--- a/src/HOL/Word/BinBoolList.thy Wed Jun 30 21:29:58 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1139 +0,0 @@
-(*
- Author: Jeremy Dawson, NICTA
-
- contains 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 BinBoolList
-imports BinOperations
-begin
-
-subsection "Arithmetic in terms of bool lists"
-
-(* 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_Pls_minus_simp [simp]:
- "0 < n ==> bin_to_bl_aux n Int.Pls bl =
- bin_to_bl_aux (n - 1) Int.Pls (False # bl)"
- by (cases n) auto
-
-lemma bin_to_bl_aux_Min_minus_simp [simp]:
- "0 < n ==> bin_to_bl_aux n Int.Min bl =
- bin_to_bl_aux (n - 1) Int.Min (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 = bit.B1) # bl)"
- by (cases n) auto
-
-lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
- "0 < n ==> bin_to_bl_aux n (Int.Bit0 w) bl =
- bin_to_bl_aux (n - 1) w (False # bl)"
- by (cases n) auto
-
-lemma bin_to_bl_aux_Bit1_minus_simp [simp]:
- "0 < n ==> bin_to_bl_aux n (Int.Bit1 w) bl =
- bin_to_bl_aux (n - 1) w (True # bl)"
- by (cases n) auto
-
-(** 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: "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: "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: "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: "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
-
-declare
- bin_to_bl_0 [simp]
- size_bin_to_bl [simp]
- bin_bl_bin [simp]
- bl_bin_bl [simp]
-
-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: "bl_to_bin (False # bl) = bl_to_bin bl"
- unfolding bl_to_bin_def by auto
-
-lemma bl_to_bin_Nil: "bl_to_bin [] = Int.Pls"
- unfolding bl_to_bin_def by auto
-
-lemma bin_to_bl_Pls_aux:
- "bin_to_bl_aux n Int.Pls bl = replicate n False @ bl"
- by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
-
-lemma bin_to_bl_Pls: "bin_to_bl n Int.Pls = replicate n False"
- unfolding bin_to_bl_def by (simp add : bin_to_bl_Pls_aux)
-
-lemma bin_to_bl_Min_aux [rule_format] :
- "ALL bl. bin_to_bl_aux n Int.Min bl = replicate n True @ bl"
- by (induct n) (auto simp: replicate_app_Cons_same)
-
-lemma bin_to_bl_Min: "bin_to_bl n Int.Min = replicate n True"
- unfolding bin_to_bl_def by (simp add : bin_to_bl_Min_aux)
-
-lemma bl_to_bin_rep_F:
- "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
- apply (simp add: bin_to_bl_Pls_aux [symmetric] bin_bl_bin')
- apply (simp add: bl_to_bin_def)
- done
-
-lemma bin_to_bl_trunc:
- "n <= m ==> bin_to_bl n (bintrunc m w) = bin_to_bl n w"
- by (auto intro: bl_to_bin_inj)
-
-declare
- bin_to_bl_trunc [simp]
- bl_to_bin_False [simp]
- bl_to_bin_Nil [simp]
-
-lemma bin_to_bl_aux_bintr [rule_format] :
- "ALL m bin bl. 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)
- apply clarsimp
- apply clarsimp
- apply (case_tac "m")
- apply (clarsimp simp: bin_to_bl_Pls_aux)
- apply (erule thin_rl)
- apply (induct_tac n)
- apply auto
- done
-
-lemmas bin_to_bl_bintr =
- bin_to_bl_aux_bintr [where bl = "[]", folded bin_to_bl_def]
-
-lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = Int.Pls"
- 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"
- unfolding bin_to_bl_def len_bin_to_bl_aux by auto
-
-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) = Int.Pls"
- 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) = Int.Min)"
- 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) = Int.Min)"
- unfolding bin_to_bl_def by (rule bl_sbin_sign_aux)
-
-lemma bin_nth_of_bl_aux [rule_format]:
- "\<forall>w. 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_tac bl)
- apply clarsimp
- apply clarsimp
