author  haftmann 
Fri, 17 Jun 2005 16:12:49 +0200  
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parent 14765  bafb24c150c1 
child 18413  50c0c118e96d 
permissions  rwrr 
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(* *********************************************************************** *) 
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(* *) 

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(* Title: SList.thy (Extended List Theory) *) 

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(* Based on: $Id$ *) 
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(* Author: Lawrence C Paulson, Cambridge University Computer Laboratory*) 
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(* Author: B. Wolff, University of Bremen *) 

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(* Purpose: Enriched theory of lists *) 

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(* mutual indirect recursive datatypes *) 

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(* *) 

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(* *********************************************************************** *) 

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(* Definition of type 'a list (strict lists) by a least fixed point 
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We use list(A) == lfp(%Z. {NUMB(0)} <+> A <*> Z) 
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and not list == lfp(%Z. {NUMB(0)} <+> range(Leaf) <*> Z) 
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so that list can serve as a "functor" for defining other recursive types. 

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This enables the conservative construction of mutual recursive datatypes 

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such as 

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datatype 'a m = Node 'a * ('a m) list 

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Tidied by lcp. Still needs removal of nat_rec. 

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*) 
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theory SList imports NatArith Sexp Hilbert_Choice begin 
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(*Hilbert_Choice is needed for the function "inv"*) 
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(* *********************************************************************** *) 
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(* *) 

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(* Building up data type *) 

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(* *) 

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(* *********************************************************************** *) 

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(* Defining the Concrete Constructors *) 
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constdefs 
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NIL :: "'a item" 
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"NIL == In0(Numb(0))" 
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CONS :: "['a item, 'a item] => 'a item" 
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"CONS M N == In1(Scons M N)" 
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consts 
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list :: "'a item set => 'a item set" 
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inductive "list(A)" 
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intros 
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NIL_I: "NIL: list A" 
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CONS_I: "[ a: A; M: list A ] ==> CONS a M : list A" 
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typedef (List) 
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'a list = "list(range Leaf) :: 'a item set" 
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by (blast intro: list.NIL_I) 
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constdefs 
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List_case :: "['b, ['a item, 'a item]=>'b, 'a item] => 'b" 
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"List_case c d == Case(%x. c)(Split(d))" 
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List_rec :: "['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b" 
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"List_rec M c d == wfrec (trancl pred_sexp) 
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(%g. List_case c (%x y. d x y (g y))) M" 
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(* *********************************************************************** *) 
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(* *) 

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(* Abstracting data type *) 

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(* *) 

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(* *********************************************************************** *) 

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(*Declaring the abstract list constructors*) 

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constdefs 
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Nil :: "'a list" 
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"Nil == Abs_List(NIL)" 
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"Cons" :: "['a, 'a list] => 'a list" (infixr "#" 65) 
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"x#xs == Abs_List(CONS (Leaf x)(Rep_List xs))" 
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(* list Recursion  the trancl is Essential; see list.ML *) 
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list_rec :: "['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b" 
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"list_rec l c d == 
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List_rec(Rep_List l) c (%x y r. d(inv Leaf x)(Abs_List y) r)" 
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list_case :: "['b, ['a, 'a list]=>'b, 'a list] => 'b" 
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"list_case a f xs == list_rec xs a (%x xs r. f x xs)" 

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(* list Enumeration *) 
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consts 
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"[]" :: "'a list" ("[]") 
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syntax 
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"@list" :: "args => 'a list" ("[(_)]") 
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translations 

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"[x, xs]" == "x#[xs]" 

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"[x]" == "x#[]" 

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"[]" == "Nil" 

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"case xs of Nil => a  y#ys => b" == "list_case(a, %y ys. b, xs)" 

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(* *********************************************************************** *) 

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(* *) 

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(* Generalized Map Functionals *) 

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(* *) 

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(* *********************************************************************** *) 

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(* Generalized Map Functionals *) 

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constdefs 
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Rep_map :: "('b => 'a item) => ('b list => 'a item)" 
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"Rep_map f xs == list_rec xs NIL(%x l r. CONS(f x) r)" 
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Abs_map :: "('a item => 'b) => 'a item => 'b list" 
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"Abs_map g M == List_rec M Nil (%N L r. g(N)#r)" 
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(**** Function definitions ****) 

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constdefs 

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null :: "'a list => bool" 

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"null xs == list_rec xs True (%x xs r. False)" 

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hd :: "'a list => 'a" 

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"hd xs == list_rec xs (@x. True) (%x xs r. x)" 

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tl :: "'a list => 'a list" 

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"tl xs == list_rec xs (@xs. True) (%x xs r. xs)" 

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(* a total version of tl: *) 

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ttl :: "'a list => 'a list" 

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"ttl xs == list_rec xs [] (%x xs r. xs)" 

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member :: "['a, 'a list] => bool" (infixl "mem" 55) 
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"x mem xs == list_rec xs False (%y ys r. if y=x then True else r)" 
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list_all :: "('a => bool) => ('a list => bool)" 

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"list_all P xs == list_rec xs True(%x l r. P(x) & r)" 

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map :: "('a=>'b) => ('a list => 'b list)" 

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"map f xs == list_rec xs [] (%x l r. f(x)#r)" 

