src/HOL/Word/Bool_List_Representation.thy

tuned op's

(*  Title:      HOL/Word/Bool_List_Representation.thy
    Author:     Jeremy Dawson, NICTA

Theorems to do with integers, expressed using Pls, Min, BIT,
theorems linking them to lists of booleans, and repeated splitting
and concatenation.
*)

section "Bool lists and integers"

theory Bool_List_Representation
  imports Bits_Int
begin

definition map2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
  where "map2 f as bs = map (case_prod f) (zip as bs)"

lemma map2_Nil [simp, code]: "map2 f [] ys = []"
  by (auto simp: map2_def)

lemma map2_Nil2 [simp, code]: "map2 f xs [] = []"
  by (auto simp: map2_def)

lemma map2_Cons [simp, code]: "map2 f (x # xs) (y # ys) = f x y # map2 f xs ys"
  by (auto simp: map2_def)


subsection \<open>Operations on lists of booleans\<close>

primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int"
  where
    Nil: "bl_to_bin_aux [] w = w"
  | Cons: "bl_to_bin_aux (b # bs) w = bl_to_bin_aux bs (w BIT b)"

definition bl_to_bin :: "bool list \<Rightarrow> int"
  where "bl_to_bin bs = bl_to_bin_aux bs 0"

primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> 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 (bin_rest w) ((bin_last w) # bl)"

definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list"
  where "bin_to_bl n w = bin_to_bl_aux n w []"

primrec bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool list"
  where
    Suc: "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
  | Z: "bl_of_nth 0 f = []"

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


subsection "Arithmetic in terms of bool lists"

text \<open>
  Arithmetic operations in terms of the reversed bool list,
  assuming input list(s) the same length, and don't extend them.
\<close>

primrec rbl_succ :: "bool list \<Rightarrow> 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 \<Rightarrow> 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 \<Rightarrow> bool list \<Rightarrow> bool list"
  where \<comment> "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 \<noteq> hd x) # (if hd x \<and> y then rbl_succ ws else ws))"

primrec rbl_mult :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list"
  where \<comment> "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 butlast_power: "(butlast ^^ n) bl = take (length bl - n) bl"
  by (induct n) (auto simp: butlast_take)

lemma bin_to_bl_aux_zero_minus_simp [simp]:
  "0 < n \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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_Bit_minus_simp [simp]:
  "0 < n \<Longrightarrow> bin_to_bl_aux n (w BIT b) bl = bin_to_bl_aux (n - 1) w (b # bl)"
  by (cases n) auto

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

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

text \<open>Link between \<open>bin\<close> and \<open>bool list\<close>.\<close>

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

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

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

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

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 \<Longrightarrow> length bs = length cs \<Longrightarrow> 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)

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 \<le> m \<Longrightarrow> bin_to_bl n (bintrunc 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 (bintrunc 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
  done

lemma bin_to_bl_bintr:
  "bin_to_bl n (bintrunc 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 (induct bs arbitrary: w) auto

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 (sbintrunc n w) = -1)"
  apply (induct n arbitrary: w bs)
   apply clarsimp
   apply (cases w rule: bin_exhaust)
   apply simp
  done

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

lemma bin_nth_of_bl_aux:
  "bin_nth (bl_to_bin_aux bl w) n =
    (n < size bl \<and> rev bl ! n \<or> n \<ge> length bl \<and> bin_nth w (n - size bl))"
  apply (induct bl arbitrary: w)
   apply clarsimp
  apply clarsimp
  apply (cut_tac x=n and y="size bl" in linorder_less_linear)
  apply (erule disjE, simp add: nth_append)+
  apply auto
  done

lemma bin_nth_of_bl: "bin_nth (bl_to_bin bl) n = (n < length bl \<and> rev bl ! n)"
  by (simp add: bl_to_bin_def bin_nth_of_bl_aux)

