src/HOL/List.thy
author haftmann
Fri Oct 20 10:44:38 2006 +0200 (2006-10-20)
changeset 21061 580dfc999ef6
parent 21046 fe1db2f991a7
child 21079 747d716e98d0
permissions -rw-r--r--
added normal post setup; cleaned up "execution" constants
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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 FunDef
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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[_../_])")
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  "[i..j] == [i..<(Suc j)]"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_:_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x:xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat"
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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primrec
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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primrec
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  "last(x#xs) = (if xs=[] then x else last xs)"
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primrec
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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primrec
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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primrec
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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primrec
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  append_Nil:"[]@ys = ys"
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  append_Cons: "(x#xs)@ys = x#(xs@ys)"
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primrec
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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primrec
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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primrec
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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primrec
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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primrec
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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  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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primrec
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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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primrec
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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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primrec
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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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primrec
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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primrec
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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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"
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  rotate1_def: "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
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  rotate_def:  "rotate n = rotate1 ^ n"
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool"
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  list_all2_def: "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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  sublist :: "'a list => nat set => 'a list"
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  sublist_def: "sublist xs A =
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    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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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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lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
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using append_same_eq [of "[]"] by auto
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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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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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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
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by (simp add: hd_append split: list.split)
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
nipkow@13145
   339
by (simp split: list.split)
wenzelm@13114
   340
wenzelm@13142
   341
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
nipkow@13145
   342
by (simp add: tl_append split: list.split)
wenzelm@13114
   343
wenzelm@13114
   344
nipkow@14300
   345
lemma Cons_eq_append_conv: "x#xs = ys@zs =
nipkow@14300
   346
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
nipkow@14300
   347
by(cases ys) auto
nipkow@14300
   348
nipkow@15281
   349
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
nipkow@15281
   350
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
nipkow@15281
   351
by(cases ys) auto
nipkow@15281
   352
nipkow@14300
   353
wenzelm@13142
   354
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
wenzelm@13114
   355
wenzelm@13114
   356
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
nipkow@13145
   357
by simp
wenzelm@13114
   358
wenzelm@13142
   359
lemma Cons_eq_appendI:
nipkow@13145
   360
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
nipkow@13145
   361
by (drule sym) simp
wenzelm@13114
   362
wenzelm@13142
   363
lemma append_eq_appendI:
nipkow@13145
   364
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
nipkow@13145
   365
by (drule sym) simp
wenzelm@13114
   366
wenzelm@13114
   367
wenzelm@13142
   368
text {*
nipkow@13145
   369
Simplification procedure for all list equalities.
nipkow@13145
   370
Currently only tries to rearrange @{text "@"} to see if
nipkow@13145
   371
- both lists end in a singleton list,
nipkow@13145
   372
- or both lists end in the same list.
wenzelm@13142
   373
*}
wenzelm@13142
   374
wenzelm@13142
   375
ML_setup {*
nipkow@3507
   376
local
nipkow@3507
   377
wenzelm@13122
   378
val append_assoc = thm "append_assoc";
wenzelm@13122
   379
val append_Nil = thm "append_Nil";
wenzelm@13122
   380
val append_Cons = thm "append_Cons";
wenzelm@13122
   381
val append1_eq_conv = thm "append1_eq_conv";
wenzelm@13122
   382
val append_same_eq = thm "append_same_eq";
wenzelm@13122
   383
wenzelm@13114
   384
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
wenzelm@13462
   385
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
wenzelm@13462
   386
  | last (Const("List.op @",_) $ _ $ ys) = last ys
wenzelm@13462
   387
  | last t = t;
wenzelm@13114
   388
wenzelm@13114
   389
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
wenzelm@13462
   390
  | list1 _ = false;
wenzelm@13114
   391
wenzelm@13114
   392
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
wenzelm@13462
   393
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
wenzelm@13462
   394
  | butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys
wenzelm@13462
   395
  | butlast xs = Const("List.list.Nil",fastype_of xs);
wenzelm@13114
   396
wenzelm@16973
   397
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons];
wenzelm@16973
   398
wenzelm@20044
   399
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
wenzelm@13462
   400
  let
wenzelm@13462
   401
    val lastl = last lhs and lastr = last rhs;
wenzelm@13462
   402
    fun rearr conv =
wenzelm@13462
   403
      let
wenzelm@13462
   404
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
wenzelm@13462
   405
        val Type(_,listT::_) = eqT
wenzelm@13462
   406
        val appT = [listT,listT] ---> listT
wenzelm@13462
   407
        val app = Const("List.op @",appT)
wenzelm@13462
   408
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
wenzelm@13480
   409
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
wenzelm@20044
   410
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
wenzelm@17877
   411
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
skalberg@15531
   412
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
wenzelm@13114
   413
wenzelm@13462
   414
  in
wenzelm@13462
   415
    if list1 lastl andalso list1 lastr then rearr append1_eq_conv
wenzelm@13462
   416
    else if lastl aconv lastr then rearr append_same_eq
skalberg@15531
   417
    else NONE
wenzelm@13462
   418
  end;
wenzelm@13462
   419
wenzelm@13114
   420
in
wenzelm@13462
   421
wenzelm@13462
   422
val list_eq_simproc =
haftmann@21061
   423
  Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq);
wenzelm@13462
   424
wenzelm@13114
   425
end;
wenzelm@13114
   426
wenzelm@13114
   427
Addsimprocs [list_eq_simproc];
wenzelm@13114
   428
*}
wenzelm@13114
   429
wenzelm@13114
   430
nipkow@15392
   431
subsubsection {* @{text map} *}
wenzelm@13114
   432
wenzelm@13142
   433
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
nipkow@13145
   434
by (induct xs) simp_all
wenzelm@13114
   435
wenzelm@13142
   436
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
nipkow@13145
   437
by (rule ext, induct_tac xs) auto
wenzelm@13114
   438
wenzelm@13142
   439
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
nipkow@13145
   440
by (induct xs) auto
wenzelm@13114
   441
wenzelm@13142
   442
lemma map_compose: "map (f o g) xs = map f (map g xs)"
nipkow@13145
   443
by (induct xs) (auto simp add: o_def)
wenzelm@13114
   444
wenzelm@13142
   445
lemma rev_map: "rev (map f xs) = map f (rev xs)"
nipkow@13145
   446
by (induct xs) auto
wenzelm@13114
   447
nipkow@13737
   448
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
nipkow@13737
   449
by (induct xs) auto
nipkow@13737
   450
krauss@19770
   451
lemma map_cong [fundef_cong, recdef_cong]:
nipkow@13145
   452
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
nipkow@13145
   453
-- {* a congruence rule for @{text map} *}
nipkow@13737
   454
by simp
wenzelm@13114
   455
wenzelm@13142
   456
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
nipkow@13145
   457
by (cases xs) auto
wenzelm@13114
   458
wenzelm@13142
   459
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
nipkow@13145
   460
by (cases xs) auto
wenzelm@13114
   461
paulson@18447
   462
lemma map_eq_Cons_conv:
nipkow@14025
   463
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
nipkow@13145
   464
by (cases xs) auto
wenzelm@13114
   465
paulson@18447
   466
lemma Cons_eq_map_conv:
nipkow@14025
   467
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
nipkow@14025
   468
by (cases ys) auto
nipkow@14025
   469
paulson@18447
   470
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
paulson@18447
   471
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
paulson@18447
   472
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
paulson@18447
   473
nipkow@14111
   474
lemma ex_map_conv:
nipkow@14111
   475
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
paulson@18447
   476
by(induct ys, auto simp add: Cons_eq_map_conv)
nipkow@14111
   477
nipkow@15110
   478
lemma map_eq_imp_length_eq:
nipkow@15110
   479
  "!!xs. map f xs = map f ys ==> length xs = length ys"
nipkow@15110
   480
apply (induct ys)
nipkow@15110
   481
 apply simp
nipkow@15110
   482
apply(simp (no_asm_use))
nipkow@15110
   483
apply clarify
nipkow@15110
   484
apply(simp (no_asm_use))
nipkow@15110
   485
apply fast
nipkow@15110
   486
done
nipkow@15110
   487
nipkow@15110
   488
lemma map_inj_on:
nipkow@15110
   489
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
nipkow@15110
   490
  ==> xs = ys"
nipkow@15110
   491
apply(frule map_eq_imp_length_eq)
nipkow@15110
   492
apply(rotate_tac -1)
nipkow@15110
   493
apply(induct rule:list_induct2)
nipkow@15110
   494
 apply simp
nipkow@15110
   495
apply(simp)
nipkow@15110
   496
apply (blast intro:sym)
nipkow@15110
   497
done
nipkow@15110
   498
nipkow@15110
   499
lemma inj_on_map_eq_map:
nipkow@15110
   500
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@15110
   501
by(blast dest:map_inj_on)
nipkow@15110
   502
wenzelm@13114
   503
lemma map_injective:
nipkow@14338
   504
 "!!xs. map f xs = map f ys ==> inj f ==> xs = ys"
nipkow@14338
   505
by (induct ys) (auto dest!:injD)
wenzelm@13114
   506
nipkow@14339
   507
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
nipkow@14339
   508
by(blast dest:map_injective)
nipkow@14339
   509
wenzelm@13114
   510
lemma inj_mapI: "inj f ==> inj (map f)"
nipkow@17589
   511
by (iprover dest: map_injective injD intro: inj_onI)
wenzelm@13114
   512
wenzelm@13114
   513
lemma inj_mapD: "inj (map f) ==> inj f"
paulson@14208
   514
apply (unfold inj_on_def, clarify)
nipkow@13145
   515
apply (erule_tac x = "[x]" in ballE)
paulson@14208
   516
 apply (erule_tac x = "[y]" in ballE, simp, blast)
nipkow@13145
   517
apply blast
nipkow@13145
   518
done
wenzelm@13114
   519
nipkow@14339
   520
lemma inj_map[iff]: "inj (map f) = inj f"
nipkow@13145
   521
by (blast dest: inj_mapD intro: inj_mapI)
wenzelm@13114
   522
nipkow@15303
   523
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
nipkow@15303
   524
apply(rule inj_onI)
nipkow@15303
   525
apply(erule map_inj_on)
nipkow@15303
   526
apply(blast intro:inj_onI dest:inj_onD)
nipkow@15303
   527
done
nipkow@15303
   528
kleing@14343
   529
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
kleing@14343
   530
by (induct xs, auto)
wenzelm@13114
   531
nipkow@14402
   532
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
nipkow@14402
   533
by (induct xs) auto
nipkow@14402
   534
nipkow@15110
   535
lemma map_fst_zip[simp]:
nipkow@15110
   536
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
nipkow@15110
   537
by (induct rule:list_induct2, simp_all)
nipkow@15110
   538
nipkow@15110
   539
lemma map_snd_zip[simp]:
nipkow@15110
   540
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
nipkow@15110
   541
by (induct rule:list_induct2, simp_all)
nipkow@15110
   542
nipkow@15110
   543
nipkow@15392
   544
subsubsection {* @{text rev} *}
wenzelm@13114
   545
wenzelm@13142
   546
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
nipkow@13145
   547
by (induct xs) auto
wenzelm@13114
   548
wenzelm@13142
   549
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
nipkow@13145
   550
by (induct xs) auto
wenzelm@13114
   551
kleing@15870
   552
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
kleing@15870
   553
by auto
kleing@15870
   554
wenzelm@13142
   555
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
nipkow@13145
   556
by (induct xs) auto
wenzelm@13114
   557
wenzelm@13142
   558
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
nipkow@13145
   559
by (induct xs) auto
wenzelm@13114
   560
kleing@15870
   561
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
kleing@15870
   562
by (cases xs) auto
kleing@15870
   563
kleing@15870
   564
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
kleing@15870
   565
by (cases xs) auto
kleing@15870
   566
haftmann@21061
   567
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
haftmann@21061
   568
apply (induct xs arbitrary: ys, force)
paulson@14208
   569
apply (case_tac ys, simp, force)
nipkow@13145
   570
done
wenzelm@13114
   571
nipkow@15439
   572
lemma inj_on_rev[iff]: "inj_on rev A"
nipkow@15439
   573
by(simp add:inj_on_def)
nipkow@15439
   574
wenzelm@13366
   575
lemma rev_induct [case_names Nil snoc]:
wenzelm@13366
   576
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
berghofe@15489
   577
apply(simplesubst rev_rev_ident[symmetric])
nipkow@13145
   578
apply(rule_tac list = "rev xs" in list.induct, simp_all)
nipkow@13145
   579
done
wenzelm@13114
   580
nipkow@13145
   581
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility"
wenzelm@13114
   582
wenzelm@13366
   583
lemma rev_exhaust [case_names Nil snoc]:
wenzelm@13366
   584
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
nipkow@13145
   585
by (induct xs rule: rev_induct) auto
wenzelm@13114
   586
wenzelm@13366
   587
lemmas rev_cases = rev_exhaust
wenzelm@13366
   588
nipkow@18423
   589
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
nipkow@18423
   590
by(rule rev_cases[of xs]) auto
nipkow@18423
   591
wenzelm@13114
   592
nipkow@15392
   593
subsubsection {* @{text set} *}
wenzelm@13114
   594
wenzelm@13142
   595
lemma finite_set [iff]: "finite (set xs)"
nipkow@13145
   596
by (induct xs) auto
wenzelm@13114
   597
wenzelm@13142
   598
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
nipkow@13145
   599
by (induct xs) auto
wenzelm@13114
   600
nipkow@17830
   601
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
nipkow@17830
   602
by(cases xs) auto
oheimb@14099
   603
wenzelm@13142
   604
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
nipkow@13145
   605
by auto
wenzelm@13114
   606
oheimb@14099
   607
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
oheimb@14099
   608
by auto
oheimb@14099
   609
wenzelm@13142
   610
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
nipkow@13145
   611
by (induct xs) auto
wenzelm@13114
   612
nipkow@15245
   613
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
nipkow@15245
   614
by(induct xs) auto
nipkow@15245
   615
wenzelm@13142
   616
lemma set_rev [simp]: "set (rev xs) = set xs"
nipkow@13145
   617
by (induct xs) auto
wenzelm@13114
   618
wenzelm@13142
   619
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
nipkow@13145
   620
by (induct xs) auto
wenzelm@13114
   621
wenzelm@13142
   622
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
nipkow@13145
   623
by (induct xs) auto
wenzelm@13114
   624
nipkow@15425
   625
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
paulson@14208
   626
apply (induct j, simp_all)
paulson@14208
   627
apply (erule ssubst, auto)
nipkow@13145
   628
done
wenzelm@13114
   629
wenzelm@13142
   630
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
paulson@15113
   631
proof (induct xs)
paulson@15113
   632
  case Nil show ?case by simp
paulson@15113
   633
  case (Cons a xs)
paulson@15113
   634
  show ?case
paulson@15113
   635
  proof 
paulson@15113
   636
    assume "x \<in> set (a # xs)"
paulson@15113
   637
    with prems show "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   638
      by (simp, blast intro: Cons_eq_appendI)
paulson@15113
   639
  next
paulson@15113
   640
    assume "\<exists>ys zs. a # xs = ys @ x # zs"
paulson@15113
   641
    then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
paulson@15113
   642
    show "x \<in> set (a # xs)" 
paulson@15113
   643
      by (cases ys, auto simp add: eq)
paulson@15113
   644
  qed
paulson@15113
   645
qed
wenzelm@13142
   646
nipkow@18049
   647
lemma in_set_conv_decomp_first:
nipkow@18049
   648
 "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
nipkow@18049
   649
proof (induct xs)
nipkow@18049
   650
  case Nil show ?case by simp
nipkow@18049
   651
next
nipkow@18049
   652
  case (Cons a xs)
nipkow@18049
   653
  show ?case
nipkow@18049
   654
  proof cases
nipkow@18049
   655
    assume "x = a" thus ?case using Cons by force
nipkow@18049
   656
  next
nipkow@18049
   657
    assume "x \<noteq> a"
nipkow@18049
   658
    show ?case
nipkow@18049
   659
    proof
nipkow@18049
   660
      assume "x \<in> set (a # xs)"
nipkow@18049
   661
      from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   662
	by(fastsimp intro!: Cons_eq_appendI)
nipkow@18049
   663
    next
nipkow@18049
   664
      assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys"
nipkow@18049
   665
      then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast
nipkow@18049
   666
      show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq)
nipkow@18049
   667
    qed
nipkow@18049
   668
  qed
nipkow@18049
   669
qed
nipkow@18049
   670
nipkow@18049
   671
lemmas split_list       = in_set_conv_decomp[THEN iffD1, standard]
nipkow@18049
   672
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard]
nipkow@18049
   673
nipkow@18049
   674
paulson@13508
   675
lemma finite_list: "finite A ==> EX l. set l = A"
paulson@13508
   676
apply (erule finite_induct, auto)
paulson@13508
   677
apply (rule_tac x="x#l" in exI, auto)
paulson@13508
   678
done
paulson@13508
   679
kleing@14388
   680
lemma card_length: "card (set xs) \<le> length xs"
kleing@14388
   681
by (induct xs) (auto simp add: card_insert_if)
wenzelm@13114
   682
paulson@15168
   683
nipkow@15392
   684
subsubsection {* @{text filter} *}
wenzelm@13114
   685
wenzelm@13142
   686
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
nipkow@13145
   687
by (induct xs) auto
wenzelm@13114
   688
nipkow@15305
   689
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
nipkow@15305
   690
by (induct xs) simp_all
nipkow@15305
   691
wenzelm@13142
   692
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
nipkow@13145
   693
by (induct xs) auto
wenzelm@13114
   694
nipkow@16998
   695
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
nipkow@16998
   696
by (induct xs) (auto simp add: le_SucI)
nipkow@16998
   697
nipkow@18423
   698
lemma sum_length_filter_compl:
nipkow@18423
   699
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
nipkow@18423
   700
by(induct xs) simp_all
nipkow@18423
   701
wenzelm@13142
   702
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
nipkow@13145
   703
by (induct xs) auto
wenzelm@13114
   704
wenzelm@13142
   705
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
nipkow@13145
   706
by (induct xs) auto
wenzelm@13114
   707
nipkow@16998
   708
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
nipkow@16998
   709
  by (induct xs) simp_all
nipkow@16998
   710
nipkow@16998
   711
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
nipkow@16998
   712
apply (induct xs)
nipkow@16998
   713
 apply auto
nipkow@16998
   714
apply(cut_tac P=P and xs=xs in length_filter_le)
nipkow@16998
   715
apply simp
nipkow@16998
   716
done
wenzelm@13114
   717
nipkow@16965
   718
lemma filter_map:
nipkow@16965
   719
  "filter P (map f xs) = map f (filter (P o f) xs)"
nipkow@16965
   720
by (induct xs) simp_all
nipkow@16965
   721
nipkow@16965
   722
lemma length_filter_map[simp]:
nipkow@16965
   723
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
nipkow@16965
   724
by (simp add:filter_map)
nipkow@16965
   725
wenzelm@13142
   726
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
nipkow@13145
   727
by auto
wenzelm@13114
   728
nipkow@15246
   729
lemma length_filter_less:
nipkow@15246
   730
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
nipkow@15246
   731
proof (induct xs)
nipkow@15246
   732
  case Nil thus ?case by simp
nipkow@15246
   733
next
nipkow@15246
   734
  case (Cons x xs) thus ?case
nipkow@15246
   735
    apply (auto split:split_if_asm)
nipkow@15246
   736
    using length_filter_le[of P xs] apply arith
nipkow@15246
   737
  done
nipkow@15246
   738
qed
wenzelm@13114
   739
nipkow@15281
   740
lemma length_filter_conv_card:
nipkow@15281
   741
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
nipkow@15281
   742
proof (induct xs)
nipkow@15281
   743
  case Nil thus ?case by simp
nipkow@15281
   744
next
nipkow@15281
   745
  case (Cons x xs)
nipkow@15281
   746
  let ?S = "{i. i < length xs & p(xs!i)}"
nipkow@15281
   747
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
nipkow@15281
   748
  show ?case (is "?l = card ?S'")
nipkow@15281
   749
  proof (cases)
nipkow@15281
   750
    assume "p x"
nipkow@15281
   751
    hence eq: "?S' = insert 0 (Suc ` ?S)"
nipkow@15281
   752
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   753
    have "length (filter p (x # xs)) = Suc(card ?S)"
nipkow@15281
   754
      using Cons by simp
nipkow@15281
   755
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
nipkow@15281
   756
      by (simp add: card_image inj_Suc)
nipkow@15281
   757
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   758
      by (simp add:card_insert_if) (simp add:image_def)
nipkow@15281
   759
    finally show ?thesis .
nipkow@15281
   760
  next
nipkow@15281
   761
    assume "\<not> p x"
nipkow@15281
   762
    hence eq: "?S' = Suc ` ?S"
nipkow@15281
   763
      by(auto simp add: nth_Cons image_def split:nat.split elim:lessE)
nipkow@15281
   764
    have "length (filter p (x # xs)) = card ?S"
nipkow@15281
   765
      using Cons by simp
nipkow@15281
   766
    also have "\<dots> = card(Suc ` ?S)" using fin
nipkow@15281
   767
      by (simp add: card_image inj_Suc)
nipkow@15281
   768
    also have "\<dots> = card ?S'" using eq fin
nipkow@15281
   769
      by (simp add:card_insert_if)
nipkow@15281
   770
    finally show ?thesis .
nipkow@15281
   771
  qed
nipkow@15281
   772
qed
nipkow@15281
   773
nipkow@17629
   774
lemma Cons_eq_filterD:
nipkow@17629
   775
 "x#xs = filter P ys \<Longrightarrow>
nipkow@17629
   776
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
wenzelm@19585
   777
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
nipkow@17629
   778
proof(induct ys)
nipkow@17629
   779
  case Nil thus ?case by simp
nipkow@17629
   780
next
nipkow@17629
   781
  case (Cons y ys)
nipkow@17629
   782
  show ?case (is "\<exists>x. ?Q x")
nipkow@17629
   783
  proof cases
nipkow@17629
   784
    assume Py: "P y"
nipkow@17629
   785
    show ?thesis
nipkow@17629
   786
    proof cases
nipkow@17629
   787
      assume xy: "x = y"
nipkow@17629
   788
      show ?thesis
nipkow@17629
   789
      proof from Py xy Cons(2) show "?Q []" by simp qed
nipkow@17629
   790
    next
nipkow@17629
   791
      assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp
nipkow@17629
   792
    qed
nipkow@17629
   793
  next
nipkow@17629
   794
    assume Py: "\<not> P y"
nipkow@17629
   795
    with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp
nipkow@17629
   796
    show ?thesis (is "? us. ?Q us")
nipkow@17629
   797
    proof show "?Q (y#us)" using 1 by simp qed
nipkow@17629
   798
  qed
nipkow@17629
   799
qed
nipkow@17629
   800
nipkow@17629
   801
lemma filter_eq_ConsD:
nipkow@17629
   802
 "filter P ys = x#xs \<Longrightarrow>
nipkow@17629
   803
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
nipkow@17629
   804
by(rule Cons_eq_filterD) simp
nipkow@17629
   805
nipkow@17629
   806
lemma filter_eq_Cons_iff:
nipkow@17629
   807
 "(filter P ys = x#xs) =
nipkow@17629
   808
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   809
by(auto dest:filter_eq_ConsD)
nipkow@17629
   810
nipkow@17629
   811
lemma Cons_eq_filter_iff:
nipkow@17629
   812
 "(x#xs = filter P ys) =
nipkow@17629
   813
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
nipkow@17629
   814
by(auto dest:Cons_eq_filterD)
nipkow@17629
   815
krauss@19770
   816
lemma filter_cong[fundef_cong, recdef_cong]:
nipkow@17501
   817
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
nipkow@17501
   818
apply simp
nipkow@17501
   819
apply(erule thin_rl)
nipkow@17501
   820
by (induct ys) simp_all
nipkow@17501
   821
nipkow@15281
   822
nipkow@15392
   823
subsubsection {* @{text concat} *}
wenzelm@13114
   824
wenzelm@13142
   825
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
nipkow@13145
   826
by (induct xs) auto
wenzelm@13114
   827
paulson@18447
   828
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   829
by (induct xss) auto
wenzelm@13114
   830
paulson@18447
   831
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
nipkow@13145
   832
by (induct xss) auto
wenzelm@13114
   833
wenzelm@13142
   834
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)"
nipkow@13145
   835
by (induct xs) auto
wenzelm@13114
   836
wenzelm@13142
   837
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
nipkow@13145
   838
by (induct xs) auto
wenzelm@13114
   839
wenzelm@13142
   840
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
nipkow@13145
   841
by (induct xs) auto
wenzelm@13114
   842
wenzelm@13142
   843
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
nipkow@13145
   844
by (induct xs) auto
wenzelm@13114
   845
wenzelm@13114
   846
nipkow@15392
   847
subsubsection {* @{text nth} *}
wenzelm@13114
   848
wenzelm@13142
   849
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
nipkow@13145
   850
by auto
wenzelm@13114
   851
wenzelm@13142
   852
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
nipkow@13145
   853
by auto
wenzelm@13114
   854
wenzelm@13142
   855
declare nth.simps [simp del]
wenzelm@13114
   856
wenzelm@13114
   857
lemma nth_append:
nipkow@13145
   858
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
paulson@14208
   859
apply (induct "xs", simp)
paulson@14208
   860
apply (case_tac n, auto)
nipkow@13145
   861
done
wenzelm@13114
   862
nipkow@14402
   863
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
nipkow@14402
   864
by (induct "xs") auto
nipkow@14402
   865
nipkow@14402
   866
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
nipkow@14402
   867
by (induct "xs") auto
nipkow@14402
   868
wenzelm@13142
   869
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)"
paulson@14208
   870
apply (induct xs, simp)
paulson@14208
   871
apply (case_tac n, auto)
nipkow@13145
   872
done
wenzelm@13114
   873
nipkow@18423
   874
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
nipkow@18423
   875
by(cases xs) simp_all
nipkow@18423
   876
nipkow@18049
   877
nipkow@18049
   878
lemma list_eq_iff_nth_eq:
nipkow@18049
   879
 "!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
nipkow@18049
   880
apply(induct xs)
nipkow@18049
   881
 apply simp apply blast
nipkow@18049
   882
apply(case_tac ys)
nipkow@18049
   883
 apply simp
nipkow@18049
   884
apply(simp add:nth_Cons split:nat.split)apply blast
nipkow@18049
   885
done
nipkow@18049
   886
wenzelm@13142
   887
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
paulson@15251
   888
apply (induct xs, simp, simp)
nipkow@13145
   889
apply safe
paulson@14208
   890
apply (rule_tac x = 0 in exI, simp)
paulson@14208
   891
 apply (rule_tac x = "Suc i" in exI, simp)
paulson@14208
   892
apply (case_tac i, simp)
nipkow@13145
   893
apply (rename_tac j)
paulson@14208
   894
apply (rule_tac x = j in exI, simp)
nipkow@13145
   895
done
wenzelm@13114
   896
nipkow@17501
   897
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
nipkow@17501
   898
by(auto simp:set_conv_nth)
nipkow@17501
   899
nipkow@13145
   900
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
nipkow@13145
   901
by (auto simp add: set_conv_nth)
wenzelm@13114
   902
wenzelm@13142
   903
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
nipkow@13145
   904
by (auto simp add: set_conv_nth)
wenzelm@13114
   905
wenzelm@13114
   906
lemma all_nth_imp_all_set:
nipkow@13145
   907
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
nipkow@13145
   908
by (auto simp add: set_conv_nth)
wenzelm@13114
   909
wenzelm@13114
   910
lemma all_set_conv_all_nth:
nipkow@13145
   911
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
nipkow@13145
   912
by (auto simp add: set_conv_nth)
wenzelm@13114
   913
wenzelm@13114
   914
nipkow@15392
   915
subsubsection {* @{text list_update} *}
wenzelm@13114
   916
wenzelm@13142
   917
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs"
nipkow@13145
   918
by (induct xs) (auto split: nat.split)
wenzelm@13114
   919
wenzelm@13114
   920
lemma nth_list_update:
nipkow@13145
   921
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
nipkow@13145
   922
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   923
wenzelm@13142
   924
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
nipkow@13145
   925
by (simp add: nth_list_update)
wenzelm@13114
   926
wenzelm@13142
   927
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j"
nipkow@13145
   928
by (induct xs) (auto simp add: nth_Cons split: nat.split)
wenzelm@13114
   929
wenzelm@13142
   930
lemma list_update_overwrite [simp]:
nipkow@13145
   931
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]"
nipkow@13145
   932
by (induct xs) (auto split: nat.split)
wenzelm@13114
   933
nipkow@14402
   934
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs"
paulson@14208
   935
apply (induct xs, simp)
nipkow@14187
   936
apply(simp split:nat.splits)
nipkow@14187
   937
done
nipkow@14187
   938
nipkow@17501
   939
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
nipkow@17501
   940
apply (induct xs)
nipkow@17501
   941
 apply simp
nipkow@17501
   942
apply (case_tac i)
nipkow@17501
   943
apply simp_all
nipkow@17501
   944
done
nipkow@17501
   945
wenzelm@13114
   946
lemma list_update_same_conv:
nipkow@13145
   947
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
nipkow@13145
   948
by (induct xs) (auto split: nat.split)
wenzelm@13114
   949
nipkow@14187
   950
lemma list_update_append1:
nipkow@14187
   951
 "!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
paulson@14208
   952
apply (induct xs, simp)
nipkow@14187
   953
apply(simp split:nat.split)
nipkow@14187
   954
done
nipkow@14187
   955
kleing@15868
   956
lemma list_update_append:
kleing@15868
   957
  "!!n. (xs @ ys) [n:= x] = 
kleing@15868
   958
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
kleing@15868
   959
by (induct xs) (auto split:nat.splits)
kleing@15868
   960
nipkow@14402
   961
lemma list_update_length [simp]:
nipkow@14402
   962
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
nipkow@14402
   963
by (induct xs, auto)
nipkow@14402
   964
wenzelm@13114
   965
lemma update_zip:
nipkow@13145
   966
"!!i xy xs. length xs = length ys ==>
nipkow@13145
   967
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])"
nipkow@13145
   968
by (induct ys) (auto, case_tac xs, auto split: nat.split)
wenzelm@13114
   969
wenzelm@13114
   970
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)"
nipkow@13145
   971
by (induct xs) (auto split: nat.split)
wenzelm@13114
   972
wenzelm@13114
   973
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A"
nipkow@13145
   974
by (blast dest!: set_update_subset_insert [THEN subsetD])
wenzelm@13114
   975
kleing@15868
   976
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])"
kleing@15868
   977
by (induct xs) (auto split:nat.splits)
kleing@15868
   978
wenzelm@13114
   979
nipkow@15392
   980
subsubsection {* @{text last} and @{text butlast} *}
wenzelm@13114
   981
wenzelm@13142
   982
lemma last_snoc [simp]: "last (xs @ [x]) = x"
nipkow@13145
   983
by (induct xs) auto
wenzelm@13114
   984
wenzelm@13142
   985
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs"
nipkow@13145
   986
by (induct xs) auto
wenzelm@13114
   987
nipkow@14302
   988
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x"
nipkow@14302
   989
by(simp add:last.simps)
nipkow@14302
   990
nipkow@14302
   991
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs"
nipkow@14302
   992
by(simp add:last.simps)
nipkow@14302
   993
nipkow@14302
   994
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)"
nipkow@14302
   995
by (induct xs) (auto)
nipkow@14302
   996
nipkow@14302
   997
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs"
nipkow@14302
   998
by(simp add:last_append)
nipkow@14302
   999
nipkow@14302
  1000
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys"
nipkow@14302
  1001
by(simp add:last_append)
nipkow@14302
  1002
nipkow@17762
  1003
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs"
nipkow@17762
  1004
by(rule rev_exhaust[of xs]) simp_all
nipkow@17762
  1005
nipkow@17762
  1006
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs"
nipkow@17762
  1007
by(cases xs) simp_all
nipkow@17762
  1008
nipkow@17765
  1009
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as"
nipkow@17765
  1010
by (induct as) auto
nipkow@17762
  1011
wenzelm@13142
  1012
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1"
nipkow@13145
  1013
by (induct xs rule: rev_induct) auto
wenzelm@13114
  1014
wenzelm@13114
  1015
lemma butlast_append:
nipkow@13145
  1016
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)"
nipkow@13145
  1017
by (induct xs) auto
wenzelm@13114
  1018
wenzelm@13142
  1019
lemma append_butlast_last_id [simp]:
nipkow@13145
  1020
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs"
nipkow@13145
  1021
by (induct xs) auto
wenzelm@13114
  1022
wenzelm@13142
  1023
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs"
nipkow@13145
  1024
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1025
wenzelm@13114
  1026
lemma in_set_butlast_appendI:
nipkow@13145
  1027
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))"
nipkow@13145
  1028
by (auto dest: in_set_butlastD simp add: butlast_append)
wenzelm@13114
  1029
nipkow@17501
  1030
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs"
nipkow@17501
  1031
apply (induct xs)
nipkow@17501
  1032
 apply simp
nipkow@17501
  1033
apply (auto split:nat.split)
nipkow@17501
  1034
done
nipkow@17501
  1035
nipkow@17589
  1036
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)"
nipkow@17589
  1037
by(induct xs)(auto simp:neq_Nil_conv)
nipkow@17589
  1038
nipkow@15392
  1039
subsubsection {* @{text take} and @{text drop} *}
wenzelm@13114
  1040
wenzelm@13142
  1041
lemma take_0 [simp]: "take 0 xs = []"
nipkow@13145
  1042
by (induct xs) auto
wenzelm@13114
  1043
wenzelm@13142
  1044
lemma drop_0 [simp]: "drop 0 xs = xs"
nipkow@13145
  1045
by (induct xs) auto
wenzelm@13114
  1046
wenzelm@13142
  1047
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs"
nipkow@13145
  1048
by simp
wenzelm@13114
  1049
wenzelm@13142
  1050
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs"
nipkow@13145
  1051
by simp
wenzelm@13114
  1052
wenzelm@13142
  1053
declare take_Cons [simp del] and drop_Cons [simp del]
wenzelm@13114
  1054
nipkow@15110
  1055
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)"
nipkow@15110
  1056
by(clarsimp simp add:neq_Nil_conv)
nipkow@15110
  1057
nipkow@14187
  1058
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)"
nipkow@14187
  1059
by(cases xs, simp_all)
nipkow@14187
  1060
nipkow@14187
  1061
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)"
nipkow@14187
  1062
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split)
nipkow@14187
  1063
nipkow@14187
  1064
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y"
paulson@14208
  1065
apply (induct xs, simp)
nipkow@14187
  1066
apply(simp add:drop_Cons nth_Cons split:nat.splits)
nipkow@14187
  1067
done
nipkow@14187
  1068
nipkow@13913
  1069
lemma take_Suc_conv_app_nth:
nipkow@13913
  1070
 "!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]"
paulson@14208
  1071
apply (induct xs, simp)
paulson@14208
  1072
apply (case_tac i, auto)
nipkow@13913
  1073
done
nipkow@13913
  1074
mehta@14591
  1075
lemma drop_Suc_conv_tl:
mehta@14591
  1076
  "!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs"
mehta@14591
  1077
apply (induct xs, simp)
mehta@14591
  1078
apply (case_tac i, auto)
mehta@14591
  1079
done
mehta@14591
  1080
wenzelm@13142
  1081
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n"
nipkow@13145
  1082
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1083
wenzelm@13142
  1084
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)"
nipkow@13145
  1085
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1086
wenzelm@13142
  1087
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs"
nipkow@13145
  1088
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1089
wenzelm@13142
  1090
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []"
nipkow@13145
  1091
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1092
wenzelm@13142
  1093
lemma take_append [simp]:
nipkow@13145
  1094
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"
nipkow@13145
  1095
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1096
wenzelm@13142
  1097
lemma drop_append [simp]:
nipkow@13145
  1098
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
nipkow@13145
  1099
by (induct n) (auto, case_tac xs, auto)
wenzelm@13114
  1100
wenzelm@13142
  1101
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs"
paulson@14208
  1102
apply (induct m, auto)
paulson@14208
  1103
apply (case_tac xs, auto)
nipkow@15236
  1104
apply (case_tac n, auto)
nipkow@13145
  1105
done
wenzelm@13114
  1106
wenzelm@13142
  1107
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs"
paulson@14208
  1108
apply (induct m, auto)
paulson@14208
  1109
apply (case_tac xs, auto)
nipkow@13145
  1110
done
wenzelm@13114
  1111
wenzelm@13114
  1112
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)"
paulson@14208
  1113
apply (induct m, auto)
paulson@14208
  1114
apply (case_tac xs, auto)
nipkow@13145
  1115
done
wenzelm@13114
  1116
nipkow@14802
  1117
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)"
nipkow@14802
  1118
apply(induct xs)
nipkow@14802
  1119
 apply simp
nipkow@14802
  1120
apply(simp add: take_Cons drop_Cons split:nat.split)
nipkow@14802
  1121
done
nipkow@14802
  1122
wenzelm@13142
  1123
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs"
paulson@14208
  1124
apply (induct n, auto)
paulson@14208
  1125
apply (case_tac xs, auto)
nipkow@13145
  1126
done
wenzelm@13114
  1127
nipkow@15110
  1128
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])"
nipkow@15110
  1129
apply(induct xs)
nipkow@15110
  1130
 apply simp
nipkow@15110
  1131
apply(simp add:take_Cons split:nat.split)
nipkow@15110
  1132
done
nipkow@15110
  1133
nipkow@15110
  1134
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)"
nipkow@15110
  1135
apply(induct xs)
nipkow@15110
  1136
apply simp
nipkow@15110
  1137
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1138
done
nipkow@15110
  1139
wenzelm@13114
  1140
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)"
paulson@14208
  1141
apply (induct n, auto)
paulson@14208
  1142
apply (case_tac xs, auto)
nipkow@13145
  1143
done
wenzelm@13114
  1144
wenzelm@13142
  1145
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)"
paulson@14208
  1146
apply (induct n, auto)
paulson@14208
  1147
apply (case_tac xs, auto)
nipkow@13145
  1148
done
wenzelm@13114
  1149
wenzelm@13114
  1150
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)"
paulson@14208
  1151
apply (induct xs, auto)
paulson@14208
  1152
apply (case_tac i, auto)
nipkow@13145
  1153
done
wenzelm@13114
  1154
wenzelm@13114
  1155
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)"
paulson@14208
  1156
apply (induct xs, auto)
paulson@14208
  1157
apply (case_tac i, auto)
nipkow@13145
  1158
done
wenzelm@13114
  1159
wenzelm@13142
  1160
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i"
paulson@14208
  1161
apply (induct xs, auto)
paulson@14208
  1162
apply (case_tac n, blast)
paulson@14208
  1163
apply (case_tac i, auto)
nipkow@13145
  1164
done
wenzelm@13114
  1165
wenzelm@13142
  1166
lemma nth_drop [simp]:
nipkow@13145
  1167
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)"
paulson@14208
  1168
apply (induct n, auto)
paulson@14208
  1169
apply (case_tac xs, auto)
nipkow@13145
  1170
done
nipkow@3507
  1171
nipkow@18423
  1172
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n"
nipkow@18423
  1173
by(simp add: hd_conv_nth)
nipkow@18423
  1174
nipkow@14025
  1175
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs"
nipkow@14025
  1176
by(induct xs)(auto simp:take_Cons split:nat.split)
nipkow@14025
  1177
nipkow@14025
  1178
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs"
nipkow@14025
  1179
by(induct xs)(auto simp:drop_Cons split:nat.split)
nipkow@14025
  1180
nipkow@14187
  1181
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1182
using set_take_subset by fast
nipkow@14187
  1183
nipkow@14187
  1184
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs"
nipkow@14187
  1185
using set_drop_subset by fast
nipkow@14187
  1186
wenzelm@13114
  1187
lemma append_eq_conv_conj:
nipkow@13145
  1188
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
paulson@14208
  1189
apply (induct xs, simp, clarsimp)
paulson@14208
  1190
apply (case_tac zs, auto)
nipkow@13145
  1191
done
wenzelm@13142
  1192
paulson@14050
  1193
lemma take_add [rule_format]: 
paulson@14050
  1194
    "\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)"
paulson@14050
  1195
apply (induct xs, auto) 
paulson@14050
  1196
apply (case_tac i, simp_all) 
paulson@14050
  1197
done
paulson@14050
  1198
nipkow@14300
  1199
lemma append_eq_append_conv_if:
nipkow@14300
  1200
 "!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) =
nipkow@14300
  1201
  (if size xs\<^isub>1 \<le> size ys\<^isub>1
nipkow@14300
  1202
   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
nipkow@14300
  1203
   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)"
nipkow@14300
  1204
apply(induct xs\<^isub>1)
nipkow@14300
  1205
 apply simp
nipkow@14300
  1206
apply(case_tac ys\<^isub>1)
nipkow@14300
  1207
apply simp_all
nipkow@14300
  1208
done
nipkow@14300
  1209
nipkow@15110
  1210
lemma take_hd_drop:
nipkow@15110
  1211
  "!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs"
nipkow@15110
  1212
apply(induct xs)
nipkow@15110
  1213
apply simp
nipkow@15110
  1214
apply(simp add:drop_Cons split:nat.split)
nipkow@15110
  1215
done
nipkow@15110
  1216
nipkow@17501
  1217
lemma id_take_nth_drop:
nipkow@17501
  1218
 "i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" 
nipkow@17501
  1219
proof -
nipkow@17501
  1220
  assume si: "i < length xs"
nipkow@17501
  1221
  hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto
nipkow@17501
  1222
  moreover
nipkow@17501
  1223
  from si have "take (Suc i) xs = take i xs @ [xs!i]"
nipkow@17501
  1224
    apply (rule_tac take_Suc_conv_app_nth) by arith
nipkow@17501
  1225
  ultimately show ?thesis by auto
nipkow@17501
  1226
qed
nipkow@17501
  1227
  
nipkow@17501
  1228
lemma upd_conv_take_nth_drop:
nipkow@17501
  1229
 "i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1230
proof -
nipkow@17501
  1231
  assume i: "i < length xs"
nipkow@17501
  1232
  have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]"
nipkow@17501
  1233
    by(rule arg_cong[OF id_take_nth_drop[OF i]])
nipkow@17501
  1234
  also have "\<dots> = take i xs @ a # drop (Suc i) xs"
nipkow@17501
  1235
    using i by (simp add: list_update_append)
nipkow@17501
  1236
  finally show ?thesis .
nipkow@17501
  1237
qed
nipkow@17501
  1238
wenzelm@13114
  1239
nipkow@15392
  1240
subsubsection {* @{text takeWhile} and @{text dropWhile} *}
wenzelm@13114
  1241
wenzelm@13142
  1242
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs"
nipkow@13145
  1243
by (induct xs) auto
wenzelm@13114
  1244
wenzelm@13142
  1245
lemma takeWhile_append1 [simp]:
nipkow@13145
  1246
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs"
nipkow@13145
  1247
by (induct xs) auto
wenzelm@13114
  1248
wenzelm@13142
  1249
lemma takeWhile_append2 [simp]:
nipkow@13145
  1250
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys"
nipkow@13145
  1251
by (induct xs) auto
wenzelm@13114
  1252
wenzelm@13142
  1253
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs"
nipkow@13145
  1254
by (induct xs) auto
wenzelm@13114
  1255
wenzelm@13142
  1256
lemma dropWhile_append1 [simp]:
nipkow@13145
  1257
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"
nipkow@13145
  1258
by (induct xs) auto
wenzelm@13114
  1259
wenzelm@13142
  1260
lemma dropWhile_append2 [simp]:
nipkow@13145
  1261
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys"
nipkow@13145
  1262
by (induct xs) auto
wenzelm@13114
  1263
wenzelm@13142
  1264
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x"
nipkow@13145
  1265
by (induct xs) (auto split: split_if_asm)
wenzelm@13114
  1266
nipkow@13913
  1267
lemma takeWhile_eq_all_conv[simp]:
nipkow@13913
  1268
 "(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1269
by(induct xs, auto)
nipkow@13913
  1270
nipkow@13913
  1271
lemma dropWhile_eq_Nil_conv[simp]:
nipkow@13913
  1272
 "(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)"
nipkow@13913
  1273
by(induct xs, auto)
nipkow@13913
  1274
nipkow@13913
  1275
lemma dropWhile_eq_Cons_conv:
nipkow@13913
  1276
 "(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)"
nipkow@13913
  1277
by(induct xs, auto)
nipkow@13913
  1278
nipkow@17501
  1279
text{* The following two lemmmas could be generalized to an arbitrary
nipkow@17501
  1280
property. *}
nipkow@17501
  1281
nipkow@17501
  1282
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1283
 takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
nipkow@17501
  1284
by(induct xs) (auto simp: takeWhile_tail[where l="[]"])
nipkow@17501
  1285
nipkow@17501
  1286
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow>
nipkow@17501
  1287
  dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)"
nipkow@17501
  1288
apply(induct xs)
nipkow@17501
  1289
 apply simp
nipkow@17501
  1290
apply auto
nipkow@17501
  1291
apply(subst dropWhile_append2)
nipkow@17501
  1292
apply auto
nipkow@17501
  1293
done
nipkow@17501
  1294
nipkow@18423
  1295
lemma takeWhile_not_last:
nipkow@18423
  1296
 "\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs"
nipkow@18423
  1297
apply(induct xs)
nipkow@18423
  1298
 apply simp
nipkow@18423
  1299
apply(case_tac xs)
nipkow@18423
  1300
apply(auto)
nipkow@18423
  1301
done
nipkow@18423
  1302
krauss@19770
  1303
lemma takeWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1304
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1305
  ==> takeWhile P l = takeWhile Q k"
wenzelm@20503
  1306
  by (induct k arbitrary: l) (simp_all)
krauss@18336
  1307
krauss@19770
  1308
lemma dropWhile_cong [fundef_cong, recdef_cong]:
krauss@18336
  1309
  "[| l = k; !!x. x : set l ==> P x = Q x |] 
krauss@18336
  1310
  ==> dropWhile P l = dropWhile Q k"
wenzelm@20503
  1311
  by (induct k arbitrary: l, simp_all)
krauss@18336
  1312
wenzelm@13114
  1313
nipkow@15392
  1314
subsubsection {* @{text zip} *}
wenzelm@13114
  1315
wenzelm@13142
  1316
lemma zip_Nil [simp]: "zip [] ys = []"
nipkow@13145
  1317
by (induct ys) auto
wenzelm@13114
  1318
wenzelm@13142
  1319
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
nipkow@13145
  1320
by simp
wenzelm@13114
  1321
wenzelm@13142
  1322
declare zip_Cons [simp del]
wenzelm@13114
  1323
nipkow@15281
  1324
lemma zip_Cons1:
nipkow@15281
  1325
 "zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)"
nipkow@15281
  1326
by(auto split:list.split)
nipkow@15281
  1327
wenzelm@13142
  1328
lemma length_zip [simp]:
nipkow@13145
  1329
"!!xs. length (zip xs ys) = min (length xs) (length ys)"
paulson@14208
  1330
apply (induct ys, simp)
paulson@14208
  1331
apply (case_tac xs, auto)
nipkow@13145
  1332
done
wenzelm@13114
  1333
wenzelm@13114
  1334
lemma zip_append1:
nipkow@13145
  1335
"!!xs. zip (xs @ ys) zs =
nipkow@13145
  1336
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
paulson@14208
  1337
apply (induct zs, simp)
paulson@14208
  1338
apply (case_tac xs, simp_all)
nipkow@13145
  1339
done
wenzelm@13114
  1340
wenzelm@13114
  1341
lemma zip_append2:
nipkow@13145
  1342
"!!ys. zip xs (ys @ zs) =
nipkow@13145
  1343
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs"
paulson@14208
  1344
apply (induct xs, simp)
paulson@14208
  1345
apply (case_tac ys, simp_all)
nipkow@13145
  1346
done
wenzelm@13114
  1347
wenzelm@13142
  1348
lemma zip_append [simp]:
wenzelm@13142
  1349
 "[| length xs = length us; length ys = length vs |] ==>
nipkow@13145
  1350
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs"
nipkow@13145
  1351
by (simp add: zip_append1)
wenzelm@13114
  1352
wenzelm@13114
  1353
lemma zip_rev:
nipkow@14247
  1354
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)"
nipkow@14247
  1355
by (induct rule:list_induct2, simp_all)
wenzelm@13114
  1356
wenzelm@13142
  1357
lemma nth_zip [simp]:
nipkow@13145
  1358
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)"
paulson@14208
  1359
apply (induct ys, simp)
nipkow@13145
  1360
apply (case_tac xs)
nipkow@13145
  1361
 apply (simp_all add: nth.simps split: nat.split)
nipkow@13145
  1362
done
wenzelm@13114
  1363
wenzelm@13114
  1364
lemma set_zip:
nipkow@13145
  1365
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}"
nipkow@13145
  1366
by (simp add: set_conv_nth cong: rev_conj_cong)
wenzelm@13114
  1367
wenzelm@13114
  1368
lemma zip_update:
nipkow@13145
  1369
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]"
nipkow@13145
  1370
by (rule sym, simp add: update_zip)
wenzelm@13114
  1371
wenzelm@13142
  1372
lemma zip_replicate [simp]:
nipkow@13145
  1373
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)"
paulson@14208
  1374
apply (induct i, auto)
paulson@14208
  1375
apply (case_tac j, auto)
nipkow@13145
  1376
done
wenzelm@13114
  1377
nipkow@19487
  1378
lemma take_zip:
nipkow@19487
  1379
 "!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)"
nipkow@19487
  1380
apply (induct n)
nipkow@19487
  1381
 apply simp
nipkow@19487
  1382
apply (case_tac xs, simp)
nipkow@19487
  1383
apply (case_tac ys, simp_all)
nipkow@19487
  1384
done
nipkow@19487
  1385
nipkow@19487
  1386
lemma drop_zip:
nipkow@19487
  1387
 "!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
nipkow@19487
  1388
apply (induct n)
nipkow@19487
  1389
 apply simp
nipkow@19487
  1390
apply (case_tac xs, simp)
nipkow@19487
  1391
apply (case_tac ys, simp_all)
nipkow@19487
  1392
done
nipkow@19487
  1393
wenzelm@13142
  1394
nipkow@15392
  1395
subsubsection {* @{text list_all2} *}
wenzelm@13114
  1396
kleing@14316
  1397
lemma list_all2_lengthD [intro?]: 
kleing@14316
  1398
  "list_all2 P xs ys ==> length xs = length ys"
haftmann@19607
  1399
  by (simp add: list_all2_def)
haftmann@19607
  1400
haftmann@19787
  1401
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])"
haftmann@19607
  1402
  by (simp add: list_all2_def)
haftmann@19607
  1403
haftmann@19787
  1404
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])"
haftmann@19787
  1405
  by (simp add: list_all2_def)
haftmann@19607
  1406
haftmann@19607
  1407
lemma list_all2_Cons [iff, code]:
haftmann@19607
  1408
  "list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)"
haftmann@19607
  1409
  by (auto simp add: list_all2_def)
wenzelm@13114
  1410
wenzelm@13114
  1411
lemma list_all2_Cons1:
nipkow@13145
  1412
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
nipkow@13145
  1413
by (cases ys) auto
wenzelm@13114
  1414
wenzelm@13114
  1415
lemma list_all2_Cons2:
nipkow@13145
  1416
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
nipkow@13145
  1417
by (cases xs) auto
wenzelm@13114
  1418
wenzelm@13142
  1419
lemma list_all2_rev [iff]:
nipkow@13145
  1420
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys"
nipkow@13145
  1421
by (simp add: list_all2_def zip_rev cong: conj_cong)
wenzelm@13114
  1422
kleing@13863
  1423
lemma list_all2_rev1:
kleing@13863
  1424
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)"
kleing@13863
  1425
by (subst list_all2_rev [symmetric]) simp
kleing@13863
  1426
wenzelm@13114
  1427
lemma list_all2_append1:
nipkow@13145
  1428
"list_all2 P (xs @ ys) zs =
nipkow@13145
  1429
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and>
nipkow@13145
  1430
list_all2 P xs us \<and> list_all2 P ys vs)"
nipkow@13145
  1431
apply (simp add: list_all2_def zip_append1)
nipkow@13145
  1432
apply (rule iffI)
nipkow@13145
  1433
 apply (rule_tac x = "take (length xs) zs" in exI)
nipkow@13145
  1434
 apply (rule_tac x = "drop (length xs) zs" in exI)
paulson@14208
  1435
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1436
apply (simp add: ball_Un)
nipkow@13145
  1437
done
wenzelm@13114
  1438
wenzelm@13114
  1439
lemma list_all2_append2:
nipkow@13145
  1440
"list_all2 P xs (ys @ zs) =
nipkow@13145
  1441
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and>
nipkow@13145
  1442
list_all2 P us ys \<and> list_all2 P vs zs)"
nipkow@13145
  1443
apply (simp add: list_all2_def zip_append2)
nipkow@13145
  1444
apply (rule iffI)
nipkow@13145
  1445
 apply (rule_tac x = "take (length ys) xs" in exI)
nipkow@13145
  1446
 apply (rule_tac x = "drop (length ys) xs" in exI)
paulson@14208
  1447
 apply (force split: nat_diff_split simp add: min_def, clarify)
nipkow@13145
  1448
apply (simp add: ball_Un)
nipkow@13145
  1449
done
wenzelm@13114
  1450
kleing@13863
  1451
lemma list_all2_append:
nipkow@14247
  1452
  "length xs = length ys \<Longrightarrow>
nipkow@14247
  1453
  list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
nipkow@14247
  1454
by (induct rule:list_induct2, simp_all)
kleing@13863
  1455
kleing@13863
  1456
lemma list_all2_appendI [intro?, trans]:
kleing@13863
  1457
  "\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)"
kleing@13863
  1458
  by (simp add: list_all2_append list_all2_lengthD)
kleing@13863
  1459
wenzelm@13114
  1460
lemma list_all2_conv_all_nth:
nipkow@13145
  1461
"list_all2 P xs ys =
nipkow@13145
  1462
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))"
nipkow@13145
  1463
by (force simp add: list_all2_def set_zip)
wenzelm@13114
  1464
berghofe@13883
  1465
lemma list_all2_trans:
berghofe@13883
  1466
  assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c"
berghofe@13883
  1467
  shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs"
berghofe@13883
  1468
        (is "!!bs cs. PROP ?Q as bs cs")
berghofe@13883
  1469
proof (induct as)
berghofe@13883
  1470
  fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs"
berghofe@13883
  1471
  show "!!cs. PROP ?Q (x # xs) bs cs"
berghofe@13883
  1472
  proof (induct bs)
berghofe@13883
  1473
    fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs"
berghofe@13883
  1474
    show "PROP ?Q (x # xs) (y # ys) cs"
berghofe@13883
  1475
      by (induct cs) (auto intro: tr I1 I2)
berghofe@13883
  1476
  qed simp
berghofe@13883
  1477
qed simp
berghofe@13883
  1478
kleing@13863
  1479
lemma list_all2_all_nthI [intro?]:
kleing@13863
  1480
  "length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b"
kleing@13863
  1481
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1482
paulson@14395
  1483
lemma list_all2I:
paulson@14395
  1484
  "\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b"
paulson@14395
  1485
  by (simp add: list_all2_def)
paulson@14395
  1486
kleing@14328
  1487
lemma list_all2_nthD:
kleing@13863
  1488
  "\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
kleing@13863
  1489
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1490
nipkow@14302
  1491
lemma list_all2_nthD2:
nipkow@14302
  1492
  "\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)"
nipkow@14302
  1493
  by (frule list_all2_lengthD) (auto intro: list_all2_nthD)
nipkow@14302
  1494
kleing@13863
  1495
lemma list_all2_map1: 
kleing@13863
  1496
  "list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs"
kleing@13863
  1497
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1498
kleing@13863
  1499
lemma list_all2_map2: 
kleing@13863
  1500
  "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
kleing@13863
  1501
  by (auto simp add: list_all2_conv_all_nth)
kleing@13863
  1502
kleing@14316
  1503
lemma list_all2_refl [intro?]:
kleing@13863
  1504
  "(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs"
kleing@13863
  1505
  by (simp add: list_all2_conv_all_nth)
kleing@13863
  1506
kleing@13863
  1507
lemma list_all2_update_cong:
kleing@13863
  1508
  "\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1509
  by (simp add: list_all2_conv_all_nth nth_list_update)
kleing@13863
  1510
kleing@13863
  1511
lemma list_all2_update_cong2:
kleing@13863
  1512
  "\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])"
kleing@13863
  1513
  by (simp add: list_all2_lengthD list_all2_update_cong)
kleing@13863
  1514
nipkow@14302
  1515
lemma list_all2_takeI [simp,intro?]:
nipkow@14302
  1516
  "\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)"
nipkow@14302
  1517
  apply (induct xs)
nipkow@14302
  1518
   apply simp
nipkow@14302
  1519
  apply (clarsimp simp add: list_all2_Cons1)
nipkow@14302
  1520
  apply (case_tac n)
nipkow@14302
  1521
  apply auto
nipkow@14302
  1522
  done
nipkow@14302
  1523
nipkow@14302
  1524
lemma list_all2_dropI [simp,intro?]:
kleing@13863
  1525
  "\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)"
paulson@14208
  1526
  apply (induct as, simp)
kleing@13863
  1527
  apply (clarsimp simp add: list_all2_Cons1)
paulson@14208
  1528
  apply (case_tac n, simp, simp)
kleing@13863
  1529
  done
kleing@13863
  1530
kleing@14327
  1531
lemma list_all2_mono [intro?]:
kleing@13863
  1532
  "\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y"
paulson@14208
  1533
  apply (induct x, simp)
paulson@14208
  1534
  apply (case_tac y, auto)
kleing@13863
  1535
  done
kleing@13863
  1536
wenzelm@13142
  1537
nipkow@15392
  1538
subsubsection {* @{text foldl} and @{text foldr} *}
wenzelm@13142
  1539
wenzelm@13142
  1540
lemma foldl_append [simp]:
nipkow@13145
  1541
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
nipkow@13145
  1542
by (induct xs) auto
wenzelm@13142
  1543
nipkow@14402
  1544
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
nipkow@14402
  1545
by (induct xs) auto
nipkow@14402
  1546
krauss@19770
  1547
lemma foldl_cong [fundef_cong, recdef_cong]:
krauss@18336
  1548
  "[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] 
krauss@18336
  1549
  ==> foldl f a l = foldl g b k"
wenzelm@20503
  1550
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1551
krauss@19770
  1552
lemma foldr_cong [fundef_cong, recdef_cong]:
krauss@18336
  1553
  "[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] 
krauss@18336
  1554
  ==> foldr f l a = foldr g k b"
wenzelm@20503
  1555
  by (induct k arbitrary: a b l) simp_all
krauss@18336
  1556
nipkow@14402
  1557
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)"
nipkow@14402
  1558
by (induct xs) auto
nipkow@14402
  1559
nipkow@14402
  1560
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a"
nipkow@14402
  1561
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"])
nipkow@14402
  1562
wenzelm@13142
  1563
text {*
nipkow@13145
  1564
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more
nipkow@13145
  1565
difficult to use because it requires an additional transitivity step.
wenzelm@13142
  1566
*}
wenzelm@13142
  1567
wenzelm@13142
  1568
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns"
nipkow@13145
  1569
by (induct ns) auto
wenzelm@13142
  1570
wenzelm@13142
  1571
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns"
nipkow@13145
  1572
by (force intro: start_le_sum simp add: in_set_conv_decomp)
wenzelm@13142
  1573
wenzelm@13142
  1574
lemma sum_eq_0_conv [iff]:
nipkow@13145
  1575
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))"
nipkow@13145
  1576
by (induct ns) auto
wenzelm@13114
  1577
wenzelm@13114
  1578
nipkow@15392
  1579
subsubsection {* @{text upto} *}
wenzelm@13114
  1580
nipkow@17090
  1581
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])"
nipkow@17090
  1582
-- {* simp does not terminate! *}
nipkow@13145
  1583
by (induct j) auto
wenzelm@13142
  1584
nipkow@15425
  1585
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []"
nipkow@13145
  1586
by (subst upt_rec) simp
wenzelm@13114
  1587
nipkow@15425
  1588
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)"
nipkow@15281
  1589
by(induct j)simp_all
nipkow@15281
  1590
nipkow@15281
  1591
lemma upt_eq_Cons_conv:
nipkow@15425
  1592
 "!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)"
nipkow@15281
  1593
apply(induct j)
nipkow@15281
  1594
 apply simp
nipkow@15281
  1595
apply(clarsimp simp add: append_eq_Cons_conv)
nipkow@15281
  1596
apply arith
nipkow@15281
  1597
done
nipkow@15281
  1598
nipkow@15425
  1599
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]"
nipkow@13145
  1600
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *}
nipkow@13145
  1601
by simp
wenzelm@13114
  1602
nipkow@15425
  1603
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]"
nipkow@13145
  1604
apply(rule trans)
nipkow@13145
  1605
apply(subst upt_rec)
paulson@14208
  1606
 prefer 2 apply (rule refl, simp)
nipkow@13145
  1607
done
wenzelm@13114
  1608
nipkow@15425
  1609
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]"
nipkow@13145
  1610
-- {* LOOPS as a simprule, since @{text "j <= j"}. *}
nipkow@13145
  1611
by (induct k) auto
wenzelm@13114
  1612
nipkow@15425
  1613
lemma length_upt [simp]: "length [i..<j] = j - i"
nipkow@13145
  1614
by (induct j) (auto simp add: Suc_diff_le)
wenzelm@13114
  1615
nipkow@15425
  1616
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k"
nipkow@13145
  1617
apply (induct j)
nipkow@13145
  1618
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split)
nipkow@13145
  1619
done
wenzelm@13114
  1620
nipkow@17906
  1621
nipkow@17906
  1622
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i"
nipkow@17906
  1623
by(simp add:upt_conv_Cons)
nipkow@17906
  1624
nipkow@17906
  1625
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1"
nipkow@17906
  1626
apply(cases j)
nipkow@17906
  1627
 apply simp
nipkow@17906
  1628
by(simp add:upt_Suc_append)
nipkow@17906
  1629
nipkow@15425
  1630
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]"
paulson@14208
  1631
apply (induct m, simp)
nipkow@13145
  1632
apply (subst upt_rec)
nipkow@13145
  1633
apply (rule sym)
nipkow@13145
  1634
apply (subst upt_rec)
nipkow@13145
  1635
apply (simp del: upt.simps)
nipkow@13145
  1636
done
nipkow@3507
  1637
nipkow@17501
  1638
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]"
nipkow@17501
  1639
apply(induct j)
nipkow@17501
  1640
apply auto
nipkow@17501
  1641
done
nipkow@17501
  1642
nipkow@15425
  1643
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]"
nipkow@13145
  1644
by (induct n) auto
wenzelm@13114
  1645
nipkow@15425
  1646
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)"
nipkow@13145
  1647
apply (induct n m rule: diff_induct)
nipkow@13145
  1648
prefer 3 apply (subst map_Suc_upt[symmetric])
nipkow@13145
  1649
apply (auto simp add: less_diff_conv nth_upt)
nipkow@13145
  1650
done
wenzelm@13114
  1651
berghofe@13883
  1652
lemma nth_take_lemma:
berghofe@13883
  1653
  "!!xs ys. k <= length xs ==> k <= length ys ==>
berghofe@13883
  1654
     (!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys"
berghofe@13883
  1655
apply (atomize, induct k)
paulson@14208
  1656
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify)
nipkow@13145
  1657
txt {* Both lists must be non-empty *}
paulson@14208
  1658
apply (case_tac xs, simp)
paulson@14208
  1659
apply (case_tac ys, clarify)
nipkow@13145
  1660
 apply (simp (no_asm_use))
nipkow@13145
  1661
apply clarify
nipkow@13145
  1662
txt {* prenexing's needed, not miniscoping *}
nipkow@13145
  1663
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps)
nipkow@13145
  1664
apply blast
nipkow@13145
  1665
done
wenzelm@13114
  1666
wenzelm@13114
  1667
lemma nth_equalityI:
wenzelm@13114
  1668
 "[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys"
nipkow@13145
  1669
apply (frule nth_take_lemma [OF le_refl eq_imp_le])
nipkow@13145
  1670
apply (simp_all add: take_all)
nipkow@13145
  1671
done
wenzelm@13142
  1672
kleing@13863
  1673
(* needs nth_equalityI *)
kleing@13863
  1674
lemma list_all2_antisym:
kleing@13863
  1675
  "\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> 
kleing@13863
  1676
  \<Longrightarrow> xs = ys"
kleing@13863
  1677
  apply (simp add: list_all2_conv_all_nth) 
paulson@14208
  1678
  apply (rule nth_equalityI, blast, simp)
kleing@13863
  1679
  done
kleing@13863
  1680
wenzelm@13142
  1681
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys"
nipkow@13145
  1682
-- {* The famous take-lemma. *}
nipkow@13145
  1683
apply (drule_tac x = "max (length xs) (length ys)" in spec)
nipkow@13145
  1684
apply (simp add: le_max_iff_disj take_all)
nipkow@13145
  1685
done
wenzelm@13142
  1686
wenzelm@13142
  1687
nipkow@15302
  1688
lemma take_Cons':
nipkow@15302
  1689
     "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
nipkow@15302
  1690
by (cases n) simp_all
nipkow@15302
  1691
nipkow@15302
  1692
lemma drop_Cons':
nipkow@15302
  1693
     "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
nipkow@15302
  1694
by (cases n) simp_all
nipkow@15302
  1695
nipkow@15302
  1696
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))"
nipkow@15302
  1697
by (cases n) simp_all
nipkow@15302
  1698
paulson@18622
  1699
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard]
paulson@18622
  1700
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard]
paulson@18622
  1701
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard]
paulson@18622
  1702
paulson@18622
  1703
declare take_Cons_number_of [simp] 
paulson@18622
  1704
        drop_Cons_number_of [simp] 
paulson@18622
  1705
        nth_Cons_number_of [simp] 
nipkow@15302
  1706
nipkow@15302
  1707
nipkow@15392
  1708
subsubsection {* @{text "distinct"} and @{text remdups} *}
wenzelm@13142
  1709
wenzelm@13142
  1710
lemma distinct_append [simp]:
nipkow@13145
  1711
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
nipkow@13145
  1712
by (induct xs) auto
wenzelm@13142
  1713
nipkow@15305
  1714
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs"
nipkow@15305
  1715
by(induct xs) auto
nipkow@15305
  1716
wenzelm@13142
  1717
lemma set_remdups [simp]: "set (remdups xs) = set xs"
nipkow@13145
  1718
by (induct xs) (auto simp add: insert_absorb)
wenzelm@13142
  1719
wenzelm@13142
  1720
lemma distinct_remdups [iff]: "distinct (remdups xs)"
nipkow@13145
  1721
by (induct xs) auto
wenzelm@13142
  1722
paulson@15072
  1723
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])"
paulson@15251
  1724
  by (induct x, auto) 
paulson@15072
  1725
paulson@15072
  1726
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])"
paulson@15251
  1727
  by (induct x, auto)
paulson@15072
  1728
nipkow@15245
  1729
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs"
nipkow@15245
  1730
by (induct xs) auto
nipkow@15245
  1731
nipkow@15245
  1732
lemma length_remdups_eq[iff]:
nipkow@15245
  1733
  "(length (remdups xs) = length xs) = (remdups xs = xs)"
nipkow@15245
  1734
apply(induct xs)
nipkow@15245
  1735
 apply auto
nipkow@15245
  1736
apply(subgoal_tac "length (remdups xs) <= length xs")
nipkow@15245
  1737
 apply arith
nipkow@15245
  1738
apply(rule length_remdups_leq)
nipkow@15245
  1739
done
nipkow@15245
  1740
nipkow@18490
  1741
nipkow@18490
  1742
lemma distinct_map:
nipkow@18490
  1743
  "distinct(map f xs) = (distinct xs & inj_on f (set xs))"
nipkow@18490
  1744
by (induct xs) auto
nipkow@18490
  1745
nipkow@18490
  1746
wenzelm@13142
  1747
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)"
nipkow@13145
  1748
by (induct xs) auto
wenzelm@13114
  1749
nipkow@17501
  1750
lemma distinct_upt[simp]: "distinct[i..<j]"
nipkow@17501
  1751
by (induct j) auto
nipkow@17501
  1752
nipkow@17501
  1753
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)"
nipkow@17501
  1754
apply(induct xs)
nipkow@17501
  1755
 apply simp
nipkow@17501
  1756
apply (case_tac i)
nipkow@17501
  1757
 apply simp_all
nipkow@17501
  1758
apply(blast dest:in_set_takeD)
nipkow@17501
  1759
done
nipkow@17501
  1760
nipkow@17501
  1761
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)"
nipkow@17501
  1762
apply(induct xs)
nipkow@17501
  1763
 apply simp
nipkow@17501
  1764
apply (case_tac i)
nipkow@17501
  1765
 apply simp_all
nipkow@17501
  1766
done
nipkow@17501
  1767
nipkow@17501
  1768
lemma distinct_list_update:
nipkow@17501
  1769
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}"
nipkow@17501
  1770
shows "distinct (xs[i:=a])"
nipkow@17501
  1771
proof (cases "i < length xs")
nipkow@17501
  1772
  case True
nipkow@17501
  1773
  with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}"
nipkow@17501
  1774
    apply (drule_tac id_take_nth_drop) by simp
nipkow@17501
  1775
  with d True show ?thesis
nipkow@17501
  1776
    apply (simp add: upd_conv_take_nth_drop)
nipkow@17501
  1777
    apply (drule subst [OF id_take_nth_drop]) apply assumption
nipkow@17501
  1778
    apply simp apply (cases "a = xs!i") apply simp by blast
nipkow@17501
  1779
next
nipkow@17501
  1780
  case False with d show ?thesis by auto
nipkow@17501
  1781
qed
nipkow@17501
  1782
nipkow@17501
  1783
nipkow@17501
  1784
text {* It is best to avoid this indexed version of distinct, but
nipkow@17501
  1785
sometimes it is useful. *}
nipkow@17501
  1786
wenzelm@13142
  1787
lemma distinct_conv_nth:
nipkow@17501
  1788
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)"
paulson@15251
  1789
apply (induct xs, simp, simp)
paulson@14208
  1790
apply (rule iffI, clarsimp)
nipkow@13145
  1791
 apply (case_tac i)
paulson@14208
  1792
apply (case_tac j, simp)
nipkow@13145
  1793
apply (simp add: set_conv_nth)
nipkow@13145
  1794
 apply (case_tac j)
paulson@14208
  1795
apply (clarsimp simp add: set_conv_nth, simp)
nipkow@13145
  1796
apply (rule conjI)
nipkow@13145
  1797
 apply (clarsimp simp add: set_conv_nth)
nipkow@17501
  1798
 apply (erule_tac x = 0 in allE, simp)
paulson@14208
  1799
 apply (erule_tac x = "Suc i" in allE, simp, clarsimp)
nipkow@17501
  1800
apply (erule_tac x = "Suc i" in allE, simp)
paulson@14208
  1801
apply (erule_tac x = "Suc j" in allE, simp)
nipkow@13145
  1802
done
wenzelm@13114
  1803
nipkow@18490
  1804
lemma nth_eq_iff_index_eq:
nipkow@18490
  1805
 "\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)"
nipkow@18490
  1806
by(auto simp: distinct_conv_nth)
nipkow@18490
  1807
nipkow@15110
  1808
lemma distinct_card: "distinct xs ==> card (set xs) = size xs"
kleing@14388
  1809
  by (induct xs) auto
kleing@14388
  1810
nipkow@15110
  1811
lemma card_distinct: "card (set xs) = size xs ==> distinct xs"
kleing@14388
  1812
proof (induct xs)
kleing@14388
  1813
  case Nil thus ?case by simp
kleing@14388
  1814
next
kleing@14388
  1815
  case (Cons x xs)
kleing@14388
  1816
  show ?case
kleing@14388
  1817
  proof (cases "x \<in> set xs")
kleing@14388
  1818
    case False with Cons show ?thesis by simp
kleing@14388
  1819
  next
kleing@14388
  1820
    case True with Cons.prems
kleing@14388
  1821
    have "card (set xs) = Suc (length xs)" 
kleing@14388
  1822
      by (simp add: card_insert_if split: split_if_asm)
kleing@14388
  1823
    moreover have "card (set xs) \<le> length xs" by (rule card_length)
kleing@14388
  1824
    ultimately have False by simp
kleing@14388
  1825
    thus ?thesis ..
kleing@14388
  1826
  qed
kleing@14388
  1827
qed
kleing@14388
  1828
nipkow@18490
  1829
nipkow@18490
  1830
lemma length_remdups_concat:
nipkow@18490
  1831
 "length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)"
nipkow@18490
  1832
by(simp add: distinct_card[symmetric])
nipkow@17906
  1833
nipkow@17906
  1834
nipkow@15392
  1835
subsubsection {* @{text remove1} *}
nipkow@15110
  1836
nipkow@18049
  1837
lemma remove1_append:
nipkow@18049
  1838
  "remove1 x (xs @ ys) =
nipkow@18049
  1839
  (if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)"
nipkow@18049
  1840
by (induct xs) auto
nipkow@18049
  1841
nipkow@15110
  1842
lemma set_remove1_subset: "set(remove1 x xs) <= set xs"
nipkow@15110
  1843
apply(induct xs)
nipkow@15110
  1844
 apply simp
nipkow@15110
  1845
apply simp
nipkow@15110
  1846
apply blast
nipkow@15110
  1847
done
nipkow@15110
  1848
paulson@17724
  1849
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}"
nipkow@15110
  1850
apply(induct xs)
nipkow@15110
  1851
 apply simp
nipkow@15110
  1852
apply simp
nipkow@15110
  1853
apply blast
nipkow@15110
  1854
done
nipkow@15110
  1855
nipkow@18049
  1856
lemma remove1_filter_not[simp]:
nipkow@18049
  1857
  "\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs"
nipkow@18049
  1858
by(induct xs) auto
nipkow@18049
  1859
nipkow@15110
  1860
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)"
nipkow@15110
  1861
apply(insert set_remove1_subset)
nipkow@15110
  1862
apply fast
nipkow@15110
  1863
done
nipkow@15110
  1864
nipkow@15110
  1865
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)"
nipkow@15110
  1866
by (induct xs) simp_all
nipkow@15110
  1867
wenzelm@13114
  1868
nipkow@15392
  1869
subsubsection {* @{text replicate} *}
wenzelm@13114
  1870
wenzelm@13142
  1871
lemma length_replicate [simp]: "length (replicate n x) = n"
nipkow@13145
  1872
by (induct n) auto
nipkow@13124
  1873
wenzelm@13142
  1874
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)"
nipkow@13145
  1875
by (induct n) auto
wenzelm@13114
  1876
wenzelm@13114
  1877
lemma replicate_app_Cons_same:
nipkow@13145
  1878
"(replicate n x) @ (x # xs) = x # replicate n x @ xs"
nipkow@13145
  1879
by (induct n) auto
wenzelm@13114
  1880
wenzelm@13142
  1881
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x"
paulson@14208
  1882
apply (induct n, simp)
nipkow@13145
  1883
apply (simp add: replicate_app_Cons_same)
nipkow@13145
  1884
done
wenzelm@13114
  1885
wenzelm@13142
  1886
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x"
nipkow@13145
  1887
by (induct n) auto
wenzelm@13114
  1888
nipkow@16397
  1889
text{* Courtesy of Matthias Daum: *}
nipkow@16397
  1890
lemma append_replicate_commute:
nipkow@16397
  1891
  "replicate n x @ replicate k x = replicate k x @ replicate n x"
nipkow@16397
  1892
apply (simp add: replicate_add [THEN sym])
nipkow@16397
  1893
apply (simp add: add_commute)
nipkow@16397
  1894
done
nipkow@16397
  1895
wenzelm@13142
  1896
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x"
nipkow@13145
  1897
by (induct n) auto
wenzelm@13114
  1898
wenzelm@13142
  1899
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x"
nipkow@13145
  1900
by (induct n) auto
wenzelm@13114
  1901
wenzelm@13142
  1902
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x"
nipkow@13145
  1903
by (atomize (full), induct n) auto
wenzelm@13114
  1904
wenzelm@13142
  1905
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x"
paulson@14208
  1906
apply (induct n, simp)
nipkow@13145
  1907
apply (simp add: nth_Cons split: nat.split)
nipkow@13145
  1908
done
wenzelm@13114
  1909
nipkow@16397
  1910
text{* Courtesy of Matthias Daum (2 lemmas): *}
nipkow@16397
  1911
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x"
nipkow@16397
  1912
apply (case_tac "k \<le> i")
nipkow@16397
  1913
 apply  (simp add: min_def)
nipkow@16397
  1914
apply (drule not_leE)
nipkow@16397
  1915
apply (simp add: min_def)
nipkow@16397
  1916
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x")
nipkow@16397
  1917
 apply  simp
nipkow@16397
  1918
apply (simp add: replicate_add [symmetric])
nipkow@16397
  1919
done
nipkow@16397
  1920
nipkow@16397
  1921
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x"
nipkow@16397
  1922
apply (induct k)
nipkow@16397
  1923
 apply simp
nipkow@16397
  1924
apply clarsimp
nipkow@16397
  1925
apply (case_tac i)
nipkow@16397
  1926
 apply simp
nipkow@16397
  1927
apply clarsimp
nipkow@16397
  1928
done
nipkow@16397
  1929
nipkow@16397
  1930
wenzelm@13142
  1931
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}"
nipkow@13145
  1932
by (induct n) auto
wenzelm@13114
  1933
wenzelm@13142
  1934
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}"
nipkow@13145
  1935
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc)
wenzelm@13114
  1936
wenzelm@13142
  1937
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})"
nipkow@13145
  1938
by auto
wenzelm@13114
  1939
wenzelm@13142
  1940
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y"
nipkow@13145
  1941
by (simp add: set_replicate_conv_if split: split_if_asm)
wenzelm@13114
  1942
wenzelm@13114
  1943
nipkow@15392
  1944
subsubsection{*@{text rotate1} and @{text rotate}*}
nipkow@15302
  1945
nipkow@15302
  1946
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]"
nipkow@15302
  1947
by(simp add:rotate1_def)
nipkow@15302
  1948
nipkow@15302
  1949
lemma rotate0[simp]: "rotate 0 = id"
nipkow@15302
  1950
by(simp add:rotate_def)
nipkow@15302
  1951
nipkow@15302
  1952
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)"
nipkow@15302
  1953
by(simp add:rotate_def)
nipkow@15302
  1954
nipkow@15302
  1955
lemma rotate_add:
nipkow@15302
  1956
  "rotate (m+n) = rotate m o rotate n"
nipkow@15302
  1957
by(simp add:rotate_def funpow_add)
nipkow@15302
  1958
nipkow@15302
  1959
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs"
nipkow@15302
  1960
by(simp add:rotate_add)
nipkow@15302
  1961
nipkow@18049
  1962
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
nipkow@18049
  1963
by(simp add:rotate_def funpow_swap1)
nipkow@18049
  1964
nipkow@15302
  1965
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs"
nipkow@15302
  1966
by(cases xs) simp_all
nipkow@15302
  1967
nipkow@15302
  1968
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1969
apply(induct n)
nipkow@15302
  1970
 apply simp
nipkow@15302
  1971
apply (simp add:rotate_def)
nipkow@13145
  1972
done
wenzelm@13114
  1973
nipkow@15302
  1974
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]"
nipkow@15302
  1975
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1976
nipkow@15302
  1977
lemma rotate_drop_take:
nipkow@15302
  1978
  "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
nipkow@15302
  1979
apply(induct n)
nipkow@15302
  1980
 apply simp
nipkow@15302
  1981
apply(simp add:rotate_def)
nipkow@15302
  1982
apply(cases "xs = []")
nipkow@15302
  1983
 apply (simp)
nipkow@15302
  1984
apply(case_tac "n mod length xs = 0")
nipkow@15302
  1985
 apply(simp add:mod_Suc)
nipkow@15302
  1986
 apply(simp add: rotate1_hd_tl drop_Suc take_Suc)
nipkow@15302
  1987
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric]
nipkow@15302
  1988
                take_hd_drop linorder_not_le)
nipkow@13145
  1989
done
wenzelm@13114
  1990
nipkow@15302
  1991
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs"
nipkow@15302
  1992
by(simp add:rotate_drop_take)
nipkow@15302
  1993
nipkow@15302
  1994
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs"
nipkow@15302
  1995
by(simp add:rotate_drop_take)
nipkow@15302
  1996
nipkow@15302
  1997
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs"
nipkow@15302
  1998
by(simp add:rotate1_def split:list.split)
nipkow@15302
  1999
nipkow@15302
  2000
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs"
nipkow@15302
  2001
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2002
nipkow@15302
  2003
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs"
nipkow@15302
  2004
by(simp add:rotate1_def split:list.split) blast
nipkow@15302
  2005
nipkow@15302
  2006
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs"
nipkow@15302
  2007
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2008
nipkow@15302
  2009
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)"
nipkow@15302
  2010
by(simp add:rotate_drop_take take_map drop_map)
nipkow@15302
  2011
nipkow@15302
  2012
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs"
nipkow@15302
  2013
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2014
nipkow@15302
  2015
lemma set_rotate[simp]: "set(rotate n xs) = set xs"
nipkow@15302
  2016
by (induct n) (simp_all add:rotate_def)
nipkow@15302
  2017
nipkow@15302
  2018
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])"
nipkow@15302
  2019
by(simp add:rotate1_def split:list.split)
nipkow@15302
  2020
nipkow@15302
  2021
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])"
nipkow@15302
  2022
by (induct n) (simp_all add:rotate_def)
wenzelm@13114
  2023
nipkow@15439
  2024
lemma rotate_rev:
nipkow@15439
  2025
  "rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)"
nipkow@15439
  2026
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2027
apply(cases "length xs = 0")
nipkow@15439
  2028
 apply simp
nipkow@15439
  2029
apply(cases "n mod length xs = 0")
nipkow@15439
  2030
 apply simp
nipkow@15439
  2031
apply(simp add:rotate_drop_take rev_drop rev_take)
nipkow@15439
  2032
done
nipkow@15439
  2033
nipkow@18423
  2034
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)"
nipkow@18423
  2035
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth)
nipkow@18423
  2036
apply(subgoal_tac "length xs \<noteq> 0")
nipkow@18423
  2037
 prefer 2 apply simp
nipkow@18423
  2038
using mod_less_divisor[of "length xs" n] by arith
nipkow@18423
  2039
wenzelm@13114
  2040
nipkow@15392
  2041
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *}
wenzelm@13114
  2042
wenzelm@13142
  2043
lemma sublist_empty [simp]: "sublist xs {} = []"
nipkow@13145
  2044
by (auto simp add: sublist_def)
wenzelm@13114
  2045
wenzelm@13142
  2046
lemma sublist_nil [simp]: "sublist [] A = []"
nipkow@13145
  2047
by (auto simp add: sublist_def)
wenzelm@13114
  2048
nipkow@15281
  2049
lemma length_sublist:
nipkow@15281
  2050
  "length(sublist xs I) = card{i. i < length xs \<and> i : I}"
nipkow@15281
  2051
by(simp add: sublist_def length_filter_conv_card cong:conj_cong)
nipkow@15281
  2052
nipkow@15281
  2053
lemma sublist_shift_lemma_Suc:
nipkow@15281
  2054
  "!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) =
nipkow@15281
  2055
         map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))"
nipkow@15281
  2056
apply(induct xs)
nipkow@15281
  2057
 apply simp
nipkow@15281
  2058
apply (case_tac "is")
nipkow@15281
  2059
 apply simp
nipkow@15281
  2060
apply simp
nipkow@15281
  2061
done
nipkow@15281
  2062
wenzelm@13114
  2063
lemma sublist_shift_lemma:
nipkow@15425
  2064
     "map fst [p:zip xs [i..<i + length xs] . snd p : A] =
nipkow@15425
  2065
      map fst [p:zip xs [0..<length xs] . snd p + i : A]"
nipkow@13145
  2066
by (induct xs rule: rev_induct) (simp_all add: add_commute)
wenzelm@13114
  2067
wenzelm@13114
  2068
lemma sublist_append:
paulson@15168
  2069
     "sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}"
nipkow@13145
  2070
apply (unfold sublist_def)
paulson@14208
  2071
apply (induct l' rule: rev_induct, simp)
nipkow@13145
  2072
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma)
nipkow@13145
  2073
apply (simp add: add_commute)
nipkow@13145
  2074
done
wenzelm@13114
  2075
wenzelm@13114
  2076
lemma sublist_Cons:
nipkow@13145
  2077
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}"
nipkow@13145
  2078
apply (induct l rule: rev_induct)
nipkow@13145
  2079
 apply (simp add: sublist_def)
nipkow@13145
  2080
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append)
nipkow@13145
  2081
done
wenzelm@13114
  2082
nipkow@15281
  2083
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}"
nipkow@15281
  2084
apply(induct xs)
nipkow@15281
  2085
 apply simp
nipkow@15281
  2086
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE)
nipkow@15281
  2087
 apply(erule lessE)
nipkow@15281
  2088
  apply auto
nipkow@15281
  2089
apply(erule lessE)
nipkow@15281
  2090
apply auto
nipkow@15281
  2091
done
nipkow@15281
  2092
nipkow@15281
  2093
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs"
nipkow@15281
  2094
by(auto simp add:set_sublist)
nipkow@15281
  2095
nipkow@15281
  2096
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)"
nipkow@15281
  2097
by(auto simp add:set_sublist)
nipkow@15281
  2098
nipkow@15281
  2099
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs"
nipkow@15281
  2100
by(auto simp add:set_sublist)
nipkow@15281
  2101
wenzelm@13142
  2102
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])"
nipkow@13145
  2103
by (simp add: sublist_Cons)
wenzelm@13114
  2104
nipkow@15281
  2105
nipkow@15281
  2106
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)"
nipkow@15281
  2107
apply(induct xs)
nipkow@15281
  2108
 apply simp
nipkow@15281
  2109
apply(auto simp add:sublist_Cons)
nipkow@15281
  2110
done
nipkow@15281
  2111
nipkow@15281
  2112
nipkow@15045
  2113
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l"
paulson@14208
  2114
apply (induct l rule: rev_induct, simp)
nipkow@13145
  2115
apply (simp split: nat_diff_split add: sublist_append)
nipkow@13145
  2116
done
wenzelm@13114
  2117
nipkow@17501
  2118
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow>
nipkow@17501
  2119
  filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s"
nipkow@17501
  2120
proof (induct xs)
nipkow@17501
  2121
  case Nil thus ?case by simp
nipkow@17501
  2122
next
nipkow@17501
  2123
  case (Cons a xs)
nipkow@17501
  2124
  moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto
nipkow@17501
  2125
  ultimately show ?case by(simp add: sublist_Cons cong:filter_cong)
nipkow@17501
  2126
qed
nipkow@17501
  2127
wenzelm@13114
  2128
nipkow@19390
  2129
subsubsection {* @{const splice} *}
nipkow@19390
  2130
haftmann@19607
  2131
lemma splice_Nil2 [simp, code]:
nipkow@19390
  2132
 "splice xs [] = xs"
nipkow@19390
  2133
by (cases xs) simp_all
nipkow@19390
  2134
haftmann@19607
  2135
lemma splice_Cons_Cons [simp, code]:
nipkow@19390
  2136
 "splice (x#xs) (y#ys) = x # y # splice xs ys"
nipkow@19390
  2137
by simp
nipkow@19390
  2138
haftmann@19607
  2139
declare splice.simps(2) [simp del, code del]
nipkow@19390
  2140
nipkow@15392
  2141
subsubsection{*Sets of Lists*}
nipkow@15392
  2142
nipkow@15392
  2143
subsubsection {* @{text lists}: the list-forming operator over sets *}
nipkow@15302
  2144
nipkow@15302
  2145
consts lists :: "'a set => 'a list set"
nipkow@15302
  2146
inductive "lists A"
nipkow@15302
  2147
 intros
nipkow@15302
  2148
  Nil [intro!]: "[]: lists A"
nipkow@15302
  2149
  Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A"
nipkow@15302
  2150
nipkow@15302
  2151
inductive_cases listsE [elim!]: "x#l : lists A"
nipkow@15302
  2152
nipkow@15302
  2153
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B"
nipkow@15302
  2154
by (unfold lists.defs) (blast intro!: lfp_mono)
nipkow@15302
  2155
nipkow@15302
  2156
lemma lists_IntI:
nipkow@15302
  2157
  assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l
nipkow@15302
  2158
  by induct blast+
nipkow@15302
  2159
nipkow@15302
  2160
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B"
nipkow@15302
  2161
proof (rule mono_Int [THEN equalityI])
nipkow@15302
  2162
  show "mono lists" by (simp add: mono_def lists_mono)
nipkow@15302
  2163
  show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI)
kleing@14388
  2164
qed
kleing@14388
  2165
nipkow@15302
  2166
lemma append_in_lists_conv [iff]:
nipkow@15302
  2167
     "(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)"
nipkow@15302
  2168
by (induct xs) auto
nipkow@15302
  2169
nipkow@15302
  2170
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)"
nipkow@15302
  2171
-- {* eliminate @{text lists} in favour of @{text set} *}
nipkow@15302
  2172
by (induct xs) auto
nipkow@15302
  2173
nipkow@15302
  2174
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A"
nipkow@15302
  2175
by (rule in_lists_conv_set [THEN iffD1])
nipkow@15302
  2176
nipkow@15302
  2177
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A"
nipkow@15302
  2178
by (rule in_lists_conv_set [THEN iffD2])
nipkow@15302
  2179
nipkow@15302
  2180
lemma lists_UNIV [simp]: "lists UNIV = UNIV"
nipkow@15302
  2181
by auto
nipkow@15302
  2182
nipkow@17086
  2183
nipkow@17086
  2184
nipkow@17086
  2185
subsubsection{* Inductive definition for membership *}
nipkow@17086
  2186
nipkow@17086
  2187
consts ListMem :: "('a \<times> 'a list)set"
nipkow@17086
  2188
inductive ListMem
nipkow@17086
  2189
intros
nipkow@17086
  2190
 elem:  "(x,x#xs) \<in> ListMem"
nipkow@17086
  2191
 insert:  "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem"
nipkow@17086
  2192
nipkow@17086
  2193
lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)"
nipkow@17086
  2194
apply (rule iffI)
nipkow@17086
  2195
 apply (induct set: ListMem)
nipkow@17086
  2196
  apply auto
nipkow@17086
  2197
apply (induct xs)
nipkow@17086
  2198
 apply (auto intro: ListMem.intros)
nipkow@17086
  2199
done
nipkow@17086
  2200
nipkow@17086
  2201
nipkow@17086
  2202
nipkow@15392
  2203
subsubsection{*Lists as Cartesian products*}
nipkow@15302
  2204
nipkow@15302
  2205
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from
nipkow@15302
  2206
@{term A} and tail drawn from @{term Xs}.*}
nipkow@15302
  2207
nipkow@15302
  2208
constdefs
nipkow@15302
  2209
  set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set"
nipkow@15302
  2210
  "set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}"
nipkow@15302
  2211
paulson@17724
  2212
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A"
nipkow@15302
  2213
by (auto simp add: set_Cons_def)
nipkow@15302
  2214
nipkow@15302
  2215
text{*Yields the set of lists, all of the same length as the argument and
nipkow@15302
  2216
with elements drawn from the corresponding element of the argument.*}
nipkow@15302
  2217
nipkow@15302
  2218
consts  listset :: "'a set list \<Rightarrow> 'a list set"
nipkow@15302
  2219
primrec
nipkow@15302
  2220
   "listset []    = {[]}"
nipkow@15302
  2221
   "listset(A#As) = set_Cons A (listset As)"
nipkow@15302
  2222
nipkow@15302
  2223
paulson@15656
  2224
subsection{*Relations on Lists*}
paulson@15656
  2225
paulson@15656
  2226
subsubsection {* Length Lexicographic Ordering *}
paulson@15656
  2227
paulson@15656
  2228
text{*These orderings preserve well-foundedness: shorter lists 
paulson@15656
  2229
  precede longer lists. These ordering are not used in dictionaries.*}
paulson@15656
  2230
paulson@15656
  2231
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set"
paulson@15656
  2232
        --{*The lexicographic ordering for lists of the specified length*}
nipkow@15302
  2233
primrec
paulson@15656
  2234
  "lexn r 0 = {}"
paulson@15656
  2235
  "lexn r (Suc n) =
paulson@15656
  2236
    (prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int
paulson@15656
  2237
    {(xs,ys). length xs = Suc n \<and> length ys = Suc n}"
nipkow@15302
  2238
nipkow@15302
  2239
constdefs
paulson@15656
  2240
  lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
paulson@15656
  2241
    "lex r == \<Union>n. lexn r n"
paulson@15656
  2242
        --{*Holds only between lists of the same length*}
paulson@15656
  2243
nipkow@15693
  2244
  lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set"
nipkow@15693
  2245
    "lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))"
paulson@15656
  2246
        --{*Compares lists by their length and then lexicographically*}
nipkow@15302
  2247
nipkow@15302
  2248
nipkow@15302
  2249
lemma wf_lexn: "wf r ==> wf (lexn r n)"
nipkow@15302
  2250
apply (induct n, simp, simp)
nipkow@15302
  2251
apply(rule wf_subset)
nipkow@15302
  2252
 prefer 2 apply (rule Int_lower1)
nipkow@15302
  2253
apply(rule wf_prod_fun_image)
nipkow@15302
  2254
 prefer 2 apply (rule inj_onI, auto)
nipkow@15302
  2255
done
nipkow@15302
  2256
nipkow@15302
  2257
lemma lexn_length:
nipkow@15302
  2258
     "!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n"
nipkow@15302
  2259
by (induct n) auto
nipkow@15302
  2260
nipkow@15302
  2261
lemma wf_lex [intro!]: "wf r ==> wf (lex r)"
nipkow@15302
  2262
apply (unfold lex_def)
nipkow@15302
  2263
apply (rule wf_UN)
nipkow@15302
  2264
apply (blast intro: wf_lexn, clarify)
nipkow@15302
  2265
apply (rename_tac m n)
nipkow@15302
  2266
apply (subgoal_tac "m \<noteq> n")
nipkow@15302
  2267
 prefer 2 apply blast
nipkow@15302
  2268
apply (blast dest: lexn_length not_sym)
nipkow@15302
  2269
done
nipkow@15302
  2270
nipkow@15302
  2271
lemma lexn_conv:
paulson@15656
  2272
  "lexn r n =
paulson@15656
  2273
    {(xs,ys). length xs = n \<and> length ys = n \<and>
paulson@15656
  2274
    (\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}"
nipkow@18423
  2275
apply (induct n, simp)
nipkow@15302
  2276
apply (simp add: image_Collect lex_prod_def, safe, blast)
nipkow@15302
  2277
 apply (rule_tac x = "ab # xys" in exI, simp)
nipkow@15302
  2278
apply (case_tac xys, simp_all, blast)
nipkow@15302
  2279
done
nipkow@15302
  2280
nipkow@15302
  2281
lemma lex_conv:
paulson@15656
  2282
  "lex r =
paulson@15656
  2283
    {(xs,ys). length xs = length ys \<and>
paulson@15656
  2284
    (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}"
nipkow@15302
  2285
by (force simp add: lex_def lexn_conv)
nipkow@15302
  2286
nipkow@15693
  2287
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)"
nipkow@15693
  2288
by (unfold lenlex_def) blast
nipkow@15693
  2289
nipkow@15693
  2290
lemma lenlex_conv:
nipkow@15693
  2291
    "lenlex r = {(xs,ys). length xs < length ys |
paulson@15656
  2292
                 length xs = length ys \<and> (xs, ys) : lex r}"
nipkow@19623
  2293
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def)
nipkow@15302
  2294
nipkow@15302
  2295
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r"
nipkow@15302
  2296
by (simp add: lex_conv)
nipkow@15302
  2297
nipkow@15302
  2298
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r"
nipkow@15302
  2299
by (simp add:lex_conv)
nipkow@15302
  2300
paulson@18447
  2301
lemma Cons_in_lex [simp]:
paulson@15656
  2302
    "((x # xs, y # ys) : lex r) =
paulson@15656
  2303
      ((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)"
nipkow@15302
  2304
apply (simp add: lex_conv)
nipkow@15302
  2305
apply (rule iffI)
nipkow@15302
  2306
 prefer 2 apply (blast intro: Cons_eq_appendI, clarify)
nipkow@15302
  2307
apply (case_tac xys, simp, simp)
nipkow@15302
  2308
apply blast
nipkow@15302
  2309
done
nipkow@15302
  2310
nipkow@15302
  2311
paulson@15656
  2312
subsubsection {* Lexicographic Ordering *}
paulson@15656
  2313
paulson@15656
  2314
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b".
paulson@15656
  2315
    This ordering does \emph{not} preserve well-foundedness.
nipkow@17090
  2316
     Author: N. Voelker, March 2005. *} 
paulson@15656
  2317
paulson@15656
  2318
constdefs 
paulson@15656
  2319
  lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" 
paulson@15656
  2320
  "lexord  r == {(x,y). \<exists> a v. y = x @ a # v \<or> 
paulson@15656
  2321
            (\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}"
paulson@15656
  2322
paulson@15656
  2323
lemma lexord_Nil_left[simp]:  "([],y) \<in> lexord r = (\<exists> a x. y = a # x)"
paulson@15656
  2324
  by (unfold lexord_def, induct_tac y, auto) 
paulson@15656
  2325
paulson@15656
  2326
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r"
paulson@15656
  2327
  by (unfold lexord_def, induct_tac x, auto)
paulson@15656
  2328
paulson@15656
  2329
lemma lexord_cons_cons[simp]:
paulson@15656
  2330
     "((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))"
paulson@15656
  2331
  apply (unfold lexord_def, safe, simp_all)
paulson@15656
  2332
  apply (case_tac u, simp, simp)
paulson@15656
  2333
  apply (case_tac u, simp, clarsimp, blast, blast, clarsimp)
paulson@15656
  2334
  apply (erule_tac x="b # u" in allE)
paulson@15656
  2335
  by force
paulson@15656
  2336
paulson@15656
  2337
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons
paulson@15656
  2338
paulson@15656
  2339
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r"
paulson@15656
  2340
  by (induct_tac x, auto)  
paulson@15656
  2341
paulson@15656
  2342
lemma lexord_append_left_rightI:
paulson@15656
  2343
     "(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r"
paulson@15656
  2344
  by (induct_tac u, auto)
paulson@15656
  2345
paulson@15656
  2346
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r"
paulson@15656
  2347
  by (induct x, auto)
paulson@15656
  2348
paulson@15656
  2349
lemma lexord_append_leftD:
paulson@15656
  2350
     "\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r"
paulson@15656
  2351
  by (erule rev_mp, induct_tac x, auto)
paulson@15656
  2352
paulson@15656
  2353
lemma lexord_take_index_conv: 
paulson@15656
  2354
   "((x,y) : lexord r) = 
paulson@15656
  2355
    ((length x < length y \<and> take (length x) y = x) \<or> 
paulson@15656
  2356
     (\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))"
paulson@15656
  2357
  apply (unfold lexord_def Let_def, clarsimp) 
paulson@15656
  2358
  apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2)
paulson@15656
  2359
  apply auto 
paulson@15656
  2360
  apply (rule_tac x="hd (drop (length x) y)" in exI)
paulson@15656
  2361
  apply (rule_tac x="tl (drop (length x) y)" in exI)
paulson@15656
  2362
  apply (erule subst, simp add: min_def) 
paulson@15656
  2363
  apply (rule_tac x ="length u" in exI, simp) 
paulson@15656
  2364
  apply (rule_tac x ="take i x" in exI) 
paulson@15656
  2365
  apply (rule_tac x ="x ! i" in exI) 
paulson@15656
  2366
  apply (rule_tac x ="y ! i" in exI, safe) 
paulson@15656
  2367
  apply (rule_tac x="drop (Suc i) x" in exI)
paulson@15656
  2368
  apply (drule sym, simp add: drop_Suc_conv_tl) 
paulson@15656
  2369
  apply (rule_tac x="drop (Suc i) y" in exI)
paulson@15656
  2370
  by (simp add: drop_Suc_conv_tl) 
paulson@15656
  2371
paulson@15656
  2372
-- {* lexord is extension of partial ordering List.lex *} 
paulson@15656
  2373
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)"
paulson@15656
  2374
  apply (rule_tac x = y in spec) 
paulson@15656
  2375
  apply (induct_tac x, clarsimp) 
paulson@15656
  2376
  by (clarify, case_tac x, simp, force)
paulson@15656
  2377
paulson@15656
  2378
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r"
paulson@15656
  2379
  by (induct y, auto)
paulson@15656
  2380
paulson@15656
  2381
lemma lexord_trans: 
paulson@15656
  2382
    "\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r"
paulson@15656
  2383
   apply (erule rev_mp)+
paulson@15656
  2384
   apply (rule_tac x = x in spec) 
paulson@15656
  2385
  apply (rule_tac x = z in spec) 
paulson@15656
  2386
  apply ( induct_tac y, simp, clarify)
paulson@15656
  2387
  apply (case_tac xa, erule ssubst) 
paulson@15656
  2388
  apply (erule allE, erule allE) -- {* avoid simp recursion *} 
paulson@15656
  2389
  apply (case_tac x, simp, simp) 
paulson@15656
  2390
  apply (case_tac x, erule allE, erule allE, simp) 
paulson@15656
  2391
  apply (erule_tac x = listb in allE) 
paulson@15656
  2392
  apply (erule_tac x = lista in allE, simp)
paulson@15656
  2393
  apply (unfold trans_def)
paulson@15656
  2394
  by blast
paulson@15656
  2395
paulson@15656
  2396
lemma lexord_transI:  "trans r \<Longrightarrow> trans (lexord r)"
paulson@15656
  2397
  by (rule transI, drule lexord_trans, blast) 
paulson@15656
  2398
paulson@15656
  2399
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r"
paulson@15656
  2400
  apply (rule_tac x = y in spec) 
paulson@15656
  2401
  apply (induct_tac x, rule allI) 
paulson@15656
  2402
  apply (case_tac x, simp, simp) 
paulson@15656
  2403
  apply (rule allI, case_tac x, simp, simp) 
paulson@15656
  2404
  by blast
paulson@15656
  2405
paulson@15656
  2406
nipkow@15392
  2407
subsubsection{*Lifting a Relation on List Elements to the Lists*}
nipkow@15302
  2408
nipkow@15302
  2409
consts  listrel :: "('a * 'a)set => ('a list * 'a list)set"
nipkow@15302
  2410
nipkow@15302
  2411
inductive "listrel(r)"
nipkow@15302
  2412
 intros
nipkow@15302
  2413
   Nil:  "([],[]) \<in> listrel r"
nipkow@15302
  2414
   Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r"
nipkow@15302
  2415
nipkow@15302
  2416
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r"
nipkow@15302
  2417
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r"
nipkow@15302
  2418
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r"
nipkow@15302
  2419
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r"
nipkow@15302
  2420
nipkow@15302
  2421
nipkow@15302
  2422
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s"
nipkow@15302
  2423
apply clarify  
nipkow@15302
  2424
apply (erule listrel.induct)
nipkow@15302
  2425
apply (blast intro: listrel.intros)+
nipkow@15302
  2426
done
nipkow@15302
  2427
nipkow@15302
  2428
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A"
nipkow@15302
  2429
apply clarify 
nipkow@15302
  2430
apply (erule listrel.induct, auto) 
nipkow@15302
  2431
done
nipkow@15302
  2432
nipkow@15302
  2433
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" 
nipkow@15302
  2434
apply (simp add: refl_def listrel_subset Ball_def)
nipkow@15302
  2435
apply (rule allI) 
nipkow@15302
  2436
apply (induct_tac x) 
nipkow@15302
  2437
apply (auto intro: listrel.intros)
nipkow@15302
  2438
done
nipkow@15302
  2439
nipkow@15302
  2440
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" 
nipkow@15302
  2441
apply (auto simp add: sym_def)
nipkow@15302
  2442
apply (erule listrel.induct) 
nipkow@15302
  2443
apply (blast intro: listrel.intros)+
nipkow@15302
  2444
done
nipkow@15302
  2445
nipkow@15302
  2446
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" 
nipkow@15302
  2447
apply (simp add: trans_def)
nipkow@15302
  2448
apply (intro allI) 
nipkow@15302
  2449
apply (rule impI) 
nipkow@15302
  2450
apply (erule listrel.induct) 
nipkow@15302
  2451
apply (blast intro: listrel.intros)+
nipkow@15302
  2452
done
nipkow@15302
  2453
nipkow@15302
  2454
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)"
nipkow@15302
  2455
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) 
nipkow@15302
  2456
nipkow@15302
  2457
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}"
nipkow@15302
  2458
by (blast intro: listrel.intros)
nipkow@15302
  2459
nipkow@15302
  2460
lemma listrel_Cons:
nipkow@15302
  2461
     "listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})";
nipkow@15302
  2462
by (auto simp add: set_Cons_def intro: listrel.intros) 
nipkow@15302
  2463
nipkow@15302
  2464
nipkow@15392
  2465
subsection{*Miscellany*}
nipkow@15392
  2466
nipkow@15392
  2467
subsubsection {* Characters and strings *}
wenzelm@13366
  2468
wenzelm@13366
  2469
datatype nibble =
wenzelm@13366
  2470
    Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7
wenzelm@13366
  2471
  | Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF
wenzelm@13366
  2472
wenzelm@13366
  2473
datatype char = Char nibble nibble
wenzelm@13366
  2474
  -- "Note: canonical order of character encoding coincides with standard term ordering"
wenzelm@13366
  2475
wenzelm@13366
  2476
types string = "char list"
wenzelm@13366
  2477
wenzelm@13366
  2478
syntax
wenzelm@13366
  2479
  "_Char" :: "xstr => char"    ("CHR _")
wenzelm@13366
  2480
  "_String" :: "xstr => string"    ("_")
wenzelm@13366
  2481
wenzelm@13366
  2482
parse_ast_translation {*
wenzelm@13366
  2483
  let
wenzelm@13366
  2484
    val constants = Syntax.Appl o map Syntax.Constant;
wenzelm@13366
  2485
wenzelm@13366
  2486
    fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10));
wenzelm@13366
  2487
    fun mk_char c =
wenzelm@13366
  2488
      if Symbol.is_ascii c andalso Symbol.is_printable c then
wenzelm@13366
  2489
        constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)]
wenzelm@13366
  2490
      else error ("Printable ASCII character expected: " ^ quote c);
wenzelm@13366
  2491
wenzelm@13366
  2492
    fun mk_string [] = Syntax.Constant "Nil"
wenzelm@13366
  2493
      | mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs];
wenzelm@13366
  2494
wenzelm@13366
  2495
    fun char_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2496
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2497
          [c] => mk_char c
wenzelm@13366
  2498
        | _ => error ("Single character expected: " ^ xstr))
wenzelm@13366
  2499
      | char_ast_tr asts = raise AST ("char_ast_tr", asts);
wenzelm@13366
  2500
wenzelm@13366
  2501
    fun string_ast_tr [Syntax.Variable xstr] =
wenzelm@13366
  2502
        (case Syntax.explode_xstr xstr of
wenzelm@13366
  2503
          [] => constants [Syntax.constrainC, "Nil", "string"]
wenzelm@13366
  2504
        | cs => mk_string cs)
wenzelm@13366
  2505
      | string_ast_tr asts = raise AST ("string_tr", asts);
wenzelm@13366
  2506
  in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end;
wenzelm@13366
  2507
*}
wenzelm@13366
  2508
berghofe@15064
  2509
ML {*
haftmann@20184
  2510
structure HOList =
haftmann@20184
  2511
struct
haftmann@20184
  2512
haftmann@20184
  2513
local
haftmann@20184
  2514
  val thy = the_context ();
haftmann@20184
  2515
in
haftmann@20184
  2516
  val typ_string = Type (Sign.intern_type thy "string", []);
haftmann@20184
  2517
  fun typ_list ty = Type (Sign.intern_type thy "list", [ty]);
haftmann@20184
  2518
  fun term_list ty f [] = Const (Sign.intern_const thy "Nil", typ_list ty)
haftmann@20184
  2519
    | term_list ty f (x::xs) = Const (Sign.intern_const thy "Cons",
haftmann@20184
  2520
        ty --> typ_list ty --> typ_list ty) $ f x $ term_list ty f xs;
haftmann@20184
  2521
end;
haftmann@20184
  2522
berghofe@15064
  2523
fun int_of_nibble h =
berghofe@15064
  2524
  if "0" <= h andalso h <= "9" then ord h - ord "0"
berghofe@15064
  2525
  else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10
berghofe@15064
  2526
  else raise Match;
berghofe@15064
  2527
berghofe@15064
  2528
fun nibble_of_int i =
berghofe@15064
  2529
  if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10);
haftmann@20181
  2530
haftmann@20453
  2531
fun dest_char (Const ("List.char.Char", _) $ c1 $ c2) =
haftmann@20453
  2532
      let
haftmann@20453
  2533
        fun dest_nibble (Const (s, _)) = (int_of_nibble o unprefix "List.nibble.Nibble") s
haftmann@20453
  2534
          | dest_nibble _ = raise Match;
haftmann@20453
  2535
      in
haftmann@20453
  2536
        (SOME (dest_nibble c1 * 16 + dest_nibble c2)
haftmann@20453
  2537
        handle Fail _ => NONE | Match => NONE)
haftmann@20453
  2538
      end
haftmann@20453
  2539
  | dest_char _ = NONE;
haftmann@20453
  2540
haftmann@20453
  2541
val print_list = Pretty.enum "," "[" "]";
haftmann@20453
  2542
haftmann@20453
  2543
fun print_char c =
haftmann@20453
  2544
  let
haftmann@20453
  2545
    val i = ord c
haftmann@20453
  2546
  in if i < 32
haftmann@20453
  2547
    then prefix "\\" (string_of_int i)
haftmann@20453
  2548
    else c
haftmann@20453
  2549
  end;
haftmann@20453
  2550
haftmann@20453
  2551
val print_string = quote;
haftmann@20453
  2552
haftmann@20184
  2553
fun term_string s =
haftmann@20181
  2554
  let
haftmann@20181
  2555
    val ty_n = Type ("List.nibble", []);
haftmann@20181
  2556
    val ty_c = Type ("List.char", []);
haftmann@20181
  2557
    val ty_l = Type ("List.list", [ty_c]);
haftmann@20181
  2558
    fun mk_nib n = Const ("List.nibble.Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)), ty_n);
haftmann@20181
  2559
    fun mk_char c =
haftmann@20181
  2560
      if Symbol.is_ascii c andalso Symbol.is_printable c then
haftmann@20181
  2561
        Const ("List.char.Char", ty_n --> ty_n --> ty_c) $ mk_nib (ord c div 16) $ mk_nib (ord c mod 16)
haftmann@20181
  2562
      else error ("Printable ASCII character expected: " ^ quote c);
haftmann@20181
  2563
    fun mk_string c t = Const ("List.list.Cons", ty_c --> ty_l --> ty_l)
haftmann@20181
  2564
      $ mk_char c $ t;
haftmann@20181
  2565
  in fold_rev mk_string (explode s) (Const ("List.list.Nil", ty_l)) end;
haftmann@20184
  2566
haftmann@20184
  2567
end;
berghofe@15064
  2568
*}
berghofe@15064
  2569
wenzelm@13366
  2570
print_ast_translation {*
wenzelm@13366
  2571
  let
wenzelm@13366
  2572
    fun dest_nib (Syntax.Constant c) =
wenzelm@13366
  2573
        (case explode c of
haftmann@20184
  2574
          ["N", "i", "b", "b", "l", "e", h] => HOList.int_of_nibble h
wenzelm@13366
  2575
        | _ => raise Match)
wenzelm@13366
  2576
      | dest_nib _ = raise Match;
wenzelm@13366
  2577
wenzelm@13366
  2578
    fun dest_chr c1 c2 =
wenzelm@13366
  2579
      let val c = chr (dest_nib c1 * 16 + dest_nib c2)
wenzelm@13366
  2580
      in if Symbol.is_printable c then c else raise Match end;
wenzelm@13366
  2581
wenzelm@13366
  2582
    fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2
wenzelm@13366
  2583
      | dest_char _ = raise Match;
wenzelm@13366
  2584
wenzelm@13366
  2585
    fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)];
wenzelm@13366
  2586
wenzelm@13366
  2587
    fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]]
wenzelm@13366
  2588
      | char_ast_tr' _ = raise Match;
wenzelm@13366
  2589
wenzelm@13366
  2590
    fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String",
wenzelm@13366
  2591
            xstr (map dest_char (Syntax.unfold_ast "_args" args))]
wenzelm@13366
  2592
      | list_ast_tr' ts = raise Match;
wenzelm@13366
  2593
  in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end;
wenzelm@13366
  2594
*}
wenzelm@13366
  2595
haftmann@20453
  2596
haftmann@21061
  2597
subsection {* Code generator *}
haftmann@21061
  2598
haftmann@21061
  2599
subsubsection {* Setup *}
berghofe@15064
  2600
berghofe@16770
  2601
types_code
berghofe@16770
  2602
  "list" ("_ list")
berghofe@16770
  2603
attach (term_of) {*
berghofe@15064
  2604
val term_of_list = HOLogic.mk_list;
berghofe@16770
  2605
*}
berghofe@16770
  2606
attach (test) {*
berghofe@15064
  2607
fun gen_list' aG i j = frequency
berghofe@15064
  2608
  [(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] ()
berghofe@15064
  2609
and gen_list aG i = gen_list' aG i i;
berghofe@16770
  2610
*}
berghofe@16770
  2611
  "char" ("string")
berghofe@16770
  2612
attach (term_of) {*
berghofe@15064
  2613
val nibbleT = Type ("List.nibble", []);
berghofe@15064
  2614
berghofe@15064
  2615
fun term_of_char c =
berghofe@15064
  2616
  Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $
haftmann@20184
  2617
    Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c div 16), nibbleT) $
haftmann@20184
  2618
    Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c mod 16), nibbleT);
berghofe@16770
  2619
*}
berghofe@16770
  2620
attach (test) {*
berghofe@15064
  2621
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z")));
berghofe@15064
  2622
*}
berghofe@15064
  2623
berghofe@15064
  2624
consts_code "Cons" ("(_ ::/ _)")
berghofe@15064
  2625
haftmann@20453
  2626
code_type list
haftmann@20453
  2627
  (SML "_ list")
haftmann@20453
  2628
  (Haskell target_atom "[_]")
haftmann@20453
  2629
haftmann@20453
  2630
code_const Nil
haftmann@20453
  2631
  (SML target_atom "[]")
haftmann@20453
  2632
  (Haskell target_atom "[]")
haftmann@20453
  2633
haftmann@20453
  2634
code_type char
haftmann@20453
  2635
  (SML target_atom "char")
haftmann@20453
  2636
  (Haskell target_atom "Char")
haftmann@20453
  2637
haftmann@20453
  2638
code_const Char
haftmann@20453
  2639
  (SML target_atom "(__,/ __)")
haftmann@20453
  2640
  (Haskell target_atom "(__,/ __)")
haftmann@20453
  2641
haftmann@20588
  2642
code_instance list :: eq and char :: eq
haftmann@20588
  2643
  (Haskell - and -)
haftmann@20588
  2644
haftmann@21046
  2645
code_const "Code_Generator.eq \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@20588
  2646
  (Haskell infixl 4 "==")
haftmann@20588
  2647
haftmann@21046
  2648
code_const "Code_Generator.eq \<Colon> char \<Rightarrow> char \<Rightarrow> bool"
haftmann@20588
  2649
  (Haskell infixl 4 "==")
haftmann@20588
  2650
haftmann@20453
  2651
setup {*
haftmann@20453
  2652
let
haftmann@20453
  2653
haftmann@20453
  2654
fun list_codegen thy defs gr dep thyname b t =
haftmann@20453
  2655
  let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false)
haftmann@20453
  2656
    (gr, HOLogic.dest_list t)
haftmann@20453
  2657
  in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE;
haftmann@20453
  2658
haftmann@20453
  2659
fun char_codegen thy defs gr dep thyname b t =
haftmann@20453
  2660
  case (Option.map chr o HOList.dest_char) t 
haftmann@20453
  2661
   of SOME c =>
haftmann@20453
  2662
        if Symbol.is_printable c
haftmann@20453
  2663
        then SOME (gr, (Pretty.quote o Pretty.str) c)
haftmann@20453
  2664
        else NONE
haftmann@20453
  2665
    | NONE => NONE;
haftmann@20453
  2666
haftmann@20453
  2667
in
haftmann@20453
  2668
haftmann@20453
  2669
  Codegen.add_codegen "list_codegen" list_codegen
haftmann@20453
  2670
  #> Codegen.add_codegen "char_codegen" char_codegen
haftmann@20699
  2671
  #> CodegenSerializer.add_pretty_list "SML" "List.list.Nil" "List.list.Cons"
haftmann@20453
  2672
       HOList.print_list NONE (7, "::")
haftmann@20699
  2673
  #> CodegenSerializer.add_pretty_list "Haskell" "List.list.Nil" "List.list.Cons"
haftmann@20453
  2674
       HOList.print_list (SOME (HOList.print_char, HOList.print_string)) (5, ":")
haftmann@20453
  2675
  #> CodegenPackage.add_appconst
haftmann@20453
  2676
       ("List.char.Char", CodegenPackage.appgen_char HOList.dest_char)
haftmann@20453
  2677
haftmann@20453
  2678
end;
haftmann@20453
  2679
*}
berghofe@15064
  2680
haftmann@21061
  2681
haftmann@21061
  2682
subsubsection {* Generation of efficient code *}
haftmann@21061
  2683
haftmann@21061
  2684
consts
haftmann@21061
  2685
  mem :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl 55)
haftmann@21061
  2686
  null:: "'a list \<Rightarrow> bool"
haftmann@21061
  2687
  list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2688
  list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool"
haftmann@21061
  2689
  list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)"
haftmann@21061
  2690
  itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
haftmann@21061
  2691
  filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2692
  map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list"
haftmann@21061
  2693
haftmann@21061
  2694
primrec
haftmann@21061
  2695
  "x mem [] = False"
haftmann@21061
  2696
  "x mem (y#ys) = (if y = x then True else x mem ys)"
haftmann@21061
  2697
haftmann@21061
  2698
primrec
haftmann@21061
  2699
  "null [] = True"
haftmann@21061
  2700
  "null (x#xs) = False"
haftmann@21061
  2701
haftmann@21061
  2702
primrec
haftmann@21061
  2703
  "list_inter [] bs = []"
haftmann@21061
  2704
  "list_inter (a#as) bs =
haftmann@21061
  2705
     (if a \<in> set bs then a # list_inter as bs else list_inter as bs)"
haftmann@21061
  2706
haftmann@21061
  2707
primrec
haftmann@21061
  2708
  "list_all P [] = True"
haftmann@21061
  2709
  "list_all P (x#xs) = (P x \<and> list_all P xs)"
haftmann@21061
  2710
haftmann@21061
  2711
primrec
haftmann@21061
  2712
  "list_ex P [] = False"
haftmann@21061
  2713
  "list_ex P (x#xs) = (P x \<or> list_ex P xs)"
haftmann@21061
  2714
haftmann@21061
  2715
primrec
haftmann@21061
  2716
  "filtermap f [] = []"
haftmann@21061
  2717
  "filtermap f (x#xs) =
haftmann@21061
  2718
     (case f x of None \<Rightarrow> filtermap f xs
haftmann@21061
  2719
      | Some y \<Rightarrow> y # filtermap f xs)"
haftmann@21061
  2720
haftmann@21061
  2721
primrec
haftmann@21061
  2722
  "map_filter f P [] = []"
haftmann@21061
  2723
  "map_filter f P (x#xs) =
haftmann@21061
  2724
     (if P x then f x # map_filter f P xs else map_filter f P xs)"
haftmann@21061
  2725
haftmann@21061
  2726
primrec
haftmann@21061
  2727
  "itrev [] ys = ys"
haftmann@21061
  2728
  "itrev (x#xs) ys = itrev xs (x#ys)"
haftmann@21061
  2729
haftmann@21061
  2730
text {*
haftmann@21061
  2731
  Only use @{text mem} for generating executable code.  Otherwise
haftmann@21061
  2732
  use @{prop "x : set xs"} instead --- it is much easier to reason about.
haftmann@21061
  2733
  The same is true for @{const list_all} and @{const list_ex}: write
haftmann@21061
  2734
  @{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL
haftmann@21061
  2735
  quantifiers are aleady known to the automatic provers. In fact,
haftmann@21061
  2736
  the declarations in the code subsection make sure that @{text "\<in>"}, @{text "\<forall>x\<in>set xs"}
haftmann@21061
  2737
  and @{text "\<exists>x\<in>set xs"} are implemented efficiently.
haftmann@21061
  2738
haftmann@21061
  2739
  Efficient emptyness check is implemented by @{const null}.
haftmann@21061
  2740
haftmann@21061
  2741
  The functions @{const itrev}, @{const filtermap} and @{const map_filter}
haftmann@21061
  2742
  are just there to generate efficient code. Do not use them
haftmann@21061
  2743
  for modelling and proving.
haftmann@21061
  2744
*}
haftmann@21061
  2745
haftmann@21061
  2746
lemma mem_iff [normal post]: