src/HOL/List.thy
author nipkow
Sun, 21 Jan 2007 13:27:41 +0100
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Added simproc list_neq (prompted by an application)
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(*  Title:      HOL/List.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* The datatype of finite lists *}
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theory List
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imports PreList
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  "@" :: "'a list => 'a list => 'a list"    (infixr 65)
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  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[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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  "last(x#xs) = (if xs=[] then x else last xs)"
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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  append_Nil: "[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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  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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primrec
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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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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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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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primrec
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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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primrec
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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 {* @{const Nil} and @{const Cons} *}
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lemma not_Cons_self [simp]:
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  "xs \<noteq> x # xs"
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by (induct xs) auto
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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by (induct xs) auto
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lemma length_induct:
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  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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by (rule measure_induct [of length]) iprover
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subsubsection {* @{const length} *}
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text {*
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  Needs to come before @{text "@"} because of theorem @{text
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  append_eq_append_conv}.
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*}
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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by (induct xs) auto
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lemma length_map [simp]: "length (map f xs) = length xs"
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by (induct xs) auto
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lemma length_rev [simp]: "length (rev xs) = length xs"
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by (induct xs) auto
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lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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by (cases xs) auto
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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by (induct xs) auto
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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by (induct xs) auto
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lemma length_Suc_conv:
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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by (induct xs) auto
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lemma Suc_length_conv:
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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apply (induct xs, simp, simp)
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apply blast
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done
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lemma impossible_Cons [rule_format]: 
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  "length xs <= length ys --> xs = x # ys = False"
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apply (induct xs)
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apply auto
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done
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lemma list_induct2[consumes 1]: "\<And>ys.
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 \<lbrakk> length xs = length ys;
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   P [] [];
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   \<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
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 \<Longrightarrow> P xs ys"
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apply(induct xs)
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 apply simp
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apply(case_tac ys)
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 apply simp
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apply(simp)
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done
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lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
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apply(rule Eq_FalseI)
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by auto
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(*
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Reduces xs=ys to False if xs and ys cannot be of the same length.
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This is the case if the atomic sublists of one are a submultiset
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of those of the other list and there are fewer Cons's in one than the other.
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*)
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ML_setup {*
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local
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val neq_if_length_neq = thm "neq_if_length_neq";
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fun len (Const("List.list.Nil",_)) acc = acc
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  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
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  | len (Const("List.op @",_) $ xs $ ys) acc = len xs (len ys acc)
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  | len (Const("List.rev",_) $ xs) acc = len xs acc
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  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
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  | len t (ts,n) = (t::ts,n);
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val list_ss = simpset_of(the_context());
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fun list_eq ss (Const(_,eqT) $ lhs $ rhs) =
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  let
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    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
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    fun prove_neq() =
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      let
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        val Type(_,listT::_) = eqT;
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        val size = Const("Nat.size", listT --> HOLogic.natT);
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        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
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        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
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        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
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          (K (simp_tac (Simplifier.inherit_context ss list_ss) 1));
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      in SOME (thm RS neq_if_length_neq) end
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  in
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    if m < n andalso gen_submultiset (op aconv) (ls,rs) orelse
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       n < m andalso gen_submultiset (op aconv) (rs,ls)
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    then prove_neq() else NONE
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  end;
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in
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val list_neq_simproc =
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  Simplifier.simproc (the_context ()) "list_neq" ["(xs::'a list) = ys"] (K list_eq);
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end;
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Addsimprocs [list_neq_simproc];
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*}
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subsubsection {* @{text "@"} -- append *}
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
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by (induct xs) auto
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lemma append_Nil2 [simp]: "xs @ [] = xs"
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by (induct xs) auto
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
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by (induct xs) auto
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
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by (induct xs) auto
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
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by (induct xs) auto
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lemma append_eq_append_conv [simp]:
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 "!!ys. length xs = length ys \<or> length us = length vs
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 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
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apply (induct xs)
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 apply (case_tac ys, simp, force)
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apply (case_tac ys, force, simp)
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done
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lemma append_eq_append_conv2: "!!ys zs ts.