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constdefs 
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append :: "['a list, 'a list] => 'a list" (infixr "@" 65) 
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"xs@ys == list_rec xs ys (%x l r. x#r)" 
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filter :: "['a => bool, 'a list] => 'a list" 
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"filter P xs == list_rec xs [] (%x xs r. if P(x)then x#r else r)" 

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foldl :: "[['b,'a] => 'b, 'b, 'a list] => 'b" 

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"foldl f a xs == list_rec xs (%a. a)(%x xs r.%a. r(f a x))(a)" 

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foldr :: "[['a,'b] => 'b, 'b, 'a list] => 'b" 

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"foldr f a xs == list_rec xs a (%x xs r. (f x r))" 

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length :: "'a list => nat" 

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"length xs== list_rec xs 0 (%x xs r. Suc r)" 

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drop :: "['a list,nat] => 'a list" 

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"drop t n == (nat_rec(%x. x)(%m r xs. r(ttl xs)))(n)(t)" 

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copy :: "['a, nat] => 'a list" (* make list of n copies of x *) 

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"copy t == nat_rec [] (%m xs. t # xs)" 

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flat :: "'a list list => 'a list" 

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"flat == foldr (op @) []" 

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nth :: "[nat, 'a list] => 'a" 

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"nth == nat_rec hd (%m r xs. r(tl xs))" 

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rev :: "'a list => 'a list" 

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"rev xs == list_rec xs [] (%x xs xsa. xsa @ [x])" 

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(* miscellaneous definitions *) 
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zipWith :: "['a * 'b => 'c, 'a list * 'b list] => 'c list" 

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"zipWith f S == (list_rec (fst S) (%T.[]) 

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(%x xs r. %T. if null T then [] 

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else f(x,hd T) # r(tl T)))(snd(S))" 

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zip :: "'a list * 'b list => ('a*'b) list" 
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"zip == zipWith (%s. s)" 

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unzip :: "('a*'b) list => ('a list * 'b list)" 
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"unzip == foldr(% (a,b)(c,d).(a#c,b#d))([],[])" 

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consts take :: "['a list,nat] => 'a list" 
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primrec 

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take_0: "take xs 0 = []" 
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take_Suc: "take xs (Suc n) = list_case [] (%x l. x # take l n) xs" 
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consts enum :: "[nat,nat] => nat list" 
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primrec 

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enum_0: "enum i 0 = []" 
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enum_Suc: "enum i (Suc j) = (if i <= j then enum i j @ [j] else [])" 
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syntax 
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(* Special syntax for list_all and filter *) 

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"@Alls" :: "[idt, 'a list, bool] => bool" ("(2Alls _:_./ _)" 10) 

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"@filter" :: "[idt, 'a list, bool] => 'a list" ("(1[_:_ ./ _])") 
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translations 
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"[x:xs. P]" == "filter(%x. P) xs" 
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"Alls x:xs. P"== "list_all(%x. P)xs" 

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lemma ListI: "x : list (range Leaf) ==> x : List" 
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by (simp add: List_def) 
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lemma ListD: "x : List ==> x : list (range Leaf)" 
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by (simp add: List_def) 
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lemma list_unfold: "list(A) = usum {Numb(0)} (uprod A (list(A)))" 
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by (fast intro!: list.intros [unfolded NIL_def CONS_def] 
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elim: list.cases [unfolded NIL_def CONS_def]) 
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(*This justifies using list in other recursive type definitions*) 
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lemma list_mono: "A<=B ==> list(A) <= list(B)" 
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apply (unfold list.defs ) 
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apply (rule lfp_mono) 
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apply (assumption  rule basic_monos)+ 
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done 
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(*Type checking  list creates wellfounded sets*) 
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lemma list_sexp: "list(sexp) <= sexp" 
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apply (unfold NIL_def CONS_def list.defs) 
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apply (rule lfp_lowerbound) 
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apply (fast intro: sexp.intros sexp_In0I sexp_In1I) 
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done 
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236 

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237 
(* A <= sexp ==> list(A) <= sexp *) 
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lemmas list_subset_sexp = subset_trans [OF list_mono list_sexp] 
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239 

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240 

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241 
(*Induction for the type 'a list *) 
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242 
lemma list_induct: 
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243 
"[ P(Nil); 
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!!x xs. P(xs) ==> P(x # xs) ] ==> P(l)" 
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245 
apply (unfold Nil_def Cons_def) 
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246 
apply (rule Rep_List_inverse [THEN subst]) 
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247 
(*types force good instantiation*) 
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apply (rule Rep_List [unfolded List_def, THEN list.induct], simp) 
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249 
apply (erule Abs_List_inverse [unfolded List_def, THEN subst], blast) 
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250 
done 
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251 

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252 

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253 
(*** Isomorphisms ***) 
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254 

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255 
lemma inj_on_Abs_list: "inj_on Abs_List (list(range Leaf))" 
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256 
apply (rule inj_on_inverseI) 
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257 
apply (erule Abs_List_inverse [unfolded List_def]) 
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258 
done 
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259 

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260 
(** Distinctness of constructors **) 
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261 

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262 
lemma CONS_not_NIL [iff]: "CONS M N ~= NIL" 
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263 
by (simp add: NIL_def CONS_def) 
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264 

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lemmas NIL_not_CONS [iff] = CONS_not_NIL [THEN not_sym] 
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266 
lemmas CONS_neq_NIL = CONS_not_NIL [THEN notE, standard] 
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267 
lemmas NIL_neq_CONS = sym [THEN CONS_neq_NIL] 
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268 

