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Isabelle is a generic theorem prover with a fragment of higher-order logic as a metalogic for deﬁning object logics. Isabelle also provides proof terms. We formalize this metalogic and the language of proof terms in Isabelle/HOL, deﬁne an executable (but inefﬁcient) proof term checker and prove its correctness w.r.t. the metalogic. We integrate the proof checker with Isabelle and run it on a range of logics and theories to check the correctness of all the proofs in those theories. Keywords Theorem proving · Higher-order logic · Isabelle · Proofchecker · Metalogic 1 Introduction One of the selling points of proof assistants is their trustworthiness. Yet in practice, soundness problems do come up in most proof assistants. Harrison [15] distinguishes errors in the logic and errors in the implementation (and cites examples). Our work contributes to the solution of both problems for the proof assistant Isabelle [35]. Isabelle is a generic theorem prover: it implements M, a fragment of intuitionistic higher-order logic, as a metalogic for deﬁning object logics. Its most developed object logic is HOL, and the resulting proof assistant is called Isabelle/HOL [27, 28]. The latter is the basis for our formalizations. Our ﬁrst contribution is the ﬁrst complete formalization of Isabelle’s metalogic. Thus our work applies to all Isabelle object logics, e.g., not only HOL but also ZF. Of course Paulson [36] describes M precisely, but only on paper. More importantly, his description does not yet cover polymorphism and type classes, which were introduced later [29]. The published account of Isabelle’s proof terms [7] is also silent about type classes, yet type classes are a signiﬁcant complication. We do, however, not formalize the theory extension mechanisms (e.g., for constant deﬁnitions) on top of the logic. Our second contribution is a veriﬁed (against M) and executable checker for Isabelle’s proof terms. We have integrated the proof checker with Isabelle. Thus, we can guarantee that Simon Roßkopf rosskops@in.tum.de https://www21.in.tum.de/∼rosskops/ Tobias Nipkow https://www21.in.tum.de/∼nipkow/ Department of Informatics, Technical University of Munich, Munich, Germany 0123456789().: V,-vol 123 1 Page 2 of 21 S. Roßkopf, T. Nipkow every theorem whose proof our proof checker accepts is provable in our deﬁnition of M.So far we are able to check the correctness of moderately sized theories across the full range of logics implemented in Isabelle. Although Isabelle follows the LCF-architecture (theorems that can only be manufactured by inference rules) it is based on an infrastructure optimized for performance. In particular, this includes multithreading, which is used in the kernel and has once led to a soundness issue. Therefore we opt for the “certiﬁcate checking” approach (via proof terms) instead of verifying the implementation. This is the ﬁrst work that deals directly with what is implemented in Isabelle as opposed to a study of the metalogic that Isabelle is meant to implement. Instead of reading the imple- mentation, you can now read and build on the more abstract formalization in this paper. The correspondence of the two can be established for each proof by running the proof checker. Our formalization reﬂects the ML implementation of Isabelle’s terms and types and some other data structures. Thus, a few implementation choices shine through, e.g., De Bruijn indices. This is necessary because we want to integrate our proof checker as directly as possible with Isabelle, with as little unveriﬁed glue code as possible, for example, no trans- lation between De Bruijn indices and named variables. We refer to this as our intentional implementation bias. In principle, however, one could extend our formalization with different representations (e.g., named terms) and prove suitable isomorphisms. Our work is purely proof theoretic; semantics is out of scope. This paper is an extended version of a conference paper [30] presented at CADE 28. In addition to the material covered in the conference paper, it includes • A section describing some useful derived rules in our inference system (Sect. 7) • A more detailed description of the executable proofchecker and its veriﬁcation (Sect. 8) • An updated formalization, including an updated, more natural formalization of the order- sorted signatures (Sect. 4), generic variable types, explicitly ﬁnite data structures, and updated proof terms. 1.1 Related Work Harrison [15] was the ﬁrst to verify some of HOL’s metatheory and an implementation of a HOL kernel in HOL itself. Kumar et al. [20] formalized HOL including deﬁnition principles, proved its soundness and synthesized a veriﬁed kernel of a HOL prover down to the machine language level. Abrahamsson [1] veriﬁed a proof checker for the OpenTheory [17] proof exchange format for HOL. Wenzel [43] showed how to interpret type classes as predicates on types. We follow his approach of reﬂecting type classes in the logic but cannot remove them completely because of our intentional implementation bias (see above). Kuncar ˇ and Popescu [21–24] focus on the subtleties of deﬁnition principles for HOL with overloading and prove that under certain conditions, type and constant deﬁnitions preserve consistency. Åman Pohjola et al. [3] formalize some of this work by Kuncar ˇ and Popescu [21, 24]. Adams [2] presents HOL Zero, a basic theorem prover for HOL that addresses the problem of how to ensure that parser and pretty-printer do not misrepresent formulas. Let us now move away from Isabelle and HOL. Barras veriﬁed fragments of Coq in Coq [4, 5]. Sozeau et al. [41] present the ﬁrst implementation of a type checker for the kernel of Coq that is proved correct in Coq with respect to a formal speciﬁcation. Carneiro [8]has https://mailmanbroy.in.tum.de/pipermail/isabelle-dev/2016-December/007251.html. 123 A Formalization and Proof... Page 3 of 21 1 implemented a highly performant proof checker for a multi-sorted ﬁrst-order logic and is in the process of verifying it in its own logic. Davis developed the bootstrapping theorem prover Milawa [9] and, together with Myreen, showed its soundness down to machine code [10]. We formalize a logic with bound variables, and there is a large body of related work that deals with this issue (e.g., [11, 18, 42]) and a range of logics and systems with special support for handling bound variables (e.g., [38–40]). We found that De Bruijn indices worked reasonably well for us. 2 Preliminaries Isabelle types are built from type variables, e.g., a and (postﬁx) type constructors, e.g., alist; the function type arrow is ⇒. Isabelle also has a type class system explained later. The notation t ::τ means that term t has type τ . Isabelle/HOL provides types aset and alist of sets and lists of elements of type a. They come with the following vocabulary: function set (conversion from lists to sets), (#) (list constructor), (@) (append), |xs| (length of list xs), xs ! i (the ith element of xs starting at 0), list-all2 p [x , ..., x ][y , ..., y ] 1 m 1 n = (m = n ∧ px y ∧ ... ∧ px y ), (> > =) (monadic bind) and other self-explanatory 1 1 n n notation. There is also the predeﬁned data type datatype aoption = None | Some a The type τ τ abbreviates τ ⇒ τ opti on, i.e., partial functions, which we call maps. 1 2 1 2 Maps have a domain and a range: dom m ={a | ma = None} ran m ={b |∃a. ma = Some b} It must be noted that in our formalization, we are not using sets/maps directly, but subtypes for ﬁnite sets/maps. This simpliﬁes some proofs and code generation; however, there is less material about them readily available. Luckily, we can easily make use of material for general sets/maps using Isabelle’s Lifting and Transfer packages[16]. Logical equivalence is written = instead of ←→. 3 Types and Terms A name is simply a string. Variables have (Isabelle/HOL level) type v; their inner structure is immaterial for the presentation of the logic. We only require v to be inﬁnite, to always guarantee a supply of fresh variables. We encode this using a type class for inﬁnite types. The logic has three layers: terms are classiﬁed by types as usual, but in addition, types are classiﬁed by sorts.A sort is simply a set of classes and classes are just strings. We discuss sorts in detail later. Types (typically denoted by T , U , …) are deﬁned like this: datatype v typ = Ty name ( v typ list ) | Tv v sort where Ty κ [T ,...,T ] represents the Isabelle type (T ,...,T )κ and Tv aS represents a 1 n 1 n type variable a of sort S—sorts are directly attached to type variables and contribute to their identity. The notation T → U is short for Ty "fun" [T ,U ],where "fun" is the name of the function type constructor. Isabelle’s terms are simply typed lambda terms in De Bruijn notation: 123 1 Page 4 of 21 S. Roßkopf, T. Nipkow datatype v term = Ct name ( v typ) | Fv v( v typ) | Bv nat | Abs ( v typ)( v term) | (·)( v term)( v term) A term (typically r, s, t, u …) can be a typed constant Ct cT or free variable Fv v T , a bound variable Bv n (a De Bruijn index), a typed abstraction Abs Tt or an application t · u.Wecall an occurrence of a bound variable Bv i in some term t loose if the occurrence is not enclosed in at least i + 1 abstractions. The term-has-type proposition has the syntax Ts t : T where Ts is a list of types, the context for the type of the bound variables. i < |Ts| Ct _ T : T _ Fv _ T : T τ τ Ts Bv i : Ts ! i T # Ts t : T Ts Abs Tt : T → T Ts u :UTs t : U → T τ τ Ts t · u : T We deﬁne t : T =[] t : T . τ τ Function fv :: v term ⇒ ( v × v typ) set collects the free variables in a term. Because bound variables are indices, fv t is simply the set of all (v, T ) such that Fv v T occurs in t. The type is an integral part of a variable. A type substitution is a function of type v ⇒ sort ⇒ v typ. It assigns a type to each type variable and sort pair. We write $$ T or $$ t for the overloaded function which applies a type substitution to all type variables (and their sort) occurring in a type or term. The type instance relation is deﬁned like this: T T = (∃. $$ T = T ) 1 2 2 1 We also need to β-contract a term Abs Tt · u to something like “t with Bv 0 replaced by u.” We deﬁne a function subst-bv such that subst-bv ut is that β-contractum. The deﬁnition of subst-bv is shown in the Appendix and can also be found in the literature (e.g., [33]). In order to abstract over a free (term) variable, there is a function bind-fv (v, T ) t that (roughly speaking) replaces all occurrences of Fv v T in t by Bv 0. Again, see the Appendix for the deﬁnition. This produces (if Fv v T occurs in t) a term with a loose Bv 0. Function Abs-fv binds it with an abstraction: Abs-fv v Tt = Abs T (bind-fv (v, T ) t ) While this section described the syntax of types and terms, they are not necessarily wellformed and should be considered pretypes/preterms. The wellformedness checks are described later. 4 Classes and Sorts Isabelle has a built-in system of type classes [32] as in Haskell 98 except that class con- straints are directly attached to variable names: our Tv a [C ,D,...] corresponds to Haskell’s (Ca, Da, ...) => ... a ....A sort is Isabelle’s terminology for a set of (class) names, e.g., {C ,D,...}, which represent a conjunction of class constraints. In our work, variables S, S etc. stand for sorts. Apart from the usual application in object logics, type classes also serve an important metalogical purpose: they allow us to restrict, for example, quantiﬁcation in object logics to object-level types and rule out meta-level propositions. 123 A Formalization and Proof... Page 5 of 21 1 Isabelle’s type class system was ﬁrst presented in a programming language context [31, 34]. We give the ﬁrst machine-checked formalization. The central data structure is a so- called order-sorted signature. Intuitively, it is composed of a set of classes, a partial subclass ordering on them and a set of type constructor signatures. A type constructor signature κ :: (S , ..., S ) c for a type constructor κ states that applying κ to types T , ..., T such 1 k 1 k that T has sort S (deﬁned below) produces a type of class c. Formally: i i type_synonym osig = (name set × (name × name) set × (name × sort list × class) set ) The projection functions are called classes, subclass and tcsigs. The subclass ordering sub can be extended to a subsort ordering as follows: S ≤ S = (∀c ∈S . ∃c ∈S . c ≤ c ) 1 sub 2 2 2 1 1 1 sub 2 The smaller sort needs to subsume all the classes in the larger sort. In particular {c }≤ {c } iff (c , c ) ∈ sub. 1 sub 2 1 2 Now we can deﬁne a predicate has-sort that checks whether, in the context of some order-sorted signature (cl,sub,tcs), a type fulﬁlls a given sort constraint: S ≤ S sub has-sort (cl, sub, tcs)(Tv aS) S ∀c∈S. ∃Ss.(κ, Ss, c) ∈ tcs ∧ list-all2 (has-sort (cl, sub, tcs)) Ts Ss has-sort (cl, sub, tcs)(Ty κ Ts) S The rule for type variables uses the subsort relation and is obvious. A type (T , ..., T )κ 1 n has sort {c , ...