src/HOL/List.thy
author blanchet
Wed, 06 Oct 2010 17:44:21 +0200
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(*  Title:      HOL/List.thy
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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 Plain Quotient Presburger Code_Numeral Recdef
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uses ("Tools/list_code.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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syntax
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  -- {* list Enumeration *}
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  "_list" :: "args => 'a list"    ("[(_)]")
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translations
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  "[x, xs]" == "x#[xs]"
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  "[x]" == "x#[]"
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subsection {* Basic list processing functions *}
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primrec
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  hd :: "'a list \<Rightarrow> 'a" where
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  "hd (x # xs) = x"
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primrec
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  tl :: "'a list \<Rightarrow> 'a list" where
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    "tl [] = []"
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  | "tl (x # xs) = xs"
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primrec
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  last :: "'a list \<Rightarrow> 'a" where
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  "last (x # xs) = (if xs = [] then x else last xs)"
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primrec
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  butlast :: "'a list \<Rightarrow> 'a list" where
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    "butlast []= []"
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  | "butlast (x # xs) = (if xs = [] then [] else x # butlast xs)"
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primrec
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  set :: "'a list \<Rightarrow> 'a set" where
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    "set [] = {}"
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  | "set (x # xs) = insert x (set xs)"
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primrec
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  map :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b list" where
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    "map f [] = []"
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  | "map f (x # xs) = f x # map f xs"
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primrec
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  append :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixr "@" 65) 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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primrec
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  rev :: "'a list \<Rightarrow> 'a list" where
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    "rev [] = []"
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  | "rev (x # xs) = rev xs @ [x]"
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primrec
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  filter:: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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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syntax
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  -- {* Special syntax for filter *}
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  "_filter" :: "[pttrn, 'a list, bool] => 'a list"    ("(1[_<-_./ _])")
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translations
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  "[x<-xs . P]"== "CONST filter (%x. P) xs"
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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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primrec
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  foldl :: "('b \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> 'b" where
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    foldl_Nil: "foldl f a [] = a"
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  | foldl_Cons: "foldl f a (x # xs) = foldl f (f a x) xs"
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primrec
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  foldr :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where
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    "foldr f [] a = a"
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  | "foldr f (x # xs) a = f x (foldr f xs a)"
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primrec
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  concat:: "'a list list \<Rightarrow> 'a list" where
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    "concat [] = []"
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  | "concat (x # xs) = x @ concat xs"
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definition (in monoid_add)
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  listsum :: "'a list \<Rightarrow> 'a" where
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  "listsum xs = foldr plus xs 0"
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primrec
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  drop:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    drop_Nil: "drop n [] = []"
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  | drop_Cons: "drop n (x # xs) = (case n of 0 \<Rightarrow> x # xs | Suc m \<Rightarrow> drop m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  take:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    take_Nil:"take n [] = []"
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  | take_Cons: "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  nth :: "'a list => nat => 'a" (infixl "!" 100) where
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  nth_Cons: "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
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  -- {*Warning: simpset does not contain this definition, but separate
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       theorems for @{text "n = 0"} and @{text "n = Suc k"} *}
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primrec
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  list_update :: "'a list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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    "list_update [] i v = []"
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  | "list_update (x # xs) i v = (case i of 0 \<Rightarrow> v # xs | Suc j \<Rightarrow> x # list_update xs j v)"
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nonterminals lupdbinds lupdbind
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syntax
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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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  "_LUpdate xs (_lupdbinds b bs)" == "_LUpdate (_LUpdate xs b) bs"
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  "xs[i:=x]" == "CONST list_update xs i x"
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primrec
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  takeWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    "takeWhile P [] = []"
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  | "takeWhile P (x # xs) = (if P x then x # takeWhile P xs else [])"
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primrec
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  dropWhile :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    "dropWhile P [] = []"
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  | "dropWhile P (x # xs) = (if P x then dropWhile P xs else x # xs)"
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primrec
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  zip :: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" where
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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 :: "nat \<Rightarrow> nat \<Rightarrow> nat list" ("(1[_..</_'])") where
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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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definition
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  insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "insert x xs = (if x \<in> set xs then xs else x # xs)"
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hide_const (open) insert
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hide_fact (open) insert_def
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primrec
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  remove1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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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  removeAll :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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    "removeAll x [] = []"
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  | "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
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inductive
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  distinct :: "'a list \<Rightarrow> bool" where
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    Nil: "distinct []"
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  | insert: "x \<notin> set xs \<Longrightarrow> distinct xs \<Longrightarrow> distinct (x # xs)"
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lemma distinct_simps [simp, code]:
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  "distinct [] \<longleftrightarrow> True"
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  "distinct (x # xs) \<longleftrightarrow> x \<notin> set xs \<and> distinct xs"
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  by (auto intro: distinct.intros elim: distinct.cases)
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primrec
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  remdups :: "'a list \<Rightarrow> 'a list" where
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    "remdups [] = []"
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  | "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
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primrec
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  replicate :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where
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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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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 \<Rightarrow> nat" where
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  "length \<equiv> size"
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definition
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  rotate1 :: "'a list \<Rightarrow> 'a list" where
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  "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])"
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definition
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  rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "rotate n = rotate1 ^^ n"
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definition
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  list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" where
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  "list_all2 P xs ys =
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    (length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))"
