)
  apply (case_tac w rule: bin_exhaust)
  apply clarsimp
  done

lemma bl_or_aux_bin:
  "map2 (∨) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v OR w) (map2 (∨) bs cs)"
  by (induct n arbitrary: v w bs cs) simp_all

lemma bl_and_aux_bin:
  "map2 (∧) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v AND w) (map2 (∧) bs cs)"
  by (induction n arbitrary: v w bs cs) simp_all

lemma bl_not_aux_bin: "map Not (bin_to_bl_aux n w cs) = bin_to_bl_aux n (NOT w) (map Not cs)"
  by (induct n arbitrary: w cs) auto

lemma bl_not_bin: "map Not (bin_to_bl n w) = bin_to_bl n (NOT w)"
  by (simp add: bin_to_bl_def bl_not_aux_bin)

lemma bl_and_bin: "map2 (∧) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v AND w)"
  by (simp add: bin_to_bl_def bl_and_aux_bin)

lemma bl_or_bin: "map2 (∨) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v OR w)"
  by (simp add: bin_to_bl_def bl_or_aux_bin)

lemma bl_xor_bin: "map2 (≠) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
  using bl_xor_aux_bin by (simp add: bin_to_bl_def)


subsection ‹Type \<^typ>‹'a word››

lift_definition of_bl :: ‹bool list ⇒ 'a::len word›
  is bl_to_bin .

lift_definition to_bl :: ‹'a::len word ⇒ bool list›
  is ‹bin_to_bl LENGTH('a)›
  by (simp add: bl_to_bin_inj)

lemma to_bl_eq:
  ‹to_bl w = bin_to_bl (LENGTH('a)) (uint w)›
  for w :: ‹'a::len word›
  by transfer simp

lemma bit_of_bl_iff [bit_simps]:
  ‹bit (of_bl bs :: 'a word) n ⟷ rev bs ! n ∧ n < LENGTH('a::len) ∧ n < length bs›
  by transfer (simp add: bin_nth_of_bl ac_simps)

lemma rev_to_bl_eq:
  ‹rev (to_bl w) = map (bit w) [0..<LENGTH('a)]›
  for w :: ‹'a::len word›
  apply (rule nth_equalityI)
   apply (simp add: to_bl.rep_eq)
  apply (simp add: bin_nth_bl bit_word.rep_eq to_bl.rep_eq)
  done

lemma to_bl_eq_rev:
  ‹to_bl w = map (bit w) (rev [0..<LENGTH('a)])›
  for w :: ‹'a::len word›
  using rev_to_bl_eq [of w]
  apply (subst rev_is_rev_conv [symmetric])
  apply (simp add: rev_map)
  done

lemma of_bl_rev_eq:
  ‹of_bl (rev bs) = horner_sum of_bool 2 bs›
  apply (rule bit_word_eqI)
  apply (simp add: bit_of_bl_iff)
  apply transfer
  apply (simp add: bit_horner_sum_bit_iff ac_simps)
  done

lemma of_bl_eq:
  ‹of_bl bs = horner_sum of_bool 2 (rev bs)›
  using of_bl_rev_eq [of ‹rev bs›] by simp

lemma bshiftr1_eq:
  ‹bshiftr1 b w = of_bl (b # butlast (to_bl w))›
  apply (rule bit_word_eqI)
  apply (auto simp add: bit_simps to_bl_eq_rev nth_append rev_nth nth_butlast not_less simp flip: bit_Suc)
  apply (metis Suc_pred len_gt_0 less_eq_decr_length_iff not_bit_length verit_la_disequality)
  done

lemma length_to_bl_eq:
  ‹length (to_bl w) = LENGTH('a)›
  for w :: ‹'a::len word›
  by transfer simp

lemma word_rotr_eq:
  ‹word_rotr n w = of_bl (rotater n (to_bl w))›
  apply (rule bit_word_eqI)
  subgoal for n
    apply (cases ‹n < LENGTH('a)›)
     apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff rotater_rev length_to_bl_eq nth_rotate rev_to_bl_eq ac_simps)
    done
  done