lemma bin_nth_bl: "n < m \<Longrightarrow> bin_nth w n = nth (rev (bin_to_bl m w)) n"
  apply (induct n arbitrary: m w)
   apply clarsimp
   apply (case_tac m, clarsimp)
   apply (clarsimp simp: bin_to_bl_def)
   apply (simp add: bin_to_bl_aux_alt)
  apply clarsimp
  apply (case_tac m, clarsimp)
  apply (clarsimp simp: bin_to_bl_def)
  apply (simp add: bin_to_bl_aux_alt)
  done

lemma nth_rev: "n < length xs \<Longrightarrow> rev xs ! n = xs ! (length xs - 1 - n)"
  apply (induct xs)
   apply simp
  apply (clarsimp simp add: nth_append nth.simps split: nat.split)
  apply (rule_tac f = "\<lambda>n. xs ! n" in arg_cong)
  apply arith
  done

lemma nth_rev_alt: "n < length ys \<Longrightarrow> ys ! n = rev ys ! (length ys - Suc n)"
  by (simp add: nth_rev)

lemma nth_bin_to_bl_aux:
  "n < m + length bl \<Longrightarrow> (bin_to_bl_aux m w bl) ! n =
    (if n < m then bin_nth w (m - 1 - n) else bl ! (n - m))"
  apply (induct m arbitrary: w n bl)
   apply clarsimp
  apply clarsimp
  apply (case_tac w rule: bin_exhaust)
  apply simp
  done

lemma nth_bin_to_bl: "n < m \<Longrightarrow> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)"
  by (simp add: bin_to_bl_def nth_bin_to_bl_aux)

lemma bl_to_bin_lt2p_aux: "bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
  apply (induct bs arbitrary: w)
   apply clarsimp
  apply clarsimp
  apply (drule meta_spec, erule xtrans(8) [rotated], simp add: Bit_def)+
  done

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 \<ge> w * (2 ^ length bs)"
  apply (induct bs arbitrary: w)
   apply clarsimp
  apply clarsimp
   apply (drule meta_spec, erule order_trans [rotated],
          simp add: Bit_B0_2t Bit_B1_2t algebra_simps)+
   apply (simp add: Bit_def)
  done

lemma bl_to_bin_ge0: "bl_to_bin bs \<ge> 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) (bin_rest w)"
  apply (unfold bin_to_bl_def)
  apply (cases w rule: bin_exhaust)
  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) (bin_rest (bl_to_bin bl))"
  using butlast_rest_bin [where w="bl_to_bin bl" and n="length bl"] by simp

lemma butlast_rest_bl2bin_aux:
  "bl \<noteq> [] \<Longrightarrow> bl_to_bin_aux (butlast bl) w = bin_rest (bl_to_bin_aux bl w)"
  by (induct bl arbitrary: w) auto

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

lemma trunc_bl2bin_aux:
  "bintrunc m (bl_to_bin_aux bl w) =
    bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (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
  qed
qed

lemma trunc_bl2bin: "bintrunc 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]: "bintrunc (length bl) (bl_to_bin bl) = bl_to_bin bl"
  by (simp add: trunc_bl2bin)

lemma bl2bin_drop: "bl_to_bin (drop k bl) = bintrunc (length bl - k) (bl_to_bin bl)"
  apply (rule trans)
   prefer 2
   apply (rule trunc_bl2bin [symmetric])
  apply (cases "k \<le> length bl")
   apply auto
  done

lemma nth_rest_power_bin: "bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
  apply (induct k arbitrary: n)
   apply clarsimp
  apply clarsimp
  apply (simp only: bin_nth.Suc [symmetric] add_Suc)
  done

lemma take_rest_power_bin: "m \<le> n \<Longrightarrow> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)"
  apply (rule nth_equalityI)
   apply simp
  apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
  done

lemma hd_butlast: "size xs > 1 \<Longrightarrow> hd (butlast xs) = hd xs"
  by (cases xs) auto

