author | wenzelm |
Wed, 08 Nov 2006 13:48:29 +0100 | |
changeset 21243 | afffe1f72143 |
parent 21211 | 5370cfbf3070 |
child 21404 | eb85850d3eb7 |
permissions | -rw-r--r-- |
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(* Title: HOL/List.thy |
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ID: $Id$ |
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Author: Tobias Nipkow |
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*) |
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header {* The datatype of finite lists *} |
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theory List |
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imports PreList FunDef |
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begin |
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datatype 'a list = |
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Nil ("[]") |
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| Cons 'a "'a list" (infixr "#" 65) |
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subsection{*Basic list processing functions*} |
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consts |
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"@" :: "'a list => 'a list => 'a list" (infixr 65) |
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filter:: "('a => bool) => 'a list => 'a list" |
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concat:: "'a list list => 'a list" |
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foldl :: "('b => 'a => 'b) => 'b => 'a list => 'b" |
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foldr :: "('a => 'b => 'b) => 'a list => 'b => 'b" |
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hd:: "'a list => 'a" |
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tl:: "'a list => 'a list" |
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last:: "'a list => 'a" |
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butlast :: "'a list => 'a list" |
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set :: "'a list => 'a set" |
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map :: "('a=>'b) => ('a list => 'b list)" |
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nth :: "'a list => nat => 'a" (infixl "!" 100) |
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list_update :: "'a list => nat => 'a => 'a list" |
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take:: "nat => 'a list => 'a list" |
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drop:: "nat => 'a list => 'a list" |
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takeWhile :: "('a => bool) => 'a list => 'a list" |
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dropWhile :: "('a => bool) => 'a list => 'a list" |
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rev :: "'a list => 'a list" |
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zip :: "'a list => 'b list => ('a * 'b) list" |
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upt :: "nat => nat => nat list" ("(1[_..</_'])") |
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remdups :: "'a list => 'a list" |
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remove1 :: "'a => 'a list => 'a list" |
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"distinct":: "'a list => bool" |
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replicate :: "nat => 'a => 'a list" |
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splice :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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abbreviation |
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upto:: "nat => nat => nat list" ("(1[_../_])") |
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"[i..j] == [i..<(Suc j)]" |
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nonterminals lupdbinds lupdbind |
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syntax |
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-- {* list Enumeration *} |
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"@list" :: "args => 'a list" ("[(_)]") |
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-- {* Special syntax for filter *} |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list" ("(1[_:_./ _])") |
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-- {* list update *} |
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"_lupdbind":: "['a, 'a] => lupdbind" ("(2_ :=/ _)") |
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"" :: "lupdbind => lupdbinds" ("_") |
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"_lupdbinds" :: "[lupdbind, lupdbinds] => lupdbinds" ("_,/ _") |
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"_LUpdate" :: "['a, lupdbinds] => 'a" ("_/[(_)]" [900,0] 900) |
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translations |
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"[x, xs]" == "x#[xs]" |
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"[x]" == "x#[]" |
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"[x:xs . P]"== "filter (%x. P) xs" |
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"_LUpdate xs (_lupdbinds b bs)"== "_LUpdate (_LUpdate xs b) bs" |
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"xs[i:=x]" == "list_update xs i x" |
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syntax (xsymbols) |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") |
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syntax (HTML output) |
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"@filter" :: "[pttrn, 'a list, bool] => 'a list"("(1[_\<in>_ ./ _])") |
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text {* |
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Function @{text size} is overloaded for all datatypes. Users may |
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refer to the list version as @{text length}. *} |
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abbreviation |
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length :: "'a list => nat" |
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"length == size" |
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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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"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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"set [] = {}" |
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"set (x#xs) = insert x (set xs)" |
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primrec |
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"map f [] = []" |
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"map f (x#xs) = f(x)#map f xs" |
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primrec |
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append_Nil:"[]@ys = ys" |
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append_Cons: "(x#xs)@ys = x#(xs@ys)" |
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primrec |
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"rev([]) = []" |
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"rev(x#xs) = rev(xs) @ [x]" |
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primrec |
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"filter P [] = []" |
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"filter P (x#xs) = (if P x then x#filter P xs else filter P xs)" |
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primrec |
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foldl_Nil:"foldl f a [] = a" |
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foldl_Cons: "foldl f a (x#xs) = foldl f (f a x) xs" |
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primrec |
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"foldr f [] a = a" |
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"foldr f (x#xs) a = f x (foldr f xs a)" |
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primrec |
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"concat([]) = []" |
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"concat(x#xs) = x @ concat(xs)" |
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primrec |
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drop_Nil:"drop n [] = []" |
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drop_Cons: "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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take_Nil:"take n [] = []" |
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take_Cons: "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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nth_Cons:"(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "n = 0"} and @{text "n = Suc k"} *} |
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primrec |
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"[][i:=v] = []" |
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"(x#xs)[i:=v] = (case i of 0 => v # xs | Suc j => x # xs[j:=v])" |
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primrec |
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"takeWhile P [] = []" |
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"takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" |
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primrec |
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"dropWhile P [] = []" |
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"dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" |
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primrec |
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"zip xs [] = []" |
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zip_Cons: "zip xs (y#ys) = (case xs of [] => [] | z#zs => (z,y)#zip zs ys)" |
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-- {*Warning: simpset does not contain this definition, but separate |
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theorems for @{text "xs = []"} and @{text "xs = z # zs"} *} |
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primrec |
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upt_0: "[i..<0] = []" |
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upt_Suc: "[i..<(Suc j)] = (if i <= j then [i..<j] @ [j] else [])" |
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primrec |
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"distinct [] = True" |
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"distinct (x#xs) = (x ~: set xs \<and> distinct xs)" |
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primrec |
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"remdups [] = []" |
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"remdups (x#xs) = (if x : set xs then remdups xs else x # remdups xs)" |
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primrec |
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"remove1 x [] = []" |
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"remove1 x (y#xs) = (if x=y then xs else y # remove1 x xs)" |
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primrec |
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replicate_0: "replicate 0 x = []" |
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replicate_Suc: "replicate (Suc n) x = x # replicate n x" |
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definition |
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rotate1 :: "'a list \<Rightarrow> 'a list" |
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rotate1_def: "rotate1 xs = (case xs of [] \<Rightarrow> [] | x#xs \<Rightarrow> xs @ [x])" |
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rotate :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
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rotate_def: "rotate n = rotate1 ^ n" |
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list_all2 :: "('a => 'b => bool) => 'a list => 'b list => bool" |
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list_all2_def: "list_all2 P xs ys = |
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(length xs = length ys \<and> (\<forall>(x, y) \<in> set (zip xs ys). P x y))" |
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sublist :: "'a list => nat set => 'a list" |
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sublist_def: "sublist xs A = |
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map fst (filter (\<lambda>p. snd p \<in> A) (zip xs [0..<size xs]))" |
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primrec |
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"splice [] ys = ys" |
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"splice (x#xs) ys = (if ys=[] then x#xs else x # hd ys # splice xs (tl ys))" |
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-- {*Warning: simpset does not contain the second eqn but a derived one. *} |
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subsubsection {* @{const Nil} and @{const Cons} *} |
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lemma not_Cons_self [simp]: |
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"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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"(\<And>xs. \<forall>ys. length ys < length xs \<longrightarrow> P ys \<Longrightarrow> P xs) \<Longrightarrow> P xs" |
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by (rule measure_induct [of length]) iprover |
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subsubsection {* @{const length} *} |
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text {* |
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Needs to come before @{text "@"} because of theorem @{text |
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append_eq_append_conv}. |
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*} |
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lemma length_append [simp]: "length (xs @ ys) = length xs + length ys" |
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by (induct xs) auto |
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lemma length_map [simp]: "length (map f xs) = length xs" |
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by (induct xs) auto |
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lemma length_rev [simp]: "length (rev xs) = length xs" |
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by (induct xs) auto |
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lemma length_tl [simp]: "length (tl xs) = length xs - 1" |
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by (cases xs) auto |
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lemma length_0_conv [iff]: "(length xs = 0) = (xs = [])" |
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by (induct xs) auto |
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lemma length_greater_0_conv [iff]: "(0 < length xs) = (xs \<noteq> [])" |
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by (induct xs) auto |
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lemma length_Suc_conv: |
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"(length xs = Suc n) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" |
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by (induct xs) auto |
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lemma Suc_length_conv: |
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"(Suc n = length xs) = (\<exists>y ys. xs = y # ys \<and> length ys = n)" |
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apply (induct xs, simp, simp) |
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apply blast |
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done |
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lemma impossible_Cons [rule_format]: |
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"length xs <= length ys --> xs = x # ys = False" |
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apply (induct xs) |
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apply auto |
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done |
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lemma list_induct2[consumes 1]: "\<And>ys. |
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\<lbrakk> length xs = length ys; |
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P [] []; |
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\<And>x xs y ys. \<lbrakk> length xs = length ys; P xs ys \<rbrakk> \<Longrightarrow> P (x#xs) (y#ys) \<rbrakk> |
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\<Longrightarrow> P xs ys" |
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apply(induct xs) |
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apply simp |
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apply(case_tac ys) |
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apply simp |
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apply(simp) |
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done |
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subsubsection {* @{text "@"} -- append *} |
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lemma append_assoc [simp]: "(xs @ ys) @ zs = xs @ (ys @ zs)" |
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by (induct xs) auto |
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lemma append_Nil2 [simp]: "xs @ [] = xs" |
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by (induct xs) auto |
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|
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lemma append_is_Nil_conv [iff]: "(xs @ ys = []) = (xs = [] \<and> ys = [])" |
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by (induct xs) auto |
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|
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lemma Nil_is_append_conv [iff]: "([] = xs @ ys) = (xs = [] \<and> ys = [])" |
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by (induct xs) auto |
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|
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lemma append_self_conv [iff]: "(xs @ ys = xs) = (ys = [])" |
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by (induct xs) auto |
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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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apply (induct xs) |
