author | nipkow |
Tue, 30 Mar 2004 08:45:39 +0200 | |
changeset 14495 | e2a1c31cf6d3 |
parent 14402 | 4201e1916482 |
child 14538 | 1d9d75a8efae |
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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51 |
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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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81 |
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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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||
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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)" |
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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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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 |
287 |
done |
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||
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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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lemma list_induct2[consumes 1]: "\<And>ys. |
295 |
\<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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298 |
\<Longrightarrow> P xs ys" |
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299 |
apply(induct xs) |
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300 |
apply simp |
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301 |
apply(case_tac ys) |
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302 |
apply simp |
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303 |
apply(simp) |
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304 |
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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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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329 |
apply (induct xs) |
14208 | 330 |
apply (case_tac ys, simp, force) |
331 |
apply (case_tac ys, force, simp) |
|
13145 | 332 |
done |
13142 | 333 |
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14495 | 334 |
lemma append_eq_append_conv2: "!!ys zs ts. |
335 |
(xs @ ys = zs @ ts) = |
|
336 |
(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" |
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337 |
apply (induct xs) |
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338 |
apply fastsimp |
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339 |
apply(case_tac zs) |
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340 |
apply simp |
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341 |
apply fastsimp |
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342 |
done |
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343 |
||
13142 | 344 |
lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" |
13145 | 345 |
by simp |
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347 |
lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" |
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13145 | 348 |
by simp |
13114 | 349 |
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lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)" |
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by simp |
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lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])" |
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using append_same_eq [of _ _ "[]"] by auto |
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|
13142 | 356 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" |
13145 | 357 |
using append_same_eq [of "[]"] by auto |
13114 | 358 |
|
13142 | 359 |
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" |
13145 | 360 |
by (induct xs) auto |
13114 | 361 |
|
13142 | 362 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" |
13145 | 363 |
by (induct xs) auto |
13114 | 364 |
|
13142 | 365 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" |
13145 | 366 |
by (simp add: hd_append split: list.split) |
13114 | 367 |
|
13142 | 368 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" |
13145 | 369 |
by (simp split: list.split) |
13114 | 370 |
|
13142 | 371 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" |
13145 | 372 |
by (simp add: tl_append split: list.split) |
13114 | 373 |
|
374 |
||
14300 | 375 |
lemma Cons_eq_append_conv: "x#xs = ys@zs = |
376 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" |
|
377 |
by(cases ys) auto |
|
378 |
||
379 |
||
13142 | 380 |
text {* Trivial rules for solving @{text "@"}-equations automatically. *} |
13114 | 381 |
|
382 |
lemma eq_Nil_appendI: "xs = ys ==> xs = [] @ ys" |
|
13145 | 383 |
by simp |
13114 | 384 |
|
13142 | 385 |
lemma Cons_eq_appendI: |
13145 | 386 |
"[| x # xs1 = ys; xs = xs1 @ zs |] ==> x # xs = ys @ zs" |
387 |
by (drule sym) simp |
|
13114 | 388 |
|
13142 | 389 |
lemma append_eq_appendI: |
13145 | 390 |
"[| xs @ xs1 = zs; ys = xs1 @ us |] ==> xs @ ys = zs @ us" |
391 |
by (drule sym) simp |
|
13114 | 392 |
|
393 |
||
13142 | 394 |
text {* |
13145 | 395 |
Simplification procedure for all list equalities. |
396 |
Currently only tries to rearrange @{text "@"} to see if |
|
397 |
- both lists end in a singleton list, |
|
398 |
- or both lists end in the same list. |
|
13142 | 399 |
*} |
400 |
||
401 |
ML_setup {* |
|
3507 | 402 |
local |
403 |
||
13122 | 404 |
val append_assoc = thm "append_assoc"; |
405 |
val append_Nil = thm "append_Nil"; |
|
406 |
val append_Cons = thm "append_Cons"; |
