src/HOL/Word/Bool_List_Representation.thy
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(* 
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  Author: Jeremy Dawson, NICTA
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  Theorems to do with integers, expressed using Pls, Min, BIT,
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  theorems linking them to lists of booleans, and repeated splitting 
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  and concatenation.
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*) 
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header "Bool lists and integers"
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theory Bool_List_Representation
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imports Bit_Int
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begin
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subsection {* Operations on lists of booleans *}
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primrec bl_to_bin_aux :: "bool list \<Rightarrow> int \<Rightarrow> int" where
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  Nil: "bl_to_bin_aux [] w = w"
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  | Cons: "bl_to_bin_aux (b # bs) w = 
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      bl_to_bin_aux bs (w BIT (if b then 1 else 0))"
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definition bl_to_bin :: "bool list \<Rightarrow> int" where
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  bl_to_bin_def : "bl_to_bin bs = bl_to_bin_aux bs Int.Pls"
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lemma [code]:
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  "bl_to_bin bs = bl_to_bin_aux bs 0"
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  by (simp add: bl_to_bin_def Pls_def)
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primrec bin_to_bl_aux :: "nat \<Rightarrow> int \<Rightarrow> bool list \<Rightarrow> bool list" where
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  Z: "bin_to_bl_aux 0 w bl = bl"
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  | Suc: "bin_to_bl_aux (Suc n) w bl =
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      bin_to_bl_aux n (bin_rest w) ((bin_last w = 1) # bl)"
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definition bin_to_bl :: "nat \<Rightarrow> int \<Rightarrow> bool list" where
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  bin_to_bl_def : "bin_to_bl n w = bin_to_bl_aux n w []"
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primrec bl_of_nth :: "nat \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool list" where
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  Suc: "bl_of_nth (Suc n) f = f n # bl_of_nth n f"
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  | Z: "bl_of_nth 0 f = []"
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primrec takefill :: "'a \<Rightarrow> nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  Z: "takefill fill 0 xs = []"
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  | Suc: "takefill fill (Suc n) xs = (
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      case xs of [] => fill # takefill fill n xs
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        | y # ys => y # takefill fill n ys)"
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definition map2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list" where
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  "map2 f as bs = map (split f) (zip as bs)"
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lemma map2_Nil [simp]: "map2 f [] ys = []"
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  unfolding map2_def by auto
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lemma map2_Nil2 [simp]: "map2 f xs [] = []"
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  unfolding map2_def by auto
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lemma map2_Cons [simp]:
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  "map2 f (x # xs) (y # ys) = f x y # map2 f xs ys"
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  unfolding map2_def by auto
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subsection "Arithmetic in terms of bool lists"
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text {* 
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  Arithmetic operations in terms of the reversed bool list,
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  assuming input list(s) the same length, and don't extend them. 
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*}
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primrec rbl_succ :: "bool list => bool list" where
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  Nil: "rbl_succ Nil = Nil"
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  | Cons: "rbl_succ (x # xs) = (if x then False # rbl_succ xs else True # xs)"
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primrec rbl_pred :: "bool list => bool list" where
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  Nil: "rbl_pred Nil = Nil"
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  | Cons: "rbl_pred (x # xs) = (if x then False # xs else True # rbl_pred xs)"
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primrec rbl_add :: "bool list => bool list => bool list" where
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  -- "result is length of first arg, second arg may be longer"
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  Nil: "rbl_add Nil x = Nil"
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  | Cons: "rbl_add (y # ys) x = (let ws = rbl_add ys (tl x) in 
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    (y ~= hd x) # (if hd x & y then rbl_succ ws else ws))"
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primrec rbl_mult :: "bool list => bool list => bool list" where
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  -- "result is length of first arg, second arg may be longer"
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  Nil: "rbl_mult Nil x = Nil"
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  | Cons: "rbl_mult (y # ys) x = (let ws = False # rbl_mult ys x in 
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    if y then rbl_add ws x else ws)"
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lemma butlast_power:
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  "(butlast ^^ n) bl = take (length bl - n) bl"
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  by (induct n) (auto simp: butlast_take)
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lemma bin_to_bl_aux_Pls_minus_simp [simp]:
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  "0 < n ==> bin_to_bl_aux n Int.Pls bl = 
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    bin_to_bl_aux (n - 1) Int.Pls (False # bl)"
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  by (cases n) auto
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lemma bin_to_bl_aux_Min_minus_simp [simp]:
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  "0 < n ==> bin_to_bl_aux n Int.Min bl = 
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    bin_to_bl_aux (n - 1) Int.Min (True # bl)"
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  by (cases n) auto
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lemma bin_to_bl_aux_Bit_minus_simp [simp]:
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  "0 < n ==> bin_to_bl_aux n (w BIT b) bl = 
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    bin_to_bl_aux (n - 1) w ((b = 1) # bl)"
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  by (cases n) auto
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lemma bin_to_bl_aux_Bit0_minus_simp [simp]:
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  "0 < n ==> bin_to_bl_aux n (Int.Bit0 w) bl = 
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    bin_to_bl_aux (n - 1) w (False # bl)"
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  by (cases n) auto
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lemma bin_to_bl_aux_Bit1_minus_simp [simp]:
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  "0 < n ==> bin_to_bl_aux n (Int.Bit1 w) bl = 
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    bin_to_bl_aux (n - 1) w (True # bl)"
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  by (cases n) auto
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text {* Link between bin and bool list. *}
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lemma bl_to_bin_aux_append: 
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  "bl_to_bin_aux (bs @ cs) w = bl_to_bin_aux cs (bl_to_bin_aux bs w)"
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  by (induct bs arbitrary: w) auto
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lemma bin_to_bl_aux_append: 
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  "bin_to_bl_aux n w bs @ cs = bin_to_bl_aux n w (bs @ cs)"
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  by (induct n arbitrary: w bs) auto
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lemma bl_to_bin_append: 
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  "bl_to_bin (bs @ cs) = bl_to_bin_aux cs (bl_to_bin bs)"
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  unfolding bl_to_bin_def by (rule bl_to_bin_aux_append)
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lemma bin_to_bl_aux_alt: 
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  "bin_to_bl_aux n w bs = bin_to_bl n w @ bs" 
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  unfolding bin_to_bl_def by (simp add : bin_to_bl_aux_append)
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lemma bin_to_bl_0: "bin_to_bl 0 bs = []"
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  unfolding bin_to_bl_def by auto
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lemma size_bin_to_bl_aux: 
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  "size (bin_to_bl_aux n w bs) = n + length bs"
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  by (induct n arbitrary: w bs) auto
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lemma size_bin_to_bl: "size (bin_to_bl n w) = n" 
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  unfolding bin_to_bl_def by (simp add : size_bin_to_bl_aux)
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lemma bin_bl_bin': 
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  "bl_to_bin (bin_to_bl_aux n w bs) = 
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    bl_to_bin_aux bs (bintrunc n w)"
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  by (induct n arbitrary: w bs) (auto simp add : bl_to_bin_def)
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lemma bin_bl_bin: "bl_to_bin (bin_to_bl n w) = bintrunc n w"
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  unfolding bin_to_bl_def bin_bl_bin' by auto
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lemma bl_bin_bl':
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  "bin_to_bl (n + length bs) (bl_to_bin_aux bs w) = 
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    bin_to_bl_aux n w bs"
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  apply (induct bs arbitrary: w n)
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   apply auto
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    apply (simp_all only : add_Suc [symmetric])
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    apply (auto simp add : bin_to_bl_def)
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  done
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lemma bl_bin_bl: "bin_to_bl (length bs) (bl_to_bin bs) = bs"
