src/HOL/List.thy
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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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uses "Tools/string_syntax.ML"
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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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  append :: "'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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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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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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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  allpairs :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> 'c list"
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abbreviation
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  upto:: "nat => nat => nat list"  ("(1[_../_])") where
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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" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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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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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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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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primrec
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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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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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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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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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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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"allpairs f [] ys = []"
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"allpairs f (x # xs) ys = map (f x) ys @ allpairs f xs ys"
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subsubsection {* List comprehehsion *}
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text{* Input syntax for Haskell-like list comprehension
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notation. Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"}, the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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There are a number of differences to Haskell.  The general synatx is
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@{text"[e. p \<leftarrow> xs, \<dots>]"} rather than \verb![x| x <- xs, ...]!. The
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first qualifier must be a generator (@{text"p \<leftarrow> xs"}). Patterns in
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generators can only be tuples (at the moment). Finally, guards are
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translated into filters, which simplifies theorem proving.
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*}
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(*
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The print translation from internal form to list comprehensions would
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be nice. Unfortunately one cannot just turn the translations around
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because in the final equality @{text p} occurs twice on the
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right-hand side. Due to this restriction, the translation must be hand-coded.
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A more substantial extension would be proper theorem proving
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support. For example, it would be nice if
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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produced something like
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
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*)
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nonterminals lc_qual
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syntax
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"_listcompr" :: "'a \<Rightarrow> pttrn \<Rightarrow> 'b list \<Rightarrow> lc_qual \<Rightarrow> 'a list"  ("[_ . _ \<leftarrow> __")
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"_listcompr" :: "'a \<Rightarrow> pttrn \<Rightarrow> 'b list \<Rightarrow> lc_qual \<Rightarrow> 'a list"  ("[_ . _ <- __")
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"_lc_end" :: "lc_qual" ("]")
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual \<Rightarrow> lc_qual" (",_ \<leftarrow> __")
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"_lc_gen" :: "pttrn \<Rightarrow> 'a list \<Rightarrow> lc_qual \<Rightarrow> lc_qual" (",_ <- __")
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"_lc_test" :: "bool \<Rightarrow> lc_qual \<Rightarrow> lc_qual" (",__")
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translations
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"[e. p\<leftarrow>xs]" => "map (%p. e) xs"
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"_listcompr e p xs (_lc_gen q ys Q)" =>
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 "concat (map (%p. _listcompr e q ys Q) xs)"
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"_listcompr e p xs (_lc_test P Q)" => "_listcompr e p (filter (%p. P) xs) Q"
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(* Some examples:
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term "[(x,y). x \<leftarrow> xs, x<y]"
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term "[(x,y). x \<leftarrow> xs, x<y, z\<leftarrow>zs]"
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term "[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x<y]"
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term "[(x,y,z). x \<leftarrow> xs, y \<leftarrow> ys, z\<leftarrow> zs]"
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term "[x. x \<leftarrow> xs, x < a, x > b]"
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*)
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subsubsection {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "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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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const 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_allpairs[simp]:
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 "length(allpairs f xs ys) = length xs * length ys"
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by(induct 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)
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apply 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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   342
done
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   343
22493
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lemma list_induct2': 
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   345
  "\<lbrakk> P [] [];
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  \<And>x xs. P (x#xs) [];
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  \<And>y ys. P [] (y#ys);
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   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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   349
 \<Longrightarrow> P xs ys"
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   350
by (induct xs arbitrary: ys) (case_tac x, auto)+
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   351
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   352
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
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apply(rule Eq_FalseI)
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by auto
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   355
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   356
(*
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   357
Reduces xs=ys to False if xs and ys cannot be of the same length.
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   358
This is the case if the atomic sublists of one are a submultiset
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   359
of those of the other list and there are fewer Cons's in one than the other.