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apply (case_tac ys, simp, force) |
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apply (case_tac ys, force, simp) |
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done |
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lemma append_eq_append_conv2: "!!ys zs ts. |
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(xs @ ys = zs @ ts) = |
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(EX us. xs = zs @ us & us @ ys = ts | xs @ us = zs & ys = us@ ts)" |
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apply (induct xs) |
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apply fastsimp |
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apply(case_tac zs) |
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apply simp |
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apply fastsimp |
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done |
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lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)" |
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by simp |
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lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)" |
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by simp |
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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 | 326 |
lemma self_append_conv2 [iff]: "(ys = xs @ ys) = (xs = [])" |
13145 | 327 |
using append_same_eq [of "[]"] by auto |
13114 | 328 |
|
13142 | 329 |
lemma hd_Cons_tl [simp]: "xs \<noteq> [] ==> hd xs # tl xs = xs" |
13145 | 330 |
by (induct xs) auto |
13114 | 331 |
|
13142 | 332 |
lemma hd_append: "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)" |
13145 | 333 |
by (induct xs) auto |
13114 | 334 |
|
13142 | 335 |
lemma hd_append2 [simp]: "xs \<noteq> [] ==> hd (xs @ ys) = hd xs" |
13145 | 336 |
by (simp add: hd_append split: list.split) |
13114 | 337 |
|
13142 | 338 |
lemma tl_append: "tl (xs @ ys) = (case xs of [] => tl ys | z#zs => zs @ ys)" |
13145 | 339 |
by (simp split: list.split) |
13114 | 340 |
|
13142 | 341 |
lemma tl_append2 [simp]: "xs \<noteq> [] ==> tl (xs @ ys) = tl xs @ ys" |
13145 | 342 |
by (simp add: tl_append split: list.split) |
13114 | 343 |
|
344 |
||
14300 | 345 |
lemma Cons_eq_append_conv: "x#xs = ys@zs = |
346 |
(ys = [] & x#xs = zs | (EX ys'. x#ys' = ys & xs = ys'@zs))" |
|
347 |
by(cases ys) auto |
|
348 |
||
15281 | 349 |
lemma append_eq_Cons_conv: "(ys@zs = x#xs) = |
350 |
(ys = [] & zs = x#xs | (EX ys'. ys = x#ys' & ys'@zs = xs))" |
|
351 |
by(cases ys) auto |
|
352 |
||
14300 | 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 |
|
16973 | 397 |
val rearr_ss = HOL_basic_ss addsimps [append_assoc, append_Nil, append_Cons]; |
398 |
||
20044
92cc2f4c7335
simprocs: no theory argument -- use simpset context instead;
wenzelm
parents:
19890
diff
changeset
|
399 |
fun list_eq ss (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
13462 | 400 |
let |
401 |
val lastl = last lhs and lastr = last rhs; |
|
402 |
fun rearr conv = |
|
403 |
let |
|
404 |
val lhs1 = butlast lhs and rhs1 = butlast rhs; |
|
405 |
val Type(_,listT::_) = eqT |
|
406 |
val appT = [listT,listT] ---> listT |
|
407 |
val app = Const("List.op @",appT) |
|
408 |
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
|
409 |
val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (F,F2)); |
20044
92cc2f4c7335
simprocs: no theory argument -- use simpset context instead;
wenzelm
parents:
19890
diff
changeset
|
410 |
val thm = Goal.prove (Simplifier.the_context ss) [] [] eq |
17877
67d5ab1cb0d8
Simplifier.inherit_context instead of Simplifier.inherit_bounds;
wenzelm
parents:
17830
diff
changeset
|
411 |
(K (simp_tac (Simplifier.inherit_context ss rearr_ss) 1)); |
15531 | 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 |
|
15531 | 417 |
else NONE |
13462 | 418 |
end; |
419 |
||
13114 | 420 |
in |
13462 | 421 |
|
422 |
val list_eq_simproc = |
|
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
423 |
Simplifier.simproc (the_context ()) "list_eq" ["(xs::'a list) = ys"] (K list_eq); |
13462 | 424 |
|
13114 | 425 |
end; |
426 |
||
427 |
Addsimprocs [list_eq_simproc]; |
|
428 |
*} |
|
429 |
||
430 |
||
15392 | 431 |
subsubsection {* @{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 |
||
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
451 |
lemma map_cong [fundef_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 |
|
18447 | 462 |
lemma map_eq_Cons_conv: |
14025 | 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 |
|
18447 | 466 |
lemma Cons_eq_map_conv: |
14025 | 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 |
||
18447 | 470 |
lemmas map_eq_Cons_D = map_eq_Cons_conv [THEN iffD1] |
471 |
lemmas Cons_eq_map_D = Cons_eq_map_conv [THEN iffD1] |
|
472 |
declare map_eq_Cons_D [dest!] Cons_eq_map_D [dest!] |
|
473 |
||
14111 | 474 |
lemma ex_map_conv: |
475 |
"(EX xs. ys = map f xs) = (ALL y : set ys. EX x. y = f x)" |
|
18447 | 476 |
by(induct ys, auto simp add: Cons_eq_map_conv) |
14111 | 477 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
478 |
lemma map_eq_imp_length_eq: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
479 |
"!!xs. map f xs = map f ys ==> length xs = length ys" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
480 |
apply (induct ys) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
481 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
482 |
apply(simp (no_asm_use)) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
483 |
apply clarify |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
484 |
apply(simp (no_asm_use)) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
485 |
apply fast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
486 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
487 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
488 |
lemma map_inj_on: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
489 |
"[| map f xs = map f ys; inj_on f (set xs Un set ys) |] |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
490 |
==> xs = ys" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
491 |
apply(frule map_eq_imp_length_eq) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
492 |
apply(rotate_tac -1) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
493 |
apply(induct rule:list_induct2) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
494 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
495 |
apply(simp) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
496 |
apply (blast intro:sym) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
497 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
498 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
499 |
lemma inj_on_map_eq_map: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
500 |
"inj_on f (set xs Un set ys) \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
501 |
by(blast dest:map_inj_on) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
502 |
|
13114 | 503 |
lemma map_injective: |
14338 | 504 |
"!!xs. map f xs = map f ys ==> inj f ==> xs = ys" |
505 |
by (induct ys) (auto dest!:injD) |
|
13114 | 506 |
|
14339 | 507 |
lemma inj_map_eq_map[simp]: "inj f \<Longrightarrow> (map f xs = map f ys) = (xs = ys)" |
508 |
by(blast dest:map_injective) |
|
509 |
||
13114 | 510 |
lemma inj_mapI: "inj f ==> inj (map f)" |
17589 | 511 |
by (iprover dest: map_injective injD intro: inj_onI) |
13114 | 512 |
|
513 |
lemma inj_mapD: "inj (map f) ==> inj f" |
|
14208 | 514 |
apply (unfold inj_on_def, clarify) |
13145 | 515 |
apply (erule_tac x = "[x]" in ballE) |
14208 | 516 |
apply (erule_tac x = "[y]" in ballE, simp, blast) |
13145 | 517 |
apply blast |
518 |
done |
|
13114 | 519 |
|
14339 | 520 |
lemma inj_map[iff]: "inj (map f) = inj f" |
13145 | 521 |
by (blast dest: inj_mapD intro: inj_mapI) |
13114 | 522 |
|
15303 | 523 |
lemma inj_on_mapI: "inj_on f (\<Union>(set ` A)) \<Longrightarrow> inj_on (map f) A" |
524 |
apply(rule inj_onI) |
|
525 |
apply(erule map_inj_on) |
|
526 |
apply(blast intro:inj_onI dest:inj_onD) |
|
527 |
done |
|
528 |
||
14343 | 529 |
lemma map_idI: "(\<And>x. x \<in> set xs \<Longrightarrow> f x = x) \<Longrightarrow> map f xs = xs" |
530 |
by (induct xs, auto) |
|
13114 | 531 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
532 |
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
|
533 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
534 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
535 |
lemma map_fst_zip[simp]: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
536 |
"length xs = length ys \<Longrightarrow> map fst (zip xs ys) = xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
537 |
by (induct rule:list_induct2, simp_all) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
538 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
539 |
lemma map_snd_zip[simp]: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
540 |
"length xs = length ys \<Longrightarrow> map snd (zip xs ys) = ys" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
541 |
by (induct rule:list_induct2, simp_all) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
542 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
543 |
|
15392 | 544 |
subsubsection {* @{text rev} *} |
13114 | 545 |
|
13142 | 546 |
lemma rev_append [simp]: "rev (xs @ ys) = rev ys @ rev xs" |
13145 | 547 |
by (induct xs) auto |
13114 | 548 |
|
13142 | 549 |
lemma rev_rev_ident [simp]: "rev (rev xs) = xs" |
13145 | 550 |
by (induct xs) auto |
13114 | 551 |
|
15870 | 552 |
lemma rev_swap: "(rev xs = ys) = (xs = rev ys)" |
553 |
by auto |
|
554 |
||
13142 | 555 |
lemma rev_is_Nil_conv [iff]: "(rev xs = []) = (xs = [])" |
13145 | 556 |
by (induct xs) auto |
13114 | 557 |
|
13142 | 558 |
lemma Nil_is_rev_conv [iff]: "([] = rev xs) = (xs = [])" |
13145 | 559 |
by (induct xs) auto |
13114 | 560 |
|
15870 | 561 |
lemma rev_singleton_conv [simp]: "(rev xs = [x]) = (xs = [x])" |
562 |
by (cases xs) auto |
|
563 |
||
564 |
lemma singleton_rev_conv [simp]: "([x] = rev xs) = (xs = [x])" |
|
565 |
by (cases xs) auto |
|
566 |
||
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
567 |
lemma rev_is_rev_conv [iff]: "(rev xs = rev ys) = (xs = ys)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
568 |
apply (induct xs arbitrary: ys, force) |
14208 | 569 |
apply (case_tac ys, simp, force) |
13145 | 570 |
done |
13114 | 571 |
|
15439 | 572 |
lemma inj_on_rev[iff]: "inj_on rev A" |
573 |
by(simp add:inj_on_def) |
|
574 |
||
13366 | 575 |
lemma rev_induct [case_names Nil snoc]: |
576 |
"[| P []; !!x xs. P xs ==> P (xs @ [x]) |] ==> P xs" |
|
15489
d136af442665
Replaced application of subst by simplesubst in proof of rev_induct
berghofe
parents:
15439
diff
changeset
|
577 |
apply(simplesubst rev_rev_ident[symmetric]) |
13145 | 578 |
apply(rule_tac list = "rev xs" in list.induct, simp_all) |
579 |
done |
|
13114 | 580 |
|
13145 | 581 |
ML {* val rev_induct_tac = induct_thm_tac (thm "rev_induct") *}-- "compatibility" |
13114 | 582 |
|
13366 | 583 |
lemma rev_exhaust [case_names Nil snoc]: |
584 |
"(xs = [] ==> P) ==>(!!ys y. xs = ys @ [y] ==> P) ==> P" |
|
13145 | 585 |
by (induct xs rule: rev_induct) auto |
13114 | 586 |
|
13366 | 587 |
lemmas rev_cases = rev_exhaust |
588 |
||
18423 | 589 |
lemma rev_eq_Cons_iff[iff]: "(rev xs = y#ys) = (xs = rev ys @ [y])" |
590 |
by(rule rev_cases[of xs]) auto |
|
591 |
||
13114 | 592 |
|
15392 | 593 |
subsubsection {* @{text set} *} |
13114 | 594 |
|
13142 | 595 |
lemma finite_set [iff]: "finite (set xs)" |
13145 | 596 |
by (induct xs) auto |
13114 | 597 |
|
13142 | 598 |
lemma set_append [simp]: "set (xs @ ys) = (set xs \<union> set ys)" |
13145 | 599 |
by (induct xs) auto |
13114 | 600 |
|
17830 | 601 |
lemma hd_in_set[simp]: "xs \<noteq> [] \<Longrightarrow> hd xs : set xs" |
602 |
by(cases xs) auto |
|
14099 | 603 |
|
13142 | 604 |
lemma set_subset_Cons: "set xs \<subseteq> set (x # xs)" |
13145 | 605 |
by auto |
13114 | 606 |
|
14099 | 607 |
lemma set_ConsD: "y \<in> set (x # xs) \<Longrightarrow> y=x \<or> y \<in> set xs" |
608 |
by auto |
|
609 |
||
13142 | 610 |
lemma set_empty [iff]: "(set xs = {}) = (xs = [])" |
13145 | 611 |
by (induct xs) auto |
13114 | 612 |
|
15245 | 613 |
lemma set_empty2[iff]: "({} = set xs) = (xs = [])" |
614 |
by(induct xs) auto |
|
615 |
||
13142 | 616 |
lemma set_rev [simp]: "set (rev xs) = set xs" |
13145 | 617 |
by (induct xs) auto |
13114 | 618 |
|
13142 | 619 |
lemma set_map [simp]: "set (map f xs) = f`(set xs)" |
13145 | 620 |
by (induct xs) auto |
13114 | 621 |
|
13142 | 622 |
lemma set_filter [simp]: "set (filter P xs) = {x. x : set xs \<and> P x}" |
13145 | 623 |
by (induct xs) auto |
13114 | 624 |
|
15425 | 625 |
lemma set_upt [simp]: "set[i..<j] = {k. i \<le> k \<and> k < j}" |
14208 | 626 |
apply (induct j, simp_all) |
627 |
apply (erule ssubst, auto) |
|
13145 | 628 |
done |
13114 | 629 |
|
13142 | 630 |
lemma in_set_conv_decomp: "(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs)" |
15113 | 631 |
proof (induct xs) |
632 |
case Nil show ?case by simp |
|
633 |
case (Cons a xs) |
|
634 |
show ?case |
|
635 |
proof |
|
636 |
assume "x \<in> set (a # xs)" |
|
637 |
with prems show "\<exists>ys zs. a # xs = ys @ x # zs" |
|
638 |
by (simp, blast intro: Cons_eq_appendI) |
|
639 |
next |
|
640 |
assume "\<exists>ys zs. a # xs = ys @ x # zs" |
|
641 |
then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast |
|
642 |
show "x \<in> set (a # xs)" |
|
643 |
by (cases ys, auto simp add: eq) |
|
644 |
qed |
|
645 |
qed |
|
13142 | 646 |
|
18049 | 647 |
lemma in_set_conv_decomp_first: |
648 |
"(x : set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)" |
|
649 |
proof (induct xs) |
|
650 |
case Nil show ?case by simp |
|
651 |
next |
|
652 |
case (Cons a xs) |
|
653 |
show ?case |
|
654 |
proof cases |
|
655 |
assume "x = a" thus ?case using Cons by force |
|
656 |
next |
|
657 |
assume "x \<noteq> a" |
|
658 |
show ?case |
|
659 |
proof |
|
660 |
assume "x \<in> set (a # xs)" |
|
661 |
from prems show "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" |
|
662 |
by(fastsimp intro!: Cons_eq_appendI) |
|
663 |
next |
|
664 |
assume "\<exists>ys zs. a # xs = ys @ x # zs \<and> x \<notin> set ys" |
|
665 |
then obtain ys zs where eq: "a # xs = ys @ x # zs" by blast |
|
666 |
show "x \<in> set (a # xs)" by (cases ys, auto simp add: eq) |
|
667 |
qed |
|
668 |
qed |
|
669 |
qed |
|
670 |
||
671 |
lemmas split_list = in_set_conv_decomp[THEN iffD1, standard] |
|
672 |
lemmas split_list_first = in_set_conv_decomp_first[THEN iffD1, standard] |
|
673 |
||
674 |
||
13508 | 675 |
lemma finite_list: "finite A ==> EX l. set l = A" |
676 |
apply (erule finite_induct, auto) |
|
677 |
apply (rule_tac x="x#l" in exI, auto) |
|
678 |
done |
|
679 |
||
14388 | 680 |
lemma card_length: "card (set xs) \<le> length xs" |
681 |
by (induct xs) (auto simp add: card_insert_if) |
|
13114 | 682 |
|
15168 | 683 |
|
15392 | 684 |
subsubsection {* @{text filter} *} |
13114 | 685 |
|
13142 | 686 |
lemma filter_append [simp]: "filter P (xs @ ys) = filter P xs @ filter P ys" |
13145 | 687 |
by (induct xs) auto |
13114 | 688 |
|
15305 | 689 |
lemma rev_filter: "rev (filter P xs) = filter P (rev xs)" |
690 |
by (induct xs) simp_all |
|
691 |
||
13142 | 692 |
lemma filter_filter [simp]: "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs" |
13145 | 693 |
by (induct xs) auto |
13114 | 694 |
|
16998 | 695 |
lemma length_filter_le [simp]: "length (filter P xs) \<le> length xs" |
696 |
by (induct xs) (auto simp add: le_SucI) |
|
697 |
||
18423 | 698 |
lemma sum_length_filter_compl: |
699 |
"length(filter P xs) + length(filter (%x. ~P x) xs) = length xs" |
|
700 |
by(induct xs) simp_all |
|
701 |
||
13142 | 702 |
lemma filter_True [simp]: "\<forall>x \<in> set xs. P x ==> filter P xs = xs" |
13145 | 703 |
by (induct xs) auto |
13114 | 704 |
|
13142 | 705 |
lemma filter_False [simp]: "\<forall>x \<in> set xs. \<not> P x ==> filter P xs = []" |
13145 | 706 |
by (induct xs) auto |
13114 | 707 |
|
16998 | 708 |
lemma filter_empty_conv: "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)" |
709 |
by (induct xs) simp_all |
|
710 |
||
711 |
lemma filter_id_conv: "(filter P xs = xs) = (\<forall>x\<in>set xs. P x)" |
|
712 |
apply (induct xs) |
|
713 |
apply auto |
|
714 |
apply(cut_tac P=P and xs=xs in length_filter_le) |
|
715 |
apply simp |
|
716 |
done |
|
13114 | 717 |
|
16965 | 718 |
lemma filter_map: |
719 |
"filter P (map f xs) = map f (filter (P o f) xs)" |
|
720 |
by (induct xs) simp_all |
|
721 |
||
722 |
lemma length_filter_map[simp]: |
|
723 |
"length (filter P (map f xs)) = length(filter (P o f) xs)" |
|
724 |
by (simp add:filter_map) |
|
725 |
||
13142 | 726 |
lemma filter_is_subset [simp]: "set (filter P xs) \<le> set xs" |
13145 | 727 |
by auto |
13114 | 728 |
|
15246 | 729 |
lemma length_filter_less: |
730 |
"\<lbrakk> x : set xs; ~ P x \<rbrakk> \<Longrightarrow> length(filter P xs) < length xs" |
|
731 |
proof (induct xs) |
|
732 |
case Nil thus ?case by simp |
|
733 |
next |
|
734 |
case (Cons x xs) thus ?case |
|
735 |
apply (auto split:split_if_asm) |
|
736 |
using length_filter_le[of P xs] apply arith |
|
737 |
done |
|
738 |
qed |
|
13114 | 739 |
|
15281 | 740 |
lemma length_filter_conv_card: |
741 |
"length(filter p xs) = card{i. i < length xs & p(xs!i)}" |
|
742 |
proof (induct xs) |
|
743 |
case Nil thus ?case by simp |
|
744 |
next |
|
745 |
case (Cons x xs) |
|
746 |
let ?S = "{i. i < length xs & p(xs!i)}" |
|
747 |
have fin: "finite ?S" by(fast intro: bounded_nat_set_is_finite) |
|
748 |
show ?case (is "?l = card ?S'") |
|
749 |
proof (cases) |
|
750 |
assume "p x" |
|
751 |
hence eq: "?S' = insert 0 (Suc ` ?S)" |
|
752 |
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) |
|
753 |
have "length (filter p (x # xs)) = Suc(card ?S)" |
|
754 |
using Cons by simp |
|
755 |
also have "\<dots> = Suc(card(Suc ` ?S))" using fin |
|
756 |
by (simp add: card_image inj_Suc) |
|
757 |
also have "\<dots> = card ?S'" using eq fin |
|
758 |
by (simp add:card_insert_if) (simp add:image_def) |
|
759 |
finally show ?thesis . |
|
760 |
next |
|
761 |
assume "\<not> p x" |
|
762 |
hence eq: "?S' = Suc ` ?S" |
|
763 |
by(auto simp add: nth_Cons image_def split:nat.split elim:lessE) |
|
764 |