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269 
lemma Cons_not_Nil [iff]: "x # xs ~= Nil" 
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270 
apply (unfold Nil_def Cons_def) 
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271 
apply (rule CONS_not_NIL [THEN inj_on_Abs_list [THEN inj_on_contraD]]) 
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272 
apply (simp_all add: list.intros rangeI Rep_List [unfolded List_def]) 
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273 
done 
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274 

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lemmas Nil_not_Cons [iff] = Cons_not_Nil [THEN not_sym, standard] 
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276 
lemmas Cons_neq_Nil = Cons_not_Nil [THEN notE, standard] 
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lemmas Nil_neq_Cons = sym [THEN Cons_neq_Nil] 
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278 

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279 
(** Injectiveness of CONS and Cons **) 
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280 

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281 
lemma CONS_CONS_eq [iff]: "(CONS K M)=(CONS L N) = (K=L & M=N)" 
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282 
by (simp add: CONS_def) 
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283 

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284 
(*For reasoning about abstract list constructors*) 
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285 
declare Rep_List [THEN ListD, intro] ListI [intro] 
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286 
declare list.intros [intro,simp] 
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287 
declare Leaf_inject [dest!] 
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288 

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289 
lemma Cons_Cons_eq [iff]: "(x#xs=y#ys) = (x=y & xs=ys)" 
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290 
apply (simp add: Cons_def) 
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291 
apply (subst Abs_List_inject) 
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292 
apply (auto simp add: Rep_List_inject) 
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293 
done 
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294 

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295 
lemmas Cons_inject2 = Cons_Cons_eq [THEN iffD1, THEN conjE, standard] 
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296 

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297 
lemma CONS_D: "CONS M N: list(A) ==> M: A & N: list(A)" 
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298 
apply (erule setup_induction) 
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299 
apply (erule list.induct, blast+) 
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300 
done 
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301 

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302 
lemma sexp_CONS_D: "CONS M N: sexp ==> M: sexp & N: sexp" 
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303 
apply (simp add: CONS_def In1_def) 
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304 
apply (fast dest!: Scons_D) 
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305 
done 
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306 

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307 

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308 
(*Reasoning about constructors and their freeness*) 
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309 

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310 

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311 
lemma not_CONS_self: "N: list(A) ==> !M. N ~= CONS M N" 
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312 
by (erule list.induct, simp_all) 
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313 

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314 
lemma not_Cons_self2: "\<forall>x. l ~= x#l" 
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315 
by (induct_tac "l" rule: list_induct, simp_all) 
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316 

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317 

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318 
lemma neq_Nil_conv2: "(xs ~= []) = (\<exists>y ys. xs = y#ys)" 
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319 
by (induct_tac "xs" rule: list_induct, auto) 
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320 

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321 
(** Conversion rules for List_case: case analysis operator **) 
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322 

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323 
lemma List_case_NIL [simp]: "List_case c h NIL = c" 
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324 
by (simp add: List_case_def NIL_def) 
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325 

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326 
lemma List_case_CONS [simp]: "List_case c h (CONS M N) = h M N" 
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327 
by (simp add: List_case_def CONS_def) 
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328 

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329 

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330 
(*** List_rec  by wf recursion on pred_sexp ***) 
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331 

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332 
(* The trancl(pred_sexp) is essential because pred_sexp_CONS_I1,2 would not 
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333 
hold if pred_sexp^+ were changed to pred_sexp. *) 
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334 

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335 
lemma List_rec_unfold_lemma: 
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336 
"(%M. List_rec M c d) == 
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337 
wfrec (trancl pred_sexp) (%g. List_case c (%x y. d x y (g y)))" 
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338 
by (simp add: List_rec_def) 
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339 

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340 
lemmas List_rec_unfold = 
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341 
def_wfrec [OF List_rec_unfold_lemma wf_pred_sexp [THEN wf_trancl], 
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342 
standard] 
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343 

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344 

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345 
(** pred_sexp lemmas **) 
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346 

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347 
lemma pred_sexp_CONS_I1: 
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348 
"[ M: sexp; N: sexp ] ==> (M, CONS M N) : pred_sexp^+" 
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349 
by (simp add: CONS_def In1_def) 
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350 

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351 
lemma pred_sexp_CONS_I2: 
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352 
"[ M: sexp; N: sexp ] ==> (N, CONS M N) : pred_sexp^+" 
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353 
by (simp add: CONS_def In1_def) 
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354 

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355 
lemma pred_sexp_CONS_D: 
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356 
"(CONS M1 M2, N) : pred_sexp^+ ==> 
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357 
(M1,N) : pred_sexp^+ & (M2,N) : pred_sexp^+" 
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358 
apply (frule pred_sexp_subset_Sigma [THEN trancl_subset_Sigma, THEN subsetD]) 
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359 
apply (blast dest!: sexp_CONS_D intro: pred_sexp_CONS_I1 pred_sexp_CONS_I2 
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360 
trans_trancl [THEN transD]) 
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361 
done 
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362 

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363 

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364 
(** Conversion rules for List_rec **) 
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365 

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366 
lemma List_rec_NIL [simp]: "List_rec NIL c h = c" 
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367 
apply (rule List_rec_unfold [THEN trans]) 
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368 
apply (simp add: List_case_NIL) 
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369 
done 
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parents:
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changeset