|
407 |
val append1_eq_conv = thm "append1_eq_conv"; |
|
408 |
val append_same_eq = thm "append_same_eq"; |
|
409 |
||
13114 | 410 |
fun last (cons as Const("List.list.Cons",_) $ _ $ xs) = |
13462 | 411 |
(case xs of Const("List.list.Nil",_) => cons | _ => last xs) |
412 |
| last (Const("List.op @",_) $ _ $ ys) = last ys |
|
413 |
| last t = t; |
|
13114 | 414 |
|
415 |
fun list1 (Const("List.list.Cons",_) $ _ $ Const("List.list.Nil",_)) = true |
|
13462 | 416 |
| list1 _ = false; |
13114 | 417 |
|
418 |
fun butlast ((cons as Const("List.list.Cons",_) $ x) $ xs) = |
|
13462 | 419 |
(case xs of Const("List.list.Nil",_) => xs | _ => cons $ butlast xs) |
420 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys |
|
421 |
| butlast xs = Const("List.list.Nil",fastype_of xs); |
|
13114 | 422 |
|
423 |
val rearr_tac = |
|
13462 | 424 |
simp_tac (HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]); |
13114 | 425 |
|
426 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
|
13462 | 427 |
let |
428 |
val lastl = last lhs and lastr = last rhs; |
|
429 |
fun rearr conv = |
|
430 |
let |
|
431 |
val lhs1 = butlast lhs and rhs1 = butlast rhs; |
|
432 |
val Type(_,listT::_) = eqT |
|
433 |
val appT = [listT,listT] ---> listT |
|
434 |
val app = Const("List.op @",appT) |
|
435 |
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
|
436 |
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
|
437 |
val thm = Tactic.prove sg [] [] eq (K (rearr_tac 1)); |
13462 | 438 |
in Some ((conv RS (thm RS trans)) RS eq_reflection) end; |
13114 | 439 |
|
13462 | 440 |
in |
441 |
if list1 lastl andalso list1 lastr then rearr append1_eq_conv |
|
442 |
else if lastl aconv lastr then rearr append_same_eq |
|
443 |
else None |
|
444 |
end; |
|
445 |
||
13114 | 446 |
in |
13462 | 447 |
|
448 |
val list_eq_simproc = |
|
449 |
Simplifier.simproc (Theory.sign_of (the_context ())) "list_eq" ["(xs::'a list) = ys"] list_eq; |
|
450 |
||
13114 | 451 |
end; |
452 |
||
453 |
Addsimprocs [list_eq_simproc]; |
|
454 |
*} |
|
455 |
||
456 |
||
13142 | 457 |
subsection {* @{text map} *} |
13114 | 458 |
|
13142 | 459 |
lemma map_ext: "(!!x. x : set xs --> f x = g x) ==> map f xs = map g xs" |
13145 | 460 |
by (induct xs) simp_all |
13114 | 461 |
|
13142 | 462 |
lemma map_ident [simp]: "map (\<lambda>x. x) = (\<lambda>xs. xs)" |
13145 | 463 |
by (rule ext, induct_tac xs) auto |
13114 | 464 |
|
13142 | 465 |
lemma map_append [simp]: "map f (xs @ ys) = map f xs @ map f ys" |
13145 | 466 |
by (induct xs) auto |
13114 | 467 |
|
13142 | 468 |
lemma map_compose: "map (f o g) xs = map f (map g xs)" |
13145 | 469 |
by (induct xs) (auto simp add: o_def) |
13114 | 470 |
|
13142 | 471 |
lemma rev_map: "rev (map f xs) = map f (rev xs)" |
13145 | 472 |
by (induct xs) auto |
13114 | 473 |
|
13737 | 474 |
lemma map_eq_conv[simp]: "(map f xs = map g xs) = (!x : set xs. f x = g x)" |
475 |
by (induct xs) auto |
|
476 |
||
13366 | 477 |
lemma map_cong [recdef_cong]: |
13145 | 478 |
"xs = ys ==> (!!x. x : set ys ==> f x = g x) ==> map f xs = map g ys" |
479 |
-- {* a congruence rule for @{text map} *} |
|
13737 | 480 |
by simp |
13114 | 481 |
|
13142 | 482 |
lemma map_is_Nil_conv [iff]: "(map f xs = []) = (xs = [])" |
13145 | 483 |
by (cases xs) auto |
13114 | 484 |
|
13142 | 485 |
lemma Nil_is_map_conv [iff]: "([] = map f xs) = (xs = [])" |
13145 | 486 |
by (cases xs) auto |
13114 | 487 |
|
14025 | 488 |
lemma map_eq_Cons_conv[iff]: |
489 |
"(map f xs = y#ys) = (\<exists>z zs. xs = z#zs \<and> f z = y \<and> map f zs = ys)" |
|
13145 | 490 |
by (cases xs) auto |
13114 | 491 |
|
14025 | 492 |
lemma Cons_eq_map_conv[iff]: |
493 |
"(x#xs = map f ys) = (\<exists>z zs. ys = z#zs \<and> x = f z \<and> xs = map f zs)" |
|
494 |
by (cases ys) auto |
|
495 |
||
14111 | 496 |
lemma ex_map_conv: |
497 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" |
|
498 |
by(induct ys, auto) |
|
499 |
||
13114 | 500 |
lemma map_injective: |
14338 | 501 |
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys" |
502 |
by (induct ys) (auto dest!:injD) |
|
13114 | 503 |
|
14339 | 504 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" |
505 |
by(blast dest:map_injective) |
|
506 |
||
13114 | 507 |
lemma inj_mapI: "inj f ==> inj (map f)" |
13585 | 508 |
by (rules dest: map_injective injD intro: inj_onI) |
13114 | 509 |
|
510 |
lemma inj_mapD: "inj (map f) ==> inj f" |
|
14208 | 511 |
apply (unfold inj_on_def, clarify) |
13145 | 512 |
apply (erule_tac x = "[x]" in ballE) |
14208 | 513 |
apply (erule_tac x = "[y]" in ballE, simp, blast) |
13145 | 514 |
apply blast |
515 |
done |
|
13114 | 516 |
|
14339 | 517 |
lemma inj_map[iff]: "inj (map f) = inj f" |
13145 | 518 |
by (blast dest: inj_mapD intro: inj_mapI) |
13114 | 519 |
|
14343 | 520 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" |
521 |
by (induct xs, auto) |
|
13114 | 522 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
523 |
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
|
524 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
525 |
|
13142 | 526 |
subsection {* @{text rev} *} |
13114 | 527 |
|
13142 | 528 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" |
13145 | 529 |
by (induct xs) auto |
13114 | 530 |
|
13142 | 531 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" |
13145 | 532 |
by (induct xs) auto |
13114 | 533 |
|
13142 | 534 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" |
13145 | 535 |
by (induct xs) auto |
13114 | 536 |
|
13142 | 537 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" |
13145 | 538 |
by (induct xs) auto |
