*To*: Makarius <makarius at sketis.net>, "cl-isabelle-users at lists.cam.ac.uk" <cl-isabelle-users at lists.cam.ac.uk>*Subject*: Re: [isabelle] Isabelle Foundation & Certification*From*: Andrei Popescu <a.popescu at mdx.ac.uk>*Date*: Wed, 30 Sep 2015 01:50:16 +0100*Accept-language*: en-US, en-GB*Acceptlanguage*: en-US, en-GB*Thread-index*: AdD7GE2zSoO5olBXQwWkm6UweBHZsA==*Thread-topic*: [isabelle] Isabelle Foundation & Certification

Hi Makarius, I thank you very much for looking at the technical contribution of the paper beyond matters of terminology and "attitude." I find your summary of our technical contribution to be excellent. In order to discuss your comments concerning our "attitude," I would first like to state my own view on the situation, which is not entirely divergent from your view. Isabelle/HOL is not a standard object logic of Isabelle/Pure, in the usual sense of object logics represented in a logical framework. The culprit is typedef, which is implemented in an ad hoc manner, and not as an object-logic judgment. This is necessary due to the nature of typedef. But this is already a warning that whatever guarantees one provides at the generic level of Isabelle/Pure may simply not be good enough for Isabelle/HOL. By contrast, the constant definition mechanism from Isabelle/Pure is imported to Isabelle/HOL following a uniform "object-logic inside meta-logic" shallow embedding scheme. This scheme also applies, for example, to the representation of bindings and the function space. In your â97 paper you prove the following very nice result: In Isabelle/Pure, under suitable orthogonality and well-foundedness conditions, the addition of overloaded constant definitions to any theory T is "meta-safe," in that it is syntactically conservative and anything provable in the extended theory can also be represented and proved in the original T in a canonical way, by ârealizingâ the new constants as terms in the old signature. This indeed is a very strong conservativity result, which also implies preservation of consistency. It applies to the object logics of Isabelle/Pure, which can be regarded as theories. As you also remind us in your review, another pleasant feature of this result is T not needing to be a definitional theory. But: Isabelle/HOL of course only partly benefits from this result. In fact, compared to Gordon-HOL, which offers model-theoretic safety for both constant and type definitions, I would say that in these conditions Isabelle/HOL would be only half-way safe, and quite non-democratically so. On the one hand, there is this luxurious (meta-)safety guarantee for constant definitions, and on the other hand there is *nothing* for typedef. Instead of stating this problem as a *pressing open problem* (already in â97), you had decided to introduce this oxymoron: "axiomatic typedef." Unfortunately, nobody (except perhaps for Florian who used to âjokeâ about typedef being axiomatic) has noticed this ideological move away from the tradition of the HOL systems. (Here, I have a confession. When reading your paper, it simply did not cross my mind that you could have just abandoned typedef :-( and instead I came up with the following explanation: Your approach is to prove as much as you can in Isabelle/Pure for constants, namely, meta-safety, and in particular syntactic conservativity. Then you would move to the object level of Isabelle/HOL where typedef is model-theoretically safe, in particular syntactically conservative for definitional theories. Putting together these two properties, you would get consistency of definitional theories of Isabelle/HOL. As we know now, this almost works but not quite, since typedefs are allowed to depend on unresolved overloading.) Next I comment on your review: >> Technically, the key misunderstanding is the role of "the kernel" (in the >> words of the paper). HOL88 as the predecessor of all the other HOL systems >> pioneered an add-on to the original LCF kernel design to have checked >> definitions as primitive rules. In contrast, Isabelle is closer to LCF in >> separating logical inferences and a-priori unchecked (axiomatic) theory >> specifications: there is no notion of definitions in the inference kernel of Isabelle. In the paper, we use the word "kernel" to mean "logic kernel" (as I hoped would be clear from the paper's general context and from the usage of the phrase âlogic kernelâ for one of the two occurrences of the word). The logic kernel consists of the inference rules and the definitional mechanisms, regardless of where they are located in the implementation. A small logic kernel is an implementation-independent virtue of Isabelle/HOL as well as of all the Gordon-HOL systems. >> Nonetheless, there is a potential for