370 

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371 
lemma List_rec_CONS [simp]: 
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372 
"[ M: sexp; N: sexp ] 
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373 
==> List_rec (CONS M N) c h = h M N (List_rec N c h)" 
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374 
apply (rule List_rec_unfold [THEN trans]) 
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375 
apply (simp add: pred_sexp_CONS_I2) 
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376 
done 
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parents:
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diff
changeset

377 

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parents:
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changeset

378 

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379 
(*** list_rec  by List_rec ***) 
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changeset

380 

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381 
lemmas Rep_List_in_sexp = 
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382 
subsetD [OF range_Leaf_subset_sexp [THEN list_subset_sexp] 
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383 
Rep_List [THEN ListD]] 
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parents:
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diff
changeset

384 

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parents:
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diff
changeset

385 

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parents:
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changeset

386 
lemma list_rec_Nil [simp]: "list_rec Nil c h = c" 
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387 
by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Nil_def) 
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parents:
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diff
changeset

388 

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parents:
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diff
changeset

389 

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changeset

390 
lemma list_rec_Cons [simp]: "list_rec (a#l) c h = h a l (list_rec l c h)" 
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391 
by (simp add: list_rec_def ListI [THEN Abs_List_inverse] Cons_def 
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392 
Rep_List_inverse Rep_List [THEN ListD] inj_Leaf Rep_List_in_sexp) 
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parents:
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diff
changeset

393 

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parents:
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diff
changeset

394 

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395 
(*Type checking. Useful?*) 
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396 
lemma List_rec_type: 
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397 
"[ M: list(A); 
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398 
A<=sexp; 
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parents:
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399 
c: C(NIL); 
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400 
!!x y r. [ x: A; y: list(A); r: C(y) ] ==> h x y r: C(CONS x y) 
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401 
] ==> List_rec M c h : C(M :: 'a item)" 
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changeset

402 
apply (erule list.induct, simp) 
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parents:
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403 
apply (insert list_subset_sexp) 
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parents:
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diff
changeset

404 
apply (subst List_rec_CONS, blast+) 
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parents:
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diff
changeset

405 
done 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

406 

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parents:
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diff
changeset

407 

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conversion of Induct/{Slist,Sexp} to Isar scripts
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parents:
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diff
changeset

408 

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parents:
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changeset

409 
(** Generalized map functionals **) 
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410 

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411 
lemma Rep_map_Nil [simp]: "Rep_map f Nil = NIL" 
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412 
by (simp add: Rep_map_def) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
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parents:
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diff
changeset

413 

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parents:
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changeset

414 
lemma Rep_map_Cons [simp]: 
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415 
"Rep_map f(x#xs) = CONS(f x)(Rep_map f xs)" 
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416 
by (simp add: Rep_map_def) 
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parents:
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changeset

417 

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418 
lemma Rep_map_type: "(!!x. f(x): A) ==> Rep_map f xs: list(A)" 
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419 
apply (simp add: Rep_map_def) 
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parents:
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diff
changeset

420 
apply (rule list_induct, auto) 
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parents:
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421 
done 
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paulson
parents:
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diff
changeset

422 

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423 
lemma Abs_map_NIL [simp]: "Abs_map g NIL = Nil" 
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424 
by (simp add: Abs_map_def) 
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parents:
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changeset

425 

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parents:
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426 
lemma Abs_map_CONS [simp]: 
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427 
"[ M: sexp; N: sexp ] ==> Abs_map g (CONS M N) = g(M) # Abs_map g N" 
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changeset

428 
by (simp add: Abs_map_def) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

429 

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paulson
parents:
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changeset

430 
(*Eases the use of primitive recursion. NOTE USE OF == *) 
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431 
lemma def_list_rec_NilCons: 
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432 
"[ !!xs. f(xs) == list_rec xs c h ] 
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parents:
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433 
==> f [] = c & f(x#xs) = h x xs (f xs)" 
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paulson
parents:
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changeset

434 
by simp 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

435 

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paulson
parents:
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diff
changeset

436 

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paulson
parents:
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diff
changeset

437 

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parents:
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changeset

438 
lemma Abs_map_inverse: 
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paulson
parents:
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439 
"[ M: list(A); A<=sexp; !!z. z: A ==> f(g(z)) = z ] 
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paulson
parents:
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440 
==> Rep_map f (Abs_map g M) = M" 
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paulson
parents:
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441 
apply (erule list.induct, simp_all) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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changeset

442 
apply (insert list_subset_sexp) 
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paulson
parents:
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diff
changeset

443 
apply (subst Abs_map_CONS, blast) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

444 
apply blast 
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paulson
parents:
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diff
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445 
apply simp 
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paulson
parents:
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446 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

447 

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paulson
parents:
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changeset

448 
(*Rep_map_inverse is obtained via Abs_Rep_map and map_ident*) 
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paulson
parents:
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changeset

449 

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parents:
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450 
(** list_case **) 
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paulson
parents:
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changeset

451 

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paulson
parents:
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452 
(* setting up rewrite sets *) 
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parents:
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453 

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454 
text{*Better to have a single theorem with a conjunctive conclusion.*} 
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paulson
parents:
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455 
declare def_list_rec_NilCons [OF list_case_def, simp] 
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paulson
parents:
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changeset

456 

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paulson
parents:
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457 
(** list_case **) 
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paulson
parents:
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diff
changeset