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  unfolding bl_to_bin_def
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  apply (rule box_equals)
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    apply (rule bl_bin_bl')
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   prefer 2
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   apply (rule bin_to_bl_aux.Z)
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  apply simp
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  done
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declare 
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  bin_to_bl_0 [simp] 
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  size_bin_to_bl [simp] 
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  bin_bl_bin [simp] 
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  bl_bin_bl [simp]
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lemma bl_to_bin_inj:
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  "bl_to_bin bs = bl_to_bin cs ==> length bs = length cs ==> bs = cs"
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  apply (rule_tac box_equals)
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    defer
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    apply (rule bl_bin_bl)
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   apply (rule bl_bin_bl)
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  apply simp
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  done
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lemma bl_to_bin_False: "bl_to_bin (False # bl) = bl_to_bin bl"
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  unfolding bl_to_bin_def by auto
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lemma bl_to_bin_Nil: "bl_to_bin [] = Int.Pls"
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  unfolding bl_to_bin_def by auto
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lemma bin_to_bl_Pls_aux: 
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  "bin_to_bl_aux n Int.Pls bl = replicate n False @ bl"
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  by (induct n arbitrary: bl) (auto simp: replicate_app_Cons_same)
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lemma bin_to_bl_Pls: "bin_to_bl n Int.Pls = replicate n False"
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  unfolding bin_to_bl_def by (simp add : bin_to_bl_Pls_aux)
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lemma bin_to_bl_Min_aux [rule_format] : 
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  "ALL bl. bin_to_bl_aux n Int.Min bl = replicate n True @ bl"
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  by (induct n) (auto simp: replicate_app_Cons_same)
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lemma bin_to_bl_Min: "bin_to_bl n Int.Min = replicate n True"
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  unfolding bin_to_bl_def by (simp add : bin_to_bl_Min_aux)
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lemma bl_to_bin_rep_F: 
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  "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
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  apply (simp add: bin_to_bl_Pls_aux [symmetric] bin_bl_bin')
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  apply (simp add: bl_to_bin_def)
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  done
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lemma bin_to_bl_trunc:
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  "n <= m ==> bin_to_bl n (bintrunc m w) = bin_to_bl n w"
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  by (auto intro: bl_to_bin_inj)
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declare 
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  bin_to_bl_trunc [simp] 
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  bl_to_bin_False [simp] 
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  bl_to_bin_Nil [simp]
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lemma bin_to_bl_aux_bintr [rule_format] :
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  "ALL m bin bl. bin_to_bl_aux n (bintrunc m bin) bl = 
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    replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl"
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  apply (induct n)
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   apply clarsimp
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  apply clarsimp
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  apply (case_tac "m")
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   apply (clarsimp simp: bin_to_bl_Pls_aux) 
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   apply (erule thin_rl)
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   apply (induct_tac n)   
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    apply auto
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  done
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lemmas bin_to_bl_bintr = 
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  bin_to_bl_aux_bintr [where bl = "[]", folded bin_to_bl_def]
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lemma bl_to_bin_rep_False: "bl_to_bin (replicate n False) = Int.Pls"
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  by (induct n) auto
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lemma len_bin_to_bl_aux: 
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  "length (bin_to_bl_aux n w bs) = n + length bs"
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  by (induct n arbitrary: w bs) auto
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lemma len_bin_to_bl [simp]: "length (bin_to_bl n w) = n"
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  unfolding bin_to_bl_def len_bin_to_bl_aux by auto
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lemma sign_bl_bin': 
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  "bin_sign (bl_to_bin_aux bs w) = bin_sign w"
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  by (induct bs arbitrary: w) auto
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lemma sign_bl_bin: "bin_sign (bl_to_bin bs) = Int.Pls"
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  unfolding bl_to_bin_def by (simp add : sign_bl_bin')
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lemma bl_sbin_sign_aux: 
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  "hd (bin_to_bl_aux (Suc n) w bs) = 
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    (bin_sign (sbintrunc n w) = Int.Min)"
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  apply (induct n arbitrary: w bs)
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   apply clarsimp
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   apply (cases w rule: bin_exhaust)
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   apply (simp split add : bit.split)
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  apply clarsimp
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  done
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lemma bl_sbin_sign: 
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  "hd (bin_to_bl (Suc n) w) = (bin_sign (sbintrunc n w) = Int.Min)"
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  unfolding bin_to_bl_def by (rule bl_sbin_sign_aux)
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lemma bin_nth_of_bl_aux [rule_format]: 
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  "\<forall>w. bin_nth (bl_to_bin_aux bl w) n = 
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    (n < size bl & rev bl ! n | n >= length bl & bin_nth w (n - size bl))"
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  apply (induct_tac bl)
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   apply clarsimp
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  apply clarsimp
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  apply (cut_tac x=n and y="size list" in linorder_less_linear)
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  apply (erule disjE, simp add: nth_append)+
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  apply auto
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  done
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lemma bin_nth_of_bl: "bin_nth (bl_to_bin bl) n = (n < length bl & rev bl ! n)"
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  unfolding bl_to_bin_def by (simp add : bin_nth_of_bl_aux)
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lemma bin_nth_bl [rule_format] : "ALL m w. n < m --> 
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    bin_nth w n = nth (rev (bin_to_bl m w)) n"
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  apply (induct n)
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   apply clarsimp
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   apply (case_tac m, clarsimp)
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   apply (clarsimp simp: bin_to_bl_def)
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   apply (simp add: bin_to_bl_aux_alt)
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  apply clarsimp
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  apply (case_tac m, clarsimp)
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  apply (clarsimp simp: bin_to_bl_def)
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  apply (simp add: bin_to_bl_aux_alt)
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  done
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lemma nth_rev [rule_format] : 
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  "n < length xs --> rev xs ! n = xs ! (length xs - 1 - n)"
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  apply (induct_tac "xs")
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   apply simp
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  apply (clarsimp simp add : nth_append nth.simps split add : nat.split)
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  apply (rule_tac f = "%n. list ! n" in arg_cong) 
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  apply arith
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  done
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lemmas nth_rev_alt = nth_rev [where xs = "rev ys", simplified, standard]
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lemma nth_bin_to_bl_aux [rule_format] : 
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   307
  "ALL w n bl. n < m + length bl --> (bin_to_bl_aux m w bl) ! n = 
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diff changeset
   308
    (if n < m then bin_nth w (m - 1 - n) else bl ! (n - m))"
27105
5f139027c365 slightly tuning of some proofs involving case distinction and induction on natural numbers and similar
haftmann
parents: 26584
diff changeset
   309
  apply (induct m)
24333
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diff changeset
   310
   apply clarsimp
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parents:
diff changeset
   311
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   312
  apply (case_tac w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   313
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   314
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   315
  
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   316
lemma nth_bin_to_bl: "n < m ==> (bin_to_bl m w) ! n = bin_nth w (m - Suc n)"
e77ea0ea7f2c * HOL-Word:
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diff changeset
   317
  unfolding bin_to_bl_def by (simp add : nth_bin_to_bl_aux)
e77ea0ea7f2c * HOL-Word:
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diff changeset