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   360
*)
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   361
ML_setup {*
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   362
local
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   363
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   364
fun len (Const("List.list.Nil",_)) acc = acc
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  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
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  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
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  | len (Const("List.rev",_) $ xs) acc = len xs acc
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  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
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  | len t (ts,n) = (t::ts,n);
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   370
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   371
fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
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   372
  let
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   373
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
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   374
    fun prove_neq() =
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   375
      let
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   376
        val Type(_,listT::_) = eqT;
22994
02440636214f abstract size function in hologic.ML
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   377
        val size = HOLogic.size_const listT;
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   378
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
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   379
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
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   380
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
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haftmann
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   381
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
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parents: 22551
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   382
      in SOME (thm RS @{thm neq_if_length_neq}) end
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   383
  in
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dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
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   384
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   385
       n < m andalso submultiset (op aconv) (rs,ls)
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diff changeset
   386
    then prove_neq() else NONE
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   387
  end;
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   388
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   389
in
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   390
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   391
val list_neq_simproc =
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parents: 22551
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   392
  Simplifier.simproc @{theory} "list_neq" ["(xs::'a list) = ys"] (K list_eq);
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   393
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   394
end;
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   395
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   396
Addsimprocs [list_neq_simproc];
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   397
*}
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   398
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diff changeset
   399
15392
290bc97038c7 First step in reorganizing Finite_Set
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   400
subsubsection {* @{text "@"} -- append *}
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wenzelm
parents: 12887
diff changeset
   401
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diff changeset
   402
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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   403
by (induct xs) auto
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f2b00262bdfc converted;
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parents: 12887
diff changeset
   404
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   405
lemma append_Nil2 [simp]: "xs @ [] = xs"
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   406
by (induct xs) auto
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157be29ad5ba Improved length = size translation.
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diff changeset
   407
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diff changeset
   408
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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   409
by (induct xs) auto
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   410
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   411
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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   412
by (induct xs) auto
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   413
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   414
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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   415
by (induct xs) auto
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   416
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   417
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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   418
by (induct xs) auto
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   419
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
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   420
lemma append_eq_append_conv [simp]:
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parents: 13863
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   421
 "!!ys. length xs = length ys \<or> length us = length vs
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
   422
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
   423
apply (induct xs)
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diff changeset
   424
 apply (case_tac ys, simp, force)
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   425
apply (case_tac ys, force, simp)
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   426
done
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parents: 13124
diff changeset
   427
14495
e2a1c31cf6d3 Added append_eq_append_conv2
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   428
lemma append_eq_append_conv2: "!!ys zs ts.
e2a1c31cf6d3 Added append_eq_append_conv2
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   429
 (xs @ ys = zs @ ts) =
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   430
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   431
apply (induct xs)
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   432
 apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   433
apply(case_tac zs)
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   434
 apply simp
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   435
apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   436
done
e2a1c31cf6d3 Added append_eq_append_conv2
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diff changeset
   437
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diff changeset
   438
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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diff changeset
   439
by simp
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diff changeset
   440
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   441
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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diff changeset
   442
by simp
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diff changeset
   443
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diff changeset
   444
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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diff changeset
   445
by simp
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wenzelm
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diff changeset
   446
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diff changeset
   447
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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diff changeset
   448
using append_same_eq [of _ _ "[]"] by auto
3507
157be29ad5ba Improved length = size translation.