13114 | 539 |
|
13142 | 540 |
lemma rev_is_rev_conv [iff]: "!!ys. (rev xs = rev ys) = (xs = ys)" |
14208 | 541 |
apply (induct xs, force) |
542 |
apply (case_tac ys, simp, force) |
|
13145 | 543 |
done |
13114 | 544 |
|
13366 | 545 |
lemma rev_induct [case_names Nil snoc]: |
546 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" |
|
13145 | 547 |
apply(subst rev_rev_ident[symmetric]) |
548 |
apply(rule_tac list = "rev xs" in list.induct, simp_all) |
|
549 |
done |
|
13114 | 550 |
|
13145 | 551 |
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" |
13114 | 552 |
|
13366 | 553 |
lemma rev_exhaust [case_names Nil snoc]: |
554 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" |
|
13145 | 555 |
by (induct xs rule: rev_induct) auto |
13114 | 556 |
|
13366 | 557 |
lemmas rev_cases = rev_exhaust |
558 |
||
13114 | 559 |
|
13142 | 560 |
subsection {* @{text set} *} |
13114 | 561 |
|
13142 | 562 |
lemma finite_set [iff]: "finite (set xs)" |
13145 | 563 |
by (induct xs) auto |
13114 | 564 |
|
13142 | 565 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" |
13145 | 566 |
by (induct xs) auto |
13114 | 567 |
|
14099 | 568 |
lemma hd_in_set: "l = x#xs \<Longrightarrow> x\<in>set l" |
14208 | 569 |
by (case_tac l, auto) |
14099 | 570 |
|
13142 | 571 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" |
13145 | 572 |
by auto |
13114 | 573 |
|
14099 | 574 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" |
575 |
by auto |
|
576 |
||
13142 | 577 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" |
13145 | 578 |
by (induct xs) auto |
13114 | 579 |
|
13142 | 580 |
lemma set_rev [simp]: "set (rev xs) = set xs" |
13145 | 581 |
by (induct xs) auto |
13114 | 582 |
|
13142 | 583 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)" |
13145 | 584 |
by (induct xs) auto |
13114 | 585 |
|
13142 | 586 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" |
13145 | 587 |
by (induct xs) auto |
13114 | 588 |
|
13142 | 589 |
lemma set_upt [simp]: "set[i..j(] = {k. i \<le> k \<and> k < j}" |
14208 | 590 |
apply (induct j, simp_all) |
591 |
apply (erule ssubst, auto) |
|
13145 | 592 |
done |
13114 | 593 |
|
13142 | 594 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" |
14208 | 595 |
apply (induct xs, simp, simp) |
13145 | 596 |
apply (rule iffI) |
597 |
apply (blast intro: eq_Nil_appendI Cons_eq_appendI) |
|
598 |
apply (erule exE)+ |
|
14208 | 599 |
apply (case_tac ys, auto) |
13145 | 600 |
done |
13142 | 601 |
|
602 |
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" |
|
13145 | 603 |
-- {* eliminate @{text lists} in favour of @{text set} *} |
604 |
by (induct xs) auto |
|
13142 | 605 |
|
606 |
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" |
|
13145 | 607 |
by (rule in_lists_conv_set [THEN iffD1]) |
13142 | 608 |
|
609 |
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" |
|
13145 | 610 |
by (rule in_lists_conv_set [THEN iffD2]) |
13114 | 611 |
|
13508 | 612 |
lemma finite_list: "finite A ==> EX l. set l = A" |
613 |
apply (erule finite_induct, auto) |
|
614 |
apply (rule_tac x="x#l" in exI, auto) |
|
615 |
done |
|
616 |
||
14388 | 617 |
lemma card_length: "card (set xs) \<le> length xs" |
618 |
by (induct xs) (auto simp add: card_insert_if) |
|
13114 | 619 |
|
13142 | 620 |
subsection {* @{text mem} *} |
13114 | 621 |
|
622 |
lemma set_mem_eq: "(x mem xs) = (x : set xs)" |
|
13145 | 623 |
by (induct xs) auto |
13114 | 624 |
|
625 |
||
13142 | 626 |
subsection {* @{text list_all} *} |
13114 | 627 |
|
13142 | 628 |
lemma list_all_conv: "list_all P xs = (\<forall>x \<in> set xs. P x)" |
13145 | 629 |
by (induct xs) auto |
13114 | 630 |
|
13142 | 631 |
lemma list_all_append [simp]: |
13145 | 632 |
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" |
633 |
by (induct xs) auto |
|
13114 | 634 |
|
635 |
||
13142 | 636 |
subsection {* @{text filter} *} |
13114 | 637 |
|
13142 | 638 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" |
13145 | 639 |
by (induct xs) auto |
13114 | 640 |
|
13142 | 641 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" |
13145 | 642 |
by (induct xs) auto |
13114 | 643 |
|
13142 | 644 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" |
13145 | 645 |
by (induct xs) auto |
13114 | 646 |
|
13142 | 647 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" |
13145 | 648 |
by (induct xs) auto |
13114 | 649 |
|
13142 | 650 |
lemma length_filter [simp]: "length (filter P xs) \<le> length xs" |
13145 | 651 |
by (induct xs) (auto simp add: le_SucI) |
13114 | 652 |
|
13142 | 653 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" |
13145 | 654 |
by auto |
13114 | 655 |
|
656 |
||
13142 | 657 |
subsection {* @{text concat} *} |
13114 | 658 |
|
13142 | 659 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" |
13145 | 660 |
by (induct xs) auto |
13114 | 661 |
|
13142 | 662 |
lemma concat_eq_Nil_conv [iff]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 663 |
by (induct xss) auto |
13114 | 664 |
|
13142 | 665 |
lemma Nil_eq_concat_conv [iff]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 666 |
by (induct xss) auto |
13114 | 667 |
|
13142 | 668 |
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" |
13145 | 669 |
by (induct xs) auto |
13114 | 670 |
|
13142 | 671 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" |
13145 | 672 |
by (induct xs) auto |
13114 | 673 |
|
13142 | 674 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" |
13145 | 675 |
by (induct xs) auto |
13114 | 676 |
|
13142 | 677 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" |
13145 | 678 |
by (induct xs) auto |
13114 | 679 |
|
680 |