   318
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parents: 26086
diff changeset
   319
lemma bl_to_bin_lt2p_aux [rule_format]: 
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parents: 26086
diff changeset
   320
  "\<forall>w. bl_to_bin_aux bs w < (w + 1) * (2 ^ length bs)"
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parents: 26086
diff changeset
   321
  apply (induct bs)
24333
e77ea0ea7f2c * HOL-Word:
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diff changeset
   322
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   323
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   324
  apply safe
26557
9e7f95903b24 more new primrec
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parents: 26086
diff changeset
   325
  apply (erule allE, erule xtr8 [rotated],
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28298
diff changeset
   326
         simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
24333
e77ea0ea7f2c * HOL-Word:
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diff changeset
   327
  done
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   328
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   329
lemma bl_to_bin_lt2p: "bl_to_bin bs < (2 ^ length bs)"
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   330
  apply (unfold bl_to_bin_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   331
  apply (rule xtr1)
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   332
   prefer 2
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   333
   apply (rule bl_to_bin_lt2p_aux)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   334
  apply simp
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   335
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   336
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   337
lemma bl_to_bin_ge2p_aux [rule_format] : 
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parents: 26086
diff changeset
   338
  "\<forall>w. bl_to_bin_aux bs w >= w * (2 ^ length bs)"
24333
e77ea0ea7f2c * HOL-Word:
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diff changeset
   339
  apply (induct bs)
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   340
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   341
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   342
  apply safe
28059
295a8fc92684 fixed names of class assumptions
haftmann
parents: 27677
diff changeset
   343
   apply (erule allE, erule preorder_class.order_trans [rotated],
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 28298
diff changeset
   344
          simp add: numeral_simps algebra_simps cong add : number_of_False_cong)+
24333
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   345
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   346
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   347
lemma bl_to_bin_ge0: "bl_to_bin bs >= 0"
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   348
  apply (unfold bl_to_bin_def)
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   349
  apply (rule xtr4)
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   350
   apply (rule bl_to_bin_ge2p_aux)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   351
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   352
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   353
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   354
lemma butlast_rest_bin: 
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   355
  "butlast (bin_to_bl n w) = bin_to_bl (n - 1) (bin_rest w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   356
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   357
  apply (cases w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   358
  apply (cases n, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   359
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   360
  apply (auto simp add: bin_to_bl_aux_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   361
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   362
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   363
lemmas butlast_bin_rest = butlast_rest_bin
25350
a5fcf6d12a53 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   364
  [where w="bl_to_bin bl" and n="length bl", simplified, standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   365
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9e7f95903b24 more new primrec
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parents: 26086
diff changeset
   366
lemma butlast_rest_bl2bin_aux:
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haftmann
parents: 26086
diff changeset
   367
  "bl ~= [] \<Longrightarrow>
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   368
    bl_to_bin_aux (butlast bl) w = bin_rest (bl_to_bin_aux bl w)"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   369
  by (induct bl arbitrary: w) auto
24333
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   370
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   371
lemma butlast_rest_bl2bin: 
e77ea0ea7f2c * HOL-Word:
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parents:
diff changeset
   372
  "bl_to_bin (butlast bl) = bin_rest (bl_to_bin bl)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   373
  apply (unfold bl_to_bin_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   374
  apply (cases bl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   375
   apply (auto simp add: butlast_rest_bl2bin_aux)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   376
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   377
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   378
lemma trunc_bl2bin_aux [rule_format]: 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   379
  "ALL w. bintrunc m (bl_to_bin_aux bl w) = 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   380
    bl_to_bin_aux (drop (length bl - m) bl) (bintrunc (m - length bl) w)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   381
  apply (induct_tac bl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   382
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   383
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   384
  apply safe
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   385
   apply (case_tac "m - size list")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   386
    apply (simp add : diff_is_0_eq [THEN iffD1, THEN Suc_diff_le])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   387
   apply simp
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   388
   apply (rule_tac f = "%nat. bl_to_bin_aux list (Int.Bit1 (bintrunc nat w))" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   389
                   in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   390
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   391
  apply (case_tac "m - size list")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   392
   apply (simp add: diff_is_0_eq [THEN iffD1, THEN Suc_diff_le])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   393
  apply simp
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   394
  apply (rule_tac f = "%nat. bl_to_bin_aux list (Int.Bit0 (bintrunc nat w))" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   395
                  in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   396
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   397
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   398
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   399
lemma trunc_bl2bin: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   400
  "bintrunc m (bl_to_bin bl) = bl_to_bin (drop (length bl - m) bl)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   401
  unfolding bl_to_bin_def by (simp add : trunc_bl2bin_aux)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   402
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   403
lemmas trunc_bl2bin_len [simp] =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   404
  trunc_bl2bin [of "length bl" bl, simplified, standard]  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   405
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   406
lemma bl2bin_drop: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   407
  "bl_to_bin (drop k bl) = bintrunc (length bl - k) (bl_to_bin bl)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   408
  apply (rule trans)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   409
   prefer 2
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   410
   apply (rule trunc_bl2bin [symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   411
  apply (cases "k <= length bl")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   412
   apply auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   413
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   414
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   415
lemma nth_rest_power_bin [rule_format] :
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   416
  "ALL n. bin_nth ((bin_rest ^^ k) w) n = bin_nth w (n + k)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   417
  apply (induct k, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   418
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   419
  apply (simp only: bin_nth.Suc [symmetric] add_Suc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   420
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   421
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   422
lemma take_rest_power_bin:
30971
7fbebf75b3ef funpow and relpow with shared "^^" syntax
haftmann
parents: 30952
diff changeset
   423
  "m <= n ==> take m (bin_to_bl n w) = bin_to_bl m ((bin_rest ^^ (n - m)) w)" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   424
  apply (rule nth_equalityI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   425
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   426
  apply (clarsimp simp add: nth_bin_to_bl nth_rest_power_bin)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   427
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   428
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   429
lemma hd_butlast: "size xs > 1 ==> hd (butlast xs) = hd xs"
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   430
  by (cases xs) auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   431
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   432
lemma last_bin_last': 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 32642
diff changeset
   433
  "size xs > 0 \<Longrightarrow> last xs = (bin_last (bl_to_bin_aux xs w) = 1)" 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   434
  by (induct xs arbitrary: w) auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   435
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   436
lemma last_bin_last: 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 32642
diff changeset
   437
  "size xs > 0 ==> last xs = (bin_last (bl_to_bin xs) = 1)" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   438
  unfolding bl_to_bin_def by (erule last_bin_last')
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   439
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   440
lemma bin_last_last: 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 32642
diff changeset
   441
  "bin_last w = (if last (bin_to_bl (Suc n) w) then 1 else 0)" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   442
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   443
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   444
  apply (auto simp add: bin_to_bl_aux_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   445
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   446
