src/HOL/List.thy
author nipkow
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  null:: "'a list => bool"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  rotate1 :: "'a list \<Rightarrow> 'a list"
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  sublist :: "'a list => nat set => 'a list"
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(* For efficiency *)
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  mem :: "'a => 'a list => bool"    (infixl 55)
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  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
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  list_all:: "('a => bool) => ('a list => bool)"
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  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
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  map_filter :: "('a => 'b) => ('a => bool) => 'a list => 'b list"
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abbreviation
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  upto:: "nat => nat => nat list"    ("(1[_../_])")
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  "[i..j] == [i..<(Suc j)]"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat"
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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  "null([]) = True"
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  "null(x#xs) = False"
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  "last(x#xs) = (if xs=[] then x else last xs)"
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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  append_Nil:"[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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defs
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rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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rotate_def:  "rotate n == rotate1 ^ n"
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list_all2_def:
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 "list_all2 P xs ys ==
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  length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)"
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sublist_def:
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 "sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))"
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"splice [] ys = ys"
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"splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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  -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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  "x mem [] = False"
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  "x mem (y#ys) = (if y=x then True else x mem ys)"
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 "list_inter [] bs = []"
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 "list_inter (a#as) bs =
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  (if a \<in> set bs then a#(list_inter as bs) else list_inter as bs)"
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  "list_all P [] = True"
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  "list_all P (x#xs) = (P(x) \<and> list_all P xs)"
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"list_ex P [] = False"
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"list_ex P (x#xs) = (P x \<or> list_ex P xs)"
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 "filtermap f [] = []"
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 "filtermap f (x#xs) =
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    (case f x of None \<Rightarrow> filtermap f xs
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     | Some y \<Rightarrow> y # (filtermap f xs))"
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  "map_filter f P [] = []"
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  "map_filter f P (x#xs) = (if P x then f x # map_filter f P xs else 
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               map_filter f P xs)"
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"itrev [] ys = ys"
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"itrev (x#xs) ys = itrev xs (x#ys)"
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{text length} *}
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text {*
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Needs to come before @{text "@"} because of theorem @{text
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append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs, auto)
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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subsubsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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13883
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lemma append_eq_append_conv [simp]:
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parents: 13863
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 "!!ys. length xs = length ys \<or> length us = length vs
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
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parents: 13863
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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berghofe
parents: 13863
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apply (induct xs)
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paulson
parents: 14187
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 apply (case_tac ys, simp, force)
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paulson
parents: 14187
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apply (case_tac ys, force, simp)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
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apply (induct xs)
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parents: 14402
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 apply fastsimp
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apply(case_tac zs)
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 apply simp
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apply fastsimp
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   356
done
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parents: 14402
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   357
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   358
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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   359
by simp
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   360
1ebd8ed5a1a0 tuned document;
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parents: 13124
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   361
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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   362
by simp
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parents: 12887
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   363
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parents: 13124
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   364
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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parents: 13142
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   365
by simp
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parents: 12887
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parents: 13124
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   367
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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parents: 13142
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   368
using append_same_eq [of _ _ "[]"] by auto
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parents: 3465
diff changeset
   369
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parents: 13124
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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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parents: 13142
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   371
using append_same_eq [of "[]"] by auto
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parents: 12887
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   372
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parents: 13124
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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   374
by (induct xs) auto
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parents: 12887
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   375
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parents: 13124
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   376
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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   377
by (induct xs) auto
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parents: 12887
diff changeset
   378
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parents: 13124
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   379
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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parents: 13142
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   380
by (simp add: hd_append split: list.split)
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parents: 12887
diff changeset
   381
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parents: 13124
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   382
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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parents: 13142
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   383
by (simp split: list.split)
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parents: 12887
diff changeset
   384
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parents: 13124
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   385
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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parents: 13142
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   386
by (simp add: tl_append split: list.split)
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wenzelm
parents: 12887
diff changeset
   387
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   388
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parents: 14247
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   389
lemma Cons_eq_append_conv: "x#xs = ys@zs =
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nipkow
parents: 14247
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   390
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
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nipkow
parents: 14247
diff changeset
   391
by(cases ys) auto
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parents: 14247
diff changeset
   392
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   393
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   394
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   395
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   396
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parents: 14247
diff changeset
   397
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wenzelm
parents: 13124
diff changeset
   398
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
13114
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wenzelm
parents: 12887
diff changeset
   399
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   400
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
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nipkow
parents: 13142
diff changeset
   401
by simp
13114
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wenzelm
parents: 12887
diff changeset
   402
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   403
lemma Cons_eq_appendI:
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nipkow
parents: 13142
diff changeset
   404
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
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nipkow
parents: 13142
diff changeset
   405
by (drule sym) simp
13114
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wenzelm
parents: 12887
diff changeset
   406
13142
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wenzelm
parents: 13124
diff changeset
   407
lemma append_eq_appendI:
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parents: 13142
diff changeset
   408
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
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nipkow
parents: 13142
diff changeset
   409
by (drule sym) simp
13114
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wenzelm
parents: 12887
diff changeset
   410
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   411
13142
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wenzelm
parents: 13124
diff changeset
   412
text {*
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parents: 13142
diff changeset
   413
Simplification procedure for all list equalities.