||
13142 | 681 |
subsection {* @{text nth} *} |
13114 | 682 |
|
13142 | 683 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" |
13145 | 684 |
by auto |
13114 | 685 |
|
13142 | 686 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" |
13145 | 687 |
by auto |
13114 | 688 |
|
13142 | 689 |
declare nth.simps [simp del] |
13114 | 690 |
|
691 |
lemma nth_append: |
|
13145 | 692 |
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" |
14208 | 693 |
apply (induct "xs", simp) |
694 |
apply (case_tac n, auto) |
|
13145 | 695 |
done |
13114 | 696 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
697 |
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
|
698 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
699 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
700 |
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
|
701 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
702 |
|
13142 | 703 |
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" |
14208 | 704 |
apply (induct xs, simp) |
705 |
apply (case_tac n, auto) |
|
13145 | 706 |
done |
13114 | 707 |
|
13142 | 708 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" |
14208 | 709 |
apply (induct_tac xs, simp, simp) |
13145 | 710 |
apply safe |
14208 | 711 |
apply (rule_tac x = 0 in exI, simp) |
712 |
apply (rule_tac x = "Suc i" in exI, simp) |
|
713 |
apply (case_tac i, simp) |
|
13145 | 714 |
apply (rename_tac j) |
14208 | 715 |
apply (rule_tac x = j in exI, simp) |
13145 | 716 |
done |
13114 | 717 |
|
13145 | 718 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" |
719 |
by (auto simp add: set_conv_nth) |
|
13114 | 720 |
|
13142 | 721 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" |
13145 | 722 |
by (auto simp add: set_conv_nth) |
13114 | 723 |
|
724 |
lemma all_nth_imp_all_set: |
|
13145 | 725 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x" |
726 |
by (auto simp add: set_conv_nth) |
|
13114 | 727 |
|
728 |
lemma all_set_conv_all_nth: |
|
13145 | 729 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" |
730 |
by (auto simp add: set_conv_nth) |
|
13114 | 731 |
|
732 |
||
13142 | 733 |
subsection {* @{text list_update} *} |
13114 | 734 |
|
13142 | 735 |
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" |
13145 | 736 |
by (induct xs) (auto split: nat.split) |
13114 | 737 |
|
738 |
lemma nth_list_update: |
|
13145 | 739 |
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" |
740 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
|
13114 | 741 |
|
13142 | 742 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" |
13145 | 743 |
by (simp add: nth_list_update) |
13114 | 744 |
|
13142 | 745 |
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" |
13145 | 746 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
13114 | 747 |
|
13142 | 748 |
lemma list_update_overwrite [simp]: |
13145 | 749 |
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" |
750 |
by (induct xs) (auto split: nat.split) |
|
13114 | 751 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
752 |
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" |
14208 | 753 |
apply (induct xs, simp) |
14187 | 754 |
apply(simp split:nat.splits) |
755 |
done |
|
756 |
||
13114 | 757 |
lemma list_update_same_conv: |
13145 | 758 |
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" |
759 |
by (induct xs) (auto split: nat.split) |
|
13114 | 760 |
|
14187 | 761 |
lemma list_update_append1: |
762 |
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" |
|
14208 | 763 |
apply (induct xs, simp) |
14187 | 764 |
apply(simp split:nat.split) |
765 |
done |
|
766 |
||
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
767 |
lemma list_update_length [simp]: |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
768 |
"(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
|
769 |
by (induct xs, auto) |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
770 |
|
13114 | 771 |
lemma update_zip: |
13145 | 772 |
"!!i xy xs. length xs = length ys ==> |
773 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" |
|
774 |
by (induct ys) (auto, case_tac xs, auto split: nat.split) |
|
13114 | 775 |
|
776 |
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" |
|
13145 | 777 |
by (induct xs) (auto split: nat.split) |
13114 | 778 |
|
779 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" |
|
13145 | 780 |
by (blast dest!: set_update_subset_insert [THEN subsetD]) |
13114 | 781 |
|
782 |
||
13142 | 783 |
subsection {* @{text last} and @{text butlast} *} |
13114 | 784 |
|
13142 | 785 |
lemma last_snoc [simp]: "last (xs @ [x]) = x" |
13145 | 786 |
by (induct xs) auto |
13114 | 787 |
|
13142 | 788 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" |
13145 | 789 |
by (induct xs) auto |
13114 | 790 |
|
14302 | 791 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" |
792 |
by(simp add:last.simps) |
|
793 |
||
794 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" |
|
795 |
by(simp add:last.simps) |
|
796 |
||
797 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" |
|
798 |
by (induct xs) (auto) |
|
799 |
||
800 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" |
|
801 |
by(simp add:last_append) |
|
802 |
||
803 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" |
|
804 |
by(simp add:last_append) |
|
805 |
||
806 |
||
807 |
||
13142 | 808 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" |
13145 | 809 |
by (induct xs rule: rev_induct) auto |
13114 | 810 |
|
811 |
lemma butlast_append: |
|
13145 | 812 |
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" |
813 |
by (induct xs) auto |
|
