(*
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 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 (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 \<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_def: "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_def : "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 [] => 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 \<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 ==> 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 ==> 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 \<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 add : 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 ==> (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: "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 (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 ~= [] \<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)"
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 \<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 ==> 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)"
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 \<and>) (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 \<or>) (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 (\<lambda>x y. x \<noteq> 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) \<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 ==> 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 \<Longrightarrow>
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 \<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':
"!!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 \<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 ==> 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 \<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:
"(!!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 \<le> length xs \<Longrightarrow> 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 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 = 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 \<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)"
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] :
"\<lbrakk>n > 0; rev sw = bin_rsplit n (nw, w)\<rbrakk> \<Longrightarrow>
(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] :
"\<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 -
{ fix i j j' k k' m :: nat and R
assume d: "(i::nat) \<le> j \<or> m < j'"
assume R1: "i * k \<le> j * k \<Longrightarrow> R"
assume R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> 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 \<Longrightarrow> sc - n + (n + lb * n) \<le> m * n \<longleftrightarrow> sc + lb * n \<le> 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 \<noteq> 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 \<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 <= n")
apply auto
done
lemma bin_rsplit_len:
"n\<noteq>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 \<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 then show ?thesis using `length bs = length cs` by simp
next
case False
from "1.hyps" `m \<noteq> 0` `n \<noteq> 0` 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
show ?thesis using `length bs = length cs` `n \<noteq> 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\<noteq>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