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   447
(** links between bit-wise operations and operations on bool lists **)
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   448
    
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   449
lemma bl_xor_aux_bin [rule_format] : "ALL v w bs cs. 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   450
    map2 (%x y. x ~= y) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   451
    bin_to_bl_aux n (v XOR w) (map2 (%x y. x ~= y) bs cs)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   452
  apply (induct_tac n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   453
   apply safe
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   454
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   455
  apply (case_tac v rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   456
  apply (case_tac w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   457
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   458
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   459
  apply (case_tac ba, safe, simp_all)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   460
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   461
    
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   462
lemma bl_or_aux_bin [rule_format] : "ALL v w bs cs. 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   463
    map2 (op | ) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   464
    bin_to_bl_aux n (v OR w) (map2 (op | ) bs cs)" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   465
  apply (induct_tac n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   466
   apply safe
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   467
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   468
  apply (case_tac v rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   469
  apply (case_tac w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   470
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   471
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   472
  apply (case_tac ba, safe, simp_all)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   473
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   474
    
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   475
lemma bl_and_aux_bin [rule_format] : "ALL v w bs cs. 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   476
    map2 (op & ) (bin_to_bl_aux n v bs) (bin_to_bl_aux n w cs) = 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   477
    bin_to_bl_aux n (v AND w) (map2 (op & ) bs cs)" 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   478
  apply (induct_tac n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   479
   apply safe
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   480
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   481
  apply (case_tac v rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   482
  apply (case_tac w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   483
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   484
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   485
  apply (case_tac ba, safe, simp_all)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   486
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   487
    
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   488
lemma bl_not_aux_bin [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   489
  "ALL w cs. map Not (bin_to_bl_aux n w cs) = 
24353
9a7a9b19e925 use overloaded bitwise operators at type int
huffman
parents: 24350
diff changeset
   490
    bin_to_bl_aux n (NOT w) (map Not cs)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   491
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   492
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   493
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   494
  apply (case_tac w rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   495
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   496
   apply auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   497
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   498
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   499
lemmas bl_not_bin = bl_not_aux_bin
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   500
  [where cs = "[]", unfolded bin_to_bl_def [symmetric] map.simps]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   501
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   502
lemmas bl_and_bin = bl_and_aux_bin [where bs="[]" and cs="[]", 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   503
                                    unfolded map2_Nil, folded bin_to_bl_def]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   504
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   505
lemmas bl_or_bin = bl_or_aux_bin [where bs="[]" and cs="[]", 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   506
                                  unfolded map2_Nil, folded bin_to_bl_def]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   507
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   508
lemmas bl_xor_bin = bl_xor_aux_bin [where bs="[]" and cs="[]", 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   509
                                    unfolded map2_Nil, folded bin_to_bl_def]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   510
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   511
lemma drop_bin2bl_aux [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   512
  "ALL m bin bs. drop m (bin_to_bl_aux n bin bs) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   513
    bin_to_bl_aux (n - m) bin (drop (m - n) bs)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   514
  apply (induct n, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   515
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   516
  apply (case_tac bin rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   517
  apply (case_tac "m <= n", simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   518
  apply (case_tac "m - n", simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   519
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   520
  apply (rule_tac f = "%nat. drop nat bs" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   521
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   522
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   523
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   524
lemma drop_bin2bl: "drop m (bin_to_bl n bin) = bin_to_bl (n - m) bin"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   525
  unfolding bin_to_bl_def by (simp add : drop_bin2bl_aux)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   526
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   527
lemma take_bin2bl_lem1 [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   528
  "ALL w bs. take m (bin_to_bl_aux m w bs) = bin_to_bl m w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   529
  apply (induct m, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   530
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   531
  apply (simp add: bin_to_bl_aux_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   532
  apply (simp add: bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   533
  apply (simp add: bin_to_bl_aux_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   534
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   535
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   536
lemma take_bin2bl_lem [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   537
  "ALL w bs. take m (bin_to_bl_aux (m + n) w bs) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   538
    take m (bin_to_bl (m + n) w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   539
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   540
   apply clarify
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   541
   apply (simp_all (no_asm) add: bin_to_bl_def take_bin2bl_lem1)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   542
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   543
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   544
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   545
lemma bin_split_take [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   546
  "ALL b c. bin_split n c = (a, b) --> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   547
    bin_to_bl m a = take m (bin_to_bl (m + n) c)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   548
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   549
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   550
  apply (clarsimp simp: Let_def split: ls_splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   551
  apply (simp add: bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   552
  apply (simp add: take_bin2bl_lem)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   553
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   554
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   555
lemma bin_split_take1: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   556
  "k = m + n ==> bin_split n c = (a, b) ==> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   557
    bin_to_bl m a = take m (bin_to_bl k c)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   558
  by (auto elim: bin_split_take)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   559
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   560
lemma nth_takefill [rule_format] : "ALL m l. m < n --> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   561
    takefill fill n l ! m = (if m < length l then l ! m else fill)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   562
  apply (induct n, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   563
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   564
  apply (case_tac m)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   565
   apply (simp split: list.split)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   566
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   567
  apply (erule allE)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   568
  apply (erule (1) impE)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   569
  apply (simp split: list.split)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   570
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   571
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   572
lemma takefill_alt [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   573
  "ALL l. takefill fill n l = take n l @ replicate (n - length l) fill"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   574
  by (induct n) (auto split: list.split)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   575
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   576
lemma takefill_replicate [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   577
  "takefill fill n (replicate m fill) = replicate n fill"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   578
  by (simp add : takefill_alt replicate_add [symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   579
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   580
lemma takefill_le' [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   581
  "ALL l n. n = m + k --> takefill x m (takefill x n l) = takefill x m l"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   582
  by (induct m) (auto split: list.split)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   583
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   584
lemma length_takefill [simp]: "length (takefill fill n l) = n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   585
