author | paulson |
Fri, 26 Sep 2003 10:34:57 +0200 | |
changeset 14208 | 144f45277d5a |
parent 14187 | 26dfcd0ac436 |
child 14247 | cb32eb89bddd |
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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License: GPL (GNU GENERAL PUBLIC LICENSE) |
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*) |
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||
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header {* The datatype of finite lists *} |
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theory List = PreList: |
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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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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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fold_rel :: "('a * 'c * 'a) set => ('a * 'c list * 'a) set" |
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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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o2l :: "'a option => 'a list" |
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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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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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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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postfix :: "'a list => 'a list => bool" |
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syntax (xsymbols) |
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postfix :: "'a list => 'a list => bool" ("(_/ \<sqsupseteq> _)" [51, 51] 50) |
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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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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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96 |
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primrec |
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"hd(x#xs) = x" |
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primrec |
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"tl([]) = []" |
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"tl(x#xs) = xs" |
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primrec |
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"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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"o2l None = []" |
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"o2l (Some x) = [x]" |
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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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"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 *} |
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-- {* but separate 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 *} |
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-- {* but separate 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 *} |
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-- {* but separate 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] = |
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(case i of 0 => v # xs |
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| 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 *} |
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-- {* but separate 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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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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postfix_def: "postfix xs ys == \<exists>zs. xs = zs @ ys" |
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defs |
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list_all2_def: |
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"list_all2 P xs ys == length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y)" |
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subsection {* Lexicographic orderings on lists *} |
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consts |
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lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" |
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primrec |
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"lexn r 0 = {}" |
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"lexn r (Suc n) = |
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(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int |
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{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" |
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constdefs |
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lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
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"lex r == \<Union>n. lexn r n" |
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lexico :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
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"lexico r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" |
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sublist :: "'a list => nat set => 'a list" |
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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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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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subsection {* @{text lists}: the list-forming operator over sets *} |
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consts lists :: "'a set => 'a list set" |
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inductive "lists A" |
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intros |
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Nil [intro!]: "[]: lists A" |
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Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" |
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inductive_cases listsE [elim!]: "x#l : lists A" |
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lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B" |
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by (unfold lists.defs) (blast intro!: lfp_mono) |
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lemma lists_IntI: |
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assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l |
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by induct blast+ |
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lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" |
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apply (rule mono_Int [THEN equalityI]) |
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apply (simp add: mono_def lists_mono) |
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apply (blast intro!: lists_IntI) |
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done |
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lemma append_in_lists_conv [iff]: |
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"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" |
251 |
by (induct xs) auto |
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||