13114 | 814 |
|
13142 | 815 |
lemma append_butlast_last_id [simp]: |
13145 | 816 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" |
817 |
by (induct xs) auto |
|
13114 | 818 |
|
13142 | 819 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" |
13145 | 820 |
by (induct xs) (auto split: split_if_asm) |
13114 | 821 |
|
822 |
lemma in_set_butlast_appendI: |
|
13145 | 823 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" |
824 |
by (auto dest: in_set_butlastD simp add: butlast_append) |
|
13114 | 825 |
|
13142 | 826 |
|
827 |
subsection {* @{text take} and @{text drop} *} |
|
13114 | 828 |
|
13142 | 829 |
lemma take_0 [simp]: "take 0 xs = []" |
13145 | 830 |
by (induct xs) auto |
13114 | 831 |
|
13142 | 832 |
lemma drop_0 [simp]: "drop 0 xs = xs" |
13145 | 833 |
by (induct xs) auto |
13114 | 834 |
|
13142 | 835 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" |
13145 | 836 |
by simp |
13114 | 837 |
|
13142 | 838 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" |
13145 | 839 |
by simp |
13114 | 840 |
|
13142 | 841 |
declare take_Cons [simp del] and drop_Cons [simp del] |
13114 | 842 |
|
14187 | 843 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" |
844 |
by(cases xs, simp_all) |
|
845 |
||
846 |
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" |
|
847 |
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) |
|
848 |
||
849 |
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" |
|
14208 | 850 |
apply (induct xs, simp) |
14187 | 851 |
apply(simp add:drop_Cons nth_Cons split:nat.splits) |
852 |
done |
|
853 |
||
13913 | 854 |
lemma take_Suc_conv_app_nth: |
855 |
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" |
|
14208 | 856 |
apply (induct xs, simp) |
857 |
apply (case_tac i, auto) |
|
13913 | 858 |
done |
859 |
||
13142 | 860 |
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" |
13145 | 861 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 862 |
|
13142 | 863 |
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" |
13145 | 864 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 865 |
|
13142 | 866 |
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" |
13145 | 867 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 868 |
|
13142 | 869 |
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" |
13145 | 870 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 871 |
|
13142 | 872 |
lemma take_append [simp]: |
13145 | 873 |
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" |
874 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 875 |
|
13142 | 876 |
lemma drop_append [simp]: |
13145 | 877 |
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" |
878 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 879 |
|
13142 | 880 |
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" |
14208 | 881 |
apply (induct m, auto) |
882 |
apply (case_tac xs, auto) |
|
883 |
apply (case_tac na, auto) |
|
13145 | 884 |
done |
13114 | 885 |
|
13142 | 886 |
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" |
14208 | 887 |
apply (induct m, auto) |
888 |
apply (case_tac xs, auto) |
|
13145 | 889 |
done |
13114 | 890 |
|
891 |
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" |
|
14208 | 892 |
apply (induct m, auto) |
893 |
apply (case_tac xs, auto) |
|
13145 | 894 |
done |
13114 | 895 |
|
13142 | 896 |
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" |
14208 | 897 |
apply (induct n, auto) |
898 |
apply (case_tac xs, auto) |
|
13145 | 899 |
done |
13114 | 900 |
|
901 |
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" |
|
14208 | 902 |
apply (induct n, auto) |
903 |
apply (case_tac xs, auto) |
|
13145 | 904 |
done |
13114 | 905 |
|
13142 | 906 |
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" |
14208 | 907 |
apply (induct n, auto) |
908 |
apply (case_tac xs, auto) |
|
13145 | 909 |
done |
13114 | 910 |
|
911 |
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" |
|
14208 | 912 |
apply (induct xs, auto) |
913 |
apply (case_tac i, auto) |
|
13145 | 914 |
done |
13114 | 915 |
|
916 |
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" |
|
14208 | 917 |
apply (induct xs, auto) |
918 |
apply (case_tac i, auto) |
|
13145 | 919 |
done |
13114 | 920 |
|
13142 | 921 |
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" |
14208 | 922 |
apply (induct xs, auto) |
923 |
apply (case_tac n, blast) |
|
924 |
apply (case_tac i, auto) |
|
13145 | 925 |
done |
13114 | 926 |
|
13142 | 927 |
lemma nth_drop [simp]: |
13145 | 928 |
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" |
14208 | 929 |
apply (induct n, auto) |
930 |
apply (case_tac xs, auto) |
|
13145 | 931 |
done |
3507 | 932 |
|
14025 | 933 |
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" |
934 |
by(induct xs)(auto simp:take_Cons split:nat.split) |
|
935 |
||
936 |
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" |
|
937 |
by(induct xs)(auto simp:drop_Cons split:nat.split) |
|
938 |
||
14187 | 939 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" |
940 |
using set_take_subset by fast |
|
941 |
||
942 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" |
|
943 |
using set_drop_subset by fast |
|
944 |
||
13114 | 945 |
lemma append_eq_conv_conj: |
13145 | 946 |
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" |
14208 | 947 |
apply (induct xs, simp, clarsimp) |
948 |
apply (case_tac zs, auto) |
|
13145 | 949 |
done |
13142 | 950 |
|
14050 | 951 |
lemma take_add [rule_format]: |
952 |
"\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" |
|
953 |
apply (induct xs, auto) |
|
954 |
apply (case_tac i, simp_all) |
|
955 |
