src/ZF/QPair.thy

UNIV now a constant; UNION1, INTER1 now translations and no longer have
separate rules for themselves

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

Quine-inspired ordered pairs and disjoint sums, for non-well-founded data
structures in ZF.  Does not precisely follow Quine's construction.  Thanks
to Thomas Forster for suggesting this approach!

W. V. Quine, On Ordered Pairs and Relations, in Selected Logic Papers,
1966.
*)

QPair = Sum +

global

consts
  QPair     :: [i, i] => i                      ("<(_;/ _)>")
  qfst,qsnd :: i => i
  qsplit    :: [[i, i] => 'a, i] => 'a::logic  (*for pattern-matching*)
  qconverse :: i => i
  QSigma    :: [i, i => i] => i

  "<+>"     :: [i,i]=>i                         (infixr 65)
  QInl,QInr :: i=>i
  qcase     :: [i=>i, i=>i, i]=>i

syntax
  "@QSUM"   :: [idt, i, i] => i               ("(3QSUM _:_./ _)" 10)
  "<*>"     :: [i, i] => i                      (infixr 80)

translations
  "QSUM x:A. B"  => "QSigma(A, %x. B)"
  "A <*> B"      => "QSigma(A, _K(B))"

local

defs
  QPair_def       "<a;b> == a+b"
  qfst_def        "qfst(p) == THE a. EX b. p=<a;b>"
  qsnd_def        "qsnd(p) == THE b. EX a. p=<a;b>"
  qsplit_def      "qsplit(c,p) == c(qfst(p), qsnd(p))"

  qconverse_def   "qconverse(r) == {z. w:r, EX x y. w=<x;y> & z=<y;x>}"
  QSigma_def      "QSigma(A,B)  ==  UN x:A. UN y:B(x). {<x;y>}"

  qsum_def        "A <+> B      == ({0} <*> A) Un ({1} <*> B)"
  QInl_def        "QInl(a)      == <0;a>"
  QInr_def        "QInr(b)      == <1;b>"
  qcase_def       "qcase(c,d)   == qsplit(%y z. cond(y, d(z), c(z)))"
end

ML

val print_translation =
  [("QSigma", dependent_tr' ("@QSUM", "op <*>"))];