Nabble - Mizar

New Mizar version/vocabularies

2009-11-24T05:28:26Z

Dear All,
as Artur Kornilowicz previously wrote on November 10, the revision
acting mainly on MML vocabularies took place, and it can be rather
painful for active authors, especially for those who continue
their work on older articles.
After the new version (7.11.04) of the Mizar system is installed,
you may encounter an internal error during the accomodation pass.
Nothing really to be afraid of, but probably old, incompatible
accomodation results are still stored somewhere in "text" directory.
Usually removing all files (OK, .miz should remain ;)) from
"text" directory will heal the situation. Or, alternatively, just
force the accomodation.
As for the accomodator errors, you will probably get some *801
errors in the "vocabularies" section, marked indentifiers should be
deleted. In parallel, *830 in "notations" section means that marked
identifier is completely unused (usually ZFMISC_1, NUMBERS etc.),
so adding this to "vocabularies" will help.
After this, "findvoc" utility for symbols which are not properly
recognized, will help.
Note, that any article send to the Library Committee for inclusion
in the MML should be verified by the newest official version.
Regards,
Adam Grabowski
Library Committee of the Association of Mizar Users

New Mizar articles

2009-11-24T04:47:04Z

Dear All,
together with the newest official version of the Mizar system
(7.11.04, MML version 4.130.1076) the following new articles are
available:

1060. TOPDIM_1
Small Inductive Dimension of Topological Spaces
by Karol P\c{a}k
Received June 29, 2009
1061. GROUP_11
On Rough Subgroup of a Group
by Xiquan Liang and Dailu Li
Received August 7, 2009
1062. TOPDIM_2
Small Inductive Dimension of Topological Spaces, Part II
by Karol P\c{a}k
Received August 7, 2009
1063. DILWORTH
Dilworth's Decomposition Theorem for Posets
by Piotr Rudnicki
Received September 17, 2009
1064. INTEGR1C
Complex Integral
by Masahiko Yamazaki, Hiroshi Yamazaki, Katsumi Wasaki
and Yasunari Shidama
Received October 10, 2009
1065. INTERVA1
On the Lattice of Intervals and Rough Sets
by Adam Grabowski and Magdalena Jastrz\c{e}bska
Received October 10, 2009
1066. FUNCT_9
Basic Properties of Periodic Functions
by Bo Li, Yanhong Men, Dailu Li and Xiquan Liang
Received October 10, 2009
1067. EUCLID_8
Vector Function and its Differentiation Formulas in 3-dimensional
Euclidean Spaces
by Xiquan Liang, Piqing Zhao and Ou Bai
Received October 10, 2009
1068. ORDINAL5
Epsilon Numbers and Cantor Normal Form
by Grzegorz Bancerek
Received October 20, 2009
1069. C0SP2
Banach Algebra of Continuous Functionals
by Kanazashi Katuhiko, Noboru Endou and Yasunari Shidama
Received October 20, 2009
1070. ALGSTR_4
Free Magmas
by Marco Riccardi
Received October 20, 2009
1071. INTEGR12
Integrability Formulas -- Part I
by Bo Li and Na Ma
Received November 7, 2009
1072. PDIFF_4
Partial Differentiation of Real Ternary Functions
by Takao Inou\'e, Bing Xie and Xiquan Liang
Received November 7, 2009
1073. POSET_1
Fix-point Theorem for Continuous Functions on Chain-complete Posets
by Kazuhisa Ishida and Yasunari Shidama
Received November 10, 2009
1074. GRNILP_1
Nilpotent Groups
by Dailu Li, Xiquan Liang and Yanhong Men
Received November 10, 2009
1075. DIFF_3
Difference and Difference Quotient -- Part III
by Xiquan Liang and Ling Tang
Received November 17, 2009
1076. ABCMIZ_A
A Model of Mizar Concepts -- Unification
by Grzegorz Bancerek
Received November 20, 2009

Regards,
Adam Grabowski
Library Committee of the Association of Mizar Users

Re: new vocabularies

2009-11-10T13:45:20Z

Hello,

I am very sorry for my previous email. It was not finished.

I've mentioned symbols [0] and |( since during the revision they had
caused some troubles.

[0] is widely used in induction schemes and when, for example, P[0] occurs
in an article where vocabulary PBOOLE (containing [0]) is imported Mizar
scanner divides P[0] into two valid tokens P and [0] and reports an error.
A solution is to write P[0] as P[ 0 ] forceing scanner to split [0] into
more tokens.

The same goes with |(, when |( is used like (TOP-REAL n)|(A\/B).

Best regards
Artur

On Tue, 10 Nov 2009, Artur Kornilowicz wrote:

> Dear Mizar Users,
>
> recently a quite big change of MML has been done - namely each symbol
> introduced in an _old_ vocabulary has been moved to the _new_ vocabulary
> named as the .miz file in which the symbol was used first.
>
> As a result of the revision vocabulary HIDDEN consists of only 3 symbols:
>
> R<>
> Rin
> Vstrict
>
>
> P[0]
> |(
>
>
> We are sorry for inconvenience. In a case of any problems with adjusting your
> articles please ask Mizar User Service (mus@...) or me directly
> about help.
>
> Artur

new vocabularies

2009-11-10T08:45:43Z

Dear Mizar Users,

recently a quite big change of MML has been done - namely each symbol
introduced in an _old_ vocabulary has been moved to the _new_ vocabulary
named as the .miz file in which the symbol was used first.

As a result of the revision vocabulary HIDDEN consists of only 3 symbols:

R<>
Rin
Vstrict

P[0]
|(

We are sorry for inconvenience. In a case of any problems with adjusting
your articles please ask Mizar User Service (mus@...) or me
directly about help.