done |
|
956 |
||
14300 | 957 |
lemma append_eq_append_conv_if: |
958 |
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = |
|
959 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1 |
|
960 |
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 |
|
961 |
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)" |
|
962 |
apply(induct xs\<^isub>1) |
|
963 |
apply simp |
|
964 |
apply(case_tac ys\<^isub>1) |
|
965 |
apply simp_all |
|
966 |
done |
|
967 |
||
13114 | 968 |
|
13142 | 969 |
subsection {* @{text takeWhile} and @{text dropWhile} *} |
13114 | 970 |
|
13142 | 971 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" |
13145 | 972 |
by (induct xs) auto |
13114 | 973 |
|
13142 | 974 |
lemma takeWhile_append1 [simp]: |
13145 | 975 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" |
976 |
by (induct xs) auto |
|
13114 | 977 |
|
13142 | 978 |
lemma takeWhile_append2 [simp]: |
13145 | 979 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" |
980 |
by (induct xs) auto |
|
13114 | 981 |
|
13142 | 982 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" |
13145 | 983 |
by (induct xs) auto |
13114 | 984 |
|
13142 | 985 |
lemma dropWhile_append1 [simp]: |
13145 | 986 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" |
987 |
by (induct xs) auto |
|
13114 | 988 |
|
13142 | 989 |
lemma dropWhile_append2 [simp]: |
13145 | 990 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" |
991 |
by (induct xs) auto |
|
13114 | 992 |
|
13142 | 993 |
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" |
13145 | 994 |
by (induct xs) (auto split: split_if_asm) |
13114 | 995 |
|
13913 | 996 |
lemma takeWhile_eq_all_conv[simp]: |
997 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" |
|
998 |
by(induct xs, auto) |
|
999 |
||
1000 |
lemma dropWhile_eq_Nil_conv[simp]: |
|
1001 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" |
|
1002 |
by(induct xs, auto) |
|
1003 |
||
1004 |
lemma dropWhile_eq_Cons_conv: |
|
1005 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" |
|
1006 |
by(induct xs, auto) |
|
1007 |
||
13114 | 1008 |
|
13142 | 1009 |
subsection {* @{text zip} *} |
13114 | 1010 |
|
13142 | 1011 |
lemma zip_Nil [simp]: "zip [] ys = []" |
13145 | 1012 |
by (induct ys) auto |
13114 | 1013 |
|
13142 | 1014 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" |
13145 | 1015 |
by simp |
13114 | 1016 |
|
13142 | 1017 |
declare zip_Cons [simp del] |
13114 | 1018 |
|
13142 | 1019 |
lemma length_zip [simp]: |
13145 | 1020 |
"!!xs. length (zip xs ys) = min (length xs) (length ys)" |
14208 | 1021 |
apply (induct ys, simp) |
1022 |
apply (case_tac xs, auto) |
|
13145 | 1023 |
done |
13114 | 1024 |
|
1025 |
lemma zip_append1: |
|
13145 | 1026 |
"!!xs. zip (xs @ ys) zs = |
1027 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" |
|
14208 | 1028 |
apply (induct zs, simp) |
1029 |
apply (case_tac xs, simp_all) |
|
13145 | 1030 |
done |
13114 | 1031 |
|
1032 |
lemma zip_append2: |
|
13145 | 1033 |
"!!ys. zip xs (ys @ zs) = |
1034 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" |
|
14208 | 1035 |
apply (induct xs, simp) |
1036 |
apply (case_tac ys, simp_all) |
|
13145 | 1037 |
done |
13114 | 1038 |
|
13142 | 1039 |
lemma zip_append [simp]: |
1040 |
"[| length xs = length us; length ys = length vs |] ==> |
|
13145 | 1041 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" |
1042 |
by (simp add: zip_append1) |
|
13114 | 1043 |
|
1044 |
lemma zip_rev: |
|
14247 | 1045 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" |
1046 |
by (induct rule:list_induct2, simp_all) |
|
13114 | 1047 |
|
13142 | 1048 |
lemma nth_zip [simp]: |
13145 | 1049 |
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" |
14208 | 1050 |
apply (induct ys, simp) |
13145 | 1051 |
apply (case_tac xs) |
1052 |
apply (simp_all add: nth.simps split: nat.split) |
|
1053 |
done |
|
13114 | 1054 |
|
1055 |
lemma set_zip: |
|
13145 | 1056 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" |
1057 |
by (simp add: set_conv_nth cong: rev_conj_cong) |
|
13114 | 1058 |
|
1059 |
lemma zip_update: |
|
13145 | 1060 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" |
1061 |
by (rule sym, simp add: update_zip) |
|
13114 | 1062 |
|
13142 | 1063 |
lemma zip_replicate [simp]: |
13145 | 1064 |
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" |
14208 | 1065 |
apply (induct i, auto) |
1066 |
apply (case_tac j, auto) |
|
13145 | 1067 |
done |
13114 | 1068 |
|
13142 | 1069 |
|
1070 |
subsection {* @{text list_all2} *} |
|
13114 | 1071 |
|
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1072 |
lemma list_all2_lengthD [intro?]: |
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1073 |
"list_all2 P xs ys ==> length xs = length ys" |
13145 | 1074 |
by (simp add: list_all2_def) |
13114 | 1075 |
|
13142 | 1076 |
lemma list_all2_Nil [iff]: "list_all2 P [] ys = (ys = [])" |
13145 | 1077 |
by (simp add: list_all2_def) |
13114 | 1078 |
|
13142 | 1079 |
lemma list_all2_Nil2[iff]: "list_all2 P xs [] = (xs = [])" |
13145 | 1080 |
by (simp add: list_all2_def) |
13114 | 1081 |
|
13142 | 1082 |
lemma list_all2_Cons [iff]: |
13145 | 1083 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" |
1084 |
by (auto simp add: list_all2_def) |
|
13114 | 1085 |
|
1086 |
lemma list_all2_Cons1: |
|
13145 | 1087 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" |
1088 |
by (cases ys) auto |
|
13114 | 1089 |
|
1090 |
lemma list_all2_Cons2: |
|
13145 | 1091 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" |
1092 |
by (cases xs) auto |
|
13114 | 1093 |
|
13142 | 1094 |
lemma list_all2_rev [iff]: |
