CS 641 Lecture -*- Outline -*-

* predicates and sets (7.1)

  The goal is to formalize predicates in HOL.

  Sets are identified with predicates,
  so this also formalizes set theory in HOL.

  Inference rules are special cases of the inference rules
  for complete Boolean lattices; hence have rules for general meets
  and joins.

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        ASCII NOTATION

In the following, we write

  ``S'' for a type (\Sigma)
  ``s'' for an element of S (\sigma)

  ``P(S)'' for predicate space over S

  ``\in'' for set membership
                       _
  ``\meet'' for meet (| |),
  ``\intersect'' for intersection,

  ``\join'' for join (|_|),
  ``\union'' for union,

  ``\subseteq'' for non-strict subset
                (predicate implication),
  ``\supseteq'' for nonstrict superset
                (predicate consequence)

  ``\bot'' for bottom element of lattice,
  ``\top'' for top element of lattice,
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** predicates and sets (7.1)

------------------------------------------
            PREDICATES

Def: Let S be a type.  Then
          ^
     P(S) = S -> Bool
is the space of *predicates* over S.

      PREDICATE-SET ISOMORPHISM

Def: Let p: P(S) be a predicate.
Then the *subset of S determined by p* is
        ^
    A_p = { s \in S | p.s <==> T }.

Def: let A \subset S be a set.
Then the *characteristic predicate* for A
is


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        ... p_A : S -> Bool
            such that p_A.s <==> s \in A

        Q: How should P(S) be ordered?
           point-wise, of course!

*** ordering and operations on predicates
------------------------------------------
          PREDICATE ORDERING

Def: Let p, q: P(S).  Then
                   ^
     p \subseteq Q = 


Corollary 7.1.  (P(S),\subseteq) is a
complete Boolean and atomic lattice.

------------------------------------------
         ... (\forall s : s \in S : p.s ==> q.s)

         Q: Why does this corollary hold?
            because Boolean is a complete Boolean and atomic lattice.

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       OPERATIONS ON PREDICATES

For all s: S, p, q: P(S), define

   false.s == F       (\bot of predicates)

    true.s == T       (\top of predicates)

(p \intersect q).s   (\meet of predicates)
      == p.s /\ q.s       

(p \union q).s       (\join of predicates)
      == p.s \/ q.s       

  (!p).s == !(p.s)       (! of predicates)

(p ==> q).s            (==> of predicates)
      == p.s ==> q.s

(p <==> q).s          (<==> of predicates)
      == (p.s <==> q.s)
------------------------------------------

        Q: What's the relation between \subseteq and ==> on predicates?
           p \subseteq q <==> (\forall s :: (p ==> s).s)

        Q: What's the lattice ordering on predicates?
           \subseteq

        Q: What's the relation between == and <==> on predicates?
           p == q <==> (\forall s :: (p <==> s).s)

*** sets as predicates

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                CORRESPONDENCE 

 characteristic       | corresponding
 predicates           | sets
 =====================|===================
 p_A \subseteq p_A    | A \subset B
                      | {}
 true                 |
 p_A \intersect p_B   |
                      | A \union B
 !(p_A)               |
                      | { s: S | t }
                      | s \in A
------------------------------------------

        fill this in       

*** properties of predicates

    Get all the properties of complete, atomic Boolean lattices
    for both predicates and sets.

    I.e., have the translation of Figure 6.1.

** images and indexed sets
*** images and preimages
------------------------------------------
         IMAGES AND PREIMAGES

Def. Let f: S -> G.  Let A \subseteq S.
Then the *image of A under f* is
        ^
 im.f.A =                          (image)
  {y: G | (\exists x : x \in A : y = f.x)}

Overloaded notation: write f.A for im.f.A.

Def. Let f: S -> G.  Then
 the *range of f* is
       ^
 ran.f = f.S                       (range)

Def. Let f: S -> G.  Let B \subseteq G.
Then the *preimage of f* is
           ^
 preim.f.B = { x | f.x \in B }  (preimage)

                 -1
Notation: write f  .B for preim.f.B
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        Q: Is im.f : P(S) -> P(G) monotonic?  strict?  a join homomorphism?
           yes
        Q: Is preim.f : P(G) -> P(S) monotonic?  strict?  a join homomorphism?
           yes, and more, it's also a meet-homomorphism, and a
           negation homomorphism.

        See table 7.1

*** indexed sets
------------------------------------------
       INDEXED SETS

Def.  Let a: S -> G and I \subset S,
and let t be a term of type G.
Then                   ^
     { t | i \in I } = im.(\i . t).I

Example,

  Let a.0 = Nat
      a.i = Nat -> a.(i-1) for i > 0

  Then
      { a.i | i \in Nat }
       == im.a.Nat



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       ... == { Nat,
                Nat -> Nat,
                Nat -> Nat -> Nat,
                 ... }