Artur

Re: I am trying to figure out...

2009-10-30T07:37:08Z

Dear Andrzej,

>The empty set has quite precise meaning. And what is
>
> the empty non empty set?

Ah yes, this new "the" thing won't work with empty types.
And the same holds for permissive definitions (that's the
name right, when there's an assumption in a func definition?)
If you would define

definition
assume contradiction;
func foo -> A means ...
end;

then A has better be a non empty type, right? So one should
have Mizar requiring something like a "cluster" for _those_
two cases. But I really would like the non-emptiness of a
type to be a _registration_ then, instead of it being part
of the definition.

>>In Coq I can do
>
>Why we should mimic Coq?

I just was trying to argue that it's unproblematic.
Because it's unproblematic there.

>You can always use a permissive definition.

What does this exactly mean in this case? To do

definition
let X be non empty set;
mode Element of X means ...
end;

or to do

definition
let X be set;
assume X <> {};
mode Element of X means ...
end;

?

However in both cases you won't be able to prove that

for x,X being set holds (x in X iff x is Element of X)

right? So then it still is _wrong._

>So, actually it is a technicality.

You can work around the problems with the non-emptiness of
types, sure. But you can do arithmetic in Roman numerals
too. My claim is that the non-emptiness of types in Mizar
is like Roman numerals for arithmetic. III + IV = VII!
What's the problem?

>If we allow for types with the empty denotation, then
>we should additionally check if all free variables have
>non empty types. Quite unnnatural, isn't it?

So if you make CHECKER first replace

for x being A holds ...

by

for x st x is A holds ...

(and similar for ex) in the cases that it cannot establish
the non-emptiness of A by way of a cluster, before running
its usual algorithms, _that_ wouldn't be so big a change,
maybe? So then you wouldn't need to change your algorithms.
And maybe it still might behave reasonably?

>In general we cannot infer
>
> for x being theta holds alfa
> -------------------------------
> alfa
>
>when x does not occur in alfa
>
>I guess that in Coq such inferences are not allowed in general.

Exactly. In Coq need to exhibit an element of theta
for that. And "the theta" doesn't exist :-)

Freek

Re: sorry to bother again :(

2009-10-26T23:18:56Z

Hi Adem,

first, please check is Mizar understands that

1 is Element of NAT

in your environment.

setA consist of pairs of such elements.

I guess you need

constructors SUBSET_1, NUMBERS;
requirements SUBSET, NUMERALS;

in your environment.

If it does not work, feel free to ask again.

Best,
Artur

On Mon, 26 Oct 2009, Ozyavas, Adem wrote:

> Dear All,
>
> Sorry to bother again with some details. I have the following introduction and question.
>
> Let A be the following set (it is not exactly the Mizar syntax:)):
>
> ::starts here
>
> defpred P[Element of NAT,Element of NAT] means ...
>
> setA = {[a,b] where a is Element of NAT, b is Element of NAT: P[a,b]}
>
> theorem setA is non empty
> proof
> ex a,b being Element of NAT st [a,b] in setA
> proof
> take 1 , 2 ;
> P[1,2] proof ... end;
> hence [1,2] in setA;
> ::> #4
> end;
> then consider a,b being Element of NAT such that L1: [a,b] in setA;
> thus thesis by L1; ::BOOLE is in the requirements
> end;
>
> ::ends here
>
>
> After proving that property P holds for particular objects (1 and 2 in this case), that is, P[1,2];
> what other conditions Mizar expects for the "[1,2] in setA" to hold?
>
> I have found the solution for finite sets in the ENUMSET1.miz article but not for this case where
> set's elements are those that satisfy a property.
>
> Thank you again and sorry my troubles...
>
> Adem Ozyavas
> Texas Tech University

==========================================================================

Artur Kornilowicz e-mail: arturk@...

Dept. of Programming and Formal Methods http://math.uwb.edu.pl/~arturk/
Institute of Computer Science tel. +48 (85) 745-7662
University of Bialystok fax. +48 (85) 745-7662
Sosnowa 64, 15-887 Bialystok, Poland

sorry to bother again :(

2009-10-26T20:20:18Z

Dear All,

Sorry to bother again with some details. I have the following introduction and question.

Let A be the following set (it is not exactly the Mizar syntax:)):

::starts here

defpred P[Element of NAT,Element of NAT] means ...

setA = {[a,b] where a is Element of NAT, b is Element of NAT: P[a,b]}

theorem setA is non empty
proof
ex a,b being Element of NAT st [a,b] in setA
proof
take 1 , 2 ;
P[1,2] proof ... end;
hence [1,2] in setA;
::> #4
end;
then consider a,b being Element of NAT such that L1: [a,b] in setA;
thus thesis by L1; ::BOOLE is in the requirements
end;

::ends here

After proving that property P holds for particular objects (1 and 2 in this case), that is, P[1,2];
what other conditions Mizar expects for the "[1,2] in setA" to hold?

I have found the solution for finite sets in the ENUMSET1.miz article but not for this case where
set's elements are those that satisfy a property.

Thank you again and sorry my troubles...

Adem Ozyavas
Texas Tech University

Re: I am trying to figure out...

2009-10-26T00:34:05Z

Dear Andrzej, Freek,

2009/10/26 <trybulec@...>

> Cytowanie freek <freek@...>:
> We've had this discussion before, I'm sure. My excuses
>> for being triggered.
>>
>
>
> No problem, we get better understanding the different approaches. I
> appreciate it.

This discussion comes every two years and brings little new arguments.
If you want to progress, you should create a wiki page summarizing the
discussion of the
topic, and everyone should check first if his argument has already
been answered or not.
Best,
Josef

Fwd: Re: Radon-Nikodym theorem

2009-10-25T16:29:11Z

I am not au courant, anybody knows something about it?

