src/HOL/List.thy
author chaieb
Wed, 29 Aug 2007 11:10:59 +0200
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removed unused theorems ; added lifting properties for foldr and foldl
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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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  ("Tools/function_package/lexicographic_order.ML")
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  ("Tools/function_package/fundef_datatype.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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  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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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[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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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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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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function (*authentic syntax for append -- revert to primrec
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  as soon as "authentic" primrec is available*)
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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  append_Nil: "[] @ ys = ys"
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  | append_Cons: "(x # xs) @ ys = x # (xs @ ys)"
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by (auto, case_tac a, auto)
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termination by (relation "measure (size o fst)") auto
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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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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primrec
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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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primrec
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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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primrec
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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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primrec
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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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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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension notation.
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![e| x <- xs, ...]!.
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The qualifiers after the dot are
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\begin{description}
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\item[generators] @{text"p \<leftarrow> xs"},
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 where @{text p} is a pattern and @{text xs} an expression of list type,
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\item[guards] @{text"b"}, where @{text b} is a boolean expression, or
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\item[local bindings] @{text"let x = e"}.
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\end{description}
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To avoid misunderstandings, the translation is not reversed upon
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output. You can add the inverse translations in your own theory if you
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desire.
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It is easy to write short list comprehensions which stand for complex
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expressions. During proofs, they may become unreadable (and
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mangled). In such cases it can be advisable to introduce separate
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definitions for the list comprehensions in question.  *}
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(*
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Proper theorem proving support would be nice. For example, 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 lc_quals
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syntax
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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"_lc_let" :: "letbinds => lc_qual"  ("let _")
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"_lc_end" :: "lc_quals" ("]")
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"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
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"_lc_abs" :: "'a => 'b list => 'b list"
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translations
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"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
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"[e. P]" => "if P then [e] else []"
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"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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 => "if P then (_listcompr e Q Qs) else []"
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"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
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 => "_Let b (_listcompr e Q Qs)"
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syntax (xsymbols)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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syntax (HTML output)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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parse_translation (advanced) {*
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let
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   fun abs_tr0 ctxt p es =
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    let
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      val x = Free (Name.variant (add_term_free_names (p$es, [])) "x", dummyT);
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      val case1 = Syntax.const "_case1" $ p $ es;
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      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
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                                        $ Syntax.const @{const_name Nil};
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      val cs = Syntax.const "_case2" $ case1 $ case2
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      val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
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                 ctxt [x, cs]
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    in lambda x ft end;
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  fun abs_tr ctxt [x as Free (s, T), r] =
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        let val thy = ProofContext.theory_of ctxt;
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            val s' = Sign.intern_const thy s
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        in if Sign.declared_const thy s' then abs_tr0 ctxt x r else lambda x r
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        end
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   | abs_tr ctxt [p,es] = abs_tr0 ctxt p es
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in [("_lc_abs", abs_tr)] end
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*}
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(*
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term "[(x,y,z). b]"
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term "[(x,y). Cons True x \<leftarrow> xs]"
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term "[(x,y,z). Cons x [] \<leftarrow> xs]"
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term "[(x,y,z). x<a, x>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs]"
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term "[(x,y,z). x\<leftarrow>xs, x>b]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x>b, x=d]"
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term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
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term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
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term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
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term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
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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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diff changeset
   332
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   333
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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   334
by (induct xs) auto
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diff changeset
   335
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   336
lemma length_induct:
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   337
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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58eeffd73be1 renamed rules to iprover
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   338
by (rule measure_induct [of length]) iprover
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diff changeset
   339
f2b00262bdfc converted;
wenzelm
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diff changeset
   340
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
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diff changeset
   341
subsubsection {* @{const length} *}
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diff changeset
   342
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   343
text {*
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   344
  Needs to come before @{text "@"} because of theorem @{text
580dfc999ef6 added normal post setup; cleaned up "execution" constants
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   345
  append_eq_append_conv}.