lemma last_bin_last': "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin_aux xs w)"
  by (induct xs arbitrary: w) auto

lemma last_bin_last: "size xs > 0 \<Longrightarrow> last xs \<longleftrightarrow> bin_last (bl_to_bin xs)"
  unfolding bl_to_bin_def by (erule last_bin_last')

lemma bin_last_last: "bin_last w \<longleftrightarrow> last (bin_to_bl (Suc n) w)"
  by (simp add: bin_to_bl_def) (auto simp: bin_to_bl_aux_alt)


subsection \<open>Links between bit-wise operations and operations on bool lists\<close>

lemma bl_xor_aux_bin:
  "map2 (\<lambda>x y. x \<noteq> y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v XOR w) (map2 (\<lambda>x y. x \<noteq> y) bs cs)"
  apply (induct n arbitrary: v w bs cs)
   apply simp
  apply (case_tac v rule: bin_exhaust)
  apply (case_tac w rule: bin_exhaust)
  apply clarsimp
  apply (case_tac b)
   apply auto
  done

lemma bl_or_aux_bin:
  "map2 (op \<or>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v OR w) (map2 (op \<or>) bs cs)"
  apply (induct n arbitrary: v w bs cs)
   apply simp
  apply (case_tac v rule: bin_exhaust)
  apply (case_tac w rule: bin_exhaust)
  apply clarsimp
  done

lemma bl_and_aux_bin:
  "map2 (op \<and>) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) =
    bin_to_bl_aux n (v AND w) (map2 (op \<and>) bs cs)"
  apply (induct n arbitrary: v w bs cs)
   apply simp
  apply (case_tac v rule: bin_exhaust)
  apply (case_tac w rule: bin_exhaust)
  apply clarsimp
  done

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 (op \<and>) (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 (op \<or>) (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 (\<lambda>x y. x \<noteq> y) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
  by (simp only: bin_to_bl_def bl_xor_aux_bin map2_Nil)

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 (induct n arbitrary: m bin bs, clarsimp)
  apply clarsimp
  apply (case_tac bin rule: bin_exhaust)
  apply (case_tac "m \<le> n", simp)
  apply (case_tac "m - n", simp)
  apply simp
  apply (rule_tac f = "\<lambda>nat. drop nat bs" in arg_cong)
  apply simp
  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) \<Longrightarrow> 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)
  done

lemma bin_split_take1:
  "k = m + n \<Longrightarrow> bin_split n c = (a, b) \<Longrightarrow> bin_to_bl m a = take m (bin_to_bl k c)"
  by (auto elim: bin_split_take)

lemma nth_takefill: "m < n \<Longrightarrow> 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 \<Longrightarrow> 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 \<Longrightarrow> 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 \<le> n \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
  by (auto simp: le_iff_add takefill_le')

lemma take_takefill [simp]: "m \<le> n \<Longrightarrow> 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 \<Longrightarrow> takefill fill l xs = xs"
  by (induct xs arbitrary: l) auto

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

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 nth_rev 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_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_bin_bl_rtf takefill_alt rev_take)

lemma tf_rev:
  "n + k = m + length bl \<Longrightarrow> 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 nth_rev)
  apply (rule_tac f = "\<lambda>n. bl ! n" in arg_cong)
  apply arith
  done

lemma takefill_minus: "0 < n \<Longrightarrow> 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 \<open>Links with function \<open>bl_to_bin\<close>\<close>

lemma bl_to_bin_aux_cat:
  "\<And>nv v. bl_to_bin_aux bs (bin_cat w nv v) =
    bin_cat w (nv + length bs) (bl_to_bin_aux bs v)"
  by (induct bs) (simp, simp add: bin_cat_Suc_Bit [symmetric] del: bin_cat.simps)

lemma bin_to_bl_aux_cat:
  "\<And>w bs. bin_to_bl_aux (nv + nw) (bin_cat v nw w) bs =
    bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
  by (induct nw) auto

lemma bl_to_bin_aux_alt: "bl_to_bin_aux bs w = bin_cat w (length bs) (bl_to_bin bs)"
  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) (bin_cat v nw w) =
    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) = bin_cat (bl_to_bin bsa) (length bs) (bl_to_bin bs)"
  by (simp only: bl_to_bin_aux_app_cat bl_to_bin_def)

