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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Declares the type ('a, 'b) node, a subtype of (nat=>'b+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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setup arith_setup
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(** lists, trees will be sets of nodes **)
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typedef (Node)
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  ('a, 'b) node = "{p. EX f x k. p = (f::nat=>'b+nat, x::'a+nat) & f k = Inr 0}"
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types
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  'a item = ('a, unit) node set
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  ('a, 'b) dtree = ('a, 'b) node set
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consts
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  apfst     :: "['a=>'c, 'a*'b] => 'c*'b"
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  Push      :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))"
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  Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node"
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  ndepth    :: ('a, 'b) node => nat
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  Atom      :: "('a + nat) => ('a, 'b) dtree"
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  Leaf      :: 'a => ('a, 'b) dtree
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  Numb      :: nat => ('a, 'b) dtree
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  Scons     :: [('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree
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  In0,In1   :: ('a, 'b) dtree => ('a, 'b) dtree
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  Lim       :: ('b => ('a, 'b) dtree) => ('a, 'b) dtree
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  Funs      :: "'u set => ('t => 'u) set"
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  ntrunc    :: [nat, ('a, 'b) dtree] => ('a, 'b) dtree
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  uprod     :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
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  usum      :: [('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set
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  Split     :: [[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
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  Case      :: [[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c
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  dprod     :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
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                => (('a, 'b) dtree * ('a, 'b) dtree)set"
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  dsum      :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] 
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                => (('a, 'b) dtree * ('a, 'b) dtree)set"
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defs
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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 (x,y). (f(x),y))"
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  Push_def   "Push == (%b h. nat_case 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. Inr 0, x))})"
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  Scons_def  "Scons M N == (Push_Node (Inr 1) `` M) Un (Push_Node (Inr 2) `` 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) == Scons (Numb 0) M"
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  In1_def    "In1(M) == Scons (Numb 1) M"
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  (*Function spaces*)
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  Lim_def "Lim f == Union {z. ? x. z = Push_Node (Inl x) `` (f x)}"
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  Funs_def "Funs S == {f. range f <= S}"
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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 = Inr 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  "uprod A B == UN x:A. UN y:B. { Scons x y }"
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  usum_def   "usum 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 = Scons 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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  (** equality for the "universe" **)
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  dprod_def  "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}"
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  dsum_def   "dsum 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