author  wenzelm 
Fri, 10 Oct 1997 19:02:28 +0200  
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permissions  rwrr 
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(* Title: HOL/LList.thy 
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ID: $Id$ 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1994 University of Cambridge 
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Definition of type 'a llist by a greatest fixed point 
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Shares NIL, CONS, List_case with List.thy 
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Still needs filter and flatten functions  hard because they need 
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bounds on the amount of lookahead required. 
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Could try (but would it work for the gfp analogue of term?) 
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LListD_Fun_def "LListD_Fun(A) == (%Z. diag({Numb(0)}) <++> diag(A) <**> Z)" 
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A nice but complex example would be [ML for the Working Programmer, page 176] 
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from(1) = enumerate (Lmap (Lmap(pack), makeqq(from(1),from(1)))) 
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Previous definition of llistD_Fun was explicit: 
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llistD_Fun_def 
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"llistD_Fun(r) == 
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{(LNil,LNil)} Un 
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(UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))``r)" 
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*) 
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LList = Gfp + SList + 
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types 
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'a llist 
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arities 
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llist :: (term)term 
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consts 
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list_Fun :: ['a item set, 'a item set] => 'a item set 
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LListD_Fun :: 
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"[('a item * 'a item)set, ('a item * 'a item)set] => 
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('a item * 'a item)set" 
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llist :: 'a item set => 'a item set 
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LListD :: "('a item * 'a item)set => ('a item * 'a item)set" 
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llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set" 
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Rep_llist :: 'a llist => 'a item 
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Abs_llist :: 'a item => 'a llist 
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LNil :: 'a llist 
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LCons :: ['a, 'a llist] => 'a llist 
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llist_case :: ['b, ['a, 'a llist]=>'b, 'a llist] => 'b 
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LList_corec_fun :: "[nat, 'a=>unit+('b item * 'a), 'a] => 'b item" 
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LList_corec :: "['a, 'a => unit + ('b item * 'a)] => 'b item" 
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llist_corec :: "['a, 'a => unit + ('b * 'a)] => 'b llist" 
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Lmap :: ('a item => 'b item) => ('a item => 'b item) 
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lmap :: ('a=>'b) => ('a llist => 'b llist) 
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iterates :: ['a => 'a, 'a] => 'a llist 
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Lconst :: 'a item => 'a item 
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Lappend :: ['a item, 'a item] => 'a item 
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lappend :: ['a llist, 'a llist] => 'a llist 
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coinductive "llist(A)" 
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intrs 
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NIL_I "NIL: llist(A)" 
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CONS_I "[ a: A; M: llist(A) ] ==> CONS a M : llist(A)" 
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coinductive "LListD(r)" 
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intrs 
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NIL_I "(NIL, NIL) : LListD(r)" 
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CONS_I "[ (a,b): r; (M,N) : LListD(r) 
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] ==> (CONS a M, CONS b N) : LListD(r)" 
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translations 
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"case p of LNil => a  LCons x l => b" == "llist_case a (%x l. b) p" 
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defs 
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(*Now used exclusively for abbreviating the coinduction rule*) 
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list_Fun_def "list_Fun A X == 
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{z. z = NIL  (? M a. z = CONS a M & a : A & M : X)}" 
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LListD_Fun_def "LListD_Fun r X == 
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{z. z = (NIL, NIL)  
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(? M N a b. z = (CONS a M, CONS b N) & 
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(a, b) : r & (M, N) : X)}" 
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(*defining the abstract constructors*) 
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LNil_def "LNil == Abs_llist(NIL)" 
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LCons_def "LCons x xs == Abs_llist(CONS (Leaf x) (Rep_llist xs))" 
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llist_case_def 
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"llist_case c d l == 
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List_case c (%x y. d (inv Leaf x) (Abs_llist y)) (Rep_llist l)" 
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LList_corec_fun_def 
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"LList_corec_fun k f == 
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nat_rec (%x. {}) 
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(%j r x. case f x of Inl u => NIL 
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 Inr(z,w) => CONS z (r w)) 
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k" 
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LList_corec_def 
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"LList_corec a f == UN k. LList_corec_fun k f a" 
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llist_corec_def 
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"llist_corec a f == 
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Abs_llist(LList_corec a 
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(%z. case f z of Inl x => Inl(x) 
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 Inr(v,w) => Inr(Leaf(v), w)))" 
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llistD_Fun_def 
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"llistD_Fun(r) == 
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prod_fun Abs_llist Abs_llist `` 
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LListD_Fun (diag(range Leaf)) 
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(prod_fun Rep_llist Rep_llist `` r)" 
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Lconst_def "Lconst(M) == lfp(%N. CONS M N)" 
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Lmap_def 
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"Lmap f M == LList_corec M (List_case (Inl ()) (%x M'. Inr((f(x), M'))))" 
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lmap_def 
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"lmap f l == llist_corec l (%z. case z of LNil => (Inl ()) 
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 LCons y z => Inr(f(y), z))" 
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iterates_def "iterates f a == llist_corec a (%x. Inr((x, f(x))))" 
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(*Append generates its result by applying f, where 
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f((NIL,NIL)) = Inl(()) 
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f((NIL, CONS N1 N2)) = Inr((N1, (NIL,N2)) 
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f((CONS M1 M2, N)) = Inr((M1, (M2,N)) 
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*) 
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Lappend_def 
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"Lappend M N == LList_corec (M,N) 
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(split(List_case (List_case (Inl ()) (%N1 N2. Inr((N1, (NIL,N2))))) 
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(%M1 M2 N. Inr((M1, (M2,N))))))" 
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lappend_def 
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"lappend l n == llist_corec (l,n) 
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(split(llist_case (llist_case (Inl ()) (%n1 n2. Inr((n1, (LNil,n2))))) 
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(%l1 l2 n. Inr((l1, (l2,n))))))" 
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rules 
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(*faking a type definition...*) 
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Rep_llist "Rep_llist(xs): llist(range(Leaf))" 
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Rep_llist_inverse "Abs_llist(Rep_llist(xs)) = xs" 
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Abs_llist_inverse "M: llist(range(Leaf)) ==> Rep_llist(Abs_llist(M)) = M" 
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end 