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definition
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  sublist :: "'a list => nat set => 'a list" where
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  "sublist xs A = map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))"
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primrec
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  splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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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]" by simp}\\
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@{lemma "length [a,b,c] = 3" by simp}\\
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@{lemma "set [a,b,c] = {a,b,c}" by simp}\\
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@{lemma "map f [a,b,c] = [f a, f b, f c]" by simp}\\
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@{lemma "rev [a,b,c] = [c,b,a]" by simp}\\
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@{lemma "hd [a,b,c,d] = a" by simp}\\
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@{lemma "tl [a,b,c,d] = [b,c,d]" by simp}\\
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@{lemma "last [a,b,c,d] = d" by simp}\\
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@{lemma "butlast [a,b,c,d] = [a,b,c]" by simp}\\
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@{lemma[source] "filter (\<lambda>n::nat. n<2) [0,2,1] = [0,1]" by simp}\\
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@{lemma "concat [[a,b],[c,d,e],[],[f]] = [a,b,c,d,e,f]" by simp}\\
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@{lemma "foldl f x [a,b,c] = f (f (f x a) b) c" by simp}\\
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@{lemma "foldr f [a,b,c] x = f a (f b (f c x))" by simp}\\
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@{lemma "zip [a,b,c] [x,y,z] = [(a,x),(b,y),(c,z)]" by simp}\\
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@{lemma "zip [a,b] [x,y,z] = [(a,x),(b,y)]" by simp}\\
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@{lemma "splice [a,b,c] [x,y,z] = [a,x,b,y,c,z]" by simp}\\
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@{lemma "splice [a,b,c,d] [x,y] = [a,x,b,y,c,d]" by simp}\\
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@{lemma "take 2 [a,b,c,d] = [a,b]" by simp}\\
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@{lemma "take 6 [a,b,c,d] = [a,b,c,d]" by simp}\\
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@{lemma "drop 2 [a,b,c,d] = [c,d]" by simp}\\
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@{lemma "drop 6 [a,b,c,d] = []" by simp}\\
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@{lemma "takeWhile (%n::nat. n<3) [1,2,3,0] = [1,2]" by simp}\\
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@{lemma "dropWhile (%n::nat. n<3) [1,2,3,0] = [3,0]" by simp}\\
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@{lemma "distinct [2,0,1::nat]" by simp}\\
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@{lemma "remdups [2,0,2,1::nat,2] = [0,1,2]" by simp}\\
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@{lemma "List.insert 2 [0::nat,1,2] = [0,1,2]" by (simp add: List.insert_def)}\\
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@{lemma "List.insert 3 [0::nat,1,2] = [3,0,1,2]" by (simp add: List.insert_def)}\\
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@{lemma "remove1 2 [2,0,2,1::nat,2] = [0,2,1,2]" by simp}\\
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@{lemma "removeAll 2 [2,0,2,1::nat,2] = [0,1]" by simp}\\
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@{lemma "nth [a,b,c,d] 2 = c" by simp}\\
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@{lemma "[a,b,c,d][2 := x] = [a,b,x,d]" by simp}\\
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@{lemma "sublist [a,b,c,d,e] {0,2,3} = [a,c,d]" by (simp add:sublist_def)}\\
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@{lemma "rotate1 [a,b,c,d] = [b,c,d,a]" by (simp add:rotate1_def)}\\
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@{lemma "rotate 3 [a,b,c,d] = [d,a,b,c]" by (simp add:rotate1_def rotate_def nat_number')}\\
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@{lemma "replicate 4 a = [a,a,a,a]" by (simp add:nat_number')}\\
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@{lemma "[2..<5] = [2,3,4]" by (simp add:nat_number')}\\
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@{lemma "listsum [1,2,3::nat] = 6" by (simp add: listsum_def)}
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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 characteristic 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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inductive sorted :: "'a list \<Rightarrow> bool" where
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  Nil [iff]: "sorted []"
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| Cons: "\<forall>y\<in>set xs. x \<le> y \<Longrightarrow> sorted xs \<Longrightarrow> sorted (x # xs)"
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lemma sorted_single [iff]:
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  "sorted [x]"
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  by (rule sorted.Cons) auto
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lemma sorted_many:
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  "x \<le> y \<Longrightarrow> sorted (y # zs) \<Longrightarrow> sorted (x # y # zs)"
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  by (rule sorted.Cons) (cases "y # zs" rule: sorted.cases, auto)
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lemma sorted_many_eq [simp, code]:
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  "sorted (x # y # zs) \<longleftrightarrow> x \<le> y \<and> sorted (y # zs)"
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  by (auto intro: sorted_many elim: sorted.cases)
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lemma [code]:
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  "sorted [] \<longleftrightarrow> True"
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  "sorted [x] \<longleftrightarrow> True"
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  by simp_all
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primrec insort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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"insort_key f x [] = [x]" |
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"insort_key f x (y#ys) = (if f x \<le> f y then (x#y#ys) else y#(insort_key f x ys))"
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definition sort_key :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b list \<Rightarrow> 'b list" where
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"sort_key f xs = foldr (insort_key f) xs []"
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abbreviation "sort \<equiv> sort_key (\<lambda>x. x)"
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abbreviation "insort \<equiv> insort_key (\<lambda>x. x)"
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definition insort_insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where
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  "insort_insert x xs = (if x \<in> set xs then xs else insort x 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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translations
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"[e. p<-xs]" => "concat(map (_lc_abs p [e]) xs)"
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"_listcompr e (_lc_gen p xs) (_lc_quals Q Qs)"
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 => "concat (map (_lc_abs p (_listcompr e Q Qs)) xs)"
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"[e. P]" => "if P then [e] else []"
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"_listcompr e (_lc_test P) (_lc_quals Q Qs)"
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 => "if P then (_listcompr e Q Qs) else []"
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"_listcompr e (_lc_let b) (_lc_quals Q Qs)"
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 => "_Let b (_listcompr e Q Qs)"
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*)
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syntax (xsymbols)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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syntax (HTML output)
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"_lc_gen" :: "'a \<Rightarrow> 'a list \<Rightarrow> lc_qual" ("_ \<leftarrow> _")
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parse_translation (advanced) {*
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let
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  val NilC = Syntax.const @{const_syntax Nil};
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  val ConsC = Syntax.const @{const_syntax Cons};
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  val mapC = Syntax.const @{const_syntax map};
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  val concatC = Syntax.const @{const_syntax concat};
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  val IfC = Syntax.const @{const_syntax If};
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   390
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  fun singl x = ConsC $ x $ NilC;
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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 (fold Term.add_free_names [p, e] []) "x", dummyT);
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      val e = if opti then singl e else e;
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      val case1 = Syntax.const @{syntax_const "_case1"} $ p $ e;
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      val case2 =
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        Syntax.const @{syntax_const "_case1"} $
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          Syntax.const @{const_syntax dummy_pattern} $ NilC;
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      val cs = Syntax.const @{syntax_const "_case2"} $ case1 $ case2;
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      val ft = Datatype_Case.case_tr false Datatype.info_of_constr ctxt [x, cs];
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    in lambda x ft end;
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  fun abs_tr ctxt (p as Free (s, T)) e opti =
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        let
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   407
          val thy = ProofContext.theory_of ctxt;
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          val s' = Sign.intern_const thy s;
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   409
        in
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   410
          if Sign.declared_const thy s'
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          then (pat_tr ctxt p e opti, false)
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          else (lambda p e, true)
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   413
        end
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    | abs_tr ctxt p e opti = (pat_tr ctxt p e opti, false);
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  fun lc_tr ctxt [e, Const (@{syntax_const "_lc_test"}, _) $ b, qs] =
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   417
        let
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          val res =
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            (case qs of
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              Const (@{syntax_const "_lc_end"}, _) => singl e
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            | Const (@{syntax_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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   423
    | lc_tr ctxt
