author | kleing |
Thu, 28 Apr 2005 12:04:34 +0200 | |
changeset 15870 | 4320bce5873f |
parent 15868 | 9634b3f9d910 |
child 16397 | c047008f88d4 |
permissions | -rw-r--r-- |
13462 | 1 |
(* Title: HOL/List.thy |
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ID: $Id$ |
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Author: Tobias Nipkow |
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*) |
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header {* The datatype of finite lists *} |
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theory List |
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imports PreList |
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begin |
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|
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datatype 'a list = |
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Nil ("[]") |
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| Cons 'a "'a list" (infixr "#" 65) |
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subsection{*Basic list processing functions*} |
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|
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consts |
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"@" :: "'a list => 'a list => 'a list" (infixr 65) |
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filter:: "('a => bool) => 'a list => 'a list" |
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concat:: "'a list list => 'a list" |
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foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" |
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foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" |
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hd:: "'a list => 'a" |
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tl:: "'a list => 'a list" |
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last:: "'a list => 'a" |
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butlast :: "'a list => 'a list" |
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set :: "'a list => 'a set" |
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list_all:: "('a => bool) => ('a list => bool)" |
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" |
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list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" |
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map :: "('a=>'b) => ('a list => 'b list)" |
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mem :: "'a => 'a list => bool" (infixl 55) |
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nth :: "'a list => nat => 'a" (infixl "!" 100) |
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list_update :: "'a list => nat => 'a => 'a list" |
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take:: "nat => 'a list => 'a list" |
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drop:: "nat => 'a list => 'a list" |
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takeWhile :: "('a => bool) => 'a list => 'a list" |
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dropWhile :: "('a => bool) => 'a list => 'a list" |
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rev :: "'a list => 'a list" |
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zip :: "'a list => 'b list => ('a * 'b) list" |
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upt :: "nat => nat => nat list" ("(1[_..</_'])") |
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remdups :: "'a list => 'a list" |
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remove1 :: "'a => 'a list => 'a list" |
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null:: "'a list => bool" |
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"distinct":: "'a list => bool" |
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replicate :: "nat => 'a => 'a list" |
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rotate1 :: "'a list \<Rightarrow> 'a list" |
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rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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sublist :: "'a list => nat set => 'a list" |
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||
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nonterminals lupdbinds lupdbind |
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syntax |
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-- {* list Enumeration *} |
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"@list" :: "args => 'a list" ("[(_)]") |
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-- {* Special syntax for filter *} |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") |
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-- {* list update *} |
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"_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") |
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"" :: "lupdbind => lupdbinds" ("_") |
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"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") |
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"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) |
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upto:: "nat => nat => nat list" ("(1[_../_])") |
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translations |
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"[x, xs]" == "x#[xs]" |
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"[x]" == "x#[]" |
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"[x:xs . P]"== "filter (%x. P) xs" |
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"_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" |
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"xs[i:=x]" == "list_update xs i x" |
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"[i..j]" == "[i..<(Suc j)]" |
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syntax (xsymbols) |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") |
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syntax (HTML output) |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") |
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85 |
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86 |
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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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syntax length :: "'a list => nat" |
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translations "length" => "size :: _ list => nat" |
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typed_print_translation {* |
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let |
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fun size_tr' _ (Type ("fun", (Type ("list", _) :: _))) [t] = |
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Syntax.const "length" $ t |
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| size_tr' _ _ _ = raise Match; |
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in [("size", size_tr')] end |
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*} |
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101 |
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primrec |
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"hd(x#xs) = x" |
105 |
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primrec |
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"tl([]) = []" |
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"tl(x#xs) = xs" |
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primrec |
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"null([]) = True" |
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"null(x#xs) = False" |
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primrec |
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"last(x#xs) = (if xs=[] then x else last xs)" |
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primrec |
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"butlast []= []" |
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"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" |
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primrec |
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"x mem [] = False" |
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"x mem (y#ys) = (if y=x then True else x mem ys)" |
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primrec |
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"set [] = {}" |
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"set (x#xs) = insert x (set xs)" |
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primrec |
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list_all_Nil:"list_all P [] = True" |
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list_all_Cons: "list_all P (x#xs) = (P(x) \<and> list_all P xs)" |
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primrec |
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"list_ex P [] = False" |
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"list_ex P (x#xs) = (P x \<or> list_ex P xs)" |
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primrec |
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"map f [] = []" |
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"map f (x#xs) = f(x)#map f xs" |
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primrec |
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append_Nil:"[]@ys = ys" |
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append_Cons: "(x#xs)@ys = x#(xs@ys)" |
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primrec |
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"rev([]) = []" |
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"rev(x#xs) = rev(xs) @ [x]" |
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primrec |
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"filter P [] = []" |
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" |
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primrec |
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foldl_Nil:"foldl f a [] = a" |
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" |
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primrec |
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"foldr f [] a = a" |
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"foldr f (x#xs) a = f x (foldr f xs a)" |
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primrec |
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"concat([]) = []" |
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"concat(x#xs) = x @ concat(xs)" |
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primrec |
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drop_Nil:"drop n [] = []" |
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drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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take_Nil:"take n [] = []" |
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take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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"[][i:=v] = []" |
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"(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" |
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primrec |
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"takeWhile P [] = []" |
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" |
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primrec |
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"dropWhile P [] = []" |
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" |
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primrec |
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"zip xs [] = []" |
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zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} |
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primrec |
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upt_0: "[i..<0] = []" |
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upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" |
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primrec |
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"distinct [] = True" |
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)" |
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primrec |
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"remdups [] = []" |
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" |
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primrec |
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"remove1 x [] = []" |
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"remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" |
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primrec |
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replicate_0: "replicate 0 x = []" |
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replicate_Suc: "replicate (Suc n) x = x # replicate n x" |
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defs |
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rotate1_def: "rotate1 xs == (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])" |
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rotate_def: "rotate n == rotate1 ^ n" |
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list_all2_def: |
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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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sublist_def: |
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"sublist xs A == map fst (filter (%p. snd p : A) (zip xs [0..<size xs]))" |
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lemma not_Cons_self [simp]: "xs \<noteq> x # xs" |
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by (induct xs) auto |
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lemmas not_Cons_self2 [simp] = not_Cons_self [symmetric] |
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lemma neq_Nil_conv: "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)" |
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by (induct xs) auto |
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|
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lemma length_induct: |
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"(!!xs. \<forall>ys. length ys < length xs --> P ys ==> P xs) ==> P xs" |
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by (rule measure_induct [of length]) rules |
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subsubsection {* @{text length} *} |
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text {* |
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Needs to come before @{text "@"} because of theorem @{text |
249 |
append_eq_append_conv}. |
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*} |
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" |
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by (induct xs) auto |
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lemma length_map [simp]: "length (map f xs) = length xs" |
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by (induct xs) auto |
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lemma length_rev [simp]: "length (rev xs) = length xs" |
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by (induct xs) auto |
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lemma length_tl [simp]: "length (tl xs) = length xs - 1" |
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by (cases xs) auto |
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" |
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by (induct xs) auto |
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" |
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by (induct xs) auto |
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lemma length_Suc_conv: |
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" |
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by (induct xs) auto |
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lemma Suc_length_conv: |
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" |
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apply (induct xs, simp, simp) |
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apply blast |
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done |
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lemma impossible_Cons [rule_format]: |
281 |
"length xs <= length ys --> xs = x # ys = False" |
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apply (induct xs, auto) |
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done |
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lemma list_induct2[consumes 1]: "\<And>ys. |
286 |
\<lbrakk> length xs = length ys; |
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P [] []; |
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\<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> |
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\<Longrightarrow> P xs ys" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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293 |
apply simp |
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apply(simp) |
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295 |
done |
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subsubsection {* @{text "@"} -- append *} |
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" |
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by (induct xs) auto |
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lemma append_Nil2 [simp]: "xs @ [] = xs" |
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by (induct xs) auto |
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|
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" |
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by (induct xs) auto |
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|
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" |
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by (induct xs) auto |
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|
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" |
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by (induct xs) auto |
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|
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lemma self_append_conv [iff]: "(xs = xs @ ys) = (ys = [])" |
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by (induct xs) auto |
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|
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lemma append_eq_append_conv [simp]: |
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"!!ys. length xs = length ys \<or> length us = length vs |
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==> (xs@us = ys@vs) = (xs=ys \<and> us=vs)" |
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apply (induct xs) |
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apply (case_tac ys, simp, force) |
322 |
apply (case_tac ys, force, simp) |
|
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done |
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lemma append_eq_append_conv2: "!!ys zs ts. |
326 |
(xs @ ys = zs @ ts) = |
|
327 |
(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" |
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328 |
apply (induct xs) |
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329 |
apply fastsimp |
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330 |
apply(case_tac zs) |
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331 |
apply simp |
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332 |
apply fastsimp |
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333 |
done |
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334 |
||
13142 | 335 |
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" |
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by simp |
13142 | 337 |
|
338 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" |
|
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by simp |
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|
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" |
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by simp |
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|
13142 | 344 |
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" |
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using append_same_eq [of _ _ "[]"] by auto |
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|
13142 | 347 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" |
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using append_same_eq [of "[]"] by auto |
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|
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" |
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by (induct xs) auto |
13114 | 352 |
|
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lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" |
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by (induct xs) auto |
13114 | 355 |
|
13142 | 356 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" |
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by (simp add: hd_append split: list.split) |
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|
13142 | 359 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" |
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by (simp split: list.split) |
13114 | 361 |
|
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lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" |
13145 | 363 |
by (simp add: tl_append split: list.split) |
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|
365 |
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14300 | 366 |
lemma Cons_eq_append_conv: "x#xs = ys@zs = |
367 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" |
|
368 |
by(cases ys) auto |
|
369 |
||
15281 | 370 |
lemma append_eq_Cons_conv: "(ys@zs = x#xs) = |
371 |
(ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" |
|
372 |
by(cases ys) auto |
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373 |
||
14300 | 374 |
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13142 | 375 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *} |
13114 | 376 |
|
377 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" |
|
13145 | 378 |
by simp |
13114 | 379 |
|
13142 | 380 |
lemma Cons_eq_appendI: |
13145 | 381 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" |
382 |
by (drule sym) simp |
|
13114 | 383 |
|
13142 | 384 |
lemma append_eq_appendI: |
13145 | 385 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" |
386 |
by (drule sym) simp |
|
13114 | 387 |
|
388 |
||
13142 | 389 |
text {* |
13145 | 390 |
Simplification procedure for all list equalities. |
391 |
Currently only tries to rearrange @{text "@"} to see if |
|
392 |
- both lists end in a singleton list, |
|
393 |
- or both lists end in the same list. |
|
13142 | 394 |
*} |
395 |
||
396 |
ML_setup {* |
|
3507 | 397 |
local |
398 |
||
13122 | 399 |
val append_assoc = thm "append_assoc"; |
400 |
val append_Nil = thm "append_Nil"; |
|
401 |
val append_Cons = thm "append_Cons"; |
|
402 |
val append1_eq_conv = thm "append1_eq_conv"; |
|
403 |
val append_same_eq = thm "append_same_eq"; |
|
404 |
||
13114 | 405 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = |
13462 | 406 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs) |
407 |
| last (Const("List.op @",_) $ _ $ ys) = last ys |
|
408 |
| last t = t; |
|
13114 | 409 |
|
410 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true |
|
13462 | 411 |
| list1 _ = false; |
13114 | 412 |
|
413 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = |
|
13462 | 414 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) |
415 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys |
|
416 |
| butlast xs = Const("List.list.Nil",fastype_of xs); |
|
13114 | 417 |
|
418 |
val rearr_tac = |
|
13462 | 419 |
simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); |
13114 | 420 |
|
421 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
|
13462 | 422 |
let |
423 |
val lastl = last lhs and lastr = last rhs; |
|
424 |
fun rearr conv = |
|
425 |
let |
|
426 |
val lhs1 = butlast lhs and rhs1 = butlast rhs; |
|
427 |
val Type(_,listT::_) = eqT |
|
428 |
val appT = [listT,listT] ---> listT |
|
429 |
val app = Const("List.op @",appT) |
|
430 |
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
|
431 |
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); |
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset
|
432 |
val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1)); |
15531 | 433 |
in SOME ((conv RS (thm RS trans)) RS eq_reflection) end; |
13114 | 434 |
|
13462 | 435 |
in |
436 |
if list1 lastl andalso list1 lastr then rearr append1_eq_conv |
|
437 |
else if lastl aconv lastr then rearr append_same_eq |
|
15531 | 438 |
else NONE |
13462 | 439 |
end; |
440 |
||
13114 | 441 |
in |
13462 | 442 |
|
443 |
val list_eq_simproc = |
|
444 |
Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; |
|
445 |
||
13114 | 446 |
end; |
447 |
||
448 |
Addsimprocs [list_eq_simproc]; |
|
449 |
*} |
|
450 |
||
451 |
||
15392 | 452 |
subsubsection {* @{text map} *} |
13114 | 453 |
|
13142 | 454 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" |
13145 | 455 |
by (induct xs) simp_all |
13114 | 456 |
|
13142 | 457 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" |
13145 | 458 |
by (rule ext, induct_tac xs) auto |
13114 | 459 |
|
13142 | 460 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" |
13145 | 461 |
by (induct xs) auto |
13114 | 462 |
|
13142 | 463 |
lemma map_compose: "map (f o g) xs = map f (map g xs)" |
13145 | 464 |
by (induct xs) (auto simp add: o_def) |
13114 | 465 |
|
13142 | 466 |
lemma rev_map: "rev (map f xs) = map f (rev xs)" |
13145 | 467 |
by (induct xs) auto |
13114 | 468 |
|
13737 | 469 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" |
470 |
by (induct xs) auto |
|
471 |
||
13366 | 472 |
lemma map_cong [recdef_cong]: |
13145 | 473 |
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" |
474 |
-- {* a congruence rule for @{text map} *} |
|
13737 | 475 |
by simp |
13114 | 476 |
|
13142 | 477 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" |
13145 | 478 |
by (cases xs) auto |
13114 | 479 |
|
13142 | 480 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" |
