src/HOL/List.thy
author nipkow
Fri, 02 May 2008 18:01:02 +0200
changeset 26771 1d67ab20f358
parent 26749 397a1aeede7d
child 26795 a27607030a1c
permissions -rw-r--r--
Added documentation
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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 ATP_Linkup
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uses "Tools/string_syntax.ML"
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begin
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datatype 'a list =
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    Nil    ("[]")
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  | Cons 'a  "'a list"    (infixr "#" 65)
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subsection{*Basic list processing functions*}
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consts
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  filter:: "('a => bool) => 'a list => 'a list"
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  concat:: "'a list list => 'a list"
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  foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b"
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  foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b"
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  hd:: "'a list => 'a"
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  tl:: "'a list => 'a list"
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  last:: "'a list => 'a"
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  butlast :: "'a list => 'a list"
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  set :: "'a list => 'a set"
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  map :: "('a=>'b) => ('a list => 'b list)"
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  listsum ::  "'a list => 'a::monoid_add"
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  nth :: "'a list => nat => 'a"    (infixl "!" 100)
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  list_update :: "'a list => nat => 'a => 'a list"
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  take:: "nat => 'a list => 'a list"
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  drop:: "nat => 'a list => 'a list"
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  takeWhile :: "('a => bool) => 'a list => 'a list"
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  dropWhile :: "('a => bool) => 'a list => 'a list"
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  rev :: "'a list => 'a list"
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  zip :: "'a list => 'b list => ('a * 'b) list"
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  upt :: "nat => nat => nat list" ("(1[_..</_'])")
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  remdups :: "'a list => 'a list"
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  remove1 :: "'a => 'a list => 'a list"
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  "distinct":: "'a list => bool"
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  replicate :: "nat => 'a => 'a list"
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
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nonterminals lupdbinds lupdbind
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syntax
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  -- {* list Enumeration *}
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  "@list" :: "args => 'a list"    ("[(_)]")
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  -- {* Special syntax for filter *}
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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  -- {* list update *}
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  "_lupdbind":: "['a, 'a] => lupdbind"    ("(2_ :=/ _)")
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  "" :: "lupdbind => lupdbinds"    ("_")
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  "_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds"    ("_,/ _")
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  "_LUpdate" :: "['a, lupdbinds] => 'a"    ("_/[(_)]" [900,0] 900)
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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  "[x<-xs . P]"== "filter (%x. P) xs"
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  "_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "list_update xs i x"
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syntax (xsymbols)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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syntax (HTML output)
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  "@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<leftarrow>_ ./ _])")
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text {*
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  Function @{text size} is overloaded for all datatypes. Users may
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  refer to the list version as @{text length}. *}
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abbreviation
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  length :: "'a list => nat" where
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  "length == size"
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primrec
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  "hd(x#xs) = x"
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  "tl([]) = []"
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  "tl(x#xs) = xs"
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  "last(x#xs) = (if xs=[] then x else last xs)"
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  "butlast []= []"
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  "butlast(x#xs) = (if xs=[] then [] else x#butlast xs)"
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  "set [] = {}"
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  "set (x#xs) = insert x (set xs)"
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  "map f [] = []"
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  "map f (x#xs) = f(x)#map f xs"
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65)
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where
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  append_Nil:"[] @ ys = ys"
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  | append_Cons: "(x#xs) @ ys = x # xs @ ys"
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  "rev([]) = []"
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  "rev(x#xs) = rev(xs) @ [x]"
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  "filter P [] = []"
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  "filter P (x#xs) = (if P x then x#filter P xs else filter P xs)"
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  foldl_Nil:"foldl f a [] = a"
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  foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs"
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  "foldr f [] a = a"
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  "foldr f (x#xs) a = f x (foldr f xs a)"
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  "concat([]) = []"
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  "concat(x#xs) = x @ concat(xs)"
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primrec
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"listsum [] = 0"
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"listsum (x # xs) = x + listsum xs"
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  drop_Nil:"drop n [] = []"
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  drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  take_Nil:"take n [] = []"
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  take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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  "[][i:=v] = []"
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  "(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])"
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primrec
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  "takeWhile P [] = []"
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  "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])"
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  "dropWhile P [] = []"
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  "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)"
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  "zip xs [] = []"
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  zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "xs = []"} and @{text "xs = z # zs"} *}
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primrec
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  upt_0: "[i..<0] = []"
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  upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])"
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primrec
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  "distinct [] = True"
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  "distinct (x#xs) = (x ~: set xs \<and> distinct xs)"
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primrec
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  "remdups [] = []"