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          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
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            Const(@{syntax_const "_lc_end"}, _)] =
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        (case abs_tr ctxt p e true of
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          (f, true) => mapC $ f $ es
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        | (f, false) => concatC $ (mapC $ f $ es))
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    | lc_tr ctxt
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          [e, Const (@{syntax_const "_lc_gen"}, _) $ p $ es,
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            Const (@{syntax_const "_lc_quals"}, _) $ q $ qs] =
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        let val e' = lc_tr ctxt [e, q, qs];
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        in concatC $ (mapC $ (fst (abs_tr ctxt p e' false)) $ es) end;
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   434
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in [(@{syntax_const "_listcompr"}, lc_tr)] end
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*}
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term "[(x,y,z). b]"
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term "[(x,y,z). x\<leftarrow>xs]"
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term "[e x y. x\<leftarrow>xs, y\<leftarrow>ys]"
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   441
term "[(x,y,z). x<a, x>b]"
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term "[(x,y,z). x\<leftarrow>xs, x>b]"
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term "[(x,y,z). x<a, x\<leftarrow>xs]"
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term "[(x,y). Cons True x \<leftarrow> xs]"
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   445
term "[(x,y,z). Cons x [] \<leftarrow> xs]"
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   446
term "[(x,y,z). x<a, x>b, x=d]"
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   447
term "[(x,y,z). x<a, x>b, y\<leftarrow>ys]"
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term "[(x,y,z). x<a, x\<leftarrow>xs,y>b]"
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diff changeset
   449
term "[(x,y,z). x<a, x\<leftarrow>xs, y\<leftarrow>ys]"
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   450
term "[(x,y,z). x\<leftarrow>xs, x>b, y<a]"
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   451
term "[(x,y,z). x\<leftarrow>xs, x>b, y\<leftarrow>ys]"
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   452
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,y>x]"
7077dc80a14b tuned list comprehension
nipkow
parents: 23235
diff changeset
   453
term "[(x,y,z). x\<leftarrow>xs, y\<leftarrow>ys,z\<leftarrow>zs]"
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
   454
(*
24349
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
   455
term "[(x,y). x\<leftarrow>xs, let xx = x+x, y\<leftarrow>ys, y \<noteq> xx]"
23192
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   456
*)
ec73b9707d48 Moved list comprehension into List
nipkow
parents: 23096
diff changeset
   457
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
   458
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   459
subsubsection {* @{const Nil} and @{const Cons} *}
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   460
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   461
lemma not_Cons_self [simp]:
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   462
  "xs \<noteq> x # xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   463
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   464
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   465
lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   466
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   467
lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   468
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   469
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   470
lemma length_induct:
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   471
  "(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   472
by (rule measure_induct [of length]) iprover
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   473
37289
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   474
lemma list_nonempty_induct [consumes 1, case_names single cons]:
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   475
  assumes "xs \<noteq> []"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   476
  assumes single: "\<And>x. P [x]"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   477
  assumes cons: "\<And>x xs. xs \<noteq> [] \<Longrightarrow> P xs \<Longrightarrow> P (x # xs)"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   478
  shows "P xs"
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   479
using `xs \<noteq> []` proof (induct xs)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   480
  case Nil then show ?case by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   481
next
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   482
  case (Cons x xs) show ?case proof (cases xs)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   483
    case Nil with single show ?thesis by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   484
  next
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   485
    case Cons then have "xs \<noteq> []" by simp
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   486
    moreover with Cons.hyps have "P xs" .
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   487
    ultimately show ?thesis by (rule cons)
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   488
  qed
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   489
qed
881fa5012451 induction over non-empty lists
haftmann
parents: 37123
diff changeset
   490
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   491
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   492
subsubsection {* @{const length} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   493
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   494
text {*
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   495
  Needs to come before @{text "@"} because of theorem @{text
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   496
  append_eq_append_conv}.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   497
*}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   498
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   499
lemma length_append [simp]: "length (xs @ ys) = length xs + length ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   500
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   501
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   502
lemma length_map [simp]: "length (map f xs) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   503
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   504
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   505
lemma length_rev [simp]: "length (rev xs) = length xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   506
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   507
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   508
lemma length_tl [simp]: "length (tl xs) = length xs - 1"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   509
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   510
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   511
lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   512
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   513
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   514
lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   515
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   516
23479
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   517
lemma length_pos_if_in_set: "x : set xs \<Longrightarrow> length xs > 0"
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   518
by auto
10adbdcdc65b new lemmas
nipkow
parents: 23388
diff changeset
   519
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   520
lemma length_Suc_conv:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   521
"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   522
by (induct xs) auto
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   523
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   524
lemma Suc_length_conv:
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   525
"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   526
apply (induct xs, simp, simp)
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   527
apply blast
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   528
done
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   529
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   530
lemma impossible_Cons: "length xs <= length ys ==> xs = x # ys = False"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   531
  by (induct xs) auto
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   532
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   533
lemma list_induct2 [consumes 1, case_names Nil Cons]:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   534
  "length xs = length ys \<Longrightarrow> P [] [] \<Longrightarrow>
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   535
   (\<And>x xs y ys. length xs = length ys \<Longrightarrow> P xs ys \<Longrightarrow> P (x#xs) (y#ys))
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   536
   \<Longrightarrow> P xs ys"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   537
proof (induct xs arbitrary: ys)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   538
  case Nil then show ?case by simp
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   539
next
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   540
  case (Cons x xs ys) then show ?case by (cases ys) simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   541
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   542
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   543
lemma list_induct3 [consumes 2, case_names Nil Cons]:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   544
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> P [] [] [] \<Longrightarrow>
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   545
   (\<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))
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   546
   \<Longrightarrow> P xs ys zs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   547
proof (induct xs arbitrary: ys zs)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   548
  case Nil then show ?case by simp
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   549
next
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   550
  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
   551
    (cases zs, simp_all)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
   552
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   553
36154
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   554
lemma list_induct4 [consumes 3, case_names Nil Cons]:
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   555
  "length xs = length ys \<Longrightarrow> length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   556
   P [] [] [] [] \<Longrightarrow> (\<And>x xs y ys z zs w ws. length xs = length ys \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   557
   length ys = length zs \<Longrightarrow> length zs = length ws \<Longrightarrow> P xs ys zs ws \<Longrightarrow>
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   558
   P (x#xs) (y#ys) (z#zs) (w#ws)) \<Longrightarrow> P xs ys zs ws"
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   559
proof (induct xs arbitrary: ys zs ws)
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   560
  case Nil then show ?case by simp
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   561
