src/HOL/Univ.thy
author clasohm
Fri Mar 03 12:02:25 1995 +0100 (1995-03-03)
changeset 923 ff1574a81019
child 972 e61b058d58d2
permissions -rw-r--r--
new version of HOL with curried function application
     1 (*  Title:      HOL/Univ.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1993  University of Cambridge
     5 
     6 Move LEAST to Nat.thy???  Could it be defined for all types 'a::ord?
     7 
     8 Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)
     9 
    10 Defines "Cartesian Product" and "Disjoint Sum" as set operations.
    11 Could <*> be generalized to a general summation (Sigma)?
    12 *)
    13 
    14 Univ = Arith + Sum +
    15 
    16 (** lists, trees will be sets of nodes **)
    17 
    18 subtype (Node)
    19   'a node = "{p. EX f x k. p = <f::nat=>nat, x::'a+nat> & f(k)=0}"
    20 
    21 types
    22   'a item = "'a node set"
    23 
    24 consts
    25   Least     :: "(nat=>bool) => nat"    (binder "LEAST " 10)
    26 
    27   apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
    28   Push      :: "[nat, nat=>nat] => (nat=>nat)"
    29 
    30   Push_Node :: "[nat, 'a node] => 'a node"
    31   ndepth    :: "'a node => nat"
    32 
    33   Atom      :: "('a+nat) => 'a item"
    34   Leaf      :: "'a => 'a item"
    35   Numb      :: "nat => 'a item"
    36   "$"       :: "['a item, 'a item]=> 'a item"   (infixr 60)
    37   In0,In1   :: "'a item => 'a item"
    38 
    39   ntrunc    :: "[nat, 'a item] => 'a item"
    40 
    41   "<*>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 80)
    42   "<+>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 70)
    43 
    44   Split  :: "[['a item, 'a item]=>'b, 'a item] => 'b"
    45   Case   :: "[['a item]=>'b, ['a item]=>'b, 'a item] => 'b"
    46 
    47   diag   :: "'a set => ('a * 'a)set"
    48   "<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
    49 \           => ('a item * 'a item)set" (infixr 80)
    50   "<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
    51 \           => ('a item * 'a item)set" (infixr 70)
    52 
    53 defs
    54 
    55   (*least number operator*)
    56   Least_def      "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
    57 
    58   Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
    59 
    60   (*crude "lists" of nats -- needed for the constructions*)
    61   apfst_def  "apfst == (%f. split(%x y. <f(x),y>))"
    62   Push_def   "Push == (%b h. nat_case (Suc b) h)"
    63 
    64   (** operations on S-expressions -- sets of nodes **)
    65 
    66   (*S-expression constructors*)
    67   Atom_def   "Atom == (%x. {Abs_Node(<%k.0, x>)})"
    68   Scons_def  "M$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"
    69 
    70   (*Leaf nodes, with arbitrary or nat labels*)
    71   Leaf_def   "Leaf == Atom o Inl"
    72   Numb_def   "Numb == Atom o Inr"
    73 
    74   (*Injections of the "disjoint sum"*)
    75   In0_def    "In0(M) == Numb(0) $ M"
    76   In1_def    "In1(M) == Numb(Suc(0)) $ M"
    77 
    78   (*the set of nodes with depth less than k*)
    79   ndepth_def "ndepth(n) == split (%f x. LEAST k. f(k)=0) (Rep_Node n)"
    80   ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
    81 
    82   (*products and sums for the "universe"*)
    83   uprod_def  "A<*>B == UN x:A. UN y:B. { (x$y) }"
    84   usum_def   "A<+>B == In0``A Un In1``B"
    85 
    86   (*the corresponding eliminators*)
    87   Split_def  "Split c M == @u. ? x y. M = x$y & u = c x y"
    88 
    89   Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) \
    90 \                              | (? y . M = In1(y) & u = d(y))"
    91 
    92 
    93   (** diagonal sets and equality for the "universe" **)
    94 
    95   diag_def   "diag(A) == UN x:A. {<x,x>}"
    96 
    97   dprod_def  "r<**>s == UN u:r. split (%x x'. \
    98 \                       UN v:s. split (%y y'. {<x$y,x'$y'>}) v) u"
    99 
   100   dsum_def   "r<++>s == (UN u:r. split (%x x'. {<In0(x),In0(x')>}) u) Un \
   101 \                       (UN v:s. split (%y y'. {<In1(y),In1(y')>}) v)"
   102 
   103 end