13145 | 1095 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" |
1096 |
by (simp add: list_all2_def zip_rev cong: conj_cong) |
|
13114 | 1097 |
|
13863 | 1098 |
lemma list_all2_rev1: |
1099 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" |
|
1100 |
by (subst list_all2_rev [symmetric]) simp |
|
1101 |
||
13114 | 1102 |
lemma list_all2_append1: |
13145 | 1103 |
"list_all2 P (xs @ ys) zs = |
1104 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> |
|
1105 |
list_all2 P xs us \<and> list_all2 P ys vs)" |
|
1106 |
apply (simp add: list_all2_def zip_append1) |
|
1107 |
apply (rule iffI) |
|
1108 |
apply (rule_tac x = "take (length xs) zs" in exI) |
|
1109 |
apply (rule_tac x = "drop (length xs) zs" in exI) |
|
14208 | 1110 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1111 |
apply (simp add: ball_Un) |
1112 |
done |
|
13114 | 1113 |
|
1114 |
lemma list_all2_append2: |
|
13145 | 1115 |
"list_all2 P xs (ys @ zs) = |
1116 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> |
|
1117 |
list_all2 P us ys \<and> list_all2 P vs zs)" |
|
1118 |
apply (simp add: list_all2_def zip_append2) |
|
1119 |
apply (rule iffI) |
|
1120 |
apply (rule_tac x = "take (length ys) xs" in exI) |
|
1121 |
apply (rule_tac x = "drop (length ys) xs" in exI) |
|
14208 | 1122 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1123 |
apply (simp add: ball_Un) |
1124 |
done |
|
13114 | 1125 |
|
13863 | 1126 |
lemma list_all2_append: |
14247 | 1127 |
"length xs = length ys \<Longrightarrow> |
1128 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" |
|
1129 |
by (induct rule:list_induct2, simp_all) |
|
13863 | 1130 |
|
1131 |
lemma list_all2_appendI [intro?, trans]: |
|
1132 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" |
|
1133 |
by (simp add: list_all2_append list_all2_lengthD) |
|
1134 |
||
13114 | 1135 |
lemma list_all2_conv_all_nth: |
13145 | 1136 |
"list_all2 P xs ys = |
1137 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" |
|
1138 |
by (force simp add: list_all2_def set_zip) |
|
13114 | 1139 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1140 |
lemma list_all2_trans: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1141 |
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
|
1142 |
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
|
1143 |
(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
|
1144 |
proof (induct as) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1145 |
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
|
1146 |
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
|
1147 |
proof (induct bs) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1148 |
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
|
1149 |
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
|
1150 |
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
|
1151 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1152 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1153 |
|
13863 | 1154 |
lemma list_all2_all_nthI [intro?]: |
1155 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" |
|
1156 |
by (simp add: list_all2_conv_all_nth) |
|
1157 |
||
14395 | 1158 |
lemma list_all2I: |
1159 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" |
|
1160 |
by (simp add: list_all2_def) |
|
1161 |
||
14328 | 1162 |
lemma list_all2_nthD: |
13863 | 1163 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
1164 |
by (simp add: list_all2_conv_all_nth) |
|
1165 |
||
14302 | 1166 |
lemma list_all2_nthD2: |
1167 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
|
1168 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD) |
|
1169 |
||
13863 | 1170 |
lemma list_all2_map1: |
1171 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" |
|
1172 |
by (simp add: list_all2_conv_all_nth) |
|
1173 |
||
1174 |
lemma list_all2_map2: |
|
1175 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" |
|
1176 |
by (auto simp add: list_all2_conv_all_nth) |
|
1177 |
||
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1178 |
lemma list_all2_refl [intro?]: |
13863 | 1179 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" |
1180 |
by (simp add: list_all2_conv_all_nth) |
|
1181 |
||
1182 |
lemma list_all2_update_cong: |
|
1183 |
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1184 |
by (simp add: list_all2_conv_all_nth nth_list_update) |
|
1185 |
||
1186 |
lemma list_all2_update_cong2: |
|
1187 |
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1188 |
by (simp add: list_all2_lengthD list_all2_update_cong) |
|
1189 |
||
14302 | 1190 |
lemma list_all2_takeI [simp,intro?]: |
1191 |
"\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" |
|
1192 |
apply (induct xs) |
|
1193 |
apply simp |
|
1194 |
apply (clarsimp simp add: list_all2_Cons1) |
|
1195 |
apply (case_tac n) |
|
1196 |
apply auto |
|
1197 |
done |
|
1198 |
||
1199 |
lemma list_all2_dropI [simp,intro?]: |
|
13863 | 1200 |
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" |
14208 | 1201 |
apply (induct as, simp) |
13863 | 1202 |
apply (clarsimp simp add: list_all2_Cons1) |
14208 | 1203 |
apply (case_tac n, simp, simp) |
13863 | 1204 |
done |
1205 |
||
14327 | 1206 |
lemma list_all2_mono [intro?]: |
13863 | 1207 |
"\<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 | 1208 |
apply (induct x, simp) |
1209 |
apply (case_tac y, auto) |
|
13863 | 1210 |
done |
1211 |
||
13142 | 1212 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1213 |
subsection {* @{text foldl} and @{text foldr} *} |
13142 | 1214 |
|
1215 |
lemma foldl_append [simp]: |
|
13145 | 1216 |
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" |