next
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   562
  case (Cons x xs ys zs ws) then show ?case by ((cases ys, simp_all), (cases zs,simp_all)) (cases ws, simp_all)
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   563
qed
11c6106d7787 Respectfullness and preservation of list_rel
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 35828
diff changeset
   564
22493
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   565
lemma list_induct2': 
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   566
  "\<lbrakk> P [] [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   567
  \<And>x xs. P (x#xs) [];
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   568
  \<And>y ys. P [] (y#ys);
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   569
   \<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
   570
 \<Longrightarrow> P xs ys"
db930e490fe5 added another rule for simultaneous induction, and lemmas for zip
krauss
parents: 22422
diff changeset
   571
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
   572
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   573
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
   574
by (rule Eq_FalseI) auto
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   575
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   576
simproc_setup list_neq ("(xs::'a list) = ys") = {*
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   577
(*
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   578
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
   579
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
   580
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
   581
*)
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   582
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   583
let
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   584
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   585
fun len (Const(@{const_name Nil},_)) acc = acc
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   586
  | len (Const(@{const_name Cons},_) $ _ $ xs) (ts,n) = len xs (ts,n+1)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   587
  | len (Const(@{const_name append},_) $ xs $ ys) acc = len xs (len ys acc)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   588
  | len (Const(@{const_name rev},_) $ xs) acc = len xs acc
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   589
  | len (Const(@{const_name map},_) $ _ $ xs) acc = len xs acc
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   590
  | len t (ts,n) = (t::ts,n);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   591
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   592
fun list_neq _ ss ct =
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   593
  let
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   594
    val (Const(_,eqT) $ lhs $ rhs) = Thm.term_of ct;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   595
    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
   596
    fun prove_neq() =
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   597
      let
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   598
        val Type(_,listT::_) = eqT;
22994
02440636214f abstract size function in hologic.ML
haftmann
parents: 22940
diff changeset
   599
        val size = HOLogic.size_const listT;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   600
        val eq_len = HOLogic.mk_eq (size $ lhs, size $ rhs);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   601
        val neq_len = HOLogic.mk_Trueprop (HOLogic.Not $ eq_len);
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   602
        val thm = Goal.prove (Simplifier.the_context ss) [] [] neq_len
22633
haftmann
parents: 22551
diff changeset
   603
          (K (simp_tac (Simplifier.inherit_context ss @{simpset}) 1));
haftmann
parents: 22551
diff changeset
   604
      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
   605
  in
23214
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   606
    if m < n andalso submultiset (op aconv) (ls,rs) orelse
dc23c062b58c renamed gen_submultiset to submultiset;
wenzelm
parents: 23212
diff changeset
   607
       n < m andalso submultiset (op aconv) (rs,ls)
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   608
    then prove_neq() else NONE
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   609
  end;
24037
0a41d2ebc0cd proper simproc_setup for "list_neq";
wenzelm
parents: 23983
diff changeset
   610
in list_neq end;
22143
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   611
*}
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   612
cf58486ca11b Added simproc list_neq (prompted by an application)
nipkow
parents: 21911
diff changeset
   613
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   614
subsubsection {* @{text "@"} -- append *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   615
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   616
lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   617
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   618
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   619
lemma append_Nil2 [simp]: "xs @ [] = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   620
by (induct xs) auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   621
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   622
lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   623
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   624
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   625
lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   626
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   627
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   628
lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   629
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   630
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   631
lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   632
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   633
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35827
diff changeset
   634
lemma append_eq_append_conv [simp, no_atp]:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   635
 "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
   636
 ==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)"
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   637
apply (induct xs arbitrary: ys)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   638
 apply (case_tac ys, simp, force)
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   639
apply (case_tac ys, force, simp)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   640
done
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   641
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   642
lemma append_eq_append_conv2: "(xs @ ys = zs @ ts) =
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   643
  (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
   644
apply (induct xs arbitrary: ys zs ts)
14495
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   645
 apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   646
apply(case_tac zs)
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   647
 apply simp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   648
apply fastsimp
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   649
done
e2a1c31cf6d3 Added append_eq_append_conv2
nipkow
parents: 14402
diff changeset
   650
34910
b23bd3ee4813 same_append_eq / append_same_eq are now used for simplifying induction rules.
berghofe
parents: 34064
diff changeset
   651
lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   652
by simp
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   653
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   654
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   655
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   656
34910
b23bd3ee4813 same_append_eq / append_same_eq are now used for simplifying induction rules.
berghofe
parents: 34064
diff changeset
   657
lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   658
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   659
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   660
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   661
using append_same_eq [of _ _ "[]"] by auto
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   662
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   663
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   664
using append_same_eq [of "[]"] by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   665
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35827
diff changeset
   666
lemma hd_Cons_tl [simp,no_atp]: "xs \<noteq> [] ==> hd xs # tl xs = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   667
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   668
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   669
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   670
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   671
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   672
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   673
by (simp add: hd_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   674
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   675
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
   676
by (simp split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   677
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   678
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   679
by (simp add: tl_append split: list.split)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   680
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   681
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   682
lemma Cons_eq_append_conv: "x#xs = ys@zs =
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   683
 (ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))"
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   684
by(cases ys) auto
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   685
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   686
lemma append_eq_Cons_conv: "(ys@zs = x#xs) =
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   687
 (ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   688
by(cases ys) auto
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
   689
14300
bf8b8c9425c3 *** empty log message ***
nipkow
parents: 14247
diff changeset
   690
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   691
text {* Trivial rules for solving @{text "@"}-equations automatically. *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   692
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   693
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   694
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   695
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   696
lemma Cons_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   697
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   698
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   699
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   700
lemma append_eq_appendI:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   701
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us"
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   702
by (drule sym) simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   703
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   704
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   705
text {*
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   706
Simplification procedure for all list equalities.
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   707
Currently only tries to rearrange @{text "@"} to see if
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   708
- both lists end in a singleton list,
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   709
- or both lists end in the same list.