Regards,
Andrzej Trybulec

----- Przekazana wiadomość od freek@... -----
Data: Sun, 25 Oct 2009 21:29:06 +0100
Od: freek <freek@...>
Odpowiedz-Do:freek <freek@...>
Temat: Re: Radon-Nikodym theorem
Do: William Faris <faris@...>

Dear Bill Faris,

> Is there a proof in one of the current systems of the
> Radon-Nikodym theorem?

Not that I know of, but that doesn't mean _too_ much.

(I'll CC this answer to John Harrison, Andrzej Trybulec
and Bas Spitters, in case one of them _does_ know. I saw
that it's in Bas' PhD thesis, and I know Bas is working on
formalizing integration in Coq: so maybe he already has it.)

Just to satisfy my curiosity: how dificult is a textbook
proof of this theorem, on top of the basic theory about
Lebesgue integration? About one page?

Freek

----- Koniec przekazanej wiadomości -----

Re: I am trying to figure out...

2009-10-25T16:24:39Z

Cytowanie freek <freek@...>:
>> Mizar does not allow for meaningless types like
>>
>> an empty non empty set
>> an element of the empty set
>> a function from a non empty set into the empty set
>
> I disagree that they're meaningless! They're just empty.
A rose by any other name ...
> Is the empty _set_ a meaningless _set?_ No, of course it
> isn't. And likewise an empty type is not a meaningless type.

The empty set has quite precise meaning. And what is

the empty non empty set?
>
> In Coq I can do

Why we should mimic Coq?

>>
>> x is element of A
>>
>> and x in A?
>
> For the same reasons that one now has types in Mizar:
>
> 1. to have the system automatically infer that certain
> predicates hold (using the types in definitions
> and registrations)
>
> 2. to disambiguate overloaded syntax
>

OK It is a valid argument.

>> However we need arguments to change it, not to keep it
>> as it is. I do not claim that it is not possible that we
>> eventually find such arguments.
>
> For me the argument is that the types can have much more
> natural defintions. Now, often when I want to define a
> type, Mizar won't allow me to write what should be there,
> and then I just give up.
>

You can always use a permissive definition. Like we do with

Element of ...
or
Function of ..., ...

So, actually it is a technicality.

>
>> The problem is that it is difficult to anticipate the
>> consequences of changing the approach.
>

I ment something like this:

In checker to get the contradiction between a universal formula we
remove quatifiers and substitute for bound variables free variables.

Than we try to find a substitution for free variables that contradicts
the other premises.

If we allow for types with the empty denotation, then we should
additionally check if all free variables have non empty types. Quite
unnnatural, isn't it?

In general we cannot infer

for x being theta holds alfa
-------------------------------
alfa

when x does not occur in alfa

I guess that in Coq such inferences are not allowed in general.

It means that we cannot remove fictitious quantifiers.

> We've had this discussion before, I'm sure. My excuses
> for being triggered.

No problem, we get better understanding the different approaches. I
appreciate it.

Regards,
Andrzej

Re: I am trying to figure out...

2009-10-25T13:13:02Z

Hi Andrzej,

>Mizar does not allow for meaningless types like
>
> an empty non empty set
> an element of the empty set
> a function from a non empty set into the empty set

I disagree that they're meaningless! They're just empty.
Is the empty _set_ a meaningless _set?_ No, of course it
isn't. And likewise an empty type is not a meaningless type.

In Coq I can do

Coq < Check unit->Empty_set.
unit -> Empty_set
: Set

Coq <

and there is no problem there at all. ("Empty_set" is an
empty type, and "unit" is a type with exactly one inhabitant,
called "tt":

Coq < Check tt.
tt
: unit

Coq <

.)

>If a meaningless type \theta is allowed that
>
> for x being \theta holds ...
>
>is understood as
>
> for x st x is \theta holds ...
>
>so actulally the type is eliminated by a predicate and the
>whole type system becomes syntactic sugar. And why we need
>two predicates
>
> x is element of A
>
>and x in A?

For the same reasons that one now has types in Mizar:

1. to have the system automatically infer that certain
predicates hold (using the types in definitions
and registrations)

2. to disambiguate overloaded syntax

>However we need arguments to change it, not to keep it
>as it is. I do not claim that it is not possible that we
>eventually find such arguments.

For me the argument is that the types can have much more
natural defintions. Now, often when I want to define a
type, Mizar won't allow me to write what should be there,
and then I just give up.

In the HOL systems, one doesn't get empty types either,
but there one doesn't have dependent types, so there it
doesn't hurt. But Mizar, rightly so, _has_ dependent types,
and there often one gets specific parameters for which the
type naturally becomes empty.

>The problem is that it is difficult to anticipate the
>consequences of changing the approach.

You are right that no one will prove "existence" correctness
conditions anymore, and that for that reason there might
be more mistakes in definitions. But that is balanced for
me by the fact that finally types can be defined in the
natural way. In the _right_ way. Like "Element of", yes.

We've had this discussion before, I'm sure. My excuses
for being triggered.

Freek

Re: I am trying to figure out...

2009-10-25T08:48:11Z

Hi:

Cytowanie freek <freek@...>:

>
> If Mizar's type system wasn't broken (in the sense that
> it doesn't allow types to be empty; well, maybe "broken"
> is too strong a word, but the Mizar type system _is_
> severly limited because of this) then one could define
> "Element of" in the natural way. And then this would be
> a provable theorem (and then in fact it would be automatic
> when one has requirement SUBSET, I guess.)
>

Mizar does not allow for meaningless types like

an empty non empty set
an element of the empty set
a function from a non empty set into the empty set

Does it means that it is "broken"? I may agree that it is more limited;
I would rather say that it is more demanding. Then suspending the
requirement might be exciting: the life becomes easier (and the quality
of work done becomes lower: some questions are left unanswered).

If a meaningless type \theta is allowed that for x being
\theta holds ...

is understood as

for x st x is \theta holds ...

so actulally the type is eliminated by a predicate and the whole type
system becomes syntactic sugar. And why we need two predicates

x is element of A

and x in A?

I do not argue that the solution adopted by Mizar is the only good solution
or the best one. However we need arguments to change it, not to keep it
as it is. I do not claim that it is not possible that we eventually
find such arguments. The problem is that it is difficult to anticipate
the consequences of changing the approach.

Regards,
Andrzej

Re: I am trying to figure out...

2009-10-24T07:20:00Z

Adam:

>> There are two user-defined sets A and B. The first theorem to be proved is:
>>
>> for a,A being set st a being Element of A holds a in A; ::Q1
>
> It holds if (and only if) A is non-empty.

If Mizar's type system wasn't broken (in the sense that
it doesn't allow types to be empty; well, maybe "broken"
is too strong a word, but the Mizar type system _is_
severly limited because of this) then one could define
"Element of" in the natural way. And then this would be
a provable theorem (and then in fact it would be automatic
when one has requirement SUBSET, I guess.)

Freek

Re: I am trying to figure out...

2009-10-24T00:34:43Z

Hi Adem,

On Sat, 24 Oct 2009, Ozyavas, Adem wrote:

> There are two user-defined sets A and B. The first theorem to be proved is:
>
> for a,A being set st a being Element of A holds a in A; ::Q1

It holds if (and only if) A is non-empty.

> and the second:
>
> for f being Function of A,B holds dom f = A; ::Q2
>
> For Q2 I tried to use
>
> definition
> let X,Y;
> let R be Relation of X,Y;
> attr R is quasi_total means
> :: FUNCT_2:def 1
>
> X = dom R if Y = {} implies X = {}
> otherwise R = {};
> end;
>
> and looked into the theorems in FUNCT_2 article. The hypothesis "Y = {}
> implies X = {}" seems a lot of work and I was wondering if there is
> another way.

As in the former case, the statement is clearly accepted if Y is
non-empty. The condition is formulated in this form just to allow
defining functions from X to Y even in a slightly more general situation.

Best,

Adam Naumowicz

=======================================================================
Dept. of Programming and Formal Methods Fax: +48(85)7457662
Institute of Informatics Tel: +48(85)7457559 (office)
University of Bialystok E-mail: adamn@...
Sosnowa 64, 15-887 Bialystok, Poland http://math.uwb.edu.pl/~adamn/
=======================================================================

I am trying to figure out...

2009-10-24T00:06:30Z

Hello All,

I would like to thank for your very helpful comments and time..
I have tried and still trying to figrue out the answers to 2 questions formulated below.

There are two user-defined sets A and B. The first theorem to be proved is:

for a,A being set st a being Element of A holds a in A; ::Q1

and the second:

for f being Function of A,B holds dom f = A; ::Q2

For Q2 I tried to use

definition
let X,Y;
let R be Relation of X,Y;
attr R is quasi_total means
:: FUNCT_2:def 1

X = dom R if Y = {} implies X = {}
otherwise R = {};
end;

and looked into the theorems in FUNCT_2 article. The hypothesis "Y = {} implies X = {}" seems a lot of work and I was wondering if there is another way.

Thank you again very much and have a nice weekend.

Adem Ozyavas

Re: Seeking description of consistency correctness-condition

2009-10-23T17:56:24Z

Artur,

Thanks! That is _exactly_ what I was seeking.

I had seen the lecture notes at http://markun.cs.shinshu-u.ac.jp/kiso/projects/proofchecker/mizar/Mizar4/index-e.html but the page you cited for me (below) is not mentioned in those notes (or anywhere else according to google links).

-Greg Frascadore

Artur Kornilowicz wrote:

Hello,

please look at

http://markun.cs.shinshu-u.ac.jp/kiso/projects/proofchecker/mizar/skeletons/toc.html

In Section '3. Definitions' you can find an answer.

Best,

Artur

On Fri, 23 Oct 2009, Greg Frascadore wrote:

>
> Hello,
>
> I'm looking for a description of the 'consistency' correctness-
> condition and the situations where it is required. Particularly, a
> statement of the template formula one must prove to establish consistency.
>
> After looking through some old papers like 'Muzalewski1993 PCMizar',
> and Matuszewski+Rudnicki 'Mizar: the first 30 years', I either couldn't
> find it, or only a brief mention that it is required for 'definitions
> per cases'.
>
> Searching the MML, consistency is usually demonstrated simply as:
> consistency; (due to disjointness of if/otherwise ?)
>
> But in the non-trivial cases, the condition being proved isn't
> obvious to me.
>
> Thanks,
>
> -Greg Frascadore
>
> --
> View this message in context: http://www.nabble.com/Seeking-description-of-consistency-correctness-condition-tp26027415p26027415.html
> Sent from the Mizar mailing list archive at Nabble.com.
>
>

==========================================================================

Artur Kornilowicz e-mail: arturk@math.uwb.edu.pl

Dept. of Programming and Formal Methods http://math.uwb.edu.pl/~arturk/
Institute of Computer Science tel. +48 (85) 745-7662
University of Bialystok fax. +48 (85) 745-7662
Sosnowa 64, 15-887 Bialystok, Poland

Re: Seeking description of consistency correctness-condition

2009-10-23T07:38:37Z

Hello,

please look at

http://markun.cs.shinshu-u.ac.jp/kiso/projects/proofchecker/mizar/skeletons/toc.html

In Section '3. Definitions' you can find an answer.

Best,

Artur

On Fri, 23 Oct 2009, Greg Frascadore wrote:

>
> Hello,
>
> I'm looking for a description of the 'consistency' correctness-
> condition and the situations where it is required. Particularly, a
> statement of the template formula one must prove to establish consistency.
>
> After looking through some old papers like 'Muzalewski1993 PCMizar',
> and Matuszewski+Rudnicki 'Mizar: the first 30 years', I either couldn't
> find it, or only a brief mention that it is required for 'definitions
> per cases'.
>
> Searching the MML, consistency is usually demonstrated simply as:
> consistency; (due to disjointness of if/otherwise ?)
>
> But in the non-trivial cases, the condition being proved isn't
> obvious to me.
>
> Thanks,
>
> -Greg Frascadore
>
> --
> View this message in context: http://www.nabble.com/Seeking-description-of-consistency-correctness-condition-tp26027415p26027415.html
> Sent from the Mizar mailing list archive at Nabble.com.
>
>

Seeking description of consistency correctness-condition

2009-10-23T07:29:45Z

Hello,

I'm looking for a description of the 'consistency' correctness-
condition and the situations where it is required. Particularly, a
statement of the template formula one must prove to establish consistency.

After looking through some old papers like 'Muzalewski1993 PCMizar',
and Matuszewski+Rudnicki 'Mizar: the first 30 years', I either couldn't
find it, or only a brief mention that it is required for 'definitions
per cases'.

Searching the MML, consistency is usually demonstrated simply as:
consistency; (due to disjointness of if/otherwise ?)

But in the non-trivial cases, the condition being proved isn't
obvious to me.

Thanks,

-Greg Frascadore

Re: "mode Element of ...", some questions...

2009-10-21T04:45:40Z

On Tue, 20 Oct 2009, Ozyavas, Adem wrote:

> Dear All,
>
> I would like to ask a question but before that please let me present the following Mizar fragment:
>
>
> definition
> func setA -> set equals
> {k where k is Element of NAT: k < 15};
> correctness;
> end;
>
> definition
> let e be Element of setA;
> func f(e) -> Element of setA means
> ...
> end;
>
> f(1) = ...;
> ::>*103 (Unknown Functor)
>
>
> 1 is Element of setA;
> ::> *4 (This inference is not accepted)
>
>
> My questions are
> (1) how to make Mizar accept "1 is Element of setA"? I checked the MML for "mode Element of ..." examples and could not find anything that could help me.

As '1' is a numeral and not a 'regular' functor, you cannot redefine it,
or register new attributes for it.

> (2) This relates to the Question (1). How can I make sure that 1 is known to be an Element of setA so that 103 error does not occur?

But you can always use a 'permissive' definition of f(), so that it
accepts all 'Element of NAT', but is meaningful only for these in 'setA',
e.g.

definition
let e be Element of NAT;
assume e in setA;
:: or e < 15
func f(e) -> Element of setA means
...
end;

Then f(1) is a perfectly legal expression;

Best,
Adam Naumowicz

=======================================================================
Dept. of Programming and Formal Methods Fax: +48(85)7457662
Institute of Informatics Tel: +48(85)7457559 (office)
University of Bialystok E-mail: adamn@...
Sosnowa 64, 15-887 Bialystok, Poland http://math.uwb.edu.pl/~adamn/
=======================================================================

Re: "mode Element of ...", some questions...

2009-10-20T22:58:43Z

Hi Adem,

there is a couple of solutions to your problem:

first, at global level (e.g. theorems) you can write

theorem
for x being Element of setA st x = 1 holds f(x) ...;

Inside proofs you can write

reconsider j = 1 as Element of setA;
f(j) = ...;

You can't use 'j' in theorems formulations because local constants can't
be imported to the library.

Another solution is to use

definition
let x, y be set;
assume x in y;
func In (x, y) -> Element of y equals
:: FUNCT_7:def 1
x;
end;

Having this, Mizar automatically knows that
In(1,setA) is Element of setA;

So you can formulate facts as

f(In(1,setA)) = ...;

I believe it helps.

Best regards
Artur

On Tue, 20 Oct 2009, Ozyavas, Adem wrote:

> Dear All,
>
> I would like to ask a question but before that please let me present the following Mizar fragment:
>
>
> definition
> func setA -> set equals
> {k where k is Element of NAT: k < 15};
> correctness;
> end;
>
> definition
> let e be Element of setA;
> func f(e) -> Element of setA means
> ...
> end;
>
> f(1) = ...;
> ::>*103 (Unknown Functor)
>
>
> 1 is Element of setA;
> ::> *4 (This inference is not accepted)
>
>
> My questions are (1) how to make Mizar accept "1 is Element of setA"? I
> checked the MML for "mode Element of ..." examples and could not find
> anything that could help me. (2) This relates to the Question (1). How
> can I make sure that 1 is known to be an Element of setA so that 103
> error does not occur?
>
>
> Thank you all and I always appreciate the group's help.
> Regards,
>
> Adem Ozyavas
> Texas Tech University

"mode Element of ...", some questions...

2009-10-20T21:36:24Z

Dear All,

I would like to ask a question but before that please let me present the following Mizar fragment:

definition
func setA -> set equals
{k where k is Element of NAT: k < 15};
correctness;
end;

definition
let e be Element of setA;
func f(e) -> Element of setA means
...
end;

f(1) = ...;
::>*103 (Unknown Functor)

1 is Element of setA;
::> *4 (This inference is not accepted)

My questions are
(1) how to make Mizar accept "1 is Element of setA"? I checked the MML for "mode Element of ..." examples and could not find anything that could help me.
(2) This relates to the Question (1). How can I make sure that 1 is known to be an Element of setA so that 103 error does not occur?

Thank you all and I always appreciate the group's help.
Regards,

Adem Ozyavas
Texas Tech University

Re: monospace font for html

2009-09-21T20:46:49Z

Hi,

On Mon, 21 Sep 2009, Josef Urban wrote:

> Freek Wiedijk has suggested using a monospace font for the Mizar HTML (like
> this: http://hair-dryer.cs.ru.nl/~urban/a9.html). His argument is that Mizar
> is code, and code should be rendered like this. I also like it better than
> the current font (http://mizar.uwb.edu.pl/version/current/html/abcmiz_0.html).

I second the suggestion, because that'a what we actually do when we
present snippets of Mizar code in publications, presentations etc.

Best,
Adam Naumowicz

=======================================================================
Dept. of Programming and Formal Methods Fax: +48(85)7457662
Institute of Informatics Tel: +48(85)7457559 (office)
University of Bialystok E-mail: adamn@...
Sosnowa 64, 15-887 Bialystok, Poland http://math.uwb.edu.pl/~adamn/
=======================================================================

monospace font for html

2009-09-21T07:37:08Z

Hi,

Freek Wiedijk has suggested using a monospace font for the Mizar HTML (like this: http://hair-dryer.cs.ru.nl/~urban/a9.html). His argument is that Mizar is code, and code should be rendered like this. I also like it better than the current font (http://mizar.uwb.edu.pl/version/current/html/abcmiz_0.html).

So if there are no protests, I will make it the default for the next versions of the Mizar HTML.

Best,
Josef Urban

RE: Inductive predicates

2009-09-02T21:35:06Z

Dear Adem,

On Wed, 2 Sep 2009, Ozyavas, Adem wrote:

> I wrote an evaluator for a functional language called SequenceL in
> Mizar. I have some recursive functions in my evaluator whose domains are
> very complicated so that their well-definedness proofs are too hard. I
> am thinking that writing them as recursive predicates and doing some
> proofs about those predicates would be much easier.

I'm not sure if this "workaround" is a good strategy here. It might
happen that when you want to have your article submitted to MML, the
reviewers might complain why some definitions are in the form of
predicates while they should rather be functors...

I believe you should still try to complete the correctness proofs using
this forum, or the Mizar User Service for help to sort out the
particular difficulties in the underlying proofs. But I don't really
know how hard the proofs are, of course :-)

Best,

Adam

=======================================================================
Dept. of Programming and Formal Methods Fax: +48(85)7457662
Institute of Informatics Tel: +48(85)7457559 (office)
University of Bialystok E-mail: adamn@...
Sosnowa 64, 15-887 Bialystok, Poland http://math.uwb.edu.pl/~adamn/
=======================================================================

RE: Inductive predicates

2009-09-02T11:44:11Z

Dear Professor Andrzej Trybulec and All,

I wrote an evaluator for a functional language called SequenceL in Mizar. I have some recursive functions in my evaluator whose domains are very complicated so that their well-definedness proofs are too hard. I am thinking that writing them as recursive predicates and doing some proofs about those predicates would be much easier. That is why I wanted to start writing a recursive factorial as a recursive predicate as an exercise. I also wanted to see a definition from experts. I am hoping that I am on the right track.

Thanks and Regards,
Adem Ozyavas
________________________________________
From: owner-mizar-forum@... [owner-mizar-forum@...] On Behalf Of trybulec@... [trybulec@...]
Sent: Wednesday, September 02, 2009 3:58 AM
To: mizar-forum@...
Subject: RE: [mizar] Inductive predicates

You're right. The private constructors, like introduced by defpred,
cannot be used in the exportable (public) part of the aricle. Just
expand F and G in the
definition of the predicate.

Why you need such a predicate? The factorial is already define in MML:

NEWTON:def 2

Regards,
Andrzej Trybulec

Cytowanie "Ozyavas, Adem" <adem.ozyavas@...>:

> Dear All,
>
> I asked Dr.Grzegorz Bancerek if there is inductive predicate
> definitions in Mizar and below is Dr.Grzegorz Bancerek's answer.
> I tried to write the recursive definition of factorial using
> Dr.Grzegorz Bancerek's
> template predicate definition.
> When I tried to define F and G functions "outside of predicate
> definition" Mizar
> is complaining that (I think all definitions are global) the
> definition of pred is global and F and G are local as follows:
>
> deffunc F ...
> deffunc G ...
>
> definition
> let X; let x,y be Element of X;
> pred x fac y means :: x! = y
> for Y st
> (for a,b st [a,b] = [0,1] holds [a,b] in Y) &
> (for a,b st [a,b] in Y holds [F(a),G(b)] in Y)
> holds [x,y] in Y;
> end;
>
>
> How can one define the recursive factorial as predicate in Mizar with
> F and G functions?
>
> Thank you...
>
> Adem Ozyavas
>

RE: Inductive predicates

2009-09-02T01:58:40Z

You're right. The private constructors, like introduced by defpred,
cannot be used in the exportable (public) part of the aricle. Just
expand F and G in the
definition of the predicate.

Why you need such a predicate? The factorial is already define in MML:

NEWTON:def 2

Regards,
Andrzej Trybulec

Cytowanie "Ozyavas, Adem" <adem.ozyavas@...>:

RE: Inductive predicates

2009-09-01T20:34:37Z

Dear All,

I asked Dr.Grzegorz Bancerek if there is inductive predicate definitions in Mizar and below is Dr.Grzegorz Bancerek's answer.
I tried to write the recursive definition of factorial using Dr.Grzegorz Bancerek's
template predicate definition.
When I tried to define F and G functions "outside of predicate definition" Mizar
is complaining that (I think all definitions are global) the definition of pred is global and F and G are local as follows:

deffunc F ...
deffunc G ...

definition
let X; let x,y be Element of X;
pred x fac y means :: x! = y
for Y st
(for a,b st [a,b] = [0,1] holds [a,b] in Y) &
(for a,b st [a,b] in Y holds [F(a),G(b)] in Y)
holds [x,y] in Y;
end;

How can one define the recursive factorial as predicate in Mizar with F and G functions?

Thank you...

Adem Ozyavas

> Hi,

> there is no regular recursive definition of predicates in Mizar.
> Instead you should use something like

> definition
> let X; let x,y be Element of X;
> pred x rec y means
> for Y st
> (for a,b st START_CONDITION holds [a,b] in Y) &
> (for a,b st [a,b] in Y holds [F(a),G(b)] in Y)
> holds [x,y] in Y;
> end;

>Grzegorz

>On Sun, 16 Aug 2009, Ozyavas, Adem wrote:

> Dear All,
>
> Is there any way to write inductive predicates in Mizar?
>
> My purpose is to have inductive predicates instead of recursive functions to avoid existence and uniqueness proofs of recursive functions and leaving the fact that this predicate has functional property as a later task. I have searched the library and could not find any examples.
>
> Thanks...
>
> Adem Ozyavas

Re: Inductive predicates

2009-08-17T09:29:03Z

Hi,

there is no regular recursive definition of predicates in Mizar.
Instead you should use something like

definition
let X; let x,y be Element of X;
pred x rec y means
for Y st
(for a,b st START_CONDITION holds [a,b] in Y) &
(for a,b st [a,b] in Y holds [F(a),G(b)] in Y)
holds [x,y] in Y;
end;

Grzegorz

On Sun, 16 Aug 2009, Ozyavas, Adem wrote:

> Dear All,
>
> Is there any way to write inductive predicates in Mizar?
>
> My purpose is to have inductive predicates instead of recursive functions to avoid existence and uniqueness proofs of recursive functions and leaving the fact that this predicate has functional property as a later task. I have searched the library and could not find any examples.
>
> Thanks...
>
> Adem Ozyavas

===============================================================
Grzegorz Bancerek
e-mail: bancerek@... (bancerek@...)
http://merak.pb.bialystok.pl/~bancerek/
Dept. of Theoretical CS
Faculty of Computer Science fax. +48 (85) 746-9057
Bialystok Technical University tel. +48 (85) 746-9056
http://www.pb.bialystok.pl

Inductive predicates

2009-08-16T14:18:46Z

Dear All,

Is there any way to write inductive predicates in Mizar?

My purpose is to have inductive predicates instead of recursive functions to avoid existence and uniqueness proofs of recursive functions and leaving the fact that this predicate has functional property as a later task. I have searched the library and could not find any examples.

Thanks...

Adem Ozyavas

New Mizar articles

2009-06-19T01:35:31Z

Dear All,
together with the latest official version of the Mizar system (7.11.02)
and the Mizar Mathematical Library (4.125.1059) the following articles
are available:

1047. POLYEQ_5
Solution of Cubic and Quartic Equations
by Marco Riccardi
Received March 3, 2009
1048. NAT_5
The Perfect Number Theorem and Wilson's Theorem
by Marco Riccardi
Received March 3, 2009
1049. RANDOM_1
Probability on Finite Set and Real Valued Random Variables
by Hiroyuki Okazaki and Yasunari Shidama
Received March 17, 2009
1050. MESFUN9C
Lebesgue's Convergence Theorem of Complex-Valued Function
by Keiko Narita, Noboru Endou and Yasunari Shidama
Received March 17, 2009
1051. METRIZTS
Basic Properties of Metrizable Topological Spaces
by Karol P\c{a}k
Received March 31, 2009
1052. GR_CY_3
Properties of Primes and Multiplicative Group of a Field
by Kenichi Arai and Hiroyuki Okazaki
Received April 7, 2009
1053. CFDIFF_2
Cauchy-Riemann Differential Equations of Complex Functions
by Hiroshi Yamazaki, Yasunari Shidama, Chanapat Pacharapokin
and Yatsuka Nakamura
Received April 7, 2009
1054. MEASURE8
The Hopf Extension Theorem of Measure
by Noboru Endou, Hiroyuki Okazaki and Yasunari Shidama
Received April 7, 2009
1055. REWRITE3
Labelled State Transition Systems
by Micha{\l} Trybulec
Received May 5, 2009
1056. DIST_1
Probability on Finite and Discrete Set and Uniform Distribution
by Hiroyuki Okazaki
Received May 5, 2009
1057. INTEGR15
Riemann Integral of Functions $\mathbbbR$ into $\mathbbbR^n$
by Keiichi Miyajima and Yasunari Shidama
Received May 5, 2009
1058. FUNCT_8
Basic properties of even and odd functions
by Bo Li and Yanhong Men
Received May 25, 2009
1059. FSM_3
Equivalence of Epsilon, Nondeterministic [Finite] Automata
and Deterministic [Finite] Automata
by Micha{\l} Trybulec
Received May 25, 2009

Regards,
Adam Grabowski
Library Committee of the Association of Mizar Users

Re: How to prove equivalences?

2009-05-26T06:14:36Z

Hi Greg,

>So 'per directions' seems like syntactic sugar that
>'should be easy to add' :-)

I'm sure it's easy to add, but not for me :-)

So I didn't see an answer to my mail about this before.
I still would be very much interested in knowing how the
rest of the Mizar community feels about this.

Also: there's "suppose" versus "case". Maybe I'm asking
about this for nothing, as maybe this is what "case" is for?

Freek

Re: how to prove...

2009-05-17T23:20:39Z

Hi Adem,

On Mon, 18 May 2009, Ozyavas, Adem wrote:

> Dear All,
>
> How does one go about proving that a real number is not a finite sequence:
>
> for n being real number holds not n is FinSequence

Well, in general it's not true with the set-theoretic construction of real
numbers as used in MML - {} is both a real number (zero) and a finite
sequence (empty function)...

Best,

Adam Naumowicz

=======================================================================
Dept. of Programming and Formal Methods Fax: +48(85)7457662
Institute of Informatics Tel: +48(85)7457559 (office)
University of Bialystok E-mail: adamn@...
Sosnowa 64, 15-887 Bialystok, Poland http://math.uwb.edu.pl/~adamn/
=======================================================================

how to prove...

2009-05-17T22:47:59Z

Dear All,

How does one go about proving that a real number is not a finite sequence:

for n being real number holds not n is FinSequence

Thanks...

Adem Ozyavas

Mizar Twiki appears off-line

2009-05-03T05:38:23Z

The wiki appears to be down:

http://wiki.mizar.org/cgi-bin/twiki/view/Mizar

PLMMS 2009: Last CFP

2009-04-28T05:45:04Z

The ACM SIGSAM 2009 International
Workshop on Programming Languages for
Mechanized Mathematics Systems
PLMMS 2009

Munich, Germany; August 21, 2009
http://plmms09.cse.tamu.edu/

CALL FOR PAPERS

The ACM SIGSAM 2009 International Workshop on Programming Languages
for Mechanized Mathematics Systems will be co-located with TPHOLs 2009.

Important Dates

* Abstract submission : May 11, 2009 (Apia, Samoa time)
* Submission deadline: May 18, 2009 (Apia, Samoa time)
* Author notification: June 22, 2009
* Camera ready papers: July 10, 2009
* Workshop: August 21, 2009

General Information

The scope of this workshop is at the intersection of programming
languages (PL) and mechanized mathematics systems (MMS). The latter
category subsumes present-day computer algebra systems (CAS),
interactive proof assistants (PA), and automated theorem provers
(ATP), all heading towards fully integrated mechanized mathematical
assistants. Areas of interest include all aspects of PL and MMS that
meet in the following topics, but not limited to:

* Dedicated input languages for MMS: covers all aspects of languages
intended for the user to deploy or extend the system, both
algorithmic and declarative ones. Typical examples are tactic
definition languages such as Ltac in Coq, mathematical proof
languages as in Mizar or Isar, or specialized programming
languages built into CA systems.

* Mathematical modeling languages used for programming: covers the
relation of logical descriptions vs. algorithmic content. For
instance the logic of ACL2 extends a version of Lisp, that of Coq
is close to Haskell, and some portions of HOL are similar to ML
and Haskell, while Maple tries to do both simultaneously. Such
mathematical languages offer rich specification capabilities,
which are rarely available in regular programming languages. How
can programming benefit from mathematical concepts, without
limiting mathematics to the computational world view?

* Programming languages with mathematical specifications: covers
advanced mathematical concepts in programming languages that
improve the expressive power of functional specifications, type
systems, module systems etc. Programming languages with dependent
types are of particular interest here, as is intentionality vs
extensionality.

* Language elements for program verification: covers specific means
built into a language to facilitate correctness proofs using
MMS. For example, logical annotations within programs may be
turned into verification conditions to be solved in a proof
assistant eventually. How need MMS and PL to be improved to make
this work conveniently and in a mathematically appealing way?

These issues have a very colorful history. Many PL innovations first
appeared in either CA or proof systems first, before migrating into
more mainstream programming languages. This workshop is an
opportunity to present the latest innovations in MMS design that may
be relevant to future programming languages, or conversely novel PL
principles that improve upon implementation and deployment of MMS.
Why are all the languages of mainstream CA systems untyped? Why
are the (strongly typed) proof assistants so much harder to use than
a typical CAS? What forms of polymorphism exist in mathematics?
What forms of dependent types may be used in mathematical modeling?
How can MMS regain the upper hand on issues of "genericity" and
"modularity"? What are the biggest barriers to using a more
mainstream language as a host language for a CAS or PA/ATP?

PLMMS 2007 was held as a satellite event of, and PLMMS 2008 was
a CICM 2008 workshop.

Submission Details

Submitted papers should be in portable document format (PDF),
formatted using the ACM SIGPLAN style guidelines
(http://www.acm.org/sigs/sigplan/authorInformation.htm). The length
is restricted to 10 pages, and the font size 9pt. Each submission
must adhere to SIGPLAN's republication policy, as explained on the
web. Violation risks summary rejection of the offending submission.

Papers are exclusively submitted via EasyChair

http://www.easychair.org/conferences?conf=plmms09

We expect that at least one author of each accepted paper attends
PLMMS 2009 and presents her or his paper.

Accepted papers will appear in the ACM Digital Library.

Links

* http://plmms09.cse.tamu.edu/, the PLMMS 2009 workshop web site
* http://tphols.in.tum.de/, the THOPLs 2009 conference web site

Program Committee

* Clemens Ballarin, aicas GmbH
* Gabriel Dos Reis, Texas A&M University (Co-Chair)
* Jean-Christophe Filliâtre, CNRS Université Paris Sud
* Predrag Janicic, University of Belgrade
* Jaakko Järvi, Texas A&M University
* Florina Piroi, Johannes Kepler University
* Laurent Théry, INRIA Sophia Antipolis (Co-Chair)
* Makarius Wenzel, Technische Universität München