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 (xs @ ys = zs @ ts) =
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 (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
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apply (induct xs)
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 apply fastsimp
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apply(case_tac zs)
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 apply simp
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apply fastsimp
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done
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
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by simp
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
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by simp
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
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by simp
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
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using append_same_eq [of _ _ "[]"] by auto
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parents: 13124
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   384
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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   385
using append_same_eq [of "[]"] by auto
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   386
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
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by (induct xs) auto
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   389
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   390
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
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by (induct xs) auto
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parents: 12887
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   392
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   393
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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   394
by (simp add: hd_append split: list.split)
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parents: 12887
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   395
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   396
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
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   397
by (simp split: list.split)
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diff changeset
   398
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parents: 13124
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   399
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
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by (simp add: tl_append split: list.split)
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   401
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parents: 12887
diff changeset
   402
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   403
lemma Cons_eq_append_conv: "x#xs = ys@zs =
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   404
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
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   405
by(cases ys) auto
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diff changeset
   406
15281
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parents: 15251
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   407
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   408
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   409
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   410
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   411
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   412
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
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   413
f2b00262bdfc converted;
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parents: 12887
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   414
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
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   415
by simp
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   416
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   417
lemma Cons_eq_appendI:
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   418
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
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   419
by (drule sym) simp
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   420
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   421
lemma append_eq_appendI:
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   422
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
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   423
by (drule sym) simp
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   424
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diff changeset
   425
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diff changeset
   426
text {*
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   427
Simplification procedure for all list equalities.
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   428
Currently only tries to rearrange @{text "@"} to see if
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   429
- both lists end in a singleton list,
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   430
- or both lists end in the same list.
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   431
*}
1ebd8ed5a1a0 tuned document;
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diff changeset
   432
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diff changeset
   433
ML_setup {*
3507
157be29ad5ba Improved length = size translation.
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diff changeset
   434
local
157be29ad5ba Improved length = size translation.
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parents: 3465
diff changeset
   435
13122
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parents: 13114
diff changeset
   436
val append_assoc = thm "append_assoc";
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diff changeset
   437
val append_Nil = thm "append_Nil";
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   438
val append_Cons = thm "append_Cons";
wenzelm
parents: 13114
diff changeset
   439
val append1_eq_conv = thm "append1_eq_conv";
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   440
val append_same_eq = thm "append_same_eq";
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parents: 13114
diff changeset
   441
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   442
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
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diff changeset
   443
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
56610e2ba220 sane interface for simprocs;
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   444
  | last (Const("List.op @",_) $ _ $ ys) = last ys
56610e2ba220 sane interface for simprocs;
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   445
  | last t = t;
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diff changeset
   446
f2b00262bdfc converted;
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diff changeset
   447
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
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   448
  | list1 _ = false;
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   449
f2b00262bdfc converted;
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parents: 12887
diff changeset
   450
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
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diff changeset
   451
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
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diff changeset
   452
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
56610e2ba220 sane interface for simprocs;
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parents: 13366
diff changeset
   453
  | butlast xs = Const("List.list.Nil",fastype_of xs);
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parents: 12887
diff changeset
   454
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   455
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   456
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   457
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
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parents: 13366
diff changeset
   458
  let
56610e2ba220 sane interface for simprocs;
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parents: 13366
diff changeset
   459
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   460
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   461
      let
56610e2ba220 sane interface for simprocs;
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diff changeset
   462
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   463
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   464
        val appT = [listT,listT] ---> listT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   465
        val app = Const("List.op @",appT)
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   466
        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
   467
        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
   468
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   469
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   470
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
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wenzelm
parents: 12887
diff changeset
   471
13462
56610e2ba220 sane interface for simprocs;
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parents: 13366
diff changeset
   472
  in
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   473
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   474
    else if lastl aconv lastr then rearr append_same_eq
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   475
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   476
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   477
13114
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wenzelm
parents: 12887
diff changeset
   478
in
13462
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wenzelm
parents: 13366
diff changeset
   479
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   480
val list_eq_simproc =
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   481
  Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq);
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   482
13114
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wenzelm
parents: 12887
diff changeset
   483
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   484
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   485
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   486
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   487
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   488
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   489
subsubsection {* @{text map} *}
13114
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wenzelm
parents: 12887
diff changeset
   490
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   491
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
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nipkow
parents: 13142
diff changeset
   492
by (induct xs) simp_all
13114
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wenzelm
parents: 12887
diff changeset
   493
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wenzelm
parents: 13124
diff changeset
   494
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
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nipkow
parents: 13142
diff changeset
   495
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   496
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   497
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   498
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   499
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   500
lemma map_compose: "map (f o g) xs = map f (map g xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   501
by (induct xs) (auto simp add: o_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   502
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   503
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   504
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   505
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   506
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
   507
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   508
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   509
lemma map_cong [fundef_cong, recdef_cong]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   510
"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
   511
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   512
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   513
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   514
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   515
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   516
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   517
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   518
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   519