458 

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paulson
parents:
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diff
changeset

459 
lemma expand_list_case: 
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paulson
parents:
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460 
"P(list_case a f xs) = ((xs=[] > P a ) & (!y ys. xs=y#ys > P(f y ys)))" 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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changeset

461 
by (induct_tac "xs" rule: list_induct, simp_all) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

462 

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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

463 

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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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changeset

464 
(**** Function definitions ****) 
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paulson
parents:
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changeset

465 

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paulson
parents:
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changeset

466 
declare def_list_rec_NilCons [OF null_def, simp] 
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467 
declare def_list_rec_NilCons [OF hd_def, simp] 
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468 
declare def_list_rec_NilCons [OF tl_def, simp] 
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469 
declare def_list_rec_NilCons [OF ttl_def, simp] 
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470 
declare def_list_rec_NilCons [OF append_def, simp] 
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471 
declare def_list_rec_NilCons [OF member_def, simp] 
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472 
declare def_list_rec_NilCons [OF map_def, simp] 
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changeset

473 
declare def_list_rec_NilCons [OF filter_def, simp] 
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474 
declare def_list_rec_NilCons [OF list_all_def, simp] 
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paulson
parents:
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diff
changeset

475 

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paulson
parents:
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diff
changeset

476 

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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

477 
(** nth **) 
e7738aa7267f
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paulson
parents:
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diff
changeset

478 

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paulson
parents:
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diff
changeset

479 
lemma def_nat_rec_0_eta: 
e7738aa7267f
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paulson
parents:
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diff
changeset

480 
"[ !!n. f == nat_rec c h ] ==> f(0) = c" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

481 
by simp 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

482 

e7738aa7267f
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paulson
parents:
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diff
changeset

483 
lemma def_nat_rec_Suc_eta: 
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paulson
parents:
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diff
changeset

484 
"[ !!n. f == nat_rec c h ] ==> f(Suc(n)) = h n (f n)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

485 
by simp 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

486 

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paulson
parents:
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changeset

487 
declare def_nat_rec_0_eta [OF nth_def, simp] 
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paulson
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changeset

488 
declare def_nat_rec_Suc_eta [OF nth_def, simp] 
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paulson
parents:
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diff
changeset

489 

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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

490 

e7738aa7267f
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paulson
parents:
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diff
changeset

491 
(** length **) 
e7738aa7267f
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paulson
parents:
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diff
changeset

492 

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paulson
parents:
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changeset

493 
lemma length_Nil [simp]: "length([]) = 0" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

494 
by (simp add: length_def) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

495 

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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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changeset

496 
lemma length_Cons [simp]: "length(a#xs) = Suc(length(xs))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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changeset

497 
by (simp add: length_def) 
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conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

498 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

499 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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500 
(** @  append **) 
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paulson
parents:
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diff
changeset

501 

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paulson
parents:
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diff
changeset

502 
lemma append_assoc [simp]: "(xs@ys)@zs = xs@(ys@zs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
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diff
changeset

503 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

504 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

505 
lemma append_Nil2 [simp]: "xs @ [] = xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

506 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

507 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

508 
(** mem **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

509 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

510 
lemma mem_append [simp]: "x mem (xs@ys) = (x mem xs  x mem ys)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

511 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

512 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

513 
lemma mem_filter [simp]: "x mem [x:xs. P x ] = (x mem xs & P(x))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

514 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

515 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

516 
(** list_all **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

517 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

518 
lemma list_all_True [simp]: "(Alls x:xs. True) = True" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

519 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

520 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

521 
lemma list_all_conj [simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

522 
"list_all p (xs@ys) = ((list_all p xs) & (list_all p ys))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

523 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

524 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

525 
lemma list_all_mem_conv: "(Alls x:xs. P(x)) = (!x. x mem xs > P(x))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

526 
apply (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

527 
apply blast 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

528 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

529 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

530 
lemma nat_case_dist : "(! n. P n) = (P 0 & (! n. P (Suc n)))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

531 
apply auto 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

532 
apply (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

533 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

534 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

535 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

536 
lemma alls_P_eq_P_nth: "(Alls u:A. P u) = (!n. n < length A > P(nth n A))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

537 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

538 
apply (rule trans) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

539 
apply (rule_tac [2] nat_case_dist [symmetric], simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

540 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

541 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

542 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

543 
lemma list_all_imp: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

544 
"[ !x. P x > Q x; (Alls x:xs. P(x)) ] ==> (Alls x:xs. Q(x))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

545 
by (simp add: list_all_mem_conv) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

546 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

547 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

548 
(** The functional "map" and the generalized functionals **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

549 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

550 
lemma Abs_Rep_map: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

551 
"(!!x. f(x): sexp) ==> 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

552 
Abs_map g (Rep_map f xs) = map (%t. g(f(t))) xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

553 
apply (induct_tac "xs" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

554 
apply (simp_all add: Rep_map_type list_sexp [THEN subsetD]) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

555 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

556 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

557 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

558 
(** Additional mapping lemmas **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

559 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

560 
lemma map_ident [simp]: "map(%x. x)(xs) = xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

561 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

562 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

563 
lemma map_append [simp]: "map f (xs@ys) = map f xs @ map f ys" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

564 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

565 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

566 
lemma map_compose: "map(f o g)(xs) = map f (map g xs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