13145 | 481 |
by (cases xs) auto |
13114 | 482 |
|
14025 | 483 |
lemma map_eq_Cons_conv[iff]: |
484 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" |
|
13145 | 485 |
by (cases xs) auto |
13114 | 486 |
|
14025 | 487 |
lemma Cons_eq_map_conv[iff]: |
488 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" |
|
489 |
by (cases ys) auto |
|
490 |
||
14111 | 491 |
lemma ex_map_conv: |
492 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" |
|
493 |
by(induct ys, auto) |
|
494 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
495 |
lemma map_eq_imp_length_eq: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
496 |
"!!xs. map f xs = map f ys ==> length xs = length ys" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
497 |
apply (induct ys) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
498 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
499 |
apply(simp (no_asm_use)) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
500 |
apply clarify |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
501 |
apply(simp (no_asm_use)) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
502 |
apply fast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
503 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
504 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
505 |
lemma map_inj_on: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
506 |
"[| 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
|
507 |
==> xs = ys" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
508 |
apply(frule map_eq_imp_length_eq) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
509 |
apply(rotate_tac -1) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
510 |
apply(induct rule:list_induct2) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
511 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
512 |
apply(simp) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
513 |
apply (blast intro:sym) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
514 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
515 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
516 |
lemma inj_on_map_eq_map: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
517 |
"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
|
518 |
by(blast dest:map_inj_on) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
519 |
|
13114 | 520 |
lemma map_injective: |
14338 | 521 |
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys" |
522 |
by (induct ys) (auto dest!:injD) |
|
13114 | 523 |
|
14339 | 524 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" |
525 |
by(blast dest:map_injective) |
|
526 |
||
13114 | 527 |
lemma inj_mapI: "inj f ==> inj (map f)" |
13585 | 528 |
by (rules dest: map_injective injD intro: inj_onI) |
13114 | 529 |
|
530 |
lemma inj_mapD: "inj (map f) ==> inj f" |
|
14208 | 531 |
apply (unfold inj_on_def, clarify) |
13145 | 532 |
apply (erule_tac x = "[x]" in ballE) |
14208 | 533 |
apply (erule_tac x = "[y]" in ballE, simp, blast) |
13145 | 534 |
apply blast |
535 |
done |
|
13114 | 536 |
|
14339 | 537 |
lemma inj_map[iff]: "inj (map f) = inj f" |
13145 | 538 |
by (blast dest: inj_mapD intro: inj_mapI) |
13114 | 539 |
|
15303 | 540 |
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A" |
541 |
apply(rule inj_onI) |
|
542 |
apply(erule map_inj_on) |
|
543 |
apply(blast intro:inj_onI dest:inj_onD) |
|
544 |
done |
|
545 |
||
14343 | 546 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" |
547 |
by (induct xs, auto) |
|
13114 | 548 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
549 |
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
|
550 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
551 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
552 |
lemma map_fst_zip[simp]: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
553 |
"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
|
554 |
by (induct rule:list_induct2, simp_all) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
555 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
556 |
lemma map_snd_zip[simp]: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
557 |
"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
|
558 |
by (induct rule:list_induct2, simp_all) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
559 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
560 |
|
15392 | 561 |
subsubsection {* @{text rev} *} |
13114 | 562 |
|
13142 | 563 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" |
13145 | 564 |
by (induct xs) auto |
13114 | 565 |
|
13142 | 566 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" |
13145 | 567 |
by (induct xs) auto |
13114 | 568 |
|
15870 | 569 |
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" |
570 |
by auto |
|
571 |
||
13142 | 572 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" |
13145 | 573 |
by (induct xs) auto |
13114 | 574 |
|
13142 | 575 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" |
13145 | 576 |
by (induct xs) auto |
13114 | 577 |
|
15870 | 578 |
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" |
579 |
by (cases xs) auto |
|
580 |
||
581 |
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" |
|
582 |
by (cases xs) auto |
|
583 |
||
13142 | 584 |
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" |
14208 | 585 |
apply (induct xs, force) |
586 |
apply (case_tac ys, simp, force) |
|
13145 | 587 |
done |
13114 | 588 |
|
15439 | 589 |
lemma inj_on_rev[iff]: "inj_on rev A" |
590 |
by(simp add:inj_on_def) |
|
591 |
||
13366 | 592 |
lemma rev_induct [case_names Nil snoc]: |
593 |
"[| 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
|
594 |
apply(simplesubst rev_rev_ident[symmetric]) |
13145 | 595 |
apply(rule_tac list = "rev xs" in list.induct, simp_all) |
596 |
done |
|
13114 | 597 |
|
13145 | 598 |
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" |
13114 | 599 |
|
13366 | 600 |
lemma rev_exhaust [case_names Nil snoc]: |
601 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" |
|
13145 | 602 |
by (induct xs rule: rev_induct) auto |
13114 | 603 |
|
13366 | 604 |
lemmas rev_cases = rev_exhaust |
605 |
||
13114 | 606 |
|
15392 | 607 |
subsubsection {* @{text set} *} |
13114 | 608 |
|
13142 | 609 |
lemma finite_set [iff]: "finite (set xs)" |
13145 | 610 |
by (induct xs) auto |
13114 | 611 |
|
13142 | 612 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" |
13145 | 613 |
by (induct xs) auto |
13114 | 614 |
|
14099 | 615 |
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l" |
14208 | 616 |
by (case_tac l, auto) |
14099 | 617 |
|
13142 | 618 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" |
13145 | 619 |
by auto |
13114 | 620 |
|
14099 | 621 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" |
622 |
by auto |
|
623 |
||
13142 | 624 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" |
13145 | 625 |
by (induct xs) auto |
13114 | 626 |
|
15245 | 627 |
lemma set_empty2[iff]: "({} = set xs) = (xs = [])" |
628 |
by(induct xs) auto |
|
629 |
||
13142 | 630 |
lemma set_rev [simp]: "set (rev xs) = set xs" |
13145 | 631 |
by (induct xs) auto |
13114 | 632 |
|
13142 | 633 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)" |
13145 | 634 |
by (induct xs) auto |
13114 | 635 |
|
13142 | 636 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" |
13145 | 637 |
by (induct xs) auto |
13114 | 638 |
|
15425 | 639 |
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}" |
14208 | 640 |
apply (induct j, simp_all) |
641 |
apply (erule ssubst, auto) |
|
13145 | 642 |
done |
13114 | 643 |
|
13142 | 644 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" |
15113 | 645 |
proof (induct xs) |
646 |
case Nil show ?case by simp |
|
647 |
case (Cons a xs) |
|
648 |
show ?case |
|
649 |
proof |
|
650 |
assume "x \<in> set (a # xs)" |
|
651 |
with prems show "\<exists>ys zs. a # xs = ys @ x # zs" |
|
652 |
by (simp, blast intro: Cons_eq_appendI) |
|
653 |
next |
|
654 |
assume "\<exists>ys zs. a # xs = ys @ x # zs" |
|
655 |
then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast |
|
656 |
show "x \<in> set (a # xs)" |
|
657 |
by (cases ys, auto simp add: eq) |
|
658 |
qed |
|
659 |
qed |
|
13142 | 660 |
|
13508 | 661 |
lemma finite_list: "finite A ==> EX l. set l = A" |
662 |
apply (erule finite_induct, auto) |
|
663 |
apply (rule_tac x="x#l" in exI, auto) |
|
664 |
done |
|
665 |
||
14388 | 666 |
lemma card_length: "card (set xs) \<le> length xs" |
667 |
by (induct xs) (auto simp add: card_insert_if) |
|
13114 | 668 |
|
15168 | 669 |
|
15439 | 670 |
subsubsection {* @{text mem}, @{text list_all} and @{text list_ex} *} |
13114 | 671 |
|
15302 | 672 |
text{* Only use @{text mem} for generating executable code. Otherwise |
15439 | 673 |
use @{prop"x : set xs"} instead --- it is much easier to reason about. |
674 |
The same is true for @{text list_all} and @{text list_ex}: write |
|
675 |
@{text"\<forall>x\<in>set xs"} and @{text"\<exists>x\<in>set xs"} instead because the HOL |
|
676 |
quantifiers are aleady known to the automatic provers. For the purpose |
|
677 |
of generating executable code use the theorems @{text set_mem_eq}, |
|
678 |
@{text list_all_conv} and @{text list_ex_iff} to get rid off or |
|
679 |
introduce the combinators. *} |
|
15302 | 680 |
|
13114 | 681 |
lemma set_mem_eq: "(x mem xs) = (x : set xs)" |
13145 | 682 |
by (induct xs) auto |
13114 | 683 |
|
13142 | 684 |
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" |
13145 | 685 |
by (induct xs) auto |
13114 | 686 |
|
13142 | 687 |
lemma list_all_append [simp]: |
13145 | 688 |
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" |
689 |
by (induct xs) auto |
|
13114 | 690 |
|
15426 | 691 |
lemma list_all_rev [simp]: "list_all P (rev xs) = list_all P xs" |
692 |
by (simp add: list_all_conv) |
|
693 |
||
15439 | 694 |
lemma list_ex_iff: "list_ex P xs = (\<exists>x \<in> set xs. P x)" |
695 |
by (induct xs) simp_all |
|
15426 | 696 |
|
13114 | 697 |
|
15392 | 698 |
subsubsection {* @{text filter} *} |
13114 | 699 |
|
13142 | 700 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" |
13145 | 701 |
by (induct xs) auto |
13114 | 702 |
|
15305 | 703 |
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" |
704 |
by (induct xs) simp_all |
|
705 |
||
13142 | 706 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" |
13145 | 707 |
by (induct xs) auto |
13114 | 708 |
|
13142 | 709 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" |
13145 | 710 |
by (induct xs) auto |
13114 | 711 |
|
13142 | 712 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" |
13145 | 713 |
by (induct xs) auto |
13114 | 714 |
|
15246 | 715 |
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs" |
13145 | 716 |
by (induct xs) (auto simp add: le_SucI) |
13114 | 717 |
|
13142 | 718 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" |
13145 | 719 |
by auto |
13114 | 720 |
|
15246 | 721 |
lemma length_filter_less: |
722 |
"\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs" |
|
723 |
proof (induct xs) |
|
724 |
case Nil thus ?case by simp |
|
725 |
next |
|
726 |
case (Cons x xs) thus ?case |
|
727 |
apply (auto split:split_if_asm) |
|
728 |
using length_filter_le[of P xs] apply arith |
|
729 |
done |
|
730 |
qed |
|
13114 | 731 |
|
15281 | 732 |
lemma length_filter_conv_card: |
733 |
"length(filter p xs) = card{i. i < length xs & p(xs!i)}" |
|
734 |
proof (induct xs) |
|
735 |
case Nil thus ?case by simp |
|
736 |
next |
|
737 |
case (Cons x xs) |
|
738 |
let ?S = "{i. i < length xs & p(xs!i)}" |
|
739 |
have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) |
|
740 |
show ?case (is "?l = card ?S'") |
|
741 |
proof (cases) |
|
742 |
assume "p x" |
|
743 |
hence eq: "?S' = insert 0 (Suc ` ?S)" |
|
744 |
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) |
|
745 |
have "length (filter p (x # xs)) = Suc(card ?S)" |
|
746 |
using Cons by simp |
|
747 |
also have "\<dots> = Suc(card(Suc ` ?S))" using fin |
|
748 |
by (simp add: card_image inj_Suc) |
|
749 |
also have "\<dots> = card ?S'" using eq fin |
|
750 |
by (simp add:card_insert_if) (simp add:image_def) |
|
751 |
finally show ?thesis . |
|
752 |
next |
|
753 |
assume "\<not> p x" |
|
754 |
hence eq: "?S' = Suc ` ?S" |
|
755 |
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) |
|
756 |
have "length (filter p (x # xs)) = card ?S" |
|
757 |
using Cons by simp |
|
758 |
also have "\<dots> = card(Suc ` ?S)" using fin |
|
759 |
by (simp add: card_image inj_Suc) |
|
760 |
also have "\<dots> = card ?S'" using eq fin |
|
761 |
by (simp add:card_insert_if) |
|
762 |
finally show ?thesis . |
|
763 |
qed |
|
764 |
qed |
|
765 |
||
766 |
||
15392 | 767 |
subsubsection {* @{text concat} *} |
13114 | 768 |
|
13142 | 769 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" |
13145 | 770 |
by (induct xs) auto |
13114 | 771 |
|
13142 | 772 |
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 773 |
by (induct xss) auto |
13114 | 774 |
|
13142 | 775 |
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 776 |
by (induct xss) auto |
13114 | 777 |
|
13142 | 778 |
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" |
13145 | 779 |
by (induct xs) auto |
13114 | 780 |
|
13142 | 781 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" |
13145 | 782 |
by (induct xs) auto |
13114 | 783 |
|
13142 | 784 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" |
13145 | 785 |
by (induct xs) auto |
13114 | 786 |
|
13142 | 787 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" |
13145 | 788 |
by (induct xs) auto |
13114 | 789 |
|
790 |
||
15392 | 791 |
subsubsection {* @{text nth} *} |
13114 | 792 |
|
13142 | 793 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" |
13145 | 794 |
by auto |
13114 | 795 |
|
13142 | 796 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" |
13145 | 797 |
by auto |
13114 | 798 |
|
13142 | 799 |
declare nth.simps [simp del] |
13114 | 800 |
|
801 |
lemma nth_append: |
|
13145 | 802 |
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" |
14208 | 803 |
apply (induct "xs", simp) |
804 |
apply (case_tac n, auto) |
|
13145 | 805 |
done |
13114 | 806 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
807 |
lemma nth_append_length [simp]: "(xs @ x # ys) ! length xs = x" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
808 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
809 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
810 |
lemma nth_append_length_plus[simp]: "(xs @ ys) ! (length xs + n) = ys ! n" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
811 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
812 |
|
13142 | 813 |
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" |
14208 | 814 |
apply (induct xs, simp) |
815 |
apply (case_tac n, auto) |
|
13145 | 816 |
done |
13114 | 817 |
|
13142 | 818 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" |
15251 | 819 |
apply (induct xs, simp, simp) |
13145 | 820 |
apply safe |
14208 | 821 |
apply (rule_tac x = 0 in exI, simp) |
822 |
apply (rule_tac x = "Suc i" in exI, simp) |
|
823 |
apply (case_tac i, simp) |
|
13145 | 824 |
apply (rename_tac j) |
14208 | 825 |
apply (rule_tac x = j in exI, simp) |
13145 | 826 |
done |
13114 | 827 |
|
13145 | 828 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" |
829 |
by (auto simp add: set_conv_nth) |
|
13114 | 830 |
|
13142 | 831 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" |
13145 | 832 |
by (auto simp add: set_conv_nth) |
13114 | 833 |
|
834 |
lemma all_nth_imp_all_set: |
|
13145 | 835 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x" |
836 |
by (auto simp add: set_conv_nth) |
|
13114 | 837 |
|
838 |
lemma all_set_conv_all_nth: |
|
13145 | 839 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" |
840 |
by (auto simp add: set_conv_nth) |
|
13114 | 841 |
|
842 |
||
15392 | 843 |
subsubsection {* @{text list_update} *} |
13114 | 844 |
|
13142 | 845 |
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" |
13145 | 846 |
by (induct xs) (auto split: nat.split) |
13114 | 847 |
|
848 |
lemma nth_list_update: |
|
13145 | 849 |
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" |
850 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
|
13114 | 851 |
|
13142 | 852 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" |
13145 | 853 |
by (simp add: nth_list_update) |
13114 | 854 |
|
13142 | 855 |
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" |
13145 | 856 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
13114 | 857 |
|
13142 | 858 |
lemma list_update_overwrite [simp]: |
13145 | 859 |
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" |
860 |
by (induct xs) (auto split: nat.split) |
|
13114 | 861 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
862 |
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" |
14208 | 863 |
apply (induct xs, simp) |
14187 | 864 |
apply(simp split:nat.splits) |
865 |
done |
|
866 |
||
13114 | 867 |
lemma list_update_same_conv: |
13145 | 868 |
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" |
869 |
by (induct xs) (auto split: nat.split) |
|
13114 | 870 |
|
14187 | 871 |
lemma list_update_append1: |
872 |
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" |
|
14208 | 873 |
apply (induct xs, simp) |
14187 | 874 |
apply(simp split:nat.split) |
875 |
done |
|
876 |
||
15868 | 877 |
lemma list_update_append: |
878 |
"!!n. (xs @ ys) [n:= x] = |
|
879 |
(if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" |
|
880 |
by (induct xs) (auto split:nat.splits) |
|
881 |
||
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
882 |
lemma list_update_length [simp]: |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
883 |
"(xs @ x # ys)[length xs := y] = (xs @ y # ys)" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
884 |
by (induct xs, auto) |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
885 |
|
13114 | 886 |
lemma update_zip: |
13145 | 887 |
"!!i xy xs. length xs = length ys ==> |
888 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" |
|
889 |
by (induct ys) (auto, case_tac xs, auto split: nat.split) |
|
13114 | 890 |
|
891 |
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" |
|
13145 | 892 |
by (induct xs) (auto split: nat.split) |
13114 | 893 |
|
894 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" |
|
13145 | 895 |
by (blast dest!: set_update_subset_insert [THEN subsetD]) |
13114 | 896 |
|
15868 | 897 |
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])" |
898 |
by (induct xs) (auto split:nat.splits) |
|
899 |
||
13114 | 900 |
|
15392 | 901 |
subsubsection {* @{text last} and @{text butlast} *} |
13114 | 902 |
|
13142 | 903 |
lemma last_snoc [simp]: "last (xs @ [x]) = x" |
13145 | 904 |
by (induct xs) auto |
13114 | 905 |
|
13142 | 906 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" |
13145 | 907 |
by (induct xs) auto |
13114 | 908 |
|
14302 | 909 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" |
910 |
by(simp add:last.simps) |
|
911 |
||
912 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" |
|
913 |
by(simp add:last.simps) |
|
914 |
||
915 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" |
|
916 |
by (induct xs) (auto) |
|
917 |
||
918 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" |
|
919 |
by(simp add:last_append) |
|
920 |
||
921 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" |
|
922 |
by(simp add:last_append) |
|
923 |
||
924 |
||
13142 | 925 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" |
13145 | 926 |
by (induct xs rule: rev_induct) auto |
13114 | 927 |
|
928 |
lemma butlast_append: |
|
13145 | 929 |
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" |
930 |
by (induct xs) auto |
|
13114 | 931 |
|
13142 | 932 |
lemma append_butlast_last_id [simp]: |
13145 | 933 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" |
934 |
by (induct xs) auto |
|
13114 | 935 |
|
13142 | 936 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" |
13145 | 937 |
by (induct xs) (auto split: split_if_asm) |
13114 | 938 |
|
939 |
lemma in_set_butlast_appendI: |
|
13145 | 940 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" |
941 |
by (auto dest: in_set_butlastD simp add: butlast_append) |
|
13114 | 942 |
|
13142 | 943 |
|
15392 | 944 |
subsubsection {* @{text take} and @{text drop} *} |
13114 | 945 |
|
13142 | 946 |
lemma take_0 [simp]: "take 0 xs = []" |
13145 | 947 |
by (induct xs) auto |
13114 | 948 |
|
13142 | 949 |
lemma drop_0 [simp]: "drop 0 xs = xs" |
13145 | 950 |
by (induct xs) auto |
13114 | 951 |
|
13142 | 952 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" |
13145 | 953 |
by simp |
13114 | 954 |
|
13142 | 955 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" |
13145 | 956 |
by simp |
13114 | 957 |
|
13142 | 958 |
declare take_Cons [simp del] and drop_Cons [simp del] |
13114 | 959 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
960 |
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
961 |
by(clarsimp simp add:neq_Nil_conv) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
962 |
|
14187 | 963 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" |
964 |
by(cases xs, simp_all) |
|
965 |
||
966 |
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" |
|
967 |
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) |
|
968 |
||
969 |
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" |
|
14208 | 970 |
apply (induct xs, simp) |
14187 | 971 |
apply(simp add:drop_Cons nth_Cons split:nat.splits) |
972 |
done |
|
973 |
||
13913 | 974 |
lemma take_Suc_conv_app_nth: |
975 |
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" |
|
14208 | 976 |
apply (induct xs, simp) |
977 |
apply (case_tac i, auto) |
|
13913 | 978 |
done |
979 |
||
14591 | 980 |
lemma drop_Suc_conv_tl: |
981 |
"!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" |
|
982 |
apply (induct xs, simp) |
|
983 |
apply (case_tac i, auto) |
|
984 |
done |
|
985 |
||
13142 | 986 |
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" |
13145 | 987 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 988 |
|
13142 | 989 |
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" |
13145 | 990 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 991 |
|
13142 | 992 |
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" |
13145 | 993 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 994 |
|
13142 | 995 |
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" |
13145 | 996 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 997 |
|
13142 | 998 |
lemma take_append [simp]: |
13145 | 999 |
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" |
1000 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 1001 |
|
13142 | 1002 |
lemma drop_append [simp]: |
13145 | 1003 |
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" |
1004 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 1005 |
|
13142 | 1006 |
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" |
14208 | 1007 |
apply (induct m, auto) |
1008 |
apply (case_tac xs, auto) |
|
15236
f289e8ba2bb3
Proofs needed to be updated because induction now preserves name of
nipkow
parents:
15176
diff
changeset
|
1009 |
apply (case_tac n, auto) |
13145 | 1010 |
done |
13114 | 1011 |
|
13142 | 1012 |
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" |
14208 | 1013 |
apply (induct m, auto) |
1014 |
apply (case_tac xs, auto) |
|
13145 | 1015 |
done |
13114 | 1016 |
|
1017 |
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" |
|
14208 | 1018 |
apply (induct m, auto) |
1019 |
apply (case_tac xs, auto) |
|
13145 | 1020 |
done |
13114 | 1021 |
|
14802 | 1022 |
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)" |
1023 |
apply(induct xs) |
|
1024 |
apply simp |
|
1025 |
apply(simp add: take_Cons drop_Cons split:nat.split) |
|
1026 |
done |
|
1027 |
||
13142 | 1028 |
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" |
14208 | 1029 |
apply (induct n, auto) |
1030 |
apply (case_tac xs, auto) |
|
13145 | 1031 |
done |
13114 | 1032 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1033 |
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1034 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1035 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1036 |
apply(simp add:take_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1037 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1038 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1039 |
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1040 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1041 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1042 |
apply(simp add:drop_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1043 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1044 |
|
13114 | 1045 |
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" |
14208 | 1046 |
apply (induct n, auto) |
1047 |
apply (case_tac xs, auto) |
|
13145 | 1048 |
done |
13114 | 1049 |
|
13142 | 1050 |
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" |
14208 | 1051 |
apply (induct n, auto) |
1052 |
apply (case_tac xs, auto) |
|
13145 | 1053 |
done |
13114 | 1054 |
|
1055 |
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" |
|
14208 | 1056 |
apply (induct xs, auto) |
1057 |
apply (case_tac i, auto) |
|
13145 | 1058 |
done |
13114 | 1059 |
|
1060 |
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" |
|
14208 | 1061 |
apply (induct xs, auto) |
1062 |
apply (case_tac i, auto) |
|
13145 | 1063 |
done |
13114 | 1064 |
|
13142 | 1065 |
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" |
14208 | 1066 |
apply (induct xs, auto) |
1067 |
apply (case_tac n, blast) |
|
1068 |
apply (case_tac i, auto) |
|
13145 | 1069 |
done |
13114 | 1070 |
|
13142 | 1071 |
lemma nth_drop [simp]: |
13145 | 1072 |
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" |
14208 | 1073 |
apply (induct n, auto) |
1074 |
apply (case_tac xs, auto) |
|
13145 | 1075 |
done |
3507 | 1076 |
|
14025 | 1077 |
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" |
1078 |
by(induct xs)(auto simp:take_Cons split:nat.split) |
|
1079 |
||
1080 |
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" |
|
1081 |
by(induct xs)(auto simp:drop_Cons split:nat.split) |
|
1082 |
||
14187 | 1083 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" |
1084 |
using set_take_subset by fast |
|
1085 |
||
1086 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" |
|
1087 |
using set_drop_subset by fast |
|
1088 |
||
13114 | 1089 |
lemma append_eq_conv_conj: |
13145 | 1090 |
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" |
14208 | 1091 |
apply (induct xs, simp, clarsimp) |
1092 |
apply (case_tac zs, auto) |
|
13145 | 1093 |
done |
13142 | 1094 |
|
14050 | 1095 |
lemma take_add [rule_format]: |
1096 |
"\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" |
|
1097 |
apply (induct xs, auto) |
|
1098 |
apply (case_tac i, simp_all) |
|
1099 |
done |
|
1100 |
||
14300 | 1101 |
lemma append_eq_append_conv_if: |
1102 |
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = |
|
1103 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1 |
|
1104 |
then xs\<^isub>1 = take (size xs\<^isub>1) ys\<^isub>1 \<and> xs\<^isub>2 = drop (size xs\<^isub>1) ys\<^isub>1 @ ys\<^isub>2 |
|
1105 |
else take (size ys\<^isub>1) xs\<^isub>1 = ys\<^isub>1 \<and> drop (size ys\<^isub>1) xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>2)" |
|
1106 |
apply(induct xs\<^isub>1) |
|
1107 |
apply simp |
|
1108 |
apply(case_tac ys\<^isub>1) |
|
1109 |
apply simp_all |
|
1110 |
done |
|
1111 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1112 |
lemma take_hd_drop: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1113 |
"!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1114 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1115 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1116 |
apply(simp add:drop_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1117 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1118 |
|
13114 | 1119 |
|
15392 | 1120 |
subsubsection {* @{text takeWhile} and @{text dropWhile} *} |
13114 | 1121 |
|
13142 | 1122 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" |
13145 | 1123 |
by (induct xs) auto |
13114 | 1124 |
|
13142 | 1125 |
lemma takeWhile_append1 [simp]: |
13145 | 1126 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" |
1127 |
by (induct xs) auto |
|
13114 | 1128 |
|
13142 | 1129 |
lemma takeWhile_append2 [simp]: |
13145 | 1130 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" |
1131 |
by (induct xs) auto |
|
13114 | 1132 |
|
13142 | 1133 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" |
13145 | 1134 |
by (induct xs) auto |
13114 | 1135 |
|
13142 | 1136 |
lemma dropWhile_append1 [simp]: |
13145 | 1137 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" |
1138 |
by (induct xs) auto |
|
13114 | 1139 |
|
13142 | 1140 |
lemma dropWhile_append2 [simp]: |
13145 | 1141 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" |
1142 |
by (induct xs) auto |
|
13114 | 1143 |
|
13142 | 1144 |
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" |
13145 | 1145 |
by (induct xs) (auto split: split_if_asm) |
13114 | 1146 |
|
13913 | 1147 |
lemma takeWhile_eq_all_conv[simp]: |
1148 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" |
|
1149 |
by(induct xs, auto) |
|
1150 |
||
1151 |
lemma dropWhile_eq_Nil_conv[simp]: |
|
1152 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" |
|
1153 |
by(induct xs, auto) |
|
1154 |
||
1155 |
lemma dropWhile_eq_Cons_conv: |
|
1156 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" |
|
1157 |
by(induct xs, auto) |
|
1158 |
||
13114 | 1159 |
|
15392 | 1160 |
subsubsection {* @{text zip} *} |
13114 | 1161 |
|
13142 | 1162 |
lemma zip_Nil [simp]: "zip [] ys = []" |
13145 | 1163 |
by (induct ys) auto |
13114 | 1164 |
|
13142 | 1165 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" |
13145 | 1166 |
by simp |
13114 | 1167 |
|
13142 | 1168 |
declare zip_Cons [simp del] |
13114 | 1169 |
|
15281 | 1170 |
lemma zip_Cons1: |
1171 |
"zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)" |
|
1172 |
by(auto split:list.split) |
|
1173 |
||
13142 | 1174 |
lemma length_zip [simp]: |
13145 | 1175 |
"!!xs. length (zip xs ys) = min (length xs) (length ys)" |
14208 | 1176 |
apply (induct ys, simp) |
1177 |
apply (case_tac xs, auto) |
|
13145 | 1178 |
done |
13114 | 1179 |
|
1180 |
lemma zip_append1: |
|
13145 | 1181 |
"!!xs. zip (xs @ ys) zs = |
1182 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" |
|
14208 | 1183 |
apply (induct zs, simp) |
1184 |
apply (case_tac xs, simp_all) |
|
13145 | 1185 |
done |
13114 | 1186 |
|
1187 |
lemma zip_append2: |
|
13145 | 1188 |
"!!ys. zip xs (ys @ zs) = |
1189 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" |
|
14208 | 1190 |
apply (induct xs, simp) |
1191 |
apply (case_tac ys, simp_all) |
|
13145 | 1192 |
done |
13114 | 1193 |
|
13142 | 1194 |
lemma zip_append [simp]: |
1195 |
"[| length xs = length us; length ys = length vs |] ==> |
|
13145 | 1196 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" |
1197 |
by (simp add: zip_append1) |
|
13114 | 1198 |
|
1199 |
lemma zip_rev: |
|
14247 | 1200 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" |
1201 |
by (induct rule:list_induct2, simp_all) |
|
13114 | 1202 |
|
13142 | 1203 |
lemma nth_zip [simp]: |
13145 | 1204 |
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" |
14208 | 1205 |
apply (induct ys, simp) |
13145 | 1206 |
apply (case_tac xs) |
1207 |
apply (simp_all add: nth.simps split: nat.split) |
|
1208 |
done |
|
13114 | 1209 |
|
1210 |
lemma set_zip: |
|
13145 | 1211 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" |
1212 |
by (simp add: set_conv_nth cong: rev_conj_cong) |
|
13114 | 1213 |
|
1214 |
lemma zip_update: |
|
13145 | 1215 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" |
1216 |
by (rule sym, simp add: update_zip) |
|
13114 | 1217 |
|
13142 | 1218 |
lemma zip_replicate [simp]: |
13145 | 1219 |
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" |
14208 | 1220 |
apply (induct i, auto) |
1221 |
apply (case_tac j, auto) |
|
13145 | 1222 |
done |
13114 | 1223 |
|
13142 | 1224 |
|
15392 | 1225 |
subsubsection {* @{text list_all2} *} |
13114 | 1226 |
|
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1227 |
lemma list_all2_lengthD [intro?]: |
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1228 |
"list_all2 P xs ys ==> length xs = length ys" |
13145 | 1229 |
by (simp add: list_all2_def) |
13114 | 1230 |
|
13142 | 1231 |
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" |
13145 | 1232 |
by (simp add: list_all2_def) |
13114 | 1233 |
|
13142 | 1234 |
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" |
13145 | 1235 |
by (simp add: list_all2_def) |
13114 | 1236 |
|
13142 | 1237 |
lemma list_all2_Cons [iff]: |
13145 | 1238 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" |
1239 |
by (auto simp add: list_all2_def) |
|
13114 | 1240 |
|
1241 |
lemma list_all2_Cons1: |
|
13145 | 1242 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" |
1243 |
by (cases ys) auto |
|
13114 | 1244 |
|
1245 |
lemma list_all2_Cons2: |
|
13145 | 1246 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" |
1247 |
by (cases xs) auto |
|
13114 | 1248 |
|
13142 | 1249 |
lemma list_all2_rev [iff]: |
13145 | 1250 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" |
1251 |
by (simp add: list_all2_def zip_rev cong: conj_cong) |
|
13114 | 1252 |
|
13863 | 1253 |
lemma list_all2_rev1: |
1254 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" |
|
1255 |
by (subst list_all2_rev [symmetric]) simp |
|
1256 |
||
13114 | 1257 |
lemma list_all2_append1: |
13145 | 1258 |
"list_all2 P (xs @ ys) zs = |
1259 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> |
|
1260 |
list_all2 P xs us \<and> list_all2 P ys vs)" |
|
1261 |
apply (simp add: list_all2_def zip_append1) |
|
1262 |
apply (rule iffI) |
|
1263 |
apply (rule_tac x = "take (length xs) zs" in exI) |
|
1264 |
apply (rule_tac x = "drop (length xs) zs" in exI) |
|
14208 | 1265 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1266 |
apply (simp add: ball_Un) |
1267 |
done |
|
13114 | 1268 |
|
1269 |
lemma list_all2_append2: |
|
13145 | 1270 |
"list_all2 P xs (ys @ zs) = |
1271 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> |
|
1272 |
list_all2 P us ys \<and> list_all2 P vs zs)" |
|
1273 |
apply (simp add: list_all2_def zip_append2) |
|
1274 |
apply (rule iffI) |
|
1275 |
apply (rule_tac x = "take (length ys) xs" in exI) |
|
1276 |
apply (rule_tac x = "drop (length ys) xs" in exI) |
|
14208 | 1277 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1278 |
apply (simp add: ball_Un) |
1279 |
done |
|
13114 | 1280 |
|
13863 | 1281 |
lemma list_all2_append: |
14247 | 1282 |
"length xs = length ys \<Longrightarrow> |
1283 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" |
|
1284 |
by (induct rule:list_induct2, simp_all) |
|
13863 | 1285 |
|
1286 |
lemma list_all2_appendI [intro?, trans]: |
|
1287 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" |
|
1288 |
by (simp add: list_all2_append list_all2_lengthD) |
|
1289 |
||
13114 | 1290 |
lemma list_all2_conv_all_nth: |
13145 | 1291 |
"list_all2 P xs ys = |
1292 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" |
|
1293 |
by (force simp add: list_all2_def set_zip) |
|
13114 | 1294 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1295 |
lemma list_all2_trans: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1296 |
assumes tr: "!!a b c. P1 a b ==> P2 b c ==> P3 a c" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1297 |
shows "!!bs cs. list_all2 P1 as bs ==> list_all2 P2 bs cs ==> list_all2 P3 as cs" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1298 |
(is "!!bs cs. PROP ?Q as bs cs") |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1299 |
proof (induct as) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1300 |
fix x xs bs assume I1: "!!bs cs. PROP ?Q xs bs cs" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1301 |
show "!!cs. PROP ?Q (x # xs) bs cs" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1302 |
proof (induct bs) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1303 |
fix y ys cs assume I2: "!!cs. PROP ?Q (x # xs) ys cs" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1304 |
show "PROP ?Q (x # xs) (y # ys) cs" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1305 |
by (induct cs) (auto intro: tr I1 I2) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1306 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1307 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1308 |
|
13863 | 1309 |
lemma list_all2_all_nthI [intro?]: |
1310 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" |
|
1311 |
by (simp add: list_all2_conv_all_nth) |
|
1312 |
||
14395 | 1313 |
lemma list_all2I: |
1314 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" |
|
1315 |
by (simp add: list_all2_def) |
|
1316 |
||
14328 | 1317 |
lemma list_all2_nthD: |
13863 | 1318 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
1319 |
by (simp add: list_all2_conv_all_nth) |
|
1320 |
||
14302 | 1321 |
lemma list_all2_nthD2: |
1322 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
|
1323 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD) |
|
1324 |
||
13863 | 1325 |
lemma list_all2_map1: |
1326 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" |
|
1327 |
by (simp add: list_all2_conv_all_nth) |
|
1328 |
||
1329 |
lemma list_all2_map2: |
|
1330 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" |
|
1331 |
by (auto simp add: list_all2_conv_all_nth) |
|
1332 |
||
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1333 |
lemma list_all2_refl [intro?]: |
13863 | 1334 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" |
1335 |
by (simp add: list_all2_conv_all_nth) |
|
1336 |
||
1337 |
lemma list_all2_update_cong: |
|
1338 |
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1339 |
by (simp add: list_all2_conv_all_nth nth_list_update) |
|
1340 |
||
1341 |
lemma list_all2_update_cong2: |
|
1342 |
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1343 |
by (simp add: list_all2_lengthD list_all2_update_cong) |
|
1344 |
||
14302 | 1345 |
lemma list_all2_takeI [simp,intro?]: |
1346 |
"\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" |
|
1347 |
apply (induct xs) |
|
1348 |
apply simp |
|
1349 |
apply (clarsimp simp add: list_all2_Cons1) |
|
1350 |
apply (case_tac n) |
|
1351 |
apply auto |
|
1352 |
done |
|
1353 |
||
1354 |
lemma list_all2_dropI [simp,intro?]: |
|
13863 | 1355 |
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" |
14208 | 1356 |
apply (induct as, simp) |
13863 | 1357 |
apply (clarsimp simp add: list_all2_Cons1) |
14208 | 1358 |
apply (case_tac n, simp, simp) |
13863 | 1359 |
done |
1360 |
||
14327 | 1361 |
lemma list_all2_mono [intro?]: |
13863 | 1362 |
"\<And>y. list_all2 P x y \<Longrightarrow> (\<And>x y. P x y \<Longrightarrow> Q x y) \<Longrightarrow> list_all2 Q x y" |
14208 | 1363 |
apply (induct x, simp) |
1364 |
apply (case_tac y, auto) |
|
13863 | 1365 |
done |
1366 |
||
13142 | 1367 |
|
15392 | 1368 |
subsubsection {* @{text foldl} and @{text foldr} *} |
13142 | 1369 |
|
1370 |
lemma foldl_append [simp]: |
|
13145 | 1371 |
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" |
1372 |
by (induct xs) auto |
|
13142 | 1373 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1374 |
lemma foldr_append[simp]: "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1375 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1376 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1377 |
lemma foldr_foldl: "foldr f xs a = foldl (%x y. f y x) a (rev xs)" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1378 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1379 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1380 |
lemma foldl_foldr: "foldl f a xs = foldr (%x y. f y x) (rev xs) a" |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1381 |
by (simp add: foldr_foldl [of "%x y. f y x" "rev xs"]) |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1382 |
|
13142 | 1383 |
text {* |