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subsection {* @{text length} *} |
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text {* |
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Needs to come before @{text "@"} because of theorem @{text |
258 |
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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279 |
lemma length_Suc_conv: |
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" |
281 |
by (induct xs) auto |
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lemma Suc_length_conv: |
284 |
"(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 |
287 |
done |
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lemma impossible_Cons [rule_format]: |
290 |
"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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subsection {* @{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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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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318 |
apply (induct xs) |
14208 | 319 |
apply (case_tac ys, simp, force) |
320 |
apply (case_tac ys, force, simp) |
|
13145 | 321 |
done |
13142 | 322 |
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323 |
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" |
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13145 | 324 |
by simp |
13142 | 325 |
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326 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" |
|
13145 | 327 |
by simp |
13114 | 328 |
|
13142 | 329 |
lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" |
13145 | 330 |
by simp |
13114 | 331 |
|
13142 | 332 |
lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" |
13145 | 333 |
using append_same_eq [of _ _ "[]"] by auto |
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|
13142 | 335 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" |
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using append_same_eq [of "[]"] by auto |
13114 | 337 |
|
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lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" |
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by (induct xs) auto |
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|
13142 | 341 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" |
13145 | 342 |
by (induct xs) auto |
13114 | 343 |
|
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lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" |
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by (simp add: hd_append split: list.split) |
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|
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lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" |
13145 | 348 |
by (simp split: list.split) |
13114 | 349 |
|
13142 | 350 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" |
13145 | 351 |
by (simp add: tl_append split: list.split) |
13114 | 352 |
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353 |
||
13142 | 354 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *} |
13114 | 355 |
|
356 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" |
|
13145 | 357 |
by simp |
13114 | 358 |
|
13142 | 359 |
lemma Cons_eq_appendI: |
13145 | 360 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" |
361 |
by (drule sym) simp |
|
13114 | 362 |
|
13142 | 363 |
lemma append_eq_appendI: |
13145 | 364 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" |
365 |
by (drule sym) simp |
|
13114 | 366 |
|
367 |
||
13142 | 368 |
text {* |
13145 | 369 |
Simplification procedure for all list equalities. |
370 |
Currently only tries to rearrange @{text "@"} to see if |
|
371 |
- both lists end in a singleton list, |
|
372 |
- or both lists end in the same list. |
|
13142 | 373 |
*} |
374 |
||
375 |
ML_setup {* |
|
3507 | 376 |
local |
377 |
||
13122 | 378 |
val append_assoc = thm "append_assoc"; |
379 |
val append_Nil = thm "append_Nil"; |
|
380 |
val append_Cons = thm "append_Cons"; |
|
381 |
val append1_eq_conv = thm "append1_eq_conv"; |
|
382 |
val append_same_eq = thm "append_same_eq"; |
|
383 |
||
13114 | 384 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = |
13462 | 385 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs) |
386 |
| last (Const("List.op @",_) $ _ $ ys) = last ys |
|
387 |
| last t = t; |
|
13114 | 388 |
|
389 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true |
|
13462 | 390 |
| list1 _ = false; |
13114 | 391 |
|
392 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = |
|
13462 | 393 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) |
394 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys |
|
395 |
| butlast xs = Const("List.list.Nil",fastype_of xs); |
|
13114 | 396 |
|
397 |
val rearr_tac = |
|
13462 | 398 |
simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); |
13114 | 399 |
|
400 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
|
13462 | 401 |
let |
402 |
val lastl = last lhs and lastr = last rhs; |
|
403 |
fun rearr conv = |
|
404 |
let |
|
405 |
val lhs1 = butlast lhs and rhs1 = butlast rhs; |
|
406 |
val Type(_,listT::_) = eqT |
|
407 |
val appT = [listT,listT] ---> listT |
|
408 |
val app = Const("List.op @",appT) |
|
409 |
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
|
410 |
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
|
411 |
val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1)); |
13462 | 412 |
in Some ((conv RS (thm RS trans)) RS eq_reflection) end; |
13114 | 413 |
|
13462 | 414 |
in |
415 |
if list1 lastl andalso list1 lastr then rearr append1_eq_conv |
|
416 |
else if lastl aconv lastr then rearr append_same_eq |
|
417 |
else None |
|
418 |
end; |
|
419 |
||
13114 | 420 |
in |
13462 | 421 |
|
422 |
val list_eq_simproc = |
|
423 |
Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; |
|
424 |
||
13114 | 425 |
end; |
426 |
||
427 |
Addsimprocs [list_eq_simproc]; |
|
428 |
*} |
|
429 |
||
430 |
||
13142 | 431 |
subsection {* @{text map} *} |
13114 | 432 |
|
13142 | 433 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" |
13145 | 434 |
by (induct xs) simp_all |
13114 | 435 |
|
13142 | 436 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" |
13145 | 437 |
by (rule ext, induct_tac xs) auto |
13114 | 438 |
|
13142 | 439 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" |
13145 | 440 |
by (induct xs) auto |
13114 | 441 |
|
13142 | 442 |
lemma map_compose: "map (f o g) xs = map f (map g xs)" |
13145 | 443 |
by (induct xs) (auto simp add: o_def) |
13114 | 444 |
|
13142 | 445 |
lemma rev_map: "rev (map f xs) = map f (rev xs)" |
13145 | 446 |
by (induct xs) auto |
13114 | 447 |
|
13737 | 448 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" |
449 |
by (induct xs) auto |
|
450 |
||
13366 | 451 |
lemma map_cong [recdef_cong]: |
13145 | 452 |
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" |
453 |
-- {* a congruence rule for @{text map} *} |
|
13737 | 454 |
by simp |
13114 | 455 |
|
13142 | 456 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" |
13145 | 457 |
by (cases xs) auto |
13114 | 458 |
|
13142 | 459 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" |
13145 | 460 |
by (cases xs) auto |
13114 | 461 |
|
14025 | 462 |
lemma map_eq_Cons_conv[iff]: |
463 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" |
|
13145 | 464 |
by (cases xs) auto |
13114 | 465 |
|
14025 | 466 |
lemma Cons_eq_map_conv[iff]: |
467 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" |