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   710
*}
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   711
26480
544cef16045b replaced 'ML_setup' by 'ML';
wenzelm
parents: 26442
diff changeset
   712
ML {*
3507
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   713
local
157be29ad5ba Improved length = size translation.
nipkow
parents: 3465
diff changeset
   714
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   715
fun last (cons as Const(@{const_name Cons},_) $ _ $ xs) =
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   716
  (case xs of Const(@{const_name Nil},_) => cons | _ => last xs)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   717
  | last (Const(@{const_name append},_) $ _ $ ys) = last ys
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   718
  | last t = t;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   719
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   720
fun list1 (Const(@{const_name Cons},_) $ _ $ Const(@{const_name Nil},_)) = true
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   721
  | list1 _ = false;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   722
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   723
fun butlast ((cons as Const(@{const_name Cons},_) $ x) $ xs) =
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   724
  (case xs of Const(@{const_name Nil},_) => xs | _ => cons $ butlast xs)
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   725
  | butlast ((app as Const(@{const_name append},_) $ xs) $ ys) = app $ butlast ys
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   726
  | butlast xs = Const(@{const_name Nil},fastype_of xs);
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   727
22633
haftmann
parents: 22551
diff changeset
   728
val rearr_ss = HOL_basic_ss addsimps [@{thm append_assoc},
haftmann
parents: 22551
diff changeset
   729
  @{thm append_Nil}, @{thm append_Cons}];
16973
b2a894562b8f simprocs: Simplifier.inherit_bounds;
wenzelm
parents: 16965
diff changeset
   730
20044
92cc2f4c7335 simprocs: no theory argument -- use simpset context instead;
wenzelm
parents: 19890
diff changeset
   731
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) =
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   732
  let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   733
    val lastl = last lhs and lastr = last rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   734
    fun rearr conv =
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   735
      let
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   736
        val lhs1 = butlast lhs and rhs1 = butlast rhs;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   737
        val Type(_,listT::_) = eqT
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   738
        val appT = [listT,listT] ---> listT
29856
984191be0357 const_name antiquotations
huffman
parents: 29829
diff changeset
   739
        val app = Const(@{const_name append},appT)
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   740
        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
   741
        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
   742
        val thm = Goal.prove (Simplifier.the_context ss) [] [] eq
17877
67d5ab1cb0d8 Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents: 17830
diff changeset
   743
          (K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1));
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   744
      in SOME ((conv RS (thm RS trans)) RS eq_reflection) end;
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   745
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   746
  in
22633
haftmann
parents: 22551
diff changeset
   747
    if list1 lastl andalso list1 lastr then rearr @{thm append1_eq_conv}
haftmann
parents: 22551
diff changeset
   748
    else if lastl aconv lastr then rearr @{thm append_same_eq}
15531
08c8dad8e399 Deleted Library.option type.
skalberg
parents: 15489
diff changeset
   749
    else NONE
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   750
  end;
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   751
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   752
in
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   753
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   754
val list_eq_simproc =
38715
6513ea67d95d renamed Simplifier.simproc(_i) to Simplifier.simproc_global(_i) to emphasize that this is not the real thing;
wenzelm
parents: 37880
diff changeset
   755
  Simplifier.simproc_global @{theory} "list_eq" ["(xs::'a list) = ys"] (K list_eq);
13462
56610e2ba220 sane interface for simprocs;
wenzelm
parents: 13366
diff changeset
   756
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   757
end;
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   758
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   759
Addsimprocs [list_eq_simproc];
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   760
*}
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   761
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   762
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   763
subsubsection {* @{text map} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   764
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   765
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
   766
by (induct xs) simp_all
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   767
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   768
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   769
by (rule ext, induct_tac xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   770
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   771
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   772
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   773
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   774
lemma map_map [simp]: "map f (map g xs) = map (f \<circ> g) xs"
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   775
by (induct xs) auto
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
   776
35208
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   777
lemma map_comp_map[simp]: "((map f) o (map g)) = map(f o g)"
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   778
apply(rule ext)
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   779
apply(simp)
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   780
done
2b9bce05e84b added lemma
nipkow
parents: 35195
diff changeset
   781
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   782
lemma rev_map: "rev (map f xs) = map f (rev xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   783
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   784
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   785
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
   786
by (induct xs) auto
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   787
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
   788
lemma map_cong [fundef_cong, recdef_cong]:
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   789
"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
   790
-- {* a congruence rule for @{text map} *}
13737
e564c3d2d174 added a few lemmas
nipkow
parents: 13601
diff changeset
   791
by simp
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   792
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   793
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   794
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   795
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   796
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   797
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   798
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   799
lemma map_eq_Cons_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   800
 "(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
   801
by (cases xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   802
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   803
lemma Cons_eq_map_conv:
14025
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   804
 "(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
   805
by (cases ys) auto
d9b155757dc8 *** empty log message ***
nipkow
parents: 13913
diff changeset
   806
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   807
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   808
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   809
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
   810
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   811
lemma ex_map_conv:
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   812
  "(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
   813
by(induct ys, auto simp add: Cons_eq_map_conv)
14111
993471c762b8 Some new thm (ex_map_conv?)