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   346
*}
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   347
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   348
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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   349
by (induct xs) auto
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diff changeset
   350
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   351
lemma length_map [simp]: "length (map f xs) = length xs"
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   352
by (induct xs) auto
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diff changeset
   353
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   354
lemma length_rev [simp]: "length (rev xs) = length xs"
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   355
by (induct xs) auto
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diff changeset
   356
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   357
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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   358
by (cases xs) auto
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diff changeset
   359
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   360
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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   361
by (induct xs) auto
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parents: 12887
diff changeset
   362
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   363
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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   364
by (induct xs) auto
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parents: 12887
diff changeset
   365
23479
10adbdcdc65b new lemmas
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   366
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
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   367
by auto
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diff changeset
   368
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   369
lemma length_Suc_conv:
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   370
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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   371
by (induct xs) auto
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parents: 13124
diff changeset
   372
14025
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   373
lemma Suc_length_conv:
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   374
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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parents: 14187
diff changeset
   375
apply (induct xs, simp, simp)
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   376
apply blast
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   377
done
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parents: 13913
diff changeset
   378
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   379
lemma impossible_Cons [rule_format]: 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   380
  "length xs <= length ys --> xs = x # ys = False"
20503
503ac4c5ef91 induct method: renamed 'fixing' to 'arbitrary';
wenzelm
parents: 20453
diff changeset
   381
apply (induct xs)
503ac4c5ef91 induct method: renamed 'fixing' to 'arbitrary';
wenzelm
parents: 20453
diff changeset
   382
apply auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   383
done
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   384
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   385
lemma list_induct2[consumes 1]: "\<And>ys.
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parents: 14208
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   386
 \<lbrakk> length xs = length ys;
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   387
   P [] [];
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parents: 14208
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   388
   \<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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   389
 \<Longrightarrow> P xs ys"
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   390
apply(induct xs)
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parents: 14208
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   391
 apply simp
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   392
apply(case_tac ys)
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   393
 apply simp
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   394
apply(simp)
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   395
done
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parents: 12887
diff changeset
   396
22493
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   397
lemma list_induct2': 
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   398
  "\<lbrakk> P [] [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   399
  \<And>x xs. P (x#xs) [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   400
  \<And>y ys. P [] (y#ys);
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   401
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   402
 \<Longrightarrow> P xs ys"
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   403
by (induct xs arbitrary: ys) (case_tac x, auto)+
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   404
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   405
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   406
by (rule Eq_FalseI) auto
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   407
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   408
simproc_setup list_neq ("(xs::'a list) = ys") = {*
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
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diff changeset
   409
(*
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   410
Reduces xs=ys to False if xs and ys cannot be of the same length.
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   411
This is the case if the atomic sublists of one are a submultiset
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   412
of those of the other list and there are fewer Cons's in one than the other.
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   413
*)
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wenzelm
parents: 23983
diff changeset
   414
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   415
let
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parents: 21911
diff changeset
   416
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   417
fun len (Const("List.list.Nil",_)) acc = acc
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   418
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   419
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
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parents: 21911
diff changeset
   420
  | len (Const("List.rev",_) $ xs) acc = len xs acc
cf58486ca11b Added simproc list_neq (prompted by an application)
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diff changeset
   421
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   422
  | len t (ts,n) = (t::ts,n);
cf58486ca11b Added simproc list_neq (prompted by an application)
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parents: 21911
diff changeset
   423
24037
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diff changeset
   424
fun list_neq _ ss ct =
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nipkow
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diff changeset
   425
  let
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0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   426
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   427
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   428
    fun prove_neq() =
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   429
      let
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   430
        val Type(_,listT::_) = eqT;
22994
02440636214f abstract size function in hologic.ML
haftmann
parents: 22940
diff changeset
   431
        val size = HOLogic.size_const listT;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   432
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   433
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   434
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
22633
haftmann
parents: 22551
diff changeset
   435
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann
parents: 22551
diff changeset
   436
      in SOME (thm RS @{thm neq_if_length_neq}) end
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   437
  in
23214
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   438
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   439
       n < m andalso submultiset (op aconv) (rs,ls)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   440
    then prove_neq() else NONE
cf58486ca11b Added simproc list_neq (prompted by an application)
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parents: 21911
diff changeset
   441
  end;
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   442
in list_neq end;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   443
*}
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   444
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   445
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   446
subsubsection {* @{text "@"} -- append *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   447
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   448
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
13145
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nipkow
parents: 13142
diff changeset
   449
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   450
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   451
lemma append_Nil2 [simp]: "xs @ [] = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   452
by (induct xs) auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   453
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   454
interpretation semigroup_append: semigroup_add ["op @"]
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   455
by unfold_locales simp
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   456
interpretation monoid_append: monoid_add ["[]" "op @"]
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   457
by unfold_locales (simp+)
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   458
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   459
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   460
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   461
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   462
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
13145
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nipkow
parents: 13142
diff changeset
   463
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   464
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   465
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
13145
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nipkow
parents: 13142
diff changeset
   466
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   467
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   468
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   469
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   470
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24219
diff changeset
   471
lemma append_eq_append_conv [simp,noatp]:
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
   472
 "!!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
   473
 ==> (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
   474
apply (induct xs)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   475
 apply (case_tac ys, simp, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   476
apply (case_tac ys, force, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   477
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   478
14495
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   479
lemma append_eq_append_conv2: "!!ys zs ts.
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   480
 (xs @ ys = zs @ ts) =
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   481
 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   482
apply (induct xs)
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   483
 apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   484
apply(case_tac zs)
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   485
 apply simp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   486
apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   487
done
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   488
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   489
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
13145
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nipkow
parents: 13142
diff changeset
   490
by simp
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   491
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   492
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   493
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   494
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   495
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   496
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   497
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   498
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   499
using append_same_eq [of _ _ "[]"] by auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   500