|
468 |
by (cases ys) auto |
|
469 |
||
14111 | 470 |
lemma ex_map_conv: |
471 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" |
|
472 |
by(induct ys, auto) |
|
473 |
||
13114 | 474 |
lemma map_injective: |
14025 | 475 |
"!!xs. map f xs = map f ys ==> (\<forall>x y. f x = f y --> x = y) ==> xs = ys" |
476 |
by (induct ys) auto |
|
13114 | 477 |
|
478 |
lemma inj_mapI: "inj f ==> inj (map f)" |
|
13585 | 479 |
by (rules dest: map_injective injD intro: inj_onI) |
13114 | 480 |
|
481 |
lemma inj_mapD: "inj (map f) ==> inj f" |
|
14208 | 482 |
apply (unfold inj_on_def, clarify) |
13145 | 483 |
apply (erule_tac x = "[x]" in ballE) |
14208 | 484 |
apply (erule_tac x = "[y]" in ballE, simp, blast) |
13145 | 485 |
apply blast |
486 |
done |
|
13114 | 487 |
|
488 |
lemma inj_map: "inj (map f) = inj f" |
|
13145 | 489 |
by (blast dest: inj_mapD intro: inj_mapI) |
13114 | 490 |
|
491 |
||
13142 | 492 |
subsection {* @{text rev} *} |
13114 | 493 |
|
13142 | 494 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" |
13145 | 495 |
by (induct xs) auto |
13114 | 496 |
|
13142 | 497 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" |
13145 | 498 |
by (induct xs) auto |
13114 | 499 |
|
13142 | 500 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" |
13145 | 501 |
by (induct xs) auto |
13114 | 502 |
|
13142 | 503 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" |
13145 | 504 |
by (induct xs) auto |
13114 | 505 |
|
13142 | 506 |
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" |
14208 | 507 |
apply (induct xs, force) |
508 |
apply (case_tac ys, simp, force) |
|
13145 | 509 |
done |
13114 | 510 |
|
13366 | 511 |
lemma rev_induct [case_names Nil snoc]: |
512 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" |
|
13145 | 513 |
apply(subst rev_rev_ident[symmetric]) |
514 |
apply(rule_tac list = "rev xs" in list.induct, simp_all) |
|
515 |
done |
|
13114 | 516 |
|
13145 | 517 |
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" |
13114 | 518 |
|
13366 | 519 |
lemma rev_exhaust [case_names Nil snoc]: |
520 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" |
|
13145 | 521 |
by (induct xs rule: rev_induct) auto |
13114 | 522 |
|
13366 | 523 |
lemmas rev_cases = rev_exhaust |
524 |
||
13114 | 525 |
|
13142 | 526 |
subsection {* @{text set} *} |
13114 | 527 |
|
13142 | 528 |
lemma finite_set [iff]: "finite (set xs)" |
13145 | 529 |
by (induct xs) auto |
13114 | 530 |
|
13142 | 531 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" |
13145 | 532 |
by (induct xs) auto |
13114 | 533 |
|
14099 | 534 |
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l" |
14208 | 535 |
by (case_tac l, auto) |
14099 | 536 |
|
13142 | 537 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" |
13145 | 538 |
by auto |
13114 | 539 |
|
14099 | 540 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" |
541 |
by auto |
|
542 |
||
13142 | 543 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" |
13145 | 544 |
by (induct xs) auto |
13114 | 545 |
|
13142 | 546 |
lemma set_rev [simp]: "set (rev xs) = set xs" |
13145 | 547 |
by (induct xs) auto |
13114 | 548 |
|
13142 | 549 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)" |
13145 | 550 |
by (induct xs) auto |
13114 | 551 |
|
13142 | 552 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" |
13145 | 553 |
by (induct xs) auto |
13114 | 554 |
|
13142 | 555 |
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" |
14208 | 556 |
apply (induct j, simp_all) |
557 |
apply (erule ssubst, auto) |
|
13145 | 558 |
done |
13114 | 559 |
|
13142 | 560 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" |
14208 | 561 |
apply (induct xs, simp, simp) |
13145 | 562 |
apply (rule iffI) |
563 |
apply (blast intro: eq_Nil_appendI Cons_eq_appendI) |
|
564 |
apply (erule exE)+ |
|
14208 | 565 |
apply (case_tac ys, auto) |
13145 | 566 |
done |
13142 | 567 |
|
568 |
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" |
|
13145 | 569 |
-- {* eliminate @{text lists} in favour of @{text set} *} |
570 |
by (induct xs) auto |
|
13142 | 571 |
|
572 |
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" |
|
13145 | 573 |
by (rule in_lists_conv_set [THEN iffD1]) |
13142 | 574 |
|
575 |
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" |
|
13145 | 576 |
by (rule in_lists_conv_set [THEN iffD2]) |
13114 | 577 |
|
13508 | 578 |
lemma finite_list: "finite A ==> EX l. set l = A" |
579 |
apply (erule finite_induct, auto) |
|
580 |
apply (rule_tac x="x#l" in exI, auto) |
|
581 |
done |
|
582 |
||
13114 | 583 |
|
13142 | 584 |
subsection {* @{text mem} *} |
13114 | 585 |
|
586 |
lemma set_mem_eq: "(x mem xs) = (x : set xs)" |
|
13145 | 587 |
by (induct xs) auto |
13114 | 588 |
|
589 |
||
13142 | 590 |
subsection {* @{text list_all} *} |
13114 | 591 |
|
13142 | 592 |
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" |
13145 | 593 |
by (induct xs) auto |
13114 | 594 |
|
13142 | 595 |
lemma list_all_append [simp]: |
13145 | 596 |
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" |
597 |
by (induct xs) auto |
|
13114 | 598 |
|
599 |
||
13142 | 600 |
subsection {* @{text filter} *} |
13114 | 601 |
|
13142 | 602 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" |
13145 | 603 |
by (induct xs) auto |
13114 | 604 |
|
13142 | 605 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" |
13145 | 606 |
by (induct xs) auto |
13114 | 607 |
|
13142 | 608 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" |
13145 | 609 |
by (induct xs) auto |
13114 | 610 |
|
13142 | 611 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" |
13145 | 612 |
by (induct xs) auto |
13114 | 613 |
|
13142 | 614 |
lemma length_filter [simp]: "length (filter P xs) \<le> length xs" |
13145 | 615 |
by (induct xs) (auto simp add: le_SucI) |
13114 | 616 |
|
13142 | 617 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" |
13145 | 618 |
by auto |
13114 | 619 |
|
620 |
||
13142 | 621 |
subsection {* @{text concat} *} |
13114 | 622 |
|
13142 | 623 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" |
13145 | 624 |
by (induct xs) auto |
13114 | 625 |
|
13142 | 626 |
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 627 |
by (induct xss) auto |
13114 | 628 |
|
13142 | 629 |
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 630 |
by (induct xss) auto |
13114 | 631 |
|
13142 | 632 |
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" |
13145 | 633 |
by (induct xs) auto |
13114 | 634 |
|
13142 | 635 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" |
13145 | 636 |
by (induct xs) auto |
13114 | 637 |
|
13142 | 638 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" |
13145 | 639 |
by (induct xs) auto |
13114 | 640 |
|
13142 | 641 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" |
13145 | 642 |
by (induct xs) auto |
13114 | 643 |
|
644 |
||
13142 | 645 |
subsection {* @{text nth} *} |
13114 | 646 |
|
13142 | 647 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" |
13145 | 648 |
by auto |
13114 | 649 |
|
13142 | 650 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" |
13145 | 651 |
by auto |
13114 | 652 |
|
13142 | 653 |
declare nth.simps [simp del] |
13114 | 654 |
|
655 |
lemma nth_append: |
|
13145 | 656 |
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" |
14208 | 657 |
apply (induct "xs", simp) |
658 |
apply (case_tac n, auto) |
|
13145 | 659 |
done |
13114 | 660 |
|
13142 | 661 |
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" |
14208 | 662 |
apply (induct xs, simp) |