  by (simp add : takefill_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   586
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   587
lemma take_takefill':
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   588
  "!!w n.  n = k + m ==> take k (takefill fill n w) = takefill fill k w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   589
  by (induct k) (auto split add : list.split) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   590
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   591
lemma drop_takefill:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   592
  "!!w. drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   593
  by (induct k) (auto split add : list.split) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   594
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   595
lemma takefill_le [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   596
  "m \<le> n \<Longrightarrow> takefill x m (takefill x n l) = takefill x m l"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   597
  by (auto simp: le_iff_add takefill_le')
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   598
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   599
lemma take_takefill [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   600
  "m \<le> n \<Longrightarrow> take m (takefill fill n w) = takefill fill m w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   601
  by (auto simp: le_iff_add take_takefill')
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   602
 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   603
lemma takefill_append:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   604
  "takefill fill (m + length xs) (xs @ w) = xs @ (takefill fill m w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   605
  by (induct xs) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   606
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   607
lemma takefill_same': 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   608
  "l = length xs ==> takefill fill l xs = xs"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   609
  by clarify (induct xs, auto)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   610
 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   611
lemmas takefill_same [simp] = takefill_same' [OF refl]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   612
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   613
lemma takefill_bintrunc:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   614
  "takefill False n bl = rev (bin_to_bl n (bl_to_bin (rev bl)))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   615
  apply (rule nth_equalityI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   616
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   617
  apply (clarsimp simp: nth_takefill nth_rev nth_bin_to_bl bin_nth_of_bl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   618
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   619
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   620
lemma bl_bin_bl_rtf:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   621
  "bin_to_bl n (bl_to_bin bl) = rev (takefill False n (rev bl))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   622
  by (simp add : takefill_bintrunc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   623
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   624
lemmas bl_bin_bl_rep_drop = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   625
  bl_bin_bl_rtf [simplified takefill_alt,
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   626
                 simplified, simplified rev_take, simplified]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   627
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   628
lemma tf_rev:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   629
  "n + k = m + length bl ==> takefill x m (rev (takefill y n bl)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   630
    rev (takefill y m (rev (takefill x k (rev bl))))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   631
  apply (rule nth_equalityI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   632
   apply (auto simp add: nth_takefill nth_rev)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   633
  apply (rule_tac f = "%n. bl ! n" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   634
  apply arith 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   635
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   636
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   637
lemma takefill_minus:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   638
  "0 < n ==> takefill fill (Suc (n - 1)) w = takefill fill n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   639
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   640
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   641
lemmas takefill_Suc_cases = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   642
  list.cases [THEN takefill.Suc [THEN trans], standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   643
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   644
lemmas takefill_Suc_Nil = takefill_Suc_cases (1)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   645
lemmas takefill_Suc_Cons = takefill_Suc_cases (2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   646
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   647
lemmas takefill_minus_simps = takefill_Suc_cases [THEN [2] 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   648
  takefill_minus [symmetric, THEN trans], standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   649
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   650
lemmas takefill_pred_simps [simp] =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   651
  takefill_minus_simps [where n="number_of bin", simplified nobm1, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   652
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   653
(* links with function bl_to_bin *)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   654
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   655
lemma bl_to_bin_aux_cat: 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   656
  "!!nv v. bl_to_bin_aux bs (bin_cat w nv v) = 
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   657
    bin_cat w (nv + length bs) (bl_to_bin_aux bs v)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   658
  apply (induct bs)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   659
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   660
  apply (simp add: bin_cat_Suc_Bit [symmetric] del: bin_cat.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   661
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   662
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   663
lemma bin_to_bl_aux_cat: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   664
  "!!w bs. bin_to_bl_aux (nv + nw) (bin_cat v nw w) bs = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   665
    bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   666
  by (induct nw) auto 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   667
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   668
lemmas bl_to_bin_aux_alt = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
   669
  bl_to_bin_aux_cat [where nv = "0" and v = "Int.Pls", 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   670
    simplified bl_to_bin_def [symmetric], simplified]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   671
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   672
lemmas bin_to_bl_cat =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   673
  bin_to_bl_aux_cat [where bs = "[]", folded bin_to_bl_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   674
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   675
lemmas bl_to_bin_aux_app_cat = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   676
  trans [OF bl_to_bin_aux_append bl_to_bin_aux_alt]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   677
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   678
lemmas bin_to_bl_aux_cat_app =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   679
  trans [OF bin_to_bl_aux_cat bin_to_bl_aux_alt]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   680
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   681
lemmas bl_to_bin_app_cat = bl_to_bin_aux_app_cat
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
   682
  [where w = "Int.Pls", folded bl_to_bin_def]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   683
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   684
lemmas bin_to_bl_cat_app = bin_to_bl_aux_cat_app
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   685
  [where bs = "[]", folded bin_to_bl_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   686
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   687
(* bl_to_bin_app_cat_alt and bl_to_bin_app_cat are easily interderivable *)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   688
lemma bl_to_bin_app_cat_alt: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   689
  "bin_cat (bl_to_bin cs) n w = bl_to_bin (cs @ bin_to_bl n w)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   690
  by (simp add : bl_to_bin_app_cat)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   691
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   692
lemma mask_lem: "(bl_to_bin (True # replicate n False)) = 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
   693
    Int.succ (bl_to_bin (replicate n True))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   694
  apply (unfold bl_to_bin_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   695
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   696
   apply simp
31790
05c92381363c corrected and unified thm names
nipkow
parents: 30971
diff changeset
   697
  apply (simp only: Suc_eq_plus1 replicate_add
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   698
                    append_Cons [symmetric] bl_to_bin_aux_append)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   699
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   700
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   701
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   702
(* function bl_of_nth *)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   703
lemma length_bl_of_nth [simp]: "length (bl_of_nth n f) = n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   704
  by (induct n)  auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   705
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   706
lemma nth_bl_of_nth [simp]:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   707
  "m < n \<Longrightarrow> rev (bl_of_nth n f) ! m = f m"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   708
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   709
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   710
  apply (clarsimp simp add : nth_append)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   711
  apply (rule_tac f = "f" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   712
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   713
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   714
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   715
lemma bl_of_nth_inj: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   716
  "(!!k. k < n ==> f k = g k) ==> bl_of_nth n f = bl_of_nth n g"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   717
  by (induct n)  auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   718
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   719
lemma bl_of_nth_nth_le [rule_format] : "ALL xs. 
