Dear ProofPower Users,
I have a little observation which I would like to confirm. The problem
is as follows. I defined a new Z type NAME by
[NAME]
and then introduced a set ALPHABET to refer to the subsets of NAME by
ALPHABET == P NAME
At some point it showed to be necessary to construct fresh names for any
given alphabet, i.e. which are not in the alphabet. In that, I attempted
to prove the following theorem.
FORALL a : ALPHABET @ (EXISTS n : NAME @ n \not \in a)
I assumed it would be sufficient to restrict the set alphabet to finite
subsets of name, for instance using the following definition for
ALPHABET instead.
ALPHABET == F NAME
but I think this is not the case. For a new Z type, the defining axiom
we obtain is that the Z type constant (i.e. NAME) is equal to the
universe over a certain, newly introduced, HOL type (here z'NAME); a HOL
constant "Universe" is used to refer to the carrier set of a HOL type.
It guarantees that any x of the correct (HOL) type is an element of the
respective (type-instantiated) Universe, that x is not an element of the
empty set which I suppose excludes the case of an empty universe, and a
property regarding the Insert functions to reason about enumerated sets.
From this alone one cannot prove that the corresponding Universe is
infinite. The underlying HOL type of the new Z type, i.e. here "z'NAME
SET" I think defines a representation function that equates the type of
a set with functions over some new HOL type into BOOL, but again I think
this cannot be used to prove that "z'NAME SET" is infinite unless
"z'NAME" is (?!). In that, I suppose if we want a Z type to be infinite,
we need to explicitly state it, that is by an axiom along the lines of
EXISTS f : z'NAME -> z'NAME @ (OneOne f) /\ ¬ Onto f
(Similar to the infinity_axiom in the HOL theory init.)
Or alternatively (maybe more useful when doing the proofs in Z rather
than HOL) that there exists an injective Z function from N to the
carrier set of the new type. The only type in ProofPower-Z that
explicitly claims its infinite seems to be the type IND. However, there
seems no way to prove that for a newly introduce HOL type there is an
injective function from IND to the elements of that. Even further, it
seems not possible to prove that there are two distinct elements in a
newly introduced Z type. For example, the consistency axioms for
| x, y : NAME
-------------
| x =/= y
Appear not to be provable by the defining axiom for NAME, unless we
introduce some axiom stating properties of the cardinality of NAME.
If someone could confirm or refute the above that would be great.
As a side-issue, I realised the automatic proof support for finite sets
is not as well developed as that for (possibly infinite) sets. I proved
a couple of theorems that could be useful, for instance discharging that
every enumerated set is finite. It may be worth introducing a new Z
component proof context for finite sets.
Many Thanks,
Frank
--
Dr Dipl.-Inform. Frank Zeyda
Research Associate
High Integrity Systems Engineering Group
Department of Computer Science
University of York (UK)
Email: ze...@cs.york.ac.uk
Phone: 0044-(0)1904-433244
WWW: http://www-users.cs.york.ac.uk/~zeyda/
--
Dr Dipl.-Inform. Frank Zeyda
Research Associate
High Integrity Systems Engineering Group
Department of Computer Science
University of York (UK)
Email: ze...@cs.york.ac.uk
Phone: 0044-(0)1904-433244
WWW: http://www-users.cs.york.ac.uk/~zeyda/
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