src/HOL/Univ.thy
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(*  Title:      HOL/Univ.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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Move LEAST to Nat.thy???  Could it be defined for all types 'a::ord?
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Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)
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Defines "Cartesian Product" and "Disjoint Sum" as set operations.
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Could <*> be generalized to a general summation (Sigma)?
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*)
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Univ = Arith + Sum +
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(** lists, trees will be sets of nodes **)
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subtype (Node)
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  'a node = "{p. EX f x k. p = (f::nat=>nat, x::'a+nat) & f(k)=0}"
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types
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  'a item = "'a node set"
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consts
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  Least     :: "(nat=>bool) => nat"    (binder "LEAST " 10)
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  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
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  Push      :: "[nat, nat=>nat] => (nat=>nat)"
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  Push_Node :: "[nat, 'a node] => 'a node"
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  ndepth    :: "'a node => nat"
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  Atom      :: "('a+nat) => 'a item"
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  Leaf      :: "'a => 'a item"
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  Numb      :: "nat => 'a item"
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  "$"       :: "['a item, 'a item]=> 'a item"   (infixr 60)
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  In0,In1   :: "'a item => 'a item"
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  ntrunc    :: "[nat, 'a item] => 'a item"
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  "<*>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 80)
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  "<+>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 70)
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  Split  :: "[['a item, 'a item]=>'b, 'a item] => 'b"
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  Case   :: "[['a item]=>'b, ['a item]=>'b, 'a item] => 'b"
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  diag   :: "'a set => ('a * 'a)set"
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  "<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set] 
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           => ('a item * 'a item)set" (infixr 80)
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  "<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set] 
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           => ('a item * 'a item)set" (infixr 70)
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defs
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  (*least number operator*)
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  Least_def      "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"
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  Push_Node_def  "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))"
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  (*crude "lists" of nats -- needed for the constructions*)
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  apfst_def  "apfst == (%f. split(%x y. (f(x),y)))"
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  Push_def   "Push == (%b h. nat_case (Suc b) h)"
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  (** operations on S-expressions -- sets of nodes **)
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  (*S-expression constructors*)
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  Atom_def   "Atom == (%x. {Abs_Node((%k.0, x))})"
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  Scons_def  "M$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"
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  (*Leaf nodes, with arbitrary or nat labels*)
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  Leaf_def   "Leaf == Atom o Inl"
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  Numb_def   "Numb == Atom o Inr"
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  (*Injections of the "disjoint sum"*)
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  In0_def    "In0(M) == Numb(0) $ M"
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  In1_def    "In1(M) == Numb(Suc(0)) $ M"
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  (*the set of nodes with depth less than k*)
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  ndepth_def "ndepth(n) == (%(f,x). LEAST k. f(k)=0) (Rep_Node n)"
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  ntrunc_def "ntrunc k N == {n. n:N & ndepth(n)<k}"
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  (*products and sums for the "universe"*)
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  uprod_def  "A<*>B == UN x:A. UN y:B. { (x$y) }"
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  usum_def   "A<+>B == In0``A Un In1``B"
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  (*the corresponding eliminators*)
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  Split_def  "Split c M == @u. ? x y. M = x$y & u = c x y"
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  Case_def   "Case c d M == @u.  (? x . M = In0(x) & u = c(x)) 
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                              | (? y . M = In1(y) & u = d(y))"
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  (** diagonal sets and equality for the "universe" **)
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  diag_def   "diag(A) == UN x:A. {(x,x)}"
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  dprod_def  "r<**>s == UN (x,x'):r. UN (y,y'):s. {(x$y,x'$y')}"
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  dsum_def   "r<++>s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un 
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                       (UN (y,y'):s. {(In1(y),In1(y'))})"
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end