have "length (filter p (x # xs)) = card ?S" |
|
765 |
using Cons by simp |
|
766 |
also have "\<dots> = card(Suc ` ?S)" using fin |
|
767 |
by (simp add: card_image inj_Suc) |
|
768 |
also have "\<dots> = card ?S'" using eq fin |
|
769 |
by (simp add:card_insert_if) |
|
770 |
finally show ?thesis . |
|
771 |
qed |
|
772 |
qed |
|
773 |
||
17629 | 774 |
lemma Cons_eq_filterD: |
775 |
"x#xs = filter P ys \<Longrightarrow> |
|
776 |
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" |
|
19585 | 777 |
(is "_ \<Longrightarrow> \<exists>us vs. ?P ys us vs") |
17629 | 778 |
proof(induct ys) |
779 |
case Nil thus ?case by simp |
|
780 |
next |
|
781 |
case (Cons y ys) |
|
782 |
show ?case (is "\<exists>x. ?Q x") |
|
783 |
proof cases |
|
784 |
assume Py: "P y" |
|
785 |
show ?thesis |
|
786 |
proof cases |
|
787 |
assume xy: "x = y" |
|
788 |
show ?thesis |
|
789 |
proof from Py xy Cons(2) show "?Q []" by simp qed |
|
790 |
next |
|
791 |
assume "x \<noteq> y" with Py Cons(2) show ?thesis by simp |
|
792 |
qed |
|
793 |
next |
|
794 |
assume Py: "\<not> P y" |
|
795 |
with Cons obtain us vs where 1 : "?P (y#ys) (y#us) vs" by fastsimp |
|
796 |
show ?thesis (is "? us. ?Q us") |
|
797 |
proof show "?Q (y#us)" using 1 by simp qed |
|
798 |
qed |
|
799 |
qed |
|
800 |
||
801 |
lemma filter_eq_ConsD: |
|
802 |
"filter P ys = x#xs \<Longrightarrow> |
|
803 |
\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs" |
|
804 |
by(rule Cons_eq_filterD) simp |
|
805 |
||
806 |
lemma filter_eq_Cons_iff: |
|
807 |
"(filter P ys = x#xs) = |
|
808 |
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" |
|
809 |
by(auto dest:filter_eq_ConsD) |
|
810 |
||
811 |
lemma Cons_eq_filter_iff: |
|
812 |
"(x#xs = filter P ys) = |
|
813 |
(\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)" |
|
814 |
by(auto dest:Cons_eq_filterD) |
|
815 |
||
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
816 |
lemma filter_cong[fundef_cong, recdef_cong]: |
17501 | 817 |
"xs = ys \<Longrightarrow> (\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x) \<Longrightarrow> filter P xs = filter Q ys" |
818 |
apply simp |
|
819 |
apply(erule thin_rl) |
|
820 |
by (induct ys) simp_all |
|
821 |
||
15281 | 822 |
|
15392 | 823 |
subsubsection {* @{text concat} *} |
13114 | 824 |
|
13142 | 825 |
lemma concat_append [simp]: "concat (xs @ ys) = concat xs @ concat ys" |
13145 | 826 |
by (induct xs) auto |
13114 | 827 |
|
18447 | 828 |
lemma concat_eq_Nil_conv [simp]: "(concat xss = []) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 829 |
by (induct xss) auto |
13114 | 830 |
|
18447 | 831 |
lemma Nil_eq_concat_conv [simp]: "([] = concat xss) = (\<forall>xs \<in> set xss. xs = [])" |
13145 | 832 |
by (induct xss) auto |
13114 | 833 |
|
13142 | 834 |
lemma set_concat [simp]: "set (concat xs) = \<Union>(set ` set xs)" |
13145 | 835 |
by (induct xs) auto |
13114 | 836 |
|
13142 | 837 |
lemma map_concat: "map f (concat xs) = concat (map (map f) xs)" |
13145 | 838 |
by (induct xs) auto |
13114 | 839 |
|
13142 | 840 |
lemma filter_concat: "filter p (concat xs) = concat (map (filter p) xs)" |
13145 | 841 |
by (induct xs) auto |
13114 | 842 |
|
13142 | 843 |
lemma rev_concat: "rev (concat xs) = concat (map rev (rev xs))" |
13145 | 844 |
by (induct xs) auto |
13114 | 845 |
|
846 |
||
15392 | 847 |
subsubsection {* @{text nth} *} |
13114 | 848 |
|
13142 | 849 |
lemma nth_Cons_0 [simp]: "(x # xs)!0 = x" |
13145 | 850 |
by auto |
13114 | 851 |
|
13142 | 852 |
lemma nth_Cons_Suc [simp]: "(x # xs)!(Suc n) = xs!n" |
13145 | 853 |
by auto |
13114 | 854 |
|
13142 | 855 |
declare nth.simps [simp del] |
13114 | 856 |
|
857 |
lemma nth_append: |
|
13145 | 858 |
"!!n. (xs @ ys)!n = (if n < length xs then xs!n else ys!(n - length xs))" |
14208 | 859 |
apply (induct "xs", simp) |
860 |
apply (case_tac n, auto) |
|
13145 | 861 |
done |
13114 | 862 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
863 |
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
|
864 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
865 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
866 |
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
|
867 |
by (induct "xs") auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
868 |
|
13142 | 869 |
lemma nth_map [simp]: "!!n. n < length xs ==> (map f xs)!n = f(xs!n)" |
14208 | 870 |
apply (induct xs, simp) |
871 |
apply (case_tac n, auto) |
|
13145 | 872 |
done |
13114 | 873 |
|
18423 | 874 |
lemma hd_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd xs = xs!0" |
875 |
by(cases xs) simp_all |
|
876 |
||
18049 | 877 |
|
878 |
lemma list_eq_iff_nth_eq: |
|
879 |
"!!ys. (xs = ys) = (length xs = length ys \<and> (ALL i<length xs. xs!i = ys!i))" |
|
880 |
apply(induct xs) |
|
881 |
apply simp apply blast |
|
882 |
apply(case_tac ys) |
|
883 |
apply simp |
|
884 |
apply(simp add:nth_Cons split:nat.split)apply blast |
|
885 |
done |
|
886 |
||
13142 | 887 |
lemma set_conv_nth: "set xs = {xs!i | i. i < length xs}" |
15251 | 888 |
apply (induct xs, simp, simp) |
13145 | 889 |
apply safe |
14208 | 890 |
apply (rule_tac x = 0 in exI, simp) |
891 |
apply (rule_tac x = "Suc i" in exI, simp) |
|
892 |
apply (case_tac i, simp) |
|
13145 | 893 |
apply (rename_tac j) |
14208 | 894 |
apply (rule_tac x = j in exI, simp) |
13145 | 895 |
done |
13114 | 896 |
|
17501 | 897 |
lemma in_set_conv_nth: "(x \<in> set xs) = (\<exists>i < length xs. xs!i = x)" |
898 |
by(auto simp:set_conv_nth) |
|
899 |
||
13145 | 900 |
lemma list_ball_nth: "[| n < length xs; !x : set xs. P x|] ==> P(xs!n)" |
901 |
by (auto simp add: set_conv_nth) |
|
13114 | 902 |
|
13142 | 903 |
lemma nth_mem [simp]: "n < length xs ==> xs!n : set xs" |
13145 | 904 |
by (auto simp add: set_conv_nth) |
13114 | 905 |
|
906 |
lemma all_nth_imp_all_set: |
|
13145 | 907 |
"[| !i < length xs. P(xs!i); x : set xs|] ==> P x" |
908 |
by (auto simp add: set_conv_nth) |
|
13114 | 909 |
|
910 |
lemma all_set_conv_all_nth: |
|
13145 | 911 |
"(\<forall>x \<in> set xs. P x) = (\<forall>i. i < length xs --> P (xs ! i))" |
912 |
by (auto simp add: set_conv_nth) |
|
13114 | 913 |
|
914 |
||
15392 | 915 |
subsubsection {* @{text list_update} *} |
13114 | 916 |
|
13142 | 917 |
lemma length_list_update [simp]: "!!i. length(xs[i:=x]) = length xs" |
13145 | 918 |
by (induct xs) (auto split: nat.split) |
13114 | 919 |
|
920 |
lemma nth_list_update: |
|
13145 | 921 |
"!!i j. i < length xs==> (xs[i:=x])!j = (if i = j then x else xs!j)" |
922 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
|
13114 | 923 |
|
13142 | 924 |
lemma nth_list_update_eq [simp]: "i < length xs ==> (xs[i:=x])!i = x" |
13145 | 925 |
by (simp add: nth_list_update) |
13114 | 926 |
|
13142 | 927 |
lemma nth_list_update_neq [simp]: "!!i j. i \<noteq> j ==> xs[i:=x]!j = xs!j" |
13145 | 928 |
by (induct xs) (auto simp add: nth_Cons split: nat.split) |
13114 | 929 |
|
13142 | 930 |
lemma list_update_overwrite [simp]: |
13145 | 931 |
"!!i. i < size xs ==> xs[i:=x, i:=y] = xs[i:=y]" |
932 |
by (induct xs) (auto split: nat.split) |
|
13114 | 933 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
934 |
lemma list_update_id[simp]: "!!i. i < length xs ==> xs[i := xs!i] = xs" |
14208 | 935 |
apply (induct xs, simp) |
14187 | 936 |
apply(simp split:nat.splits) |
937 |
done |
|
938 |
||
17501 | 939 |
lemma list_update_beyond[simp]: "\<And>i. length xs \<le> i \<Longrightarrow> xs[i:=x] = xs" |
940 |
apply (induct xs) |
|
941 |
apply simp |
|
942 |
apply (case_tac i) |
|
943 |
apply simp_all |
|
944 |
done |
|
945 |
||
13114 | 946 |
lemma list_update_same_conv: |
13145 | 947 |
"!!i. i < length xs ==> (xs[i := x] = xs) = (xs!i = x)" |
948 |
by (induct xs) (auto split: nat.split) |
|
13114 | 949 |
|
14187 | 950 |
lemma list_update_append1: |
951 |
"!!i. i < size xs \<Longrightarrow> (xs @ ys)[i:=x] = xs[i:=x] @ ys" |
|
14208 | 952 |
apply (induct xs, simp) |
14187 | 953 |
apply(simp split:nat.split) |
954 |
done |
|
955 |
||
15868 | 956 |
lemma list_update_append: |
957 |
"!!n. (xs @ ys) [n:= x] = |
|
958 |
(if n < length xs then xs[n:= x] @ ys else xs @ (ys [n-length xs:= x]))" |
|
959 |
by (induct xs) (auto split:nat.splits) |
|
960 |
||
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
961 |
lemma list_update_length [simp]: |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
962 |
"(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
|
963 |
by (induct xs, auto) |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
964 |
|
13114 | 965 |
lemma update_zip: |
13145 | 966 |
"!!i xy xs. length xs = length ys ==> |
967 |
(zip xs ys)[i:=xy] = zip (xs[i:=fst xy]) (ys[i:=snd xy])" |
|
968 |
by (induct ys) (auto, case_tac xs, auto split: nat.split) |
|
13114 | 969 |
|
970 |
lemma set_update_subset_insert: "!!i. set(xs[i:=x]) <= insert x (set xs)" |
|
13145 | 971 |
by (induct xs) (auto split: nat.split) |
13114 | 972 |
|
973 |
lemma set_update_subsetI: "[| set xs <= A; x:A |] ==> set(xs[i := x]) <= A" |
|
13145 | 974 |
by (blast dest!: set_update_subset_insert [THEN subsetD]) |
13114 | 975 |
|
15868 | 976 |
lemma set_update_memI: "!!n. n < length xs \<Longrightarrow> x \<in> set (xs[n := x])" |
977 |
by (induct xs) (auto split:nat.splits) |
|
978 |
||
13114 | 979 |
|
15392 | 980 |
subsubsection {* @{text last} and @{text butlast} *} |
13114 | 981 |
|
13142 | 982 |
lemma last_snoc [simp]: "last (xs @ [x]) = x" |
13145 | 983 |
by (induct xs) auto |
13114 | 984 |
|
13142 | 985 |
lemma butlast_snoc [simp]: "butlast (xs @ [x]) = xs" |
13145 | 986 |
by (induct xs) auto |
13114 | 987 |
|
14302 | 988 |
lemma last_ConsL: "xs = [] \<Longrightarrow> last(x#xs) = x" |
989 |
by(simp add:last.simps) |
|
990 |
||
991 |
lemma last_ConsR: "xs \<noteq> [] \<Longrightarrow> last(x#xs) = last xs" |
|
992 |
by(simp add:last.simps) |
|
993 |
||
994 |
lemma last_append: "last(xs @ ys) = (if ys = [] then last xs else last ys)" |
|
995 |
by (induct xs) (auto) |
|
996 |
||
997 |
lemma last_appendL[simp]: "ys = [] \<Longrightarrow> last(xs @ ys) = last xs" |
|
998 |
by(simp add:last_append) |
|
999 |
||
1000 |
lemma last_appendR[simp]: "ys \<noteq> [] \<Longrightarrow> last(xs @ ys) = last ys" |
|
1001 |
by(simp add:last_append) |
|
1002 |
||
17762 | 1003 |
lemma hd_rev: "xs \<noteq> [] \<Longrightarrow> hd(rev xs) = last xs" |
1004 |
by(rule rev_exhaust[of xs]) simp_all |
|
1005 |
||
1006 |
lemma last_rev: "xs \<noteq> [] \<Longrightarrow> last(rev xs) = hd xs" |
|
1007 |
by(cases xs) simp_all |
|
1008 |
||
17765 | 1009 |
lemma last_in_set[simp]: "as \<noteq> [] \<Longrightarrow> last as \<in> set as" |
1010 |
by (induct as) auto |
|
17762 | 1011 |
|
13142 | 1012 |
lemma length_butlast [simp]: "length (butlast xs) = length xs - 1" |
13145 | 1013 |
by (induct xs rule: rev_induct) auto |
13114 | 1014 |
|
1015 |
lemma butlast_append: |
|
13145 | 1016 |
"!!ys. butlast (xs @ ys) = (if ys = [] then butlast xs else xs @ butlast ys)" |
1017 |
by (induct xs) auto |
|
13114 | 1018 |
|
13142 | 1019 |
lemma append_butlast_last_id [simp]: |
13145 | 1020 |
"xs \<noteq> [] ==> butlast xs @ [last xs] = xs" |
1021 |
by (induct xs) auto |
|
13114 | 1022 |
|
13142 | 1023 |
lemma in_set_butlastD: "x : set (butlast xs) ==> x : set xs" |
13145 | 1024 |
by (induct xs) (auto split: split_if_asm) |
13114 | 1025 |
|
1026 |
lemma in_set_butlast_appendI: |
|
13145 | 1027 |
"x : set (butlast xs) | x : set (butlast ys) ==> x : set (butlast (xs @ ys))" |
1028 |
by (auto dest: in_set_butlastD simp add: butlast_append) |
|
13114 | 1029 |
|
17501 | 1030 |
lemma last_drop[simp]: "!!n. n < length xs \<Longrightarrow> last (drop n xs) = last xs" |
1031 |
apply (induct xs) |
|
1032 |
apply simp |
|
1033 |
apply (auto split:nat.split) |
|
1034 |
done |
|
1035 |
||
17589 | 1036 |
lemma last_conv_nth: "xs\<noteq>[] \<Longrightarrow> last xs = xs!(length xs - 1)" |
1037 |
by(induct xs)(auto simp:neq_Nil_conv) |
|
1038 |
||
15392 | 1039 |
subsubsection {* @{text take} and @{text drop} *} |
13114 | 1040 |
|
13142 | 1041 |
lemma take_0 [simp]: "take 0 xs = []" |
13145 | 1042 |
by (induct xs) auto |
13114 | 1043 |
|
13142 | 1044 |
lemma drop_0 [simp]: "drop 0 xs = xs" |
13145 | 1045 |
by (induct xs) auto |
13114 | 1046 |
|
13142 | 1047 |
lemma take_Suc_Cons [simp]: "take (Suc n) (x # xs) = x # take n xs" |
13145 | 1048 |
by simp |
13114 | 1049 |
|
13142 | 1050 |
lemma drop_Suc_Cons [simp]: "drop (Suc n) (x # xs) = drop n xs" |
13145 | 1051 |
by simp |
13114 | 1052 |
|
13142 | 1053 |
declare take_Cons [simp del] and drop_Cons [simp del] |
13114 | 1054 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1055 |
lemma take_Suc: "xs ~= [] ==> take (Suc n) xs = hd xs # take n (tl xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1056 |
by(clarsimp simp add:neq_Nil_conv) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1057 |
|
14187 | 1058 |
lemma drop_Suc: "drop (Suc n) xs = drop n (tl xs)" |
1059 |
by(cases xs, simp_all) |
|
1060 |
||
1061 |
lemma drop_tl: "!!n. drop n (tl xs) = tl(drop n xs)" |
|
1062 |
by(induct xs, simp_all add:drop_Cons drop_Suc split:nat.split) |
|
1063 |
||
1064 |
lemma nth_via_drop: "!!n. drop n xs = y#ys \<Longrightarrow> xs!n = y" |
|
14208 | 1065 |
apply (induct xs, simp) |
14187 | 1066 |
apply(simp add:drop_Cons nth_Cons split:nat.splits) |
1067 |
done |
|
1068 |
||
13913 | 1069 |
lemma take_Suc_conv_app_nth: |
1070 |
"!!i. i < length xs \<Longrightarrow> take (Suc i) xs = take i xs @ [xs!i]" |
|
14208 | 1071 |
apply (induct xs, simp) |
1072 |
apply (case_tac i, auto) |
|
13913 | 1073 |
done |
1074 |
||
14591 | 1075 |
lemma drop_Suc_conv_tl: |
1076 |
"!!i. i < length xs \<Longrightarrow> (xs!i) # (drop (Suc i) xs) = drop i xs" |
|
1077 |
apply (induct xs, simp) |
|
1078 |
apply (case_tac i, auto) |
|
1079 |
done |
|
1080 |
||
13142 | 1081 |
lemma length_take [simp]: "!!xs. length (take n xs) = min (length xs) n" |
13145 | 1082 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 1083 |
|
13142 | 1084 |
lemma length_drop [simp]: "!!xs. length (drop n xs) = (length xs - n)" |
13145 | 1085 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 1086 |
|
13142 | 1087 |
lemma take_all [simp]: "!!xs. length xs <= n ==> take n xs = xs" |
13145 | 1088 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 1089 |
|
13142 | 1090 |
lemma drop_all [simp]: "!!xs. length xs <= n ==> drop n xs = []" |
13145 | 1091 |
by (induct n) (auto, case_tac xs, auto) |
13114 | 1092 |
|
13142 | 1093 |
lemma take_append [simp]: |
13145 | 1094 |
"!!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)" |
1095 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 1096 |
|
13142 | 1097 |
lemma drop_append [simp]: |
13145 | 1098 |
"!!xs. drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys" |
1099 |
by (induct n) (auto, case_tac xs, auto) |
|
13114 | 1100 |
|
13142 | 1101 |
lemma take_take [simp]: "!!xs n. take n (take m xs) = take (min n m) xs" |
14208 | 1102 |
apply (induct m, auto) |
1103 |
apply (case_tac xs, auto) |
|
15236
f289e8ba2bb3
Proofs needed to be updated because induction now preserves name of
nipkow
parents:
15176
diff
changeset
|
1104 |
apply (case_tac n, auto) |
13145 | 1105 |
done |
13114 | 1106 |
|
13142 | 1107 |
lemma drop_drop [simp]: "!!xs. drop n (drop m xs) = drop (n + m) xs" |
14208 | 1108 |
apply (induct m, auto) |
1109 |
apply (case_tac xs, auto) |
|
13145 | 1110 |
done |
13114 | 1111 |
|
1112 |
lemma take_drop: "!!xs n. take n (drop m xs) = drop m (take (n + m) xs)" |
|
14208 | 1113 |
apply (induct m, auto) |
1114 |
apply (case_tac xs, auto) |
|
13145 | 1115 |
done |
13114 | 1116 |
|
14802 | 1117 |
lemma drop_take: "!!m n. drop n (take m xs) = take (m-n) (drop n xs)" |
1118 |
apply(induct xs) |
|
1119 |
apply simp |
|
1120 |
apply(simp add: take_Cons drop_Cons split:nat.split) |
|
1121 |
done |
|
1122 |
||
13142 | 1123 |
lemma append_take_drop_id [simp]: "!!xs. take n xs @ drop n xs = xs" |
14208 | 1124 |
apply (induct n, auto) |
1125 |
apply (case_tac xs, auto) |
|
13145 | 1126 |
done |
13114 | 1127 |
|
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1128 |
lemma take_eq_Nil[simp]: "!!n. (take n xs = []) = (n = 0 \<or> xs = [])" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1129 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1130 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1131 |
apply(simp add:take_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1132 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1133 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1134 |
lemma drop_eq_Nil[simp]: "!!n. (drop n xs = []) = (length xs <= n)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1135 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1136 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1137 |
apply(simp add:drop_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1138 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1139 |
|
13114 | 1140 |
lemma take_map: "!!xs. take n (map f xs) = map f (take n xs)" |
14208 | 1141 |
apply (induct n, auto) |
1142 |
apply (case_tac xs, auto) |
|
13145 | 1143 |
done |
13114 | 1144 |
|
13142 | 1145 |
lemma drop_map: "!!xs. drop n (map f xs) = map f (drop n xs)" |
14208 | 1146 |
apply (induct n, auto) |
1147 |
apply (case_tac xs, auto) |
|
13145 | 1148 |
done |
13114 | 1149 |
|
1150 |
lemma rev_take: "!!i. rev (take i xs) = drop (length xs - i) (rev xs)" |
|
14208 | 1151 |
apply (induct xs, auto) |
1152 |
apply (case_tac i, auto) |
|
13145 | 1153 |
done |
13114 | 1154 |
|
1155 |
lemma rev_drop: "!!i. rev (drop i xs) = take (length xs - i) (rev xs)" |
|
14208 | 1156 |
apply (induct xs, auto) |
1157 |
apply (case_tac i, auto) |
|
13145 | 1158 |
done |
13114 | 1159 |
|
13142 | 1160 |
lemma nth_take [simp]: "!!n i. i < n ==> (take n xs)!i = xs!i" |
14208 | 1161 |
apply (induct xs, auto) |
1162 |
apply (case_tac n, blast) |
|
1163 |
apply (case_tac i, auto) |
|
13145 | 1164 |
done |
13114 | 1165 |
|
13142 | 1166 |
lemma nth_drop [simp]: |
13145 | 1167 |
"!!xs i. n + i <= length xs ==> (drop n xs)!i = xs!(n + i)" |
14208 | 1168 |
apply (induct n, auto) |
1169 |
apply (case_tac xs, auto) |
|
13145 | 1170 |
done |
3507 | 1171 |
|
18423 | 1172 |
lemma hd_drop_conv_nth: "\<lbrakk> xs \<noteq> []; n < length xs \<rbrakk> \<Longrightarrow> hd(drop n xs) = xs!n" |
1173 |
by(simp add: hd_conv_nth) |
|
1174 |
||
14025 | 1175 |
lemma set_take_subset: "\<And>n. set(take n xs) \<subseteq> set xs" |
1176 |
by(induct xs)(auto simp:take_Cons split:nat.split) |