- apply (cut_tac x=n and y="size list" 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 [rule_format] : "ALL m w. n < m -->
- bin_nth w n = nth (rev (bin_to_bl m w)) n"
- apply (induct n)
- 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 [rule_format] :
- "n < length xs --> rev xs ! n = xs ! (length xs - 1 - n)"
- apply (induct_tac "xs")
- apply simp
- apply (clarsimp simp add : nth_append nth.simps split add : nat.split)
- apply (rule_tac f = "%n. list ! n" in arg_cong)
- apply arith
- done
-
-lemmas nth_rev_alt = nth_rev [where xs = "rev ys", simplified, standard]
-
-lemma nth_bin_to_bl_aux [rule_format] :
- "ALL w n bl. 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)
- apply clarsimp
- apply clarsimp
- apply (case_tac w rule: bin_exhaust)
- apply clarsimp
- apply (case_tac "n - m")
- apply arith
- apply simp
- apply (rule_tac f = "%n. bl ! n" in arg_cong)
- apply arith
- 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 [rule_format]:
- "\<forall>w. bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
- apply (induct bs)
- apply clarsimp
- apply clarsimp
- apply safe
- apply (erule allE, erule xtr8 [rotated],
- simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
- done
-
-lemma bl_to_bin_lt2p: "bl_to_bin bs < (2 ^ length bs)"
- apply (unfold bl_to_bin_def)
- apply (rule xtr1)
- prefer 2
- apply (rule bl_to_bin_lt2p_aux)
- apply simp
- done
-
-lemma bl_to_bin_ge2p_aux [rule_format] :
- "\<forall>w. bl_to_bin_aux bs w >= w * (2 ^ length bs)"
- apply (induct bs)
- apply clarsimp
- apply clarsimp
- apply safe
- apply (erule allE, erule preorder_class.order_trans [rotated],
- simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
- done
-
-lemma bl_to_bin_ge0: "bl_to_bin bs >= 0"
- apply (unfold bl_to_bin_def)
- apply (rule xtr4)
- 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
-
-lemmas butlast_bin_rest = butlast_rest_bin
- [where w="bl_to_bin bl" and n="length bl", simplified, standard]
-
-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 [rule_format]:
- "ALL w. bintrunc m (bl_to_bin_aux bl w) =
- bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (m - length bl) w)"
- apply (induct_tac bl)
- apply clarsimp
- apply clarsimp
- apply safe
- apply (case_tac "m - size list")
- apply (simp add : diff_is_0_eq [THEN iffD1, THEN Suc_diff_le])
- apply simp
- apply (rule_tac f = "%nat. bl_to_bin_aux list (Int.Bit1 (bintrunc nat w))"
- in arg_cong)
- apply simp
- apply (case_tac "m - size list")
- apply (simp add: diff_is_0_eq [THEN iffD1, THEN Suc_diff_le])
- apply simp
- apply (rule_tac f = "%nat. bl_to_bin_aux list (Int.Bit0 (bintrunc nat w))"
- in arg_cong)
- apply simp
- done
-
-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)
-
-lemmas trunc_bl2bin_len [simp] =
- trunc_bl2bin [of "length bl" bl, simplified, standard]
-
-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 [rule_format] :
- "ALL n. bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
- apply (induct k, 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 = (bin_last (bl_to_bin_aux xs w) = bit.B1)"
- by (induct xs arbitrary: w) auto
-
-lemma last_bin_last:
- "size xs > 0 ==> last xs = (bin_last (bl_to_bin xs) = bit.B1)"
- 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 bit.B1 else bit.B0)"
- 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 map2_Nil [simp]: "map2 f [] ys = []"
- unfolding map2_def by auto
-
-lemma map2_Cons [simp]:
- "map2 f (x # xs) (y # ys) = f x y # map2 f xs ys"
- unfolding map2_def by auto
-
-lemma bl_xor_aux_bin [rule_format] : "ALL v w bs cs.