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

text \<open>\<open>bl_to_bin_app_cat_alt\<close> and \<open>bl_to_bin_app_cat\<close> are easily interderivable.\<close>
lemma bl_to_bin_app_cat_alt: "bin_cat (bl_to_bin cs) n w = 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 add: Bit_B0_2t Bit_B1_2t)
  done


subsection \<open>Function \<open>bl_of_nth\<close>\<close>

lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n"
  by (induct n)  auto

lemma nth_bl_of_nth [simp]: "m < n \<Longrightarrow> rev (bl_of_nth n f) ! m = f m"
  apply (induct n)
   apply simp
  apply (clarsimp simp add: nth_append)
  apply (rule_tac f = "f" in arg_cong)
  apply simp
  done

lemma bl_of_nth_inj: "(\<And>k. k < n \<Longrightarrow> f k = g k) \<Longrightarrow> bl_of_nth n f = bl_of_nth n g"
  by (induct n)  auto

lemma bl_of_nth_nth_le: "n \<le> length xs \<Longrightarrow> 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: nth_rev trans [OF drop_Suc drop_tl, symmetric])
   apply (subst hd_drop_conv_nth)
    apply force
   apply simp_all
  apply (rule_tac f = "\<lambda>n. drop n xs" in arg_cong)
  apply simp
  done

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

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_pred: "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))"
  apply (unfold bin_to_bl_def)
  apply (induct n arbitrary: bin)
   apply simp
  apply clarsimp
  apply (case_tac bin rule: bin_exhaust)
  apply (case_tac b)
   apply (clarsimp simp: bin_to_bl_aux_alt)+
  done

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 (induct n arbitrary: bin)
   apply simp
  apply clarsimp
  apply (case_tac bin rule: bin_exhaust)
  apply (case_tac b)
   apply (clarsimp simp: bin_to_bl_aux_alt)+
  done

lemma rbl_add:
  "\<And>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_app2: "length blb \<ge> length bla \<Longrightarrow> 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 \<ge> length bla \<Longrightarrow> 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_add_long:
  "m \<ge> n \<Longrightarrow> 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_app2: "length blb \<ge> length bla \<Longrightarrow> 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 \<ge> length bla \<Longrightarrow> 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_mult_gt1:
  "m \<ge> length bl \<Longrightarrow>
    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 \<Longrightarrow>
    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) (x BIT b))"
  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
  apply (unfold bin_to_bl_def)
  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: bin_to_bl_aux_alt Let_def)
     apply (auto simp: rbbl_Cons rbl_mult_Suc rbl_add)
  done

lemma rbl_add_split:
  "P (rbl_add (y # ys) (x # xs)) =
    (\<forall>ws. length ws = length ys \<longrightarrow> ws = rbl_add ys xs \<longrightarrow>
      (y \<longrightarrow> ((x \<longrightarrow> P (False # rbl_succ ws)) \<and> (\<not> x \<longrightarrow> P (True # ws)))) \<and>
      (\<not> y \<longrightarrow> P (x # ws)))"
  by (cases y) (auto simp: Let_def)

lemma rbl_mult_split:
  "P (rbl_mult (y # ys) xs) =
    (\<forall>ws. length ws = Suc (length ys) \<longrightarrow> ws = False # rbl_mult ys xs \<longrightarrow>
      (y \<longrightarrow> P (rbl_add ws xs)) \<and> (\<not> y \<longrightarrow> P ws))"
  by (auto simp: Let_def)


subsection \<open>Repeated splitting or concatenation\<close>

lemma sclem: "size (concat (map (bin_to_bl n) xs)) = length xs * n"
  by (induct xs) auto