663 |
apply (case_tac n, auto) |
|
13145 | 664 |
done |
13114 | 665 |
|
13142 | 666 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" |
14208 | 667 |
apply (induct_tac xs, simp, simp) |
13145 | 668 |
apply safe |
14208 | 669 |
apply (rule_tac x = 0 in exI, simp) |
670 |
apply (rule_tac x = "Suc i" in exI, simp) |
|
671 |
apply (case_tac i, simp) |
|
13145 | 672 |
apply (rename_tac j) |
14208 | 673 |
apply (rule_tac x = j in exI, simp) |
13145 | 674 |
done |
13114 | 675 |
|
13145 | 676 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" |
677 |
by (auto simp add: set_conv_nth) |
|
13114 | 678 |
|
13142 | 679 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" |
13145 | 680 |
by (auto simp add: set_conv_nth) |
13114 | 681 |
|
682 |
lemma all_nth_imp_all_set: |
|
13145 | 683 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x" |
684 |
by (auto simp add: set_conv_nth) |
|
13114 | 685 |
|
686 |
lemma all_set_conv_all_nth: |
|
13145 | 687 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" |
688 |
by (auto simp add: set_conv_nth) |
|
13114 | 689 |
|
690 |
||
13142 | 691 |
subsection {* @{text list_update} *} |
13114 | 692 |
|
13142 | 693 |
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" |
13145 | 694 |
by (induct xs) (auto split: nat.split) |
13114 | 695 |
|
696 |
lemma nth_list_update: |
|
13145 | 697 |
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" |
698 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
|
13114 | 699 |
|
13142 | 700 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" |
13145 | 701 |
by (simp add: nth_list_update) |
13114 | 702 |
|
13142 | 703 |
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" |
13145 | 704 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
13114 | 705 |
|
13142 | 706 |
lemma list_update_overwrite [simp]: |
13145 | 707 |
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" |
708 |
by (induct xs) (auto split: nat.split) |
|
13114 | 709 |
|
14187 | 710 |
lemma list_update_id[simp]: "!!i. i < length xs \<Longrightarrow> xs[i := xs!i] = xs" |
14208 | 711 |
apply (induct xs, simp) |
14187 | 712 |
apply(simp split:nat.splits) |
713 |
done |
|
714 |
||
13114 | 715 |
lemma list_update_same_conv: |
13145 | 716 |
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" |
717 |
by (induct xs) (auto split: nat.split) |
|
13114 | 718 |
|
14187 | 719 |
lemma list_update_append1: |
720 |
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" |
|
14208 | 721 |
apply (induct xs, simp) |
14187 | 722 |
apply(simp split:nat.split) |
723 |
done |
|
724 |
||
13114 | 725 |
lemma update_zip: |
13145 | 726 |
"!!i xy xs. length xs = length ys ==> |
727 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" |
|
728 |
by (induct ys) (auto, case_tac xs, auto split: nat.split) |
|
13114 | 729 |
|
730 |
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" |
|
13145 | 731 |
by (induct xs) (auto split: nat.split) |
13114 | 732 |
|
733 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" |
|
13145 | 734 |
by (blast dest!: set_update_subset_insert [THEN subsetD]) |
13114 | 735 |
|
736 |
||
13142 | 737 |
subsection {* @{text last} and @{text butlast} *} |
13114 | 738 |
|
13142 | 739 |
lemma last_snoc [simp]: "last (xs @ [x]) = x" |
13145 | 740 |
by (induct xs) auto |
13114 | 741 |
|
13142 | 742 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" |
13145 | 743 |
by (induct xs) auto |
13114 | 744 |
|
13142 | 745 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" |
13145 | 746 |
by (induct xs rule: rev_induct) auto |
13114 | 747 |
|
748 |
lemma butlast_append: |
|
13145 | 749 |
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" |
750 |
by (induct xs) auto |
|
13114 | 751 |
|
13142 | 752 |
lemma append_butlast_last_id [simp]: |
13145 | 753 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" |
754 |
by (induct xs) auto |
|
13114 | 755 |
|
13142 | 756 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" |
13145 | 757 |
by (induct xs) (auto split: split_if_asm) |
13114 | 758 |
|
759 |
lemma in_set_butlast_appendI: |
|
13145 | 760 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" |
761 |
by (auto dest: in_set_butlastD simp add: butlast_append) |
|
13114 | 762 |
|
13142 | 763 |
|
764 |
subsection {* @{text take} and @{text drop} *} |
|
13114 | 765 |
|
13142 | 766 |
lemma take_0 [simp]: "take 0 xs = []" |
13145 | 767 |
by (induct xs) auto |
13114 | 768 |
|
13142 | 769 |
lemma drop_0 [simp]: "drop 0 xs = xs" |
13145 | 770 |
by (induct xs) auto |
13114 | 771 |
|
13142 | 772 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" |
13145 | 773 |
by simp |
13114 | 774 |
|
13142 | 775 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" |
13145 | 776 |
by simp |
13114 | 777 |
|
13142 | 778 |
declare take_Cons [simp del] and drop_Cons [simp del] |
13114 | 779 |
|
14187 | 780 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" |
781 |
by(cases xs, simp_all) |
|
782 |
||
783 |
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" |
|
784 |
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) |
|
785 |
||
786 |
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" |
|
14208 | 787 |
apply (induct xs, simp) |
14187 | 788 |
apply(simp add:drop_Cons nth_Cons split:nat.splits) |
789 |
done |
|
790 |
||
13913 | 791 |
lemma take_Suc_conv_app_nth: |
792 |
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" |
|
14208 | 793 |
apply (induct xs, simp) |
794 |
apply (case_tac i, auto) |
|
13913 | 795 |
done |
796 |
||
13142 | 797 |
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" |
13145 | 798 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 799 |
|
13142 | 800 |
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" |
13145 | 801 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 802 |
|
13142 | 803 |
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" |
13145 | 804 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 805 |
|
13142 | 806 |
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" |
13145 | 807 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 808 |
|
13142 | 809 |
lemma take_append [simp]: |
13145 | 810 |
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" |
811 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 812 |
|
13142 | 813 |
lemma drop_append [simp]: |
13145 | 814 |
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" |
815 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 816 |
|
13142 | 817 |
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" |
14208 | 818 |
apply (induct m, auto) |
819 |
apply (case_tac xs, auto) |
|
820 |
apply (case_tac na, auto) |
|
13145 | 821 |
done |
13114 | 822 |
|
13142 | 823 |
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" |
14208 | 824 |
apply (induct m, auto) |
825 |
apply (case_tac xs, auto) |
|
13145 | 826 |
done |
13114 | 827 |
|
828 |
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" |
|
14208 | 829 |
apply (induct m, auto) |
830 |
apply (case_tac xs, auto) |
|
13145 | 831 |
done |
13114 | 832 |
|
13142 | 833 |
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" |
14208 | 834 |
apply (induct n, auto) |
835 |
apply (case_tac xs, auto) |
|
13145 | 836 |
done |
13114 | 837 |
|
838 |
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" |
|
14208 | 839 |
apply (induct n, auto) |
840 |
apply (case_tac xs, auto) |
|
13145 | 841 |
done |
13114 | 842 |
|
13142 | 843 |
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" |
14208 | 844 |
apply (induct n, auto) |
845 |