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diff changeset
   449
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diff changeset
   450
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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diff changeset
   451
using append_same_eq [of "[]"] by auto
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diff changeset
   452
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wenzelm
parents: 13124
diff changeset
   453
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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diff changeset
   454
by (induct xs) auto
13114
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wenzelm
parents: 12887
diff changeset
   455
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1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   456
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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diff changeset
   457
by (induct xs) auto
13114
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wenzelm
parents: 12887
diff changeset
   458
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1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   459
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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diff changeset
   460
by (simp add: hd_append split: list.split)
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parents: 12887
diff changeset
   461
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diff changeset
   462
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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diff changeset
   463
by (simp split: list.split)
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parents: 12887
diff changeset
   464
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parents: 13124
diff changeset
   465
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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parents: 13142
diff changeset
   466
by (simp add: tl_append split: list.split)
13114
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wenzelm
parents: 12887
diff changeset
   467
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   468
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diff changeset
   469
lemma Cons_eq_append_conv: "x#xs = ys@zs =
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diff changeset
   470
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
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diff changeset
   471
by(cases ys) auto
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parents: 14247
diff changeset
   472
15281
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parents: 15251
diff changeset
   473
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
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diff changeset
   474
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   475
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   476
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diff changeset
   477
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diff changeset
   478
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
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f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   479
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   480
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
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diff changeset
   481
by simp
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parents: 12887
diff changeset
   482
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wenzelm
parents: 13124
diff changeset
   483
lemma Cons_eq_appendI:
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diff changeset
   484
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
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nipkow
parents: 13142
diff changeset
   485
by (drule sym) simp
13114
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wenzelm
parents: 12887
diff changeset
   486
13142
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wenzelm
parents: 13124
diff changeset
   487
lemma append_eq_appendI:
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diff changeset
   488
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
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nipkow
parents: 13142
diff changeset
   489
by (drule sym) simp
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wenzelm
parents: 12887
diff changeset
   490
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   491
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   492
text {*
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parents: 13142
diff changeset
   493
Simplification procedure for all list equalities.
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   494
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   495
- both lists end in a singleton list,
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   496
- or both lists end in the same list.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   497
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   498
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   499
ML_setup {*
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   500
local
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   501
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   502
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   503
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   504
  | last (Const("List.append",_) $ _ $ ys) = last ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   505
  | last t = t;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   506
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   507
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   508
  | list1 _ = false;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   509
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   510
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   511
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   512
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   513
  | butlast xs = Const("List.list.Nil",fastype_of xs);
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   514
22633
haftmann
parents: 22551
diff changeset
   515
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann
parents: 22551
diff changeset
   516
  @{thm append_Nil}, @{thm append_Cons}];
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   517
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   518
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   519
  let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   520
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   521
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   522
      let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   523
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   524
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   525
        val appT = [listT,listT] ---> listT
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   526
        val app = Const("List.append",appT)
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   527
        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
   528
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   529
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   530
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   531
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   532
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   533
  in
22633
haftmann
parents: 22551
diff changeset
   534
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann
parents: 22551
diff changeset
   535
    else if lastl aconv lastr then rearr @{thm append_same_eq}
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   536
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   537
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   538
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   539
in
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   540
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   541
val list_eq_simproc =
22633
haftmann
parents: 22551
diff changeset
   542
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   543
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   544
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   545
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   546
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   547
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   548
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   549
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   550
subsubsection {* @{text map} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   551
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   552
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   553
by (induct xs) simp_all
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   554
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   555
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   556
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   557
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   558
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   559
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   560
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   561
lemma map_compose: "map (f o g) xs = map f (map g xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   562
by (induct xs) (auto simp add: o_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   563
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   564
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   565
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   566
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   567
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
   568
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   569
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   570
lemma map_cong [fundef_cong, recdef_cong]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   571
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   572
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   573
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   574
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   575
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   576
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   577
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   578
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   579
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   580
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   581
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   582
 "(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
   583
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   584
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   585
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   586
 "(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
   587
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   588
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   589
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   590
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   591
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   592
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   593
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   594
  "(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
   595
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   596
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   597
lemma map_eq_imp_length_eq:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   598
  "!!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
   599
apply (induct ys)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   600
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   601
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   602
apply clarify
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   603
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   604
apply fast
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   605
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   606
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   607
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   608
 "[| 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
   609
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   610
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   611
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   612
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   613
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   614
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   615
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   616
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   617
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   618
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   619
 "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
   620
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   621
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   622
lemma map_injective:
14338
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   623
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   624
by (induct ys) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   625
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   626
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
   627
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   628
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   629
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   630
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   631
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   632
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   633
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   634
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   635
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   636
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   637
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   638
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   639
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   640
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   641
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   642
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   643
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   644
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   645
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   646
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   647
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   648
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
   649
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   650
