LiFtEr: Language to Encode Induction Heuristics for Isabelle/HOL

06/19/2019 ∙ by Yutaka Nagashima, et al. ∙ Czech Technical University in Prague 0

Proof assistants, such as Isabelle/HOL, offer tools to facilitate inductive theorem proving. Isabelle experts know how to use these tools effectively; however they did not have a systematic way to encode their expertise. To address this problem, we present our domain-specific language, LiFtEr. LiFtEr allows experienced Isabelle users to encode their induction heuristics in a style independent of any problem domain. LiFtEr's interpreter mechanically checks if a given application of induction tool matches the heuristics, thus transferring experienced users' expertise to new Isabelle users.

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1 Introduction

Consider the following reverse functions, rev and itrev, from literature [18]:

primrec rev::"’a list =>’a list" where
  "rev  []      = []"
| "rev (x # xs) = rev xs @ [x]"

fun itrev::"’a list =>’a list =>’a list" where
  "itrev  []    ys = ys"
| "itrev (x#xs) ys = itrev xs (x#ys)"

where # is the list constructor, and @ appends two lists. How do you prove the following lemma?

lemma "itrev xs ys = rev xs @ ys"

Since both rev and itrev are defined recursively, it is natural to imagine that we can handle this problem by applying induction. But how do you apply induction and why? What induction heuristics do you use? In which language do you describe those heuristics?

Modern proof assistants (PAs), such as Isabelle/HOL [18], are forming the basis of trustworthy software. Klein et al., for example, verified the correctness of the seL4 micro-kernel in Isabelle/HOL [7], whereas Leroy developed a certifying C compiler, CompCert, using Coq [10]. Despite the growing number of such complete formal verification projects, the limited progress in proof automation still keeps the cost of proof development high, thus preventing the widespread adoption of complete formal verification.

A noteworthy approach in proof automation for proof assistants is the so-called hammer tools [1]. Sledgehammer [2], for example, exports proof goals in Isabelle/HOL to various external automated theorem provers (ATPs) to exploit the state-of-the-art proof automation of these backend provers; however, the discrepancies between the polymorphic higher-order logic of Isabelle/HOL and the monomorphic first-order logic of the backend provers severely impair sledgehammer’s performance when it comes to inductive theorem proving (ITP).

This is unfortunate for two reasons. First, many Isabelle users chose Isabelle/HOL precisely because its higher-order logic is expressive enough to specify mathematical objects and procedures involving recursion without introducing new axioms. Second, induction lies at the heart of mathematics and computer science. For instance, induction is often necessary for reasoning about natural numbers, recursive data-structures, such as lists and trees, computer programs containing recursion and iteration [3].

This way ITP remains as a long-standing challenge in computer science, and its automation is much needed. Facing the limited automation in ITP, Gramlich surveyed the problems in ITP and presented the following prediction in 2005:

in the near future, ITP will only be successful for very specialized domains for very restricted classes of conjectures. ITP will continue to be a very challenging engineering process.

We address this conundrum with our domain-specific language, LiFtEr. LiFtEr allows experienced Isabelle users to encode their induction heuristics in a style independent of problem domains. LiFtEr’s interpreter mechanically checks if a given application of induction is compatible with the induction heuristics written by experienced users. Our research hypothesis is that:

it is possible to encode valuable induction heuristics for Isabelle/HOL in LiFtEr and such heuristics can be valid across diverse problem domains, because LiFtEr allows for meta-reasoning on applications of induction methods, without relying on concrete proof goals, their underlying proof states, nor concrete applications of induction methods.

In the rest of the paper, we first review how induction works in Isabelle in Section 2. Then, we give the overview of LiFtEr and its syntax in Section 3. Section 4 presents six small example assertions written in LiFtEr and demonstrates how to write induction heuristics for our ongoing example about rev and itrev. Section 5 shows that the LiFtEr assertions from Section 4 are applicable to an inductive problem in a completely unrelated problem domain. Then, Section 6 reveals LiFtEr’s internal pre-processing stage, which allowed for intuitive reasoning about inductive problems. We compare LiFtEr with other work for inductive theorem proving in Section 7 before summarizing our contributions and future work in Section 8. Our working prototype is available at GitHub [16].

2 Induction in Isabelle/HOL

To handle inductive problems, modern proof assistants offer tools to apply induction. For example, Isabelle comes with the induct proof method and the induction method 111Proof methods are the Isar syntactic layer of LCF-style tactics.. For example, Nipkow et al. proved our ongoing example as following [17]:

lemma model_prf:"itrev xs ys = rev xs @ ys"
  apply(induct xs arbitrary: ys) by auto

Namely, they applied structural induction on xs while generalizing ys before applying induction by passing the string ys to the arbitrary field. The resulting sub-goals are as following:

1. !!ys. itrev [] ys = rev [] @ ys
2. !!a xs ys. (!!ys. itrev xs ys = rev xs @ ys) ==>
              itrev (a # xs) ys = rev (a # xs) @ ys

where !! is the universal quantifier and ==> is the implication in Isabelle’s meta-logic. Due to the generalization, the ys in the induction hypothesis is quantified within the hypothesis, and it is differentiated from the ys that appears in the conclusion. Had Nipkow et al. omitted arbitrary: ys, the first sub-goal would be the same, but the second sub-goal would have been as following:

 2. !!a xs. itrev xs ys = rev xs @ ys ==>
            itrev (a # xs) ys = rev (a # xs) @ ys

Since the same ys is shared by the induction hypothesis and the conclusion, the subsequent application of auto fails to discharge this sub-goal.

It is worth noting that in general there are multiple equivalently appropriate combinations of arguments to prove a given inductive problem. For instance, the following proof snippet shows an alternative proof script for our example:

lemma alt_prf:"itrev xs ys = rev xs @ ys"
  apply(induct xs ys rule:itrev.induct) by auto

Here we passed the itrev.induct rule to the rule field of the induct method and proved the lemma by recursion induction 222Recursion induction is also known as functional induction or computation induction. over itrev. This rule was derived by Isabelle automatically when we defined itrev, and it states the following:

(!!ys. P [] ys) ==>
(!!x xs ys. P xs (x # ys) ==> P (x # xs) ys) ==>
P a0 a1

Essentially, this rule states that to prove a property P of a0 and a1 we have to prove it for two cases where a0 is the empty list and the list with at least two elements. When the induct method takes this rule and xs and ys as induction variables, Isabelle produces the following sub-goals:

 1. !!ys. itrev [] ys = rev [] @ ys
 2. !!x xs ys. itrev xs (x # ys) = rev xs @ x # ys ==>
               itrev (x # xs) ys = rev (x # xs) @ ys

where the two sub-goals correspond to the two clauses in the definition of itrev.

There are other lesser-known techniques to handle difficult inductive problems using the induct method, and sometimes users have to develop useful auxiliary lemmas manually; however, for most cases the problem of how to apply induction often boils down to the the following three questions:

  • On which terms do we apply induction?

  • Which variables do we generalize?

  • Which rule do we use for recursion induction?

Isabelle experts resort to induction heuristics to answer such questions and decide what arguments to pass to the induct method; however, such reasoning still requires human engineers to carefully investigate the inductive problem at hand. Moreover, Isabelle experts’ induction heuristics are sparsely documented across various documents, and there was no way to encode their heuristics in programs. For the wide spread adoption of complete formal verification, we need a program language to encode such heuristics and the system to check if an invocation of the induct method written by an Isabelle novice complies with such heuristics.

3 Overview and Syntax of LiFtEr

datatype numb    = Numb    of int;
datatype trm     = Trm     of int;
datatype rule    = Rule    of int;
datatype trm_occ = Trm_Occ of int;
datatype pattern = All_Only_Var | All_Constr | Mixed;
datatype assrt   =
(*quantifiers*)
  All_Ind                 of trm  * assrt
| All_Arb                 of trm  * assrt
| All_Trm                 of trm  * assrt
| All_Rule                of rule * assrt
| All_Numb                of numb * assrt
| Some_Ind                of trm  * assrt
| Some_Arb                of trm  * assrt
| Some_Trm                of trm  * assrt
| Some_Rule               of rule * assrt
| Some_Numb               of numb * assrt
| All_Trm_Occ             of trm_occ * assrt
| Some_Trm_Occ            of trm_occ * assrt
| All_Trm_Occ_Of          of trm_occ * trm * assrt
| Some_Trm_Occ_Of         of trm_occ * trm * assrt
(*combinators*)
| And                     of assrt * assrt
| Or                      of assrt * assrt
| Not                     of assrt
| True
| Imply                   of assrt * assrt
(*atomic about proof goal*)
| Is_Rule_Of              of rule    * trm_occ
| Trm_Occ_Is_Of_Trm       of trm_occ * trm
| Are_Same_Trm            of trm     * trm
| Is_In_Trm_Loc           of trm_occ * trm_occ
| Is_Atom                 of trm_occ
| Is_Cnst                 of trm_occ
| Is_Recursive_Cnst       of trm_occ
| Is_Var                  of trm_occ
| Is_Free                 of trm_occ
| Is_Bound                of trm_occ
| Is_Lambda               of trm_occ
| Is_App                  of trm_occ
| Is_An_Arg_Of            of trm_occ * trm_occ
| Is_Nth_Arg_Of           of trm_occ * numb * trm_occ
| Is_Nth_Ind              of trm     * numb
| Is_Nth_Arb              of trm     * numb
| Pattern                 of numb * trm_occ * pattern
| Is_At_Deepest           of trm_occ
...
Program 1 The Syntax of LiFtEr.

We designed LiFtEr to encode induction heuristics as assertions on invocations of the induct method in Isabelle/HOL. An assertion written in LiFtEr takes the pair of proof goals at hand together with their underlying proof state and arguments passed to the induct method. When one applies a LiFtEr assertion to an invocation of the induct method, LiFtEr’s interpreter returns a boolean value as the result of the assertion applied to the proof goals and their underlying proof state.

The goal of a LiFtEr programmer is to write assertions that implement reliable heuristics. A heuristic encoded as a LiFtEr assertion is reliable when it satisfies the following two properties: first, the LiFtEr interpreter is likely to evaluate the assertion to true when the arguments of the induct method are appropriate for the given proof goal. Second, the interpreter is likely to evaluate the assertion to false when the arguments are inappropriate for the goal.

Program 1 shows the essential part of LiFtEr’s syntax. LiFtEr has five types of variables: numb, rule, trm, trm_occ, and pattern. A value of type numb is a natural number from to the maximum of one of the following two numbers: the number of terms appearing in the proof goals at hand, and the maximum arity of constants appearing in the proof goals. A value of type rule corresponds to a name of an auxiliary lemma passed to the induct method as an argument in the arbitrary field.

The difference between trm and trm_occ is crucial: a value of trm is a term appearing in the proof goals, whereas a value of trm_occ is an occurrence of such terms. It is important to distinguish terms and term occurrences because the induct method in Isabelle/HOL only allows its users to specify induction terms but it does not allow us to specify on which occurrences of such terms we intend to apply induction.

The connectives, And, Or, Not, and Imply correspond to conjunction, disjunction, negation, and implication in the classical logic, respectively; And Imply admits the principle of explosion.

LiFtEr has 12 essential quantifiers and two quantifiers as syntactic sugars. Those starting with the string All are universal quantifiers, and those with Some are existential quantifiers. Again, it is important to notice the difference between the quantifiers over trm and the ones over trm_occ: for example, All_Trm quantifies all sub-terms appearing in the proof goals, whereas All_Trm_Occ quantifies all occurrences of such sub-terms. Quantifiers that end with the string Ind quantify over all induction terms passed to the induct method as induction terms, while quantifiers that end with the string Arb quantify over all terms passed to the induct method as arguments of the arbitrary field.

Some atomic assertions judge properties of term occurrences, and some judge the syntactic structure of proof goals with respects to certain terms, their occurrences, or certain numbers. While most atomic assertions workn on the syntactic structures of proof goals, Pattern provides a means to describe a limited amount of semantic information of proof goals since it checks how terms are defined. Section 4 explains the meaning of important atomic assertions through examples.

Attentive readers may have noticed that LiFtEr’s syntax does not cover any user-defined types or constants. This absence of specific types and constants is our intentional choice to promote induction heuristics that are valid across various problem domains: it encourages LiFtEr users to write heuristics that are not specific to particular data-types or functions. And LiFtEr’s interpreter can check if an application of the induct method is compatible with a given LiFtEr heuristic even if the proof goal involves user-defined data-types and functions even though such types and functions are unknown to the LiFtEr developer or the author of the heuristic but come into existence in the future only after developing LiFtEr and such heuristic.

4 LiFtEr by Example

This section illustrates how to use those atomic assertions and quantifiers to encode induction heuristics through examples.