lemma bin_cat_foldl_lem:
  "foldl (\<lambda>u. bin_cat u n) x xs =
    bin_cat x (size xs * n) (foldl (\<lambda>u. bin_cat u n) y xs)"
  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 = "bin_cat y n a" in meta_spec)
  apply (simp add: bin_cat_assoc_sym min.absorb2)
  done

lemma bin_rcat_bl: "bin_rcat n wl = bl_to_bin (concat (map (bin_to_bl n) wl))"
  apply (unfold bin_rcat_def)
  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

lemmas bin_rsplit_aux_simps = bin_rsplit_aux.simps bin_rsplitl_aux.simps
lemmas rsplit_aux_simps = bin_rsplit_aux_simps

lemmas th_if_simp1 = if_split [where P = "op = l", THEN iffD1, THEN conjunct1, THEN mp] for l
lemmas th_if_simp2 = if_split [where P = "op = l", THEN iffD1, THEN conjunct2, THEN mp] for l

lemmas rsplit_aux_simp1s = rsplit_aux_simps [THEN th_if_simp1]

lemmas rsplit_aux_simp2ls = rsplit_aux_simps [THEN th_if_simp2]
(* these safe to [simp add] as require calculating m - n *)
lemmas bin_rsplit_aux_simp2s [simp] = rsplit_aux_simp2ls [unfolded Let_def]
lemmas rbscl = bin_rsplit_aux_simp2s (2)

lemmas rsplit_aux_0_simps [simp] =
  rsplit_aux_simp1s [OF disjI1] rsplit_aux_simp1s [OF disjI2]

lemma bin_rsplit_aux_append: "bin_rsplit_aux n m c (bs @ cs) = bin_rsplit_aux n m c bs @ cs"
  apply (induct n m c bs rule: bin_rsplit_aux.induct)
  apply (subst bin_rsplit_aux.simps)
  apply (subst bin_rsplit_aux.simps)
  apply (clarsimp split: prod.split)
  done

lemma bin_rsplitl_aux_append: "bin_rsplitl_aux n m c (bs @ cs) = bin_rsplitl_aux n m c bs @ cs"
  apply (induct n m c bs rule: bin_rsplitl_aux.induct)
  apply (subst bin_rsplitl_aux.simps)
  apply (subst bin_rsplitl_aux.simps)
  apply (clarsimp split: prod.split)
  done

lemmas rsplit_aux_apps [where bs = "[]"] =
  bin_rsplit_aux_append bin_rsplitl_aux_append

lemmas rsplit_def_auxs = bin_rsplit_def bin_rsplitl_def

lemmas rsplit_aux_alts = rsplit_aux_apps
  [unfolded append_Nil rsplit_def_auxs [symmetric]]

lemma bin_split_minus: "0 < n \<Longrightarrow> bin_split (Suc (n - 1)) w = bin_split n w"
  by auto

lemmas bin_split_minus_simp =
  bin_split.Suc [THEN [2] bin_split_minus [symmetric, THEN trans]]

lemma bin_split_pred_simp [simp]:
  "(0::nat) < numeral bin \<Longrightarrow>
    bin_split (numeral bin) w =
      (let (w1, w2) = bin_split (numeral bin - 1) (bin_rest w)
       in (w1, w2 BIT bin_last w))"
  by (simp only: bin_split_minus_simp)

lemma bin_rsplit_aux_simp_alt:
  "bin_rsplit_aux n m c bs =
    (if m = 0 \<or> n = 0 then bs
     else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
  apply (simp add: bin_rsplit_aux.simps [of n m c bs])
  apply (subst rsplit_aux_alts)
  apply (simp add: bin_rsplit_def)
  done

lemmas bin_rsplit_simp_alt =
  trans [OF bin_rsplit_def bin_rsplit_aux_simp_alt]

lemmas bthrs = bin_rsplit_simp_alt [THEN [2] trans]