lemma word_rotl_eq:
  ‹word_rotl n w = of_bl (rotate n (to_bl w))›
proof -
  have ‹rotate n (to_bl w) = rev (rotater n (rev (to_bl w)))›
    by (simp add: rotater_rev')
  then show ?thesis
    apply (simp add: word_rotl_eq_word_rotr bit_of_bl_iff length_to_bl_eq rev_to_bl_eq)
    apply (rule bit_word_eqI)
    subgoal for n
      apply (cases ‹n < LENGTH('a)›)
       apply (simp_all add: bit_word_rotr_iff bit_of_bl_iff nth_rotater)
      done
    done
qed

lemma to_bl_def': "(to_bl :: 'a::len word ⇒ bool list) = bin_to_bl (LENGTH('a)) ∘ uint"
  by transfer (simp add: fun_eq_iff)

― ‹type definitions theorem for in terms of equivalent bool list›
lemma td_bl:
  "type_definition
    (to_bl :: 'a::len word ⇒ bool list)
    of_bl
    {bl. length bl = LENGTH('a)}"
  apply (standard; transfer)
  apply (auto dest: sym)
  done

global_interpretation word_bl:
  type_definition
    "to_bl :: 'a::len word ⇒ bool list"
    of_bl
    "{bl. length bl = LENGTH('a::len)}"
  by (fact td_bl)

lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]

lemma word_size_bl: "size w = size (to_bl w)"
  by (auto simp: word_size)

lemma to_bl_use_of_bl: "to_bl w = bl ⟷ w = of_bl bl ∧ length bl = length (to_bl w)"
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])

lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)"
  for x :: "'a::len word"
  unfolding word_bl_Rep' by (rule len_gt_0)

lemma bl_not_Nil [iff]: "to_bl x ≠ []"
  for x :: "'a::len word"
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])

lemma length_bl_neq_0 [iff]: "length (to_bl x) ≠ 0"
  for x :: "'a::len word"
  by (fact length_bl_gt_0 [THEN gr_implies_not0])

lemma hd_to_bl_iff:
  ‹hd (to_bl w) ⟷ bit w (LENGTH('a) - 1)›
  for w :: ‹'a::len word›
  by (simp add: to_bl_eq_rev hd_map hd_rev)

lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
  by (simp add: hd_to_bl_iff bit_last_iff bin_sign_def)

lemma of_bl_drop':
  "lend = length bl - LENGTH('a::len) ⟹
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
  by transfer (simp flip: trunc_bl2bin)

lemma test_bit_of_bl:
  "bit (of_bl bl::'a::len word) n = (rev bl ! n ∧ n < LENGTH('a) ∧ n < length bl)"
  by transfer (simp add: bin_nth_of_bl ac_simps)

lemma no_of_bl: "(numeral bin ::'a::len word) = of_bl (bin_to_bl (LENGTH('a)) (numeral bin))"
  by transfer simp

lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
  by transfer simp

lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
  by (simp add: uint_bl word_size)

lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len word) = bin_to_bl (LENGTH('a)) bin"
  by (auto simp: uint_bl word_ubin.eq_norm word_size)

lemma to_bl_numeral [simp]:
  "to_bl (numeral bin::'a::len word) =
    bin_to_bl (LENGTH('a)) (numeral bin)"
  unfolding word_numeral_alt by (rule to_bl_of_bin)

lemma to_bl_neg_numeral [simp]:
  "to_bl (- numeral bin::'a::len word) =
    bin_to_bl (LENGTH('a)) (- numeral bin)"
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)

lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
  by (simp add: uint_bl word_size)

lemma uint_bl_bin: "bl_to_bin (bin_to_bl (LENGTH('a)) (uint x)) = uint x"
  for x :: "'a::len word"
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])

lemma ucast_bl: "ucast w = of_bl (to_bl w)"
  by transfer simp

lemma ucast_down_bl:
  ‹(ucast :: 'a::len word ⇒ 'b::len word) (of_bl bl) = of_bl bl›
    if ‹is_down (ucast :: 'a::len word ⇒ 'b::len word)›
  using that by transfer simp

lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
  by transfer (simp add: bl_to_bin_app_cat)

lemma ucast_of_bl_up:
  ‹ucast (of_bl bl :: 'a::len word) = of_bl bl›
  if ‹size bl ≤ size (of_bl bl :: 'a::len word)›
  using that
  apply transfer
  apply (rule bit_eqI)
  apply (auto simp add: bit_take_bit_iff)
  apply (subst (asm) trunc_bl2bin_len [symmetric])
  apply (auto simp only: bit_take_bit_iff)
  done

lemma word_rev_tf:
  "to_bl (of_bl bl::'a::len word) =
    rev (takefill False (LENGTH('a)) (rev bl))"
  by transfer (simp add: bl_bin_bl_rtf)

lemma word_rep_drop:
  "to_bl (of_bl bl::'a::len word) =
    replicate (LENGTH('a) - length bl) False @
    drop (length bl - LENGTH('a)) bl"
  by (simp add: word_rev_tf takefill_alt rev_take)

lemma to_bl_ucast:
  "to_bl (ucast (w::'b::len word) ::'a::len word) =
    replicate (LENGTH('a) - LENGTH('b)) False @
    drop (LENGTH('b) - LENGTH('a)) (to_bl w)"
  apply (unfold ucast_bl)
  apply (rule trans)
   apply (rule word_rep_drop)
  apply simp
  done

lemma ucast_up_app:
  ‹to_bl (ucast w :: 'b::len word) = replicate n False @ (to_bl w)›
    if ‹source_size (ucast :: 'a word ⇒ 'b word) + n = target_size (ucast :: 'a word ⇒ 'b word)›
    for w :: ‹'a::len word›
  using that
  by (auto simp add : source_size target_size to_bl_ucast)

lemma ucast_down_drop [OF refl]:
  "uc = ucast ⟹ source_size uc = target_size uc + n ⟹
    to_bl (uc w) = drop n (to_bl w)"
  by (auto simp add : source_size target_size to_bl_ucast)

lemma scast_down_drop [OF refl]:
  "sc = scast ⟹ source_size sc = target_size sc + n ⟹
    to_bl (sc w) = drop n (to_bl w)"
  apply (subgoal_tac "sc = ucast")
   apply safe
   apply simp
   apply (erule ucast_down_drop)
  apply (rule down_cast_same [symmetric])
  apply (simp add : source_size target_size is_down)
  done

lemma word_0_bl [simp]: "of_bl [] = 0"
  by transfer simp

lemma word_1_bl: "of_bl [True] = 1"
  by transfer (simp add: bl_to_bin_def)

lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
  by transfer (simp add: bl_to_bin_rep_False)

lemma to_bl_0 [simp]: "to_bl (0::'a::len word) = replicate (LENGTH('a)) False"
  by (simp add: uint_bl word_size bin_to_bl_zero)

― ‹links with ‹rbl› operations›
lemma word_succ_rbl: "to_bl w = bl ⟹ to_bl (word_succ w) = rev (rbl_succ (rev bl))"
  by transfer (simp add: rbl_succ)

lemma word_pred_rbl: "to_bl w = bl ⟹ to_bl (word_pred w) = rev (rbl_pred (rev bl))"
  by transfer (simp add: rbl_pred)

lemma word_add_rbl:
  "to_bl v = vbl ⟹ to_bl w = wbl ⟹
    to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))"
  apply transfer
  apply (drule sym)
  apply (drule sym)
  apply (simp add: rbl_add)
  done

lemma word_mult_rbl:
  "to_bl v = vbl ⟹ to_bl w = wbl ⟹
    to_bl (v * w) = rev (rbl_mult (rev vbl) (rev wbl))"
  apply transfer
  apply (drule sym)
  apply (drule sym)
  apply (simp add: rbl_mult)
  done