13145 | 1384 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more |
1385 |
difficult to use because it requires an additional transitivity step. |
|
13142 | 1386 |
*} |
1387 |
||
1388 |
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" |
|
13145 | 1389 |
by (induct ns) auto |
13142 | 1390 |
|
1391 |
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" |
|
13145 | 1392 |
by (force intro: start_le_sum simp add: in_set_conv_decomp) |
13142 | 1393 |
|
1394 |
lemma sum_eq_0_conv [iff]: |
|
13145 | 1395 |
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" |
1396 |
by (induct ns) auto |
|
13114 | 1397 |
|
1398 |
||
15392 | 1399 |
subsubsection {* @{text upto} *} |
13114 | 1400 |
|
15425 | 1401 |
lemma upt_rec: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" |
13145 | 1402 |
-- {* Does not terminate! *} |
1403 |
by (induct j) auto |
|
13142 | 1404 |
|
15425 | 1405 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []" |
13145 | 1406 |
by (subst upt_rec) simp |
13114 | 1407 |
|
15425 | 1408 |
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)" |
15281 | 1409 |
by(induct j)simp_all |
1410 |
||
1411 |
lemma upt_eq_Cons_conv: |
|
15425 | 1412 |
"!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)" |
15281 | 1413 |
apply(induct j) |
1414 |
apply simp |
|
1415 |
apply(clarsimp simp add: append_eq_Cons_conv) |
|
1416 |
apply arith |
|
1417 |
done |
|
1418 |
||
15425 | 1419 |
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]" |
13145 | 1420 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} |
1421 |
by simp |
|
13114 | 1422 |
|
15425 | 1423 |
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" |
13145 | 1424 |
apply(rule trans) |
1425 |
apply(subst upt_rec) |
|
14208 | 1426 |
prefer 2 apply (rule refl, simp) |
13145 | 1427 |
done |
13114 | 1428 |
|
15425 | 1429 |
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]" |
13145 | 1430 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *} |
1431 |
by (induct k) auto |
|
13114 | 1432 |
|
15425 | 1433 |
lemma length_upt [simp]: "length [i..<j] = j - i" |
13145 | 1434 |
by (induct j) (auto simp add: Suc_diff_le) |
13114 | 1435 |
|
15425 | 1436 |
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k" |
13145 | 1437 |
apply (induct j) |
1438 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) |
|
1439 |
done |
|
13114 | 1440 |
|
15425 | 1441 |
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]" |
14208 | 1442 |
apply (induct m, simp) |
13145 | 1443 |
apply (subst upt_rec) |
1444 |
apply (rule sym) |
|
1445 |
apply (subst upt_rec) |
|
1446 |
apply (simp del: upt.simps) |
|
1447 |
done |
|
3507 | 1448 |
|
15425 | 1449 |
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]" |
13145 | 1450 |
by (induct n) auto |
13114 | 1451 |
|
15425 | 1452 |
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)" |
13145 | 1453 |
apply (induct n m rule: diff_induct) |
1454 |
prefer 3 apply (subst map_Suc_upt[symmetric]) |
|
1455 |
apply (auto simp add: less_diff_conv nth_upt) |
|
1456 |
done |
|
13114 | 1457 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1458 |
lemma nth_take_lemma: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1459 |
"!!xs ys. k <= length xs ==> k <= length ys ==> |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1460 |
(!!i. i < k --> xs!i = ys!i) ==> take k xs = take k ys" |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1461 |
apply (atomize, induct k) |
14208 | 1462 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) |
13145 | 1463 |
txt {* Both lists must be non-empty *} |
14208 | 1464 |
apply (case_tac xs, simp) |
1465 |
apply (case_tac ys, clarify) |
|
13145 | 1466 |
apply (simp (no_asm_use)) |
1467 |
apply clarify |
|
1468 |
txt {* prenexing's needed, not miniscoping *} |
|
1469 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) |
|
1470 |
apply blast |
|
1471 |
done |
|
13114 | 1472 |
|
1473 |
lemma nth_equalityI: |
|
1474 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" |
|
13145 | 1475 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) |
1476 |
apply (simp_all add: take_all) |
|
1477 |
done |
|
13142 | 1478 |
|
13863 | 1479 |
(* needs nth_equalityI *) |
1480 |
lemma list_all2_antisym: |
|
1481 |
"\<lbrakk> (\<And>x y. \<lbrakk>P x y; Q y x\<rbrakk> \<Longrightarrow> x = y); list_all2 P xs ys; list_all2 Q ys xs \<rbrakk> |
|
1482 |
\<Longrightarrow> xs = ys" |
|
1483 |
apply (simp add: list_all2_conv_all_nth) |
|
14208 | 1484 |
apply (rule nth_equalityI, blast, simp) |
13863 | 1485 |
done |
1486 |
||
13142 | 1487 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" |
13145 | 1488 |
-- {* The famous take-lemma. *} |
1489 |
apply (drule_tac x = "max (length xs) (length ys)" in spec) |
|
1490 |
apply (simp add: le_max_iff_disj take_all) |
|
1491 |
done |
|
13142 | 1492 |
|
1493 |
||
15302 | 1494 |
lemma take_Cons': |
1495 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" |
|
1496 |
by (cases n) simp_all |
|
1497 |
||
1498 |
lemma drop_Cons': |
|
1499 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" |
|
1500 |
by (cases n) simp_all |
|
1501 |
||
1502 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" |
|
1503 |
by (cases n) simp_all |
|
1504 |
||
1505 |
lemmas [simp] = take_Cons'[of "number_of v",standard] |
|
1506 |
drop_Cons'[of "number_of v",standard] |
|
1507 |
nth_Cons'[of _ _ "number_of v",standard] |
|
1508 |
||
1509 |
||
15392 | 1510 |
subsubsection {* @{text "distinct"} and @{text remdups} *} |
13142 | 1511 |
|
1512 |
lemma distinct_append [simp]: |
|
13145 | 1513 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" |
1514 |
by (induct xs) auto |
|
13142 | 1515 |
|
15305 | 1516 |
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" |
1517 |
by(induct xs) auto |
|
1518 |
||
13142 | 1519 |
lemma set_remdups [simp]: "set (remdups xs) = set xs" |
13145 | 1520 |
by (induct xs) (auto simp add: insert_absorb) |
13142 | 1521 |
|
1522 |
lemma distinct_remdups [iff]: "distinct (remdups xs)" |
|
13145 | 1523 |
by (induct xs) auto |
13142 | 1524 |
|
15072 | 1525 |
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" |
15251 | 1526 |
by (induct x, auto) |
15072 | 1527 |
|
1528 |
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" |
|
15251 | 1529 |
by (induct x, auto) |
15072 | 1530 |
|
15245 | 1531 |
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs" |
1532 |
by (induct xs) auto |
|
1533 |
||
1534 |
lemma length_remdups_eq[iff]: |
|
1535 |
"(length (remdups xs) = length xs) = (remdups xs = xs)" |
|
1536 |
apply(induct xs) |
|
1537 |
apply auto |
|
1538 |
apply(subgoal_tac "length (remdups xs) <= length xs") |
|
1539 |
apply arith |
|
1540 |
apply(rule length_remdups_leq) |
|
1541 |
done |
|
1542 |
||
13142 | 1543 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" |
13145 | 1544 |
by (induct xs) auto |
13114 | 1545 |
|
15304 | 1546 |
lemma distinct_map_filterI: |
1547 |
"distinct(map f xs) \<Longrightarrow> distinct(map f (filter P xs))" |
|
1548 |
apply(induct xs) |
|
1549 |
apply simp |
|
1550 |
apply force |
|
1551 |
done |
|
1552 |
||
13142 | 1553 |
text {* |
13145 | 1554 |
It is best to avoid this indexed version of distinct, but sometimes |
1555 |
it is useful. *} |
|
13142 | 1556 |
lemma distinct_conv_nth: |
13145 | 1557 |
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)" |
15251 | 1558 |
apply (induct xs, simp, simp) |
14208 | 1559 |
apply (rule iffI, clarsimp) |
13145 | 1560 |
apply (case_tac i) |
14208 | 1561 |
apply (case_tac j, simp) |
13145 | 1562 |
apply (simp add: set_conv_nth) |
1563 |
apply (case_tac j) |
|
14208 | 1564 |
apply (clarsimp simp add: set_conv_nth, simp) |
13145 | 1565 |
apply (rule conjI) |
1566 |
apply (clarsimp simp add: set_conv_nth) |
|
1567 |
apply (erule_tac x = 0 in allE) |
|
14208 | 1568 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp) |
13145 | 1569 |
apply (erule_tac x = "Suc i" in allE) |
14208 | 1570 |
apply (erule_tac x = "Suc j" in allE, simp) |
13145 | 1571 |
done |
13114 | 1572 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1573 |
lemma distinct_card: "distinct xs ==> card (set xs) = size xs" |
14388 | 1574 |
by (induct xs) auto |
1575 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1576 |
lemma card_distinct: "card (set xs) = size xs ==> distinct xs" |
14388 | 1577 |
proof (induct xs) |
1578 |
case Nil thus ?case by simp |
|
1579 |
next |
|
1580 |
case (Cons x xs) |
|
1581 |
show ?case |
|
1582 |
proof (cases "x \<in> set xs") |
|
1583 |
case False with Cons show ?thesis by simp |
|
1584 |
next |
|
1585 |
case True with Cons.prems |
|
1586 |
have "card (set xs) = Suc (length xs)" |
|
1587 |
by (simp add: card_insert_if split: split_if_asm) |
|
1588 |
moreover have "card (set xs) \<le> length xs" by (rule card_length) |
|
1589 |
ultimately have False by simp |
|
1590 |
thus ?thesis .. |
|
1591 |
qed |
|
1592 |
qed |
|
1593 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1594 |
lemma inj_on_setI: "distinct(map f xs) ==> inj_on f (set xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1595 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1596 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1597 |
apply fastsimp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1598 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1599 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1600 |
lemma inj_on_set_conv: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1601 |
"distinct xs \<Longrightarrow> inj_on f (set xs) = distinct(map f xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1602 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1603 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1604 |
apply fastsimp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1605 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1606 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1607 |
|
15392 | 1608 |
subsubsection {* @{text remove1} *} |
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1609 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1610 |
lemma set_remove1_subset: "set(remove1 x xs) <= set xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1611 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1612 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1613 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1614 |
apply blast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1615 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1616 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1617 |
lemma [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1618 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1619 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1620 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1621 |
apply blast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1622 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1623 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1624 |
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1625 |
apply(insert set_remove1_subset) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1626 |
apply fast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1627 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1628 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1629 |
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1630 |
by (induct xs) simp_all |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1631 |
|
13114 | 1632 |
|
15392 | 1633 |
subsubsection {* @{text replicate} *} |
13114 | 1634 |
|
13142 | 1635 |
lemma length_replicate [simp]: "length (replicate n x) = n" |
13145 | 1636 |
by (induct n) auto |
13124 | 1637 |
|
13142 | 1638 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" |
13145 | 1639 |
by (induct n) auto |
13114 | 1640 |
|
1641 |
lemma replicate_app_Cons_same: |
|
13145 | 1642 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs" |
1643 |
by (induct n) auto |
|
13114 | 1644 |
|
13142 | 1645 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" |
14208 | 1646 |
apply (induct n, simp) |
13145 | 1647 |
apply (simp add: replicate_app_Cons_same) |
1648 |
done |
|
13114 | 1649 |
|
13142 | 1650 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" |
13145 | 1651 |
by (induct n) auto |
13114 | 1652 |
|
13142 | 1653 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" |
13145 | 1654 |
by (induct n) auto |
13114 | 1655 |
|
13142 | 1656 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x" |
13145 | 1657 |
by (induct n) auto |
13114 | 1658 |
|
13142 | 1659 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" |
13145 | 1660 |
by (atomize (full), induct n) auto |
13114 | 1661 |
|
13142 | 1662 |
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" |
14208 | 1663 |
apply (induct n, simp) |
13145 | 1664 |
apply (simp add: nth_Cons split: nat.split) |
1665 |
done |
|
13114 | 1666 |
|
13142 | 1667 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" |
13145 | 1668 |
by (induct n) auto |
13114 | 1669 |
|
13142 | 1670 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" |
13145 | 1671 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) |
13114 | 1672 |
|
13142 | 1673 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" |
13145 | 1674 |
by auto |
13114 | 1675 |
|
13142 | 1676 |
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" |
13145 | 1677 |
by (simp add: set_replicate_conv_if split: split_if_asm) |
13114 | 1678 |