|
1177 |
||
1178 |
lemma set_drop_subset: "\<And>n. set(drop n xs) \<subseteq> set xs" |
|
1179 |
by(induct xs)(auto simp:drop_Cons split:nat.split) |
|
1180 |
||
14187 | 1181 |
lemma in_set_takeD: "x : set(take n xs) \<Longrightarrow> x : set xs" |
1182 |
using set_take_subset by fast |
|
1183 |
||
1184 |
lemma in_set_dropD: "x : set(drop n xs) \<Longrightarrow> x : set xs" |
|
1185 |
using set_drop_subset by fast |
|
1186 |
||
13114 | 1187 |
lemma append_eq_conv_conj: |
13145 | 1188 |
"!!zs. (xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)" |
14208 | 1189 |
apply (induct xs, simp, clarsimp) |
1190 |
apply (case_tac zs, auto) |
|
13145 | 1191 |
done |
13142 | 1192 |
|
14050 | 1193 |
lemma take_add [rule_format]: |
1194 |
"\<forall>i. i+j \<le> length(xs) --> take (i+j) xs = take i xs @ take j (drop i xs)" |
|
1195 |
apply (induct xs, auto) |
|
1196 |
apply (case_tac i, simp_all) |
|
1197 |
done |
|
1198 |
||
14300 | 1199 |
lemma append_eq_append_conv_if: |
1200 |
"!! ys\<^isub>1. (xs\<^isub>1 @ xs\<^isub>2 = ys\<^isub>1 @ ys\<^isub>2) = |
|
1201 |
(if size xs\<^isub>1 \<le> size ys\<^isub>1 |
|
1202 |
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 |
|
1203 |
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)" |
|
1204 |
apply(induct xs\<^isub>1) |
|
1205 |
apply simp |
|
1206 |
apply(case_tac ys\<^isub>1) |
|
1207 |
apply simp_all |
|
1208 |
done |
|
1209 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1210 |
lemma take_hd_drop: |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1211 |
"!!n. n < length xs \<Longrightarrow> take n xs @ [hd (drop n xs)] = take (n+1) xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1212 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1213 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1214 |
apply(simp add:drop_Cons split:nat.split) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1215 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1216 |
|
17501 | 1217 |
lemma id_take_nth_drop: |
1218 |
"i < length xs \<Longrightarrow> xs = take i xs @ xs!i # drop (Suc i) xs" |
|
1219 |
proof - |
|
1220 |
assume si: "i < length xs" |
|
1221 |
hence "xs = take (Suc i) xs @ drop (Suc i) xs" by auto |
|
1222 |
moreover |
|
1223 |
from si have "take (Suc i) xs = take i xs @ [xs!i]" |
|
1224 |
apply (rule_tac take_Suc_conv_app_nth) by arith |
|
1225 |
ultimately show ?thesis by auto |
|
1226 |
qed |
|
1227 |
||
1228 |
lemma upd_conv_take_nth_drop: |
|
1229 |
"i < length xs \<Longrightarrow> xs[i:=a] = take i xs @ a # drop (Suc i) xs" |
|
1230 |
proof - |
|
1231 |
assume i: "i < length xs" |
|
1232 |
have "xs[i:=a] = (take i xs @ xs!i # drop (Suc i) xs)[i:=a]" |
|
1233 |
by(rule arg_cong[OF id_take_nth_drop[OF i]]) |
|
1234 |
also have "\<dots> = take i xs @ a # drop (Suc i) xs" |
|
1235 |
using i by (simp add: list_update_append) |
|
1236 |
finally show ?thesis . |
|
1237 |
qed |
|
1238 |
||
13114 | 1239 |
|
15392 | 1240 |
subsubsection {* @{text takeWhile} and @{text dropWhile} *} |
13114 | 1241 |
|
13142 | 1242 |
lemma takeWhile_dropWhile_id [simp]: "takeWhile P xs @ dropWhile P xs = xs" |
13145 | 1243 |
by (induct xs) auto |
13114 | 1244 |
|
13142 | 1245 |
lemma takeWhile_append1 [simp]: |
13145 | 1246 |
"[| x:set xs; ~P(x)|] ==> takeWhile P (xs @ ys) = takeWhile P xs" |
1247 |
by (induct xs) auto |
|
13114 | 1248 |
|
13142 | 1249 |
lemma takeWhile_append2 [simp]: |
13145 | 1250 |
"(!!x. x : set xs ==> P x) ==> takeWhile P (xs @ ys) = xs @ takeWhile P ys" |
1251 |
by (induct xs) auto |
|
13114 | 1252 |
|
13142 | 1253 |
lemma takeWhile_tail: "\<not> P x ==> takeWhile P (xs @ (x#l)) = takeWhile P xs" |
13145 | 1254 |
by (induct xs) auto |
13114 | 1255 |
|
13142 | 1256 |
lemma dropWhile_append1 [simp]: |
13145 | 1257 |
"[| x : set xs; ~P(x)|] ==> dropWhile P (xs @ ys) = (dropWhile P xs)@ys" |
1258 |
by (induct xs) auto |
|
13114 | 1259 |
|
13142 | 1260 |
lemma dropWhile_append2 [simp]: |
13145 | 1261 |
"(!!x. x:set xs ==> P(x)) ==> dropWhile P (xs @ ys) = dropWhile P ys" |
1262 |
by (induct xs) auto |
|
13114 | 1263 |
|
13142 | 1264 |
lemma set_take_whileD: "x : set (takeWhile P xs) ==> x : set xs \<and> P x" |
13145 | 1265 |
by (induct xs) (auto split: split_if_asm) |
13114 | 1266 |
|
13913 | 1267 |
lemma takeWhile_eq_all_conv[simp]: |
1268 |
"(takeWhile P xs = xs) = (\<forall>x \<in> set xs. P x)" |
|
1269 |
by(induct xs, auto) |
|
1270 |
||
1271 |
lemma dropWhile_eq_Nil_conv[simp]: |
|
1272 |
"(dropWhile P xs = []) = (\<forall>x \<in> set xs. P x)" |
|
1273 |
by(induct xs, auto) |
|
1274 |
||
1275 |
lemma dropWhile_eq_Cons_conv: |
|
1276 |
"(dropWhile P xs = y#ys) = (xs = takeWhile P xs @ y # ys & \<not> P y)" |
|
1277 |
by(induct xs, auto) |
|
1278 |
||
17501 | 1279 |
text{* The following two lemmmas could be generalized to an arbitrary |
1280 |
property. *} |
|
1281 |
||
1282 |
lemma takeWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> |
|
1283 |
takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))" |
|
1284 |
by(induct xs) (auto simp: takeWhile_tail[where l="[]"]) |
|
1285 |
||
1286 |
lemma dropWhile_neq_rev: "\<lbrakk>distinct xs; x \<in> set xs\<rbrakk> \<Longrightarrow> |
|
1287 |
dropWhile (\<lambda>y. y \<noteq> x) (rev xs) = x # rev (takeWhile (\<lambda>y. y \<noteq> x) xs)" |
|
1288 |
apply(induct xs) |
|
1289 |
apply simp |
|
1290 |
apply auto |
|
1291 |
apply(subst dropWhile_append2) |
|
1292 |
apply auto |
|
1293 |
done |
|
1294 |
||
18423 | 1295 |
lemma takeWhile_not_last: |
1296 |
"\<lbrakk> xs \<noteq> []; distinct xs\<rbrakk> \<Longrightarrow> takeWhile (\<lambda>y. y \<noteq> last xs) xs = butlast xs" |
|
1297 |
apply(induct xs) |
|
1298 |
apply simp |
|
1299 |
apply(case_tac xs) |
|
1300 |
apply(auto) |
|
1301 |
done |
|
1302 |
||
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
1303 |
lemma takeWhile_cong [fundef_cong, recdef_cong]: |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1304 |
"[| l = k; !!x. x : set l ==> P x = Q x |] |
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1305 |
==> takeWhile P l = takeWhile Q k" |
20503 | 1306 |
by (induct k arbitrary: l) (simp_all) |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1307 |
|
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
1308 |
lemma dropWhile_cong [fundef_cong, recdef_cong]: |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1309 |
"[| l = k; !!x. x : set l ==> P x = Q x |] |
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1310 |
==> dropWhile P l = dropWhile Q k" |
20503 | 1311 |
by (induct k arbitrary: l, simp_all) |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1312 |
|
13114 | 1313 |
|
15392 | 1314 |
subsubsection {* @{text zip} *} |
13114 | 1315 |
|
13142 | 1316 |
lemma zip_Nil [simp]: "zip [] ys = []" |
13145 | 1317 |
by (induct ys) auto |
13114 | 1318 |
|
13142 | 1319 |
lemma zip_Cons_Cons [simp]: "zip (x # xs) (y # ys) = (x, y) # zip xs ys" |
13145 | 1320 |
by simp |
13114 | 1321 |
|
13142 | 1322 |
declare zip_Cons [simp del] |
13114 | 1323 |
|
15281 | 1324 |
lemma zip_Cons1: |
1325 |
"zip (x#xs) ys = (case ys of [] \<Rightarrow> [] | y#ys \<Rightarrow> (x,y)#zip xs ys)" |
|
1326 |
by(auto split:list.split) |
|
1327 |
||
13142 | 1328 |
lemma length_zip [simp]: |
13145 | 1329 |
"!!xs. length (zip xs ys) = min (length xs) (length ys)" |
14208 | 1330 |
apply (induct ys, simp) |
1331 |
apply (case_tac xs, auto) |
|
13145 | 1332 |
done |
13114 | 1333 |
|
1334 |
lemma zip_append1: |
|
13145 | 1335 |
"!!xs. zip (xs @ ys) zs = |
1336 |
zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)" |
|
14208 | 1337 |
apply (induct zs, simp) |
1338 |
apply (case_tac xs, simp_all) |
|
13145 | 1339 |
done |
13114 | 1340 |
|
1341 |
lemma zip_append2: |
|
13145 | 1342 |
"!!ys. zip xs (ys @ zs) = |
1343 |
zip (take (length ys) xs) ys @ zip (drop (length ys) xs) zs" |
|
14208 | 1344 |
apply (induct xs, simp) |
1345 |
apply (case_tac ys, simp_all) |
|
13145 | 1346 |
done |
13114 | 1347 |
|
13142 | 1348 |
lemma zip_append [simp]: |
1349 |
"[| length xs = length us; length ys = length vs |] ==> |
|
13145 | 1350 |
zip (xs@ys) (us@vs) = zip xs us @ zip ys vs" |
1351 |
by (simp add: zip_append1) |
|
13114 | 1352 |
|
1353 |
lemma zip_rev: |
|
14247 | 1354 |
"length xs = length ys ==> zip (rev xs) (rev ys) = rev (zip xs ys)" |
1355 |
by (induct rule:list_induct2, simp_all) |
|
13114 | 1356 |
|
13142 | 1357 |
lemma nth_zip [simp]: |
13145 | 1358 |
"!!i xs. [| i < length xs; i < length ys|] ==> (zip xs ys)!i = (xs!i, ys!i)" |
14208 | 1359 |
apply (induct ys, simp) |
13145 | 1360 |
apply (case_tac xs) |
1361 |
apply (simp_all add: nth.simps split: nat.split) |
|
1362 |
done |
|
13114 | 1363 |
|
1364 |
lemma set_zip: |
|
13145 | 1365 |
"set (zip xs ys) = {(xs!i, ys!i) | i. i < min (length xs) (length ys)}" |
1366 |
by (simp add: set_conv_nth cong: rev_conj_cong) |
|
13114 | 1367 |
|
1368 |
lemma zip_update: |
|
13145 | 1369 |
"length xs = length ys ==> zip (xs[i:=x]) (ys[i:=y]) = (zip xs ys)[i:=(x,y)]" |
1370 |
by (rule sym, simp add: update_zip) |
|
13114 | 1371 |
|
13142 | 1372 |
lemma zip_replicate [simp]: |
13145 | 1373 |
"!!j. zip (replicate i x) (replicate j y) = replicate (min i j) (x,y)" |
14208 | 1374 |
apply (induct i, auto) |
1375 |
apply (case_tac j, auto) |
|
13145 | 1376 |
done |
13114 | 1377 |
|
19487 | 1378 |
lemma take_zip: |
1379 |
"!!xs ys. take n (zip xs ys) = zip (take n xs) (take n ys)" |
|
1380 |
apply (induct n) |
|
1381 |
apply simp |
|
1382 |
apply (case_tac xs, simp) |
|
1383 |
apply (case_tac ys, simp_all) |
|
1384 |
done |
|
1385 |
||
1386 |
lemma drop_zip: |
|
1387 |
"!!xs ys. drop n (zip xs ys) = zip (drop n xs) (drop n ys)" |
|
1388 |
apply (induct n) |
|
1389 |
apply simp |
|
1390 |
apply (case_tac xs, simp) |
|
1391 |
apply (case_tac ys, simp_all) |
|
1392 |
done |
|
1393 |
||
13142 | 1394 |
|
15392 | 1395 |
subsubsection {* @{text list_all2} *} |
13114 | 1396 |
|
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1397 |
lemma list_all2_lengthD [intro?]: |
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1398 |
"list_all2 P xs ys ==> length xs = length ys" |
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1399 |
by (simp add: list_all2_def) |
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1400 |
|
19787 | 1401 |
lemma list_all2_Nil [iff, code]: "list_all2 P [] ys = (ys = [])" |
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1402 |
by (simp add: list_all2_def) |
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1403 |
|
19787 | 1404 |
lemma list_all2_Nil2 [iff, code]: "list_all2 P xs [] = (xs = [])" |
1405 |
by (simp add: list_all2_def) |
|
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1406 |
|
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1407 |
lemma list_all2_Cons [iff, code]: |
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1408 |
"list_all2 P (x # xs) (y # ys) = (P x y \<and> list_all2 P xs ys)" |
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
1409 |
by (auto simp add: list_all2_def) |
13114 | 1410 |
|
1411 |
lemma list_all2_Cons1: |
|
13145 | 1412 |
"list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)" |
1413 |
by (cases ys) auto |
|
13114 | 1414 |
|
1415 |
lemma list_all2_Cons2: |
|
13145 | 1416 |
"list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)" |
1417 |
by (cases xs) auto |
|
13114 | 1418 |
|
13142 | 1419 |
lemma list_all2_rev [iff]: |
13145 | 1420 |
"list_all2 P (rev xs) (rev ys) = list_all2 P xs ys" |
1421 |
by (simp add: list_all2_def zip_rev cong: conj_cong) |
|
13114 | 1422 |
|
13863 | 1423 |
lemma list_all2_rev1: |
1424 |
"list_all2 P (rev xs) ys = list_all2 P xs (rev ys)" |
|
1425 |
by (subst list_all2_rev [symmetric]) simp |
|
1426 |
||
13114 | 1427 |
lemma list_all2_append1: |
13145 | 1428 |
"list_all2 P (xs @ ys) zs = |
1429 |
(EX us vs. zs = us @ vs \<and> length us = length xs \<and> length vs = length ys \<and> |
|
1430 |
list_all2 P xs us \<and> list_all2 P ys vs)" |
|
1431 |
apply (simp add: list_all2_def zip_append1) |
|
1432 |
apply (rule iffI) |
|
1433 |
apply (rule_tac x = "take (length xs) zs" in exI) |
|
1434 |
apply (rule_tac x = "drop (length xs) zs" in exI) |
|
14208 | 1435 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1436 |
apply (simp add: ball_Un) |
1437 |
done |
|
13114 | 1438 |
|
1439 |
lemma list_all2_append2: |
|
13145 | 1440 |
"list_all2 P xs (ys @ zs) = |
1441 |
(EX us vs. xs = us @ vs \<and> length us = length ys \<and> length vs = length zs \<and> |
|
1442 |
list_all2 P us ys \<and> list_all2 P vs zs)" |
|
1443 |
apply (simp add: list_all2_def zip_append2) |
|
1444 |
apply (rule iffI) |
|
1445 |
apply (rule_tac x = "take (length ys) xs" in exI) |
|
1446 |
apply (rule_tac x = "drop (length ys) xs" in exI) |
|
14208 | 1447 |
apply (force split: nat_diff_split simp add: min_def, clarify) |
13145 | 1448 |
apply (simp add: ball_Un) |
1449 |
done |
|
13114 | 1450 |
|
13863 | 1451 |
lemma list_all2_append: |
14247 | 1452 |
"length xs = length ys \<Longrightarrow> |
1453 |
list_all2 P (xs@us) (ys@vs) = (list_all2 P xs ys \<and> list_all2 P us vs)" |
|
1454 |
by (induct rule:list_induct2, simp_all) |
|
13863 | 1455 |
|
1456 |
lemma list_all2_appendI [intro?, trans]: |
|
1457 |
"\<lbrakk> list_all2 P a b; list_all2 P c d \<rbrakk> \<Longrightarrow> list_all2 P (a@c) (b@d)" |
|
1458 |
by (simp add: list_all2_append list_all2_lengthD) |
|
1459 |
||
13114 | 1460 |
lemma list_all2_conv_all_nth: |
13145 | 1461 |
"list_all2 P xs ys = |
1462 |
(length xs = length ys \<and> (\<forall>i < length xs. P (xs!i) (ys!i)))" |
|
1463 |
by (force simp add: list_all2_def set_zip) |
|
13114 | 1464 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1465 |
lemma list_all2_trans: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1466 |
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
|
1467 |
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
|
1468 |
(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
|
1469 |
proof (induct as) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1470 |
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
|
1471 |
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
|
1472 |
proof (induct bs) |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1473 |
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
|
1474 |
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
|
1475 |
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
|
1476 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1477 |
qed simp |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1478 |
|
13863 | 1479 |
lemma list_all2_all_nthI [intro?]: |
1480 |
"length a = length b \<Longrightarrow> (\<And>n. n < length a \<Longrightarrow> P (a!n) (b!n)) \<Longrightarrow> list_all2 P a b" |
|
1481 |
by (simp add: list_all2_conv_all_nth) |
|
1482 |
||
14395 | 1483 |
lemma list_all2I: |
1484 |
"\<forall>x \<in> set (zip a b). split P x \<Longrightarrow> length a = length b \<Longrightarrow> list_all2 P a b" |
|
1485 |
by (simp add: list_all2_def) |
|
1486 |
||
14328 | 1487 |
lemma list_all2_nthD: |
13863 | 1488 |
"\<lbrakk> list_all2 P xs ys; p < size xs \<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
1489 |
by (simp add: list_all2_conv_all_nth) |
|
1490 |
||
14302 | 1491 |
lemma list_all2_nthD2: |
1492 |
"\<lbrakk>list_all2 P xs ys; p < size ys\<rbrakk> \<Longrightarrow> P (xs!p) (ys!p)" |
|
1493 |
by (frule list_all2_lengthD) (auto intro: list_all2_nthD) |
|
1494 |
||
13863 | 1495 |
lemma list_all2_map1: |
1496 |
"list_all2 P (map f as) bs = list_all2 (\<lambda>x y. P (f x) y) as bs" |
|
1497 |
by (simp add: list_all2_conv_all_nth) |
|
1498 |
||
1499 |
lemma list_all2_map2: |
|
1500 |
"list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs" |
|
1501 |
by (auto simp add: list_all2_conv_all_nth) |
|
1502 |
||
14316
91b897b9a2dc
added some [intro?] and [trans] for list_all2 lemmas
kleing
parents:
14302
diff
changeset
|
1503 |
lemma list_all2_refl [intro?]: |
13863 | 1504 |
"(\<And>x. P x x) \<Longrightarrow> list_all2 P xs xs" |
1505 |
by (simp add: list_all2_conv_all_nth) |
|
1506 |
||
1507 |
lemma list_all2_update_cong: |
|
1508 |
"\<lbrakk> i<size xs; list_all2 P xs ys; P x y \<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1509 |
by (simp add: list_all2_conv_all_nth nth_list_update) |
|
1510 |
||
1511 |
lemma list_all2_update_cong2: |
|
1512 |
"\<lbrakk>list_all2 P xs ys; P x y; i < length ys\<rbrakk> \<Longrightarrow> list_all2 P (xs[i:=x]) (ys[i:=y])" |
|
1513 |
by (simp add: list_all2_lengthD list_all2_update_cong) |
|
1514 |
||
14302 | 1515 |
lemma list_all2_takeI [simp,intro?]: |
1516 |
"\<And>n ys. list_all2 P xs ys \<Longrightarrow> list_all2 P (take n xs) (take n ys)" |
|
1517 |
apply (induct xs) |
|
1518 |
apply simp |
|
1519 |
apply (clarsimp simp add: list_all2_Cons1) |
|
1520 |
apply (case_tac n) |
|
1521 |
apply auto |
|
1522 |
done |
|
1523 |
||
1524 |
lemma list_all2_dropI [simp,intro?]: |
|
13863 | 1525 |
"\<And>n bs. list_all2 P as bs \<Longrightarrow> list_all2 P (drop n as) (drop n bs)" |
14208 | 1526 |
apply (induct as, simp) |
13863 | 1527 |
apply (clarsimp simp add: list_all2_Cons1) |
14208 | 1528 |
apply (case_tac n, simp, simp) |
13863 | 1529 |
done |
1530 |
||
14327 | 1531 |
lemma list_all2_mono [intro?]: |
13863 | 1532 |
"\<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 | 1533 |
apply (induct x, simp) |
1534 |
apply (case_tac y, auto) |
|
13863 | 1535 |
done |
1536 |
||
13142 | 1537 |
|
15392 | 1538 |
subsubsection {* @{text foldl} and @{text foldr} *} |
13142 | 1539 |
|
1540 |
lemma foldl_append [simp]: |
|
13145 | 1541 |
"!!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys" |
1542 |
by (induct xs) auto |
|
13142 | 1543 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1544 |
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
|
1545 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1546 |
|
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
1547 |
lemma foldl_cong [fundef_cong, recdef_cong]: |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1548 |
"[| a = b; l = k; !!a x. x : set l ==> f a x = g a x |] |
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1549 |
==> foldl f a l = foldl g b k" |
20503 | 1550 |
by (induct k arbitrary: a b l) simp_all |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1551 |
|
19770
be5c23ebe1eb
HOL/Tools/function_package: Added support for mutual recursive definitions.