45475
b2b087c20e45 removed some old-style semicolons
huffman
parents: 44939
diff changeset
   720
    length xs >= n --> bl_of_nth n (nth (rev xs)) = drop (length xs - n) xs"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   721
  apply (induct n, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   722
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   723
  apply (rule trans [OF _ hd_Cons_tl])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   724
   apply (frule Suc_le_lessD)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   725
   apply (simp add: nth_rev trans [OF drop_Suc drop_tl, symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   726
   apply (subst hd_drop_conv_nth)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   727
     apply force
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   728
    apply simp_all
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   729
  apply (rule_tac f = "%n. drop n xs" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   730
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   731
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   732
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   733
lemmas bl_of_nth_nth [simp] = order_refl [THEN bl_of_nth_nth_le, simplified]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   734
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   735
lemma size_rbl_pred: "length (rbl_pred bl) = length bl"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   736
  by (induct bl) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   737
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   738
lemma size_rbl_succ: "length (rbl_succ bl) = length bl"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   739
  by (induct bl) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   740
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   741
lemma size_rbl_add:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   742
  "!!cl. length (rbl_add bl cl) = length bl"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   743
  by (induct bl) (auto simp: Let_def size_rbl_succ)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   744
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   745
lemma size_rbl_mult: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   746
  "!!cl. length (rbl_mult bl cl) = length bl"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   747
  by (induct bl) (auto simp add : Let_def size_rbl_add)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   748
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   749
lemmas rbl_sizes [simp] = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   750
  size_rbl_pred size_rbl_succ size_rbl_add size_rbl_mult
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   751
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   752
lemmas rbl_Nils =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   753
  rbl_pred.Nil rbl_succ.Nil rbl_add.Nil rbl_mult.Nil
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   754
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   755
lemma rbl_pred: 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
   756
  "!!bin. rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (Int.pred bin))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   757
  apply (induct n, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   758
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   759
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   760
  apply (case_tac bin rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   761
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   762
   apply (clarsimp simp: bin_to_bl_aux_alt)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   763
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   764
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   765
lemma rbl_succ: 
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
   766
  "!!bin. rbl_succ (rev (bin_to_bl n bin)) = rev (bin_to_bl n (Int.succ bin))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   767
  apply (induct n, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   768
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   769
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   770
  apply (case_tac bin rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   771
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   772
   apply (clarsimp simp: bin_to_bl_aux_alt)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   773
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   774
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   775
lemma rbl_add: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   776
  "!!bina binb. rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   777
    rev (bin_to_bl n (bina + binb))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   778
  apply (induct n, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   779
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   780
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   781
  apply (case_tac bina rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   782
  apply (case_tac binb rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   783
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   784
   apply (case_tac [!] "ba")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   785
     apply (auto simp: rbl_succ succ_def bin_to_bl_aux_alt Let_def add_ac)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   786
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   787
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   788
lemma rbl_add_app2: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   789
  "!!blb. length blb >= length bla ==> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   790
    rbl_add bla (blb @ blc) = rbl_add bla blb"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   791
  apply (induct bla, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   792
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   793
  apply (case_tac blb, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   794
  apply (clarsimp simp: Let_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   795
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   796
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   797
lemma rbl_add_take2: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   798
  "!!blb. length blb >= length bla ==> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   799
    rbl_add bla (take (length bla) blb) = rbl_add bla blb"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   800
  apply (induct bla, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   801
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   802
  apply (case_tac blb, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   803
  apply (clarsimp simp: Let_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   804
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   805
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   806
lemma rbl_add_long: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   807
  "m >= n ==> rbl_add (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   808
    rev (bin_to_bl n (bina + binb))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   809
  apply (rule box_equals [OF _ rbl_add_take2 rbl_add])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   810
   apply (rule_tac f = "rbl_add (rev (bin_to_bl n bina))" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   811
   apply (rule rev_swap [THEN iffD1])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   812
   apply (simp add: rev_take drop_bin2bl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   813
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   814
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   815
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   816
lemma rbl_mult_app2:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   817
  "!!blb. length blb >= length bla ==> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   818
    rbl_mult bla (blb @ blc) = rbl_mult bla blb"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   819
  apply (induct bla, simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   820
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   821
  apply (case_tac blb, clarsimp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   822
  apply (clarsimp simp: Let_def rbl_add_app2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   823
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   824
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   825
lemma rbl_mult_take2: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   826
  "length blb >= length bla ==> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   827
    rbl_mult bla (take (length bla) blb) = rbl_mult bla blb"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   828
  apply (rule trans)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   829
   apply (rule rbl_mult_app2 [symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   830
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   831
  apply (rule_tac f = "rbl_mult bla" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   832
  apply (rule append_take_drop_id)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   833
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   834
    
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   835
lemma rbl_mult_gt1: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   836
  "m >= length bl ==> rbl_mult bl (rev (bin_to_bl m binb)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   837
    rbl_mult bl (rev (bin_to_bl (length bl) binb))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   838
  apply (rule trans)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   839
   apply (rule rbl_mult_take2 [symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   840
   apply simp_all
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   841
  apply (rule_tac f = "rbl_mult bl" in arg_cong) 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   842
  apply (rule rev_swap [THEN iffD1])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   843
  apply (simp add: rev_take drop_bin2bl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   844
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   845
    
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   846
lemma rbl_mult_gt: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   847
  "m > n ==> rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl m binb)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   848
    rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   849
  by (auto intro: trans [OF rbl_mult_gt1])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   850
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   851
lemmas rbl_mult_Suc = lessI [THEN rbl_mult_gt]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   852
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   853
lemma rbbl_Cons: 
37654
8e33b9d04a82 use existing bit type from theory Bit
haftmann
parents: 32642
diff changeset
   854
  "b # rev (bin_to_bl n x) = rev (bin_to_bl (Suc n) (x BIT If b 1 0))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   855
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   856
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   857
  apply (simp add: bin_to_bl_aux_alt)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   858
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   859
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   860
lemma rbl_mult: "!!bina binb. 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   861
    rbl_mult (rev (bin_to_bl n bina)) (rev (bin_to_bl n binb)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   862
    rev (bin_to_bl n (bina * binb))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   863
  apply (induct n)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   864
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   865
  apply (unfold bin_to_bl_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   866
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   867
  apply (case_tac bina rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   868
  apply (case_tac binb rule: bin_exhaust)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   869
  apply (case_tac b)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   870
   apply (case_tac [!] "ba")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   871
     apply (auto simp: bin_to_bl_aux_alt Let_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   872
     apply (auto simp: rbbl_Cons rbl_mult_Suc rbl_add)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   873
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   874
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   875
lemma rbl_add_split: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   876
  "P (rbl_add (y # ys) (x # xs)) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   877
    (ALL ws. length ws = length ys --> ws = rbl_add ys xs --> 
26008
24c82bef5696 eliminated escaped white space;
wenzelm
parents: 25919
diff changeset
   878
    (y --> ((x --> P (False # rbl_succ ws)) & (~ x -->  P (True # ws)))) &
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   879
    (~ y --> P (x # ws)))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   880
  apply (auto simp add: Let_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   881
   apply (case_tac [!] "y")
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   882
     apply auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   883
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   884
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   885
lemma rbl_mult_split: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   886
  "P (rbl_mult (y # ys) xs) = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   887
    (ALL ws. length ws = Suc (length ys) --> ws = False # rbl_mult ys xs --> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   888
    (y --> P (rbl_add ws xs)) & (~ y -->  P ws))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   889
  by (clarsimp simp add : Let_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   890
  
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   891
lemma and_len: "xs = ys ==> xs = ys & length xs = length ys"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   892
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   893
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   894
lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   895
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   896
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   897
lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   898
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   899
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   900
lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   901
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   902
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   903
lemma if_Not_x: "(if p then ~ x else x) = (p = (~ x))"
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   904
  by auto
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   905
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   906
lemma if_x_Not: "(if p then x else ~ x) = (p = x)"
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   907
  by auto
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   908
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   909
lemma if_same_and: "(If p x y & If p u v) = (if p then x & u else y & v)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   910
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   911
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   912
lemma if_same_eq: "(If p x y  = (If p u v)) = (if p then x = (u) else y = (v))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   913
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   914
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   915
lemma if_same_eq_not:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   916
  "(If p x y  = (~ If p u v)) = (if p then x = (~u) else y = (~v))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   917
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   918
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   919
(* note - if_Cons can cause blowup in the size, if p is complex,
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   920
  so make a simproc *)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   921
lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   922
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   923
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   924
lemma if_single:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   925
  "(if xc then [xab] else [an]) = [if xc then xab else an]"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   926
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   927
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   928
lemma if_bool_simps:
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   929
  "If p True y = (p | y) & If p False y = (~p & y) & 
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   930
    If p y True = (p --> y) & If p y False = (p & y)"
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   931
  by auto
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   932
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   933
lemmas if_simps = if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
   934
25350
a5fcf6d12a53 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   935
lemmas seqr = eq_reflection [where x = "size w", standard]
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   936
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   937
lemmas tl_Nil = tl.simps (1)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   938
lemmas tl_Cons = tl.simps (2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   939
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   940
24350
4d74f37c6367 headers for document generation
huffman
parents: 24333
diff changeset
   941
subsection "Repeated splitting or concatenation"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   942
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   943
lemma sclem:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   944
  "size (concat (map (bin_to_bl n) xs)) = length xs * n"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   945
  by (induct xs) auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   946
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   947
lemma bin_cat_foldl_lem [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   948
  "ALL x. foldl (%u. bin_cat u n) x xs = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   949
    bin_cat x (size xs * n) (foldl (%u. bin_cat u n) y xs)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   950
  apply (induct xs)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   951
   apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   952
  apply clarify
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   953
  apply (simp (no_asm))
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   954
  apply (frule asm_rl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   955
  apply (drule spec)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   956
  apply (erule trans)
32439
7a91c7bcfe7e tuned proofs
nipkow
parents: 31790
diff changeset
   957
  apply (drule_tac x = "bin_cat y n a" in spec)
32642
026e7c6a6d08 be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
haftmann
parents: 32439
diff changeset
   958
  apply (simp add : bin_cat_assoc_sym min_max.inf_absorb2)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   959
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   960
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   961
lemma bin_rcat_bl:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   962
  "(bin_rcat n wl) = bl_to_bin (concat (map (bin_to_bl n) wl))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   963
  apply (unfold bin_rcat_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   964
  apply (rule sym)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   965
  apply (induct wl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   966
   apply (auto simp add : bl_to_bin_append)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   967
  apply (simp add : bl_to_bin_aux_alt sclem)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   968
  apply (simp add : bin_cat_foldl_lem [symmetric])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   969
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   970
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   971
lemmas bin_rsplit_aux_simps = bin_rsplit_aux.simps bin_rsplitl_aux.simps
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   972
lemmas rsplit_aux_simps = bin_rsplit_aux_simps
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   973
25350
a5fcf6d12a53 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   974
lemmas th_if_simp1 = split_if [where P = "op = l",
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   975
  THEN iffD1, THEN conjunct1, THEN mp, standard]
25350
a5fcf6d12a53 eliminated illegal schematic variables in where/of;
wenzelm
parents: 25134
diff changeset
   976
lemmas th_if_simp2 = split_if [where P = "op = l",
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   977
  THEN iffD1, THEN conjunct2, THEN mp, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   978
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   979
lemmas rsplit_aux_simp1s = rsplit_aux_simps [THEN th_if_simp1]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   980
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   981
lemmas rsplit_aux_simp2ls = rsplit_aux_simps [THEN th_if_simp2]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   982
(* these safe to [simp add] as require calculating m - n *)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   983
lemmas bin_rsplit_aux_simp2s [simp] = rsplit_aux_simp2ls [unfolded Let_def]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   984
lemmas rbscl = bin_rsplit_aux_simp2s (2)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   985
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   986
lemmas rsplit_aux_0_simps [simp] = 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   987
  rsplit_aux_simp1s [OF disjI1] rsplit_aux_simp1s [OF disjI2]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   988
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   989
lemma bin_rsplit_aux_append:
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   990
  "bin_rsplit_aux n m c (bs @ cs) = bin_rsplit_aux n m c bs @ cs"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   991
  apply (induct n m c bs rule: bin_rsplit_aux.induct)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   992
  apply (subst bin_rsplit_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   993
  apply (subst bin_rsplit_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   994
  apply (clarsimp split: ls_splits)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   995
  apply auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   996
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   997
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
   998
lemma bin_rsplitl_aux_append:
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
   999
  "bin_rsplitl_aux n m c (bs @ cs) = bin_rsplitl_aux n m c bs @ cs"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1000
  apply (induct n m c bs rule: bin_rsplitl_aux.induct)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1001
  apply (subst bin_rsplitl_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1002
  apply (subst bin_rsplitl_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1003
  apply (clarsimp split: ls_splits)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1004
  apply auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1005
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1006
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1007
lemmas rsplit_aux_apps [where bs = "[]"] =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1008
  bin_rsplit_aux_append bin_rsplitl_aux_append
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1009
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1010
lemmas rsplit_def_auxs = bin_rsplit_def bin_rsplitl_def
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1011
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1012
lemmas rsplit_aux_alts = rsplit_aux_apps 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1013
  [unfolded append_Nil rsplit_def_auxs [symmetric]]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1014
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1015
lemma bin_split_minus: "0 < n ==> bin_split (Suc (n - 1)) w = bin_split n w"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1016
  by auto
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1017
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1018
lemmas bin_split_minus_simp =
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1019
  bin_split.Suc [THEN [2] bin_split_minus [symmetric, THEN trans], standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1020
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1021
lemma bin_split_pred_simp [simp]: 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1022
  "(0::nat) < number_of bin \<Longrightarrow>
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1023
  bin_split (number_of bin) w =
25919
8b1c0d434824 joined theories IntDef, Numeral, IntArith to theory Int
haftmann
parents: 25350
diff changeset
  1024
  (let (w1, w2) = bin_split (number_of (Int.pred bin)) (bin_rest w)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1025
   in (w1, w2 BIT bin_last w))" 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1026
  by (simp only: nobm1 bin_split_minus_simp)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1027
24465
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
  1028
declare bin_split_pred_simp [simp]
70f0214b3ecc revert to Word library version from 2007/08/20
huffman