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nipkow
parents: 13142
diff changeset
   414
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   415
- both lists end in a singleton list,
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   416
- or both lists end in the same list.
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parents: 13124
diff changeset
   417
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   418
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   419
ML_setup {*
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   420
local
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   421
13122
wenzelm
parents: 13114
diff changeset
   422
val append_assoc = thm "append_assoc";
wenzelm
parents: 13114
diff changeset
   423
val append_Nil = thm "append_Nil";
wenzelm
parents: 13114
diff changeset
   424
val append_Cons = thm "append_Cons";
wenzelm
parents: 13114
diff changeset
   425
val append1_eq_conv = thm "append1_eq_conv";
wenzelm
parents: 13114
diff changeset
   426
val append_same_eq = thm "append_same_eq";
wenzelm
parents: 13114
diff changeset
   427
13114
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wenzelm
parents: 12887
diff changeset
   428
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   429
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   430
  | last (Const("List.op @",_) $ _ $ ys) = last ys
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   431
  | last t = t;
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parents: 12887
diff changeset
   432
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   433
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
13462
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wenzelm
parents: 13366
diff changeset
   434
  | list1 _ = false;
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wenzelm
parents: 12887
diff changeset
   435
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   436
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   437
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   438
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   439
  | butlast xs = Const("List.list.Nil",fastype_of xs);
13114
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wenzelm
parents: 12887
diff changeset
   440
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   441
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   442
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   443
fun list_eq sg ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   444
  let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   445
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   446
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   447
      let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   448
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   449
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   450
        val appT = [listT,listT] ---> listT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   451
        val app = Const("List.op @",appT)
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   452
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
13480
bb72bd43c6c3 use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents: 13462
diff changeset
   453
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
17956
369e2af8ee45 Goal.prove;
wenzelm
parents: 17906
diff changeset
   454
        val thm = Goal.prove sg [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   455
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   456
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
13114
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wenzelm
parents: 12887
diff changeset
   457
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   458
  in
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   459
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   460
    else if lastl aconv lastr then rearr append_same_eq
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   461
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   462
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   463
13114
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wenzelm
parents: 12887
diff changeset
   464
in
13462
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wenzelm
parents: 13366
diff changeset
   465
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   466
val list_eq_simproc =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   467
  Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   468
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   469
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   470
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   471
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   472
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   473
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   474
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   475
subsubsection {* @{text map} *}
13114
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wenzelm
parents: 12887
diff changeset
   476
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   477
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
13145
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nipkow
parents: 13142
diff changeset
   478
by (induct xs) simp_all
13114
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wenzelm
parents: 12887
diff changeset
   479
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   480
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
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nipkow
parents: 13142
diff changeset
   481
by (rule ext, induct_tac xs) auto
13114
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wenzelm
parents: 12887
diff changeset
   482
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   483
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
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nipkow
parents: 13142
diff changeset
   484
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   485
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   486
lemma map_compose: "map (f o g) xs = map f (map g xs)"
13145
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nipkow
parents: 13142
diff changeset
   487
by (induct xs) (auto simp add: o_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   488
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   489
lemma rev_map: "rev (map f xs) = map f (rev xs)"
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nipkow
parents: 13142
diff changeset
   490
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   491
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   492
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   493
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   494
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   495
lemma map_cong [recdef_cong]:
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nipkow
parents: 13142
diff changeset
   496
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
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nipkow
parents: 13142
diff changeset
   497
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   498
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   499
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   500
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
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nipkow
parents: 13142
diff changeset
   501
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   502
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   503
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   504
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   505
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   506
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   507
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   508
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   509
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   510
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   511
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   512
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   513
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   514
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   515
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   516
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   517
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   518
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   519
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   520
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   521
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   522
lemma map_eq_imp_length_eq:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   523
  "!!xs. map f xs = map f ys ==> length xs = length ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   524
apply (induct ys)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   525
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   526
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   527
apply clarify
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   528
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   529
apply fast
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   530
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   531
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   532
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   533
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   534
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   535
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   536
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   537
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   538
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   539
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   540
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   541
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   542
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   543
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   544
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   545
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   546
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   547
lemma map_injective:
14338
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   548
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   549
by (induct ys) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   550
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   551
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   552
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   553
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   554
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   555
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   556
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   557
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   558
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   559
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   560
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   561
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   562
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   563
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   564
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   565
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   566
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   567
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   568
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   569
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   570
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   571
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   572
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   573
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   574
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   575
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   576
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   577
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   578
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   579
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   580
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   581
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   582
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   583
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   584
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   585
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   586
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   587
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   588
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   589
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   590
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   591
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   592
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   593
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   594
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   595
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   596
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   597
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   598
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   599
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   600
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   601
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   602
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   603
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   604
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   605
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   606
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   607
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   608
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   609
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   610
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   611
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   612
apply (induct xs, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   613
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   614
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   615
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   616
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   617
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   618
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   619
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   620
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
15489
d136af442665 Replaced application of subst by simplesubst in proof of rev_induct
berghofe
parents: 15439
diff changeset
   621
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   622
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   623
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   624
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   625
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   626
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   627
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   628
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   629
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   630
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   631
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   632
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   633