krauss
parents:
19623
diff
changeset
|
1552 |
lemma foldr_cong [fundef_cong, recdef_cong]: |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1553 |
"[| a = b; l = k; !!a x. x : set l ==> f x a = g x a |] |
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1554 |
==> foldr f l a = foldr g k b" |
20503 | 1555 |
by (induct k arbitrary: a b l) simp_all |
18336
1a2e30b37ed3
Added recdef congruence rules for bounded quantifiers and commonly used
krauss
parents:
18049
diff
changeset
|
1556 |
|
14402
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1557 |
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
|
1558 |
by (induct xs) auto |
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1559 |
|
4201e1916482
moved lemmas from MicroJava/Comp/AuxLemmas.thy to List.thy
nipkow
parents:
14395
diff
changeset
|
1560 |
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
|
1561 |
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
|
1562 |
|
13142 | 1563 |
text {* |
13145 | 1564 |
Note: @{text "n \<le> foldl (op +) n ns"} looks simpler, but is more |
1565 |
difficult to use because it requires an additional transitivity step. |
|
13142 | 1566 |
*} |
1567 |
||
1568 |
lemma start_le_sum: "!!n::nat. m <= n ==> m <= foldl (op +) n ns" |
|
13145 | 1569 |
by (induct ns) auto |
13142 | 1570 |
|
1571 |
lemma elem_le_sum: "!!n::nat. n : set ns ==> n <= foldl (op +) 0 ns" |
|
13145 | 1572 |
by (force intro: start_le_sum simp add: in_set_conv_decomp) |
13142 | 1573 |
|
1574 |
lemma sum_eq_0_conv [iff]: |
|
13145 | 1575 |
"!!m::nat. (foldl (op +) m ns = 0) = (m = 0 \<and> (\<forall>n \<in> set ns. n = 0))" |
1576 |
by (induct ns) auto |
|
13114 | 1577 |
|
1578 |
||
15392 | 1579 |
subsubsection {* @{text upto} *} |
13114 | 1580 |
|
17090 | 1581 |
lemma upt_rec[code]: "[i..<j] = (if i<j then i#[Suc i..<j] else [])" |
1582 |
-- {* simp does not terminate! *} |
|
13145 | 1583 |
by (induct j) auto |
13142 | 1584 |
|
15425 | 1585 |
lemma upt_conv_Nil [simp]: "j <= i ==> [i..<j] = []" |
13145 | 1586 |
by (subst upt_rec) simp |
13114 | 1587 |
|
15425 | 1588 |
lemma upt_eq_Nil_conv[simp]: "([i..<j] = []) = (j = 0 \<or> j <= i)" |
15281 | 1589 |
by(induct j)simp_all |
1590 |
||
1591 |
lemma upt_eq_Cons_conv: |
|
15425 | 1592 |
"!!x xs. ([i..<j] = x#xs) = (i < j & i = x & [i+1..<j] = xs)" |
15281 | 1593 |
apply(induct j) |
1594 |
apply simp |
|
1595 |
apply(clarsimp simp add: append_eq_Cons_conv) |
|
1596 |
apply arith |
|
1597 |
done |
|
1598 |
||
15425 | 1599 |
lemma upt_Suc_append: "i <= j ==> [i..<(Suc j)] = [i..<j]@[j]" |
13145 | 1600 |
-- {* Only needed if @{text upt_Suc} is deleted from the simpset. *} |
1601 |
by simp |
|
13114 | 1602 |
|
15425 | 1603 |
lemma upt_conv_Cons: "i < j ==> [i..<j] = i # [Suc i..<j]" |
13145 | 1604 |
apply(rule trans) |
1605 |
apply(subst upt_rec) |
|
14208 | 1606 |
prefer 2 apply (rule refl, simp) |
13145 | 1607 |
done |
13114 | 1608 |
|
15425 | 1609 |
lemma upt_add_eq_append: "i<=j ==> [i..<j+k] = [i..<j]@[j..<j+k]" |
13145 | 1610 |
-- {* LOOPS as a simprule, since @{text "j <= j"}. *} |
1611 |
by (induct k) auto |
|
13114 | 1612 |
|
15425 | 1613 |
lemma length_upt [simp]: "length [i..<j] = j - i" |
13145 | 1614 |
by (induct j) (auto simp add: Suc_diff_le) |
13114 | 1615 |
|
15425 | 1616 |
lemma nth_upt [simp]: "i + k < j ==> [i..<j] ! k = i + k" |
13145 | 1617 |
apply (induct j) |
1618 |
apply (auto simp add: less_Suc_eq nth_append split: nat_diff_split) |
|
1619 |
done |
|
13114 | 1620 |
|
17906 | 1621 |
|
1622 |
lemma hd_upt[simp]: "i < j \<Longrightarrow> hd[i..<j] = i" |
|
1623 |
by(simp add:upt_conv_Cons) |
|
1624 |
||
1625 |
lemma last_upt[simp]: "i < j \<Longrightarrow> last[i..<j] = j - 1" |
|
1626 |
apply(cases j) |
|
1627 |
apply simp |
|
1628 |
by(simp add:upt_Suc_append) |
|
1629 |
||
15425 | 1630 |
lemma take_upt [simp]: "!!i. i+m <= n ==> take m [i..<n] = [i..<i+m]" |
14208 | 1631 |
apply (induct m, simp) |
13145 | 1632 |
apply (subst upt_rec) |
1633 |
apply (rule sym) |
|
1634 |
apply (subst upt_rec) |
|
1635 |
apply (simp del: upt.simps) |
|
1636 |
done |
|
3507 | 1637 |
|
17501 | 1638 |
lemma drop_upt[simp]: "drop m [i..<j] = [i+m..<j]" |
1639 |
apply(induct j) |
|
1640 |
apply auto |
|
1641 |
done |
|
1642 |
||
15425 | 1643 |
lemma map_Suc_upt: "map Suc [m..<n] = [Suc m..n]" |
13145 | 1644 |
by (induct n) auto |
13114 | 1645 |
|
15425 | 1646 |
lemma nth_map_upt: "!!i. i < n-m ==> (map f [m..<n]) ! i = f(m+i)" |
13145 | 1647 |
apply (induct n m rule: diff_induct) |
1648 |
prefer 3 apply (subst map_Suc_upt[symmetric]) |
|
1649 |
apply (auto simp add: less_diff_conv nth_upt) |
|
1650 |
done |
|
13114 | 1651 |
|
13883
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1652 |
lemma nth_take_lemma: |
0451e0fb3f22
Re-structured some proofs in order to get rid of rule_format attribute.
berghofe
parents:
13863
diff
changeset
|
1653 |
"!!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
|
1654 |
(!!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
|
1655 |
apply (atomize, induct k) |
14208 | 1656 |
apply (simp_all add: less_Suc_eq_0_disj all_conj_distrib, clarify) |
13145 | 1657 |
txt {* Both lists must be non-empty *} |
14208 | 1658 |
apply (case_tac xs, simp) |
1659 |
apply (case_tac ys, clarify) |
|
13145 | 1660 |
apply (simp (no_asm_use)) |
1661 |
apply clarify |
|
1662 |
txt {* prenexing's needed, not miniscoping *} |
|
1663 |
apply (simp (no_asm_use) add: all_simps [symmetric] del: all_simps) |
|
1664 |
apply blast |
|
1665 |
done |
|
13114 | 1666 |
|
1667 |
lemma nth_equalityI: |
|
1668 |
"[| length xs = length ys; ALL i < length xs. xs!i = ys!i |] ==> xs = ys" |
|
13145 | 1669 |
apply (frule nth_take_lemma [OF le_refl eq_imp_le]) |
1670 |
apply (simp_all add: take_all) |
|
1671 |
done |
|
13142 | 1672 |
|
13863 | 1673 |
(* needs nth_equalityI *) |
1674 |
lemma list_all2_antisym: |
|
1675 |
"\<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> |
|
1676 |
\<Longrightarrow> xs = ys" |
|
1677 |
apply (simp add: list_all2_conv_all_nth) |
|
14208 | 1678 |
apply (rule nth_equalityI, blast, simp) |
13863 | 1679 |
done |
1680 |
||
13142 | 1681 |
lemma take_equalityI: "(\<forall>i. take i xs = take i ys) ==> xs = ys" |
13145 | 1682 |
-- {* The famous take-lemma. *} |
1683 |
apply (drule_tac x = "max (length xs) (length ys)" in spec) |
|
1684 |
apply (simp add: le_max_iff_disj take_all) |
|
1685 |
done |
|
13142 | 1686 |
|
1687 |
||
15302 | 1688 |
lemma take_Cons': |
1689 |
"take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)" |
|
1690 |
by (cases n) simp_all |
|
1691 |
||
1692 |
lemma drop_Cons': |
|
1693 |
"drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)" |
|
1694 |
by (cases n) simp_all |
|
1695 |
||
1696 |
lemma nth_Cons': "(x # xs)!n = (if n = 0 then x else xs!(n - 1))" |
|
1697 |
by (cases n) simp_all |
|
1698 |
||
18622 | 1699 |
lemmas take_Cons_number_of = take_Cons'[of "number_of v",standard] |
1700 |
lemmas drop_Cons_number_of = drop_Cons'[of "number_of v",standard] |
|
1701 |
lemmas nth_Cons_number_of = nth_Cons'[of _ _ "number_of v",standard] |
|
1702 |
||
1703 |
declare take_Cons_number_of [simp] |
|
1704 |
drop_Cons_number_of [simp] |
|
1705 |
nth_Cons_number_of [simp] |
|
15302 | 1706 |
|
1707 |
||
15392 | 1708 |
subsubsection {* @{text "distinct"} and @{text remdups} *} |
13142 | 1709 |
|
1710 |
lemma distinct_append [simp]: |
|
13145 | 1711 |
"distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})" |
1712 |
by (induct xs) auto |
|
13142 | 1713 |
|
15305 | 1714 |
lemma distinct_rev[simp]: "distinct(rev xs) = distinct xs" |
1715 |
by(induct xs) auto |
|
1716 |
||
13142 | 1717 |
lemma set_remdups [simp]: "set (remdups xs) = set xs" |
13145 | 1718 |
by (induct xs) (auto simp add: insert_absorb) |
13142 | 1719 |
|
1720 |
lemma distinct_remdups [iff]: "distinct (remdups xs)" |
|
13145 | 1721 |
by (induct xs) auto |
13142 | 1722 |
|
15072 | 1723 |
lemma remdups_eq_nil_iff [simp]: "(remdups x = []) = (x = [])" |
15251 | 1724 |
by (induct x, auto) |
15072 | 1725 |
|
1726 |
lemma remdups_eq_nil_right_iff [simp]: "([] = remdups x) = (x = [])" |
|
15251 | 1727 |
by (induct x, auto) |
15072 | 1728 |
|
15245 | 1729 |
lemma length_remdups_leq[iff]: "length(remdups xs) <= length xs" |
1730 |
by (induct xs) auto |
|
1731 |
||
1732 |
lemma length_remdups_eq[iff]: |
|
1733 |
"(length (remdups xs) = length xs) = (remdups xs = xs)" |
|
1734 |
apply(induct xs) |
|
1735 |
apply auto |
|
1736 |
apply(subgoal_tac "length (remdups xs) <= length xs") |
|
1737 |
apply arith |
|
1738 |
apply(rule length_remdups_leq) |
|
1739 |
done |
|
1740 |
||
18490 | 1741 |
|
1742 |
lemma distinct_map: |
|
1743 |
"distinct(map f xs) = (distinct xs & inj_on f (set xs))" |
|
1744 |
by (induct xs) auto |
|
1745 |
||
1746 |
||
13142 | 1747 |
lemma distinct_filter [simp]: "distinct xs ==> distinct (filter P xs)" |
13145 | 1748 |
by (induct xs) auto |
13114 | 1749 |
|
17501 | 1750 |
lemma distinct_upt[simp]: "distinct[i..<j]" |
1751 |
by (induct j) auto |
|
1752 |
||
1753 |
lemma distinct_take[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (take i xs)" |
|
1754 |
apply(induct xs) |
|
1755 |
apply simp |
|
1756 |
apply (case_tac i) |
|
1757 |
apply simp_all |
|
1758 |
apply(blast dest:in_set_takeD) |
|
1759 |
done |
|
1760 |
||
1761 |
lemma distinct_drop[simp]: "\<And>i. distinct xs \<Longrightarrow> distinct (drop i xs)" |
|
1762 |
apply(induct xs) |
|
1763 |
apply simp |
|
1764 |
apply (case_tac i) |
|
1765 |
apply simp_all |
|
1766 |
done |
|
1767 |
||
1768 |
lemma distinct_list_update: |
|
1769 |
assumes d: "distinct xs" and a: "a \<notin> set xs - {xs!i}" |
|
1770 |
shows "distinct (xs[i:=a])" |
|
1771 |
proof (cases "i < length xs") |
|
1772 |
case True |
|
1773 |
with a have "a \<notin> set (take i xs @ xs ! i # drop (Suc i) xs) - {xs!i}" |
|
1774 |
apply (drule_tac id_take_nth_drop) by simp |
|
1775 |
with d True show ?thesis |
|
1776 |
apply (simp add: upd_conv_take_nth_drop) |
|
1777 |
apply (drule subst [OF id_take_nth_drop]) apply assumption |
|
1778 |
apply simp apply (cases "a = xs!i") apply simp by blast |
|
1779 |
next |
|
1780 |
case False with d show ?thesis by auto |
|
1781 |
qed |
|
1782 |
||
1783 |
||
1784 |
text {* It is best to avoid this indexed version of distinct, but |
|
1785 |
sometimes it is useful. *} |
|
1786 |
||
13142 | 1787 |
lemma distinct_conv_nth: |
17501 | 1788 |
"distinct xs = (\<forall>i < size xs. \<forall>j < size xs. i \<noteq> j --> xs!i \<noteq> xs!j)" |
15251 | 1789 |
apply (induct xs, simp, simp) |
14208 | 1790 |
apply (rule iffI, clarsimp) |
13145 | 1791 |
apply (case_tac i) |
14208 | 1792 |
apply (case_tac j, simp) |
13145 | 1793 |
apply (simp add: set_conv_nth) |
1794 |
apply (case_tac j) |
|
14208 | 1795 |
apply (clarsimp simp add: set_conv_nth, simp) |
13145 | 1796 |
apply (rule conjI) |
1797 |
apply (clarsimp simp add: set_conv_nth) |
|
17501 | 1798 |
apply (erule_tac x = 0 in allE, simp) |
14208 | 1799 |
apply (erule_tac x = "Suc i" in allE, simp, clarsimp) |
17501 | 1800 |
apply (erule_tac x = "Suc i" in allE, simp) |
14208 | 1801 |
apply (erule_tac x = "Suc j" in allE, simp) |
13145 | 1802 |
done |
13114 | 1803 |
|
18490 | 1804 |
lemma nth_eq_iff_index_eq: |
1805 |
"\<lbrakk> distinct xs; i < length xs; j < length xs \<rbrakk> \<Longrightarrow> (xs!i = xs!j) = (i = j)" |
|
1806 |
by(auto simp: distinct_conv_nth) |
|
1807 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1808 |
lemma distinct_card: "distinct xs ==> card (set xs) = size xs" |
14388 | 1809 |
by (induct xs) auto |
1810 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1811 |
lemma card_distinct: "card (set xs) = size xs ==> distinct xs" |
14388 | 1812 |
proof (induct xs) |
1813 |
case Nil thus ?case by simp |
|
1814 |
next |
|
1815 |
case (Cons x xs) |
|
1816 |
show ?case |
|
1817 |
proof (cases "x \<in> set xs") |
|
1818 |
case False with Cons show ?thesis by simp |
|
1819 |
next |
|
1820 |
case True with Cons.prems |
|
1821 |
have "card (set xs) = Suc (length xs)" |
|
1822 |
by (simp add: card_insert_if split: split_if_asm) |
|
1823 |
moreover have "card (set xs) \<le> length xs" by (rule card_length) |
|
1824 |
ultimately have False by simp |
|
1825 |
thus ?thesis .. |
|
1826 |
qed |
|
1827 |
qed |
|
1828 |
||
18490 | 1829 |
|
1830 |
lemma length_remdups_concat: |
|
1831 |
"length(remdups(concat xss)) = card(\<Union>xs \<in> set xss. set xs)" |
|
1832 |
by(simp add: distinct_card[symmetric]) |
|
17906 | 1833 |
|
1834 |
||
15392 | 1835 |
subsubsection {* @{text remove1} *} |
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1836 |
|
18049 | 1837 |
lemma remove1_append: |
1838 |
"remove1 x (xs @ ys) = |
|
1839 |
(if x \<in> set xs then remove1 x xs @ ys else xs @ remove1 x ys)" |
|
1840 |
by (induct xs) auto |
|
1841 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1842 |
lemma set_remove1_subset: "set(remove1 x xs) <= set xs" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1843 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1844 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1845 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1846 |
apply blast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1847 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1848 |
|
17724 | 1849 |
lemma set_remove1_eq [simp]: "distinct xs ==> set(remove1 x xs) = set xs - {x}" |
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1850 |
apply(induct xs) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1851 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1852 |
apply simp |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1853 |
apply blast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1854 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1855 |
|
18049 | 1856 |
lemma remove1_filter_not[simp]: |
1857 |
"\<not> P x \<Longrightarrow> remove1 x (filter P xs) = filter P xs" |
