src/HOL/Univ.thy

HOL.thy:

"@" is no longer introduced as a "binder" but has its own explicit
translation rule "@x.b" == "Eps(%x.b)". If x is a proper pattern, further
translation rules for abstractions with patterns take care of the rest. This
is very modular and avoids problems with "binders" such as "!" mentioned
below.

let now allows pttrn (let (x,y) = t in u) instead of just idt (let x = t in u)

Set.thy:

UN, INT, ALL, EX, etc all use "pttrn" instead of idt. Same change as for "@"
above, except that "@" was a "binder" originally.

Prod.thy:

Added new syntax for pttrn which allows arbitrarily nested tuples. Two
translation rules take care of %pttrn. Unfortunately they cannot be
reversed. Hence a little ML-code is used as well.

Note that now "! (x,y). ..." is syntactically valid but leads to a
translation error. This is because "!" is introduced as a "binder" which
means that its translation into lambda-terms is not done by a rewrite rule
(aka macro) but by some fixed ML-code which comes after the rewriting stage
and does not know how to handle patterns. This looks like a minor blemish
since patterns in unbounded quantifiers are not that useful (well, except
maybe in unique existence ...). Ideally, there should be two syntactic
categories:

idts, as we know and love it, which does not admit patterns.
patterns, which is what idts has become now.

There is one more point where patterns are now allowed but don't make sense:

{e | idts . P}

where idts is the list of local variables.

Univ.thy: converted the defs for <++> and <**> into pattern form. It worked
perfectly.

(*  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) == (%(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 (x,x'):r. UN (y,y'):s. {(x$y,x'$y')}"

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

end