lemma rtb_rbl_ariths:
  "rev (to_bl w) = ys ⟹ rev (to_bl (word_succ w)) = rbl_succ ys"
  "rev (to_bl w) = ys ⟹ rev (to_bl (word_pred w)) = rbl_pred ys"
  "rev (to_bl v) = ys ⟹ rev (to_bl w) = xs ⟹ rev (to_bl (v * w)) = rbl_mult ys xs"
  "rev (to_bl v) = ys ⟹ rev (to_bl w) = xs ⟹ rev (to_bl (v + w)) = rbl_add ys xs"
  by (auto simp: rev_swap [symmetric] word_succ_rbl word_pred_rbl word_mult_rbl word_add_rbl)

lemma of_bl_length_less:
  ‹(of_bl x :: 'a::len word) < 2 ^ k›
    if ‹length x = k› ‹k < LENGTH('a)›
proof -
  from that have ‹length x < LENGTH('a)›
    by simp
  then have ‹(of_bl x :: 'a::len word) < 2 ^ length x›
    apply (simp add: of_bl_eq)
    apply transfer
    apply (simp add: take_bit_horner_sum_bit_eq)
    apply (subst length_rev [symmetric])
    apply (simp only: horner_sum_of_bool_2_less)
    done
  with that show ?thesis
    by simp
qed

lemma word_eq_rbl_eq: "x = y ⟷ rev (to_bl x) = rev (to_bl y)"
  by simp

lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)"
  by transfer (simp add: bl_not_bin)

lemma bl_word_xor: "to_bl (v XOR w) = map2 (≠) (to_bl v) (to_bl w)"
  by transfer (simp flip: bl_xor_bin)

lemma bl_word_or: "to_bl (v OR w) = map2 (∨) (to_bl v) (to_bl w)"
  by transfer (simp flip: bl_or_bin)

lemma bl_word_and: "to_bl (v AND w) = map2 (∧) (to_bl v) (to_bl w)"
  by transfer (simp flip: bl_and_bin)

lemma bin_nth_uint': "bit (uint w) n ⟷ rev (bin_to_bl (size w) (uint w)) ! n ∧ n < size w"
  apply (unfold word_size)
  apply (safe elim!: bin_nth_uint_imp)
   apply (frule bin_nth_uint_imp)
   apply (fast dest!: bin_nth_bl)+
  done

lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]

lemma test_bit_bl: "bit w n ⟷ rev (to_bl w) ! n ∧ n < size w"
  by transfer (auto simp add: bin_nth_bl)

lemma to_bl_nth: "n < size w ⟹ to_bl w ! n = bit w (size w - Suc n)"
  by (simp add: word_size rev_nth test_bit_bl)

lemma map_bit_interval_eq:
  ‹map (bit w) [0..<n] = takefill False n (rev (to_bl w))› for w :: ‹'a::len word›
proof (rule nth_equalityI)
  show ‹length (map (bit w) [0..<n]) =
    length (takefill False n (rev (to_bl w)))›
    by simp
  fix m
  assume ‹m < length (map (bit w) [0..<n])›
  then have ‹m < n›
    by simp
  then have ‹bit w m ⟷ takefill False n (rev (to_bl w)) ! m›
    by (auto simp add: nth_takefill not_less rev_nth to_bl_nth word_size dest: bit_imp_le_length)
  with ‹m < n ›show ‹map (bit w) [0..<n] ! m ⟷ takefill False n (rev (to_bl w)) ! m›
    by simp
qed

lemma to_bl_unfold:
  ‹to_bl w = rev (map (bit w) [0..<LENGTH('a)])› for w :: ‹'a::len word›
  by (simp add: map_bit_interval_eq takefill_bintrunc to_bl_def flip: bin_to_bl_def)

lemma nth_rev_to_bl:
  ‹rev (to_bl w) ! n ⟷ bit w n›
  if ‹n < LENGTH('a)› for w :: ‹'a::len word›
  using that by (simp add: to_bl_unfold)

lemma nth_to_bl:
  ‹to_bl w ! n ⟷ bit w (LENGTH('a) - Suc n)›
  if ‹n < LENGTH('a)› for w :: ‹'a::len word›
  using that by (simp add: to_bl_unfold rev_nth)

lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
  by (auto simp: of_bl_def bl_to_bin_rep_F)