567 
apply (simp add: o_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

568 
apply (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

569 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

570 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

571 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

572 
lemma mem_map_aux1 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

573 
"x mem (map f q) > (\<exists>y. y mem q & x = f y)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

574 
by (induct_tac "q" rule: list_induct, simp_all, blast) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

575 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

576 
lemma mem_map_aux2 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

577 
"(\<exists>y. y mem q & x = f y) > x mem (map f q)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

578 
by (induct_tac "q" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

579 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

580 
lemma mem_map: "x mem (map f q) = (\<exists>y. y mem q & x = f y)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

581 
apply (rule iffI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

582 
apply (erule mem_map_aux1) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

583 
apply (erule mem_map_aux2) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

584 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

585 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

586 
lemma hd_append [rule_format]: "A ~= [] > hd(A @ B) = hd(A)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

587 
by (induct_tac "A" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

588 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

589 
lemma tl_append [rule_format]: "A ~= [] > tl(A @ B) = tl(A) @ B" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

590 
by (induct_tac "A" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

591 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

592 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

593 
(** take **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

594 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

595 
lemma take_Suc1 [simp]: "take [] (Suc x) = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

596 
by simp 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

597 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

598 
lemma take_Suc2 [simp]: "take(a#xs)(Suc x) = a#take xs x" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

599 
by simp 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

600 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

601 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

602 
(** drop **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

603 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

604 
lemma drop_0 [simp]: "drop xs 0 = xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

605 
by (simp add: drop_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

606 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

607 
lemma drop_Suc1 [simp]: "drop [] (Suc x) = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

608 
apply (simp add: drop_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

609 
apply (induct_tac "x", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

610 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

611 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

612 
lemma drop_Suc2 [simp]: "drop(a#xs)(Suc x) = drop xs x" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

613 
by (simp add: drop_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

614 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

615 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

616 
(** copy **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

617 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

618 
lemma copy_0 [simp]: "copy x 0 = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

619 
by (simp add: copy_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

620 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

621 
lemma copy_Suc [simp]: "copy x (Suc y) = x # copy x y" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

622 
by (simp add: copy_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

623 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

624 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

625 
(** fold **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

626 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

627 
lemma foldl_Nil [simp]: "foldl f a [] = a" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

628 
by (simp add: foldl_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

629 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

630 
lemma foldl_Cons [simp]: "foldl f a(x#xs) = foldl f (f a x) xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

631 
by (simp add: foldl_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

632 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

633 
lemma foldr_Nil [simp]: "foldr f a [] = a" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

634 
by (simp add: foldr_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

635 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

636 
lemma foldr_Cons [simp]: "foldr f z(x#xs) = f x (foldr f z xs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

637 
by (simp add: foldr_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

638 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

639 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

640 
(** flat **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

641 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

642 
lemma flat_Nil [simp]: "flat [] = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

643 
by (simp add: flat_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

644 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

645 
lemma flat_Cons [simp]: "flat (x # xs) = x @ flat xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

646 
by (simp add: flat_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

647 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

648 
(** rev **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

649 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

650 
lemma rev_Nil [simp]: "rev [] = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

651 
by (simp add: rev_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

652 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

653 
lemma rev_Cons [simp]: "rev (x # xs) = rev xs @ [x]" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

654 
by (simp add: rev_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

655 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

656 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

657 
(** zip **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

658 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

659 
lemma zipWith_Cons_Cons [simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

660 
"zipWith f (a#as,b#bs) = f(a,b) # zipWith f (as,bs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

661 
by (simp add: zipWith_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

662 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

663 
lemma zipWith_Nil_Nil [simp]: "zipWith f ([],[]) = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

664 
by (simp add: zipWith_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

665 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

666 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

667 
lemma zipWith_Cons_Nil [simp]: "zipWith f (x,[]) = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

668 
apply (simp add: zipWith_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

669 
apply (induct_tac "x" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

670 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

671 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

672 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

673 
lemma zipWith_Nil_Cons [simp]: "zipWith f ([],x) = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

674 
by (simp add: zipWith_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

675 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

676 
lemma unzip_Nil [simp]: "unzip [] = ([],[])" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

677 
by (simp add: unzip_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

678 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

679 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

680 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

681 
(** SOME LIST THEOREMS **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

682 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

683 
(* SQUIGGOL LEMMAS *) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

684 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

685 
lemma map_compose_ext: "map(f o g) = ((map f) o (map g))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

686 
apply (simp add: o_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

687 
apply (rule ext) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

688 
apply (simp add: map_compose [symmetric] o_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

689 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

690 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

691 
lemma map_flat: "map f (flat S) = flat(map (map f) S)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

692 
by (induct_tac "S" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

693 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

694 
lemma list_all_map_eq: "(Alls u:xs. f(u) = g(u)) > map f xs = map g xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

695 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

696 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

697 
lemma filter_map_d: "filter p (map f xs) = map f (filter(p o f)(xs))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

698 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

699 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

700 
lemma filter_compose: "filter p (filter q xs) = filter(%x. p x & q x) xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

701 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

702 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

703 
(* "filter(p, filter(q,xs)) = filter(q, filter(p,xs))", 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

704 
"filter(p, filter(p,xs)) = filter(p,xs)" BIRD's thms.*) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

705 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

706 
lemma filter_append [rule_format, simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

707 
"\<forall>B. filter p (A @ B) = (filter p A @ filter p B)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