- 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_tac n)
- apply safe
- 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 [rule_format] : "ALL v w bs cs.
- 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_tac n)
- apply safe
- 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 [rule_format] : "ALL v w bs cs.
- 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_tac n)
- apply safe
- 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_not_aux_bin [rule_format] :
- "ALL w cs. map Not (bin_to_bl_aux n w cs) =
- bin_to_bl_aux n (NOT w) (map Not cs)"
- apply (induct n)
- apply clarsimp
- apply clarsimp
- apply (case_tac w rule: bin_exhaust)
- apply (case_tac b)
- apply auto
- done
-
-lemmas bl_not_bin = bl_not_aux_bin
- [where cs = "[]", unfolded bin_to_bl_def [symmetric] map.simps]
-
-lemmas bl_and_bin = bl_and_aux_bin [where bs="[]" and cs="[]",
- unfolded map2_Nil, folded bin_to_bl_def]
-
-lemmas bl_or_bin = bl_or_aux_bin [where bs="[]" and cs="[]",
- unfolded map2_Nil, folded bin_to_bl_def]
-
-lemmas bl_xor_bin = bl_xor_aux_bin [where bs="[]" and cs="[]",
- unfolded map2_Nil, folded bin_to_bl_def]
-
-lemma drop_bin2bl_aux [rule_format] :
- "ALL m bin bs. drop m (bin_to_bl_aux n bin bs) =
- bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
- apply (induct n, 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 [rule_format] :
- "ALL w bs. take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
- apply (induct m, 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 [rule_format] :
- "ALL w bs. take m (bin_to_bl_aux (m + n) w bs) =
- take m (bin_to_bl (m + n) w)"
- apply (induct n)
- apply clarify
- apply (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1)
- apply simp
- done
-
-lemma bin_split_take [rule_format] :
- "ALL b c. bin_split n c = (a, b) -->
- bin_to_bl m a = take m (bin_to_bl (m + n) c)"
- apply (induct n)
- apply clarsimp
- apply (clarsimp simp: Let_def split: ls_splits)
- 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 [rule_format] : "ALL m l. m < n -->
- takefill fill n l ! m = (if m < length l then l ! m else fill)"
- apply (induct n, clarsimp)
- apply clarsimp
- apply (case_tac m)
- apply (simp split: list.split)
- apply clarsimp
- apply (erule allE)+
- apply (erule (1) impE)
- apply (simp split: list.split)
- done
-
-lemma takefill_alt [rule_format] :
- "ALL l. takefill fill n l = take n l @ replicate (n - length l) fill"
- by (induct n) (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' [rule_format] :
- "ALL l n. n = m + k --> takefill x m (takefill x n l) = takefill x m l"
- by (induct m) (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)
-
-lemmas bl_bin_bl_rep_drop =
- bl_bin_bl_rtf [simplified takefill_alt,
- simplified, simplified rev_take, simplified]
-
-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], standard]
-
-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], standard]
-
-lemmas takefill_pred_simps [simp] =
- takefill_minus_simps [where n="number_of bin", simplified nobm1, standard]
-
-(* 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
-
-lemmas bl_to_bin_aux_alt =
- bl_to_bin_aux_cat [where nv = "0" and v = "Int.Pls",
- simplified bl_to_bin_def [symmetric], simplified]
-
-lemmas bin_to_bl_cat =
- bin_to_bl_aux_cat [where bs = "[]", folded bin_to_bl_def]
-
-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]
-
-lemmas bl_to_bin_app_cat = bl_to_bin_aux_app_cat
- [where w = "Int.Pls", folded bl_to_bin_def]
-
-lemmas bin_to_bl_cat_app = bin_to_bl_aux_cat_app
- [where bs = "[]", folded bin_to_bl_def]
-
-(* 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)) =
- Int.succ (bl_to_bin (replicate n True))"
- apply (unfold bl_to_bin_def)
- apply (induct n)
- apply simp
- apply (simp only: Suc_eq_plus1 replicate_add
- append_Cons [symmetric] bl_to_bin_aux_append)
- apply simp
- done
-
-(* 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 [rule_format] : "ALL xs.