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   520
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   521
 "(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
   522
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   523
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   524
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   525
 "(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
   526
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   527
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   528
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   529
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   530
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   531
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   532
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   533
  "(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
   534
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   535
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   536
lemma map_eq_imp_length_eq:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   537
  "!!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
   538
apply (induct ys)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   539
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   540
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   541
apply clarify
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   542
apply(simp (no_asm_use))
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   543
apply fast
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   544
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   545
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   546
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   547
 "[| 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
   548
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   549
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   550
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   551
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   552
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   553
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   554
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   555
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   556
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   557
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   558
 "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
   559
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   560
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   561
lemma map_injective:
14338
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   562
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
a1add2de7601 *** empty log message ***
nipkow
parents: 14328
diff changeset
   563
by (induct ys) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   564
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   565
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
   566
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   567
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   568
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   569
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   570
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   571
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   572
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   573
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   574
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   575
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   576
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   577
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   578
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   579
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   580
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   581
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   582
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   583
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   584
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   585
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   586
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   587
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
   588
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   589
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   590
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
   591
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   592
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   593
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   594
  "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
   595
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   596
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   597
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   598
  "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
   599
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   600
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   601
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   602
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   603
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   604
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   605
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   606
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   607
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   608
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   609
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   610
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   611
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   612
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   613
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   614
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   615
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   616
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   617
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   618
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   619
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   620
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   621
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   622
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   623
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   624
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   625
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
   626
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   627
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   628
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   629
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   630
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   631
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   632
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   633
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   634
  "[| 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
   635
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   636
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   637
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   638
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   639
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   640
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   641
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   642
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   643
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   644
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   645
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   646
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   647
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   648
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   649
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   650
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   651
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   652
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   653
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   654
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   655
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   656
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   657
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   658
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   659
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   660
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   661
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   662
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   663
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   664
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   665
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
   666
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   667
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   668
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   669
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   670
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   671
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   672
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   673
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   674
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   675
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   676
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   677
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   678
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   679
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   680
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
   681
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   682
15425
6356d2523f73 [ .. (] -> [ ..< ]
nipkow
parents: 15392
diff changeset
   683
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   684
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   685
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   686
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   687
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   688
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
   689
proof (induct xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   690
  case Nil show ?case by simp
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   691
  case (Cons a xs)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   692
  show ?case
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   693
  proof 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   694
    assume "x \<in> set (a # xs)"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   695
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   696
      by (simp, blast intro: Cons_eq_appendI)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   697
  next
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   698
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   699
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   700
    show "x \<in> set (a # xs)" 
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   701
      by (cases ys, auto simp add: eq)
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   702
  qed
fafcd72b9d4b some structured proofs
paulson
parents: 15110
diff changeset
   703
qed
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   704
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   705
lemma in_set_conv_decomp_first:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   706
 "(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
   707
proof (induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   708
  case Nil show ?case by simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   709
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   710
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   711
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   712
  proof cases
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   713
    assume "x = a" thus ?case using Cons by force
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   714
  next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   715
    assume "x \<noteq> a"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   716
    show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   717
    proof
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   718
      assume "x \<in> set (a # xs)"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   719
      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
   720
	by(fastsimp intro!: Cons_eq_appendI)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   721
    next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   722
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   723
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   724
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   725
    qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   726
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   727
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   728