lemma bin_rsplit_size_sign' [rule_format]:
  "n > 0 \<Longrightarrow> rev sw = bin_rsplit n (nw, w) \<Longrightarrow> \<forall>v\<in>set sw. bintrunc n v = v"
  apply (induct sw arbitrary: nw w)
   apply clarsimp
  apply clarsimp
  apply (drule bthrs)
  apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
  apply clarify
  apply (drule split_bintrunc)
  apply simp
  done

lemmas bin_rsplit_size_sign = bin_rsplit_size_sign' [OF asm_rl
  rev_rev_ident [THEN trans] set_rev [THEN equalityD2 [THEN subsetD]]]

lemma bin_nth_rsplit [rule_format] :
  "n > 0 \<Longrightarrow> m < n \<Longrightarrow>
    \<forall>w k nw.
      rev sw = bin_rsplit n (nw, w) \<longrightarrow>
      k < size sw \<longrightarrow> bin_nth (sw ! k) m = bin_nth w (k * n + m)"
  apply (induct sw)
   apply clarsimp
  apply clarsimp
  apply (drule bthrs)
  apply (simp (no_asm_use) add: Let_def split: prod.split_asm if_split_asm)
  apply clarify
  apply (erule allE, erule impE, erule exI)
  apply (case_tac k)
   apply clarsimp
   prefer 2
   apply clarsimp
   apply (erule allE)
   apply (erule (1) impE)
   apply (drule bin_nth_split, erule conjE, erule allE, erule trans, simp add: ac_simps)+
  done

lemma bin_rsplit_all: "0 < nw \<Longrightarrow> nw \<le> n \<Longrightarrow> bin_rsplit n (nw, w) = [bintrunc n w]"
  by (auto simp: bin_rsplit_def rsplit_aux_simp2ls split: prod.split dest!: split_bintrunc)

lemma bin_rsplit_l [rule_format]:
  "\<forall>bin. bin_rsplitl n (m, bin) = bin_rsplit n (m, bintrunc m bin)"
  apply (rule_tac a = "m" in wf_less_than [THEN wf_induct])
  apply (simp (no_asm) add: bin_rsplitl_def bin_rsplit_def)
  apply (rule allI)
  apply (subst bin_rsplitl_aux.simps)
  apply (subst bin_rsplit_aux.simps)
  apply (clarsimp simp: Let_def split: prod.split)
  apply (drule bin_split_trunc)
  apply (drule sym [THEN trans], assumption)
  apply (subst rsplit_aux_alts(1))
  apply (subst rsplit_aux_alts(2))
  apply clarsimp
  unfolding bin_rsplit_def bin_rsplitl_def
  apply simp
  done

lemma bin_rsplit_rcat [rule_format]:
  "n > 0 \<longrightarrow> bin_rsplit n (n * size ws, bin_rcat n ws) = map (bintrunc n) ws"
  apply (unfold bin_rsplit_def bin_rcat_def)
  apply (rule_tac xs = ws in rev_induct)
   apply clarsimp
  apply clarsimp
  apply (subst rsplit_aux_alts)
  unfolding bin_split_cat
  apply simp
  done

lemma bin_rsplit_aux_len_le [rule_format] :
  "\<forall>ws m. n \<noteq> 0 \<longrightarrow> ws = bin_rsplit_aux n nw w bs \<longrightarrow>
    length ws \<le> m \<longleftrightarrow> nw + length bs * n \<le> m * n"
proof -
  have *: R
    if d: "i \<le> j \<or> m < j'"
    and R1: "i * k \<le> j * k \<Longrightarrow> R"
    and R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R"
    for i j j' k k' m :: nat and R
    using d
    apply safe
    apply (rule R1, erule mult_le_mono1)
    apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]])
    done
  have **: "0 < sc \<Longrightarrow> sc - n + (n + lb * n) \<le> m * n \<longleftrightarrow> sc + lb * n \<le> m * n"
    for sc m n lb :: nat
    apply safe
     apply arith
    apply (case_tac "sc \<ge> n")
     apply arith
    apply (insert linorder_le_less_linear [of m lb])
    apply (erule_tac k=n and k'=n in *)
     apply arith
    apply simp
    done
  show ?thesis
    apply (induct n nw w bs rule: bin_rsplit_aux.induct)
    apply (subst bin_rsplit_aux.simps)
    apply (simp add: ** Let_def split: prod.split)
    done
qed