apply (case_tac xs, auto) |
|
13145 | 846 |
done |
13114 | 847 |
|
848 |
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" |
|
14208 | 849 |
apply (induct xs, auto) |
850 |
apply (case_tac i, auto) |
|
13145 | 851 |
done |
13114 | 852 |
|
853 |
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" |
|
14208 | 854 |
apply (induct xs, auto) |
855 |
apply (case_tac i, auto) |
|
13145 | 856 |
done |
13114 | 857 |
|
13142 | 858 |
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" |
14208 | 859 |
apply (induct xs, auto) |
860 |
apply (case_tac n, blast) |
|
861 |
apply (case_tac i, auto) |
|
13145 | 862 |
done |
13114 | 863 |
|
13142 | 864 |
lemma nth_drop [simp]: |
13145 | 865 |
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" |
14208 | 866 |
apply (induct n, auto) |
867 |
apply (case_tac xs, auto) |
|
13145 | 868 |
done |
3507 | 869 |
|
14025 | 870 |
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" |
871 |
by(induct xs)(auto simp:take_Cons split:nat.split) |
|
872 |
||
873 |
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" |
|
874 |
by(induct xs)(auto simp:drop_Cons split:nat.split) |
|
875 |
||
14187 | 876 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" |
877 |
using set_take_subset by fast |
|
878 |
||
879 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" |
|
880 |
using set_drop_subset by fast |
|
881 |
||
13114 | 882 |
lemma append_eq_conv_conj: |
13145 | 883 |
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" |
14208 | 884 |
apply (induct xs, simp, clarsimp) |
885 |
apply (case_tac zs, auto) |
|
13145 | 886 |
done |
13142 | 887 |
|
14050 | 888 |
lemma take_add [rule_format]: |
889 |
"\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" |
|
890 |
apply (induct xs, auto) |
|
891 |
apply (case_tac i, simp_all) |
|
892 |
done |
|
893 |
||
13114 | 894 |
|
13142 | 895 |
subsection {* @{text takeWhile} and @{text dropWhile} *} |
13114 | 896 |
|
13142 | 897 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" |
13145 | 898 |
by (induct xs) auto |
13114 | 899 |
|
13142 | 900 |
lemma takeWhile_append1 [simp]: |
13145 | 901 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" |
902 |
by (induct xs) auto |
|
13114 | 903 |
|
13142 | 904 |
lemma takeWhile_append2 [simp]: |
13145 | 905 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" |
906 |
by (induct xs) auto |
|
13114 | 907 |
|
13142 | 908 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" |
13145 | 909 |
by (induct xs) auto |
13114 | 910 |
|
13142 | 911 |
lemma dropWhile_append1 [simp]: |
13145 | 912 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" |
913 |
by (induct xs) auto |
|
13114 | 914 |
|
13142 | 915 |
lemma dropWhile_append2 [simp]: |
13145 | 916 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" |
917 |
by (induct xs) auto |
|
13114 | 918 |
|
13142 | 919 |
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" |
13145 | 920 |
by (induct xs) (auto split: split_if_asm) |
13114 | 921 |
|
13913 | 922 |
lemma takeWhile_eq_all_conv[simp]: |
923 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" |
|
924 |
by(induct xs, auto) |
|
925 |
||
926 |
lemma dropWhile_eq_Nil_conv[simp]: |
|
927 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" |
|
928 |
by(induct xs, auto) |
|
929 |
||
930 |
lemma dropWhile_eq_Cons_conv: |
|
931 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" |
|
932 |
by(induct xs, auto) |
|
933 |
||
13114 | 934 |
|
13142 | 935 |
subsection {* @{text zip} *} |
13114 | 936 |
|
13142 | 937 |
lemma zip_Nil [simp]: "zip [] ys = []" |
13145 | 938 |
by (induct ys) auto |
13114 | 939 |
|
13142 | 940 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" |
13145 | 941 |
by simp |
13114 | 942 |
|
13142 | 943 |
declare zip_Cons [simp del] |
13114 | 944 |
|
13142 | 945 |
lemma length_zip [simp]: |
13145 | 946 |
"!!xs. length (zip xs ys) = min (length xs) (length ys)" |
14208 | 947 |
apply (induct ys, simp) |
948 |
apply (case_tac xs, auto) |
|
13145 | 949 |
done |
13114 | 950 |
|
951 |
lemma zip_append1: |
|
13145 | 952 |
"!!xs. zip (xs @ ys) zs = |
953 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" |
|
14208 | 954 |
apply (induct zs, simp) |
955 |
apply (case_tac xs, simp_all) |
|
13145 | 956 |
done |
13114 | 957 |
|
958 |
lemma zip_append2: |
|
13145 | 959 |
"!!ys. zip xs (ys @ zs) = |
960 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" |
|
14208 | 961 |
apply (induct xs, simp) |
962 |
apply (case_tac ys, simp_all) |
|
13145 | 963 |
done |
13114 | 964 |
|
13142 | 965 |
lemma zip_append [simp]: |
966 |
"[| length xs = length us; length ys = length vs |] ==> |
|
13145 | 967 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" |
968 |
by (simp add: zip_append1) |
|
13114 | 969 |
|
970 |
lemma zip_rev: |
|
13145 | 971 |
"!!xs. length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" |
14208 | 972 |
apply (induct ys, simp) |
973 |
apply (case_tac xs, simp_all) |
|
13145 | 974 |
done |
13114 | 975 |
|
13142 | 976 |
lemma nth_zip [simp]: |
13145 | 977 |
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" |
14208 | 978 |
apply (induct ys, simp) |
13145 | 979 |
apply (case_tac xs) |
980 |
apply (simp_all add: nth.simps split: nat.split) |
|
981 |
done |
|
13114 | 982 |
|
983 |
lemma set_zip: |
|
13145 | 984 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" |
985 |
by (simp add: set_conv_nth cong: rev_conj_cong) |
|
13114 | 986 |
|
987 |
lemma zip_update: |
|
13145 | 988 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" |
989 |
by (rule sym, simp add: update_zip) |
|
13114 | 990 |
|
13142 | 991 |
lemma zip_replicate [simp]: |
13145 | 992 |
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" |
14208 | 993 |
apply (induct i, auto) |
994 |
apply (case_tac j, auto) |
|
13145 | 995 |
done |
13114 | 996 |
|
13142 | 997 |
|
998 |
subsection {* @{text list_all2} *} |
|
13114 | 999 |
|
1000 |
lemma list_all2_lengthD: "list_all2 P xs ys ==> length xs = length ys" |
|
13145 | 1001 |
by (simp add: list_all2_def) |
13114 | 1002 |
|
13142 | 1003 |
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" |
13145 | 1004 |
by (simp add: list_all2_def) |
13114 | 1005 |
|
13142 | 1006 |
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" |
13145 | 1007 |
by (simp add: list_all2_def) |
13114 | 1008 |
|
13142 | 1009 |
lemma list_all2_Cons [iff]: |
13145 | 1010 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" |
1011 |
by (auto simp add: list_all2_def) |
|
13114 | 1012 |
|
1013 |
lemma list_all2_Cons1: |
|
13145 | 1014 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" |
1015 |
by (cases ys) auto |
|
13114 | 1016 |
|
1017 |
lemma list_all2_Cons2: |
|
13145 | 1018 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" |
1019 |
by (cases xs) auto |
|
13114 | 1020 |
|
13142 | 1021 |
lemma list_all2_rev [iff]: |
13145 | 1022 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" |
1023 |
by (simp add: list_all2_def zip_rev cong: conj_cong) |
|
13114 | 1024 |
|
13863 | 1025 |
lemma list_all2_rev1: |
1026 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" |
|
1027 |
by (subst list_all2_rev [symmetric]) simp |
|
1028 |
||
13114 | 1029 |
lemma list_all2_append1: |
13145 | 1030 |
"list_all2 P (xs @ ys) zs = |
1031 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> |
|
1032 |
list_all2 P xs us \<and> list_all2 P ys vs)" |
|
1033 |
apply (simp add: list_all2_def zip_append1) |
|
1034 |
apply (rule iffI) |
|
1035 |
apply (rule_tac x = "take (length xs) zs" in exI) |
|
1036 |
