
diff -r 17e1085d07b2 -r df789294c77a src/HOL/Word/BinBoolList.thy
--- a/src/HOL/Word/BinBoolList.thy	Wed Jun 30 16:41:03 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1174 +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 {* 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 (if b then 1 else 0))"
-
-definition bl_to_bin :: "bool list \<Rightarrow> int" where
-  bl_to_bin_def : "bl_to_bin bs = bl_to_bin_aux bs Int.Pls"
-
-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 = 1) # 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)"
-
-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]: "map2 f [] ys = []"
-  unfolding map2_def by auto
-
-lemma map2_Nil2 [simp]: "map2 f xs [] = []"
-  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
-
-
-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 = 1) # 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) = 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 [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 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)
-  
-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