lemma [code abstract]:
  ‹Word.the_int (of_bl bs :: 'a word) = horner_sum of_bool 2 (take LENGTH('a::len) (rev bs))›
  apply (simp add: of_bl_eq flip: take_bit_horner_sum_bit_eq)
  apply transfer
  apply simp
  done

lemma [code]:
  ‹to_bl w = map (bit w) (rev [0..<LENGTH('a::len)])›
  for w :: ‹'a::len word›
  by (fact to_bl_eq_rev)

lemma word_reverse_eq_of_bl_rev_to_bl:
  ‹word_reverse w = of_bl (rev (to_bl w))›
  by (rule bit_word_eqI)
    (auto simp add: bit_word_reverse_iff bit_of_bl_iff nth_to_bl)

lemmas word_reverse_no_def [simp] =
  word_reverse_eq_of_bl_rev_to_bl [of "numeral w"] for w

lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
  by (rule nth_equalityI) (simp_all add: nth_rev_to_bl word_reverse_def word_rep_drop flip: of_bl_eq)

lemma to_bl_n1 [simp]: "to_bl (-1::'a::len word) = replicate (LENGTH('a)) True"
  apply (rule word_bl.Abs_inverse')
   apply simp
  apply (rule word_eqI)
  apply (clarsimp simp add: word_size)
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
  done

lemma rbl_word_or: "rev (to_bl (x OR y)) = map2 (∨) (rev (to_bl x)) (rev (to_bl y))"
  by (simp add: zip_rev bl_word_or rev_map)

lemma rbl_word_and: "rev (to_bl (x AND y)) = map2 (∧) (rev (to_bl x)) (rev (to_bl y))"
  by (simp add: zip_rev bl_word_and rev_map)

lemma rbl_word_xor: "rev (to_bl (x XOR y)) = map2 (≠) (rev (to_bl x)) (rev (to_bl y))"
  by (simp add: zip_rev bl_word_xor rev_map)

lemma rbl_word_not: "rev (to_bl (NOT x)) = map Not (rev (to_bl x))"
  by (simp add: bl_word_not rev_map)

lemma bshiftr1_numeral [simp]:
  ‹bshiftr1 b (numeral w :: 'a word) = of_bl (b # butlast (bin_to_bl LENGTH('a::len) (numeral w)))›
  by (rule bit_word_eqI) (auto simp add: bit_simps rev_nth nth_append nth_butlast nth_bin_to_bl simp flip: bit_Suc)

lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)"
  unfolding bshiftr1_eq by (rule word_bl.Abs_inverse) simp

lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])"
  apply (rule bit_word_eqI)
  apply (simp add: bit_simps)
  subgoal for n
    apply (cases n)
     apply simp_all
    done
  done

lemma shiftl1_bl: "shiftl1 w = of_bl (to_bl w @ [False])"
  apply (rule bit_word_eqI)
  apply (simp add: bit_simps)
  subgoal for n
    apply (cases n)
     apply (simp_all add: nth_rev_to_bl)
    done
  done

lemma bl_shiftl1: "to_bl (shiftl1 w) = tl (to_bl w) @ [False]"
  for w :: "'a::len word"
  by (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons') (fast intro!: Suc_leI)

lemma to_bl_double_eq:
  ‹to_bl (2 * w) = tl (to_bl w) @ [False]›
  using bl_shiftl1 [of w] by (simp add: shiftl1_def ac_simps)

― ‹Generalized version of ‹bl_shiftl1›. Maybe this one should replace it?›
lemma bl_shiftl1': "to_bl (shiftl1 w) = tl (to_bl w @ [False])"
  by (simp add: shiftl1_bl word_rep_drop drop_Suc del: drop_append)

lemma shiftr1_bl:
  ‹shiftr1 w = of_bl (butlast (to_bl w))›
proof (rule bit_word_eqI)
  fix n
  assume ‹n < LENGTH('a)›
  show ‹bit (shiftr1 w) n ⟷ bit (of_bl (butlast (to_bl w)) :: 'a word) n›
  proof (cases ‹n = LENGTH('a) - 1›)
    case True
    then show ?thesis
      by (simp add: bit_shiftr1_iff bit_of_bl_iff)
  next
    case False
    with ‹n < LENGTH('a)›
    have ‹n < LENGTH('a) - 1›
      by simp
    with ‹n < LENGTH('a)› show ?thesis
      by (simp add: bit_shiftr1_iff bit_of_bl_iff rev_nth nth_butlast
        word_size to_bl_nth)
  qed
qed

lemma bl_shiftr1: "to_bl (shiftr1 w) = False # butlast (to_bl w)"
  for w :: "'a::len word"
  by (simp add: shiftr1_bl word_rep_drop len_gt_0 [THEN Suc_leI])