confusion due to slight deviation of >> terminology and concepts, compared to papers on the subject e.g. by >> Wenzel/Haftmann and the Isabelle documentation (isar-ref manual). >> ... >> In contrast, the specifications of a "definitional theory" of the paper >> would be called in the Isabelle documentation "unchecked definitions", >> "definitional axioms", or "axiomatizations", to emphasize the missing aspect >> of global well-formedness that this paper is mainly about. For the journal version, we will switch to the terminology from the Isabelle documentation. I believe "definitional axioms" is the most suggestive term here. And "axiomatizations" is too wide. >> A "well-formed definitional theory" of the paper would be just called "definitional theory" >> in Isabelle terminology, and much stronger properties intended than just consistency. >> Instead of merely ensuring the existence of a model, the requirement is to preserve derivability of old propositions precisely, and >> allow to reduce new derivations of new propositions into old derivations by >> "expanding" definitions in some way. If and how this works depends on >> fine-points of the underlying definitional mechanisms: for overloaded >> constant definitions in Isabelle/Pure this works [13, 35], but for type >> definitions the situation is a-priori quite different, e.g. see [13] section 4.3. >> The deeper reason for this are the distinctive categories of terms and >> types in HOL, and the lack of equational reasoning on types. Indeed, consistency is a crucial, but rather weak property. With a bit of extra care, we could have stated our result as more than consistency, namely, something similar to the preservation of standard models in the sense of Gordon-HOL (which you revisit in your '97 paper). >> A suitable semantic treatment is required to re-unify the view on terms and >> types, as is done in the present paper. But this leads to weaker results. >> The paper does briefly mention "a suitable notion of conservativeness" as >> future work, but due to lack of further explanations, many readers will >> probably miss key point behind this. We do cite your paper there, so the reader can look up the relevant notion. However, I think that for most of the interesting theories, typedef cannot be safe in any sense even remotely similar to what you prove for constant definitions. In fact, typedef cannot be proved to preserve consistency for non-definitional theories. This is shown by an example in your '97 paper, which I slightly generalize below: Given any finite consistent definitional theory T, we should always be able find a number N such that there exists a set-theoretic model of T where no type has precisely N elements. So T + "no type 'a has precisely N elements" is consistent, but defining a type with N elements makes it inconsistent. >> There is another aspect of "consistency" versus "conservativity" that is >> easily missed in the paper. Section 3.5 states "Isabelle/HOL theory >> development proceeds by: 1 ... 2 ... 3 ...". The three kinds of operations >> ensure that every stage of theory development is "definitional" in the sense >> of the paper -- and "well-formedness" can be established later. But what >> happens when the user starts adding non-definitional axioms in between? >> Strictly speaking, the main result does not apply and the whole theory >> looses its good definitional properties. >> ... >> For HOL applications one might argue that genuine axiomatic extensions are >> not done in practice these days. You are right, our result is restricted to definitional theories. But typedef seems brittle in the face of amendments to the standard HOL model theory. And such amendments are easily possible if one steps outside definitional theories. >> Nonetheless some exotic applications like >> HOLZF (by Obua) do exist. Without stronger results under which conditions >> typedefs "make sense" or "are OK" (using words from the paper), such >> ambitious users would have to revisit the whole model theory of HOL again. IMO, Obua and other initiators of interesting experiments, besides forming an absolute minority of the users, are logical grown-ups who typically know what they are doing. And they are in plain wilderness in the first place, when proving consistency. >> In contrast, the traditional explanation of (constant) definitions in >> Isabelle is more modular in that respect: given an arbitrary base theory --- >> one that somehow makes sense to the user in a particular application, even >> after making unchecked axiomatizations --- definitions merely introduce >> names for existing entities and thus preserve the key properties of the base >> theory. This agnostic approach of "definitional theory extensions" >> (occording to Isabelle terminolgy) is particularly important from the >> perspective of the Isabelle/Pure framework, where the interpretation of >> object-logics is not yet known. Agreed. But I am happy that the less generic Isabelle/HOL typedef feature now receives some consideration as well. >> For example, the delicate notion of "build-in types" vs. "non-built-in >> types" is absent in the implementation. The notes on the patch provide some >> explanations, why it works out nonetheless. It would be nice to see this >> refinement applied to the main work of part (1), to trim it further down to >> the very essence of symbolic specifications with schematic polymorphism and >> overloading. This is Ondraâs territory, so he should comment here. But my understanding was that what Ondra does in the patch is actually *more conservative* than what we have in the paper. The paper's approach is more refined. >> The proposed change to Isabelle2014 is called "correction patch", as if >> something would be broken that is fixed by the change. This misunderstanding >> leads directly into the general discussion of this work below. Regardless of what the documentation states (explicitly or implicitly), I view the prevention of an undesirable proof of False as a correction. But I understand your point of view. >> These are sections 1 "Introduction", section 2 "Related Work", section 6 >> "Conclusion", i.e. the important pieces that put the technical contribution >> into proper perspective. This is what most people read, and what attendants >> of a conference presentation who are busy with their e-mails or smart-phone >> usually take home. I really hope this is not the case. The paper's overall message is quite positive: Gordon-style HOL can be consistently combined with delayed overloading of constants, and hence (factoring in work by you and others) with Haskell-style type classes. >> The results of the present paper certainly improve the >> situation, but after applying the "correction patch" (i.e. the additional >> feature of fully-checked typedefs) no known applications of Isabelle + AFP >> are rejected. This is good, but it also indicates relatively low practical impact. This is indeed very good (and expected). But I would argue that being able to rely on the definitional mechanisms to not introduce inconsistencies is of intrinsic practical interest. Burkhart's and Holger's posts have illustrated this. While you may be tempted to blame our paper for this, I would say that a look at any piece of documentation stating that type definitions are completely unchecked, in conjunction with the realization that datatypes are based on typedef, would have achieved the same effect. >> In recent years such applications have shown up >> nonetheless, e.g. a type of polynomials that refers to a generic constant >> "zero" (see src/HOL/Library/Polynominal.thy e.g. in Isabelle2014 or >> Isabelle2015). This indicates that the extra expressive power of typedefs >> over type class parameters occasionally matters. Thank you for pointing out this very interesting (and by no means exotic) example. >> It is good that the new explanation of overloading treats constants and >> types uniformly, but there is also an investment into a new model theory >> that deviates from the other HOL systems. I actually like the new >> interpretation better than the one by A. Pitts, but without the mainstream >> HOLs following this upgrade, a translation of Isabelle/HOL theories into >> e.g. HOL Zero or OpenTheory for separate checking would mean that overloaded >> typedefs are axiomatic specifications without any well-formedness checks at all. Very good point. This issue can be addressed by a suitable translation between the two types of semantics. >> The older Example 3 is presented somewhat dramatically "when Obua took over >> the, he found that the overloaded were almost completely unchecked". Here is >> a quote from the relevant documentation (Isabelle2004 or Isabelle2005, "ref" >> manual, section 6.1.1 "Definitions"): >> Definitions are intended to express abbreviations. The simplest form of a >> definition is f == t, where f is a constant. [...] >> Isabelle makes the following checks on definitions: [...] >> These checks are intended to catch the sort of errors that might be made >> accidentally. Misspellings, for instance, might result in additional >> variables appearing on the right-hand side. More elaborate checks could be >> made, but the cost might be overly strict rules on declaration order, etc. >> Why do users and especially power users tend to ignore documentation? There >> are many more fine points about Isabelle that could be misunderstood in >> similar ways; and not all of them are as explicitly documented. I believe people tend to make the assumption that definitions should be at least checked for consistency in order to deserve the status of definition. Likewise, if, for the notion of âtype definitionâ, a long and technical documentation eventually states that it does not in fact represent definitions but (completely unchecked) axioms, then I would argue that there is a âmoralâ inconsistency in the documentation. This situation reminds me of the following joke. Someone calls a radio station in the USSR and asks: âIs it true that Ivan Ilyich was given a brand-new car by the Soviet state?â The (reasonable, but rather too technical) answer is the following: âYes, this is true. However, it was not a car, but a bicycle; and it was not given to him, but taken away from him.â >> A potential explanation why users of Isabelle might consider the system to >> be "absolutely right" and "unbreakable" is that in everyday life serious >> incidents are never encountered, even after several years of intensive use. Again, they donât consider the system to be absolutely right, but merely want to be able to trust that in principle (that is, save for possible implementation errors which should be fixed whenever detected) definitions to not introduce inconsistency. Several Isabellers (e.g, Peter, Lars, Jasmin) and non-Isabellers (e.g, Mark and Ramana) have made similar points on this thread. >> My impression is that the "Consistency Club" is presently empty: none of the >> major ITP systems are absolutely well-defined and correctly implemented in >> every respect. Note that HOL Light is often quoted is being really small and >> formally well-understood, but it resides in the unsafe environment of the >> OCaml toplevel: users need to refrain from playing dirty tricks in ML. We need to distinguish between a logical system and its implementation. Then we would probably agree on the consistency issue. For example, I would argue that now, according to the updated documentation, Isabelle/HOL has joined the consistency club, in the following sense: its logic, given by the inference rules plus type and constant definitions (and excluding the addition of arbitrary axioms), is known/reasonably-believed-to-be consistent, in that there exists a mathematical argument for its consistency. >> Formalizations of complete systems like CakeML + classic HOL look very >> promising, but the community is not there yet. For big environments like >> Isabelle or Coq full formalization appears futile to me. A better strategy >> is to connect to some smaller and fully-proven proof checker. For the >> present paper, this means the authors either need to formalize the extended >> model theory of HOL with full overloading themselves, or convince the >> colleagues [18] to do such an upgrade of their formalization. This is indeed a very interesting prospect. I did not get a chance to answer to Ramanaâs post about this, but I am also very interested in achieving such a degree of certification. Especially since what we propose in the paper is not so far from preservation of standard HOL models, the main difference being the "macro-style" interpretation of polymorphism. >> Then we only need OpenTheory to follow suite, and a proper >> Isabelle/OpenTheory implementation as constructive proof that Isabelle/HOL >> actually fits under the same "umbrella" of HOL-based systems. Then we could >> proceed to the next order of magnitude in overally reliability. A great goal, by all means. The work of the "consistency club" members shows that this may not be infeasible. And I am glad that in the end you do accept the prospect of Isabelle/HOL being declared as a "HOL-based system." But again, I find it strange that you so strongly reject this terminological inclusion today, as in another place you write: >> Unfortunately, the story being told here do not quite fit to Isabelle. >> The misunderstanding already starts in the first paragraph, where >> Isabelle/HOL is included into the "umbrella term" of "HOL-based >> provers". All the best, Andrei -----Original Message----- From: cl-isabelle-users-bounces at lists.cam.ac.uk [mailto:cl-isabelle-users-bounces at lists.cam.ac.uk] On Behalf Of Makarius Sent: 28 September 2015 15:56 To: cl-isabelle-users at lists.cam.ac.uk Subject: Re: [isabelle] Isabelle Foundation & Certification Here is now my virtual review of the ITP paper behind this thread. It came out a bit longer than anticipated, although I included only half of the thoughts that came to me, while reading it 3 times. Makarius ------------------------------------------------------------------------------ Title: A Consistent Foundation for Isabelle/HOL Authors: OndÅej KunÄar, Andrei Popescu http://www21.in.tum.de/~kuncar/documents/kuncar-popescu-itp2015.pdf Overall evaluation: accept Reviewer's confidence: high Review: The paper revisits the question how overloaded definitions (of constants and type constructores) may be understood within the HOL logic. This work consists of three parts: (1) Technical treatment of the HOL logic, with a notion of "definitional theory" and its well-formedness, interpretation of this extended HOL logic in a substantially reformed version of the original HOL88 interpretation due to A. Pitts. (2) A proposed change to the Isabelle2014 code base, to upgrade its 'typedef' mechanism to perform the additional well-formedness checks described in the paper. The text refers to http://www21.in.tum.de/~kuncar/documents/patch.html which provides some changesets and one page of explanations how this fits into the setup of the paper. (3) General motivation and discussion of perceived problems of Isabelle/HOL, which the paper sets out to resolve. This part is apt to cause confusion in various ways, both to users and non-users of Isabelle. The style of presentation considerably subtracts from this otherwise fine paper. Part (1) -------- These are sections 3, 4, 5 in the paper. Section 5 is the main contribution. The classic HOL logic of HOL88 (due to M. Gordon and semantics by A. Pitts) is taken as starting point. There is a nice explanation in 4.3, why overloaded definitions of constants or types does not work out in the interpretation of A. Pitts (HOL88 book from 1993). Thus a new interpretation is introduced as follows: * Types are not immediately interpreted away into sets, but preserved as syntactic entities to keep sufficient structure that allows well-founded recursion. The paper calls this "a natural syntactic-semantic blend, which avoids stunt performance in set theory". * Type interpretation starts out on ground types. Polymorphism is interpreted only in the final stage, treating it as strictly schematic over all ground interpretations. * Models for ground interpretation are constructed in stages, following the well-founded dependency relation of the given collection of definitions. Here types and constants are interpreted simultaneously for each stage, while in the HOL88 semantics by A. Pitts all types are interpreted before all constants. This model construction is fit together with other recent work by the first author [19], opening the perspective of fully checked definitional theories in Isabelle/HOL. More about this in part (2). Taking the little space of a paper in LNCS proceedings into account, the technical treatment of the problem is worked out very carefully. Numerous fine points and additional side-conditions are included in the considerations. Nonetheless, there is a potential for confusion due to slight deviation of terminology and concepts, compared to papers on the subject e.g. by Wenzel/Haftmann and the Isabelle documentation (isar-ref manual). In particular: Section 3.4 defines a "definitional theory" as a set of overloaded definitions (for constants or types) such that certain *local* syntactic conditions hold. Section 4.2 defines a "well-formed definitional theory" as a definitional theory with well-founded dependency relation as *global* condition. Section 4.5 establishes the main theorem that a well-formed definitional theory is consistent -- by demonstrating that it has a model in the above sense, and concluding that False cannot be derived in the inference system -- due to soundness of the interpretation. In contrast, the specifications of a "definitional theory" of the paper would be called in the Isabelle documentation "unchecked definitions", "definitional axioms", or "axiomatizations", to emphasize the missing aspect of global well-formedness that this paper is mainly about. A "well-formed definitional theory" of the paper would be just called "definitional theory" in Isabelle terminology, and much stronger properties intended than just consistency. Instead of merely ensuring the existence of a model, the requirement is to preserve derivability of old propositions precisely, and allow to reduce new derivations of new propositions into old derivations by "expanding" definitions in some way. If and how this works depends on fine-points of the underlying definitional mechanisms: for overloaded constant definitions in Isabelle/Pure this works [13, 35], but for type definitions the situation is a-priori quite different, e.g. see [13] section 4.3. The deeper reason for this are the distinctive categories of terms and types in HOL, and the lack of equational reasoning on types. A suitable semantic treatment is required to re-unify the view on terms and types, as is done in the present paper. But this leads to weaker results. The paper does briefly mention "a suitable notion of conservativeness" as future work, but due to lack of further explanations, many readers will probably miss key point behind this. There is another aspect of "consistency" versus "conservativity" that is easily missed in the paper. Section 3.5 states "Isabelle/HOL theory development proceeds by: 1 ... 2 ... 3 ...". The three kinds of operations ensure that every stage of theory development is "definitional" in the sense of the paper -- and "well-formedness" can be established later. But what happens when the user starts adding non-definitional axioms in between? Strictly speaking, the main result does not apply and the whole theory looses its good definitional properties. In contrast, the traditional explanation of (constant) definitions in Isabelle is more modular in that respect: given an arbitrary base theory --- one that somehow makes sense to the user in a particular application, even after making unchecked axiomatizations --- definitions merely introduce names for existing entities and thus preserve the key properties of the base theory. This agnostic approach of "definitional theory extensions" (occording to Isabelle terminolgy) is particularly important from the perspective of the Isabelle/Pure framework, where the interpretation of object-logics is not yet known. For HOL applications one might argue that genuine axiomatic extensions are not done in practice these days. Nonetheless some exotic applications like HOLZF (by Obua) do exist. Without stronger results under which conditions typedefs "make sense" or "are OK" (using words from the paper), such ambitious users would have to revisit the whole model theory of HOL again. Part (2) -------- The patch to Isabelle2014 is rather minimal: the existing infrastructure to check overloaded constant definitions (going back to Isabelle2007, and slightly improved by O. Kuncar [19] is generalized to cover type definitions as well. So Isabelle/Pure is upgraded to provide a general service for "definitional specification items" that are identified in the name space for constants or types, but without looking at actual content. While the explanations for the patch states "the situation is from a technical (implementation) point view a little bit more complicated", the outcome is actually simpler than in the paper. Being forced to strip away accidental aspects of the HOL object-logic does occasionally have advantages. For example, the delicate notion of "build-in types" vs. "non-built-in types" is absent in the implementation. The notes on the patch provide some explanations, why it works out nonetheless. It would be nice to see this refinement applied to the main work of part (1), to trim it further down to the very essence of symbolic specifications with schematic polymorphism and overloading. The proposed change to Isabelle2014 is called "correction patch", as if something would be broken that is fixed by the change. This misunderstanding leads directly into the general discussion of this work below. Part (3) -------- These are sections 1 "Introduction", section 2 "Related Work", section 6 "Conclusion", i.e. the important pieces that put the technical contribution into proper perspective. This is what most people read, and what attendants of a conference presentation who are busy with their e-mails or smart-phone usually take home. Unfortunately, the story being told here do not quite fit to Isabelle. The misunderstanding already starts in the first paragraph, where Isabelle/HOL is included into the "umbrella term" of "HOL-based provers". HOL4, HOL-Light, ProofPower, HOL Zero are fine systems, but Isabelle/HOL is not as closely related to them as the "HOL" name might suggest. In many respects Isabelle is actually closer to Coq. Technically, the key misunderstanding is the role of "the kernel" (in the words of the paper). HOL88 as the predecessor of all the other HOL systems pioneered an add-on to the original LCF kernel design to have checked definitions as primitive rules. In contrast, Isabelle is closer to LCF in separating logical inferences and a-priori unchecked (axiomatic) theory specifications: there is no notion of definitions in the inference kernel of Isabelle. Since this is a bit impractical for big applications, more and more checks on theory content have been added over the years (e.g. 2004, 2005, 2007, 2010, 2014), but that process was never formally closed. And this is what the paper is actually about: complete emulation of HOL88-style definitions in Isabelle/HOL (by providing additional services in the Isabelle/Pure framework). We can now revisit the critical examples in the light of the Isabelle architecture and tradition, instead of the one of HOL88. Example 2 is presented as the main failure point to motivate this work, but it is merely an illustration what the isar-ref manual, section 11.5 "Typedef axiomatization" covers abstractly: 'typedef' is a certain axiom scheme of Isabelle/HOL with some local sanity checks. There is no claim of it being "definitional" (in Isabelle terminology, "well-formed definitional" in the paper). So the proof of False is not a problem of the Isabelle inference kernel, but the consequence of producing a malformed axiomatic theory. A legitimate discussion would be about the chances of users making such mistakes in practice. The results of the present paper certainly improve the situation, but after applying the "correction patch" (i.e. the additional feature of fully-checked typedefs) no known applications of Isabelle + AFP are rejected. This is good, but it also indicates relatively low practical impact. Other interesting information that is missing: How often do type definitions depend on open parameters due to overloading, notably type class parameters? The original concept of overloading in Isabelle did not expect types to depend on unresolved overloading at all: such closed type definitions could be interpreted within the original setting of HOL88, after expansion of all constant definitions. In recent years such applications have shown up nonetheless, e.g. a type of polynomials that refers to a generic constant "zero" (see src/HOL/Library/Polynominal.thy e.g. in Isabelle2014 or Isabelle2015). This indicates that the extra expressive power of typedefs over type class parameters occasionally matters. It is good that the new explanation of overloading treats constants and types uniformly, but there is also an investment into a new model theory that deviates from the other HOL systems. I actually like the new interpretation better than the one by A. Pitts, but without the mainstream HOLs following this upgrade, a translation of Isabelle/HOL theories into e.g. HOL Zero or OpenTheory for separate checking would mean that overloaded typedefs are axiomatic specifications without any well-formedness checks at all. The older Example 3 is presented somewhat dramatically "when Obua took over the, he found that the overloaded were almost completely unchecked". Here is a quote from the relevant documentation (Isabelle2004 or Isabelle2005, "ref" manual, section 6.1.1 "Definitions"): Definitions are intended to express abbreviations. The simplest form of a definition is f == t, where f is a constant. [...] Isabelle makes the following checks on definitions: [...] These checks are intended to catch the sort of errors that might be made accidentally. Misspellings, for instance, might result in additional variables appearing on the right-hand side. More elaborate checks could be made, but the cost might be overly strict rules on declaration order, etc. Why do users and especially power users tend to ignore documentation? There are many more fine points about Isabelle that could be misunderstood in similar ways; and not all of them are as explicitly documented. A potential explanation why users of Isabelle might consider the system to be "absolutely right" and "unbreakable" is that in everyday life serious incidents are never encountered, even after several years of intensive use. Thus the sense for the degree of reliability is lost. We desparately need for fine-grained assesment of the overall quality of ITP systems than the "Consistency Club" versus "Inconsistency Club" in the paper. My impression is that the "Consistency Club" is presently empty: none of the major ITP systems are absolutely well-defined and correctly implemented in every respect. Note that HOL Light is often quoted is being really small and formally well-understood, but it resides in the unsafe environment of the OCaml toplevel: users need to refrain from playing dirty tricks in ML. Formalizations of complete systems like CakeML + classic HOL look very promising, but the community is not there yet. For big environments like Isabelle or Coq full formalization appears futile to me. A better strategy is to connect to some smaller and fully-proven proof checker. For the present paper, this means the authors either need to formalize the extended model theory of HOL with full overloading themselves, or convince the colleagues [18] to do such an upgrade of their formalization. Then we only need OpenTheory to follow suite, and a proper Isabelle/OpenTheory implementation as constructive proof that Isabelle/HOL actually fits under the same "umbrella" of HOL-based systems. Then we could proceed to the next order of magnitude in overally reliability. ------------------------------------------------------------------------------

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