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   501
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   502
using append_same_eq [of "[]"] by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   503
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24219
diff changeset
   504
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   505
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   506
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   507
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   508
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   509
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   510
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   511
by (simp add: hd_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   512
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   513
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   514
by (simp split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   515
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   516
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   517
by (simp add: tl_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   518
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   519
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   520
lemma Cons_eq_append_conv: "x#xs = ys@zs =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   521
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   522
by(cases ys) auto
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   523
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   524
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   525
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   526
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   527
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   528
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   529
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   530
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   531
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   532
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   533
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   534
lemma Cons_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   535
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   536
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   537
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   538
lemma append_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   539
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   540
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   541
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   542
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   543
text {*
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   544
Simplification procedure for all list equalities.
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   545
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   546
- both lists end in a singleton list,
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   547
- or both lists end in the same list.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   548
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   549
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   550
ML_setup {*
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   551
local
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   552
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   553
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   554
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   555
  | last (Const("List.append",_) $ _ $ ys) = last ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   556
  | last t = t;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   557
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   558
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   559
  | list1 _ = false;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   560
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   561
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   562
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   563
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   564
  | butlast xs = Const("List.list.Nil",fastype_of xs);
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   565
22633
haftmann
parents: 22551
diff changeset
   566
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann
parents: 22551
diff changeset
   567
  @{thm append_Nil}, @{thm append_Cons}];
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   568
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   569
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   570
  let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   571
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   572
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   573
      let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   574
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   575
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   576
        val appT = [listT,listT] ---> listT
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   577
        val app = Const("List.append",appT)
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   578
        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
   579
        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
   580
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   581
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   582
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   583
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   584
  in
22633
haftmann
parents: 22551
diff changeset
   585
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann
parents: 22551
diff changeset
   586
    else if lastl aconv lastr then rearr @{thm append_same_eq}
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   587
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   588
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   589
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   590
in
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   591
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   592
val list_eq_simproc =
22633
haftmann
parents: 22551
diff changeset
   593
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   594
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   595
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   596
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   597
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   598
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   599
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   600
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   601
subsubsection {* @{text map} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   602
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   603
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
   604
by (induct xs) simp_all
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   605
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   606
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   607
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   608
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   609
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   610
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   611
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   612
lemma map_compose: "map (f o g) xs = map f (map g xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   613
by (induct xs) (auto simp add: o_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   614
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   615
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   616
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   617
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   618
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
   619
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   620
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   621
lemma map_cong [fundef_cong, recdef_cong]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   622
"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
   623
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   624
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   625
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   626
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   627
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   628
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   629
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   630
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   631
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   632
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   633
 "(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
   634
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   635
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   636
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   637
 "(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
   638
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   639
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   640
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   641
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   642
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   643
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   644
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   645
  "(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
   646
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   647
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   648
lemma map_eq_imp_length_eq:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   649
  "!!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
   650
apply (induct ys)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   651
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   652
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   653
apply clarify
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   654
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   655
apply fast
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   656
done
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_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   659
 "[| 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
   660
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   661
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   662
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   663
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   664
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   665
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   666
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   667
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   668
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   669
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   670
 "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
   671
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   672
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   673
lemma map_injective:
14338
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   674
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   675
by (induct ys) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   676
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   677
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
   678
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   679
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   680
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   681
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   682
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   683
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   684
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   685
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   686
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   687
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   688
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   689
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   690
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   691
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   692
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   693
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   694
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   695
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   696
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   697
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   698
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   699
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
   700
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   701
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   702
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
   703
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   704
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   705
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   706
  "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
   707
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   708
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   709
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   710