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  "remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)"
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primrec
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  "remove1 x [] = []"
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  "remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)"
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primrec
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  replicate_0: "replicate 0 x = []"
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  replicate_Suc: "replicate (Suc n) x = x # replicate n x"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  [code func del]: "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  "splice [] ys = ys"
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  "splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))"
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    -- {*Warning: simpset does not contain the second eqn but a derived one. *}
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text{*
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\begin{figure}[htbp]
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\fbox{
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\begin{tabular}{l}
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@{lemma "[a,b]@[c,d] = [a,b,c,d]" simp}\\
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@{lemma "length [a,b,c] = 3" simp}\\
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@{lemma "set [a,b,c] = {a,b,c}" simp}\\
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@{lemma "map f [a,b,c] = [f a, f b, f c]" simp}\\
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@{lemma "rev [a,b,c] = [c,b,a]" simp}\\
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@{lemma "hd [a,b,c,d] = a" simp}\\
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@{lemma "tl [a,b,c,d] = [b,c,d]" simp}\\
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@{lemma "last [a,b,c,d] = d" simp}\\
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@{lemma "butlast [a,b,c,d] = [a,b,c]" simp}\\
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@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" simp}\\
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@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" simp}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" simp}\\
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@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" simp}\\
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@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" simp}\\
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@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" simp}\\
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@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" simp}\\
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@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" simp}\\
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@{lemma "take 2 [a,b,c,d] = [a,b]" simp}\\
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@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" simp}\\
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@{lemma "drop 2 [a,b,c,d] = [c,d]" simp}\\
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@{lemma "drop 6 [a,b,c,d] = []" simp}\\
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@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" simp}\\
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@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" simp}\\
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@{lemma "distinct [2,0,1::nat]" simp}\\
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@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" simp}\\
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@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" (simp add:sublist_def)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" (simp add:rotate1_def)}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" (simp add:rotate1_def rotate_def nat_number)}\\
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@{lemma "replicate 4 a = [a,a,a,a]" (simp add:nat_number)}\\
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@{lemma "[2..<5] = [2,3,4]" (simp add:nat_number)}\\
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@{lemma "listsum [1,2,3::nat] = 6" simp}
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\end{tabular}}
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\caption{Characteristic examples}
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\label{fig:Characteristic}
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\end{figure}
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Figure~\ref{fig:Characteristic} shows charachteristic examples
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that should give an intuitive understanding of the above functions.
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*}
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text{* The following simple sort functions are intended for proofs,
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not for efficient implementations. *}
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context linorder
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begin
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fun sorted :: "'a list \<Rightarrow> bool" where
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"sorted [] \<longleftrightarrow> True" |
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"sorted [x] \<longleftrightarrow> True" |
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"sorted (x#y#zs) \<longleftrightarrow> x <= y \<and> sorted (y#zs)"
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primrec insort :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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"insort x [] = [x]" |
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"insort x (y#ys) = (if x <= y then (x#y#ys) else y#(insort x ys))"
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primrec sort :: "'a list \<Rightarrow> 'a list" where
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"sort [] = []" |
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"sort (x#xs) = insort x (sort xs)"
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end
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subsubsection {* List comprehension *}
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text{* Input syntax for Haskell-like list comprehension notation.
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Typical example: @{text"[(x,y). x \<leftarrow> xs, y \<leftarrow> ys, x \<noteq> y]"},
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the list of all pairs of distinct elements from @{text xs} and @{text ys}.
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The syntax is as in Haskell, except that @{text"|"} becomes a dot
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(like in Isabelle's set comprehension): @{text"[e. x \<leftarrow> xs, \<dots>]"} rather than
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\verb![e| x <- xs, ...]!.
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The qualifiers after the dot are
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\begin{description}
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\item[generators] @{text"p \<leftarrow> xs"},
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 where @{text p} is a pattern and @{text xs} an expression of list type, or
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\item[guards] @{text"b"}, where @{text b} is a boolean expression.
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%\item[local bindings] @ {text"let x = e"}.
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\end{description}
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Just like in Haskell, list comprehension is just a shorthand. To avoid
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misunderstandings, the translation into desugared form is not reversed
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upon output. Note that the translation of @{text"[e. x \<leftarrow> xs]"} is
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optmized to @{term"map (%x. e) xs"}.
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It is easy to write short list comprehensions which stand for complex
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expressions. During proofs, they may become unreadable (and
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mangled). In such cases it can be advisable to introduce separate
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definitions for the list comprehensions in question.  *}
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(*
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Proper theorem proving support would be nice. For example, if
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@{text"set[f x y. x \<leftarrow> xs, y \<leftarrow> ys, P x y]"}
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produced something like
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@{term"{z. EX x: set xs. EX y:set ys. P x y \<and> z = f x y}"}.