|
1858 |
by(induct xs) auto |
|
1859 |
||
15110
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1860 |
lemma notin_set_remove1[simp]: "x ~: set xs ==> x ~: set(remove1 y xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1861 |
apply(insert set_remove1_subset) |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1862 |
apply fast |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1863 |
done |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1864 |
|
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1865 |
lemma distinct_remove1[simp]: "distinct xs ==> distinct(remove1 x xs)" |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1866 |
by (induct xs) simp_all |
78b5636eabc7
Added a number of new thms and the new function remove1
nipkow
parents:
15072
diff
changeset
|
1867 |
|
13114 | 1868 |
|
15392 | 1869 |
subsubsection {* @{text replicate} *} |
13114 | 1870 |
|
13142 | 1871 |
lemma length_replicate [simp]: "length (replicate n x) = n" |
13145 | 1872 |
by (induct n) auto |
13124 | 1873 |
|
13142 | 1874 |
lemma map_replicate [simp]: "map f (replicate n x) = replicate n (f x)" |
13145 | 1875 |
by (induct n) auto |
13114 | 1876 |
|
1877 |
lemma replicate_app_Cons_same: |
|
13145 | 1878 |
"(replicate n x) @ (x # xs) = x # replicate n x @ xs" |
1879 |
by (induct n) auto |
|
13114 | 1880 |
|
13142 | 1881 |
lemma rev_replicate [simp]: "rev (replicate n x) = replicate n x" |
14208 | 1882 |
apply (induct n, simp) |
13145 | 1883 |
apply (simp add: replicate_app_Cons_same) |
1884 |
done |
|
13114 | 1885 |
|
13142 | 1886 |
lemma replicate_add: "replicate (n + m) x = replicate n x @ replicate m x" |
13145 | 1887 |
by (induct n) auto |
13114 | 1888 |
|
16397 | 1889 |
text{* Courtesy of Matthias Daum: *} |
1890 |
lemma append_replicate_commute: |
|
1891 |
"replicate n x @ replicate k x = replicate k x @ replicate n x" |
|
1892 |
apply (simp add: replicate_add [THEN sym]) |
|
1893 |
apply (simp add: add_commute) |
|
1894 |
done |
|
1895 |
||
13142 | 1896 |
lemma hd_replicate [simp]: "n \<noteq> 0 ==> hd (replicate n x) = x" |
13145 | 1897 |
by (induct n) auto |
13114 | 1898 |
|
13142 | 1899 |
lemma tl_replicate [simp]: "n \<noteq> 0 ==> tl (replicate n x) = replicate (n - 1) x" |
13145 | 1900 |
by (induct n) auto |
13114 | 1901 |
|
13142 | 1902 |
lemma last_replicate [simp]: "n \<noteq> 0 ==> last (replicate n x) = x" |
13145 | 1903 |
by (atomize (full), induct n) auto |
13114 | 1904 |
|
13142 | 1905 |
lemma nth_replicate[simp]: "!!i. i < n ==> (replicate n x)!i = x" |
14208 | 1906 |
apply (induct n, simp) |
13145 | 1907 |
apply (simp add: nth_Cons split: nat.split) |
1908 |
done |
|
13114 | 1909 |
|
16397 | 1910 |
text{* Courtesy of Matthias Daum (2 lemmas): *} |
1911 |
lemma take_replicate[simp]: "take i (replicate k x) = replicate (min i k) x" |
|
1912 |
apply (case_tac "k \<le> i") |
|
1913 |
apply (simp add: min_def) |
|
1914 |
apply (drule not_leE) |
|
1915 |
apply (simp add: min_def) |
|
1916 |
apply (subgoal_tac "replicate k x = replicate i x @ replicate (k - i) x") |
|
1917 |
apply simp |
|
1918 |
apply (simp add: replicate_add [symmetric]) |
|
1919 |
done |
|
1920 |
||
1921 |
lemma drop_replicate[simp]: "!!i. drop i (replicate k x) = replicate (k-i) x" |
|
1922 |
apply (induct k) |
|
1923 |
apply simp |
|
1924 |
apply clarsimp |
|
1925 |
apply (case_tac i) |
|
1926 |
apply simp |
|
1927 |
apply clarsimp |
|
1928 |
done |
|
1929 |
||
1930 |
||
13142 | 1931 |
lemma set_replicate_Suc: "set (replicate (Suc n) x) = {x}" |
13145 | 1932 |
by (induct n) auto |
13114 | 1933 |
|
13142 | 1934 |
lemma set_replicate [simp]: "n \<noteq> 0 ==> set (replicate n x) = {x}" |
13145 | 1935 |
by (fast dest!: not0_implies_Suc intro!: set_replicate_Suc) |
13114 | 1936 |
|
13142 | 1937 |
lemma set_replicate_conv_if: "set (replicate n x) = (if n = 0 then {} else {x})" |
13145 | 1938 |
by auto |
13114 | 1939 |
|
13142 | 1940 |
lemma in_set_replicateD: "x : set (replicate n y) ==> x = y" |
13145 | 1941 |
by (simp add: set_replicate_conv_if split: split_if_asm) |
13114 | 1942 |
|
1943 |
||
15392 | 1944 |
subsubsection{*@{text rotate1} and @{text rotate}*} |
15302 | 1945 |
|
1946 |
lemma rotate_simps[simp]: "rotate1 [] = [] \<and> rotate1 (x#xs) = xs @ [x]" |
|
1947 |
by(simp add:rotate1_def) |
|
1948 |
||
1949 |
lemma rotate0[simp]: "rotate 0 = id" |
|
1950 |
by(simp add:rotate_def) |
|
1951 |
||
1952 |
lemma rotate_Suc[simp]: "rotate (Suc n) xs = rotate1(rotate n xs)" |
|
1953 |
by(simp add:rotate_def) |
|
1954 |
||
1955 |
lemma rotate_add: |
|
1956 |
"rotate (m+n) = rotate m o rotate n" |
|
1957 |
by(simp add:rotate_def funpow_add) |
|
1958 |
||
1959 |
lemma rotate_rotate: "rotate m (rotate n xs) = rotate (m+n) xs" |
|
1960 |
by(simp add:rotate_add) |
|
1961 |
||
18049 | 1962 |
lemma rotate1_rotate_swap: "rotate1 (rotate n xs) = rotate n (rotate1 xs)" |
1963 |
by(simp add:rotate_def funpow_swap1) |
|
1964 |
||
15302 | 1965 |
lemma rotate1_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate1 xs = xs" |
1966 |
by(cases xs) simp_all |
|
1967 |
||
1968 |
lemma rotate_length01[simp]: "length xs <= 1 \<Longrightarrow> rotate n xs = xs" |
|
1969 |
apply(induct n) |
|
1970 |
apply simp |
|
1971 |
apply (simp add:rotate_def) |
|
13145 | 1972 |
done |
13114 | 1973 |
|
15302 | 1974 |
lemma rotate1_hd_tl: "xs \<noteq> [] \<Longrightarrow> rotate1 xs = tl xs @ [hd xs]" |
1975 |
by(simp add:rotate1_def split:list.split) |
|
1976 |
||
1977 |
lemma rotate_drop_take: |
|
1978 |
"rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs" |
|
1979 |
apply(induct n) |
|
1980 |
apply simp |
|
1981 |
apply(simp add:rotate_def) |
|
1982 |
apply(cases "xs = []") |
|
1983 |
apply (simp) |
|
1984 |
apply(case_tac "n mod length xs = 0") |
|
1985 |
apply(simp add:mod_Suc) |
|
1986 |
apply(simp add: rotate1_hd_tl drop_Suc take_Suc) |
|
1987 |
apply(simp add:mod_Suc rotate1_hd_tl drop_Suc[symmetric] drop_tl[symmetric] |
|
1988 |
take_hd_drop linorder_not_le) |
|
13145 | 1989 |
done |
13114 | 1990 |
|
15302 | 1991 |
lemma rotate_conv_mod: "rotate n xs = rotate (n mod length xs) xs" |
1992 |
by(simp add:rotate_drop_take) |
|
1993 |
||
1994 |
lemma rotate_id[simp]: "n mod length xs = 0 \<Longrightarrow> rotate n xs = xs" |
|
1995 |
by(simp add:rotate_drop_take) |
|
1996 |
||
1997 |
lemma length_rotate1[simp]: "length(rotate1 xs) = length xs" |
|
1998 |
by(simp add:rotate1_def split:list.split) |
|
1999 |
||
2000 |
lemma length_rotate[simp]: "!!xs. length(rotate n xs) = length xs" |
|
2001 |
by (induct n) (simp_all add:rotate_def) |
|
2002 |
||
2003 |
lemma distinct1_rotate[simp]: "distinct(rotate1 xs) = distinct xs" |
|
2004 |
by(simp add:rotate1_def split:list.split) blast |
|
2005 |
||
2006 |
lemma distinct_rotate[simp]: "distinct(rotate n xs) = distinct xs" |
|
2007 |
by (induct n) (simp_all add:rotate_def) |
|
2008 |
||
2009 |
lemma rotate_map: "rotate n (map f xs) = map f (rotate n xs)" |
|
2010 |
by(simp add:rotate_drop_take take_map drop_map) |
|
2011 |
||
2012 |
lemma set_rotate1[simp]: "set(rotate1 xs) = set xs" |
|
2013 |
by(simp add:rotate1_def split:list.split) |
|
2014 |
||
2015 |
lemma set_rotate[simp]: "set(rotate n xs) = set xs" |
|
2016 |
by (induct n) (simp_all add:rotate_def) |
|
2017 |
||
2018 |
lemma rotate1_is_Nil_conv[simp]: "(rotate1 xs = []) = (xs = [])" |
|
2019 |
by(simp add:rotate1_def split:list.split) |
|
2020 |
||
2021 |
lemma rotate_is_Nil_conv[simp]: "(rotate n xs = []) = (xs = [])" |
|
2022 |
by (induct n) (simp_all add:rotate_def) |
|
13114 | 2023 |
|
15439 | 2024 |
lemma rotate_rev: |
2025 |
"rotate n (rev xs) = rev(rotate (length xs - (n mod length xs)) xs)" |
|
2026 |
apply(simp add:rotate_drop_take rev_drop rev_take) |
|
2027 |
apply(cases "length xs = 0") |
|
2028 |
apply simp |
|
2029 |
apply(cases "n mod length xs = 0") |
|
2030 |
apply simp |
|
2031 |
apply(simp add:rotate_drop_take rev_drop rev_take) |
|
2032 |
done |
|
2033 |
||
18423 | 2034 |
lemma hd_rotate_conv_nth: "xs \<noteq> [] \<Longrightarrow> hd(rotate n xs) = xs!(n mod length xs)" |
2035 |
apply(simp add:rotate_drop_take hd_append hd_drop_conv_nth hd_conv_nth) |
|
2036 |
apply(subgoal_tac "length xs \<noteq> 0") |
|
2037 |
prefer 2 apply simp |
|
2038 |
using mod_less_divisor[of "length xs" n] by arith |
|
2039 |
||
13114 | 2040 |
|
15392 | 2041 |
subsubsection {* @{text sublist} --- a generalization of @{text nth} to sets *} |
13114 | 2042 |
|
13142 | 2043 |
lemma sublist_empty [simp]: "sublist xs {} = []" |
13145 | 2044 |
by (auto simp add: sublist_def) |
13114 | 2045 |
|
13142 | 2046 |
lemma sublist_nil [simp]: "sublist [] A = []" |
13145 | 2047 |
by (auto simp add: sublist_def) |
13114 | 2048 |
|
15281 | 2049 |
lemma length_sublist: |
2050 |
"length(sublist xs I) = card{i. i < length xs \<and> i : I}" |
|
2051 |
by(simp add: sublist_def length_filter_conv_card cong:conj_cong) |
|
2052 |
||
2053 |
lemma sublist_shift_lemma_Suc: |
|
2054 |
"!!is. map fst (filter (%p. P(Suc(snd p))) (zip xs is)) = |
|
2055 |
map fst (filter (%p. P(snd p)) (zip xs (map Suc is)))" |
|
2056 |
apply(induct xs) |
|
2057 |
apply simp |
|
2058 |
apply (case_tac "is") |
|
2059 |
apply simp |
|
2060 |
apply simp |
|
2061 |
done |
|
2062 |
||
13114 | 2063 |
lemma sublist_shift_lemma: |
15425 | 2064 |
"map fst [p:zip xs [i..<i + length xs] . snd p : A] = |
2065 |
map fst [p:zip xs [0..<length xs] . snd p + i : A]" |
|
13145 | 2066 |
by (induct xs rule: rev_induct) (simp_all add: add_commute) |
13114 | 2067 |
|
2068 |
lemma sublist_append: |
|
15168 | 2069 |
"sublist (l @ l') A = sublist l A @ sublist l' {j. j + length l : A}" |
13145 | 2070 |
apply (unfold sublist_def) |
14208 | 2071 |
apply (induct l' rule: rev_induct, simp) |
13145 | 2072 |
apply (simp add: upt_add_eq_append[of 0] zip_append sublist_shift_lemma) |
2073 |
apply (simp add: add_commute) |
|
2074 |
done |
|
13114 | 2075 |
|
2076 |
lemma sublist_Cons: |
|
13145 | 2077 |
"sublist (x # l) A = (if 0:A then [x] else []) @ sublist l {j. Suc j : A}" |
2078 |
apply (induct l rule: rev_induct) |
|
2079 |
apply (simp add: sublist_def) |
|
2080 |
apply (simp del: append_Cons add: append_Cons[symmetric] sublist_append) |
|
2081 |
done |
|
13114 | 2082 |
|
15281 | 2083 |
lemma set_sublist: "!!I. set(sublist xs I) = {xs!i|i. i<size xs \<and> i \<in> I}" |
2084 |
apply(induct xs) |
|
2085 |
apply simp |
|
2086 |
apply(auto simp add:sublist_Cons nth_Cons split:nat.split elim: lessE) |
|
2087 |
apply(erule lessE) |
|
2088 |
apply auto |
|
2089 |
apply(erule lessE) |
|
2090 |
apply auto |
|
2091 |
done |
|
2092 |
||
2093 |
lemma set_sublist_subset: "set(sublist xs I) \<subseteq> set xs" |
|
2094 |
by(auto simp add:set_sublist) |
|
2095 |
||
2096 |
lemma notin_set_sublistI[simp]: "x \<notin> set xs \<Longrightarrow> x \<notin> set(sublist xs I)" |
|
2097 |
by(auto simp add:set_sublist) |
|
2098 |
||
2099 |
lemma in_set_sublistD: "x \<in> set(sublist xs I) \<Longrightarrow> x \<in> set xs" |
|
2100 |
by(auto simp add:set_sublist) |
|
2101 |
||
13142 | 2102 |
lemma sublist_singleton [simp]: "sublist [x] A = (if 0 : A then [x] else [])" |
13145 | 2103 |
by (simp add: sublist_Cons) |
13114 | 2104 |
|
15281 | 2105 |
|
2106 |
lemma distinct_sublistI[simp]: "!!I. distinct xs \<Longrightarrow> distinct(sublist xs I)" |
|
2107 |
apply(induct xs) |
|
2108 |
apply simp |
|
2109 |
apply(auto simp add:sublist_Cons) |
|
2110 |
done |
|
2111 |
||
2112 |
||
15045 | 2113 |
lemma sublist_upt_eq_take [simp]: "sublist l {..<n} = take n l" |
14208 | 2114 |
apply (induct l rule: rev_induct, simp) |
13145 | 2115 |
apply (simp split: nat_diff_split add: sublist_append) |
2116 |
done |
|
13114 | 2117 |
|
17501 | 2118 |
lemma filter_in_sublist: "\<And>s. distinct xs \<Longrightarrow> |
2119 |
filter (%x. x \<in> set(sublist xs s)) xs = sublist xs s" |
|
2120 |
proof (induct xs) |
|
2121 |
case Nil thus ?case by simp |
|
2122 |
next |
|
2123 |
case (Cons a xs) |
|
2124 |
moreover hence "!x. x: set xs \<longrightarrow> x \<noteq> a" by auto |
|
2125 |
ultimately show ?case by(simp add: sublist_Cons cong:filter_cong) |
|
2126 |
qed |
|
2127 |
||
13114 | 2128 |
|
19390 | 2129 |
subsubsection {* @{const splice} *} |
2130 |
||
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
2131 |
lemma splice_Nil2 [simp, code]: |
19390 | 2132 |
"splice xs [] = xs" |
2133 |
by (cases xs) simp_all |
|
2134 |
||
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
2135 |
lemma splice_Cons_Cons [simp, code]: |
19390 | 2136 |
"splice (x#xs) (y#ys) = x # y # splice xs ys" |
2137 |
by simp |
|
2138 |
||
19607
07eeb832f28d
introduced characters for code generator; some improved code lemmas for some list functions
haftmann
parents:
19585
diff
changeset
|
2139 |
declare splice.simps(2) [simp del, code del] |
19390 | 2140 |
|
15392 | 2141 |
subsubsection{*Sets of Lists*} |
2142 |
||
2143 |
subsubsection {* @{text lists}: the list-forming operator over sets *} |
|
15302 | 2144 |
|
2145 |
consts lists :: "'a set => 'a list set" |
|
2146 |
inductive "lists A" |
|
2147 |
intros |
|
2148 |
Nil [intro!]: "[]: lists A" |
|
2149 |
Cons [intro!]: "[| a: A;l: lists A|] ==> a#l : lists A" |
|
2150 |
||
2151 |
inductive_cases listsE [elim!]: "x#l : lists A" |
|
2152 |
||
2153 |
lemma lists_mono [mono]: "A \<subseteq> B ==> lists A \<subseteq> lists B" |
|
2154 |
by (unfold lists.defs) (blast intro!: lfp_mono) |
|
2155 |
||
2156 |
lemma lists_IntI: |
|
2157 |
assumes l: "l: lists A" shows "l: lists B ==> l: lists (A Int B)" using l |
|
2158 |
by induct blast+ |
|
2159 |
||
2160 |
lemma lists_Int_eq [simp]: "lists (A \<inter> B) = lists A \<inter> lists B" |
|
2161 |
proof (rule mono_Int [THEN equalityI]) |
|
2162 |
show "mono lists" by (simp add: mono_def lists_mono) |
|
2163 |
show "lists A \<inter> lists B \<subseteq> lists (A \<inter> B)" by (blast intro: lists_IntI) |
|
14388 | 2164 |
qed |
2165 |
||