  "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
   711
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   712
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   713
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   714
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   715
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   716
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   717
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   718
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   719
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   720
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   721
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   722
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   723
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   724
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   725
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   726
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   727
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   728
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   729
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   730
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   731
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   732
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   733
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   734
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   735
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   736
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   737
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
   738
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   739
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   740
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   741
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   742
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   743
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   744
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   745
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   746
  "[| 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
   747
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   748
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   749
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   750
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   751
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   752
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   753
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   754
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   755
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   756
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   757
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   758
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   759
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   760
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   761
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   762
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   763
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   764
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   765
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   766
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   767
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   768
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   769
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   770
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   771
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   772
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   773
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   774
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   775
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   776
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   777
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
   778
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   779
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   780
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   781
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   782
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   783
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   784
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   785
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   786
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   787
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   788
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   789
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   790
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   791
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   792
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
   793
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   794
15425
6356d2523f73 [ .. (] -> [ ..< ]
nipkow
parents: 15392
diff changeset
   795
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   796
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   797
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   798
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   799
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   800
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
   801
proof (induct xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   802
  case Nil show ?case by simp
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   803
  case (Cons a xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   804
  show ?case
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   805
  proof 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   806
    assume "x \<in> set (a # xs)"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   807
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   808
      by (simp, blast intro: Cons_eq_appendI)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   809
  next
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   810
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   811
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   812
    show "x \<in> set (a # xs)" 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   813
      by (cases ys, auto simp add: eq)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   814
  qed
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   815
qed
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   816
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   817
lemma in_set_conv_decomp_first:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   818
 "(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
   819
proof (induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   820
  case Nil show ?case by simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   821
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   822
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   823
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   824
  proof cases
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   825
    assume "x = a" thus ?case using Cons by force
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   826
  next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   827
    assume "x \<noteq> a"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   828
    show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   829
    proof
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   830
      assume "x \<in> set (a # xs)"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   831
      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
   832
	by(fastsimp intro!: Cons_eq_appendI)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   833
    next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   834
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   835
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   836
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   837
    qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   838
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   839
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   840
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   841
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   842
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   843
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   844
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   845
lemma finite_list: "finite A ==> EX l. set l = A"
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   846
apply (erule finite_induct, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   847
apply (rule_tac x="x#l" in exI, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   848
done
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   849
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   850
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   851
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   852
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
   853
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   854
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   855
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   856
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   857
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   858
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   859
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   860
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   861
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   862
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
   863
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   864
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   865
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   866
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   867
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   868
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   869
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   870
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   871
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   872
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
   873
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   874
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   875
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
   876
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   877
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   878
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   879
by (induct xs) simp_all
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   880
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   881
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   882
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   883
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   884
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   885
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   886
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   887
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   888
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   889
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   890
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   891
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   892
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   893
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   894
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   895
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   896
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   897
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   898
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   899
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   900
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   901
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   902
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   903
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   904
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   905
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   906
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   907
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   908
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   909
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   910
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   911
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   912
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   913
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   914
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   915
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   916
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   917
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   918
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   919
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   920
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   921
    hence eq: "?S' = insert 0 (Suc ` ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   922
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   923
    have "length (filter p (x # xs)) = Suc(card ?S)"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
   924
