Theory Bool_List_Representation

theory Bool_List_Representation
imports Bit_Int
(* 
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.
*)


header "Bool lists and integers"

theory Bool_List_Representation
imports Bit_Int
begin

definition map2 :: "('a => 'b => 'c) => 'a list => 'b list => 'c list"
where
"map2 f as bs = map (split 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 (if b then 1 else 0))"


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 = 1) # 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 = 1) # 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 (induct n arbitrary: w bs) auto

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 split add : bit.split)
apply clarsimp
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 safe
apply (drule meta_spec, erule xtrans(8) [rotated], simp add: Bit_def)+
done

lemma bl_to_bin_lt2p: "bl_to_bin bs < (2 ^ length bs)"
apply (unfold bl_to_bin_def)
apply (rule xtrans(1))
prefer 2
apply (rule bl_to_bin_lt2p_aux)
apply simp
done

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 safe
apply (drule meta_spec, erule order_trans [rotated],
simp add: Bit_B0_2t Bit_B1_2t algebra_simps)+
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) = 1)"
by (induct xs arbitrary: w) auto

lemma last_bin_last:
"size xs > 0 ==> last xs = (bin_last (bl_to_bin xs) = 1)"
unfolding bl_to_bin_def by (erule last_bin_last')

lemma bin_last_last:
"bin_last w = (if last (bin_to_bl (Suc n) w) then 1 else 0)"
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 (case_tac ba, safe, simp_all)+
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
apply (case_tac b)
apply (case_tac ba, safe, simp_all)+
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 clarify (induct xs, 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 add_ac)
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 If b 1 0))"
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_max.inf_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 = split_if [where P = "op = l", THEN iffD1, THEN conjunct1, THEN mp] for l
lemmas th_if_simp2 = split_if [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)
apply auto
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)
apply auto
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 v)
apply clarsimp
apply clarsimp
apply (drule bthrs)
apply (simp (no_asm_use) add: Let_def split: prod.split_asm split_if_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 split_if_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 : add_ac)+
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"

apply (induct n nw w bs rule: bin_rsplit_aux.induct)
apply (subst bin_rsplit_aux.simps)
apply (simp add: lrlem Let_def split: prod.split)
done

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

end