15302 | 2166 |
lemma append_in_lists_conv [iff]: |
2167 |
"(xs @ ys : lists A) = (xs : lists A \<and> ys : lists A)" |
|
2168 |
by (induct xs) auto |
|
2169 |
||
2170 |
lemma in_lists_conv_set: "(xs : lists A) = (\<forall>x \<in> set xs. x : A)" |
|
2171 |
-- {* eliminate @{text lists} in favour of @{text set} *} |
|
2172 |
by (induct xs) auto |
|
2173 |
||
2174 |
lemma in_listsD [dest!]: "xs \<in> lists A ==> \<forall>x\<in>set xs. x \<in> A" |
|
2175 |
by (rule in_lists_conv_set [THEN iffD1]) |
|
2176 |
||
2177 |
lemma in_listsI [intro!]: "\<forall>x\<in>set xs. x \<in> A ==> xs \<in> lists A" |
|
2178 |
by (rule in_lists_conv_set [THEN iffD2]) |
|
2179 |
||
2180 |
lemma lists_UNIV [simp]: "lists UNIV = UNIV" |
|
2181 |
by auto |
|
2182 |
||
17086 | 2183 |
|
2184 |
||
2185 |
subsubsection{* Inductive definition for membership *} |
|
2186 |
||
2187 |
consts ListMem :: "('a \<times> 'a list)set" |
|
2188 |
inductive ListMem |
|
2189 |
intros |
|
2190 |
elem: "(x,x#xs) \<in> ListMem" |
|
2191 |
insert: "(x,xs) \<in> ListMem \<Longrightarrow> (x,y#xs) \<in> ListMem" |
|
2192 |
||
2193 |
lemma ListMem_iff: "((x,xs) \<in> ListMem) = (x \<in> set xs)" |
|
2194 |
apply (rule iffI) |
|
2195 |
apply (induct set: ListMem) |
|
2196 |
apply auto |
|
2197 |
apply (induct xs) |
|
2198 |
apply (auto intro: ListMem.intros) |
|
2199 |
done |
|
2200 |
||
2201 |
||
2202 |
||
15392 | 2203 |
subsubsection{*Lists as Cartesian products*} |
15302 | 2204 |
|
2205 |
text{*@{text"set_Cons A Xs"}: the set of lists with head drawn from |
|
2206 |
@{term A} and tail drawn from @{term Xs}.*} |
|
2207 |
||
2208 |
constdefs |
|
2209 |
set_Cons :: "'a set \<Rightarrow> 'a list set \<Rightarrow> 'a list set" |
|
2210 |
"set_Cons A XS == {z. \<exists>x xs. z = x#xs & x \<in> A & xs \<in> XS}" |
|
2211 |
||
17724 | 2212 |
lemma set_Cons_sing_Nil [simp]: "set_Cons A {[]} = (%x. [x])`A" |
15302 | 2213 |
by (auto simp add: set_Cons_def) |
2214 |
||
2215 |
text{*Yields the set of lists, all of the same length as the argument and |
|
2216 |
with elements drawn from the corresponding element of the argument.*} |
|
2217 |
||
2218 |
consts listset :: "'a set list \<Rightarrow> 'a list set" |
|
2219 |
primrec |
|
2220 |
"listset [] = {[]}" |
|
2221 |
"listset(A#As) = set_Cons A (listset As)" |
|
2222 |
||
2223 |
||
15656 | 2224 |
subsection{*Relations on Lists*} |
2225 |
||
2226 |
subsubsection {* Length Lexicographic Ordering *} |
|
2227 |
||
2228 |
text{*These orderings preserve well-foundedness: shorter lists |
|
2229 |
precede longer lists. These ordering are not used in dictionaries.*} |
|
2230 |
||
2231 |
consts lexn :: "('a * 'a)set => nat => ('a list * 'a list)set" |
|
2232 |
--{*The lexicographic ordering for lists of the specified length*} |
|
15302 | 2233 |
primrec |
15656 | 2234 |
"lexn r 0 = {}" |
2235 |
"lexn r (Suc n) = |
|
2236 |
(prod_fun (%(x,xs). x#xs) (%(x,xs). x#xs) ` (r <*lex*> lexn r n)) Int |
|
2237 |
{(xs,ys). length xs = Suc n \<and> length ys = Suc n}" |
|
15302 | 2238 |
|
2239 |
constdefs |
|
15656 | 2240 |
lex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
2241 |
"lex r == \<Union>n. lexn r n" |
|
2242 |
--{*Holds only between lists of the same length*} |
|
2243 |
||
15693 | 2244 |
lenlex :: "('a \<times> 'a) set => ('a list \<times> 'a list) set" |
2245 |
"lenlex r == inv_image (less_than <*lex*> lex r) (%xs. (length xs, xs))" |
|
15656 | 2246 |
--{*Compares lists by their length and then lexicographically*} |
15302 | 2247 |
|
2248 |
||
2249 |
lemma wf_lexn: "wf r ==> wf (lexn r n)" |
|
2250 |
apply (induct n, simp, simp) |
|
2251 |
apply(rule wf_subset) |
|
2252 |
prefer 2 apply (rule Int_lower1) |
|
2253 |
apply(rule wf_prod_fun_image) |
|
2254 |
prefer 2 apply (rule inj_onI, auto) |
|
2255 |
done |
|
2256 |
||
2257 |
lemma lexn_length: |
|
2258 |
"!!xs ys. (xs, ys) : lexn r n ==> length xs = n \<and> length ys = n" |
|
2259 |
by (induct n) auto |
|
2260 |
||
2261 |
lemma wf_lex [intro!]: "wf r ==> wf (lex r)" |
|
2262 |
apply (unfold lex_def) |
|
2263 |
apply (rule wf_UN) |
|
2264 |
apply (blast intro: wf_lexn, clarify) |
|
2265 |
apply (rename_tac m n) |
|
2266 |
apply (subgoal_tac "m \<noteq> n") |
|
2267 |
prefer 2 apply blast |
|
2268 |
apply (blast dest: lexn_length not_sym) |
|
2269 |
done |
|
2270 |
||
2271 |
lemma lexn_conv: |
|
15656 | 2272 |
"lexn r n = |
2273 |
{(xs,ys). length xs = n \<and> length ys = n \<and> |
|
2274 |
(\<exists>xys x y xs' ys'. xs= xys @ x#xs' \<and> ys= xys @ y # ys' \<and> (x, y):r)}" |
|
18423 | 2275 |
apply (induct n, simp) |
15302 | 2276 |
apply (simp add: image_Collect lex_prod_def, safe, blast) |
2277 |
apply (rule_tac x = "ab # xys" in exI, simp) |
|
2278 |
apply (case_tac xys, simp_all, blast) |
|
2279 |
done |
|
2280 |
||
2281 |
lemma lex_conv: |
|
15656 | 2282 |
"lex r = |
2283 |
{(xs,ys). length xs = length ys \<and> |
|
2284 |
(\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y):r)}" |
|
15302 | 2285 |
by (force simp add: lex_def lexn_conv) |
2286 |
||
15693 | 2287 |
lemma wf_lenlex [intro!]: "wf r ==> wf (lenlex r)" |
2288 |
by (unfold lenlex_def) blast |
|
2289 |
||
2290 |
lemma lenlex_conv: |
|
2291 |
"lenlex r = {(xs,ys). length xs < length ys | |
|
15656 | 2292 |
length xs = length ys \<and> (xs, ys) : lex r}" |
19623 | 2293 |
by (simp add: lenlex_def diag_def lex_prod_def inv_image_def) |
15302 | 2294 |
|
2295 |
lemma Nil_notin_lex [iff]: "([], ys) \<notin> lex r" |
|
2296 |
by (simp add: lex_conv) |
|
2297 |
||
2298 |
lemma Nil2_notin_lex [iff]: "(xs, []) \<notin> lex r" |
|
2299 |
by (simp add:lex_conv) |
|
2300 |
||
18447 | 2301 |
lemma Cons_in_lex [simp]: |
15656 | 2302 |
"((x # xs, y # ys) : lex r) = |
2303 |
((x, y) : r \<and> length xs = length ys | x = y \<and> (xs, ys) : lex r)" |
|
15302 | 2304 |
apply (simp add: lex_conv) |
2305 |
apply (rule iffI) |
|
2306 |
prefer 2 apply (blast intro: Cons_eq_appendI, clarify) |
|
2307 |
apply (case_tac xys, simp, simp) |
|
2308 |
apply blast |
|
2309 |
done |
|
2310 |
||
2311 |
||
15656 | 2312 |
subsubsection {* Lexicographic Ordering *} |
2313 |
||
2314 |
text {* Classical lexicographic ordering on lists, ie. "a" < "ab" < "b". |
|
2315 |
This ordering does \emph{not} preserve well-foundedness. |
|
17090 | 2316 |
Author: N. Voelker, March 2005. *} |
15656 | 2317 |
|
2318 |
constdefs |
|
2319 |
lexord :: "('a * 'a)set \<Rightarrow> ('a list * 'a list) set" |
|
2320 |
"lexord r == {(x,y). \<exists> a v. y = x @ a # v \<or> |
|
2321 |
(\<exists> u a b v w. (a,b) \<in> r \<and> x = u @ (a # v) \<and> y = u @ (b # w))}" |
|
2322 |
||
2323 |
lemma lexord_Nil_left[simp]: "([],y) \<in> lexord r = (\<exists> a x. y = a # x)" |
|
2324 |
by (unfold lexord_def, induct_tac y, auto) |
|
2325 |
||
2326 |
lemma lexord_Nil_right[simp]: "(x,[]) \<notin> lexord r" |
|
2327 |
by (unfold lexord_def, induct_tac x, auto) |
|
2328 |
||
2329 |
lemma lexord_cons_cons[simp]: |
|
2330 |
"((a # x, b # y) \<in> lexord r) = ((a,b)\<in> r | (a = b & (x,y)\<in> lexord r))" |
|
2331 |
apply (unfold lexord_def, safe, simp_all) |
|
2332 |
apply (case_tac u, simp, simp) |
|
2333 |
apply (case_tac u, simp, clarsimp, blast, blast, clarsimp) |
|
2334 |
apply (erule_tac x="b # u" in allE) |
|
2335 |
by force |
|
2336 |
||
2337 |
lemmas lexord_simps = lexord_Nil_left lexord_Nil_right lexord_cons_cons |
|
2338 |
||
2339 |
lemma lexord_append_rightI: "\<exists> b z. y = b # z \<Longrightarrow> (x, x @ y) \<in> lexord r" |
|
2340 |
by (induct_tac x, auto) |
|
2341 |
||
2342 |
lemma lexord_append_left_rightI: |
|
2343 |
"(a,b) \<in> r \<Longrightarrow> (u @ a # x, u @ b # y) \<in> lexord r" |
|
2344 |
by (induct_tac u, auto) |
|
2345 |
||
2346 |
lemma lexord_append_leftI: " (u,v) \<in> lexord r \<Longrightarrow> (x @ u, x @ v) \<in> lexord r" |
|
2347 |
by (induct x, auto) |
|
2348 |
||
2349 |
lemma lexord_append_leftD: |
|
2350 |
"\<lbrakk> (x @ u, x @ v) \<in> lexord r; (! a. (a,a) \<notin> r) \<rbrakk> \<Longrightarrow> (u,v) \<in> lexord r" |
|
2351 |
by (erule rev_mp, induct_tac x, auto) |
|
2352 |
||
2353 |
lemma lexord_take_index_conv: |
|
2354 |
"((x,y) : lexord r) = |
|
2355 |
((length x < length y \<and> take (length x) y = x) \<or> |
|
2356 |
(\<exists>i. i < min(length x)(length y) & take i x = take i y & (x!i,y!i) \<in> r))" |
|
2357 |
apply (unfold lexord_def Let_def, clarsimp) |
|
2358 |
apply (rule_tac f = "(% a b. a \<or> b)" in arg_cong2) |
|
2359 |
apply auto |
|
2360 |
apply (rule_tac x="hd (drop (length x) y)" in exI) |
|
2361 |
apply (rule_tac x="tl (drop (length x) y)" in exI) |
|
2362 |
apply (erule subst, simp add: min_def) |
|
2363 |
apply (rule_tac x ="length u" in exI, simp) |
|
2364 |
apply (rule_tac x ="take i x" in exI) |
|
2365 |
apply (rule_tac x ="x ! i" in exI) |
|
2366 |
apply (rule_tac x ="y ! i" in exI, safe) |
|
2367 |
apply (rule_tac x="drop (Suc i) x" in exI) |
|
2368 |
apply (drule sym, simp add: drop_Suc_conv_tl) |
|
2369 |
apply (rule_tac x="drop (Suc i) y" in exI) |
|
2370 |
by (simp add: drop_Suc_conv_tl) |
|
2371 |
||
2372 |
-- {* lexord is extension of partial ordering List.lex *} |
|
2373 |
lemma lexord_lex: " (x,y) \<in> lex r = ((x,y) \<in> lexord r \<and> length x = length y)" |
|
2374 |
apply (rule_tac x = y in spec) |
|
2375 |
apply (induct_tac x, clarsimp) |
|
2376 |
by (clarify, case_tac x, simp, force) |
|
2377 |
||
2378 |
lemma lexord_irreflexive: "(! x. (x,x) \<notin> r) \<Longrightarrow> (y,y) \<notin> lexord r" |
|
2379 |
by (induct y, auto) |
|
2380 |
||
2381 |
lemma lexord_trans: |
|
2382 |
"\<lbrakk> (x, y) \<in> lexord r; (y, z) \<in> lexord r; trans r \<rbrakk> \<Longrightarrow> (x, z) \<in> lexord r" |
|
2383 |
apply (erule rev_mp)+ |
|
2384 |
apply (rule_tac x = x in spec) |
|
2385 |
apply (rule_tac x = z in spec) |
|
2386 |
apply ( induct_tac y, simp, clarify) |
|
2387 |
apply (case_tac xa, erule ssubst) |
|
2388 |
apply (erule allE, erule allE) -- {* avoid simp recursion *} |
|
2389 |
apply (case_tac x, simp, simp) |
|
2390 |
apply (case_tac x, erule allE, erule allE, simp) |
|
2391 |
apply (erule_tac x = listb in allE) |
|
2392 |
apply (erule_tac x = lista in allE, simp) |
|
2393 |
apply (unfold trans_def) |
|
2394 |
by blast |
|
2395 |
||
2396 |
lemma lexord_transI: "trans r \<Longrightarrow> trans (lexord r)" |
|
2397 |
by (rule transI, drule lexord_trans, blast) |
|
2398 |
||
2399 |
lemma lexord_linear: "(! a b. (a,b)\<in> r | a = b | (b,a) \<in> r) \<Longrightarrow> (x,y) : lexord r | x = y | (y,x) : lexord r" |
|
2400 |
apply (rule_tac x = y in spec) |
|
2401 |
apply (induct_tac x, rule allI) |
|
2402 |
apply (case_tac x, simp, simp) |
|
2403 |
apply (rule allI, case_tac x, simp, simp) |
|
2404 |
by blast |
|
2405 |
||
2406 |
||
21103
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2407 |
subsection {* Lexicographic combination of measure functions *} |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2408 |
|
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2409 |
text {* These are useful for termination proofs *} |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2410 |
|
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2411 |
definition |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2412 |
"measures fs = inv_image (lex less_than) (%a. map (%f. f a) fs)" |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2413 |
|
21106
51599a81b308
Added "recdef_wf" and "simp" attribute to "wf_measures"
krauss
parents:
21103
diff
changeset
|
2414 |
lemma wf_measures[recdef_wf, simp]: "wf (measures fs)" |
21103
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2415 |
unfolding measures_def |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2416 |
by blast |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2417 |
|
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2418 |
lemma in_measures[simp]: |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2419 |
"(x, y) \<in> measures [] = False" |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2420 |
"(x, y) \<in> measures (f # fs) |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2421 |
= (f x < f y \<or> (f x = f y \<and> (x, y) \<in> measures fs))" |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2422 |
unfolding measures_def |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2423 |
by auto |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2424 |
|
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2425 |
lemma measures_less: "f x < f y ==> (x, y) \<in> measures (f#fs)" |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2426 |
by simp |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2427 |
|
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2428 |
lemma measures_lesseq: "f x <= f y ==> (x, y) \<in> measures fs ==> (x, y) \<in> measures (f#fs)" |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2429 |
by auto |
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2430 |
|
21211
5370cfbf3070
Preparations for making "lexicographic_order" part of "fun"
krauss
parents:
21193
diff
changeset
|
2431 |
(* install the lexicographic_order method and the "fun" command *) |
21131 | 2432 |
use "Tools/function_package/lexicographic_order.ML" |
21211
5370cfbf3070
Preparations for making "lexicographic_order" part of "fun"
krauss
parents:
21193
diff
changeset
|
2433 |
use "Tools/function_package/fundef_datatype.ML" |
5370cfbf3070
Preparations for making "lexicographic_order" part of "fun"
krauss
parents:
21193
diff
changeset
|
2434 |
setup LexicographicOrder.setup |
5370cfbf3070
Preparations for making "lexicographic_order" part of "fun"
krauss
parents:
21193
diff
changeset
|
2435 |
setup FundefDatatype.setup |
5370cfbf3070
Preparations for making "lexicographic_order" part of "fun"
krauss
parents:
21193
diff
changeset
|
2436 |
|
21103
367b4ad7c7cc
Added "measures" combinator for lexicographic combinations of multiple measures.