      using Cons `p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   925
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   926
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   927
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   928
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   929
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   930
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   931
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   932
    hence eq: "?S' = Suc ` ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   933
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   934
    have "length (filter p (x # xs)) = card ?S"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
   935
      using Cons `\<not> p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   936
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   937
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   938
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   939
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   940
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   941
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   942
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   943
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   944
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   945
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   946
  \<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
   947
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   948
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   949
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   950
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   951
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   952
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   953
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   954
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   955
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   956
    proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   957
      assume xy: "x = y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   958
      show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   959
      proof from Py xy Cons(2) show "?Q []" by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   960
    next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   961
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   962
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   963
  next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   964
    assume Py: "\<not> P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   965
    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
   966
    show ?thesis (is "? us. ?Q us")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   967
    proof show "?Q (y#us)" using 1 by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   968
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   969
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   970
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   971
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   972
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   973
  \<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
   974
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   975
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   976
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   977
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   978
  (\<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
   979
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   980
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   981
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   982
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   983
  (\<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
   984
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   985
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   986
lemma filter_cong[fundef_cong, recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   987
 "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
   988
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   989
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   990
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   991
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   992
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   993
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   994
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   995
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   996
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   997
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   998
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   999
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1000
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1001
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1002
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1003
24308
700e745994c1 removed set_concat_map and improved set_concat
nipkow
parents: 24286
diff changeset
  1004
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1005
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1006
24461
bbff04c027ec Changed "code" attribute of concat_map_singleton to "code unfold".
berghofe
parents: 24449
diff changeset
  1007
lemma concat_map_singleton[simp, code unfold]: "concat(map (%x. [f x]) xs) = map f xs"
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1008
by (induct xs) auto
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1009
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1010
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1011
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1012
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1013
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1014
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1015
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1016
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1017
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1018
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1019
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1020
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1021
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1022
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1023
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1024
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1025
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1026
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1027
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1028
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1029
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1030
lemma nth_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1031
"!!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
  1032
apply (induct "xs", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1033
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1034
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1035
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1036
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
  1037
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1038
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1039
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
  1040
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1041
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1042
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1043
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1044
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1045
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1046
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1047
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1048
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1049
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1050
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1051
lemma list_eq_iff_nth_eq:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1052
 "!!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
  1053
apply(induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1054
 apply simp apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1055
apply(case_tac ys)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1056
 apply simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1057
apply(simp add:nth_Cons split:nat.split)apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1058
done
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1059
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1060
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
  1061
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1062
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1063
apply (rule_tac x = 0 in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1064
 apply (rule_tac x = "Suc i" in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1065
apply (case_tac i, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1066
apply (rename_tac j)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1067
apply (rule_tac x = j in exI, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1068
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1069
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1070
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
  1071
by(auto simp:set_conv_nth)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1072
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1073
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1074
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1075
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1076
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1077
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1078
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1079
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1080
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1081
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1082
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1083
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1084
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1085
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1086
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1087
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1088
subsubsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1089
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1090
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1091
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1092
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1093
lemma nth_list_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1094
"!!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
  1095
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1096
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1097
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1098
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1099
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1100
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
  1101
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1102
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1103
lemma list_update_overwrite [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1104
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1105
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1106
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1107
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1108
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1109
apply(simp split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1110
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1111
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1112
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
  1113
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1114
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1115
apply (case_tac i)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1116
apply simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1117
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1118
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1119
lemma list_update_same_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1120
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1121
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1122
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1123
lemma list_update_append1:
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1124
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1125
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1126
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1127
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1128
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1129
lemma list_update_append:
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1130
  "!!n. (xs @ ys) [n:= x] = 
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1131
  (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
  1132
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1133
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1134
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1135
 "(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
  1136
by (induct xs, auto)