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   634
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   635
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   636
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   637
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   638
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   639
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   640
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   641
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   642
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   643
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   644
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   645
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   646
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   647
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   648
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   649
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   650
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   651
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   652
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   653
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   654
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   655
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   656
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   657
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   658
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   659
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   660
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   661
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   662
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   663
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   664
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   665
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   666
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   667
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   668
15425
6356d2523f73 [ .. (] -> [ ..< ]
nipkow
parents: 15392
diff changeset
   669
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   670
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   671
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   672
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   673
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   674
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
15113
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   675
proof (induct xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   676
  case Nil show ?case by simp
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   677
  case (Cons a xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   678
  show ?case
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   679
  proof 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   680
    assume "x \<in> set (a # xs)"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   681
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   682
      by (simp, blast intro: Cons_eq_appendI)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   683
  next
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   684
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   685
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   686
    show "x \<in> set (a # xs)" 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   687
      by (cases ys, auto simp add: eq)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   688
  qed
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   689
qed
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   690
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   691
lemma in_set_conv_decomp_first:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   692
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   693
proof (induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   694
  case Nil show ?case by simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   695
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   696
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   697
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   698
  proof cases
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   699
    assume "x = a" thus ?case using Cons by force
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   700
  next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   701
    assume "x \<noteq> a"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   702
    show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   703
    proof
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   704
      assume "x \<in> set (a # xs)"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   705
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   706
	by(fastsimp intro!: Cons_eq_appendI)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   707
    next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   708
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   709
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   710
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   711
    qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   712
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   713
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   714
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   715
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   716
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   717
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   718
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   719
lemma finite_list: "finite A ==> EX l. set l = A"
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   720
apply (erule finite_induct, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   721
apply (rule_tac x="x#l" in exI, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   722
done
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   723
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   724
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   725
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   726
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
   727
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   728
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   729
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   730
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   731
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   732
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   733
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   734
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   735
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   736
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   737
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   738
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   739
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   740
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   741
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   742
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   743
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   744
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   745
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   746
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   747
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   748
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   749
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   750
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   751
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   752
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   753
  by (induct xs) simp_all
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   754
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   755
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   756
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   757
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   758
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   759
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   760
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   761
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   762
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   763
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   764
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   765
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   766
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   767
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   768
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   769
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   770
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   771
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   772
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   773
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   774
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   775
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   776
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   777
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   778
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   779
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   780
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   781
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   782
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   783
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   784
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   785
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   786
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   787
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   788
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   789
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   790
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   791
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   792
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   793
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   794
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   795
    hence eq: "?S' = insert 0 (Suc ` ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   796
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   797
    have "length (filter p (x # xs)) = Suc(card ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   798
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   799
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   800
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   801
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   802
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   803
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   804
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   805
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   806
    hence eq: "?S' = Suc ` ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   807
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   808
    have "length (filter p (x # xs)) = card ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   809
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   810
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   811
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   812
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   813
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   814
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   815
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   816
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   817
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   818
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   819
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   820
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   821
  (concl is "\<exists>us vs. ?P ys us vs")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   822
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   823
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   824
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   825
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   826
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   827
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   828
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   829
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   830
    proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   831
      assume xy: "x = y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   832
      show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   833
      proof from Py xy Cons(2) show "?Q []" by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   834
    next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   835
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   836
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   837
  next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   838
    assume Py: "\<not> P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   839
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   840
    show ?thesis (is "? us. ?Q us")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   841
    proof show "?Q (y#us)" using 1 by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   842
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   843
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   844
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   845
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   846
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   847
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   848
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   849
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   850
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   851
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   852
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   853
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   854
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   855
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   856
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   857
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   858
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   859
18336
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
   860
lemma filter_cong[recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   861
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   862
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   863
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   864
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   865
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   866
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   867
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   868
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   869
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   870
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   871
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   872
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   873
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   874
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   875