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   729
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   730
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   731
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   732
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   733
lemma finite_list: "finite A ==> EX l. set l = A"
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   734
apply (erule finite_induct, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   735
apply (rule_tac x="x#l" in exI, auto)
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   736
done
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
   737
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   738
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
   739
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   740
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
   741
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   742
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   743
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   744
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   745
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   746
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   747
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   748
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
   749
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   750
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
   751
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   752
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   753
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   754
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   755
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   756
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   757
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   758
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   759
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   760
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
   761
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   762
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   763
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
   764
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   765
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   766
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   767
  by (induct xs) simp_all
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   768
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   769
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   770
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   771
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   772
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   773
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
   774
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   775
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   776
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   777
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   778
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   779
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   780
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   781
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   782
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
   783
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   784
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   785
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   786
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   787
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   788
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   789
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   790
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   791
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   792
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   793
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   794
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   795
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
   796
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   797
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   798
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   799
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   800
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   801
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   802
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   803
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   804
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   805
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   806
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   807
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   808
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   809
    hence eq: "?S' = insert 0 (Suc ` ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   810
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   811
    have "length (filter p (x # xs)) = Suc(card ?S)"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   812
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   813
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   814
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   815
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   816
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   817
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   818
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   819
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   820
    hence eq: "?S' = Suc ` ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   821
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   822
    have "length (filter p (x # xs)) = card ?S"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   823
      using Cons by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   824
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   825
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   826
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   827
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   828
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   829
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   830
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   831
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   832
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   833
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   834
  \<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
   835
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   836
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   837
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   838
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   839
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   840
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   841
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   842
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   843
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   844
    proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   845
      assume xy: "x = y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   846
      show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   847
      proof from Py xy Cons(2) show "?Q []" by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   848
    next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   849
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   850
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   851
  next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   852
    assume Py: "\<not> P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   853
    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
   854
    show ?thesis (is "? us. ?Q us")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   855
    proof show "?Q (y#us)" using 1 by simp qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   856
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   857
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   858
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   859
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   860
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   861
  \<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
   862
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   863
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   864
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   865
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   866
  (\<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
   867
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   868
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   869
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   870
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   871
  (\<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
   872
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
   873
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   874
lemma filter_cong[fundef_cong, recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   875
 "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
   876
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   877
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   878
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   879
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   880
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   881
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   882
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   883
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   884
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   885
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   886
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   887
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   888
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   889
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   890
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   891
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   892
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   893
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   894
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   895
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   896
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   897
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   898
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   899
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   900
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   901
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   902
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   903
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   904
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   905
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   906
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   907
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   908
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   909
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   910
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   911
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   912
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   913
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   914
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   915
lemma nth_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   916
"!!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
   917
apply (induct "xs", simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   918
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   919
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   920
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   921
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
   922
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   923
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   924
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
   925
by (induct "xs") auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   926
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   927
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   928
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   929
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   930
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   931
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   932
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   933
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   934
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   935
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   936
lemma list_eq_iff_nth_eq:
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   937
 "!!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
   938
apply(induct xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   939
 apply simp apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   940
apply(case_tac ys)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   941
 apply simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   942
apply(simp add:nth_Cons split:nat.split)apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   943
done
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   944
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   945
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
   946
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   947
apply safe
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   948
apply (rule_tac x = 0 in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   949
 apply (rule_tac x = "Suc i" in exI, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   950
apply (case_tac i, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   951
apply (rename_tac j)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   952
apply (rule_tac x = j in exI, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   953
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   954
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   955
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
   956
by(auto simp:set_conv_nth)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   957
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   958
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   959
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   960
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   961
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   962
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   963
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   964
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   965
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   966
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   967
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   968
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   969
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   970
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   971
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   972
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   973
subsubsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   974
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   975
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   976
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   977
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   978
lemma nth_list_update:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   979
"!!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
   980
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   981
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   982
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   983
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   984
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   985
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
   986
by (induct xs) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   987
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   988
lemma list_update_overwrite [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   989
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   990
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   991
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   992
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   993
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   994
apply(simp split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   995
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
   996
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   997
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
   998
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
   999
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1000
apply (case_tac i)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1001
apply simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1002
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1003
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1004
lemma list_update_same_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1005
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1006
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1007
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1008
lemma list_update_append1:
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1009
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1010
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1011
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1012
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1013
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1014
lemma list_update_append:
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1015
  "!!n. (xs @ ys) [n:= x] = 
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1016
  (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
  1017
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1018
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1019
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1020
 "(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
  1021
by (induct xs, auto)
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1022
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1023
lemma update_zip:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1024
"!!i xy xs. length xs = length ys ==>
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1025
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1026
by (induct ys) (auto, case_tac xs, auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1027
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1028
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1029
by (induct xs) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1030
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1031
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1032
by (blast dest!: set_update_subset_insert [THEN subsetD])
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1033
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1034
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
  1035
by (induct xs) (auto split:nat.splits)
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1036
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1037
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1038
subsubsection {* @{text last} and @{text butlast} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1039
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1040
lemma last_snoc [simp]: "last (xs @ [x]) = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1041
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1042
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1043
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1044
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1045
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1046
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1047
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1048
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1049
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1050
by(simp add:last.simps)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1051
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1052
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1053
by (induct xs) (auto)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1054
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1055
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1056
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1057
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1058
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1059
by(simp add:last_append)
6c24235e8d5d *** empty log message ***
nipkow
parents: 14300
diff changeset
  1060
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1061
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1062
by(rule rev_exhaust[of xs]) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1063
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1064
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1065
by(cases xs) simp_all
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1066
17765
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1067
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
e3cd31bc2e40 added last in set lemma
nipkow
parents: 17762
diff changeset
  1068
by (induct as) auto
17762
478869f444ca new hd/rev/last lemmas
nipkow
parents: 17724
diff changeset
  1069
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1070
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1071
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1072
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1073
lemma butlast_append:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1074
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1075
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1076
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1077
lemma append_butlast_last_id [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1078
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1079
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1080
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1081
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1082
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1083
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1084
lemma in_set_butlast_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1085
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1086
by (auto dest: in_set_butlastD simp add: butlast_append)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1087
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1088
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
  1089
apply (induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1090
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1091
apply (auto split:nat.split)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1092
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1093
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1094
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1095
by(induct xs)(auto simp:neq_Nil_conv)
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
  1096
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1097
subsubsection {* @{text take} and @{text drop} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1098
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1099
lemma take_0 [simp]: "take 0 xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1100
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1101
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1102
lemma drop_0 [simp]: "drop 0 xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1103
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1104
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1105
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1106
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1107
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1108
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1109
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1110
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1111
declare take_Cons [simp del] and drop_Cons [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1112
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1113
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
  1114
by(clarsimp simp add:neq_Nil_conv)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1115
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1116
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1117
by(cases xs, simp_all)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1118
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1119
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1120
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1121
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1122
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1123
apply (induct xs, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1124
apply(simp add:drop_Cons nth_Cons split:nat.splits)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1125
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1126
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1127
lemma take_Suc_conv_app_nth:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1128
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1129
apply (induct xs, simp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1130
apply (case_tac i, auto)
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1131
done
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1132
14591
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1133
lemma drop_Suc_conv_tl:
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1134
  "!!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
  1135
apply (induct xs, simp)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1136
apply (case_tac i, auto)
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1137
done
7be4d5dadf15 lemma drop_Suc_conv_tl added.
mehta
parents: 14589
diff changeset
  1138
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1139
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1140
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1141
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1142
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1143
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1144
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1145
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1146
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1147
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1148
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1149
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1150
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1151
lemma take_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1152
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1153
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1154
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1155
lemma drop_append [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1156
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1157
by (induct n) (auto, case_tac xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1158
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1159
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
  1160
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1161
apply (case_tac xs, auto)
15236
f289e8ba2bb3 Proofs needed to be updated because induction now preserves name of
nipkow
parents: 15176
diff changeset
  1162
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1163
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1164
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1165
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1166
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1167
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1168
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1169
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1170
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
  1171
apply (induct m, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1172
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1173
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1174
14802
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1175
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
  1176
apply(induct xs)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1177
 apply simp
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1178
apply(simp add: take_Cons drop_Cons split:nat.split)
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1179
done
e05116289ff9 added drop_take:thm
nipkow
parents: 14770
diff changeset
  1180
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1181
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1182
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1183
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1184
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1185
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1186
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
  1187
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1188
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1189
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
  1190
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1191
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1192
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
  1193
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1194
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1195
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
  1196
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1197
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1198
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1199
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1200
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1201
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1202
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1203
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1204
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1205
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1206
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1207
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1208
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1209
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1210
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1211
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1212
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1213
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1214
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1215
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1216
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1217
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1218
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1219
apply (induct xs, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1220
apply (case_tac n, blast)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1221
apply (case_tac i, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1222
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1223
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1224
lemma nth_drop [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1225
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1226
apply (induct n, auto)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1227
apply (case_tac xs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1228
done
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
  1229
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1230
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
  1231
by(simp add: hd_conv_nth)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1232
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1233
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1234
by(induct xs)(auto simp:take_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1235
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1236
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1237
by(induct xs)(auto simp:drop_Cons split:nat.split)
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
  1238
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1239
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1240
using set_take_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1241
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1242
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1243
using set_drop_subset by fast
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1244
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1245
lemma append_eq_conv_conj:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1246
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1247
apply (induct xs, simp, clarsimp)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1248
apply (case_tac zs, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1249
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1250
14050
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1251
lemma take_add [rule_format]: 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1252
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1253
apply (induct xs, auto) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1254
apply (case_tac i, simp_all) 
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1255
done
826037db30cd new theorem
paulson
parents: 14025
diff changeset
  1256
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1257
lemma append_eq_append_conv_if:
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1258
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1259
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1260
   then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1261
   else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1262
apply(induct xs\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1263
 apply simp
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1264
apply(case_tac ys\<^isub>1)
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1265
apply simp_all
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1266
done
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
  1267
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1268
lemma take_hd_drop:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1269
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1270
apply(induct xs)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1271
apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1272
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
  1273
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
  1274
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1275
lemma id_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1276
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1277
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1278
  assume si: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1279
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1280
  moreover
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1281
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1282
    apply (rule_tac take_Suc_conv_app_nth) by arith
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1283
  ultimately show ?thesis by auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1284
qed
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1285
  
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1286
lemma upd_conv_take_nth_drop:
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1287
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1288
proof -
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1289
  assume i: "i < length xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1290
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1291
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1292
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1293
    using i by (simp add: list_update_append)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1294
  finally show ?thesis .
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1295
qed
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1296
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1297
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1298
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1299
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1300
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1301
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1302
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1303
lemma takeWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1304
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1305
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1306
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1307
lemma takeWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1308
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1309
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1310
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1311
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1312
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1313
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1314
lemma dropWhile_append1 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1315
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1316
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1317
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1318
lemma dropWhile_append2 [simp]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1319
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1320
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1321
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1322
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1323
by (induct xs) (auto split: split_if_asm)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1324
13913
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1325
lemma takeWhile_eq_all_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1326
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1327
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1328
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1329
lemma dropWhile_eq_Nil_conv[simp]:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1330
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1331
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1332
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1333
lemma dropWhile_eq_Cons_conv:
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1334
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1335
by(induct xs, auto)
b3ed67af04b8 Added take/dropWhile thms
nipkow
parents: 13883
diff changeset
  1336
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1337
text{* The following two lemmmas could be generalized to an arbitrary
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1338
property. *}
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1339
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1340
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1341
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1342
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1343
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1344
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1345
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1346
apply(induct xs)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1347
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1348
apply auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1349
apply(subst dropWhile_append2)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1350
apply auto
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1351
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1352
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1353
lemma takeWhile_not_last:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1354
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1355
apply(induct xs)
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1356
 apply simp
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1357
apply(case_tac xs)