708 
by (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

709 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

710 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

711 
(* inits(xs) == map(fst,splits(xs)), 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

712 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

713 
splits([]) = [] 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

714 
splits(a # xs) = <[],xs> @ map(%x. <a # fst(x),snd(x)>, splits(xs)) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

715 
(x @ y = z) = <x,y> mem splits(z) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

716 
x mem xs & y mem ys = <x,y> mem diag(xs,ys) *) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

717 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

718 
lemma length_append: "length(xs@ys) = length(xs)+length(ys)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

719 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

720 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

721 
lemma length_map: "length(map f xs) = length(xs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

722 
by (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

723 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

724 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

725 
lemma take_Nil [simp]: "take [] n = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

726 
by (induct_tac "n", simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

727 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

728 
lemma take_take_eq [simp]: "\<forall>n. take (take xs n) n = take xs n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

729 
apply (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

730 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

731 
apply (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

732 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

733 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

734 
lemma take_take_Suc_eq1 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

735 
"\<forall>n. take (take xs(Suc(n+m))) n = take xs n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

736 
apply (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

737 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

738 
apply (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

739 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

740 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

741 
declare take_Suc [simp del] 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

742 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

743 
lemma take_take_1: "take (take xs (n+m)) n = take xs n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

744 
apply (induct_tac "m") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

745 
apply (simp_all add: take_take_Suc_eq1) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

746 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

747 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

748 
lemma take_take_Suc_eq2 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

749 
"\<forall>n. take (take xs n)(Suc(n+m)) = take xs n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

750 
apply (induct_tac "xs" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

751 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

752 
apply (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

753 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

754 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

755 
lemma take_take_2: "take(take xs n)(n+m) = take xs n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

756 
apply (induct_tac "m") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

757 
apply (simp_all add: take_take_Suc_eq2) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

758 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

759 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

760 
(* length(take(xs,n)) = min(n, length(xs)) *) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

761 
(* length(drop(xs,n)) = length(xs)  n *) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

762 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

763 
lemma drop_Nil [simp]: "drop [] n = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

764 
by (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

765 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

766 
lemma drop_drop [rule_format]: "\<forall>xs. drop (drop xs m) n = drop xs(m+n)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

767 
apply (induct_tac "m", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

768 
apply (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

769 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

770 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

771 
lemma take_drop [rule_format]: "\<forall>xs. (take xs n) @ (drop xs n) = xs" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

772 
apply (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

773 
apply (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

774 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

775 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

776 
lemma copy_copy: "copy x n @ copy x m = copy x (n+m)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

777 
by (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

778 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

779 
lemma length_copy: "length(copy x n) = n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

780 
by (induct_tac "n", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

781 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

782 
lemma length_take [rule_format, simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

783 
"\<forall>xs. length(take xs n) = min (length xs) n" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

784 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

785 
apply auto 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

786 
apply (induct_tac "xs" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

787 
apply auto 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

788 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

789 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

790 
lemma length_take_drop: "length(take A k) + length(drop A k) = length(A)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

791 
by (simp only: length_append [symmetric] take_drop) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

792 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

793 
lemma take_append [rule_format]: "\<forall>A. length(A) = n > take(A@B) n = A" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

794 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

795 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

796 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

797 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

798 
apply (induct_tac [3] "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

799 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

800 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

801 
lemma take_append2 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

802 
"\<forall>A. length(A) = n > take(A@B) (n+k) = A @ take B k" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

803 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

804 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

805 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

806 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

807 
apply (induct_tac [3] "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

808 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

809 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

810 
lemma take_map [rule_format]: "\<forall>n. take (map f A) n = map f (take A n)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

811 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

812 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

813 
apply (induct_tac "n", simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

814 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

815 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

816 
lemma drop_append [rule_format]: "\<forall>A. length(A) = n > drop(A@B)n = B" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

817 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

818 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

819 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

820 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

821 
apply (induct_tac [3] "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

822 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

823 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

824 
lemma drop_append2 [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

825 
"\<forall>A. length(A) = n > drop(A@B)(n+k) = drop B k" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

826 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

827 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

828 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

829 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

830 
apply (induct_tac [3] "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

831 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

832 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

833 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

834 
lemma drop_all [rule_format]: "\<forall>A. length(A) = n > drop A n = []" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

835 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

836 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

837 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

838 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

839 
apply (induct_tac [3] "A" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

840 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

841 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

842 
lemma drop_map [rule_format]: "\<forall>n. drop (map f A) n = map f (drop A n)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

843 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

844 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

845 
apply (induct_tac "n", simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

846 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

847 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

848 
lemma take_all [rule_format]: "\<forall>A. length(A) = n > take A n = A" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

849 
apply (induct_tac "n") 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

850 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

851 
apply (rule_tac [2] allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

852 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

853 
apply (induct_tac [3] "A" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

854 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

855 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

856 
lemma foldl_single: "foldl f a [b] = f a b" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

857 
by simp_all 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

858 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

859 
lemma foldl_append [rule_format, simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

860 
"\<forall>a. foldl f a (A @ B) = foldl f (foldl f a A) B" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

861 
by (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

862 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

863 
lemma foldl_map [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

864 
"\<forall>e. foldl f e (map g S) = foldl (%x y. f x (g y)) e S" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

865 
by (induct_tac "S" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

866 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

867 
lemma foldl_neutr_distr [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

868 
assumes r_neutr: "\<forall>a. f a e = a" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