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*)
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nonterminals lc_qual lc_quals
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syntax
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"_listcompr" :: "'a \<Rightarrow> lc_qual \<Rightarrow> lc_quals \<Rightarrow> 'a list"  ("[_ . __")
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ <- _")
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"_lc_test" :: "bool \<Rightarrow> lc_qual" ("_")
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(*"_lc_let" :: "letbinds => lc_qual"  ("let _")*)
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"_lc_end" :: "lc_quals" ("]")
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"_lc_quals" :: "lc_qual \<Rightarrow> lc_quals \<Rightarrow> lc_quals" (", __")
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"_lc_abs" :: "'a => 'b list => 'b list"
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(* These are easier than ML code but cannot express the optimized
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   translation of [e. p<-xs]
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   329
translations
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"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
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   333
"[e. P]" => "if P then [e] else []"
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   334
"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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   335
 => "if P then (_listcompr e Q Qs) else []"
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   336
"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
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 => "_Let b (_listcompr e Q Qs)"
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*)
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syntax (xsymbols)
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   341
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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   342
syntax (HTML output)
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   343
"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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   344
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parse_translation (advanced) {*
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let
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  val NilC = Syntax.const @{const_name Nil};
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  val ConsC = Syntax.const @{const_name Cons};
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  val mapC = Syntax.const @{const_name map};
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  val concatC = Syntax.const @{const_name concat};
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  val IfC = Syntax.const @{const_name If};
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  fun singl x = ConsC $ x $ NilC;
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   353
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   fun pat_tr ctxt p e opti = (* %x. case x of p => e | _ => [] *)
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    let
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      val x = Free (Name.variant (add_term_free_names (p$e, [])) "x", dummyT);
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   357
      val e = if opti then singl e else e;
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   358
      val case1 = Syntax.const "_case1" $ p $ e;
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   359
      val case2 = Syntax.const "_case1" $ Syntax.const Term.dummy_patternN
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                                        $ NilC;
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   361
      val cs = Syntax.const "_case2" $ case1 $ case2
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   362
      val ft = DatatypeCase.case_tr false DatatypePackage.datatype_of_constr
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                 ctxt [x, cs]
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    in lambda x ft end;
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   365
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  fun abs_tr ctxt (p as Free(s,T)) e opti =
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        let val thy = ProofContext.theory_of ctxt;
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            val s' = Sign.intern_const thy s
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   369
        in if Sign.declared_const thy s'
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           then (pat_tr ctxt p e opti, false)
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   371
           else (lambda p e, true)
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   372
        end
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    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
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   374
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   375
  fun lc_tr ctxt [e, Const("_lc_test",_)$b, qs] =
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        let val res = case qs of Const("_lc_end",_) => singl e
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   377
                      | Const("_lc_quals",_)$q$qs => lc_tr ctxt [e,q,qs];
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        in IfC $ b $ res $ NilC end
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    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_end",_)] =
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   380
        (case abs_tr ctxt p e true of
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   381
           (f,true) => mapC $ f $ es
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         | (f, false) => concatC $ (mapC $ f $ es))
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    | lc_tr ctxt [e, Const("_lc_gen",_) $ p $ es, Const("_lc_quals",_)$q$qs] =
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   384
        let val e' = lc_tr ctxt [e,q,qs];
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   385
        in concatC $ (mapC $ (fst(abs_tr ctxt p e' false)) $ es) end
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   386
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   387
in [("_listcompr", lc_tr)] end
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   388
*}
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   389
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   390
(*
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   391
term "[(x,y,z). b]"
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   392
term "[(x,y,z). x\<leftarrow>xs]"
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   393
term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
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   394
term "[(x,y,z). x<a, x>b]"
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   395
term "[(x,y,z). x\<leftarrow>xs, x>b]"
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   396
term "[(x,y,z). x<a, x\<leftarrow>xs]"
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   397
term "[(x,y). Cons True x \<leftarrow> xs]"
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   398
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
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   399
term "[(x,y,z). x<a, x>b, x=d]"
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   400
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
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   401
term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
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   402
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
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diff changeset
   403
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
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diff changeset
   404
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
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   405
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
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   406
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
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   407
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
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   408
*)
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diff changeset
   409
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   410
subsubsection {* @{const Nil} and @{const Cons} *}
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   411
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   412
lemma not_Cons_self [simp]:
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   413
  "xs \<noteq> x # xs"
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   414
by (induct xs) auto
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   415
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   416
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
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   417
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   418
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
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   419
by (induct xs) auto
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   420
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   421
lemma length_induct:
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   422
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
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   423
by (rule measure_induct [of length]) iprover
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   424
f2b00262bdfc converted;
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diff changeset
   425
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   426
subsubsection {* @{const length} *}
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   427
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   428
text {*
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   429
  Needs to come before @{text "@"} because of theorem @{text
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   430
  append_eq_append_conv}.
