Theory Bool_List_Representation

(* 
  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 Main Bits_Int
begin

definition map2 :: "('a ⇒ 'b ⇒ 'c) ⇒ 'a list ⇒ 'b list ⇒ 'c list"
where
  "map2 f as bs = map (case_prod f) (zip as bs)"

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

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

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


subsection ‹Operations on lists of booleans›

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 (w BIT b)"

definition bl_to_bin :: "bool list ⇒ int"
where
  bl_to_bin_def: "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 (bin_rest w) ((bin_last w) # bl)"

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

primrec bl_of_nth :: "nat ⇒ (nat ⇒ bool) ⇒ 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 ⇒ 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)"


subsection "Arithmetic in terms of bool lists"

text ‹
  Arithmetic operations in terms of the reversed bool list,
  assuming input list(s) the same length, and don't extend them. 
›

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 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 ⟹ 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_Bit_minus_simp [simp]:
  "0 < n ==> 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 ==> 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 ==> 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 ‹Link between bin and bool list.›

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" 
  unfolding bin_to_bl_def by (simp add : bin_to_bl_aux_append)

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

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" 
  unfolding bin_to_bl_def by (simp add : 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 add : bl_to_bin_def)

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

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"
  unfolding bl_to_bin_def by auto

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

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"
  unfolding bin_to_bl_def by (simp add: 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"
  unfolding bin_to_bl_def by (simp add: bin_to_bl_minus1_aux)

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

lemma bin_to_bl_trunc [simp]:
  "n <= m ==> 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 [simp]: "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"
  unfolding bl_to_bin_def by (simp add : 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 & rev bl ! n | n >= length bl & 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 & rev bl ! n)"
  unfolding bl_to_bin_def by (simp add : bin_nth_of_bl_aux)

lemma bin_nth_bl: "n < m ⟹ 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 ⟹ rev xs ! n = xs ! (length xs - 1 - n)"
  apply (induct xs)
   apply simp
  apply (clarsimp simp add : nth_append nth.simps split add : nat.split)
  apply (rule_tac f = "λn. xs ! n" in arg_cong)
  apply arith
  done

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

lemma nth_bin_to_bl_aux:
  "n < m + length bl ⟹ (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 ==> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)"
  unfolding bin_to_bl_def by (simp add : 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 (Cons b bs) with bl_to_bin_lt2p_aux[where w=1]
  show ?case unfolding bl_to_bin_def by simp
qed simp

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)"
  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 >= 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 ~= [] ⟹
    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)"
  apply (unfold bl_to_bin_def)
  apply (cases bl)
   apply (auto simp add: butlast_rest_bl2bin_aux)
  done

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 Suc Cons show ?thesis by simp
  qed
qed

lemma trunc_bl2bin: 
  "bintrunc m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
  unfolding bl_to_bin_def by (simp add : 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 <= 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, clarsimp)
  apply clarsimp
  apply (simp only: bin_nth.Suc [symmetric] add_Suc)
  done

lemma take_rest_power_bin:
  "m <= n ==> 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 ==> hd (butlast xs) = hd xs"
  by (cases xs) auto

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

lemma last_bin_last: 
  "size xs > 0 ==> last xs ⟷ bin_last (bl_to_bin xs)" 
  unfolding bl_to_bin_def by (erule last_bin_last')
  
lemma bin_last_last: 
  "bin_last w ⟷ last (bin_to_bl (Suc n) w)" 
  apply (unfold bin_to_bl_def)
  apply simp
  apply (auto simp add: bin_to_bl_aux_alt)
  done

(** links between bit-wise operations and operations on bool lists **)
    
lemma bl_xor_aux_bin:
  "map2 (%x 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 (%x y. x ~= 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 | ) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = 
    bin_to_bl_aux n (v OR w) (map2 (op | ) 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 & ) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = 
    bin_to_bl_aux n (v AND w) (map2 (op & ) 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)"
  apply (induct n arbitrary: w cs)
   apply clarsimp
  apply clarsimp
  done