869 
and r_neutl: "\<forall>a. f e a = a" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

870 
and assoc: "\<forall>a b c. f a (f b c) = f(f a b) c" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

871 
shows "\<forall>y. f y (foldl f e A) = foldl f y A" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

872 
apply (induct_tac "A" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

873 
apply (simp_all add: r_neutr r_neutl, clarify) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

874 
apply (erule all_dupE) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

875 
apply (rule trans) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

876 
prefer 2 apply assumption 
13612  877 
apply (simp (no_asm_use) add: assoc [THEN spec, THEN spec, THEN spec, THEN sym]) 
878 
apply simp 

13079
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

879 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

880 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

881 
lemma foldl_append_sym: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

882 
"[ !a. f a e = a; !a. f e a = a; 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

883 
!a b c. f a (f b c) = f(f a b) c ] 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

884 
==> foldl f e (A @ B) = f(foldl f e A)(foldl f e B)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

885 
apply (rule trans) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

886 
apply (rule foldl_append) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

887 
apply (rule sym) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

888 
apply (rule foldl_neutr_distr, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

889 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

890 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

891 
lemma foldr_append [rule_format, simp]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

892 
"\<forall>a. foldr f a (A @ B) = foldr f (foldr f a B) A" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

893 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

894 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

895 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

896 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

897 
lemma foldr_map [rule_format]: "\<forall>e. foldr f e (map g S) = foldr (f o g) e S" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

898 
apply (simp add: o_def) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

899 
apply (induct_tac "S" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

900 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

901 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

902 
lemma foldr_Un_eq_UN: "foldr op Un {} S = (UN X: {t. t mem S}.X)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

903 
by (induct_tac "S" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

904 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

905 
lemma foldr_neutr_distr: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

906 
"[ !a. f e a = a; !a b c. f a (f b c) = f(f a b) c ] 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

907 
==> foldr f y S = f (foldr f e S) y" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

908 
by (induct_tac "S" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

909 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

910 
lemma foldr_append2: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

911 
"[ !a. f e a = a; !a b c. f a (f b c) = f(f a b) c ] 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

912 
==> foldr f e (A @ B) = f (foldr f e A) (foldr f e B)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

913 
apply auto 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

914 
apply (rule foldr_neutr_distr, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

915 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

916 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

917 
lemma foldr_flat: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

918 
"[ !a. f e a = a; !a b c. f a (f b c) = f(f a b) c ] ==> 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

919 
foldr f e (flat S) = (foldr f e)(map (foldr f e) S)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

920 
apply (induct_tac "S" rule: list_induct) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

921 
apply (simp_all del: foldr_append add: foldr_append2) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

922 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

923 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

924 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

925 
lemma list_all_map: "(Alls x:map f xs .P(x)) = (Alls x:xs.(P o f)(x))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

926 
by (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

927 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

928 
lemma list_all_and: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

929 
"(Alls x:xs. P(x)&Q(x)) = ((Alls x:xs. P(x))&(Alls x:xs. Q(x)))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

930 
by (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

931 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

932 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

933 
lemma nth_map [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

934 
"\<forall>i. i < length(A) > nth i (map f A) = f(nth i A)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

935 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

936 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

937 
apply (induct_tac "i", auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

938 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

939 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

940 
lemma nth_app_cancel_right [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

941 
"\<forall>i. i < length(A) > nth i(A@B) = nth i A" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

942 
apply (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

943 
apply (rule allI) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

944 
apply (induct_tac "i", simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

945 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

946 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

947 
lemma nth_app_cancel_left [rule_format]: 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

948 
"\<forall>n. n = length(A) > nth(n+i)(A@B) = nth i B" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

949 
by (induct_tac "A" rule: list_induct, simp_all) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

950 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

951 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

952 
(** flat **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

953 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

954 
lemma flat_append [simp]: "flat(xs@ys) = flat(xs) @ flat(ys)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

955 
by (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

956 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

957 
lemma filter_flat: "filter p (flat S) = flat(map (filter p) S)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

958 
by (induct_tac "S" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

959 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

960 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

961 
(** rev **) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

962 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

963 
lemma rev_append [simp]: "rev(xs@ys) = rev(ys) @ rev(xs)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

964 
by (induct_tac "xs" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

965 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

966 
lemma rev_rev_ident [simp]: "rev(rev l) = l" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

967 
by (induct_tac "l" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

968 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

969 
lemma rev_flat: "rev(flat ls) = flat (map rev (rev ls))" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

970 
by (induct_tac "ls" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

971 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

972 
lemma rev_map_distrib: "rev(map f l) = map f (rev l)" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

973 
by (induct_tac "l" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

974 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

975 
lemma foldl_rev: "foldl f b (rev l) = foldr (%x y. f y x) b l" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

976 
by (induct_tac "l" rule: list_induct, auto) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

977 

e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

978 
lemma foldr_rev: "foldr f b (rev l) = foldl (%x y. f y x) b l" 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

979 
apply (rule sym) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

980 
apply (rule trans) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

981 
apply (rule_tac [2] foldl_rev, simp) 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

982 
done 
e7738aa7267f
conversion of Induct/{Slist,Sexp} to Isar scripts
paulson
parents:
12169
diff
changeset

983 

3120
c58423c20740
New directory to contain examples of (co)inductive definitions
paulson
parents:
diff
changeset

984 
end 