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   431
*}
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   432
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   433
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
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   434
by (induct xs) auto
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diff changeset
   435
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   436
lemma length_map [simp]: "length (map f xs) = length xs"
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   437
by (induct xs) auto
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   438
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   439
lemma length_rev [simp]: "length (rev xs) = length xs"
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   440
by (induct xs) auto
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   441
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   442
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
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   443
by (cases xs) auto
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   444
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   445
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
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   446
by (induct xs) auto
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diff changeset
   447
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   448
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
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   449
by (induct xs) auto
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diff changeset
   450
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   451
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
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   452
by auto
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   453
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   454
lemma length_Suc_conv:
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   455
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
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   456
by (induct xs) auto
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diff changeset
   457
14025
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   458
lemma Suc_length_conv:
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   459
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
14208
144f45277d5a misc tidying
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diff changeset
   460
apply (induct xs, simp, simp)
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   461
apply blast
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   462
done
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diff changeset
   463
25221
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diff changeset
   464
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
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diff changeset
   465
  by (induct xs) auto
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parents: 25215
diff changeset
   466
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
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diff changeset
   467
lemma list_induct2 [consumes 1, case_names Nil Cons]:
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   468
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
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diff changeset
   469
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
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diff changeset
   470
   \<Longrightarrow> P xs ys"
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diff changeset
   471
proof (induct xs arbitrary: ys)
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diff changeset
   472
  case Nil then show ?case by simp
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haftmann
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diff changeset
   473
next
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diff changeset
   474
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
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haftmann
parents: 26300
diff changeset
   475
qed
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haftmann
parents: 26300
diff changeset
   476
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haftmann
parents: 26300
diff changeset
   477
lemma list_induct3 [consumes 2, case_names Nil Cons]:
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haftmann
parents: 26300
diff changeset
   478
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
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haftmann
parents: 26300
diff changeset
   479
   (\<And>x xs y ys z zs. length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P xs ys zs \<Longrightarrow> P (x#xs) (y#ys) (z#zs))
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haftmann
parents: 26300
diff changeset
   480
   \<Longrightarrow> P xs ys zs"
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haftmann
parents: 26300
diff changeset
   481
proof (induct xs arbitrary: ys zs)
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haftmann
parents: 26300
diff changeset
   482
  case Nil then show ?case by simp
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haftmann
parents: 26300
diff changeset
   483
next
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   484
  case (Cons x xs ys zs) then show ?case by (cases ys, simp_all)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   485
    (cases zs, simp_all)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   486
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   487
22493
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   488
lemma list_induct2': 
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   489
  "\<lbrakk> P [] [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   490
  \<And>x xs. P (x#xs) [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   491
  \<And>y ys. P [] (y#ys);
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   492
   \<And>x xs y ys. P xs ys  \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk>
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   493
 \<Longrightarrow> P xs ys"
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   494
by (induct xs arbitrary: ys) (case_tac x, auto)+
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   495
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   496
lemma neq_if_length_neq: "length xs \<noteq> length ys \<Longrightarrow> (xs = ys) == False"
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   497
by (rule Eq_FalseI) auto
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   498
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   499
simproc_setup list_neq ("(xs::'a list) = ys") = {*
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   500
(*
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   501
Reduces xs=ys to False if xs and ys cannot be of the same length.
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   502
This is the case if the atomic sublists of one are a submultiset
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   503
of those of the other list and there are fewer Cons's in one than the other.
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   504
*)
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   505
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   506
let
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   507
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   508
fun len (Const("List.list.Nil",_)) acc = acc
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   509
  | len (Const("List.list.Cons",_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   510
  | len (Const("List.append",_) $ xs $ ys) acc = len xs (len ys acc)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   511
  | len (Const("List.rev",_) $ xs) acc = len xs acc
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   512
  | len (Const("List.map",_) $ _ $ xs) acc = len xs acc
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   513
  | len t (ts,n) = (t::ts,n);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   514
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   515
fun list_neq _ ss ct =
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   516
  let
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   517
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   518
    val (ls,m) = len lhs ([],0) and (rs,n) = len rhs ([],0);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   519
    fun prove_neq() =
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   520
      let
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   521
        val Type(_,listT::_) = eqT;
22994
02440636214f abstract size function in hologic.ML
haftmann
parents: 22940
diff changeset
   522
        val size = HOLogic.size_const listT;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   523
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   524
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   525
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
22633
haftmann
parents: 22551
diff changeset
   526
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann
parents: 22551
diff changeset
   527
      in SOME (thm RS @{thm neq_if_length_neq}) end
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   528
  in
23214
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   529
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   530
       n < m andalso submultiset (op aconv) (rs,ls)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   531
    then prove_neq() else NONE
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   532
  end;
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   533
in list_neq end;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   534
*}
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   535
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   536
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   537
subsubsection {* @{text "@"} -- append *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   538
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   539
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   540
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   541
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   542
lemma append_Nil2 [simp]: "xs @ [] = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   543
by (induct xs) auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   544
24449
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   545
interpretation semigroup_append: semigroup_add ["op @"]
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   546
by unfold_locales simp
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   547
interpretation monoid_append: monoid_add ["[]" "op @"]
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   548
by unfold_locales (simp+)
2f05cb7fed85 added (code) lemmas for setsum and foldl
nipkow
parents: 24349
diff changeset
   549
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   550
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   551
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   552
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   553
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   554
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   555
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   556
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   557
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   558
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   559
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   560
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   561
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   562
lemma append_eq_append_conv [simp, noatp]:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   563
 "length xs = length ys \<or> length us = length vs
13883
0451e0fb3f22 Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents: 13863
diff changeset
   564
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   565
apply (induct xs arbitrary: ys)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   566
 apply (case_tac ys, simp, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   567
apply (case_tac ys, force, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   568
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   569
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   570
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   571
  (EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   572
apply (induct xs arbitrary: ys zs ts)
14495
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   573
 apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   574
apply(case_tac zs)
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   575
 apply simp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   576
apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   577
done
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   578
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   579
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   580
by simp
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   581
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   582
