Univ.thy

small updates because datatype list is now used. In particular Nil -> []

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

Move LEAST to Nat.thy???  Could it be defined for all types 'a::ord?

Declares the type 'a node, a subtype of (nat=>nat) * ('a+nat)

Defines "Cartesian Product" and "Disjoint Sum" as set operations.
Could <*> be generalized to a general summation (Sigma)?
*)

Univ = Arith + Sum +

(** lists, trees will be sets of nodes **)

subtype (Node)
  'a node = "{p. EX f x k. p = <f::nat=>nat, x::'a+nat> & f(k)=0}"

types
  'a item = "'a node set"

consts
  Least     :: "(nat=>bool) => nat"    (binder "LEAST " 10)

  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
  Push      :: "[nat, nat=>nat] => (nat=>nat)"

  Push_Node :: "[nat, 'a node] => 'a node"
  ndepth    :: "'a node => nat"

  Atom      :: "('a+nat) => 'a item"
  Leaf      :: "'a => 'a item"
  Numb      :: "nat => 'a item"
  "$"       :: "['a item, 'a item]=> 'a item"   (infixr 60)
  In0,In1   :: "'a item => 'a item"

  ntrunc    :: "[nat, 'a item] => 'a item"

  "<*>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 80)
  "<+>"  :: "['a item set, 'a item set]=> 'a item set" (infixr 70)

  Split  :: "[['a item, 'a item]=>'b, 'a item] => 'b"
  Case   :: "[['a item]=>'b, ['a item]=>'b, 'a item] => 'b"

  diag   :: "'a set => ('a * 'a)set"
  "<**>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
\           => ('a item * 'a item)set" (infixr 80)
  "<++>" :: "[('a item * 'a item)set, ('a item * 'a item)set] \
\           => ('a item * 'a item)set" (infixr 70)

defs

  (*least number operator*)
  Least_def        "Least(P) == @k. P(k) & (ALL j. j<k --> ~P(j))"

  Push_Node_def    "Push_Node == (%n x. Abs_Node (apfst(Push(n),Rep_Node(x))))"

  (*crude "lists" of nats -- needed for the constructions*)
  apfst_def  "apfst == (%f. split(%x y. <f(x),y>))"
  Push_def   "Push == (%b h. nat_case(Suc(b),h))"

  (** operations on S-expressions -- sets of nodes **)

  (*S-expression constructors*)
  Atom_def   "Atom == (%x. {Abs_Node(<%k.0, x>)})"
  Scons_def  "M$N == (Push_Node(0) `` M) Un (Push_Node(Suc(0)) `` N)"

  (*Leaf nodes, with arbitrary or nat labels*)
  Leaf_def   "Leaf == Atom o Inl"
  Numb_def   "Numb == Atom o Inr"

  (*Injections of the "disjoint sum"*)
  In0_def    "In0(M) == Numb(0) $ M"
  In1_def    "In1(M) == Numb(Suc(0)) $ M"

  (*the set of nodes with depth less than k*)
  ndepth_def "ndepth(n) == split(%f x. LEAST k. f(k)=0, Rep_Node(n))"
  ntrunc_def "ntrunc(k,N) == {n. n:N & ndepth(n)<k}"

  (*products and sums for the "universe"*)
  uprod_def  "A<*>B == UN x:A. UN y:B. { (x$y) }"
  usum_def   "A<+>B == In0``A Un In1``B"

  (*the corresponding eliminators*)
  Split_def  "Split(c,M) == @u. ? x y. M = x$y & u = c(x,y)"

  Case_def   "Case(c,d,M) == @u.  (? x . M = In0(x) & u = c(x)) \
\                               | (? y . M = In1(y) & u = d(y))"


  (** diagonal sets and equality for the "universe" **)

  diag_def   "diag(A) == UN x:A. {<x,x>}"

  dprod_def  "r<**>s == UN u:r. split(%x x'. \
\                       UN v:s. split(%y y'. {<x$y,x'$y'>}, v), u)"

  dsum_def   "r<++>s == (UN u:r. split(%x x'. {<In0(x),In0(x')>}, u)) Un \
\                       (UN v:s. split(%y y'. {<In1(y),In1(y')>}, v))"

end