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1137
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1138
lemma update_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1139
"!!i xy xs. length xs = length ys ==>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1140
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1141
by (induct ys) (auto, case_tac xs, auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1142
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1143
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1144
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1145
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1146
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1147
by (blast dest!: set_update_subset_insert [THEN subsetD])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1148
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1149
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
  1150
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1151
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1152
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1153
subsubsection {* @{text last} and @{text butlast} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1154
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1155
lemma last_snoc [simp]: "last (xs @ [x]) = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1156
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1157
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1158
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1159
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1160
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1161
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1162
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1163
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1164
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1165
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1166
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1167
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1168
by (induct xs) (auto)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1169
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1170
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1171
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1172
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1173
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1174
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1175
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1176
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1177
by(rule rev_exhaust[of xs]) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1178
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1179
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1180
by(cases xs) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1181
17765
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1182
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1183
by (induct as) auto
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1184
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1185
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1186
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1187
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1188
lemma butlast_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1189
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1190
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1191
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1192
lemma append_butlast_last_id [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1193
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1194
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1195
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1196
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1197
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1198
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1199
lemma in_set_butlast_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1200
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1201
by (auto dest: in_set_butlastD simp add: butlast_append)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1202
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1203
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
  1204
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1205
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1206
apply (auto split:nat.split)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1207
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1208
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1209
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1210
by(induct xs)(auto simp:neq_Nil_conv)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1211
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1212
subsubsection {* @{text take} and @{text drop} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1213
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1214
lemma take_0 [simp]: "take 0 xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1215
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1216
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1217
lemma drop_0 [simp]: "drop 0 xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1218
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1219
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1220
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1221
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1222
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1223
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1224
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1225
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1226
declare take_Cons [simp del] and drop_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1227
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1228
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
  1229
by(clarsimp simp add:neq_Nil_conv)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1230
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1231
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1232
by(cases xs, simp_all)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1233
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1234
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1235
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1236
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1237
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1238
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1239
apply(simp add:drop_Cons nth_Cons split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1240
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1241
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1242
lemma take_Suc_conv_app_nth:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1243
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1244
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1245
apply (case_tac i, auto)
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1246
done
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1247
14591
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1248
lemma drop_Suc_conv_tl:
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1249
  "!!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
  1250
apply (induct xs, simp)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1251
apply (case_tac i, auto)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1252
done
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1253
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1254
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1255
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1256
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1257
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1258
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1259
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1260
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1261
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1262
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1263
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1264
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1265
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1266
lemma take_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1267
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1268
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1269
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1270
lemma drop_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1271
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1272
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1273
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1274
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
  1275
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1276
apply (case_tac xs, auto)
15236
f289e8ba2bb3 Proofs needed to be updated because induction now preserves name of
nipkow
parents: 15176
diff changeset
  1277
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1278
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1279
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1280
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1281
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1282
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1283
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1284
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1285
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
  1286
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1287
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1288
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1289
14802
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1290
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
  1291
apply(induct xs)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1292
 apply simp
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1293
apply(simp add: take_Cons drop_Cons split:nat.split)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1294
done
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1295
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1296
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1297
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1298
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1299
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1300
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1301
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
  1302
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1303
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1304
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
  1305
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1306
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1307
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
  1308
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1309
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1310
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
  1311
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1312
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1313
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1314
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1315
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1316
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1317
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1318
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1319
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1320
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1321
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1322
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1323
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1324
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1325
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1326
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1327
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1328
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1329
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1330
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1331
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1332
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1333
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1334
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1335
apply (case_tac n, blast)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1336
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1337
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1338
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1339
lemma nth_drop [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1340
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1341
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1342
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1343
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
  1344
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1345
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
  1346
by(simp add: hd_conv_nth)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1347
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1348
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1349
by(induct xs)(auto simp:take_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1350