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   651
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
   652
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   653
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   654
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   655
  "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
   656
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   657
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   658
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   659
  "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
   660
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   661
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   662
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   663
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   664
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   665
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   666
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   667
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   668
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   669
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   670
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   671
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   672
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   673
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   674
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   675
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   676
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   677
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   678
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   679
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   680
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   681
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   682
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   683
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   684
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   685
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   686
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   687
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   688
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   689
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   690
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   691
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   692
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   693
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   694
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   695
  "[| 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
   696
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   697
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   698
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   699
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   700
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   701
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   702
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   703
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   704
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   705
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   706
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   707
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   708
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   709
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   710
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   711
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   712
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   713
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   714
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   715
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   716
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   717
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   718
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   719
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   720
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   721
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   722
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   723
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   724
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   725
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   726
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
   727
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   728
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   729
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   730
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   731
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   732
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   733
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   734
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   735
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   736
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   737
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   738
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   739
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   740
22828
2064f0fd20c9 added allpairs
nipkow
parents: 22799
diff changeset
   741
lemma set_allpairs[simp]:
22830
72a7b6ad153b tuned allpairs
nipkow
parents: 22828
diff changeset
   742
 "set(allpairs f xs ys) = {z. EX x : set xs. EX y : set ys. z = f x y}"
22828
2064f0fd20c9 added allpairs
nipkow
parents: 22799
diff changeset
   743
by(induct xs) auto
2064f0fd20c9 added allpairs
nipkow
parents: 22799
diff changeset
   744
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   745
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
   746
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   747
15425
6356d2523f73 [ .. (] -> [ ..< ]
nipkow
parents: 15392
diff changeset
   748
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   749
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   750
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   751
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   752
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   753
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
   754
proof (induct xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   755
  case Nil show ?case by simp
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   756
  case (Cons a xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   757
  show ?case
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   758
  proof 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   759
    assume "x \<in> set (a # xs)"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   760
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   761
      by (simp, blast intro: Cons_eq_appendI)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   762
  next
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   763
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   764
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   765
    show "x \<in> set (a # xs)" 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   766
      by (cases ys, auto simp add: eq)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   767
  qed
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   768
qed
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   769
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   770
lemma in_set_conv_decomp_first:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   771
 "(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
   772
proof (induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   773
  case Nil show ?case by simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   774
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   775
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   776
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   777
  proof cases
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   778
    assume "x = a" thus ?case using Cons by force
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   779
  next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   780
    assume "x \<noteq> a"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   781
    show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   782
    proof
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   783
      assume "x \<in> set (a # xs)"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   784
      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
   785
	by(fastsimp intro!: Cons_eq_appendI)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   786
    next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   787
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   788
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   789
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   790
    qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   791
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   792
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   793
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   794
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   795
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   796
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   797
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   798
lemma finite_list: "finite A ==> EX l. set l = A"
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   799
apply (erule finite_induct, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   800
apply (rule_tac x="x#l" in exI, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   801
done
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   802
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   803
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   804
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   805
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
   806
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   807
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   808
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   809
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   810
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   811
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   812
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   813
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   814
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   815
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
   816
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   817
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   818
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   819
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   820
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   821
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   822
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   823
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   824
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   825
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
   826
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   827
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   828
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
   829
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   830
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   831
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   832
  by (induct xs) simp_all
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   833
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   834
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   835
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   836
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   837
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   838
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   839
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   840
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   841
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   842
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   843
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   844
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   845
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   846
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   847
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   848
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   849
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   850
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   851
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   852
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   853
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   854
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   855
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   856
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   857
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   858
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   859
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   860
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   861
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   862
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   863
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   864
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   865
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   866
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   867
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   868
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   869
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   870
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   871
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   872
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   873
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   874
    hence eq: "?S' = insert 0 (Suc ` ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   875
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   876
    have "length (filter p (x # xs)) = Suc(card ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   877
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   878
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   879
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   880
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   881
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   882
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   883
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   884
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   885