|
1679 |
||
15392 | 1680 |
subsubsection{*@{text rotate1} and @{text rotate}*} |
15302 | 1681 |
|
1682 |
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]" |
|
1683 |
by(simp add:rotate1_def) |
|
1684 |
||
1685 |
lemma rotate0[simp]: "rotate 0 = id" |
|
1686 |
by(simp add:rotate_def) |
|
1687 |
||
1688 |
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" |
|
1689 |
by(simp add:rotate_def) |
|
1690 |
||
1691 |
lemma rotate_add: |
|
1692 |
"rotate (m+n) = rotate m o rotate n" |
|
1693 |
by(simp add:rotate_def funpow_add) |
|
1694 |
||
1695 |
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" |
|
1696 |
by(simp add:rotate_add) |
|
1697 |
||
1698 |
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs" |
|
1699 |
by(cases xs) simp_all |
|
1700 |
||
1701 |
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs" |
|
1702 |
apply(induct n) |
|
1703 |
apply simp |
|
1704 |
apply (simp add:rotate_def) |
|
13145 | 1705 |
done |
13114 | 1706 |
|
15302 | 1707 |
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]" |
1708 |
by(simp add:rotate1_def split:list.split) |
|
1709 |
||
1710 |
lemma rotate_drop_take: |
|
1711 |
"rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" |
|
1712 |
apply(induct n) |
|
1713 |
apply simp |
|
1714 |
apply(simp add:rotate_def) |
|
1715 |
apply(cases "xs = []") |
|
1716 |
apply (simp) |
|
1717 |
apply(case_tac "n mod length xs = 0") |
|
1718 |
apply(simp add:mod_Suc) |
|
1719 |
apply(simp add: rotate1_hd_tl drop_Suc take_Suc) |
|
1720 |
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] |
|
1721 |
take_hd_drop linorder_not_le) |
|
13145 | 1722 |
done |
13114 | 1723 |
|
15302 | 1724 |
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" |
1725 |
by(simp add:rotate_drop_take) |
|
1726 |
||
1727 |
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs" |
|
1728 |
by(simp add:rotate_drop_take) |
|
1729 |
||
1730 |
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" |
|
1731 |
by(simp add:rotate1_def split:list.split) |
|
1732 |
||
1733 |
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs" |
|
1734 |
by (induct n) (simp_all add:rotate_def) |
|
1735 |
||
1736 |
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" |
|
1737 |
by(simp add:rotate1_def split:list.split) blast |
|
1738 |
||
1739 |
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" |
|
1740 |
by (induct n) (simp_all add:rotate_def) |
|
1741 |
||
1742 |
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" |
|
1743 |
by(simp add:rotate_drop_take take_map drop_map) |
|
1744 |
||
1745 |
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" |
|
1746 |
by(simp add:rotate1_def split:list.split) |
|
1747 |
||
1748 |
lemma set_rotate[simp]: "set(rotate n xs) = set xs" |
|
1749 |
by (induct n) (simp_all add:rotate_def) |
|
1750 |
||
1751 |
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" |
|
1752 |
by(simp add:rotate1_def split:list.split) |
|
1753 |
||
1754 |
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" |
|
1755 |
by (induct n) (simp_all add:rotate_def) |
|
13114 | 1756 |
|
15439 | 1757 |
lemma rotate_rev: |
1758 |
"rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" |
|
1759 |
apply(simp add:rotate_drop_take rev_drop rev_take) |
|
1760 |
apply(cases "length xs = 0") |
|
1761 |
apply simp |
|
1762 |
apply(cases "n mod length xs = 0") |
|
1763 |
apply simp |
|
1764 |
apply(simp add:rotate_drop_take rev_drop rev_take) |
|
1765 |
done |
|
1766 |
||
13114 | 1767 |
|
15392 | 1768 |
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *} |
13114 | 1769 |
|
13142 | 1770 |
lemma sublist_empty [simp]: "sublist xs {} = []" |
13145 | 1771 |
by (auto simp add: sublist_def) |
13114 | 1772 |
|
13142 | 1773 |
lemma sublist_nil [simp]: "sublist [] A = []" |
13145 | 1774 |
by (auto simp add: sublist_def) |
13114 | 1775 |
|
15281 | 1776 |
lemma length_sublist: |
1777 |
"length(sublist xs I) = card{i. i < length xs \<and> i : I}" |
|
1778 |
by(simp add: sublist_def length_filter_conv_card cong:conj_cong) |
|
1779 |
||
1780 |
lemma sublist_shift_lemma_Suc: |
|
1781 |
"!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = |
|
1782 |
map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" |
|
1783 |
apply(induct xs) |
|
1784 |
apply simp |
|
1785 |
apply (case_tac "is") |
|
1786 |
apply simp |
|
1787 |
apply simp |
|
1788 |
done |
|
1789 |
||
13114 | 1790 |
lemma sublist_shift_lemma: |
15425 | 1791 |
"map fst [p:zip xs [i..<i + length xs] . snd p : A] = |
1792 |
map fst [p:zip xs [0..<length xs] . snd p + i : A]" |
|
13145 | 1793 |
by (induct xs rule: rev_induct) (simp_all add: add_commute) |
13114 | 1794 |
|
1795 |
lemma sublist_append: |
|
15168 | 1796 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" |
13145 | 1797 |
apply (unfold sublist_def) |
14208 | 1798 |
apply (induct l' rule: rev_induct, simp) |
13145 | 1799 |
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) |
1800 |
apply (simp add: add_commute) |
|
1801 |
done |
|
13114 | 1802 |
|
1803 |
lemma sublist_Cons: |
|
13145 | 1804 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" |
1805 |
apply (induct l rule: rev_induct) |
|
1806 |
apply (simp add: sublist_def) |
|
1807 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) |
|
1808 |
done |
|
13114 | 1809 |
|
15281 | 1810 |
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}" |
1811 |
apply(induct xs) |
|
1812 |
apply simp |
|
1813 |
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE) |
|
1814 |
apply(erule lessE) |
|
1815 |
apply auto |
|
1816 |
apply(erule lessE) |
|
1817 |
apply auto |
|
1818 |
done |
|
1819 |
||
1820 |
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs" |
|
1821 |
by(auto simp add:set_sublist) |
|
1822 |
||
1823 |
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)" |
|
1824 |
by(auto simp add:set_sublist) |
|
1825 |
||
1826 |
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs" |
|
1827 |
by(auto simp add:set_sublist) |
|
1828 |
||
13142 | 1829 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" |
13145 | 1830 |
by (simp add: sublist_Cons) |
13114 | 1831 |
|
15281 | 1832 |
|
1833 |
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)" |
|
1834 |
apply(induct xs) |
|
1835 |
apply simp |
|
1836 |
apply(auto simp add:sublist_Cons) |
|
1837 |
done |
|
1838 |
||
1839 |
||
15045 | 1840 |
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l" |
14208 | 1841 |
apply (induct l rule: rev_induct, simp) |
13145 | 1842 |
apply (simp split: nat_diff_split add: sublist_append) |
1843 |
done |
|
13114 | 1844 |
|
1845 |
||
15392 | 1846 |
subsubsection{*Sets of Lists*} |
1847 |
||
1848 |
subsubsection {* @{text lists}: the list-forming operator over sets *} |
|
15302 | 1849 |
|
1850 |
consts lists :: "'a set => 'a list set" |
|
1851 |
inductive "lists A" |
|
1852 |
intros |
|
1853 |
Nil [intro!]: "[]: lists A" |
|
1854 |
Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" |
|
1855 |
||
1856 |
inductive_cases listsE [elim!]: "x#l : lists A" |
|
1857 |
||
1858 |
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B" |
|
1859 |
by (unfold lists.defs) (blast intro!: lfp_mono) |
|
1860 |
||
1861 |
lemma lists_IntI: |
|
1862 |
assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l |
|
1863 |
by induct blast+ |
|
1864 |
||
1865 |
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" |
|
1866 |
proof (rule mono_Int [THEN equalityI]) |
|
1867 |
show "mono lists" by (simp add: mono_def lists_mono) |
|
1868 |
show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI) |
|
14388 | 1869 |
qed |
1870 |
||
15302 | 1871 |
lemma append_in_lists_conv [iff]: |
1872 |
"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" |
|
1873 |
by (induct xs) auto |
|
1874 |
||
1875 |
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" |
|
1876 |
-- {* eliminate @{text lists} in favour of @{text set} *} |
|
1877 |
by (induct xs) auto |
|
1878 |
||
1879 |
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" |
|
1880 |
by (rule in_lists_conv_set [THEN iffD1]) |
|
1881 |
||
1882 |
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" |
|
1883 |
by (rule in_lists_conv_set [THEN iffD2]) |
|
1884 |
||
1885 |
lemma lists_UNIV [simp]: "lists UNIV = UNIV" |
|
1886 |
by auto |
|
1887 |
||
15392 | 1888 |
subsubsection{*Lists as Cartesian products*} |
15302 | 1889 |
|
1890 |
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from |
|
1891 |
@{term A} and tail drawn from @{term Xs}.*} |
|
1892 |
||
1893 |
constdefs |
|
1894 |
set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" |
|
1895 |
"set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}" |
|
1896 |
||
1897 |
lemma [simp]: "set_Cons A {[]} = (%x. [x])`A" |
|
1898 |
by (auto simp add: set_Cons_def) |
|
1899 |
||
1900 |
text{*Yields the set of lists, all of the same length as the argument and |
|
1901 |
with elements drawn from the corresponding element of the argument.*} |
|
1902 |
||
1903 |
consts listset :: "'a set list \<Rightarrow> 'a list set" |
|
1904 |
primrec |
|
1905 |
"listset [] = {[]}" |
|
1906 |
"listset(A#As) = set_Cons A (listset As)" |
|
1907 |
||
1908 |
||
15656 | 1909 |
subsection{*Relations on Lists*} |
1910 |
||
1911 |
subsubsection {* Length Lexicographic Ordering *} |
|
1912 |
||
1913 |
text{*These orderings preserve well-foundedness: shorter lists |
|
1914 |
precede longer lists. These ordering are not used in dictionaries.*} |
|
1915 |
||
1916 |
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" |
|
1917 |
--{*The lexicographic ordering for lists of the specified length*} |
|
15302 | 1918 |
primrec |
15656 | 1919 |
"lexn r 0 = {}" |
1920 |
"lexn r (Suc n) = |
|
1921 |
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int |
|
1922 |
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" |
|
15302 | 1923 |
|
1924 |
constdefs |
|
15656 | 1925 |
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
1926 |
"lex r == \<Union>n. lexn r n" |
|
1927 |
--{*Holds only between lists of the same length*} |
|
1928 |
||
15693 | 1929 |
lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
1930 |
"lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" |
|
15656 | 1931 |
--{*Compares lists by their length and then lexicographically*} |
15302 | 1932 |
|
1933 |
||
1934 |
lemma wf_lexn: "wf r ==> wf (lexn r n)" |
|
1935 |
apply (induct n, simp, simp) |
|
1936 |
apply(rule wf_subset) |
|
1937 |
prefer 2 apply (rule Int_lower1) |
|
1938 |
apply(rule wf_prod_fun_image) |
|
1939 |
prefer 2 apply (rule inj_onI, auto) |
|
1940 |
done |
|
1941 |
||
1942 |
lemma lexn_length: |
|
1943 |
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" |
|
1944 |
by (induct n) auto |
|
1945 |
||
1946 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)" |
|
1947 |
apply (unfold lex_def) |
|
1948 |
apply (rule wf_UN) |
|
1949 |
apply (blast intro: wf_lexn, clarify) |
|
1950 |
apply (rename_tac m n) |
|
1951 |
apply (subgoal_tac "m \<noteq> n") |
|
1952 |
prefer 2 apply blast |
|
1953 |
apply (blast dest: lexn_length not_sym) |
|
1954 |
done |
|
1955 |
||
1956 |
lemma lexn_conv: |
|
15656 | 1957 |
"lexn r n = |
1958 |
{(xs,ys). length xs = n \<and> length ys = n \<and> |
|
1959 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" |
|
15302 | 1960 |
apply (induct n, simp, blast) |
1961 |
apply (simp add: image_Collect lex_prod_def, safe, blast) |
|
1962 |
apply (rule_tac x = "ab # xys" in exI, simp) |
|
1963 |
apply (case_tac xys, simp_all, blast) |
|
1964 |
done |
|
1965 |
||
1966 |
lemma lex_conv: |
|
15656 | 1967 |
"lex r = |
1968 |
{(xs,ys). length xs = length ys \<and> |
|
1969 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" |
|
15302 | 1970 |
by (force simp add: lex_def lexn_conv) |
1971 |
||
15693 | 1972 |
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" |
1973 |
by (unfold lenlex_def) blast |
|
1974 |
||
1975 |
lemma lenlex_conv: |
|
1976 |
"lenlex r = {(xs,ys). length xs < length ys | |
|
15656 | 1977 |
length xs = length ys \<and> (xs, ys) : lex r}" |
15693 | 1978 |
by (simp add: lenlex_def diag_def lex_prod_def measure_def inv_image_def) |
15302 | 1979 |
|
1980 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" |
|
1981 |
by (simp add: lex_conv) |
|
1982 |
||
1983 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" |
|
1984 |
by (simp add:lex_conv) |
|
1985 |
||
1986 |
lemma Cons_in_lex [iff]: |
|
15656 | 1987 |
"((x # xs, y # ys) : lex r) = |
1988 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" |
|
15302 | 1989 |
apply (simp add: lex_conv) |
1990 |
apply (rule iffI) |
|
1991 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify) |
|
1992 |
apply (case_tac xys, simp, simp) |
|
1993 |
apply blast |
|
1994 |
done |
|
1995 |
||
1996 |
||
15656 | 1997 |
subsubsection {* Lexicographic Ordering *} |
1998 |
||
1999 |
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". |
|
2000 |
This ordering does \emph{not} preserve well-foundedness. |
|
2001 |
Author: N. Voelker, March 2005 *} |
|
2002 |
||
2003 |
constdefs |
|
2004 |
lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" |
|
2005 |
"lexord r == {(x,y). \<exists> a v. y = x @ a # v \<or> |
|
2006 |
(\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}" |
|
2007 |
||
2008 |
lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)" |
|
2009 |
by (unfold lexord_def, induct_tac y, auto) |
|
2010 |
||
2011 |
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r" |
|
2012 |
by (unfold lexord_def, induct_tac x, auto) |
|
2013 |
||
2014 |
lemma lexord_cons_cons[simp]: |
|
2015 |
"((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))" |
|
2016 |
apply (unfold lexord_def, safe, simp_all) |
|
2017 |
apply (case_tac u, simp, simp) |
|
2018 |
apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) |
|
2019 |
apply (erule_tac x="b # u" in allE) |
|
2020 |
by force |
|
2021 |
||
2022 |
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons |
|
2023 |
||
2024 |
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r" |
|
2025 |
by (induct_tac x, auto) |
|
2026 |
||
2027 |
lemma lexord_append_left_rightI: |
|
2028 |
"(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r" |
|
2029 |
by (induct_tac u, auto) |
|
2030 |
||
2031 |
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r" |
|
2032 |
by (induct x, auto) |
|
2033 |
||
2034 |
lemma lexord_append_leftD: |
|
2035 |
"\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r" |
|
2036 |
by (erule rev_mp, induct_tac x, auto) |
|
2037 |
||
2038 |
lemma lexord_take_index_conv: |
|
2039 |
"((x,y) : lexord r) = |
|
2040 |
((length x < length y \<and> take (length x) y = x) \<or> |
|
2041 |
(\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))" |
|
2042 |
apply (unfold lexord_def Let_def, clarsimp) |
|
2043 |
apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2) |
|
2044 |
apply auto |
|
2045 |
apply (rule_tac x="hd (drop (length x) y)" in exI) |
|
2046 |
apply (rule_tac x="tl (drop (length x) y)" in exI) |
|
2047 |
apply (erule subst, simp add: min_def) |
|
2048 |
apply (rule_tac x ="length u" in exI, simp) |
|
2049 |
apply (rule_tac x ="take i x" in exI) |
|
2050 |
apply (rule_tac x ="x ! i" in exI) |
|
2051 |
apply (rule_tac x ="y ! i" in exI, safe) |
|
2052 |
apply (rule_tac x="drop (Suc i) x" in exI) |
|
2053 |
apply (drule sym, simp add: drop_Suc_conv_tl) |
|
2054 |
apply (rule_tac x="drop (Suc i) y" in exI) |
|
2055 |
by (simp add: drop_Suc_conv_tl) |
|
2056 |
||
2057 |
-- {* lexord is extension of partial ordering List.lex *} |
|
2058 |
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)" |
|
2059 |
apply (rule_tac x = y in spec) |
|
2060 |
apply (induct_tac x, clarsimp) |
|
2061 |
by (clarify, case_tac x, simp, force) |
|
2062 |
||
2063 |
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r" |
|
2064 |
by (induct y, auto) |
|
2065 |
||
2066 |
lemma lexord_trans: |
|
2067 |
"\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r" |
|
2068 |
apply (erule rev_mp)+ |
|
2069 |
apply (rule_tac x = x in spec) |
|
2070 |
apply (rule_tac x = z in spec) |
|
2071 |
apply ( induct_tac y, simp, clarify) |
|
2072 |
apply (case_tac xa, erule ssubst) |
|
2073 |
apply (erule allE, erule allE) -- {* avoid simp recursion *} |
|
2074 |
apply (case_tac x, simp, simp) |
|
2075 |
apply (case_tac x, erule allE, erule allE, simp) |
|
2076 |
apply (erule_tac x = listb in allE) |
|
2077 |
apply (erule_tac x = lista in allE, simp) |
|
2078 |
apply (unfold trans_def) |
|
2079 |
by blast |
|
2080 |
||
2081 |
lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)" |
|
2082 |
by (rule transI, drule lexord_trans, blast) |
|
2083 |
||
2084 |
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r" |
|
2085 |
apply (rule_tac x = y in spec) |
|
2086 |
apply (induct_tac x, rule allI) |
|
2087 |
apply (case_tac x, simp, simp) |
|
2088 |
apply (rule allI, case_tac x, simp, simp) |
|
2089 |
by blast |
|
2090 |
||
2091 |
||
15392 | 2092 |
subsubsection{*Lifting a Relation on List Elements to the Lists*} |
15302 | 2093 |
|
2094 |
consts listrel :: "('a * 'a)set => ('a list * 'a list)set" |
|
2095 |
||
2096 |
inductive "listrel(r)" |
|
2097 |
intros |
|
2098 |
Nil: "([],[]) \<in> listrel r" |
|
2099 |
Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r" |
|
2100 |
||
2101 |
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r" |
|
2102 |
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r" |
|
2103 |
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r" |
|
2104 |
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r" |
|
2105 |
||
2106 |
||
2107 |
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s" |
|
2108 |
apply clarify |
|
2109 |
apply (erule listrel.induct) |
|
2110 |
apply (blast intro: listrel.intros)+ |
|
2111 |
done |
|
2112 |
||
2113 |
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A" |
|
2114 |
apply clarify |
|
2115 |
apply (erule listrel.induct, auto) |
|
2116 |
done |
|
2117 |
||
2118 |
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" |
|
2119 |
apply (simp add: refl_def listrel_subset Ball_def) |
|
2120 |
apply (rule allI) |
|
2121 |
apply (induct_tac x) |
|
2122 |
apply (auto intro: listrel.intros) |
|
2123 |
done |
|
2124 |
||
2125 |
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" |
|
2126 |
apply (auto simp add: sym_def) |
|
2127 |
apply (erule listrel.induct) |
|
2128 |
apply (blast intro: listrel.intros)+ |
|
2129 |
done |
|
2130 |
||
2131 |
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" |
|
2132 |
apply (simp add: trans_def) |
|
2133 |
apply (intro allI) |
|
2134 |
apply (rule impI) |
|
2135 |
apply (erule listrel.induct) |
|
2136 |
apply (blast intro: listrel.intros)+ |
|
2137 |
done |
|
2138 |
||
2139 |
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)" |
|
2140 |
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) |
|
2141 |
||
2142 |
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" |
|
2143 |
by (blast intro: listrel.intros) |
|
2144 |
||
2145 |
lemma listrel_Cons: |
|
2146 |
"listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"; |
|
2147 |
by (auto simp add: set_Cons_def intro: listrel.intros) |
|
2148 |
||
2149 |
||
15392 | 2150 |
subsection{*Miscellany*} |
2151 |
||
2152 |
subsubsection {* Characters and strings *} |
|
13366 | 2153 |
|
2154 |
datatype nibble = |
|
2155 |
Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 |
|
2156 |
| Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF |
|
2157 |
||
2158 |
datatype char = Char nibble nibble |
|
2159 |
-- "Note: canonical order of character encoding coincides with standard term ordering" |
|
2160 |
||
2161 |
types string = "char list" |
|
2162 |
||
2163 |
syntax |
|
2164 |
"_Char" :: "xstr => char" ("CHR _") |
|
2165 |
"_String" :: "xstr => string" ("_") |
|
2166 |
||
2167 |
parse_ast_translation {* |
|
2168 |
let |
|
2169 |
val constants = Syntax.Appl o map Syntax.Constant; |
|
2170 |
||
2171 |
fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); |
|
2172 |
fun mk_char c = |
|
2173 |
if Symbol.is_ascii c andalso Symbol.is_printable c then |
|
2174 |
constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] |
|
2175 |
else error ("Printable ASCII character expected: " ^ quote c); |
|
2176 |
||
2177 |
fun mk_string [] = Syntax.Constant "Nil" |
|
2178 |
| mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; |
|
2179 |
||
2180 |
fun char_ast_tr [Syntax.Variable xstr] = |
|
2181 |
(case Syntax.explode_xstr xstr of |
|
2182 |
[c] => mk_char c |
|
2183 |
| _ => error ("Single character expected: " ^ xstr)) |
|
2184 |
| char_ast_tr asts = raise AST ("char_ast_tr", asts); |
|
2185 |
||
2186 |
fun string_ast_tr [Syntax.Variable xstr] = |
|
2187 |
(case Syntax.explode_xstr xstr of |
|
2188 |
[] => constants [Syntax.constrainC, "Nil", "string"] |
|
2189 |
| cs => mk_string cs) |
|
2190 |
| string_ast_tr asts = raise AST ("string_tr", asts); |
|
2191 |
in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; |
|
2192 |
*} |
|
2193 |
||
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2194 |
ML {* |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2195 |
fun int_of_nibble h = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2196 |
if "0" <= h andalso h <= "9" then ord h - ord "0" |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2197 |
else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2198 |
else raise Match; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2199 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2200 |
fun nibble_of_int i = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2201 |
if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2202 |
*} |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2203 |
|
13366 | 2204 |
print_ast_translation {* |
2205 |
let |
|
2206 |
fun dest_nib (Syntax.Constant c) = |
|
2207 |
(case explode c of |
|
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2208 |
["N", "i", "b", "b", "l", "e", h] => int_of_nibble h |
13366 | 2209 |
| _ => raise Match) |
2210 |
| dest_nib _ = raise Match; |
|
2211 |
||
2212 |
fun dest_chr c1 c2 = |
|
2213 |
let val c = chr (dest_nib c1 * 16 + dest_nib c2) |
|
2214 |
in if Symbol.is_printable c then c else raise Match end; |
|
2215 |
||
2216 |
fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 |
|
2217 |
| dest_char _ = raise Match; |
|
2218 |
||
2219 |
fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; |
|
2220 |
||
2221 |
fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] |
|
2222 |
| char_ast_tr' _ = raise Match; |
|
2223 |
||
2224 |
fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", |
|
2225 |
xstr (map dest_char (Syntax.unfold_ast "_args" args))] |
|
2226 |
| list_ast_tr' ts = raise Match; |
|
2227 |
in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; |
|
2228 |
*} |
|
2229 |
||
15392 | 2230 |
subsubsection {* Code generator setup *} |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2231 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2232 |
ML {* |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2233 |
local |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2234 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2235 |
fun list_codegen thy gr dep b t = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2236 |
let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy dep false) |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2237 |
(gr, HOLogic.dest_list t) |
15531 | 2238 |
in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2239 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2240 |
fun dest_nibble (Const (s, _)) = int_of_nibble (unprefix "List.nibble.Nibble" s) |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2241 |
| dest_nibble _ = raise Match; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2242 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2243 |
fun char_codegen thy gr dep b (Const ("List.char.Char", _) $ c1 $ c2) = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2244 |
(let val c = chr (dest_nibble c1 * 16 + dest_nibble c2) |
15531 | 2245 |
in if Symbol.is_printable c then SOME (gr, Pretty.quote (Pretty.str c)) |
2246 |
else NONE |
|
15570 | 2247 |
end handle Fail _ => NONE | Match => NONE) |
15531 | 2248 |
| char_codegen thy gr dep b _ = NONE; |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2249 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2250 |
in |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2251 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2252 |
val list_codegen_setup = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2253 |
[Codegen.add_codegen "list_codegen" list_codegen, |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2254 |
Codegen.add_codegen "char_codegen" char_codegen]; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2255 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2256 |
end; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2257 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2258 |
val term_of_list = HOLogic.mk_list; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2259 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2260 |
fun gen_list' aG i j = frequency |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2261 |
[(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2262 |
and gen_list aG i = gen_list' aG i i; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2263 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2264 |
val nibbleT = Type ("List.nibble", []); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2265 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2266 |
fun term_of_char c = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2267 |
Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $ |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2268 |
Const ("List.nibble.Nibble" ^ nibble_of_int (ord c div 16), nibbleT) $ |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2269 |
Const ("List.nibble.Nibble" ^ nibble_of_int (ord c mod 16), nibbleT); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2270 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2271 |
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2272 |
*} |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2273 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2274 |
types_code |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2275 |
"list" ("_ list") |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2276 |
"char" ("string") |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2277 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2278 |
consts_code "Cons" ("(_ ::/ _)") |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2279 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2280 |
setup list_codegen_setup |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2281 |
|
13122 | 2282 |
end |