nipkow
parents: 14099
diff changeset
   814
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   815
lemma map_eq_imp_length_eq:
35510
64d2d54cbf03 Slightly generalised a theorem
paulson
parents: 35296
diff changeset
   816
  assumes "map f xs = map g ys"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   817
  shows "length xs = length ys"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   818
using assms proof (induct ys arbitrary: xs)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   819
  case Nil then show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   820
next
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   821
  case (Cons y ys) then obtain z zs where xs: "xs = z # zs" by auto
35510
64d2d54cbf03 Slightly generalised a theorem
paulson
parents: 35296
diff changeset
   822
  from Cons xs have "map f zs = map g ys" by simp
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   823
  moreover with Cons have "length zs = length ys" by blast
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   824
  with xs show ?case by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   825
qed
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   826
  
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   827
lemma map_inj_on:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   828
 "[| 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
   829
  ==> xs = ys"
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   830
apply(frule map_eq_imp_length_eq)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   831
apply(rotate_tac -1)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   832
apply(induct rule:list_induct2)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   833
 apply simp
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   834
apply(simp)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   835
apply (blast intro:sym)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   836
done
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   837
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   838
lemma inj_on_map_eq_map:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   839
 "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
   840
by(blast dest:map_inj_on)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   841
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   842
lemma map_injective:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   843
 "map f xs = map f ys ==> inj f ==> xs = ys"
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
   844
by (induct ys arbitrary: xs) (auto dest!:injD)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   845
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   846
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
   847
by(blast dest:map_injective)
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   848
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   849
lemma inj_mapI: "inj f ==> inj (map f)"
17589
58eeffd73be1 renamed rules to iprover
nipkow
parents: 17501
diff changeset
   850
by (iprover dest: map_injective injD intro: inj_onI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   851
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   852
lemma inj_mapD: "inj (map f) ==> inj f"
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   853
apply (unfold inj_on_def, clarify)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   854
apply (erule_tac x = "[x]" in ballE)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   855
 apply (erule_tac x = "[y]" in ballE, simp, blast)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   856
apply blast
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   857
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   858
14339
ec575b7bde7a *** empty log message ***
nipkow
parents: 14338
diff changeset
   859
lemma inj_map[iff]: "inj (map f) = inj f"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   860
by (blast dest: inj_mapD intro: inj_mapI)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   861
15303
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   862
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A"
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   863
apply(rule inj_onI)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   864
apply(erule map_inj_on)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   865
apply(blast intro:inj_onI dest:inj_onD)
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   866
done
eedbb8d22ca2 added lemmas
nipkow
parents: 15302
diff changeset
   867
14343
6bc647f472b9 map_idI
kleing
parents: 14339
diff changeset
   868
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
   869
by (induct xs, auto)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   870
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   871
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
   872
by (induct xs) auto
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
   873
15110
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   874
lemma map_fst_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   875
  "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
   876
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   877
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   878
lemma map_snd_zip[simp]:
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   879
  "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
   880
by (induct rule:list_induct2, simp_all)
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   881
78b5636eabc7 Added a number of new thms and the new function remove1
nipkow
parents: 15072
diff changeset
   882
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   883
subsubsection {* @{text rev} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   884
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   885
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   886
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   887
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   888
lemma rev_rev_ident [simp]: "rev (rev xs) = xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   889
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   890
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   891
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   892
by auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   893
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   894
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   895
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   896
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   897
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   898
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   899
15870
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   900
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   901
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   902
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   903
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])"
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   904
by (cases xs) auto
4320bce5873f more on rev
kleing
parents: 15868
diff changeset
   905
21061
580dfc999ef6 added normal post setup; cleaned up "execution" constants
haftmann
parents: 21046
diff changeset
   906
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
   907
apply (induct xs arbitrary: ys, force)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
   908
apply (case_tac ys, simp, force)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   909
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   910
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   911
lemma inj_on_rev[iff]: "inj_on rev A"
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   912
by(simp add:inj_on_def)
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15426
diff changeset
   913
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   914
lemma rev_induct [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   915
  "[| 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
   916
apply(simplesubst rev_rev_ident[symmetric])
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   917
apply(rule_tac list = "rev xs" in list.induct, simp_all)
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   918
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   919
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   920
lemma rev_exhaust [case_names Nil snoc]:
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   921
  "(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   922
by (induct xs rule: rev_induct) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   923
13366
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   924
lemmas rev_cases = rev_exhaust
114b7c14084a moved stuff from Main.thy;
wenzelm
parents: 13187
diff changeset
   925
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   926
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   927
by(rule rev_cases[of xs]) auto
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
   928
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   929
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
   930
subsubsection {* @{text set} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   931
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   932
lemma finite_set [iff]: "finite (set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   933
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   934
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   935
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   936
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   937
17830
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   938
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs"
695a2365d32f added hd lemma
nipkow
parents: 17765
diff changeset
   939
by(cases xs) auto
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   940
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   941
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   942
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   943
14099
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   944
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
   945
by auto
55d244f3c86d added fold_red, o2l, postfix, some thms
oheimb
parents: 14050
diff changeset
   946
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   947
lemma set_empty [iff]: "(set xs = {}) = (xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   948
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   949
15245
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   950
lemma set_empty2[iff]: "({} = set xs) = (xs = [])"
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   951
by(induct xs) auto
5a21d9a8f14b Added a few lemmas
nipkow
parents: 15236
diff changeset
   952
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   953
lemma set_rev [simp]: "set (rev xs) = set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   954
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   955
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   956
lemma set_map [simp]: "set (map f xs) = f`(set xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
   957
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   958
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   959
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
   960
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   961
32417
e87d9c78910c tuned code generation for lists
nipkow
parents: 32415
diff changeset
   962
lemma set_upt [simp]: "set[i..<j] = {i..<j}"
e87d9c78910c tuned code generation for lists
nipkow
parents: 32415
diff changeset
   963
by (induct j) (simp_all add: atLeastLessThanSuc)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
   964
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
   965
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   966
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
   967
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   968
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   969
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   970
  case Cons thus ?case by (auto intro: Cons_eq_appendI)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   971
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   972
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
   973
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
   974
  by (auto elim: split_list)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   975
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   976
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
   977
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   978
  case Nil thus ?case by simp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   979
next
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   980
  case (Cons a xs)
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   981
  show ?case
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   982
  proof cases
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
   983
    assume "x = a" thus ?case using Cons by fastsimp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
   984
  next
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   985
    assume "x \<noteq> a" thus ?case using Cons by(fastsimp intro!: Cons_eq_appendI)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   986
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   987
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   988
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   989
lemma in_set_conv_decomp_first:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   990
  "(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
   991
  by (auto dest!: split_list_first)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   992
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   993
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
   994
proof (induct xs rule:rev_induct)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   995
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   996
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   997
  case (snoc a xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   998
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
   999
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1000
    assume "x = a" thus ?case using snoc by simp (metis ex_in_conv set_empty2)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1001
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1002
    assume "x \<noteq> a" thus ?case using snoc by fastsimp
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1003
  qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1004
qed
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1005
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1006
lemma in_set_conv_decomp_last:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1007
  "(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
  1008
  by (auto dest!: split_list_last)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1009
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1010
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
  1011
proof (induct xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1012
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1013
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1014
  case Cons thus ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1015
    by(simp add:Bex_def)(metis append_Cons append.simps(1))
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1016
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1017
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1018
lemma split_list_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1019
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1020
  obtains ys x zs where "xs = ys @ x # zs" and "P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1021
using split_list_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1022
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1023
lemma split_list_first_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1024
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1025
   \<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
  1026
proof (induct xs)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1027
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1028
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1029
  case (Cons x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1030
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1031
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1032
    assume "P x"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1033
    thus ?thesis by simp
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1034
      (metis Un_upper1 contra_subsetD in_set_conv_decomp_first self_append_conv2 set_append)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1035
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1036
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1037
    hence "\<exists>x\<in>set xs. P x" using Cons(2) by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1038
    thus ?thesis using `\<not> P x` Cons(1) by (metis append_Cons set_ConsD)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1039
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1040
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1041
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1042
lemma split_list_first_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1043
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1044
  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
  1045
using split_list_first_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1046
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1047
lemma split_list_first_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1048
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1049
   (\<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
  1050
by (rule, erule split_list_first_prop) auto
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1051
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1052
lemma split_list_last_prop:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1053
  "\<exists>x \<in> set xs. P x \<Longrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1054
   \<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
  1055
proof(induct xs rule:rev_induct)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1056
  case Nil thus ?case by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1057
next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1058
  case (snoc x xs)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1059
  show ?case
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1060
  proof cases
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1061
    assume "P x" thus ?thesis by (metis emptyE set_empty)
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1062
  next
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1063
    assume "\<not> P x"
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1064
    hence "\<exists>x\<in>set xs. P x" using snoc(2) by simp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1065
    thus ?thesis using `\<not> P x` snoc(1) by fastsimp
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1066
  qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1067
qed
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1068
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1069
lemma split_list_last_propE:
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1070
  assumes "\<exists>x \<in> set xs. P x"
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1071
  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
  1072
using split_list_last_prop [OF assms] by blast
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1074
lemma split_list_last_prop_iff:
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1075
  "(\<exists>x \<in> set xs. P x) \<longleftrightarrow>
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1076
   (\<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
  1077
by (metis split_list_last_prop [where P=P] in_set_conv_decomp)
26073
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1078
0e70d3bd2eb4 more lemmas
nipkow
parents: 25966
diff changeset
  1079
lemma finite_list: "finite A ==> EX xs. set xs = A"
26734
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1080
  by (erule finite_induct)
a92057c1ee21 dropped some metis calls
haftmann
parents: 26584
diff changeset
  1081
    (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
  1082
14388
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1083
lemma card_length: "card (set xs) \<le> length xs"
04f04408b99b lemmas about card (set xs)
kleing
parents: 14343
diff changeset
  1084
by (induct xs) (auto simp add: card_insert_if)
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1085
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1086
lemma set_minus_filter_out:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1087
  "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
  1088
  by (induct xs) auto
15168
33a08cfc3ae5 new functions for sets of lists
paulson
parents: 15140
diff changeset
  1089
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
  1090
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1091
subsubsection {* @{text filter} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1092
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1093
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1094
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1095
15305
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1096
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)"
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1097
by (induct xs) simp_all
0bd9eedaa301 added lemmas
nipkow
parents: 15304
diff changeset
  1098
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1099
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
  1100
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1101
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1102
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1103
by (induct xs) (auto simp add: le_SucI)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1104
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1105
lemma sum_length_filter_compl:
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1106
  "length(filter P xs) + length(filter (%x. ~P x) xs) = length xs"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1107
by(induct xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1108
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1109
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
  1110
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1111
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1112
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
  1113
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1114
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1115
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
  1116
by (induct xs) simp_all
16998
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1117
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1118
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)"
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1119
apply (induct xs)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1120
 apply auto
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1121
apply(cut_tac P=P and xs=xs in length_filter_le)
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1122
apply simp
e0050191e2d1 Added filter lemma
nipkow
parents: 16973
diff changeset
  1123
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1124
16965
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1125
lemma filter_map:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1126
  "filter P (map f xs) = map f (filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1127
by (induct xs) simp_all
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1128
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1129
lemma length_filter_map[simp]:
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1130
  "length (filter P (map f xs)) = length(filter (P o f) xs)"
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1131
by (simp add:filter_map)
46697fa536ab added map_filter lemmas
nipkow
parents: 16770
diff changeset
  1132
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1133
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1134
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1135
15246
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1136
lemma length_filter_less:
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1137
  "\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs"
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1138
proof (induct xs)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1139
  case Nil thus ?case by simp
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1140
next
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1141
  case (Cons x xs) thus ?case
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1142
    apply (auto split:split_if_asm)
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1143
    using length_filter_le[of P xs] apply arith
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1144
  done
0984a2c2868b added and renamed
nipkow
parents: 15245
diff changeset
  1145
qed
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1146
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1147
lemma length_filter_conv_card:
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1148
 "length(filter p xs) = card{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1149
proof (induct xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1150
  case Nil thus ?case by simp
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1151
next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1152
  case (Cons x xs)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1153
  let ?S = "{i. i < length xs & p(xs!i)}"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1154
  have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1155
  show ?case (is "?l = card ?S'")
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1156
  proof (cases)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1157
    assume "p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1158
    hence eq: "?S' = insert 0 (Suc ` ?S)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1159
      by(auto simp: image_def split:nat.split dest:gr0_implies_Suc)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1160
    have "length (filter p (x # xs)) = Suc(card ?S)"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1161
      using Cons `p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1162
    also have "\<dots> = Suc(card(Suc ` ?S))" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1163
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1164
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1165
      by (simp add:card_insert_if) (simp add:image_def)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1166
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1167
  next
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1168
    assume "\<not> p x"
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1169
    hence eq: "?S' = Suc ` ?S"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1170
      by(auto simp add: image_def split:nat.split elim:lessE)
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1171
    have "length (filter p (x # xs)) = card ?S"
23388
77645da0db85 tuned proofs: avoid implicit prems;
wenzelm
parents: 23279
diff changeset
  1172
      using Cons `\<not> p x` by simp
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1173
    also have "\<dots> = card(Suc ` ?S)" using fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1174
      by (simp add: card_image inj_Suc)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1175
    also have "\<dots> = card ?S'" using eq fin
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1176
      by (simp add:card_insert_if)
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1177
    finally show ?thesis .
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1178
  qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1179
qed
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1180
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1181
lemma Cons_eq_filterD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1182
 "x#xs = filter P ys \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1183
  \<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
  1184
  (is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs")
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1185
proof(induct ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1186
  case Nil thus ?case by simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1187
next
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1188
  case (Cons y ys)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1189
  show ?case (is "\<exists>x. ?Q x")
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1190
  proof cases
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1191
    assume Py: "P y"
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1192
    show ?thesis
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1193
    proof cases
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1194
      assume "x = y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1195
      with Py Cons.prems have "?Q []" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1196
      then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1197
    next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1198
      assume "x \<noteq> y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1199
      with Py Cons.prems show ?thesis by simp
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1200
    qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1201
  next
25221
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1202
    assume "\<not> P y"
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1203
    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
  1204
    then have "?Q (y#us)" by simp
5ded95dda5df append/member: more light-weight way to declare authentic syntax;
wenzelm
parents: 25215
diff changeset
  1205
    then show ?thesis ..