lemma bin_rsplit_len_le: "n \<noteq> 0 \<longrightarrow> ws = bin_rsplit n (nw, w) \<longrightarrow> length ws \<le> m \<longleftrightarrow> nw \<le> m * n"
  by (auto simp: bin_rsplit_def bin_rsplit_aux_len_le)

lemma bin_rsplit_aux_len:
  "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit_aux n nw w cs) = (nw + n - 1) div n + length cs"
  apply (induct n nw w cs rule: bin_rsplit_aux.induct)
  apply (subst bin_rsplit_aux.simps)
  apply (clarsimp simp: Let_def split: prod.split)
  apply (erule thin_rl)
  apply (case_tac m)
   apply simp
  apply (case_tac "m \<le> n")
   apply (auto simp add: div_add_self2)
  done

lemma bin_rsplit_len: "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
  by (auto simp: bin_rsplit_def bin_rsplit_aux_len)

lemma bin_rsplit_aux_len_indep:
  "n \<noteq> 0 \<Longrightarrow> length bs = length cs \<Longrightarrow>
    length (bin_rsplit_aux n nw v bs) =
    length (bin_rsplit_aux n nw w cs)"
proof (induct n nw w cs arbitrary: v bs rule: bin_rsplit_aux.induct)
  case (1 n m w cs v bs)
  show ?case
  proof (cases "m = 0")
    case True
    with \<open>length bs = length cs\<close> show ?thesis by simp
  next
    case False
    from "1.hyps" \<open>m \<noteq> 0\<close> \<open>n \<noteq> 0\<close>
    have hyp: "\<And>v bs. length bs = Suc (length cs) \<Longrightarrow>
      length (bin_rsplit_aux n (m - n) v bs) =
      length (bin_rsplit_aux n (m - n) (fst (bin_split n w)) (snd (bin_split n w) # cs))"
      by auto
    from \<open>length bs = length cs\<close> \<open>n \<noteq> 0\<close> show ?thesis
      by (auto simp add: bin_rsplit_aux_simp_alt Let_def bin_rsplit_len split: prod.split)
  qed
qed

lemma bin_rsplit_len_indep:
  "n \<noteq> 0 \<Longrightarrow> length (bin_rsplit n (nw, v)) = length (bin_rsplit n (nw, w))"
  apply (unfold bin_rsplit_def)
  apply (simp (no_asm))
  apply (erule bin_rsplit_aux_len_indep)
  apply (rule refl)
  done


text \<open>Even more bit operations\<close>

instantiation int :: bitss
begin

definition [iff]: "i !! n \<longleftrightarrow> bin_nth i n"

definition "lsb i = i !! 0" for i :: int

definition "set_bit i n b = bin_sc n b i"

definition
  "set_bits f =
    (if \<exists>n. \<forall>n'\<ge>n. \<not> f n' then
      let n = LEAST n. \<forall>n'\<ge>n. \<not> f n'
      in bl_to_bin (rev (map f [0..<n]))
     else if \<exists>n. \<forall>n'\<ge>n. f n' then
      let n = LEAST n. \<forall>n'\<ge>n. f n'
      in sbintrunc n (bl_to_bin (True # rev (map f [0..<n])))
     else 0 :: int)"

definition "shiftl x n = x * 2 ^ n" for x :: int

definition "shiftr x n = x div 2 ^ n" for x :: int

definition "msb x \<longleftrightarrow> x < 0" for x :: int

instance ..

end

end