src/HOL/Univ.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Univ.thy	Fri Mar 03 12:02:25 1995 +0100
@@ -0,0 +1,103 @@
+(*  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