― ‹Generalized version of ‹bl_shiftr1›. Maybe this one should replace it?›
lemma bl_shiftr1': "to_bl (shiftr1 w) = butlast (False # to_bl w)"
  apply (rule word_bl.Abs_inverse')
   apply (simp del: butlast.simps)
  apply (simp add: shiftr1_bl of_bl_def)
  done

lemma bl_sshiftr1: "to_bl (sshiftr1 w) = hd (to_bl w) # butlast (to_bl w)"
  for w :: "'a::len word"
proof (rule nth_equalityI)
  fix n
  assume ‹n < length (to_bl (sshiftr1 w))›
  then have ‹n < LENGTH('a)›
    by simp
  then show ‹to_bl (sshiftr1 w) ! n ⟷ (hd (to_bl w) # butlast (to_bl w)) ! n›
    apply (cases n)
     apply (simp_all add: to_bl_nth word_size hd_conv_nth bit_sshiftr1_iff nth_butlast Suc_diff_Suc nth_to_bl)
    done
qed simp

lemma drop_shiftr: "drop n (to_bl (w >> n)) = take (size w - n) (to_bl w)"
  for w :: "'a::len word"
  apply (rule nth_equalityI)
   apply (simp_all add: word_size to_bl_nth bit_simps)
  done

lemma drop_sshiftr: "drop n (to_bl (w >>> n)) = take (size w - n) (to_bl w)"
  for w :: "'a::len word"
  apply (rule nth_equalityI)
   apply (simp_all add: word_size nth_to_bl bit_simps)
  done

lemma take_shiftr: "n ≤ size w ⟹ take n (to_bl (w >> n)) = replicate n False"
  apply (rule nth_equalityI)
   apply (auto simp add: word_size to_bl_nth bit_simps dest: bit_imp_le_length)
  done

lemma take_sshiftr':
  "n ≤ size w ⟹ hd (to_bl (w >>> n)) = hd (to_bl w) ∧
    take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))"
  for w :: "'a::len word"
  apply (cases n)
   apply (auto simp add: hd_to_bl_iff bit_simps not_less word_size)
  apply (rule nth_equalityI)
   apply (auto simp add: nth_to_bl bit_simps nth_Cons split: nat.split)
  done

lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1]
lemmas take_sshiftr = take_sshiftr' [THEN conjunct2]

lemma atd_lem: "take n xs = t ⟹ drop n xs = d ⟹ xs = t @ d"
  by (auto intro: append_take_drop_id [symmetric])

lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr]
lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr]

lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)"
  apply (rule bit_word_eqI)
  apply (auto simp add: bit_simps nth_append)
  done

lemma shiftl_bl: "w << n = of_bl (to_bl w @ replicate n False)"
  for w :: "'a::len word"
  by (simp flip: shiftl_of_bl)

lemma bl_shiftl: "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False"
  by (simp add: shiftl_bl word_rep_drop word_size)

lemma shiftr1_bl_of:
  "length bl ≤ LENGTH('a) ⟹
    shiftr1 (of_bl bl::'a::len word) = of_bl (butlast bl)"
  apply (rule bit_word_eqI)
  apply (simp add: bit_simps)
  apply (cases bl rule: rev_cases)
  apply auto
  done

lemma shiftr_bl_of:
  "length bl ≤ LENGTH('a) ⟹
     (of_bl bl::'a::len word) >> n = of_bl (take (length bl - n) bl)"
  by (rule bit_word_eqI) (auto simp add: bit_simps rev_nth)

lemma shiftr_bl: "x >> n ≡ of_bl (take (LENGTH('a) - n) (to_bl x))"
  for x :: "'a::len word"
  using shiftr_bl_of [where 'a='a, of "to_bl x"] by simp

lemma aligned_bl_add_size [OF refl]:
  "size x - n = m ⟹ n ≤ size x ⟹ drop m (to_bl x) = replicate n False ⟹
    take m (to_bl y) = replicate m False ⟹
    to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)" for x :: ‹'a::len word›
  apply (subgoal_tac "x AND y = 0")
   prefer 2
   apply (rule word_bl.Rep_eqD)
   apply (simp add: bl_word_and)
   apply (rule align_lem_and [THEN trans])
       apply (simp_all add: word_size)[5]
   apply simp
  apply (subst word_plus_and_or [symmetric])
  apply (simp add : bl_word_or)
  apply (rule align_lem_or)
     apply (simp_all add: word_size)
  done

lemma mask_bl: "mask n = of_bl (replicate n True)"
  by (auto simp add: bit_simps intro!: word_eqI)

lemma bl_and_mask':
  "to_bl (w AND mask n :: 'a::len word) =
    replicate (LENGTH('a) - n) False @
    drop (LENGTH('a) - n) (to_bl w)"
  apply (rule nth_equalityI)
   apply simp
  apply (clarsimp simp add: to_bl_nth word_size bit_simps)
  apply (auto simp add: word_size test_bit_bl nth_append rev_nth)
  done

lemma slice1_eq_of_bl:
  ‹(slice1 n w :: 'b::len word) = of_bl (takefill False n (to_bl w))›
  for w :: ‹'a::len word›
proof (rule bit_word_eqI)
  fix m
  assume ‹m < LENGTH('b)›
  show ‹bit (slice1 n w :: 'b::len word) m ⟷ bit (of_bl (takefill False n (to_bl w)) :: 'b word) m›
    by (cases ‹m ≥ n›; cases ‹LENGTH('a) ≥ n›)
      (auto simp add: bit_slice1_iff bit_of_bl_iff not_less rev_nth not_le nth_takefill nth_to_bl algebra_simps)
qed

lemma slice1_no_bin [simp]:
  "slice1 n (numeral w :: 'b word) = of_bl (takefill False n (bin_to_bl (LENGTH('b::len)) (numeral w)))"
  by (simp add: slice1_eq_of_bl) (* TODO: neg_numeral *)

lemma slice_no_bin [simp]:
  "slice n (numeral w :: 'b word) = of_bl (takefill False (LENGTH('b::len) - n)
    (bin_to_bl (LENGTH('b::len)) (numeral w)))"
  by (simp add: slice_def) (* TODO: neg_numeral *)

lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))"
  by (simp add: slice_def word_size slice1_eq_of_bl takefill_alt)

lemmas slice_take = slice_take' [unfolded word_size]

― ‹shiftr to a word of the same size is just slice,
    slice is just shiftr then ucast›
lemmas shiftr_slice = trans [OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric]]

lemma slice1_down_alt':
  "sl = slice1 n w ⟹ fs = size sl ⟹ fs + k = n ⟹
    to_bl sl = takefill False fs (drop k (to_bl w))"
  apply (simp add: slice1_eq_of_bl)
  apply transfer
  apply (simp add: bl_bin_bl_rep_drop)
  using drop_takefill
  apply force
  done

lemma slice1_up_alt':
  "sl = slice1 n w ⟹ fs = size sl ⟹ fs = n + k ⟹
    to_bl sl = takefill False fs (replicate k False @ (to_bl w))"
  apply (simp add: slice1_eq_of_bl)
  apply transfer
  apply (simp add: bl_bin_bl_rep_drop flip: takefill_append)
  apply (metis diff_add_inverse)
  done

lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size]
lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size]
lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1]
lemmas slice1_up_alts =
  le_add_diff_inverse [symmetric, THEN su1]
  le_add_diff_inverse2 [symmetric, THEN su1]

lemma slice1_tf_tf':
  "to_bl (slice1 n w :: 'a::len word) =
    rev (takefill False (LENGTH('a)) (rev (