src/HOL/Univ.thy
 author clasohm Mon Feb 05 21:27:16 1996 +0100 (1996-02-05) changeset 1475 7f5a4cd08209 parent 1396 48bcde67391b child 1531 e5eb247ad13c permissions -rw-r--r--
expanded tabs; renamed subtype to typedef;
```     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 typedef (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 (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) == (%(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 (x,x'):r. UN (y,y'):s. {(x\$y,x'\$y')}"
```
```    98
```
```    99   dsum_def   "r<++>s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un
```
```   100                        (UN (y,y'):s. {(In1(y),In1(y'))})"
```
```   101
```
```   102 end
```