17629
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1206
  qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1207
qed
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1208
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1209
lemma filter_eq_ConsD:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1210
 "filter P ys = x#xs \<Longrightarrow>
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1211
  \<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
  1212
by(rule Cons_eq_filterD) simp
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1213
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1214
lemma filter_eq_Cons_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1215
 "(filter P ys = x#xs) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1216
  (\<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
  1217
by(auto dest:filter_eq_ConsD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1218
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1219
lemma Cons_eq_filter_iff:
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1220
 "(x#xs = filter P ys) =
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1221
  (\<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
  1222
by(auto dest:Cons_eq_filterD)
f8ea8068c6d9 a few new filter lemmas
nipkow
parents: 17589
diff changeset
  1223
19770
be5c23ebe1eb HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents: 19623
diff changeset
  1224
lemma filter_cong[fundef_cong, recdef_cong]:
17501
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1225
 "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
  1226
apply simp
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1227
apply(erule thin_rl)
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1228
by (induct ys) simp_all
acbebb72e85a added a number of lemmas
nipkow
parents: 17090
diff changeset
  1229
15281
bd4611956c7b More lemmas
nipkow
parents: 15251
diff changeset
  1230
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1231
subsubsection {* List partitioning *}
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1232
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1233
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
  1234
  "partition P [] = ([], [])"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1235
  | "partition P (x # xs) = 
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1236
      (let (yes, no) = partition P xs
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1237
      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
  1238
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1239
lemma partition_filter1:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1240
    "fst (partition P xs) = filter P xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1241
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1242
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1243
lemma partition_filter2:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1244
    "snd (partition P xs) = filter (Not o P) xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1245
by (induct xs) (auto simp add: Let_def split_def)
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1246
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1247
lemma partition_P:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1248
  assumes "partition P xs = (yes, no)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1249
  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
  1250
proof -
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1251
  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
  1252
    by simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1253
  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
  1254
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1255
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1256
lemma partition_set:
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1257
  assumes "partition P xs = (yes, no)"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1258
  shows "set yes \<union> set no = set xs"
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1259
proof -
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1260
  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
  1261
    by simp_all
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1262
  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
  1263
qed
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1264
33639
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1265
lemma partition_filter_conv[simp]:
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1266
  "partition f xs = (filter f xs,filter (Not o f) xs)"
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1267
unfolding partition_filter2[symmetric]
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1268
unfolding partition_filter1[symmetric] by simp
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1269
603320b93668 New list theorems; added map_map to simpset, this is the prefered direction; allow sorting by a key
hoelzl
parents: 33593
diff changeset
  1270
declare partition.simps[simp del]
26442
57fb6a8b099e restructuring; explicit case names for rule list_induct2
haftmann
parents: 26300
diff changeset
  1271
35115
446c5063e4fd modernized translations;
wenzelm
parents: 35028
diff changeset
  1272
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1273
subsubsection {* @{text concat} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1274
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1275
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1276
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1277
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1278
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1279
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1280
18447
da548623916a removed or modified some instances of [iff]
paulson
parents: 18423
diff changeset
  1281
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1282
by (induct xss) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1283
24308
700e745994c1 removed set_concat_map and improved set_concat
nipkow
parents: 24286
diff changeset
  1284
lemma set_concat [simp]: "set (concat xs) = (UN x:set xs. set x)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1285
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1286
24476
f7ad9fbbeeaa turned list comprehension translations into ML to optimize base case
nipkow
parents: 24471
diff changeset
  1287
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
  1288
by (induct xs) auto
0dd8782fb02d Final mods for list comprehension
nipkow
parents: 24335
diff changeset
  1289
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1290
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1291
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1292
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1293
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1294
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1295
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1296
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1297
by (induct xs) auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1298
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1299
15392
290bc97038c7 First step in reorganizing Finite_Set
nipkow
parents: 15307
diff changeset
  1300
subsubsection {* @{text nth} *}
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1301
29827
c82b3e8a4daf code theorems for nth, list_update
haftmann
parents: 29822
diff changeset
  1302
lemma nth_Cons_0 [simp, code]: "(x # xs)!0 = x"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1303
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1304
29827
c82b3e8a4daf code theorems for nth, list_update
haftmann
parents: 29822
diff changeset
  1305
lemma nth_Cons_Suc [simp, code]: "(x # xs)!(Suc n) = xs!n"
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1306
by auto
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1307
13142
1ebd8ed5a1a0 tuned document;
wenzelm
parents: 13124
diff changeset
  1308
declare nth.simps [simp del]
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1309
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1310
lemma nth_append:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1311
  "(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
  1312
apply (induct xs arbitrary: n, simp)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1313
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1314
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1315
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1316
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
  1317
by (induct xs) auto
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1318
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1319
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
  1320
by (induct xs) auto
14402
4201e1916482 moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents: 14395
diff changeset
  1321
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1322
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
  1323
apply (induct xs arbitrary: n, simp)
14208
144f45277d5a misc tidying
paulson
parents: 14187
diff changeset
  1324
apply (case_tac n, auto)
13145
59bc43b51aa2 *** empty log message ***
nipkow
parents: 13142
diff changeset
  1325
done
13114
f2b00262bdfc converted;
wenzelm
parents: 12887
diff changeset
  1326
18423
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1327
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0"
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1328
by(cases xs) simp_all
d7859164447f new lemmas
nipkow
parents: 18336
diff changeset
  1329
18049
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1330
156bba334c12 A few new lemmas
nipkow
parents: 17956
diff changeset
  1331
lemma list_eq_iff_nth_eq:
24526
7fa202789bf6 tuned lemma; replaced !! by arbitrary
nipkow
parents: 24476
diff changeset
  1332
 "(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
  1333
apply(induct xs arbitrary: ys)
24632
779fc4fcbf8b metis now available in PreList
paulson
parents: 24617
diff changeset
  1334
 apply force
18049
156bba334c12 A few new lemmas
nipkow
parent