krauss
parents:
21079
diff
changeset
|
2437 |
|
15392 | 2438 |
subsubsection{*Lifting a Relation on List Elements to the Lists*} |
15302 | 2439 |
|
2440 |
consts listrel :: "('a * 'a)set => ('a list * 'a list)set" |
|
2441 |
||
2442 |
inductive "listrel(r)" |
|
2443 |
intros |
|
2444 |
Nil: "([],[]) \<in> listrel r" |
|
2445 |
Cons: "[| (x,y) \<in> r; (xs,ys) \<in> listrel r |] ==> (x#xs, y#ys) \<in> listrel r" |
|
2446 |
||
2447 |
inductive_cases listrel_Nil1 [elim!]: "([],xs) \<in> listrel r" |
|
2448 |
inductive_cases listrel_Nil2 [elim!]: "(xs,[]) \<in> listrel r" |
|
2449 |
inductive_cases listrel_Cons1 [elim!]: "(y#ys,xs) \<in> listrel r" |
|
2450 |
inductive_cases listrel_Cons2 [elim!]: "(xs,y#ys) \<in> listrel r" |
|
2451 |
||
2452 |
||
2453 |
lemma listrel_mono: "r \<subseteq> s \<Longrightarrow> listrel r \<subseteq> listrel s" |
|
2454 |
apply clarify |
|
2455 |
apply (erule listrel.induct) |
|
2456 |
apply (blast intro: listrel.intros)+ |
|
2457 |
done |
|
2458 |
||
2459 |
lemma listrel_subset: "r \<subseteq> A \<times> A \<Longrightarrow> listrel r \<subseteq> lists A \<times> lists A" |
|
2460 |
apply clarify |
|
2461 |
apply (erule listrel.induct, auto) |
|
2462 |
done |
|
2463 |
||
2464 |
lemma listrel_refl: "refl A r \<Longrightarrow> refl (lists A) (listrel r)" |
|
2465 |
apply (simp add: refl_def listrel_subset Ball_def) |
|
2466 |
apply (rule allI) |
|
2467 |
apply (induct_tac x) |
|
2468 |
apply (auto intro: listrel.intros) |
|
2469 |
done |
|
2470 |
||
2471 |
lemma listrel_sym: "sym r \<Longrightarrow> sym (listrel r)" |
|
2472 |
apply (auto simp add: sym_def) |
|
2473 |
apply (erule listrel.induct) |
|
2474 |
apply (blast intro: listrel.intros)+ |
|
2475 |
done |
|
2476 |
||
2477 |
lemma listrel_trans: "trans r \<Longrightarrow> trans (listrel r)" |
|
2478 |
apply (simp add: trans_def) |
|
2479 |
apply (intro allI) |
|
2480 |
apply (rule impI) |
|
2481 |
apply (erule listrel.induct) |
|
2482 |
apply (blast intro: listrel.intros)+ |
|
2483 |
done |
|
2484 |
||
2485 |
theorem equiv_listrel: "equiv A r \<Longrightarrow> equiv (lists A) (listrel r)" |
|
2486 |
by (simp add: equiv_def listrel_refl listrel_sym listrel_trans) |
|
2487 |
||
2488 |
lemma listrel_Nil [simp]: "listrel r `` {[]} = {[]}" |
|
2489 |
by (blast intro: listrel.intros) |
|
2490 |
||
2491 |
lemma listrel_Cons: |
|
2492 |
"listrel r `` {x#xs} = set_Cons (r``{x}) (listrel r `` {xs})"; |
|
2493 |
by (auto simp add: set_Cons_def intro: listrel.intros) |
|
2494 |
||
2495 |
||
15392 | 2496 |
subsection{*Miscellany*} |
2497 |
||
2498 |
subsubsection {* Characters and strings *} |
|
13366 | 2499 |
|
2500 |
datatype nibble = |
|
2501 |
Nibble0 | Nibble1 | Nibble2 | Nibble3 | Nibble4 | Nibble5 | Nibble6 | Nibble7 |
|
2502 |
| Nibble8 | Nibble9 | NibbleA | NibbleB | NibbleC | NibbleD | NibbleE | NibbleF |
|
2503 |
||
2504 |
datatype char = Char nibble nibble |
|
2505 |
-- "Note: canonical order of character encoding coincides with standard term ordering" |
|
2506 |
||
2507 |
types string = "char list" |
|
2508 |
||
2509 |
syntax |
|
2510 |
"_Char" :: "xstr => char" ("CHR _") |
|
2511 |
"_String" :: "xstr => string" ("_") |
|
2512 |
||
2513 |
parse_ast_translation {* |
|
2514 |
let |
|
2515 |
val constants = Syntax.Appl o map Syntax.Constant; |
|
2516 |
||
2517 |
fun mk_nib n = "Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)); |
|
2518 |
fun mk_char c = |
|
2519 |
if Symbol.is_ascii c andalso Symbol.is_printable c then |
|
2520 |
constants ["Char", mk_nib (ord c div 16), mk_nib (ord c mod 16)] |
|
2521 |
else error ("Printable ASCII character expected: " ^ quote c); |
|
2522 |
||
2523 |
fun mk_string [] = Syntax.Constant "Nil" |
|
2524 |
| mk_string (c :: cs) = Syntax.Appl [Syntax.Constant "Cons", mk_char c, mk_string cs]; |
|
2525 |
||
2526 |
fun char_ast_tr [Syntax.Variable xstr] = |
|
2527 |
(case Syntax.explode_xstr xstr of |
|
2528 |
[c] => mk_char c |
|
2529 |
| _ => error ("Single character expected: " ^ xstr)) |
|
2530 |
| char_ast_tr asts = raise AST ("char_ast_tr", asts); |
|
2531 |
||
2532 |
fun string_ast_tr [Syntax.Variable xstr] = |
|
2533 |
(case Syntax.explode_xstr xstr of |
|
2534 |
[] => constants [Syntax.constrainC, "Nil", "string"] |
|
2535 |
| cs => mk_string cs) |
|
2536 |
| string_ast_tr asts = raise AST ("string_tr", asts); |
|
2537 |
in [("_Char", char_ast_tr), ("_String", string_ast_tr)] end; |
|
2538 |
*} |
|
2539 |
||
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2540 |
ML {* |
20184 | 2541 |
structure HOList = |
2542 |
struct |
|
2543 |
||
2544 |
local |
|
2545 |
val thy = the_context (); |
|
2546 |
in |
|
2547 |
val typ_string = Type (Sign.intern_type thy "string", []); |
|
2548 |
fun typ_list ty = Type (Sign.intern_type thy "list", [ty]); |
|
2549 |
fun term_list ty f [] = Const (Sign.intern_const thy "Nil", typ_list ty) |
|
2550 |
| term_list ty f (x::xs) = Const (Sign.intern_const thy "Cons", |
|
2551 |
ty --> typ_list ty --> typ_list ty) $ f x $ term_list ty f xs; |
|
2552 |
end; |
|
2553 |
||
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2554 |
fun int_of_nibble h = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2555 |
if "0" <= h andalso h <= "9" then ord h - ord "0" |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2556 |
else if "A" <= h andalso h <= "F" then ord h - ord "A" + 10 |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2557 |
else raise Match; |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2558 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2559 |
fun nibble_of_int i = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2560 |
if i <= 9 then chr (ord "0" + i) else chr (ord "A" + i - 10); |
20181 | 2561 |
|
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2562 |
fun dest_char (Const ("List.char.Char", _) $ c1 $ c2) = |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2563 |
let |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2564 |
fun dest_nibble (Const (s, _)) = (int_of_nibble o unprefix "List.nibble.Nibble") s |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2565 |
| dest_nibble _ = raise Match; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2566 |
in |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2567 |
(SOME (dest_nibble c1 * 16 + dest_nibble c2) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2568 |
handle Fail _ => NONE | Match => NONE) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2569 |
end |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2570 |
| dest_char _ = NONE; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2571 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2572 |
val print_list = Pretty.enum "," "[" "]"; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2573 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2574 |
fun print_char c = |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2575 |
let |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2576 |
val i = ord c |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2577 |
in if i < 32 |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2578 |
then prefix "\\" (string_of_int i) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2579 |
else c |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2580 |
end; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2581 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2582 |
val print_string = quote; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2583 |
|
20184 | 2584 |
fun term_string s = |
20181 | 2585 |
let |
2586 |
val ty_n = Type ("List.nibble", []); |
|
2587 |
val ty_c = Type ("List.char", []); |
|
2588 |
val ty_l = Type ("List.list", [ty_c]); |
|
2589 |
fun mk_nib n = Const ("List.nibble.Nibble" ^ chr (n + (if n <= 9 then ord "0" else ord "A" - 10)), ty_n); |
|
2590 |
fun mk_char c = |
|
2591 |
if Symbol.is_ascii c andalso Symbol.is_printable c then |
|
2592 |
Const ("List.char.Char", ty_n --> ty_n --> ty_c) $ mk_nib (ord c div 16) $ mk_nib (ord c mod 16) |
|
2593 |
else error ("Printable ASCII character expected: " ^ quote c); |
|
2594 |
fun mk_string c t = Const ("List.list.Cons", ty_c --> ty_l --> ty_l) |
|
2595 |
$ mk_char c $ t; |
|
2596 |
in fold_rev mk_string (explode s) (Const ("List.list.Nil", ty_l)) end; |
|
20184 | 2597 |
|
2598 |
end; |
|
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2599 |
*} |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2600 |
|
13366 | 2601 |
print_ast_translation {* |
2602 |
let |
|
2603 |
fun dest_nib (Syntax.Constant c) = |
|
2604 |
(case explode c of |
|
20184 | 2605 |
["N", "i", "b", "b", "l", "e", h] => HOList.int_of_nibble h |
13366 | 2606 |
| _ => raise Match) |
2607 |
| dest_nib _ = raise Match; |
|
2608 |
||
2609 |
fun dest_chr c1 c2 = |
|
2610 |
let val c = chr (dest_nib c1 * 16 + dest_nib c2) |
|
2611 |
in if Symbol.is_printable c then c else raise Match end; |
|
2612 |
||
2613 |
fun dest_char (Syntax.Appl [Syntax.Constant "Char", c1, c2]) = dest_chr c1 c2 |
|
2614 |
| dest_char _ = raise Match; |
|
2615 |
||
2616 |
fun xstr cs = Syntax.Appl [Syntax.Constant "_xstr", Syntax.Variable (Syntax.implode_xstr cs)]; |
|
2617 |
||
2618 |
fun char_ast_tr' [c1, c2] = Syntax.Appl [Syntax.Constant "_Char", xstr [dest_chr c1 c2]] |
|
2619 |
| char_ast_tr' _ = raise Match; |
|
2620 |
||
2621 |
fun list_ast_tr' [args] = Syntax.Appl [Syntax.Constant "_String", |
|
2622 |
xstr (map dest_char (Syntax.unfold_ast "_args" args))] |
|
2623 |
| list_ast_tr' ts = raise Match; |
|
2624 |
in [("Char", char_ast_tr'), ("@list", list_ast_tr')] end; |
|
2625 |
*} |
|
2626 |
||
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2627 |
|
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2628 |
subsection {* Code generator *} |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2629 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2630 |
subsubsection {* Setup *} |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2631 |
|
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2632 |
types_code |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2633 |
"list" ("_ list") |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2634 |
attach (term_of) {* |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2635 |
val term_of_list = HOLogic.mk_list; |
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2636 |
*} |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2637 |
attach (test) {* |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2638 |
fun gen_list' aG i j = frequency |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2639 |
[(i, fn () => aG j :: gen_list' aG (i-1) j), (1, fn () => [])] () |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2640 |
and gen_list aG i = gen_list' aG i i; |
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2641 |
*} |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2642 |
"char" ("string") |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2643 |
attach (term_of) {* |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2644 |
val nibbleT = Type ("List.nibble", []); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2645 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2646 |
fun term_of_char c = |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2647 |
Const ("List.char.Char", nibbleT --> nibbleT --> Type ("List.char", [])) $ |
20184 | 2648 |
Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c div 16), nibbleT) $ |
2649 |
Const ("List.nibble.Nibble" ^ HOList.nibble_of_int (ord c mod 16), nibbleT); |
|
16770
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2650 |
*} |
1f1b1fae30e4
Auxiliary functions to be used in generated code are now defined using "attach".
berghofe
parents:
16634
diff
changeset
|
2651 |
attach (test) {* |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2652 |
fun gen_char i = chr (random_range (ord "a") (Int.min (ord "a" + i, ord "z"))); |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2653 |
*} |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2654 |
|
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2655 |
consts_code "Cons" ("(_ ::/ _)") |
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2656 |
|
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2657 |
code_type list |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2658 |
(SML "_ list") |
21113 | 2659 |
(Haskell "![_]") |
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2660 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2661 |
code_const Nil |
21113 | 2662 |
(SML "[]") |
2663 |
(Haskell "[]") |
|
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2664 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2665 |
code_type char |
21113 | 2666 |
(SML "char") |
2667 |
(Haskell "Char") |
|
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2668 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2669 |
code_const Char |
21126 | 2670 |
(SML "!((_),/ (_))") |
2671 |
(Haskell "!((_),/ (_))") |
|
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2672 |
|
20588 | 2673 |
code_instance list :: eq and char :: eq |
2674 |
(Haskell - and -) |
|
2675 |
||
21046
fe1db2f991a7
moved HOL code generator setup to Code_Generator
haftmann
parents:
20699
diff
changeset
|
2676 |
code_const "Code_Generator.eq \<Colon> 'a\<Colon>eq list \<Rightarrow> 'a list \<Rightarrow> bool" |
20588 | 2677 |
(Haskell infixl 4 "==") |
2678 |
||
21046
fe1db2f991a7
moved HOL code generator setup to Code_Generator
haftmann
parents:
20699
diff
changeset
|
2679 |
code_const "Code_Generator.eq \<Colon> char \<Rightarrow> char \<Rightarrow> bool" |
20588 | 2680 |
(Haskell infixl 4 "==") |
2681 |
||
21079 | 2682 |
code_reserved SML |
2683 |
list char |
|
2684 |
||
2685 |
code_reserved Haskell |
|
2686 |
Char |
|
2687 |
||
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2688 |
setup {* |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2689 |
let |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2690 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2691 |
fun list_codegen thy defs gr dep thyname b t = |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2692 |
let val (gr', ps) = foldl_map (Codegen.invoke_codegen thy defs dep thyname false) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2693 |
(gr, HOLogic.dest_list t) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2694 |
in SOME (gr', Pretty.list "[" "]" ps) end handle TERM _ => NONE; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2695 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2696 |
fun char_codegen thy defs gr dep thyname b t = |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2697 |
case (Option.map chr o HOList.dest_char) t |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2698 |
of SOME c => |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2699 |
if Symbol.is_printable c |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2700 |
then SOME (gr, (Pretty.quote o Pretty.str) c) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2701 |
else NONE |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2702 |
| NONE => NONE; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2703 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2704 |
in |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2705 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2706 |
Codegen.add_codegen "list_codegen" list_codegen |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2707 |
#> Codegen.add_codegen "char_codegen" char_codegen |
20699 | 2708 |
#> CodegenSerializer.add_pretty_list "SML" "List.list.Nil" "List.list.Cons" |
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2709 |
HOList.print_list NONE (7, "::") |
20699 | 2710 |
#> CodegenSerializer.add_pretty_list "Haskell" "List.list.Nil" "List.list.Cons" |
20453
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2711 |
HOList.print_list (SOME (HOList.print_char, HOList.print_string)) (5, ":") |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2712 |
#> CodegenPackage.add_appconst |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2713 |
("List.char.Char", CodegenPackage.appgen_char HOList.dest_char) |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2714 |
|
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2715 |
end; |
855f07fabd76
final syntax for some Isar code generator keywords
haftmann
parents:
20439
diff
changeset
|
2716 |
*} |
15064
4f3102b50197
- Moved code generator setup for lists from Main.thy to List.thy
berghofe
parents:
15045
diff
changeset
|
2717 |
|
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2718 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2719 |
subsubsection {* Generation of efficient code *} |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2720 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2721 |
consts |
21079 | 2722 |
memberl :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "mem" 55) |
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2723 |
null:: "'a list \<Rightarrow> bool" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2724 |
list_inter :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2725 |
list_ex :: "('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> bool" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2726 |
list_all :: "('a \<Rightarrow> bool) \<Rightarrow> ('a list \<Rightarrow> bool)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2727 |
itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2728 |
filtermap :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a list \<Rightarrow> 'b list" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2729 |
map_filter :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'b list" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2730 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2731 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2732 |
"x mem [] = False" |
21079 | 2733 |
"x mem (y#ys) = (x = y \<or> x mem ys)" |
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2734 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2735 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2736 |
"null [] = True" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2737 |
"null (x#xs) = False" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2738 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2739 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2740 |
"list_inter [] bs = []" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2741 |
"list_inter (a#as) bs = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2742 |
(if a \<in> set bs then a # list_inter as bs else list_inter as bs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2743 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2744 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2745 |
"list_all P [] = True" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2746 |
"list_all P (x#xs) = (P x \<and> list_all P xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2747 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2748 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2749 |
"list_ex P [] = False" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2750 |
"list_ex P (x#xs) = (P x \<or> list_ex P xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2751 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2752 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2753 |
"filtermap f [] = []" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2754 |
"filtermap f (x#xs) = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2755 |
(case f x of None \<Rightarrow> filtermap f xs |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2756 |
| Some y \<Rightarrow> y # filtermap f xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2757 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2758 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2759 |
"map_filter f P [] = []" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2760 |
"map_filter f P (x#xs) = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2761 |
(if P x then f x # map_filter f P xs else map_filter f P xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2762 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2763 |
primrec |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2764 |
"itrev [] ys = ys" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2765 |
"itrev (x#xs) ys = itrev xs (x#ys)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2766 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2767 |
text {* |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2768 |
Only use @{text mem} for generating executable code. Otherwise |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2769 |
use @{prop "x : set xs"} instead --- it is much easier to reason about. |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2770 |
The same is true for @{const list_all} and @{const list_ex}: write |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2771 |
@{text "\<forall>x\<in>set xs"} and @{text "\<exists>x\<in>set xs"} instead because the HOL |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2772 |
quantifiers are aleady known to the automatic provers. In fact, |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2773 |
the declarations in the code subsection make sure that @{text "\<in>"}, @{text "\<forall>x\<in>set xs"} |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2774 |
and @{text "\<exists>x\<in>set xs"} are implemented efficiently. |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2775 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2776 |
Efficient emptyness check is implemented by @{const null}. |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2777 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2778 |
The functions @{const itrev}, @{const filtermap} and @{const map_filter} |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2779 |
are just there to generate efficient code. Do not use them |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2780 |
for modelling and proving. |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2781 |
*} |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2782 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2783 |
lemma mem_iff [normal post]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2784 |
"(x mem xs) = (x \<in> set xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2785 |
by (induct xs) auto |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2786 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2787 |
lemmas in_set_code [code unfold] = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2788 |
mem_iff [symmetric, THEN eq_reflection] |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2789 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2790 |
lemma empty_null [code inline]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2791 |
"(xs = []) = null xs" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2792 |
by (cases xs) simp_all |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2793 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2794 |
lemmas null_empty [normal post] = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2795 |
empty_null [symmetric] |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2796 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2797 |
lemma list_inter_conv: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2798 |
"set (list_inter xs ys) = set xs \<inter> set ys" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2799 |
by (induct xs) auto |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2800 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2801 |
lemma list_all_iff [normal post]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2802 |
"list_all P xs = (\<forall>x \<in> set xs. P x)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2803 |
by (induct xs) auto |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2804 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2805 |
lemmas list_ball_code [code unfold] = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2806 |
list_all_iff [symmetric, THEN eq_reflection] |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2807 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2808 |
lemma list_all_append [simp]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2809 |
"list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2810 |
by (induct xs) auto |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2811 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2812 |
lemma list_all_rev [simp]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2813 |
"list_all P (rev xs) = list_all P xs" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2814 |
by (simp add: list_all_iff) |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2815 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2816 |
lemma list_ex_iff [normal post]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2817 |
"list_ex P xs = (\<exists>x \<in> set xs. P x)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2818 |
by (induct xs) simp_all |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2819 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2820 |
lemmas list_bex_code [code unfold] = |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2821 |
list_ex_iff [symmetric, THEN eq_reflection] |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2822 |
|
21193 | 2823 |
lemma itrev [simp]: |
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2824 |
"itrev xs ys = rev xs @ ys" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2825 |
by (induct xs arbitrary: ys) simp_all |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2826 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2827 |
lemma filtermap_conv: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2828 |
"filtermap f xs = map (\<lambda>x. the (f x)) (filter (\<lambda>x. f x \<noteq> None) xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2829 |
by (induct xs) (simp_all split: option.split) |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2830 |
|
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2831 |
lemma map_filter_conv [simp]: |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2832 |
"map_filter f P xs = map f (filter P xs)" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2833 |
by (induct xs) auto |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2834 |
|
21193 | 2835 |
lemma rev_code [code func, code unfold, code noinline]: |
21061
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2836 |
"rev xs == itrev xs []" |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2837 |
by simp |
580dfc999ef6
added normal post setup; cleaned up "execution" constants
haftmann
parents:
21046
diff
changeset
|
2838 |
|
13122 | 2839 |
end |