src/HOL/Induct/SList.thy

New directory to contain examples of (co)inductive definitions

(*  Title:      HOL/ex/SList.thy
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1993  University of Cambridge

Definition of type 'a list (strict lists) by a least fixed point

We use          list(A) == lfp(%Z. {NUMB(0)} <+> A <*> Z)
and not         list    == lfp(%Z. {NUMB(0)} <+> range(Leaf) <*> Z)
so that list can serve as a "functor" for defining other recursive types
*)

SList = Sexp +

types
  'a list

arities
  list :: (term) term


consts

  list        :: 'a item set => 'a item set
  Rep_list    :: 'a list => 'a item
  Abs_list    :: 'a item => 'a list
  NIL         :: 'a item
  CONS        :: ['a item, 'a item] => 'a item
  Nil         :: 'a list
  "#"         :: ['a, 'a list] => 'a list                         (infixr 65)
  List_case   :: ['b, ['a item, 'a item]=>'b, 'a item] => 'b
  List_rec    :: ['a item, 'b, ['a item, 'a item, 'b]=>'b] => 'b
  list_case   :: ['b, ['a, 'a list]=>'b, 'a list] => 'b
  list_rec    :: ['a list, 'b, ['a, 'a list, 'b]=>'b] => 'b
  Rep_map     :: ('b => 'a item) => ('b list => 'a item)
  Abs_map     :: ('a item => 'b) => 'a item => 'b list
  null        :: 'a list => bool
  hd          :: 'a list => 'a
  tl,ttl      :: 'a list => 'a list
  set_of_list :: ('a list => 'a set)
  mem         :: ['a, 'a list] => bool                            (infixl 55)
  map         :: ('a=>'b) => ('a list => 'b list)
  "@"         :: ['a list, 'a list] => 'a list                    (infixr 65)
  filter      :: ['a => bool, 'a list] => 'a list

  (* list Enumeration *)

  "[]"        :: 'a list                              ("[]")
  "@list"     :: args => 'a list                      ("[(_)]")

  (* Special syntax for filter *)
  "@filter"   :: [idt, 'a list, bool] => 'a list      ("(1[_:_ ./ _])")

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

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

  "[x:xs . P]"  == "filter (%x.P) xs"

defs
  (* Defining the Concrete Constructors *)
  NIL_def       "NIL == In0(Numb(0))"
  CONS_def      "CONS M N == In1(M $ N)"

inductive "list(A)"
  intrs
    NIL_I  "NIL: list(A)"
    CONS_I "[| a: A;  M: list(A) |] ==> CONS a M : list(A)"

rules
  (* Faking a Type Definition ... *)
  Rep_list          "Rep_list(xs): list(range(Leaf))"
  Rep_list_inverse  "Abs_list(Rep_list(xs)) = xs"
  Abs_list_inverse  "M: list(range(Leaf)) ==> Rep_list(Abs_list(M)) = M"


defs
  (* Defining the Abstract Constructors *)
  Nil_def       "Nil == Abs_list(NIL)"
  Cons_def      "x#xs == Abs_list(CONS (Leaf x) (Rep_list xs))"

  List_case_def "List_case c d == Case (%x.c) (Split d)"

  (* list Recursion -- the trancl is Essential; see list.ML *)

  List_rec_def
   "List_rec M c d == wfrec (trancl pred_sexp)
                            (%g. List_case c (%x y. d x y (g y))) M"

  list_rec_def
   "list_rec l c d == 
   List_rec (Rep_list l) c (%x y r. d (inv Leaf x) (Abs_list y) r)"

  (* Generalized Map Functionals *)

  Rep_map_def "Rep_map f xs == list_rec xs NIL (%x l r. CONS (f x) r)"
  Abs_map_def "Abs_map g M == List_rec M Nil (%N L r. g(N)#r)"

  null_def      "null(xs)            == list_rec xs True (%x xs r.False)"
  hd_def        "hd(xs)              == list_rec xs (@x.True) (%x xs r.x)"
  tl_def        "tl(xs)              == list_rec xs (@xs.True) (%x xs r.xs)"
  (* a total version of tl: *)
  ttl_def       "ttl(xs)             == list_rec xs [] (%x xs r.xs)"

  set_of_list_def "set_of_list xs    == list_rec xs {} (%x l r. insert x r)"

  mem_def       "x mem xs            == 
                   list_rec xs False (%y ys r. if y=x then True else r)"
  map_def       "map f xs            == list_rec xs [] (%x l r. f(x)#r)"
  append_def    "xs@ys               == list_rec xs ys (%x l r. x#r)"
  filter_def    "filter P xs         == 
                  list_rec xs [] (%x xs r. if P(x) then x#r else r)"

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

end