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   876
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   877
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   878
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   879
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   880
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   881
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   882
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   883
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   884
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   885
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   886
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   887
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   888
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   889
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   890
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   891
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   892
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   893
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   894
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   895
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   896
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   897
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   898
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   899
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   900
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   901
lemma nth_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   902
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   903
apply (induct "xs", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   904
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   905
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   906
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   907
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   908
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   909
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   910
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   911
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   912
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   913
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   914
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   915
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   916
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   917
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   918
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   919
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   920
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   921
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   922
lemma list_eq_iff_nth_eq:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   923
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   924
apply(induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   925
 apply simp apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   926
apply(case_tac ys)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   927
 apply simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   928
apply(simp add:nth_Cons split:nat.split)apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   929
done
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   930
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   931
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15246
diff changeset
   932
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   933
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   934
apply (rule_tac x = 0 in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   935
 apply (rule_tac x = "Suc i" in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   936
apply (case_tac i, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   937
apply (rename_tac j)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   938
apply (rule_tac x = j in exI, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   939
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   940
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   941
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   942
by(auto simp:set_conv_nth)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   943
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   944
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   945
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   946
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   947
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   948
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   949
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   950
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   951
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   952
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   953
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   954
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   955
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   956
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   957
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   958
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   959
subsubsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   960
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   961
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   962
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   963
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   964
lemma nth_list_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   965
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   966
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   967
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   968
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   969
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   970
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   971
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   972
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   973
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   974
lemma list_update_overwrite [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   975
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   976
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   977
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   978
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   979
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   980
apply(simp split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   981
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   982
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   983
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   984
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   985
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   986
apply (case_tac i)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   987
apply simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   988
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   989
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   990
lemma list_update_same_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   991
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   992
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   993
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   994
lemma list_update_append1:
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   995
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   996
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   997
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   998
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   999
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1000
lemma list_update_append:
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1001
  "!!n. (xs @ ys) [n:= x] = 
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1002
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1003
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1004
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1005
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1006
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1007
by (induct xs, auto)
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1008
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1009
lemma update_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1010
"!!i xy xs. length xs = length ys ==>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1011
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1012
by (induct ys) (auto, case_tac xs, auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1013
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1014
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1015
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1016
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1017
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1018
by (blast dest!: set_update_subset_insert [THEN subsetD])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1019
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1020
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1021
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1022
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1023
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1024
subsubsection {* @{text last} and @{text butlast} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1025
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1026
lemma last_snoc [simp]: "last (xs @ [x]) = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1027
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1028
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1029
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1030
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1031
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1032
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1033
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1034
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1035
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1036
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1037
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1038
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1039
by (induct xs) (auto)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1040
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1041
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1042
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1043
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1044
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1045
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1046
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1047
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1048
by(rule rev_exhaust[of xs]) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1049
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1050
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1051
by(cases xs) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1052
17765
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1053
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1054
by (induct as) auto
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1055
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1056
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1057
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1058
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1059
lemma butlast_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1060
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1061
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1062
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1063
lemma append_butlast_last_id [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1064
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1065
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1066
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1067
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1068
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1069
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1070
lemma in_set_butlast_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1071
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1072
by (auto dest: in_set_butlastD simp add: butlast_append)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1073
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1074
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1075
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1076
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1077
apply (auto split:nat.split)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1078
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1079
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1080
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1081
by(induct xs)(auto simp:neq_Nil_conv)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1082
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1083
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1084
subsubsection {* @{text take} and @{text drop} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1085
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1086
lemma take_0 [simp]: "take 0 xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1087
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1088
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1089
lemma drop_0 [simp]: "drop 0 xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1090
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1091
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1092
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1093
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1094
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1095
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1096
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1097
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1098
declare take_Cons [simp del] and drop_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1099
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1100
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1101
by(clarsimp simp add:neq_Nil_conv)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1102
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1103
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1104
by(cases xs, simp_all)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1105
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1106
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1107
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1108
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1109
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1110
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1111
apply(simp add:drop_Cons nth_Cons split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1112
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1113
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1114
lemma take_Suc_conv_app_nth:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1115
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1116
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1117
apply (case_tac i, auto)
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1118
done
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1119
14591
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1120
lemma drop_Suc_conv_tl:
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1121
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1122
apply (induct xs, simp)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1123
apply (case_tac i, auto)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1124
done
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1125
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1126
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1127
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1128
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1129
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1130
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1131
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1132
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1133
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1134
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1135
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1136
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1137
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1138
lemma take_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1139
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1140
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1141
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1142
lemma drop_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1143
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1144
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1145
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1146
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1147
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1148
apply (case_tac xs, auto)
15236
f289e8ba2bb3 Proofs needed to be updated because induction now preserves name of
nipkow
parents: 15176
diff changeset
  1149
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1150
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1151
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1152
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1153
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1154
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1155
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1156
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1157
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1158
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1159
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1160
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1161
14802
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1162
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1163
apply(induct xs)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1164
 apply simp
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1165
apply(simp add: take_Cons drop_Cons split:nat.split)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1166
done
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1167
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1168
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1169
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1170
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1171
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1172
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1173
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1174
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1175
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1176
apply(simp add:take_Cons split:nat.split)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1177
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1178
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1179
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1180
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1181
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1182
apply(simp add:drop_Cons split:nat.split)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1183
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1184
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1185
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1186
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1187
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1188
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1189
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1190
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1191
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1192
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1193
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1194
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1195
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1196
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1197
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1198
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1199
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1200
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1201
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1202
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1203
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1204
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1205
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1206
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1207
apply (case_tac n, blast)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1208
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1209
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1210
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1211
lemma nth_drop [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1212
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1213
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1214
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1215
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
  1216
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1217
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1218
by(simp add: hd_conv_nth)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1219
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1220
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1221
by(induct xs)(auto simp:take_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1222
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1223
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1224
by(induct xs)(auto simp:drop_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1225
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1226
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1227
using set_take_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1228
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1229
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1230
using set_drop_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1231
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1232
lemma append_eq_conv_conj:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1233
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1234
apply (induct xs, simp, clarsimp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1235
apply (case_tac zs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1236
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1237
14050
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1238
lemma take_add [rule_format]: 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1239
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1240
apply (induct xs, auto) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1241
apply (case_tac i, simp_all) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1242
done
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1243
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1244
lemma append_eq_append_conv_if:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1245
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1246
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1247
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1248
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1249
apply(induct xs\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1250
 apply simp
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1251
apply(case_tac ys\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1252
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1253
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1254
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1255
lemma take_hd_drop:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1256
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1257
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1258
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1259
apply(simp add:drop_Cons split:nat.split)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1260
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1261
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1262
lemma id_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1263
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1264
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1265
  assume si: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1266
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1267
  moreover
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1268
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1269
    apply (rule_tac take_Suc_conv_app_nth) by arith
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1270
  ultimately show ?thesis by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1271
qed
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1272
  
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1273
lemma upd_conv_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1274
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1275
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1276
  assume i: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1277
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1278
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1279
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1280
    using i by (simp add: list_update_append)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1281
  finally show ?thesis .
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1282
qed
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1283
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1284
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1285
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1286
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1287
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1288
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1289
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1290
lemma takeWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1291
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1292
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1293
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1294
lemma takeWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1295
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1296
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1297
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1298
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1299
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1300
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1301
lemma dropWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1302
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1303
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1304
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1305
lemma dropWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1306
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1307
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1308
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1309
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1310
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1311
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1312
lemma takeWhile_eq_all_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1313
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1314
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1315
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1316
lemma dropWhile_eq_Nil_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1317
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1318
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1319
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1320
lemma dropWhile_eq_Cons_conv:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1321
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1322
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1323
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1324
text{* The following two lemmmas could be generalized to an arbitrary
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1325
property. *}
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1326
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1327
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1328
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1329
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1330
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1331
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1332
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1333
apply(induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1334
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1335
apply auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1336
apply(subst dropWhile_append2)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1337
apply auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1338
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1339
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1340
lemma takeWhile_not_last:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1341
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1342
apply(induct xs)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1343
 apply simp
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1344
apply(case_tac xs)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1345
apply(auto)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1346
done
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1347
18336
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1348
lemma takeWhile_cong [recdef_cong]:
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1349
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1350
  ==> takeWhile P l = takeWhile Q k"
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1351
  by (induct k fixing: l, simp_all)
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1352
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1353
lemma dropWhile_cong [recdef_cong]:
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1354
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1355
  ==> dropWhile P l = dropWhile Q k"
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1356
  by (induct k fixing: l, simp_all)
1a2e30b37ed3 Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents: 18049
diff changeset
  1357
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1358
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1359
subsubsection {* @{text zip} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1360