- length xs >= n --> bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs";
- apply (induct n, 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
-
-lemmas bl_of_nth_nth [simp] = order_refl [THEN bl_of_nth_nth_le, simplified]
-
-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:
- "!!bin. rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (Int.pred bin))"
- apply (induct n, 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:
- "!!bin. rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (Int.succ bin))"
- apply (induct n, 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 succ_def 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 bit.B1 bit.B0))"
- 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)
-
-lemma and_len: "xs = ys ==> xs = ys & length xs = length ys"
- by auto
-
-lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
- by auto
-
-lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
- by auto
-
-lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
- by auto
-
-lemma if_Not_x: "(if p then ~ x else x) = (p = (~ x))"
- by auto
-
-lemma if_x_Not: "(if p then x else ~ x) = (p = x)"
- by auto
-
-lemma if_same_and: "(If p x y & If p u v) = (if p then x & u else y & v)"
- by auto
-
-lemma if_same_eq: "(If p x y = (If p u v)) = (if p then x = (u) else y = (v))"
- by auto
-
-lemma if_same_eq_not:
- "(If p x y = (~ If p u v)) = (if p then x = (~u) else y = (~v))"
- by auto
-
-(* note - if_Cons can cause blowup in the size, if p is complex,
- so make a simproc *)
-lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
- by auto
-
-lemma if_single:
- "(if xc then [xab] else [an]) = [if xc then xab else an]"
- by auto
-
-lemma if_bool_simps:
- "If p True y = (p | y) & If p False y = (~p & y) &
- If p y True = (p --> y) & If p y False = (p & y)"
- by auto
-
-lemmas if_simps = if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps
-
-lemmas seqr = eq_reflection [where x = "size w", standard]
-
-lemmas tl_Nil = tl.simps (1)
-lemmas tl_Cons = tl.simps (2)
-
-
-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 [rule_format] :
- "ALL x. 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)
- apply simp
- apply clarify
- apply (simp (no_asm))
- apply (frule asm_rl)
- apply (drule spec)
- apply (erule trans)
- apply (drule_tac x = "bin_cat y n a" in 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, standard]
-lemmas th_if_simp2 = split_if [where P = "op = l",
- THEN iffD1, THEN conjunct2, THEN mp, standard]
-
-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: ls_splits)
- 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: ls_splits)
- 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], standard]
-
-lemma bin_split_pred_simp [simp]:
- "(0::nat) < number_of bin \<Longrightarrow>
- bin_split (number_of bin) w =
- (let (w1, w2) = bin_split (number_of (Int.pred bin)) (bin_rest w)
- in (w1, w2 BIT bin_last w))"
- by (simp only: nobm1 bin_split_minus_simp)
-
-declare bin_split_pred_simp [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, standard]
-
-lemmas bthrs = bin_rsplit_simp_alt [THEN [2] trans]
-
-lemma bin_rsplit_size_sign' [rule_format] :
- "n > 0 ==> (ALL nw w. rev sw = bin_rsplit n (nw, w) -->
- (ALL v: set sw. bintrunc n v = v))"
- apply (induct sw)
- apply clarsimp
- apply clarsimp
- apply (drule bthrs)
- apply (simp (no_asm_use) add: Let_def split: ls_splits)
- apply clarify
- apply (erule impE, rule exI, erule exI)
- 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]],
- standard]
-
-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: ls_splits)
- 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: ls_splits)
-
-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: ls_splits)
- 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"
- apply (induct n nw w bs rule: bin_rsplit_aux.induct)
- apply (subst bin_rsplit_aux.simps)
- apply (simp add: lrlem Let_def split: ls_splits)
- done
-
-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 [rule_format] :
- "n\<noteq>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: ls_splits)
- 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: ls_splits)
- 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