parents: 24399
diff changeset
  1029
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1030
lemma bin_rsplit_aux_simp_alt:
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1031
  "bin_rsplit_aux n m c bs =
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1032
   (if m = 0 \<or> n = 0 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1033
   then bs
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1034
   else let (a, b) = bin_split n c in bin_rsplit n (m - n, a) @ b # bs)"
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1035
  unfolding bin_rsplit_aux.simps [of n m c bs]
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1036
  apply simp
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1037
  apply (subst rsplit_aux_alts)
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1038
  apply (simp add: bin_rsplit_def)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1039
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1040
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1041
lemmas bin_rsplit_simp_alt = 
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1042
  trans [OF bin_rsplit_def
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1043
            bin_rsplit_aux_simp_alt, standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1044
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1045
lemmas bthrs = bin_rsplit_simp_alt [THEN [2] trans]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1046
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1047
lemma bin_rsplit_size_sign' [rule_format] : 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1048
  "n > 0 ==> (ALL nw w. rev sw = bin_rsplit n (nw, w) --> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1049
    (ALL v: set sw. bintrunc n v = v))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1050
  apply (induct sw)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1051
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1052
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1053
  apply (drule bthrs)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1054
  apply (simp (no_asm_use) add: Let_def split: ls_splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1055
  apply clarify
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1056
  apply (erule impE, rule exI, erule exI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1057
  apply (drule split_bintrunc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1058
  apply simp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1059
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1060
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1061
lemmas bin_rsplit_size_sign = bin_rsplit_size_sign' [OF asm_rl 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1062
  rev_rev_ident [THEN trans] set_rev [THEN equalityD2 [THEN subsetD]],
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1063
  standard]
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1064
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1065
lemma bin_nth_rsplit [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1066
  "n > 0 ==> m < n ==> (ALL w k nw. rev sw = bin_rsplit n (nw, w) --> 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1067
       k < size sw --> bin_nth (sw ! k) m = bin_nth w (k * n + m))"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1068
  apply (induct sw)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1069
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1070
  apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1071
  apply (drule bthrs)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1072
  apply (simp (no_asm_use) add: Let_def split: ls_splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1073
  apply clarify
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1074
  apply (erule allE, erule impE, erule exI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1075
  apply (case_tac k)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1076
   apply clarsimp   
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1077
   prefer 2
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1078
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1079
   apply (erule allE)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1080
   apply (erule (1) impE)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1081
   apply (drule bin_nth_split, erule conjE, erule allE,
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1082
          erule trans, simp add : add_ac)+
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1083
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1084
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1085
lemma bin_rsplit_all:
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1086
  "0 < nw ==> nw <= n ==> bin_rsplit n (nw, w) = [bintrunc n w]"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1087
  unfolding bin_rsplit_def
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1088
  by (clarsimp dest!: split_bintrunc simp: rsplit_aux_simp2ls split: ls_splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1089
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1090
lemma bin_rsplit_l [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1091
  "ALL bin. bin_rsplitl n (m, bin) = bin_rsplit n (m, bintrunc m bin)"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1092
  apply (rule_tac a = "m" in wf_less_than [THEN wf_induct])
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1093
  apply (simp (no_asm) add : bin_rsplitl_def bin_rsplit_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1094
  apply (rule allI)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1095
  apply (subst bin_rsplitl_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1096
  apply (subst bin_rsplit_aux.simps)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1097
  apply (clarsimp simp: Let_def split: ls_splits)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1098
  apply (drule bin_split_trunc)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1099
  apply (drule sym [THEN trans], assumption)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1100
  apply (subst rsplit_aux_alts(1))
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1101
  apply (subst rsplit_aux_alts(2))
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1102
  apply clarsimp
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1103
  unfolding bin_rsplit_def bin_rsplitl_def
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1104
  apply simp
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1105
  done
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1106
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1107
lemma bin_rsplit_rcat [rule_format] :
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1108
  "n > 0 --> bin_rsplit n (n * size ws, bin_rcat n ws) = map (bintrunc n) ws"
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1109
  apply (unfold bin_rsplit_def bin_rcat_def)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1110
  apply (rule_tac xs = "ws" in rev_induct)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1111
   apply clarsimp
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1112
  apply clarsimp
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1113
  apply (subst rsplit_aux_alts)
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1114
  unfolding bin_split_cat
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1115
  apply simp
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1116
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1117
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1118
lemma bin_rsplit_aux_len_le [rule_format] :
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1119
  "\<forall>ws m. n \<noteq> 0 \<longrightarrow> ws = bin_rsplit_aux n nw w bs \<longrightarrow>
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1120
    length ws \<le> m \<longleftrightarrow> nw + length bs * n \<le> m * n"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1121
  apply (induct n nw w bs rule: bin_rsplit_aux.induct)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1122
  apply (subst bin_rsplit_aux.simps)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1123
  apply (simp add: lrlem Let_def split: ls_splits)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1124
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1125
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1126
lemma bin_rsplit_len_le: 
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1127
  "n \<noteq> 0 --> ws = bin_rsplit n (nw, w) --> (length ws <= m) = (nw <= m * n)"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1128
  unfolding bin_rsplit_def by (clarsimp simp add : bin_rsplit_aux_len_le)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1129
 
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1130
lemma bin_rsplit_aux_len [rule_format] :
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1131
  "n\<noteq>0 --> length (bin_rsplit_aux n nw w cs) = 
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1132
    (nw + n - 1) div n + length cs"
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1133
  apply (induct n nw w cs rule: bin_rsplit_aux.induct)
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1134
  apply (subst bin_rsplit_aux.simps)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1135
  apply (clarsimp simp: Let_def split: ls_splits)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1136
  apply (erule thin_rl)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27105
diff changeset
  1137
  apply (case_tac m)
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27105
diff changeset
  1138
  apply simp
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1139
  apply (case_tac "m <= n")
27677
646ea25ff59d (re-)added simp rules for (_ + _) div/mod _
haftmann
parents: 27651
diff changeset
  1140
  apply auto
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1141
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1142
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1143
lemma bin_rsplit_len: 
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1144
  "n\<noteq>0 ==> length (bin_rsplit n (nw, w)) = (nw + n - 1) div n"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1145
  unfolding bin_rsplit_def by (clarsimp simp add : bin_rsplit_aux_len)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1146
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1147
lemma bin_rsplit_aux_len_indep:
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1148
  "n \<noteq> 0 \<Longrightarrow> length bs = length cs \<Longrightarrow>
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1149
    length (bin_rsplit_aux n nw v bs) =
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1150
    length (bin_rsplit_aux n nw w cs)"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1151
proof (induct n nw w cs arbitrary: v bs rule: bin_rsplit_aux.induct)
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1152
  case (1 n m w cs v bs) show ?case
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1153
  proof (cases "m = 0")
28298
3eb2855e5402 avoid using implicit assumptions
huffman
parents: 28059
diff changeset
  1154
    case True then show ?thesis using `length bs = length cs` by simp
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1155
  next
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1156
    case False
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1157
    from "1.hyps" `m \<noteq> 0` `n \<noteq> 0` have hyp: "\<And>v bs. length bs = Suc (length cs) \<Longrightarrow>
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1158
      length (bin_rsplit_aux n (m - n) v bs) =
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1159
      length (bin_rsplit_aux n (m - n) (fst (bin_split n w)) (snd (bin_split n w) # cs))"
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1160
    by auto
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1161
    show ?thesis using `length bs = length cs` `n \<noteq> 0`
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1162
      by (auto simp add: bin_rsplit_aux_simp_alt Let_def bin_rsplit_len
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1163
        split: ls_splits)
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1164
  qed
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1165
qed
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1166
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1167
lemma bin_rsplit_len_indep: 
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
  1168
  "n\<noteq>0 ==> length (bin_rsplit n (nw, v)) = length (bin_rsplit n (nw, w))"
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1169
  apply (unfold bin_rsplit_def)
26557
9e7f95903b24 more new primrec
haftmann
parents: 26086
diff changeset
  1170
  apply (simp (no_asm))
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1171
  apply (erule bin_rsplit_aux_len_indep)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1172
  apply (rule refl)
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1173
  done
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1174
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1175
end