} if for every c there is a signature κ :: (S , ..., S ) c and 1 i 1 n i has-sort(cl, sub,tcs) T S for j = 1, ..., n. j j We normalize a sort by removing “superﬂuous” class constraints, i.e., retaining only those classes that are not subsumed by other classes. This gives us unique representatives for sorts which we call normalized: normalize-sort sub S ={c ∈ S |¬ (∃c ∈S. c = c ∧ (c , c) ∈ sub)} normalized-sort sub S = (normalize-sort sub S = S) We work with normalized sorts because it simpliﬁes the derivation of efﬁcient executable code later on. Now we can deﬁne wellformedness of an osig: wf-osig (cl, sub, tcs) = (wf-subclass cl sub ∧ wf-tcsigs cl sub t cs) A subclass relation is wellformed if it is a partial order where reﬂexivity is restricted to the set of classes cl.Wellformedness of type constructor signatures (wf-tcsigs) is more complex. The conditions are the following: • The following property requires a) that for any κ :: (...) c theremustbea κ :: (...) c for 1 2 every superclass c of c and b) coregularity which guarantees the existence of principal 2 1 types [14, 31]: ∀(κ, Ss , c )∈tcs. 1 1 ∀c .(c , c ) ∈ sub −→ 2 1 2 (∃Ss .(κ, Ss , c ) ∈ tcs ∧ list-all2 (sort-leq sub ) Ss Ss ) 2 2 2 1 2 • A type constructor must always take the same number of argument types: ∀κ Ss c Ss c . 1 1 2 2 (κ, Ss , c ) ∈ tcs ∧ (κ, Ss , c ) ∈ tcs −→ |Ss |=|Ss | 1 1 2 2 1 2 123 1 Page 6 of 21 S. Roßkopf, T. Nipkow • Sorts must be normalized and must exist in cl: ∀(κ, Ss, c)∈tcs. ∀S∈Ss. wf-sort cl sub S where wf-sort cl sub S = (normalized-sort sub S ∧ S ⊆ cl) • The argument sorts uniquely determine the class of the constructed type: ∀(κ, Ss , c)∈tcs. ∀Ss .(κ, Ss , c) ∈ tcs −→ Ss = Ss 1 2 2 2 1 These conditions are used in a number of places to show that the type system is well behaved. For example, has-sort is upward closed: wf-osig (cl, sub, tcs) ∧ has-sort (cl, sub, tcs) TS∧S ≤ S sub −→ has-sort (cl, sub, tcs) TS 5 Signatures A signature consists of a map from constant names to their (most general) types, a map from type constructor names to their arities, and an order-sorted signature: type_synonym vsignature = (name v typ) × (name nat ) × osig The three projection functions are called const-type, type-arity and osig.Wenow deﬁne a number of wellformedness checks w.r.t. a signature . We start with wellformedness of types, which essentially requires that all type constructors have correct arity and all type variables have wellformed sort constraints: type-arity κ = Some |Ts|∀T ∈Ts. wf-type T wf-type (Ty κ Ts) wf-sort (classes (osig )) (subclass (osig )) S wf-type (Tv aS) Wellformedness of a term essentially just says that all types in the term are wellformed and that the type T of a constant in the term must be an instance of the type T of that constant in the signature: T T . wf-type T wf-term (Bv n) wf-term (Fv v T ) const-type s = Some T wf-type T T T wf-term (Ct sT ) wf-term t wf-term u wf-term (t · u) wf-type T wf-term t wf-term (Abs Tt ) These rules only check whether a term conforms to a signature, not that the contained types are consistent. Combining wellformedness and yields welltypedness of a term: wt-term t = (wf-term t ∧ (∃T . t : T )) Wellformedness of a signature = (ct f , ar f , oss) where oss = (cl, sub, tcs) is deﬁned as follows: wf-sig = ((∀T ∈ran ct f . wf-type T ) ∧ (wf-osig oss ∧ (∀(κ, Ss, c)∈tcs. ar f κ = Some |Ss|)) 123 A Formalization and Proof... Page 7 of 21 1 In words: all types in ct f are wellformed, oss is wellformed, and the type constructors in tcs are exactly those that have a matching arity in ar f . 6 Logic Isabelle’s metalogic M is an extension of the logic described by Paulson [36]. It is a fragment of intuitionistic higher-order logic. The basic types and connectives of M are the following: Concept Representation Abbreviation Type of propositions Ty "prop" [] prop Implication Ct "imp" (prop → prop → prop) ⇒ Universal quantiﬁer Ct "all" ((T → prop) → prop) Equality Ct "eq" (T → T → prop) = The type subscripts of and ≡ are dropped in the text if they can be inferred. Readers familiar with Isabelle syntax must keep in mind that for readability we use the symbols , ⇒ and ≡ for the encodings of the respective symbols in Isabelle’s metalogic. We avoid the corresponding metalogical constants completely in favor of HOL’s ∀, −→, =, and inference rule notation. The provability judgment of M is of the form , t where is a theory, (the hypotheses) is a set of terms of type prop,and t a term of type prop. A theory is a pair of a signature and a set of axioms: type_synonym v theor y = v signature × v term set The projection functions are sig and axioms. We extend the notion of wellformedness from signatures to theories: wf-theory (, axs) = (wf-sig ∧ (∀ p∈axs. wt-term p ∧ p : prop) ∧ is-std-sig ∧ eq-axs ⊆ axs) The ﬁrst two conjuncts need no explanation. Predicate is-std-sig (not shown) requires the signature to have certain minimal content: the basic types (→, prop) and constants (≡, , ⇒)of M and the additional types and constants for type class reasoning from Sect. 6.3. Our theories also need to contain a minimal set of axioms. The set eq-axs is an axiomatic basis for equality reasoning and will be explained in Sect. 6.2. We will now discuss the inference system in three steps: the basic inference rules, equality, and type class reasoning. 6.1 Basic Inference Rules The axiom rule states that wellformed type-instances of axioms are provable: wf-theory t ∈ axioms wf-inst (sig ) $$ t where :: v ⇒ sort ⇒ v typ is a type substitution and $$ denotes its application (see Sect. 3). The types substituted into the type variables need to be wellformed and conform to the sort constraint of the type variable: 123 1 Page 8 of 21 S. Roßkopf, T. Nipkow wf-inst = (∀v S. v S = Tv v S −→ has-sort (osig ) ( v S) S ∧ wf-type ( v S)) The conjunction only needs to hold if actually changes something, i.e., if v S = Tv v S.This condition is not superﬂuous because otherwise has-sort oss (Tv v S) S and wf-type (Tv v S) only hold if S is wellformed w.r.t . Note that there are no extra rules for general instantiation of type or term variables. Type variables can only be instantiated in the axioms. Term instantiation can be performed using the introduction and elimination rules. The assumption rule allows us to prove terms already in the hypotheses: wf-term (sig ) t t : prop t ∈ Both and ⇒ are characterized by introduction and elimination rules: wf-theory, t (x , T)/ ∈ FV wf-type (sig ) T (Abs-fv xT t ) (Abs Tt ) u : T wf-term (sig ) u T τ subst-bv ut wf-theory, u wf-term (sig ) t t : prop , −{t} t ⇒ u t ⇒ u , 1 2 , ∪ 1 2 where FV = ( fv t ). t ∈ 6.2 Equality Most rules about equality are not part of the inference system but are axioms (the set eq-axs mentioned above). Consequences are obtained via the axiom rule. The ﬁrst three axioms express that = is reﬂexive, symmetric, and transitive: x ≡xx ≡ y ⇒ y ≡xx ≡ y ⇒ y ≡ z ⇒ x ≡ z The next two axioms express that terms A and B of type prop are equal iff they imply each other: A ≡ B ⇒ A ⇒ B ( A ⇒ B) ⇒ (B ⇒ A) ⇒ A ≡ B The last equality axioms are congruence rules for application and abstraction: f ≡ g ⇒ x ≡ y ⇒ ( f · x ) ≡ (g · y) (Abs T (( f · Bv 0) ≡ (g · Bv 0))) ⇒ Abs T ( f · Bv 0) ≡ Abs T (g · Bv 0) Paulson [36] gives a slightly different congruence rule for abstraction, which allows to abstract over an arbitrary, free x in f ,g. We are able to derive this rule in our inference system. Finally, there are the lambda calculus rules. There is no need for α conversion because α-equivalent terms are already identical thanks to the De Bruijn indices for bound variables. For β and η conversion the following rules are added. In contrast to the rest of this subsection, these are not expressed as axioms. 123 A Formalization and Proof... Page 9 of 21 1 wf-theory wt-term (sig ) (Abs Tt ) wf-term (sig ) u u : T (β) (Abs Tt · u) ≡ subst-bv ut wf-theory wf-term (sig ) t t : T → T (η) Abs T (t · Bv 0) ≡ t Rule (β) uses the substitution function subst-bv as explained in Sect. 3 (and deﬁned in the Appendix). Rule (η) requires a few words of explanation. We do not explicitly require that t does not contain Bv 0. This is already a consequence of the precondition that t : T → T : it implies that t is closed. For that reason, it is perfectly unproblematic to remove the abstraction above t. 6.3 Type Class Reasoning Wenzel [43] encoded class constraints of the form “type T has class C” in the term language as follows. There is a unary type constructor named "itsel f "and T itself abbreviates Ty "itsel f " [T ]. The notation TYPE is short for Ct "type" (T itself) where "type"isthe T itself name of a constant. You should view TYPE as the term-level representation of type T . T itself Next we represent the predicate “is of class C” on the term level. For this, we deﬁne some ﬁxed injective mapping const-of-class from class to constant names. For each new class C, a new constant const-of-class C of type T itself → prop is added. The term Ct (const-of-class C)(T itself → prop) · TYPE represents the statement “type T has T itself class C”. This is the inference rule deriving such propositions: wf-theory const-type (sig ) (const-of-class C ) = Some ( a itself → prop) wf-type (sig ) T has-sort (osig (sig )) T {C } Ct (const-of-class C)(T itself → prop) · TYPE T itself This is how the has-sort inference system is integrated into the logic. This concludes the presentation of M. We have shown some minimal sanity properties, incl. that all provable terms are of type prop and wellformed: t −→ t : prop ∧ wf-term (sig ) t The attentive reader will have noticed that we do not require unused hypotheses in to be wellformed and of type prop. Similarly, we only require wf-theory in rules that need it to preserve wellformedness of the terms and types involved. To restrict to wellformed theories and hypotheses, we deﬁne a top-level provability judgment that requires wellformedness: t = (wf-theory ∧ (∀h∈ . wf-term (sig ) h ∧ h : prop) ∧ , t ) 7 Admissible Rules Reasoning directly with these basic rules can be very tedious. In the following, we discuss some useful admissible rules that we frequently encountered during our formalization work and sketch their formal proofs. 123 1 Page 10 of 21 S. Roßkopf, T. Nipkow As already mentioned, our inference system has no inbuilt way of performing term sub- stitutions and one has to simulate them using -introductions and eliminations. This can be particularly annoying when performing simultaneous substitutions, as one needs to ensure that no interferences occur. We deﬁne a function subst-term which takes a list of (vari- able,term) pairs, an association list, and substitutes them simultaneously into a term and prove the following corresponding rule: wf-theory B tinst-ok (sig ) insts fst ‘ insts ∩ FV =∅ subst-term insts B where tinst-ok requires there to be only one instantiation per variable and for the terms to be wellformed and of the same type as their corresponding variable. The last condition ensures that we do not substitute into variables occurring in an assumption. To prove this rule, we start with the special case of a single instantiation, which is performed by a single -introduction followed by a -elimination. To use this result to prove our desired rule, we need to perform the simultaneous instantiations sequentially. This is not possible in general, as they might interfere with one another. To remedy this, we decompose the substitution process into two phases: First we replace all variables we want to substitute into with distinct, fresh variables. Then we modify the original instantiations, so they substitute into their corresponding new variables instead. As these are fresh, they do not occur in the substituted terms. Therefore, no interference occurs and we can perform both phases sequentially. Another useful derived result is the weakening rule, which tells us that correct inferences remain correct when adding additional (superﬂuous) assumptions: B wf-term (sig ) A A : prop ,{ A}∪ We prove this rule by rule induction on , B. Most cases do not interact with at all and are, therefore, trivial. The only interesting case is the one for -introduction, where, once again, variable capture causes trouble. To prove this case, we would like to use the corresponding -introduction rule, but the added assumption A might contain the variable we want to bind. To ﬁx this problem, we use the substitution rule proved above to rename the problematic variable in A. As we cannot use this rule to substitute into the hypotheses, we use implication rules to move A into the proposition on the right side of the turnstile. Another major source of complications is equality reasoning. We start by providing cor- responding proof rules for each equality axiom, making them easier to use. For example, the resulting rule for reﬂexivity looks like this: wf-theory ∀ A∈ . wf-term (sig ) A ∧ A : prop wt-term (sig ) t t ≡ t The proofs for all these rules are very similar. We ﬁrst prove them for an empty set of assump- tions, to, once again, avoid accidental variable capturing. For this, we use the axiom rule and the derived simultaneous term instantiation rule to substitute the correct types and terms into the axiom. Then we allow for arbitrary (well typed) assumptions using the weakening rule. By just combining these equality rules and rules of the inference system one can derive new rules, like for example, Paulson’s original congruence for abstraction. Such derivations are, however, complicated by having to propagate the wellformedness conditions of all involved 123 A Formalization and Proof... Page 11 of 21 1 objects, which makes them lengthy. As they also clutter the presentation, we will not show them for the rest of this section and assume that all involved objects in the rules are wellformed. A last derived rule, which will be useful later, is the fact that β-reduction preserves provability. Our inference system already contains a rule concerning β-reduction but it can only be applied at the top of a term. We ﬁrst deﬁne an inductive notion of a β-step, based on a formalization by Nipkow [33]. Abs Ts · t → subst-bv ts s → t s → t s → t β β β s · u → t · u u · s → u · t Abs Ts → Abs Tt β β β This allows us to state a more useful β rule, which proves equality of two terms if they just differ by a single beta step, regardless of where it occurs. t → u t ≡ u We naturally want to prove this statement by induction over (→ ). Because (→ ) is deﬁned β β by four rules, there are four cases. The ﬁrst case allows application of the β rule of the inference system, the next two use the congruence rule for applications and the respective induction hypotheses. However, the last case is a problem, as descending under an abstraction can expose previously bound variables, which means we cannot apply our proof rules. To remedy this, we replace the now loose variable with a fresh free variable and perform our reasoning with the again wellformed term. The following rule justiﬁes this transformation and is readily proved by combining basic and derived inference rules: (x,τ )/ ∈ FV ∪ fv s ∪ fv t subst-bv (Fv x τ ) s ≡ subst-bv (Fv x τ ) t Abs τ s ≡ Abs τ t However, applying this rule makes it impossible to use the induction hypothesis as adding the substitution changes the shape of the goal. The solution is to generalize the β rule to reﬂect that one might have passed abstractions and substituted the respective loose variables by new fresh variables: set vs ∩ (fv t ∪ FV ) =∅ t → u subst-bvs (map (λ(v, τ ). Fv vτ) vs) t ≡ subst-bvs (map (λ(v, τ ). Fv vτ) vs) u This rule uses the subst-bvs function, which behaves like the previously seen subst-bv function, only instantiating multiple loose variables simultaneously. We can prove that it is possible to merge the call to subst-bv, arising in the problematic case with the other sub- stitutions. Therefore, we are now able to apply the induction hypothesis. The original rule is the special case for an empty list of variables. A similar approach was taken to prove that η reduction preserves provability. Because of symmetry, we have also proved that β/η expansion preserves provability, or combined that β/η convertibility does not affect provability. 8 Proof Terms and Checker Berghofer and Nipkow [7] added proof terms to Isabelle. We present an executable checker for these proof terms that is proved sound w.r.t. the above formalization of the metalogic. 123 1 Page 12 of 21 S. Roßkopf, T. Nipkow Berghofer and Nipkow also developed a proof checker but it is unveriﬁed and checks the generated proof terms by feeding them back through Isabelle’s unveriﬁed inference kernel. It is crucial to realize that all we need to know about the proof term checker is the soundness theorem below. The internals are, from a soundness perspective, irrelevant, which is why we can get away with sketching them informally. For this reason, we will not give deﬁnitions for all involved functions in the following presentation, preferring informal descriptions (all deﬁnitions are of course part of the formalization). This is in contrast to the logic itself, which acts like a speciﬁcation, which is why we presented it in detail. This is our data type of proof terms: datatype v pr oo f ter m = PAxm name ((( v × sort ) × v typ) list ) | PThm name ((( v × sort ) × v typ) list ) | PBound nat | Abst ( v typ)( v pr oo f ter m) | AbsP ( v term)( v pr oo f ter m) | Appt ( v pr oo f ter m)( v term) | AppP ( v pr oo f ter m)( v pr oo f ter m) | OfClass ( v typ) name | Hyp ( v term) These proof terms are not designed to record proofs in our inference system, but to mirror the proof terms generated by Isabelle. Nevertheless, the constructors of our proof terms correspond roughly to the rules of the inference system. As the axiom rule of our inference system allows for type instantiations, PAxm contains an axiom and a type substitution. This substitution is encoded as an association list instead of a function. The axiom is referenced by name. During proof checking, a mapping from names to terms is provided. PThm represents a (previously proved) theorem, containing the same information as a PAxm constructor. While we treat theorems as axioms, they use a different constructor, as axioms and theorems make use of different namespaces in the implementation. AbsP and Abst correspond to introduction of ⇒ and , AppP and Appt correspond to the respective eliminations. Hyp and PBound relate to the assumption rule, where Hyp refers to a free assumption while PBound contains a De Bruijn index referring to an assumption added during the proof by an AbsP constructor. OfClass denotes a proof that a type belongs to a given type class. Isabelle looks at terms modulo αβη-equivalence and, therefore, does not save β or η steps, while they are explicit steps in our inference system. Therefore we have no constructors corresponding to the (β)and (η) rules. The remaining equality axioms are naturally handled by the PAxm constructor. In the rest of this section, we discuss how to derive an executable proof checker. Exe- cutability means that the checker is deﬁned as a set of recursive functions that Isabelle’s code generator can translate into one of a number of target languages, in particular its implemen- tation language SML [6, 12, 13]. Because of the approximate correspondence between proof term constructors and infer- ence rules, implementing the proof checker largely amounts to providing executable versions of each inference rule, as in LCF: each rule becomes a function that checks the side condi- tions, and if they are true, computes the conclusion from the premises given as arguments. However, as the checked proof terms are not for our exact inference system but the imple- mentation, there is some additional work to perform. The heart of our checker is a recursive function, where var is a concrete type of variables discussed further down: replay::var t heor y ⇒ var t yp li st ⇒ var ter m li st ⇒ var pr oo f ter m ⇒ var ter m opti on It takes a theory, a context (see below), a list of current assumptions and a proof term and returns the certiﬁed proposition for a valid proof term or None foraninvalid one.Wenow 123 A Formalization and Proof... Page 13 of 21 1 discuss some of the more involved implementation steps and illustrate them with some cases of the replay function. First, as we save only names of axioms and theorems in the proof terms, not the propositions themselves, we need a way to look these up. Therefore, our proof checker actually uses a slightly different theory type than the one show before: type_synonym v theory = v signature × ((name v term) × (name v term)) It replaces the set of axioms with two (ﬁnite) maps. These allow efﬁcient, name-based lookup of the actual axiom/theorem. We use two maps, as the Isabelle implementation uses distinct namespaces for axioms and theorems. The set of axioms in the original type can be recovered as the union of the ranges of the two maps. In the following, we will hide this implementation detail and use the original theor y type. The type var of variables is deﬁned as follows: type_synonym indexname = (name × int ) datatype var = Free name | Var indexname | Internal nat The constructors Free and Var are “inherited” from the Isabelle implementation of type term. Both represent free variables but the ones represented by Var can be instantiated by uniﬁcation. We added a third constructor, not present in Isabelle, which we use for easy generation of fresh variables by simply counting up. Such internal variables are generated only when visiting an Abst constructor ( - introduction) during the traversal. Abst constructors introduce -quantiﬁers, which even- tually bind variables present in the contained proof term. The proof terms generated by Isabelle already use De Bruijn notation for these variables, so descending under Abst con- structor can produce loose variables. For each of them, we add an internal variable to the context, which contains exactly the types of Abst-bound variables passed while descend- ing into the proof term. To obtain a fresh variable, we use the size of the current context. Conceptually, when passing an Abst constructor, our proof checker substitutes the newly generated variable for this index everywhere in the proof term, then replays the proof, and binds the variable again in the result, therefore, always working with closed terms. However, performing this substitution ﬁrst would require an extra traversal of the proof term at each Abst constructor. To avoid this, we remember all passed Abst constructors and substitute the corresponding variables simultaneously. replay vsHs (Abst Tp) = (if wf-type (sig ) T then map-option (λt . (Abs-fv (Internal |vs|) Tt )) (replay (T # vs) Hs p) else None) Such substitutions happen at the AbsP (⇒-introduction) and the Appt ( -elimination) case, by means of the subst-bvs function. Note that, somewhat counterintuitively, the innermost abstraction/lowest De Bruijn index corresponds to the highest internal name. We only show the AbsP case here: replay vsHs (AbsP tp) = (let t = subst-bvs (map-index (λiT . Fv (Internal (|vs|− Suc i )) T )vs) t ; rep = replay vs (t # Hs) p in if t : prop ∧ wf-term (sig ) t then map-option ((⇒) t ) rep else None) We have shown that the result of running replay does not contain any internal variables (as long as the inputs do not contain any). 123 1 Page 14 of 21 S. Roßkopf, T. Nipkow As already mentioned, term instantiations can be performed by means of the rules. However, the proof terms generated by Isabelle do not use these rules when instantiating term variables in axioms. Instead, all variables in an axiom are assumed to already have been universally quantiﬁed, so that only the elimination step remains. For our checker, this means we need to also -quantify all free variables when handling a PAxm constructor. A pitfall here is the order in which we quantify the free variables. The structure of the proof terms expects them to occur in the order given by a reverse inorder traversal of the axiom. As theorems are treated as just another kind of axioms, PThm is handled analogously. replay __ (PAxm nTis) = (if inst-ok (sig ) Tis then map-option (λt . all-close (subst-typ Tis t )) (axioms n) else None) To model Isabelle’s view of terms modulo αβη-equivalence, we sometimes βη normalize our terms (α-equivalence is for free thanks to De Bruijn notation) during the reconstruction of the proof. This is necessary when replaying an AppP constructor because checking the conditions of the corresponding implication elimination rule requires checking equality of two terms. For all other constructors, no equality checks are necessary. To avoid repeatedly traversing the terms we only normalize in the AppP case and work with possibly non-βη normalized terms in all other cases. replay vsHs (AppP p p )= 1 2 let rep1 = replay vsHs p > > = beta-eta-norm; rep2 = replay vsHs p > > = beta-eta-norm in case (rep1, rep2) of (Some ( A ⇒ B), Some A ) ⇒ if A= A then Some B else None |−⇒ None For our soundness proof of the checker, we need to verify that these normalizations preserve provability. For this, we show that they can be expressed as a ﬁnite number of β reduction steps, followed by a ﬁnite number of η reduction steps. These steps can then be justiﬁed using the rules presented in Sect. 7, yielding us the desired result. wf-theory ∧ (∀ A∈ . wt-term (sig ) A ∧ A : prop) ∧ t ∧ beta-eta-norm t = Some u −→ For our replay function to be executable, all constructs used by it must be executable as well. As we are continuously using explicitly ﬁnite data structures in the deﬁnition of (order sorted) signatures, Isabelle’s code generator needs no further help handling them. Still, the represen- tation of type constructor signatures in Sect. 4 as an unstructured set tcs does not facilitate efﬁcient access to relevant information. In particular, to compute has-sort oss (Ty κ Ts) S, one needs to ﬁnd the signatures for κ required to fulﬁll all class constraints in S. This means searching all of tcs for each constraint. To speed this up, we deﬁne an alternative representa- tion TCS, inspired by the Isabelle implementation, which the code generator can transparently use as a replacement. This TCS component has type name (class sort list ),and it ﬁrst groups all signatures by type constructor and then allows ﬁnding necessary argument sort constraints by passing an expected return class. More formally, TCS represents the set of all type constructor signatures κ :: (Ss) c such that TCS κ = Some dm and dm c = Some Ss. 123 A Formalization and Proof... Page 15 of 21 1 We can therefore recreate the equivalent but more intuitive, original version tcs the following way: tcs ={(κ, Ss, c) |∃dm. TC S κ = Some dm ∧ dm c = Some Ss} We also need to make the inductive wellformedness checks for sorts, types, terms, signa- tures and theories executable. Mostly, this amounts to providing recursive versions for their inductive deﬁnitions and proving them equivalent. A problematic point is the deﬁnition of the type instance relation (), which contains an (unbounded) existential quantiﬁer. To make this executable, we provide an implementation which tries to compute a suitable type substitution by matching the types. In the end, we obtain an executable proof checker check-proof Pp = (wf-theory ∧ (∀h∈hyps P. wf-term (sig ) h ∧ h : prop) ∧ replay-norm P = Some p) where replay-norm P = (replay [] (hyps P) P > > = beta-eta-norm). This ﬁnal βη nor- malization step is once again necessary to account for possible different but αβη-equivalent results. check-proof checks wellformedness of theory and the hypotheses and then checks if proof P proves the given proposition p. The latter check is important because the Isabelle theorems that we check contain both a proof and a proposition that the theorem claims to prove. As one of our main results, we can prove the correctness of our checker: check-proof Pp −→ ,hyps P The proof itself is conceptually simple and proceeds by induction over the structure of the computation of replay. For each proof constructor we need to show that there are correspond- ing inference rules in our system for each step taken by the functional version replay.Most of the proof effort goes into a large library of results about terms, types, signatures, substi- tutions, wellformedness etc. required for this proof. In particular, we need to prove derived rules characterizing all the technical operations we use, similar to Sect. 7. 9 Size and Structure of the Formalization All material presented so far has been formalized in Isabelle/HOL. The deﬁnition of the inference system (incl. types, terms etc.) resides in a separate theory Core that depends only on the basic library of Isabelle/HOL. It takes about 300 LOC and is fairly high level and readable – we presented most of it. This is at least an order or magnitude smaller than Isabelle’s inference kernel (which is not clearly delineated) – of course, the latter is optimized for performance. Its abstract type of theorems alone takes about 2,500 LOC, not counting any infrastructure of terms, types, uniﬁcation etc. The whole formalization consists of 12,000 LOC. The main components are: • Almost half the formalization (5,500 LOC) is devoted to providing a library of operations on types and terms and their properties. This includes, among others, executable func- tions for type checking, different types of substitutions, abstractions, the wellformedness checks, and β and η reductions. • Proving admissible rules of our inference system takes up 3,000 LOC. A large part of this is deriving rules for equality and the β and η reductions. Weakening rules are also derived. 123 1 Page 16 of 21 S. Roßkopf, T. Nipkow • Making the wellformedness checks for (order-sorted) signatures and theories as well as the type instance checks executable takes 1,800 LOC. • Deﬁnition and correctness proof for the checker builds on the above material and take only about 500 additional LOC. • Around 1,000 LOC are spent on preliminary material, most importantly results about ﬁnite sets and maps, transferred from existing material for general sets and maps. 10 Integration with Isabelle As explained above, Isabelle generates SML code for the proof checker. This code has its own deﬁnitions of types, terms etc. and needs to be interfaced with the corresponding data structures in Isabelle. This step requires 150 lines of handwritten SML code (glue code)that translates Isabelle’s data structures into the corresponding data structures in the generated proof checker such that we can feed them into check-proof. We cannot verify this code and therefore aim to keep it as small and simple as possible. This is the reason for the previously mentioned intentional implementation bias we introduced in our formalization. We describe now how the various data types are translated. We call a translation trivial if it merely replaces one constructor by another, possibly forgetting some information. The translation of types and terms is trivial as their structure is almost identical in the two settings. Proof term translation is trivial except for two special cases. So-called “oracles” (typically the result of unﬁnished proofs, i.e. ,“sorry” on the user level) are rejected (but none of the theories we checked contain oracles). Furthermore, translating previously proved lemmas requires some additional name handling work. Also remember that the translation of proofs is not safety critical because all that matters is that in the end we obtain a correct proof of the claimed proposition. We also provide functions to translate relevant content from the background theory: axioms (including previously proved theorems) and (order-sorted) signatures. This mostly amounts to extracting association lists from efﬁcient internal data structures. Translating the axioms also involves translating some alternative internal representation of type class constraints into their standard form presented in Sect. 6.3. The checker is integrated into Isabelle by calling it every time a new named theorem has been proved. The set of theorems proved so far is added to the axiomatic basis for this check. Cyclic dependencies between lemmas are ruled out by this ordering because every theorem is checked before being added to the axiomatic basis. However, an explicit cyclicity check is not part of the formalization (yet), which speaks only about checking single proofs. 11 Running the Proof Checker We run this modiﬁed Isabelle with our proof checker on multiple theories in various object logics contained in the Isabelle distribution. A rough overview of the scope of the covered material for some logics and the required running times can be found in the following table. The running times are the total times for running Isabelle, not just the proof checking, but the latter takes over 90% of the time. All tests were performed on an Intel Core i7-9750H CPU running at 2.60GHz and 32GB of RAM. 123 A Formalization and Proof... Page 17 of 21 1 Logic LOC Time FOL 4,500 40 s ZF 55,000 12 min HOL 29,000 110 min We can check the material in several smaller object logics in their entirety. One of the larger such logics is ﬁrst-order logic (FOL). These logics do not develop any applications but FOL comes with proof automation and theories testing that automation, in particular Pelletier’s collection of problems that were considered challenges in their day [37]. Because the proofs are found automatically, the resulting proof terms will typically be quite complex and good test material for a proof checker. The logic ZF (Zermelo-Fraenkel set theory) builds on FOL but contains real applications and is an order of magnitude larger than FOL. We are able to check all material formalized in ZF in the Isabelle distribution. Isabelle’s most frequently used and largest object logic is HOL. We managed to check some of the initial theories of the main library. These theories contain the basic logic and among others the libraries of sets, functions, orderings, lattices, groups, rings, ﬁelds and natural numbers. The formalizations are non-trivial and make heavy use of Isabelle’s type classes. Why is checking material in ZF easier than in HOL? Proﬁling revealed that the proof checker spends a lot of time in functions that access the signature, especially the wellformed- ness checks. One reason for this is inefﬁcient data structures (e.g., association lists) and thus the running time depends heavily on the size of the signature and increases with every new constant, type and class. This is aggravated by our current approach which exports the cur- rent state of the background theory and has to ensure its wellformedness before each check. Furthermore, there is no sharing of any kind in terms/types and their wellformedness checks. Because ZF is free of polymorphism and type classes, these checks are much simpler. Lastly, the presence of type classes also increases the size of the involved proofterms. These effects can also be seen purely in the HOL material. For example, despite having similar sizes and containing roughly the same number of theorems, checking the material on rings takes about 10 times as long as the one on natural numbers. 12 Trust Assumptions We need to trust the following components outside of the formalization: • The veriﬁcation (and code generation) of our proof checker in Isabelle/HOL. This is inevitable, one has to trust some theorem prover to start with. We could improve the trustworthiness of this step by porting our proofs to the veriﬁed HOL prover by Kumar et al. [20] but its code generator produces CakeML [19], not SML. • The unveriﬁed glue code in the integration of our proof checker into Isabelle (Sect. 10). Because users currently cannot examine Isabelle’s internal data structures that we start from, they have to trust Isabelle’s front end that parses and transforms some textual input ﬁle into internal data structures. One could add a (possibly veriﬁed) presentation layer that outputs those internal representations into a readable format that can be inspected, while avoiding the traps Adams [2] is concerned with. 123 1 Page 18 of 21 S. Roßkopf, T. Nipkow 13 Future Work Our primary focus will be on scaling up the proof checker to not just deal with all of HOL but with real applications (including itself!). There is a host of avenues for exploration. Just to name a few promising directions: more efﬁcient data structures than association lists (e.g., via existing frameworks [25, 26]); caching of wellformedness checks for types and terms; exploiting sharing within terms and types (tricky because our intentionally simple glue code creates copies); working with the compressed proof terms [6] that Isabelle creates by default instead of uncompressing them as we do now. We will also upgrade the formalization of our checker from individual theorems to sets of theorems, explicitly checking cyclic dependencies (which are currently prevented by the glue code, see Sect. 10). A presentation layer as discussed in Sect. 12 would not just allow the inspection of the internal representation of the theories but could also be extended to the proofs themselves, thus, permitting checkers to be interfaced with Isabelle on a textual level instead of on internal data structures. Another possible next step would be to formalize the theory extension mechanisms includ- ing veriﬁed cyclicity checks. It would also be nice to have a model-theoretic semantics for M. We believe that the work by Kuncar ˇ and Popescu [21–24] could be adapted from HOL to M for these purposes. Supplementary Information The online version contains supplementary material available at https://doi. org/10.1007/s10817-022-09648-w. Acknowledgements We thank Kevin Kappelmann, Magnus Myreen, Larry Paulson, Andrei Popescu, Makar- ius Wenzel, and the anonymous reviewers for their comments. Supported by Wirtschaftsministerium Bayern under DIK-2002-0027//DIK0185/03 and DFG GRK 2428 ConVeY Funding Open Access funding enabled and organized by Projekt DEAL. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. A Appendix subst-bv ut = subst-bv2 t 0 u subst-bv2 (Bv i ) nu = (if i < n then Bv i else if i = n then u else Bv (i − 1)) subst-bv2 (Abs Tt ) nu = Abs T (subst-bv2 t (n + 1)(lift u 0)) subst-bv2 ( f · t ) nu = subst-bv2 fn u · subst-bv2 tn u subst-bv2 t __ = t lift (Bv i ) n = (if n ≤ i then Bv (i + 1) else Bv i ) lift (Abs Tt ) n = Abs T (lift t (n + 1)) lift ( f · t ) n = lift fn · lift tn lift t _ = t 123 A Formalization and Proof... Page 19 of 21 1 bind-fv Tt = bind-fv2 T 0 t bind-fv2 var n (Fv v T ) = (if var = (v, T ) then Bv n else Fv v T ) bind-fv2 var n (Abs Tt ) = Abs T (bind-fv2 var (n + 1) t ) bind-fv2 var n ( f · u) = bind-fv2 var n f · bind-fv2 var n u bind-fv2 __ t = t tinst-ok insts ≡ distinct (map fst insts) ∧ list-all (λ((v, T ), t ). wf-term t ∧ t : T ) insts subst-term _ (Ct cT ) = Ct cT subst-term insts (Fv idn T ) = subst_ f v idn T insts subst-term _ (Bv n) = Bv n subst-term insts (Abs Tt ) = Abs T (subst-term insts t ) subst-term insts (t · u) = subst-term insts t · subst-term insts u subst-bvs st = subst-bvs2 t 0 s subst-bvs2 (Bv i ) nus = (if i < n then Bv i (else if i − n < |us| then us ! (i − n) else Bv (i −|us|)) subst-bvs2 (Abs Tt ) nus = Abs T (subst-bvs2 t (n + 1)(map (λt . lift t 0) us)) subst-bvs2 ( f · t ) nus = subst-bvs2 fn us · subst-bvs2 tn us subst-bvs2 t __ = t References 1. Abrahamsson, O.: A veriﬁed proof checker for higher-order logic. J. Log. 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Journal of Automated Reasoning – Springer Journals
Published: Mar 1, 2023
Keywords: Theorem proving; Higher-order logic; Isabelle; Proofchecker; Metalogic
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