apply (rule_tac x = "drop (length xs) zs" in exI) |
|
14208 | 1037 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1038 |
apply (simp add: ball_Un) |
1039 |
done |
|
13114 | 1040 |
|
1041 |
lemma list_all2_append2: |
|
13145 | 1042 |
"list_all2 P xs (ys @ zs) = |
1043 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> |
|
1044 |
list_all2 P us ys \<and> list_all2 P vs zs)" |
|
1045 |
apply (simp add: list_all2_def zip_append2) |
|
1046 |
apply (rule iffI) |
|
1047 |
apply (rule_tac x = "take (length ys) xs" in exI) |
|
1048 |
apply (rule_tac x = "drop (length ys) xs" in exI) |
|
14208 | 1049 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1050 |
apply (simp add: ball_Un) |
1051 |
done |
|
13114 | 1052 |
|
13863 | 1053 |
lemma list_all2_append: |
1054 |
"\<And>b. length a = length b \<Longrightarrow> |
|
1055 |
list_all2 P (a@c) (b@d) = (list_all2 P a b \<and> list_all2 P c d)" |
|
14208 | 1056 |
apply (induct a, simp) |
1057 |
apply (case_tac b, auto) |
|
13863 | 1058 |
done |
1059 |
||
1060 |
lemma list_all2_appendI [intro?, trans]: |
|
1061 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" |
|
1062 |
by (simp add: list_all2_append list_all2_lengthD) |
|
1063 |
||
13114 | 1064 |
lemma list_all2_conv_all_nth: |
13145 | 1065 |
"list_all2 P xs ys = |
1066 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" |
|
1067 |
by (force simp add: list_all2_def set_zip) |
|
13114 | 1068 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1069 |
lemma list_all2_trans: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1070 |
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
|
1071 |
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
|
1072 |
(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
|
1073 |
proof (induct as) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1074 |
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
|
1075 |
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
|
1076 |
proof (induct bs) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1077 |
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
|
1078 |
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
|
1079 |
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
|
1080 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1081 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1082 |
|
13863 | 1083 |
lemma list_all2_all_nthI [intro?]: |
1084 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" |
|
1085 |
by (simp add: list_all2_conv_all_nth) |
|
1086 |
||
1087 |
lemma list_all2_nthD [dest?]: |
|
1088 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
|
1089 |
by (simp add: list_all2_conv_all_nth) |
|
1090 |
||
1091 |
lemma list_all2_map1: |
|
1092 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" |
|
1093 |
by (simp add: list_all2_conv_all_nth) |
|
1094 |
||
1095 |
lemma list_all2_map2: |
|
1096 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" |
|
1097 |
by (auto simp add: list_all2_conv_all_nth) |
|
1098 |
||
1099 |
lemma list_all2_refl: |
|
1100 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" |
|
1101 |
by (simp add: list_all2_conv_all_nth) |
|
1102 |
||
1103 |
lemma list_all2_update_cong: |
|
1104 |
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1105 |
by (simp add: list_all2_conv_all_nth nth_list_update) |
|
1106 |
||
1107 |
lemma list_all2_update_cong2: |
|
1108 |
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1109 |
by (simp add: list_all2_lengthD list_all2_update_cong) |
|
1110 |
||
1111 |
lemma list_all2_dropI [intro?]: |
|
1112 |
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" |
|
14208 | 1113 |
apply (induct as, simp) |
13863 | 1114 |
apply (clarsimp simp add: list_all2_Cons1) |
14208 | 1115 |
apply (case_tac n, simp, simp) |
13863 | 1116 |
done |
1117 |
||
1118 |
lemma list_all2_mono [intro?]: |
|
1119 |
"\<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 | 1120 |
apply (induct x, simp) |
1121 |
apply (case_tac y, auto) |
|
13863 | 1122 |
done |
1123 |
||
13142 | 1124 |
|
1125 |
subsection {* @{text foldl} *} |
|
1126 |
||
1127 |
lemma foldl_append [simp]: |
|
13145 | 1128 |
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" |
1129 |
by (induct xs) auto |
|
13142 | 1130 |
|
1131 |
text {* |
|
13145 | 1132 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more |
1133 |
difficult to use because it requires an additional transitivity step. |
|
13142 | 1134 |
*} |
1135 |
||
1136 |
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" |
|
13145 | 1137 |
by (induct ns) auto |
13142 | 1138 |
|
1139 |
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" |
|
13145 | 1140 |
by (force intro: start_le_sum simp add: in_set_conv_decomp) |
13142 | 1141 |
|
1142 |
lemma sum_eq_0_conv [iff]: |
|
13145 | 1143 |
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" |
1144 |
by (induct ns) auto |
|
13114 | 1145 |
|
1146 |
||
14099 | 1147 |
subsection {* folding a relation over a list *} |
1148 |
||
1149 |
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*) |
|
1150 |
inductive "fold_rel R" intros |
|
1151 |
Nil: "(a, [],a) : fold_rel R" |
|
1152 |
Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R" |
|
1153 |
inductive_cases fold_rel_elim_case [elim!]: |
|
14208 | 1154 |
"(a, [] , b) : fold_rel R" |
14099 | 1155 |
"(a, x#xs, b) : fold_rel R" |
1156 |
||
1157 |
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" |
|
1158 |
by (simp add: fold_rel.Nil) |
|
1159 |
declare fold_rel.Cons [intro!] |
|
1160 |
||
1161 |
||
13142 | 1162 |
subsection {* @{text upto} *} |
13114 | 1163 |
|
13142 | 1164 |
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" |
13145 | 1165 |
-- {* Does not terminate! *} |
1166 |
by (induct j) auto |
|
13142 | 1167 |
|
1168 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" |
|
13145 | 1169 |
by (subst upt_rec) simp |
13114 | 1170 |
|
13142 | 1171 |
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" |
13145 | 1172 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} |
1173 |
by simp |
|
13114 | 1174 |
|
13142 | 1175 |
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" |
13145 | 1176 |
apply(rule trans) |
1177 |
apply(subst upt_rec) |
|
14208 | 1178 |
prefer 2 apply (rule refl, simp) |
13145 | 1179 |
done |
13114 | 1180 |
|
13142 | 1181 |
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" |
13145 | 1182 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *} |
1183 |
by (induct k) auto |
|
13114 | 1184 |
|
13142 | 1185 |
lemma length_upt [simp]: "length [i..j(] = j - i" |
13145 | 1186 |
by (induct j) (auto simp add: Suc_diff_le) |
13114 | 1187 |
|
13142 | 1188 |
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" |
13145 | 1189 |
apply (induct j) |
1190 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) |
|
1191 |
done |
|
13114 | 1192 |
|
13142 | 1193 |
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" |
14208 | 1194 |
apply (induct m, simp) |
13145 | 1195 |
apply (subst upt_rec) |
1196 |
apply (rule sym) |
|
1197 |
apply (subst upt_rec) |
|
1198 |
apply (simp del: upt.simps) |
|
1199 |
done |
|
3507 | 1200 |
|
13114 | 1201 |
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" |
13145 | 1202 |
by (induct n) auto |
13114 | 1203 |
|
1204 |
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)" |
|
13145 | 1205 |
apply (induct n m rule: diff_induct) |
1206 |
prefer 3 apply (subst map_Suc_upt[symmetric]) |
|
1207 |
apply (auto simp add: less_diff_conv nth_upt) |
|
1208 |
done |
|
13114 | 1209 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1210 |