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

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

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

lemma bl_xor_bin:
  "map2 (λx y. x ≠ y) (bin_to_bl n v) (bin_to_bl n w) = bin_to_bl n (v XOR w)"
  unfolding bin_to_bl_def by (simp only: 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 <= n", simp)
  apply (case_tac "m - n", simp)
  apply simp
  apply (rule_tac f = "%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"
  unfolding bin_to_bl_def by (simp add : 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, 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)"
  apply (induct n arbitrary: w bs)
   apply (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1)
  apply simp
  done

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)
  done

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)"
  by (auto elim: bin_split_take)
  
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, 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':
  "!!w n.  n = k + m ==> take k (takefill fill n w) = takefill fill k w"
  by (induct k) (auto split add : list.split) 

lemma drop_takefill:
  "!!w. drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
  by (induct k) (auto split add : 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 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 ==> 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 = "%n. 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)

(* links with function bl_to_bin *)

lemma bl_to_bin_aux_cat: 
  "!!nv v. bl_to_bin_aux bs (bin_cat w nv v) = 
    bin_cat w (nv + length bs) (bl_to_bin_aux bs v)"
  apply (induct bs)
   apply simp
  apply (simp add: bin_cat_Suc_Bit [symmetric] del: bin_cat.simps)
  done

lemma bin_to_bl_aux_cat: 
  "!!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"]
  unfolding bl_to_bin_def [symmetric] by simp

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)"
  unfolding bin_to_bl_def by (simp add: 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)

(* bl_to_bin_app_cat_alt and bl_to_bin_app_cat are easily interderivable *)
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

(* function bl_of_nth *)
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 ⟹ 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: 
  "(!!k. k < n ==> f k = g k) ==> bl_of_nth n f = bl_of_nth n g"
  by (induct n)  auto

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, 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 = "%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:
  "!!cl. length (rbl_add bl cl) = length bl"
  by (induct bl) (auto simp: Let_def size_rbl_succ)

lemma size_rbl_mult: 
  "!!cl. length (rbl_mult bl cl) = length bl"
  by (induct bl) (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 (induct n arbitrary: bin, simp)
  apply (unfold bin_to_bl_def)
  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 (induct n arbitrary: bin, simp)
  apply (unfold bin_to_bl_def)
  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: 
  "!!bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = 
    rev (bin_to_bl n (bina + binb))"
  apply (induct n, 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: rbl_succ bin_to_bl_aux_alt Let_def ac_simps)
  done

lemma rbl_add_app2: 
  "!!blb. length blb >= length bla ==> 
    rbl_add bla (blb @ blc) = rbl_add bla blb"
  apply (induct bla, simp)
  apply clarsimp
  apply (case_tac blb, clarsimp)
  apply (clarsimp simp: Let_def)
  done

lemma rbl_add_take2: 
  "!!blb. length blb >= length bla ==> 
    rbl_add bla (take (length bla) blb) = rbl_add bla blb"
  apply (induct bla, simp)
  apply clarsimp
  apply (case_tac blb, clarsimp)
  apply (clarsimp simp: Let_def)
  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_app2:
  "!!blb. length blb >= length bla ==> 
    rbl_mult bla (blb @ blc) = rbl_mult bla blb"
  apply (induct bla, 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_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) (x BIT b))"
  apply (unfold bin_to_bl_def)
  apply simp
  apply (simp add: bin_to_bl_aux_alt)
  done

lemma rbl_mult: "!!bina binb. 
    rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = 
    rev (bin_to_bl n (bina * binb))"
  apply (induct n)
   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)) = 
    (ALL 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)))"
  apply (auto simp add: Let_def)
   apply (case_tac [!] "y")
     apply auto
  done

lemma rbl_mult_split: 
  "P (rbl_mult (y # ys) xs) = 
    (ALL ws. length ws = Suc (length ys) --> ws = False # rbl_mult ys xs --> 
    (y --> P (rbl_add ws xs)) & (~ y -->  P ws))"
  by (clarsimp simp add : Let_def)
  

subsection "Repeated splitting or concatenation"