    hence eq: "?S' = Suc ` ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   886
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   887
    have "length (filter p (x # xs)) = card ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   888
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   889
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   890
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   891
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   892
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   893
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   894
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   895
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   896
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   897
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   898
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   899
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
19585
70a1ce3b23ae removed 'concl is' patterns;
wenzelm
parents: 19487
diff changeset
   900
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   901
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   902
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   903
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   904
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   905
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   906
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   907
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   908
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   909
    proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   910
      assume xy: "x = y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   911
      show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   912
      proof from Py xy Cons(2) show "?Q []" by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   913
    next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   914
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   915
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   916
  next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   917
    assume Py: "\<not> P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   918
    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
   919
    show ?thesis (is "? us. ?Q us")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   920
    proof show "?Q (y#us)" using 1 by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   921
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   922
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   923
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   924
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   925
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   926
  \<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
   927
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   928
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   929
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   930
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   931
  (\<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
   932
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   933
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   934
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   935
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   936
  (\<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
   937
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   938
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   939
lemma filter_cong[fundef_cong, recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   940
 "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
   941
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   942
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   943
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   944
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   945
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   946
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   947
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   948
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   949
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   950
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   951
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   952
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   953
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   954
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   955
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   956
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   957
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   958
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   959
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   960
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   961
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   962
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   963
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   964
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   965
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   966
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   967
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   968
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   969
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   970
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   971
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   972
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   973
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   974
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   975
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   976
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   977
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   978
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   979
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   980
lemma nth_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   981
"!!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
   982
apply (induct "xs", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   983
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   984
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   985
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   986
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
   987
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   988
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   989
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
   990
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   991
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   992
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   993
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   994
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   995
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   996
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   997
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   998
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   999
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1000
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1001
lemma list_eq_iff_nth_eq:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1002
 "!!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
  1003
apply(induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1004
 apply simp apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1005
apply(case_tac ys)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1006
 apply simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1007
apply(simp add:nth_Cons split:nat.split)apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1008
done
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1009
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1010
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
  1011
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1012
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1013
apply (rule_tac x = 0 in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1014
 apply (rule_tac x = "Suc i" in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1015
apply (case_tac i, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1016
apply (rename_tac j)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1017
apply (rule_tac x = j in exI, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1018
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1019
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1020
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
  1021
by(auto simp:set_conv_nth)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1022
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1023
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1024
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1025
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1026
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1027
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1028
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1029
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1030
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1031
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1032
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1033
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1034
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1035
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1036
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1037
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1038
subsubsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1039
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1040
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1041
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1042
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1043
lemma nth_list_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1044
"!!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
  1045
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1046
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1047
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1048
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1049
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1050
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
  1051
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1052
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1053
lemma list_update_overwrite [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1054
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1055
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1056
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1057
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1058
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1059
apply(simp split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1060
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1061
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1062
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
  1063
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1064
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1065
apply (case_tac i)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1066
apply simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1067
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1068
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1069
lemma list_update_same_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1070
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1071
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1072
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1073
lemma list_update_append1:
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1074
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1075
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1076
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1077
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1078
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1079
lemma list_update_append:
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1080
  "!!n. (xs @ ys) [n:= x] = 
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1081
  (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
  1082
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1083
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1084
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1085
 "(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
  1086
by (induct xs, auto)
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1087
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1088
lemma update_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1089
"!!i xy xs. length xs = length ys ==>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1090
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1091
by (induct ys) (auto, case_tac xs, auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1092
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1093
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1094
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1095
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1096
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1097
by (blast dest!: set_update_subset_insert [THEN subsetD])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1098
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1099
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
  1100
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1101
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1102
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1103
subsubsection {* @{text last} and @{text butlast} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1104
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1105
lemma last_snoc [simp]: "last (xs @ [x]) = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1106
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1107
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1108
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1109
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1110
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1111
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1112
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1113
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1114
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1115
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1116
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1117
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1118
by (induct xs) (auto)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1119
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1120