lemma nth_take_lemma: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1211 |
"!!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
|
1212 |
(!!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
|
1213 |
apply (atomize, induct k) |
14208 | 1214 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) |
13145 | 1215 |
txt {* Both lists must be non-empty *} |
14208 | 1216 |
apply (case_tac xs, simp) |
1217 |
apply (case_tac ys, clarify) |
|
13145 | 1218 |
apply (simp (no_asm_use)) |
1219 |
apply clarify |
|
1220 |
txt {* prenexing's needed, not miniscoping *} |
|
1221 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) |
|
1222 |
apply blast |
|
1223 |
done |
|
13114 | 1224 |
|
1225 |
lemma nth_equalityI: |
|
1226 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" |
|
13145 | 1227 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) |
1228 |
apply (simp_all add: take_all) |
|
1229 |
done |
|
13142 | 1230 |
|
13863 | 1231 |
(* needs nth_equalityI *) |
1232 |
lemma list_all2_antisym: |
|
1233 |
"\<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> |
|
1234 |
\<Longrightarrow> xs = ys" |
|
1235 |
apply (simp add: list_all2_conv_all_nth) |
|
14208 | 1236 |
apply (rule nth_equalityI, blast, simp) |
13863 | 1237 |
done |
1238 |
||
13142 | 1239 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" |
13145 | 1240 |
-- {* The famous take-lemma. *} |
1241 |
apply (drule_tac x = "max (length xs) (length ys)" in spec) |
|
1242 |
apply (simp add: le_max_iff_disj take_all) |
|
1243 |
done |
|
13142 | 1244 |
|
1245 |
||
1246 |
subsection {* @{text "distinct"} and @{text remdups} *} |
|
1247 |
||
1248 |
lemma distinct_append [simp]: |
|
13145 | 1249 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" |
1250 |
by (induct xs) auto |
|
13142 | 1251 |
|
1252 |
lemma set_remdups [simp]: "set (remdups xs) = set xs" |
|
13145 | 1253 |
by (induct xs) (auto simp add: insert_absorb) |
13142 | 1254 |
|
1255 |
lemma distinct_remdups [iff]: "distinct (remdups xs)" |
|
13145 | 1256 |
by (induct xs) auto |
13142 | 1257 |
|
1258 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" |
|
13145 | 1259 |
by (induct xs) auto |
13114 | 1260 |
|
13142 | 1261 |
text {* |
13145 | 1262 |
It is best to avoid this indexed version of distinct, but sometimes |
1263 |
it is useful. *} |
|
13142 | 1264 |
lemma distinct_conv_nth: |
13145 | 1265 |
"distinct xs = (\<forall>i j. i < size xs \<and> j < size xs \<and> i \<noteq> j --> xs!i \<noteq> xs!j)" |
14208 | 1266 |
apply (induct_tac xs, simp, simp) |
1267 |
apply (rule iffI, clarsimp) |
|
13145 | 1268 |
apply (case_tac i) |
14208 | 1269 |
apply (case_tac j, simp) |
13145 | 1270 |
apply (simp add: set_conv_nth) |
1271 |
apply (case_tac j) |
|
14208 | 1272 |
apply (clarsimp simp add: set_conv_nth, simp) |
13145 | 1273 |
apply (rule conjI) |
1274 |
apply (clarsimp simp add: set_conv_nth) |
|
1275 |
apply (erule_tac x = 0 in allE) |
|
14208 | 1276 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp) |
13145 | 1277 |
apply (erule_tac x = "Suc i" in allE) |
14208 | 1278 |
apply (erule_tac x = "Suc j" in allE, simp) |
13145 | 1279 |
done |
13114 | 1280 |
|
1281 |
||
13142 | 1282 |
subsection {* @{text replicate} *} |
13114 | 1283 |
|
13142 | 1284 |
lemma length_replicate [simp]: "length (replicate n x) = n" |
13145 | 1285 |
by (induct n) auto |
13124 | 1286 |
|
13142 | 1287 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" |
13145 | 1288 |
by (induct n) auto |
13114 | 1289 |
|
1290 |
lemma replicate_app_Cons_same: |
|
13145 | 1291 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs" |
1292 |
by (induct n) auto |
|
13114 | 1293 |
|
13142 | 1294 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" |
14208 | 1295 |
apply (induct n, simp) |
13145 | 1296 |
apply (simp add: replicate_app_Cons_same) |
1297 |
done |
|
13114 | 1298 |
|
13142 | 1299 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" |
13145 | 1300 |
by (induct n) auto |
13114 | 1301 |
|
13142 | 1302 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" |
13145 | 1303 |
by (induct n) auto |
13114 | 1304 |
|
13142 | 1305 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x" |
13145 | 1306 |
by (induct n) auto |
13114 | 1307 |
|
13142 | 1308 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" |
13145 | 1309 |
by (atomize (full), induct n) auto |
13114 | 1310 |
|
13142 | 1311 |
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" |
14208 | 1312 |
apply (induct n, simp) |
13145 | 1313 |
apply (simp add: nth_Cons split: nat.split) |
1314 |
done |
|
13114 | 1315 |
|
13142 | 1316 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" |
13145 | 1317 |
by (induct n) auto |
13114 | 1318 |
|
13142 | 1319 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" |
13145 | 1320 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) |
13114 | 1321 |
|
13142 | 1322 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" |
13145 | 1323 |
by auto |
13114 | 1324 |
|
13142 | 1325 |
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" |
13145 | 1326 |
by (simp add: set_replicate_conv_if split: split_if_asm) |
13114 | 1327 |
|
1328 |
||
14099 | 1329 |
subsection {* @{text postfix} *} |
1330 |
||
1331 |
lemma postfix_refl [simp, intro!]: "xs \<sqsupseteq> xs" by (auto simp add: postfix_def) |
|
1332 |
lemma postfix_trans: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> zs\<rbrakk> \<Longrightarrow> xs \<sqsupseteq> zs" |
|
1333 |
by (auto simp add: postfix_def) |
|
1334 |
lemma postfix_antisym: "\<lbrakk>xs \<sqsupseteq> ys; ys \<sqsupseteq> xs\<rbrakk> \<Longrightarrow> xs = ys" |
|
1335 |
by (auto simp add: postfix_def) |
|
1336 |
||
1337 |
lemma postfix_emptyI [simp, intro!]: "xs \<sqsupseteq> []" by (auto simp add: postfix_def) |
|
1338 |
lemma postfix_emptyD [dest!]: "[] \<sqsupseteq> xs \<Longrightarrow> xs = []"by(auto simp add:postfix_def) |
|
1339 |
lemma postfix_ConsI: "xs \<sqsupseteq> ys \<Longrightarrow> x#xs \<sqsupseteq> ys" by (auto simp add: postfix_def) |
|
1340 |
lemma postfix_ConsD: "xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def) |
|
1341 |
lemma postfix_appendI: "xs \<sqsupseteq> ys \<Longrightarrow> zs@xs \<sqsupseteq> ys" by (auto simp add: postfix_def) |
|
1342 |
lemma postfix_appendD: "xs \<sqsupseteq> zs@ys \<Longrightarrow> xs \<sqsupseteq> ys" by (auto simp add: postfix_def) |
|
1343 |
||
1344 |
lemma postfix_is_subset_lemma: "xs = zs @ ys \<Longrightarrow> set ys \<subseteq> set xs" |
|
1345 |
by (induct zs, auto) |
|
1346 |
lemma postfix_is_subset: "xs \<sqsupseteq> ys \<Longrightarrow> set ys \<subseteq> set xs" |
|
1347 |
by (unfold postfix_def, erule exE, erule postfix_is_subset_lemma) |
|
1348 |
||
1349 |
lemma postfix_ConsD2_lemma [rule_format]: "x#xs = zs @ y#ys \<longrightarrow> xs \<sqsupseteq> ys" |
|
1350 |
by (induct zs, auto intro!: postfix_appendI postfix_ConsI) |
|
1351 |
lemma postfix_ConsD2: "x#xs \<sqsupseteq> y#ys \<Longrightarrow> xs \<sqsupseteq> ys" |
|
1352 |
by (auto simp add: postfix_def dest!: postfix_ConsD2_lemma) |
|
1353 |
||
1354 |
subsection {* Lexicographic orderings on lists *} |
|
3507 | 1355 |
|
13142 | 1356 |
lemma wf_lexn: "wf r ==> wf (lexn r n)" |
14208 | 1357 |
apply (induct_tac n, simp, simp) |
13145 | 1358 |
apply(rule wf_subset) |
1359 |
prefer 2 apply (rule Int_lower1) |
|
1360 |
apply(rule wf_prod_fun_image) |
|
14208 | 1361 |
prefer 2 apply (rule inj_onI, auto) |
13145 | 1362 |
done |
13114 | 1363 |
|
1364 |
lemma lexn_length: |
|
