src/HOL/Datatype_Universe.thy
author oheimb
Wed Jan 31 10:15:55 2001 +0100 (2001-01-31)
changeset 11008 f7333f055ef6
parent 10832 e33b47e4246d
child 11451 8abfb4f7bd02
permissions -rw-r--r--
improved theory reference in comment
     1 (*  Title:      HOL/Datatype_Universe.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Declares the type ('a, 'b) node, a subtype of (nat=>'b+nat) * ('a+nat)
     7 
     8 Defines "Cartesian Product" and "Disjoint Sum" as set operations.
     9 Could <*> be generalized to a general summation (Sigma)?
    10 *)
    11 
    12 Datatype_Universe = NatArith + Sum_Type +
    13 
    14 
    15 (** lists, trees will be sets of nodes **)
    16 
    17 typedef (Node)
    18   ('a, 'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
    19 
    20 types
    21   'a item = ('a, unit) node set
    22   ('a, 'b) dtree = ('a, 'b) node set
    23 
    24 consts
    25   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
    26   Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
    27 
    28   Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
    29   ndepth    :: ('a, 'b) node => nat
    30 
    31   Atom      :: "('a + nat) => ('a, 'b) dtree"
    32   Leaf      :: 'a => ('a, 'b) dtree
    33   Numb      :: nat => ('a, 'b) dtree
    34   Scons     :: [('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree
    35   In0,In1   :: ('a, 'b) dtree => ('a, 'b) dtree
    36 
    37   Lim       :: ('b => ('a, 'b) dtree) => ('a, 'b) dtree
    38   Funs      :: "'u set => ('t => 'u) set"
    39 
    40   ntrunc    :: [nat, ('a, 'b) dtree] => ('a, 'b) dtree
    41 
    42   uprod     :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
    43   usum      :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
    44 
    45   Split     :: [[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
    46   Case      :: [[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
    47 
    48   dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
    49                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    50   dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
    51                 => (('a, 'b) dtree * ('a, 'b) dtree)set"
    52 
    53 
    54 defs
    55 
    56   Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    57 
    58   (*crude "lists" of nats -- needed for the constructions*)
    59   apfst_def  "apfst == (%f (x,y). (f(x),y))"
    60   Push_def   "Push == (%b h. nat_case b h)"
    61 
    62   (** operations on S-expressions -- sets of nodes **)
    63 
    64   (*S-expression constructors*)
    65   Atom_def   "Atom == (%x. {Abs_Node((%k. Inr 0, x))})"
    66   Scons_def  "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr 2) ` N)"
    67 
    68   (*Leaf nodes, with arbitrary or nat labels*)
    69   Leaf_def   "Leaf == Atom o Inl"
    70   Numb_def   "Numb == Atom o Inr"
    71 
    72   (*Injections of the "disjoint sum"*)
    73   In0_def    "In0(M) == Scons (Numb 0) M"
    74   In1_def    "In1(M) == Scons (Numb 1) M"
    75 
    76   (*Function spaces*)
    77   Lim_def "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}"
    78   Funs_def "Funs S == {f. range f <= S}"
    79 
    80   (*the set of nodes with depth less than k*)
    81   ndepth_def "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)"
    82   ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
    83 
    84   (*products and sums for the "universe"*)
    85   uprod_def  "uprod A B == UN x:A. UN y:B. { Scons x y }"
    86   usum_def   "usum A B == In0`A Un In1`B"
    87 
    88   (*the corresponding eliminators*)
    89   Split_def  "Split c M == @u. ? x y. M = Scons x y & u = c x y"
    90 
    91   Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) 
    92                                | (? y . M = In1(y) & u = d(y))"
    93 
    94 
    95   (** equality for the "universe" **)
    96 
    97   dprod_def  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
    98 
    99   dsum_def   "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un 
   100                           (UN (y,y'):s. {(In1(y),In1(y'))})"
   101 
   102 end