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1121
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1122
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1123
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1124
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1125
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1126
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1127
by(rule rev_exhaust[of xs]) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1128
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1129
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1130
by(cases xs) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1131
17765
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1132
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1133
by (induct as) auto
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1134
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1135
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1136
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1137
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1138
lemma butlast_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1139
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1140
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1141
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1142
lemma append_butlast_last_id [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1143
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1144
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1145
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1146
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1147
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1148
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1149
lemma in_set_butlast_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1150
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1151
by (auto dest: in_set_butlastD simp add: butlast_append)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1152
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1153
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
  1154
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1155
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1156
apply (auto split:nat.split)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1157
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1158
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1159
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1160
by(induct xs)(auto simp:neq_Nil_conv)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1161
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1162
subsubsection {* @{text take} and @{text drop} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1163
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1164
lemma take_0 [simp]: "take 0 xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1165
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1166
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1167
lemma drop_0 [simp]: "drop 0 xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1168
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1169
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1170
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1171
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1172
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1173
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1174
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1175
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1176
declare take_Cons [simp del] and drop_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1177
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1178
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
  1179
by(clarsimp simp add:neq_Nil_conv)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1180
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1181
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1182
by(cases xs, simp_all)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1183
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1184
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1185
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1186
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1187
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1188
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1189
apply(simp add:drop_Cons nth_Cons split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1190
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1191
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1192
lemma take_Suc_conv_app_nth:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1193
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1194
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1195
apply (case_tac i, auto)
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1196
done
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1197
14591
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1198
lemma drop_Suc_conv_tl:
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1199
  "!!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
  1200
apply (induct xs, simp)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1201
apply (case_tac i, auto)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1202
done
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1203
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1204
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1205
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1206
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1207
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1208
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1209
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1210
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1211
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1212
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1213
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1214
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1215
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1216
lemma take_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1217
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1218
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1219
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1220
lemma drop_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1221
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1222
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1223
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1224
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
  1225
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1226
apply (case_tac xs, auto)
15236
f289e8ba2bb3 Proofs needed to be updated because induction now preserves name of
nipkow
parents: 15176
diff changeset
  1227
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1228
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1229
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1230
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1231
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1232
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1233
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1234
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1235
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
  1236
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1237
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1238
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1239
14802
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1240
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
  1241
apply(induct xs)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1242
 apply simp
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1243
apply(simp add: take_Cons drop_Cons split:nat.split)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1244
done
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1245
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1246
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1247
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1248
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1249
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1250
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1251
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
  1252
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1253
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1254
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
  1255
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1256
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1257
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
  1258
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1259
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1260
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
  1261
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1262
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1263
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1264
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1265
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1266
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1267
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1268
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1269
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1270
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1271
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1272
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1273
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1274
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1275
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1276
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1277
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1278
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1279
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1280
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1281
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1282
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1283
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1284
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1285
apply (case_tac n, blast)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1286
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1287
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1288
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1289
lemma nth_drop [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1290
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1291
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1292
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1293
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
  1294
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1295
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
  1296
by(simp add: hd_conv_nth)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1297
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1298
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1299
by(induct xs)(auto simp:take_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1300
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1301
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1302
by(induct xs)(auto simp:drop_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1303
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1304
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1305
using set_take_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1306
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1307
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1308
using set_drop_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1309
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1310
lemma append_eq_conv_conj:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1311
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1312
apply (induct xs, simp, clarsimp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1313
apply (case_tac zs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1314
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1315
14050
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1316
lemma take_add [rule_format]: 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1317
    "\<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
  1318
apply (induct xs, auto) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1319
apply (case_tac i, simp_all) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1320
done
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1321
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1322
lemma append_eq_append_conv_if:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1323
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1324
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1325
   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
  1326
   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
  1327
apply(induct xs\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1328
 apply simp
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1329
apply(case_tac ys\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1330
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1331
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1332
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1333
lemma take_hd_drop:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1334
  "!!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
  1335
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1336
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1337
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
  1338
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1339
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1340
lemma id_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1341
 "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
  1342
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1343
  assume si: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1344
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1345
  moreover
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1346
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1347
    apply (rule_tac take_Suc_conv_app_nth) by arith
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1348
  ultimately show ?thesis by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1349
qed
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1350
  
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1351
lemma upd_conv_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1352
 "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
  1353
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1354
  assume i: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow