> > > Coq provides gcd as > > > > Fixpoint gcd a b := > > match a with > > | 0 => b > > | S a' => gcd (b mod (S a')) (S a') > > end. > > > and Axiom's definition is > > > gcd(x:NNI,y:NNI):NNI == > > zero? x => y > > gcd(y rem x, x) > > > Everything in Spad is strongly typed and function definitions are chosen > not only > > by the arguments but also by the return type (so there can be multiple > functions > > with the same name and same arguments but different return types, for > example). > > Every statement in the function is strongly type-checked. >
> That is what I referred to as a shallow embedding -- you are associating to every > axiom definition a Coq or Lean >definition which has the same behavior. > If you do that, you cannot model arbitrary while loops. You have to write > functions in Coq or Lean in a way that, from the start, they are > guaranteed to terminate. You can do this, for example, by showing > the recursive calls are decreasing along a suitable measure, or giving a > priori bounds on a while loop. If you want to translate spad functions > automatically, you'll have to write the former in such a way that the > translations have this property. You can't translate an arbitrary, a priori > partial, function and then show after the fact that it terminates for every input. This is not intended to be a shallow embedding. In order to prove Spad's gcd we have to prove zero? and rem and have those available. The idea is to initially create the definitions so that Spad code is reducible to code that the proof engine can process directly. This involves defining the Spad 'words' like 'zero?' so they properly type check. What can't be directly proven will have to be restructured/rewritten. As we discussed on Tuesday, a lot of Spad code uses loops. I'm looking at the Isabelle/HOL book you recommended for some advice on Hoare triples. That's queued behind getting up to speed on FoCaL, as mentioned by Renaud. Eventually I'd like to see the proof engine embedded directly into Axiom rather than called at compile time so that it is also available in the interpreter. Where are grad students when you need them? :-) Tim On Wed, Feb 8, 2017 at 9:44 PM, Jeremy Avigad <avi...@cmu.edu> wrote: > > > On Wed, Feb 8, 2017 at 9:23 PM, Tim Daly <axiom...@gmail.com> wrote: > >> Part of your struggle of understanding what I wrote is that I'm not yet >> fluent in the >> logic language and syntax. I'm learning as fast as I can so please be >> patient. >> >> ====================================================== >> CATEGORY SIGNATURE vs DOMAIN SEMANTICS >> >> > Presumably you will eventually want to add axioms to the structures >> that say >> > things about what eq and neq do >> >> The semantics of = is given in the Domain (the current one being defined >> is called % in Spad) >> not in the Category (well...you can... sigh) >> >> Each domain that inherits '=' from the Category BasicType needs to >> specify the meaning >> of that function for the Domain you're implementing.. >> > > In our language, we would say that every instance of the structure has all > the necessary data. For example, every group (=instance of the group > structure, or element of the type group α) has a unit, a binary operation, > and inverse operation, etc. > > >> For a Polynomial domain with some >> structural data representation you have to define what it means for two >> polynomial objects >> to be =. such as a function to compare coefficients. Part of the game >> would be to prove >> that the coefficient-compare function is correct, always returns a >> Boolean, and terminates. >> >> All a Category like BasicType does is specify that the Domain Polynomial >> should >> implement an = operation with the given signature. That is, you have to >> implement >> >> poly = poly >> >> which returns a boolean. (Note that there are other = functions in >> Polynomial such as one >> that returns an equation object but that signature is inherited from a >> different Category). >> > > Is there anything that requires that the operation you implement is > reflexive, symmetric, and transitive? Putting axioms on the structure > specifies that that has to be the case. Without such axioms, you cannot > prove anything about implementations in general. You can only prove things > about individual implementations. > > >> It looks like your 'class' syntax implements what I need. I will try this >> for the other >> Categories used in NNI. >> >> >> >> >> ======================================================= >> PROVING TERMINATION >> >> As I understood from class, for an algorithm like gcd it should be >> sufficient to construct >> a function that fulfills the signature of >> >> gcd(a:NNI,b:NNI):NNI >> >> Coq provides gcd as >> >> Fixpoint gcd a b := >> match a with >> | 0 => b >> | S a' => gcd (b mod (S a')) (S a') >> end. >> >> and Axiom's definition is >> >> gcd(x:NNI,y:NNI):NNI == >> zero? x => y >> gcd(y rem x, x) >> >> Everything in Spad is strongly typed and function definitions are chosen >> not only >> by the arguments but also by the return type (so there can be multiple >> functions >> with the same name and same arguments but different return types, for >> example). >> Every statement in the function is strongly type-checked. >> > > That is what I referred to as a shallow embedding -- you are associating > to every axiom definition a Coq or Lean definition which has the same > behavior. > > If you do that, you cannot model arbitrary while loops. You have to write > functions in Coq or Lean in a way that, from the start, they are guaranteed > to terminate. You can do this, for example, by showing the recursive calls > are decreasing along a suitable measure, or giving a priori bounds on a > while loop. If you want to translate spad functions automatically, you'll > have to write the former in such a way that the translations have this > property. You can't translate an arbitrary, a priori partial, function and > then show after the fact that it terminates for every input. > > Other approaches are possible. You can, for example, translate spad > functions to relations in Coq or Lean, and then prove that the relations > give rise to total functions. > > Best wishes, > > Jeremy > > > >> >> Thus we are guaranteed that the Spad version of gcd above (in the Domain >> NNI) >> can only be called with NNI arguments and is guaranteed to only return >> NNI results. >> >> The languages are very close in spirit if not in syntax. >> >> What Axiom does not do, for example, is prove termination. >> >> Coq, in its version, will figure out that the recursion is on 'a' and >> that it will terminate. >> >> Part of the game is to provide the same termination analysis on Spad code. >> >> >> >> >> ==================================================== >> ADDITIONAL CONSTRAINTS >> >> It would be ideal to reject code that did not fulfill all of the >> requirements >> such as specifying at the Category level definition of gcd that it not >> only >> has to have the correct signature, it also has to return the 'positive' >> divisor. For NNI this is trivially fulfilled. >> >> The Category definition should say something like >> >> gcd(%,%) -> % and gcd >= 1$% >> >> where 1$% says to use the unit from the implementing Domain. >> >> So for some domains we have: >> >> gcd(x,y) == >> x := unitCanonical x >> y := unitCanonical y >> while not zero? y repeat >> (x,y) := (y, x rem y) >> y := unitCanonical y >> x >> >> using unitCanonical to deal with things like signs. (This also adds the >> complication >> of loops which I mentioned in a previous email.) >> >> Not only the signature but the side-conditions would have to be checked. >> >> >> >> >> >> >> ==================================================== >> ALTERNATE APPROACHES >> >> Instead of a new library organization it might be possible to have a >> generator function >> in Coq that translates Coq code to Spad code. Or a translator from Spad >> code to >> Coq code. >> >> Unfortunately Coq and Lean do not seem to use function name overloading >> or inheritance (or do they?) which confuses the problem of name >> translation. >> >> Axiom has 42 functions named 'gcd', each living in a different Domain. >> >> >> >> >> >> There is no such thing as a simple job. But this one promises to be >> interesting. >> >> In any case I'll push the implementation forward. Thanks for your help. >> >> Tim >> >> >> >> >> >> >> On Wed, Feb 8, 2017 at 5:52 PM, Jeremy Avigad <avi...@cmu.edu> wrote: >> >>> Dear Tim, >>> >>> I don't understand what you mean. For one thing, theorems in both Lean >>> and Coq are marked as opaque, since you generally don't care about the >>> contents. But even if we replace "theorem" by "definition," I don't know >>> what you imagine going into the "...". >>> >>> I think what you want to do is represent Axiom categories as structures. >>> For example, the declarations below declare a BasicType structure and >>> notation for elements of that structure. You can then prove theorems about >>> arbitrary types α that have a BasicType structure. You can also extend the >>> structure as needed. >>> >>> (Presumably you will eventually want to add axioms to the structures >>> that say things about what eq and neq do. Otherwise, you are just reasoning >>> about a type with two relations.) >>> >>> Best wishes, >>> >>> Jeremy >>> >>> class BasicType (α : Type) : Type := >>> (eq : α → α → bool) (neq : α → α → bool) >>> >>> infix `?=?`:50 := BasicType.eq >>> infix `?~=?`:50 := BasicType.neq >>> >>> section >>> variables (α : Type) [BasicType α] >>> variables a b : α >>> >>> check a ?=? b >>> check a ?~=? b >>> end >>> >>> >>> >>> >>> On Wed, Feb 8, 2017 at 9:29 AM, Tim Daly <axiom...@gmail.com> wrote: >>> >>>> The game is to prove GCD in NonNegativeInteger (NNI). >>>> >>>> We would like to use the 'nat' theorems from the existing library >>>> but extract those theorems automatically from Axiom sources >>>> at build time. >>>> >>>> Axiom's NNI inherits from a dozen Category objects, one of which >>>> is BasicType which contains two signatures: >>>> >>>> ?=?: (%,%) -> Boolean ?~=?: (%,%) -> Boolean >>>> >>>> In the ideal case we would decorate BasicType with the existing >>>> definitions of = and ~= so we could create a new library structure >>>> for the logic system. So BasicType would contain >>>> >>>> theorem = (a, b : Type) : Boolean := ..... >>>> theorem ~= (a, b : Type) : Boolean := .... >>>> >>>> These theorems would be imported into NNI when it inherits the >>>> signatures from the BasicType Category. The collection of all of >>>> the theorems in NNI's Category structure would be used (hopefully >>>> exclusively) to prove GCD. In this way, all of the theorems used to >>>> prove Axiom source code would be inheritied from the Category >>>> structure. >>>> >>>> Unfortunately it appears the Coq and Lean will not take kindly to >>>> removing the existing libraries and replacing them with a new version >>>> that only contains a limited number of theorems. I'm not yet sure about >>>> FoCaL but I suspect it has the same bootstrap problem. >>>> >>>> Jeremy Avigad (Lean) made the suggestion to rename these theorems. >>>> Thus, instead of =, the supporting theorem would be 'spad=' (spad is >>>> the name of Axiom's algebra language). >>>> >>>> Initially this would make Axiom depend on the external library >>>> structure. >>>> Eventually there should be enough embedded logic to start coding Axiom >>>> theorems by changing external references from = to spad= everywhere. >>>> >>>> Axiom proofs would still depend on the external proof system but only >>>> for the correctness engine, not the library structure. This will >>>> minimize >>>> the struggle about Axiom's world view (e.g. handling excluded middle). >>>> It will also organize the logic library to more closely mirror abstract >>>> algebra. >>>> >>>> Comments, suggestions? >>>> >>>> Tim >>>> >>>> >>>> >>> >> >
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