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   583
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   584
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   585
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   586
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   587
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   588
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   589
using append_same_eq [of _ _ "[]"] by auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   590
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   591
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   592
using append_same_eq [of "[]"] by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   593
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24219
diff changeset
   594
lemma hd_Cons_tl [simp,noatp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   595
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   596
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   597
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   598
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   599
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   600
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   601
by (simp add: hd_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   602
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   603
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   604
by (simp split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   605
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   606
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   607
by (simp add: tl_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   608
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   609
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   610
lemma Cons_eq_append_conv: "x#xs = ys@zs =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   611
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   612
by(cases ys) auto
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   613
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   614
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   615
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   616
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   617
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   618
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   619
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   620
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   621
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   622
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   623
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   624
lemma Cons_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   625
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   626
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   627
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   628
lemma append_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   629
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   630
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   631
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   632
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   633
text {*
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   634
Simplification procedure for all list equalities.
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   635
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   636
- both lists end in a singleton list,
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   637
- or both lists end in the same list.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   638
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   639
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26442
diff changeset
   640
ML {*
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   641
local
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   642
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   643
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   644
  (case xs of Const("List.list.Nil",_) => cons | _ => last xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   645
  | last (Const("List.append",_) $ _ $ ys) = last ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   646
  | last t = t;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   647
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   648
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   649
  | list1 _ = false;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   650
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   651
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   652
  (case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs)
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   653
  | butlast ((app as Const("List.append",_) $ xs) $ ys) = app $ butlast ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   654
  | butlast xs = Const("List.list.Nil",fastype_of xs);
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   655
22633
haftmann
parents: 22551
diff changeset
   656
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann
parents: 22551
diff changeset
   657
  @{thm append_Nil}, @{thm append_Cons}];
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   658
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   659
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   660
  let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   661
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   662
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   663
      let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   664
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   665
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   666
        val appT = [listT,listT] ---> listT
23029
79ee75dc1e59 constant op @ now named append
haftmann
parents: 22994
diff changeset
   667
        val app = Const("List.append",appT)
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   668
        val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr)
13480
bb72bd43c6c3 use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents: 13462
diff changeset
   669
        val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2));
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   670
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   671
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   672
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   673
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   674
  in
22633
haftmann
parents: 22551
diff changeset
   675
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann
parents: 22551
diff changeset
   676
    else if lastl aconv lastr then rearr @{thm append_same_eq}
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   677
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   678
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   679
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   680
in
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   681
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   682
val list_eq_simproc =
22633
haftmann
parents: 22551
diff changeset
   683
  Simplifier.simproc @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   684
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   685
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   686
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   687
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   688
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   689
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   690
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   691
subsubsection {* @{text map} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   692
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   693
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   694
by (induct xs) simp_all
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   695
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   696
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   697
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   698
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   699
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   700
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   701
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   702
lemma map_compose: "map (f o g) xs = map f (map g xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   703
by (induct xs) (auto simp add: o_def)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   704
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   705
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   706
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   707
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   708
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)"
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   709
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   710
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   711
lemma map_cong [fundef_cong, recdef_cong]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   712
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   713
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   714
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   715
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   716
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   717
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   718
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   719
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   720
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   721
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   722
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   723
 "(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   724
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   725
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   726
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   727
 "(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)"
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   728
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   729
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   730
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   731
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   732
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   733
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   734
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   735
  "(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)"
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   736
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   737
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   738
lemma map_eq_imp_length_eq:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   739
  assumes "map f xs = map f ys"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   740
  shows "length xs = length ys"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   741
using assms proof (induct ys arbitrary: xs)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   742
  case Nil then show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   743
next
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   744
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   745
  from Cons xs have "map f zs = map f ys" by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   746
  moreover with Cons have "length zs = length ys" by blast
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   747
  with xs show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   748
qed
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   749
  
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   750
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   751
 "[| map f xs = map f ys; inj_on f (set xs Un set ys) |]
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   752
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   753
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   754
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   755
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   756
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   757
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   758
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   759
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   760
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   761
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   762
 "inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   763
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   764
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   765
lemma map_injective:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   766
 "map f xs = map f ys ==> inj f ==> xs = ys"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   767
by (induct ys arbitrary: xs) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   768
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   769
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)"
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   770
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   771
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   772
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   773
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   774
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   775
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   776
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   777
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   778
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   779
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   780
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   781
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   782
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   783
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   784
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   785
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   786
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   787
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   788
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   789
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   790
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   791
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs"
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   792
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   793
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   794
lemma map_fun_upd [simp]: "y \<notin> set xs \<Longrightarrow> map (f(y:=v)) xs = map f xs"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   795
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   796
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   797
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   798
  "length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   799
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   800
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   801
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   802
  "length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   803
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   804
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   805
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   806
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   807
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   808
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   809
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   810
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   811
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   812
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   813
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   814
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   815
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   816
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   817
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   818
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   819
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   820
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   821
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   822
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   823
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   824
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   825
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   826
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   827
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   828
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   829
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)"
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   830
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   831
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   832
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   833
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   834
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   835
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   836
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   837
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   838
  "[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs"
15489
d136af442665 Replaced application of subst by simplesubst in proof of rev_induct
berghofe
parents: 15439
diff changeset
   839
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   840
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   841
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   842
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   843
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   844
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   845
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   846
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   847
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   848
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   849
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   850
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   851
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   852
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   853
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   854
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   855
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   856
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   857
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   858
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   859
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   860
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   861
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   862
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   863
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   864
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   865
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   866
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   867
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" 
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   868
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   869
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   870
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   871
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   872
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   873
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   874
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   875
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   876
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   877
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   878
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   879
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   880
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   881
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   882
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   883
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   884
15425
6356d2523f73 [ .. (] -> [ ..< ]
nipkow
parents: 15392
diff changeset
   885
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   886
apply (induct j, simp_all)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   887
apply (erule ssubst, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   888
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   889
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   890
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   891
lemma split_list: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs"
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   892
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   893
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   894
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   895
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   896
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   897
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   898
lemma in_set_conv_decomp: "x \<in> set xs \<longleftrightarrow> (\<exists>ys zs. xs = ys @ x # zs)"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   899
  by (auto elim: split_list)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   900
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   901
lemma split_list_first: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   902
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   903
  case Nil thus ?case by simp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   904
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   905
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   906
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   907
  proof cases
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   908
    assume "x = a" thus ?case using Cons by fastsimp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   909
  next
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   910
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   911
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   912
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   913
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   914
lemma in_set_conv_decomp_first:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   915
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   916
  by (auto dest!: split_list_first)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   917
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   918
lemma split_list_last: "x : set xs \<Longrightarrow> \<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   919
proof (induct xs rule:rev_induct)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   920
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   921
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   922
  case (snoc a xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   923
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   924
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   925
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   926
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   927
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   928
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   929
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   930
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   931
lemma in_set_conv_decomp_last:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   932
  "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   933
  by (auto dest!: split_list_last)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   934
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   935
lemma split_list_prop: "\<exists>x \<in> set xs. P x \<Longrightarrow> \<exists>ys x zs. xs = ys @ x # zs & P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   936
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   937
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   938
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   939
  case Cons thus ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   940
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   941
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   942
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   943
lemma split_list_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   944
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   945
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   946
using split_list_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   947
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   948
lemma split_list_first_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   949
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   950
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y)"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   951
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   952
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   953
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   954
  case (Cons x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   955
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   956
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   957
    assume "P x"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   958
    thus ?thesis by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   959
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   960
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   961
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   962
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   963
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   964
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   965
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   966
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   967
lemma split_list_first_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   968
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   969
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>y \<in> set ys. \<not> P y"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   970
using split_list_first_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   971
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   972
lemma split_list_first_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   973
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   974
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>y \<in> set ys. \<not> P y))"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   975
by (rule, erule split_list_first_prop) auto
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   976
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   977
lemma split_list_last_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   978
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   979
   \<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z)"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   980
proof(induct xs rule:rev_induct)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   981
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   982
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   983
  case (snoc x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   984
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   985
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   986
    assume "P x" thus ?thesis by (metis emptyE set_empty)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   987
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   988
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   989
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   990
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   991
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   992
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   993
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   994
lemma split_list_last_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   995
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   996
  obtains ys x zs where "xs = ys @ x # zs" and "P x" and "\<forall>z \<in> set zs. \<not> P z"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   997
using split_list_last_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   998
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   999
lemma split_list_last_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1000
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1001
   (\<exists>ys x zs. xs = ys@x#zs \<and> P x \<and> (\<forall>z \<in> set zs. \<not> P z))"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1002
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1003
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1004
lemma finite_list: "finite A ==> EX xs. set xs = A"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1005
  by (erule finite_induct)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1006
    (auto simp add: set.simps(2) [symmetric] simp del: set.simps(2))
13508
890d736b93a5 Frederic Blanqui's new "guard" examples
paulson
parents: 13480
diff changeset
  1007
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1008
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1009
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1010
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1011
lemma set_minus_filter_out:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1012
  "set xs - {y} = set (filter (\<lambda>x. \<not> (x = y)) xs)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1013
  by (induct xs) auto
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
  1014
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1015
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1016
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1017
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1018
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1019
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1020
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1021
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1022
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1023
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1024
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1025
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1026
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1027
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1028
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1029
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1030
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1031
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1032
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1033
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1034
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1035
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1036
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1037
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1038
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1039
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" 
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1040
by (induct xs) simp_all
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1041
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1042
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1043
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1044
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1045
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1046
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1047
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1048
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1049
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1050
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1051
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1052
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1053
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1054
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1055
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1056
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1057
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1058
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1059
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1060
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1061
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1062
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1063
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1064
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1065
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1066
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1067
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1068
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1069
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1070
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1071
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1072
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1073
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1074
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1075
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1076
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1077
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1078
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1079
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1080
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1081
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1082
    hence eq: "?S' = insert 0 (Suc ` ?S)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1083
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1084
    have "length (filter p (x # xs)) = Suc(card ?S)"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1085
      using Cons `p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1086
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1087
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1088
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1089
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1090
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1091
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1092
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1093
    hence eq: "?S' = Suc ` ?S"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1094
      by(auto simp add: image_def split:nat.split elim:lessE)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1095
    have "length (filter p (x # xs)) = card ?S"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1096
      using Cons `\<not> p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1097
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1098
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1099
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1100
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1101
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1102
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1103
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1104
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1105
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1106
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1107
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
19585
70a1ce3b23ae removed 'concl is' patterns;
wenzelm
parents: 19487
diff changeset
  1108
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1109
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1110
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1111
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1112
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1113
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1114
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1115
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1116
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1117
    proof cases
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1118
      assume "x = y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1119
      with Py Cons.prems have "?Q []" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1120
      then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1121
    next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1122
      assume "x \<noteq> y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1123
      with Py Cons.prems show ?thesis by simp
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1124
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1125
  next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1126
    assume "\<not> P y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1127
    with Cons obtain us vs where "?P (y#ys) (y#us) vs" by fastsimp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1128
    then have "?Q (y#us)" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1129
    then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1130
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1131
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1132
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1133
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1134
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1135
  \<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1136
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1137
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1138
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1139
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1140
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1141
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1142
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1143
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1144
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1145
  (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1146
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1147
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
  1148
lemma filter_cong[fundef_cong, recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1149
 "xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1150
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1151
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1152
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1153
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1154
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1155
subsubsection {* List partitioning *}
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1156
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1157
primrec partition :: "('a \<Rightarrow> bool) \<Rightarrow>'a list \<Rightarrow> 'a list \<times> 'a list" where
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1158
  "partition P [] = ([], [])"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1159
  | "partition P (x # xs) = 
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1160
      (let (yes, no) = partition P xs
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1161
      in if P x then (x # yes, no) else (yes, x # no))"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1162
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1163
lemma partition_filter1:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1164
    "fst (partition P xs) = filter P xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1165
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1166
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1167
lemma partition_filter2:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1168
    "snd (partition P xs) = filter (Not o P) xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1169
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1170
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1171
lemma partition_P:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1172
  assumes "partition P xs = (yes, no)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1173
  shows "(\<forall>p \<in> set yes.  P p) \<and> (\<forall>p  \<in> set no. \<not> P p)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1174
proof -
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1175
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1176
    by simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1177
  then show ?thesis by (simp_all add: partition_filter1 partition_filter2)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1178
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1179
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1180
lemma partition_set:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1181
  assumes "partition P xs = (yes, no)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1182
  shows "set yes \<union> set no = set xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1183
proof -
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1184
  from assms have "yes = fst (partition P xs)" and "no = snd (partition P xs)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1185
    by simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1186
  then show ?thesis by (auto simp add: partition_filter1 partition_filter2) 
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1187
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1188
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1189
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1190
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1191
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1192
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1193
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1194
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1195
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1196
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1197
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1198
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1199
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1200
24308
700e745994c1 removed set_concat_map and improved set_concat
nipkow
parents: 24286
diff changeset
  1201
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1202
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1203
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
  1204
lemma concat_map_singleton[simp]: "concat(map (%x. [f x]) xs) = map f xs"
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1205
by (induct xs) auto
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1206
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1207
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1208
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1209
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1210
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1211
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1212
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1213
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1214
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1215
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1216
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1217
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1218
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1219
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1220
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1221
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1222
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1223
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1224
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1225
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1226
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1227
lemma nth_append:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1228
  "(xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1229
apply (induct xs arbitrary: n, simp)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1230
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1231
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1232
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1233
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x"
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1234
by (induct xs) auto
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1235
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1236
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n"
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1237
by (induct xs) auto
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1238
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1239
lemma nth_map [simp]: "n < length xs ==> (map f xs)!n = f(xs!n)"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1240
apply (induct xs arbitrary: n, simp)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1241
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1242
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1243
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1244
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1245
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1246
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1247
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1248
lemma list_eq_iff_nth_eq:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1249
 "(xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1250
apply(induct xs arbitrary: ys)
24632
779fc4fcbf8b metis now available in PreList
paulson
parents: 24617
diff changeset
  1251
 apply force
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1252
apply(case_tac ys)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1253
 apply simp
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1254
apply(simp add:nth_Cons split:nat.split)apply blast
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1255
done
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1256
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1257
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15246
diff changeset
  1258
apply (induct xs, simp, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1259
apply safe
24632
779fc4fcbf8b metis now available in PreList
paulson
parents: 24617
diff changeset
  1260
apply (metis nat_case_0 nth.simps zero_less_Suc)
779fc4fcbf8b metis now available in PreList
paulson
parents: 24617
diff changeset
  1261
apply (metis less_Suc_eq_0_disj nth_Cons_Suc)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1262
apply (case_tac i, simp)
24632
779fc4fcbf8b metis now available in PreList
paulson
parents: 24617
diff changeset
  1263
apply (metis diff_Suc_Suc nat_case_Suc nth.simps zero_less_diff)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1264
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1265
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1266
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)"
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1267
by(auto simp:set_conv_nth)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1268
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1269
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1270
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1271
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1272
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1273
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1274
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1275
lemma all_nth_imp_all_set:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1276
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1277
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1278
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1279
lemma all_set_conv_all_nth:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1280
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1281
by (auto simp add: set_conv_nth)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1282
25296
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1283
lemma rev_nth:
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1284
  "n < size xs \<Longrightarrow> rev xs ! n = xs ! (length xs - Suc n)"
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1285
proof (induct xs arbitrary: n)
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1286
  case Nil thus ?case by simp
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1287
next
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1288
  case (Cons x xs)
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1289
  hence n: "n < Suc (length xs)" by simp
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1290
  moreover
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1291
  { assume "n < length xs"
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1292
    with n obtain n' where "length xs - n = Suc n'"
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1293
      by (cases "length xs - n", auto)
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1294
    moreover
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1295
    then have "length xs - Suc n = n'" by simp
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1296
    ultimately
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1297
    have "xs ! (length xs - Suc n) = (x # xs) ! (length xs - n)" by simp
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1298
  }
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1299
  ultimately
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1300
  show ?case by (clarsimp simp add: Cons nth_append)
c187b7422156 rev_nth
kleing
parents: 25287
diff changeset
  1301
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1302
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1303
subsubsection {* @{text list_update} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1304
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1305
lemma length_list_update [simp]: "length(xs[i:=x]) = length xs"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1306
by (induct xs arbitrary: i) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1307
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1308
lemma nth_list_update:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1309
"i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1310
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1311
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1312
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1313
by (simp add: nth_list_update)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1314
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1315
lemma nth_list_update_neq [simp]: "i \<noteq> j ==> xs[i:=x]!j = xs!j"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1316
by (induct xs arbitrary: i j) (auto simp add: nth_Cons split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1317
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1318
lemma list_update_id[simp]: "xs[i := xs!i] = xs"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1319
by (induct xs arbitrary: i) (simp_all split:nat.splits)
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1320
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1321
lemma list_update_beyond[simp]: "length xs \<le> i \<Longrightarrow> xs[i:=x] = xs"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1322
apply (induct xs arbitrary: i)
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1323
 apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1324
apply (case_tac i)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1325
apply simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1326
done
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1327
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1328
lemma list_update_same_conv:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1329
"i < length xs ==> (xs[i := x] = xs) = (xs!i = x)"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1330
by (induct xs arbitrary: i) (auto split: nat.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1331
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1332
lemma list_update_append1:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1333
 "i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1334
apply (induct xs arbitrary: i, simp)
14187
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1335
apply(simp split:nat.split)
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1336
done
26dfcd0ac436 Added new theorems
nipkow
parents: 14111
diff changeset
  1337
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1338
lemma list_update_append:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1339
  "(xs @ ys) [n:= x] = 
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1340
  (if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))"
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1341
by (induct xs arbitrary: n) (auto split:nat.splits)
15868
9634b3f9d910 more about list_update
kleing
parents: 15693
diff changeset
  1342
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1343
lemma list_update_length [simp]:
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1344
 "(xs @ x # ys)[length xs := y] = (xs @ y # ys)"
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1345