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

lemma bin_cat_foldl_lem:
  "foldl (%u. bin_cat u n) x xs = 
    bin_cat x (size xs * n) (foldl (%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 ==> 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 ⟹
  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 ∨ n = 0 
   then bs
   else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
  unfolding bin_rsplit_aux.simps [of n m c bs]
  apply simp
  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; rev sw = bin_rsplit n (nw, w)⟧ ⟹ 
    (ALL v: 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 ==> m < n ==> (ALL w k nw. rev sw = bin_rsplit n (nw, w) --> 
       k < size sw --> 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 ==> nw <= n ==> bin_rsplit n (nw, w) = [bintrunc n w]"
  unfolding bin_rsplit_def
  by (clarsimp dest!: split_bintrunc simp: rsplit_aux_simp2ls split: prod.split)

lemma bin_rsplit_l [rule_format] :
  "ALL 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 --> 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] :
  "∀ws m. n ≠ 0 ⟶ ws = bin_rsplit_aux n nw w bs ⟶
    length ws ≤ m ⟷ nw + length bs * n ≤ m * n"
proof -
  { fix i j j' k k' m :: nat and R
    assume d: "(i::nat) ≤ j ∨ m < j'"
    assume R1: "i * k ≤ j * k ⟹ R"
    assume R2: "Suc m * k' ≤ j' * k' ⟹ R"
    have "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
  } note A = this
  { fix sc m n lb :: nat
    have "(0::nat) < sc ⟹ sc - n + (n + lb * n) ≤ m * n ⟷ sc + lb * n ≤ m * n"
      apply safe
       apply arith
      apply (case_tac "sc >= n")
       apply arith
      apply (insert linorder_le_less_linear [of m lb])
      apply (erule_tac k2=n and k'2=n in A)
       apply arith
      apply simp
      done
  } note B = this
  show ?thesis
    apply (induct n nw w bs rule: bin_rsplit_aux.induct)
    apply (subst bin_rsplit_aux.simps)
    apply (simp add: B Let_def split: prod.split)
    done
qed

lemma bin_rsplit_len_le: 
  "n ≠ 0 --> ws = bin_rsplit n (nw, w) --> (length ws <= m) = (nw <= m * n)"
  unfolding bin_rsplit_def by (clarsimp simp add : bin_rsplit_aux_len_le)
 
lemma bin_rsplit_aux_len:
  "n ≠ 0 ⟹ 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 <= n")
  apply auto
  done

lemma bin_rsplit_len: 
  "n≠0 ==> length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
  unfolding bin_rsplit_def by (clarsimp simp add : bin_rsplit_aux_len)

lemma bin_rsplit_aux_len_indep:
  "n ≠ 0 ⟹ length bs = length cs ⟹
    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 then show ?thesis using ‹length bs = length cs› by simp
  next
    case False
    from "1.hyps" ‹m ≠ 0› ‹n ≠ 0› have hyp: "⋀v bs. length bs = Suc (length cs) ⟹
      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
    show ?thesis using ‹length bs = length cs› ‹n ≠ 0›
      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≠0 ==> 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 ‹Even more bit operations›

instantiation int :: bitss
begin

definition [iff]:
  "i !! n ⟷ bin_nth i n"

definition
  "lsb i = (i :: int) !! 0"

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

definition
  "set_bits f =
  (if ∃n. ∀n'≥n. ¬ f n' then 
     let n = LEAST n. ∀n'≥n. ¬ f n'
     in bl_to_bin (rev (map f [0..<n]))
   else if ∃n. ∀n'≥n. f n' then
     let n = LEAST n. ∀n'≥n. f n'
     in sbintrunc n (bl_to_bin (True # rev (map f [0..<n])))
   else 0 :: int)"

definition
  "shiftl x n = (x :: int) * 2 ^ n"

definition
  "shiftr x n = (x :: int) div 2 ^ n"

definition
  "msb x ⟷ (x :: int) < 0"

instance ..

end

end