13145 | 1365 |
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" |
1366 |
by (induct n) auto |
|
13114 | 1367 |
|
13142 | 1368 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)" |
13145 | 1369 |
apply (unfold lex_def) |
1370 |
apply (rule wf_UN) |
|
14208 | 1371 |
apply (blast intro: wf_lexn, clarify) |
13145 | 1372 |
apply (rename_tac m n) |
1373 |
apply (subgoal_tac "m \<noteq> n") |
|
1374 |
prefer 2 apply blast |
|
1375 |
apply (blast dest: lexn_length not_sym) |
|
1376 |
done |
|
13114 | 1377 |
|
1378 |
lemma lexn_conv: |
|
13145 | 1379 |
"lexn r n = |
1380 |
{(xs,ys). length xs = n \<and> length ys = n \<and> |
|
1381 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" |
|
14208 | 1382 |
apply (induct_tac n, simp, blast) |
1383 |
apply (simp add: image_Collect lex_prod_def, safe, blast) |
|
1384 |
apply (rule_tac x = "ab # xys" in exI, simp) |
|
1385 |
apply (case_tac xys, simp_all, blast) |
|
13145 | 1386 |
done |
13114 | 1387 |
|
1388 |
lemma lex_conv: |
|
13145 | 1389 |
"lex r = |
1390 |
{(xs,ys). length xs = length ys \<and> |
|
1391 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" |
|
1392 |
by (force simp add: lex_def lexn_conv) |
|
13114 | 1393 |
|
13142 | 1394 |
lemma wf_lexico [intro!]: "wf r ==> wf (lexico r)" |
13145 | 1395 |
by (unfold lexico_def) blast |
13114 | 1396 |
|
1397 |
lemma lexico_conv: |
|
13145 | 1398 |
"lexico r = {(xs,ys). length xs < length ys | |
1399 |
length xs = length ys \<and> (xs, ys) : lex r}" |
|
1400 |
by (simp add: lexico_def diag_def lex_prod_def measure_def inv_image_def) |
|
13114 | 1401 |
|
13142 | 1402 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" |
13145 | 1403 |
by (simp add: lex_conv) |
13114 | 1404 |
|
13142 | 1405 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" |
13145 | 1406 |
by (simp add:lex_conv) |
13114 | 1407 |
|
13142 | 1408 |
lemma Cons_in_lex [iff]: |
13145 | 1409 |
"((x # xs, y # ys) : lex r) = |
1410 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" |
|
1411 |
apply (simp add: lex_conv) |
|
1412 |
apply (rule iffI) |
|
14208 | 1413 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify) |
1414 |
apply (case_tac xys, simp, simp) |
|
13145 | 1415 |
apply blast |
1416 |
done |
|
13114 | 1417 |
|
1418 |
||
13142 | 1419 |
subsection {* @{text sublist} --- a generalization of @{text nth} to sets *} |
13114 | 1420 |
|
13142 | 1421 |
lemma sublist_empty [simp]: "sublist xs {} = []" |
13145 | 1422 |
by (auto simp add: sublist_def) |
13114 | 1423 |
|
13142 | 1424 |
lemma sublist_nil [simp]: "sublist [] A = []" |
13145 | 1425 |
by (auto simp add: sublist_def) |
13114 | 1426 |
|
1427 |
lemma sublist_shift_lemma: |
|
13145 | 1428 |
"map fst [p:zip xs [i..i + length xs(] . snd p : A] = |
1429 |
map fst [p:zip xs [0..length xs(] . snd p + i : A]" |
|
1430 |
by (induct xs rule: rev_induct) (simp_all add: add_commute) |
|
13114 | 1431 |
|
1432 |
lemma sublist_append: |
|
13145 | 1433 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" |
1434 |
apply (unfold sublist_def) |
|
14208 | 1435 |
apply (induct l' rule: rev_induct, simp) |
13145 | 1436 |
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) |
1437 |
apply (simp add: add_commute) |
|
1438 |
done |
|
13114 | 1439 |
|
1440 |
lemma sublist_Cons: |
|
13145 | 1441 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" |
1442 |
apply (induct l rule: rev_induct) |
|
1443 |
apply (simp add: sublist_def) |
|
1444 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) |
|
1445 |
done |
|
13114 | 1446 |
|
13142 | 1447 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" |
13145 | 1448 |
by (simp add: sublist_Cons) |
13114 | 1449 |
|
13142 | 1450 |
lemma sublist_upt_eq_take [simp]: "sublist l {..n(} = take n l" |
14208 | 1451 |
apply (induct l rule: rev_induct, simp) |
13145 | 1452 |
apply (simp split: nat_diff_split add: sublist_append) |
1453 |
done |
|
13114 | 1454 |
|
1455 |
||
13142 | 1456 |
lemma take_Cons': |
13145 | 1457 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" |
1458 |
by (cases n) simp_all |
|
13114 | 1459 |
|
13142 | 1460 |
lemma drop_Cons': |
13145 | 1461 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" |
1462 |
by (cases n) simp_all |
|
13114 | 1463 |
|
13142 | 1464 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" |
13145 | 1465 |
by (cases n) simp_all |
13142 | 1466 |
|
13145 | 1467 |
lemmas [simp] = take_Cons'[of "number_of v",standard] |
1468 |
drop_Cons'[of "number_of v",standard] |
|
1469 |
nth_Cons'[of _ _ "number_of v",standard] |
|
3507 | 1470 |
|
13462 | 1471 |
|
13366 | 1472 |
subsection {* Characters and strings *} |
1473 |
||
1474 |
datatype nibble = |
|
1475 |
Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 |
|
1476 |
| Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF |
|
1477 |
||
1478 |
datatype char = Char nibble nibble |
|
1479 |
-- "Note: canonical order of character encoding coincides with standard term ordering" |
|
1480 |
||
1481 |
types string = "char list" |
|
1482 |
||
1483 |
syntax |
|
1484 |
"_Char" :: "xstr => char" ("CHR _") |
|
1485 |
"_String" :: "xstr => string" ("_") |
|
1486 |
||
1487 |
parse_ast_translation {* |
|
1488 |
let |
|
1489 |
val constants = Syntax.Appl o map Syntax.Constant; |
|
1490 |
||
1491 |
fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); |
|
1492 |
fun mk_char c = |
|
1493 |
if Symbol.is_ascii c andalso Symbol.is_printable c then |
|
1494 |
constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] |
|
1495 |
else error ("Printable ASCII character expected: " ^ quote c); |
|
1496 |
||
1497 |
fun mk_string [] = Syntax.Constant "Nil" |
|
1498 |
| mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; |
|
1499 |
||
1500 |
fun char_ast_tr [Syntax.Variable xstr] = |
|
1501 |
(case Syntax.explode_xstr xstr of |
|
1502 |
[c] => mk_char c |
|
1503 |
| _ => error ("Single character expected: " ^ xstr)) |
|
1504 |
| char_ast_tr asts = raise AST ("char_ast_tr", asts); |
|
1505 |
||
1506 |
fun string_ast_tr [Syntax.Variable xstr] = |
|
1507 |
(case Syntax.explode_xstr xstr of |
|
1508 |
[] => constants [Syntax.constrainC, "Nil", "string"] |
|
1509 |
| cs => mk_string cs) |
|
1510 |
| string_ast_tr asts = raise AST ("string_tr", asts); |
|
1511 |
in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; |
|
1512 |
*} |
|
1513 |
||
1514 |
print_ast_translation {* |
|
1515 |
let |
|
1516 |
fun dest_nib (Syntax.Constant c) = |
|
1517 |
(case explode c of |
|
1518 |
["N", "i", "b", "b", "l", "e", h] => |
|
1519 |
if "0" <= h andalso h <= "9" then ord h - ord "0" |
|
1520 |
else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 |
|
1521 |
else raise Match |
|
1522 |
| _ => raise Match) |
|
1523 |
| dest_nib _ = raise Match; |
|
1524 |
||
1525 |
fun dest_chr c1 c2 = |
|
1526 |
let val c = chr (dest_nib c1 * 16 + dest_nib c2) |
|
1527 |
in if Symbol.is_printable c then c else raise Match end; |
|
1528 |
||
1529 |
fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 |
|
1530 |
| dest_char _ = raise Match; |
|
1531 |
||
1532 |
fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; |
|
1533 |
||
1534 |
fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] |
|
1535 |
| char_ast_tr' _ = raise Match; |
|
1536 |
||
1537 |
fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", |
|
1538 |
xstr (map dest_char (Syntax.unfold_ast "_args" args))] |
|
1539 |
| list_ast_tr' ts = raise Match; |
|
1540 |
in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; |
|
1541 |
*} |
|
1542 |
||
13122 | 1543 |
end |