1217 |
by (induct xs) auto |
|
13142 | 1218 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1219 |
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
|
1220 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1221 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1222 |
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
|
1223 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1224 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1225 |
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
|
1226 |
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
|
1227 |
|
13142 | 1228 |
text {* |
13145 | 1229 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more |
1230 |
difficult to use because it requires an additional transitivity step. |
|
13142 | 1231 |
*} |
1232 |
||
1233 |
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" |
|
13145 | 1234 |
by (induct ns) auto |
13142 | 1235 |
|
1236 |
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" |
|
13145 | 1237 |
by (force intro: start_le_sum simp add: in_set_conv_decomp) |
13142 | 1238 |
|
1239 |
lemma sum_eq_0_conv [iff]: |
|
13145 | 1240 |
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" |
1241 |
by (induct ns) auto |
|
13114 | 1242 |
|
1243 |
||
14099 | 1244 |
subsection {* folding a relation over a list *} |
1245 |
||
1246 |
(*"fold_rel R cs \<equiv> foldl (%r c. r O {(x,y). (c,x,y):R}) Id cs"*) |
|
1247 |
inductive "fold_rel R" intros |
|
1248 |
Nil: "(a, [],a) : fold_rel R" |
|
1249 |
Cons: "[|(a,x,b) : R; (b,xs,c) : fold_rel R|] ==> (a,x#xs,c) : fold_rel R" |
|
1250 |
inductive_cases fold_rel_elim_case [elim!]: |
|
14208 | 1251 |
"(a, [] , b) : fold_rel R" |
14099 | 1252 |
"(a, x#xs, b) : fold_rel R" |
1253 |
||
1254 |
lemma fold_rel_Nil [intro!]: "a = b ==> (a, [], b) : fold_rel R" |
|
1255 |
by (simp add: fold_rel.Nil) |
|
1256 |
declare fold_rel.Cons [intro!] |
|
1257 |
||
1258 |
||
13142 | 1259 |
subsection {* @{text upto} *} |
13114 | 1260 |
|
13142 | 1261 |
lemma upt_rec: "[i..j(] = (if i<j then i#[Suc i..j(] else [])" |
13145 | 1262 |
-- {* Does not terminate! *} |
1263 |
by (induct j) auto |
|
13142 | 1264 |
|
1265 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..j(] = []" |
|
13145 | 1266 |
by (subst upt_rec) simp |
13114 | 1267 |
|
13142 | 1268 |
lemma upt_Suc_append: "i <= j ==> [i..(Suc j)(] = [i..j(]@[j]" |
13145 | 1269 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} |
1270 |
by simp |
|
13114 | 1271 |
|
13142 | 1272 |
lemma upt_conv_Cons: "i < j ==> [i..j(] = i # [Suc i..j(]" |
13145 | 1273 |
apply(rule trans) |
1274 |
apply(subst upt_rec) |
|
14208 | 1275 |
prefer 2 apply (rule refl, simp) |
13145 | 1276 |
done |
13114 | 1277 |
|
13142 | 1278 |
lemma upt_add_eq_append: "i<=j ==> [i..j+k(] = [i..j(]@[j..j+k(]" |
13145 | 1279 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *} |
1280 |
by (induct k) auto |
|
13114 | 1281 |
|
13142 | 1282 |
lemma length_upt [simp]: "length [i..j(] = j - i" |
13145 | 1283 |
by (induct j) (auto simp add: Suc_diff_le) |
13114 | 1284 |
|
13142 | 1285 |
lemma nth_upt [simp]: "i + k < j ==> [i..j(] ! k = i + k" |
13145 | 1286 |
apply (induct j) |
1287 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) |
|
1288 |
done |
|
13114 | 1289 |
|
13142 | 1290 |
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..n(] = [i..i+m(]" |
14208 | 1291 |
apply (induct m, simp) |
13145 | 1292 |
apply (subst upt_rec) |
1293 |
apply (rule sym) |
|
1294 |
apply (subst upt_rec) |
|
1295 |
apply (simp del: upt.simps) |
|
1296 |
done |
|
3507 | 1297 |
|
13114 | 1298 |
lemma map_Suc_upt: "map Suc [m..n(] = [Suc m..n]" |
13145 | 1299 |
by (induct n) auto |
13114 | 1300 |
|
1301 |
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..n(]) ! i = f(m+i)" |
|
13145 | 1302 |
apply (induct n m rule: diff_induct) |
1303 |
prefer 3 apply (subst map_Suc_upt[symmetric]) |
|
1304 |
apply (auto simp add: less_diff_conv nth_upt) |
|
1305 |
done |
|
13114 | 1306 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1307 |
lemma nth_take_lemma: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1308 |
"!!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
|
1309 |
(!!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
|
1310 |
apply (atomize, induct k) |
14208 | 1311 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) |
13145 | 1312 |
txt {* Both lists must be non-empty *} |
14208 | 1313 |
apply (case_tac xs, simp) |
1314 |
apply (case_tac ys, clarify) |
|
13145 | 1315 |
apply (simp (no_asm_use)) |
1316 |
apply clarify |
|
1317 |
txt {* prenexing's needed, not miniscoping *} |
|
1318 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) |
|
1319 |
apply blast |
|
1320 |
done |
|
13114 | 1321 |
|
1322 |
lemma nth_equalityI: |
|
1323 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" |
|
13145 | 1324 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) |
1325 |
apply (simp_all add: take_all) |
|
1326 |
done |
|
13142 | 1327 |
|
13863 | 1328 |
(* needs nth_equalityI *) |
1329 |
lemma list_all2_antisym: |
|
1330 |
"\<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> |
|
1331 |
\<Longrightarrow> xs = ys" |
|
1332 |
apply (simp add: list_all2_conv_all_nth) |
|
14208 | 1333 |
apply (rule nth_equalityI, blast, simp) |
13863 | 1334 |
done |
1335 |
||
13142 | 1336 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" |
13145 | 1337 |
-- {* The famous take-lemma. *} |
1338 |
apply (drule_tac x = "max (length xs) (length ys)" in spec) |
|
1339 |
apply (simp add: le_max_iff_disj take_all) |
|
1340 |
done |
|
13142 | 1341 |
|
1342 |
||