4.1 Example 1: Induction terms should not be constants.

Let us revise the first example about the equivalence of two reverse functions, itrev and rev. One naive induction heuristic would be “any induction term should not be a constant” 333This naive heuristic is not very reliable: there are cases where the induct method takes terms involving constants and apply induction appropriately by automatically introducing induction variables. See Concrete Semantics [17] for more details. In LiFtEr, we can encode this heuristic as the following assertion:

All_Ind (Trm 1,
 Some_Trm_Occ (Trm_Occ 1,
   Trm_Occ_Is_Of_Trm (Trm_Occ 1, Trm 1)
  And
   Not (Is_Cnst (Trm_Occ 1)))): assrt;

Note the use of All_Ind and Some_Trm_Occ: when LiFtEr handles induction terms, LiFtEr treats them as terms, but it is often necessary to analyze the occurrences of these terms in the proof goal to decide how to apply induction. In our example lemma, xs is a variable, which appears twice: once as the first argument of itrev, and once as the first argument of rev. With this mind, the above assertion reads as following:

for all induction terms, named Trm 1, there exists a term occurrence, named Trm_Occ 1, such that Trm_Occ 1 is an occurrence of Trm 1 and Trm_Occ 1 is not a constant.

Now we compare this heuristic with the model proof by Nipkow et al.

The only induction term, xs, has two occurrences in the proof goal both as variables. Therefore, if we apply this LiFtEr assertion to the model solution, LiFtEr’s interpreter acknowledges that the model solution complies with the induction heuristics defined above.

It is a common practice to analyze occurrences of specific terms when describing induction heuristics. Therefore, we introduced two pieces of syntactic sugars to avoid boilerplate code: Some_Trm_Occ_Of and All_Trm_Occ_Of. Both Some_Trm_Occ_Of and All_Trm_Occ_Of quantify over term occurrences of a particular term rather than all term occurrences in the proof goal at hand. Using Some_Trm_Occ_Of, we can shrink the above assertion from 5 lines to 3 lines as following:

All_Ind (Trm 1,
 Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
  Not (Is_Cnst (Trm_Occ 1)))): assrt;

In English, this reads as following:

for all induction terms, named Trm 1, there exists an occurrence of Trm 1, named Trm_Occ 1, such that Trm_Occ 1 is not a constant.

4.2 Example 2. Induction terms should appear at the bottom of syntax trees.

Not applying induction on a constant would sound a plausible heuristic, but such heuristic is not very useful.

In this example, we encode an induction heuristic that analyzes not only the properties of the induction terms but also the location of their occurrences within the proof goal at hand. When attacking inductive problems with many variables, it is sometimes a good attempt to apply induction on variables that appear at the bottom of the syntax tree representing the proof goal. We encode such heuristic using Is_At_Deepest as the following LiFtEr assertion:

All_Ind (Trm 1,
 Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
   Is_Atom (Trm_Occ 1)
  Imply
   Is_At_Deepest (Trm_Occ 1)));

In English, this assertion reads as following:

for all induction terms, named Trm 1, there exists an occurrence of Trm 1, named Trm_Occ 1, such that if Trm_Occ 1 is an atomic term then Trm_Occ 1 lies at the deepest layer in the syntax tree that represents the proof goal.

We used the infix operator, Imply, to add the condition that we consider only the induction terms that are atomic terms. An atomic term is either a constant, free variable, schematic variable, or variable bound by a lambda abstraction. We added this condition because it makes little sense to check if the induction term resides at the bottom of the syntax tree when an induction term is not an atomic term, but a compound term: such compound terms have sub-terms at lower layers.

LiFtEr’s interpreter acknowledges that the model solution provided by Nipkow et al. complies with this heuristic when applied to this lemma: There is only one induction term, xs, and xs appears as an argument of rev on the right-hand side of the equation in the lemma at the lowest layer of this syntax tree.

4.3 Example 3. All induction terms should be arguments of the same occurrence of a recursively defined function.

Probably, it is more meaningful to analyze where induction terms reside in the proof goal with respects to other terms in the goal. More specifically, one heuristic for promising application of induction would be “apply induction on terms that appear as arguments of the same occurrence of a recursively defined function”. We encode this heuristic using LiFtEr’s atomic assertions, Is_Atomic_Cnst and Is_An_Arg_Of, as following:

Some_Trm (Trm 1,
 Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
  All_Ind (Trm 2,
   Some_Trm_Occ_Of (Trm_Occ 2, Trm 2,
     Is_Recursive_Cnst (Trm_Occ 1)
    And
     (Trm_Occ 2 Is_An_Arg_Of Trm_Occ 1)))));

where Is_Recursive_Cnst checks if a constant is defined recursively or not, and Is_An_Arg_Of takes two term occurrences and checks if the first one is an argument of the second one.

Note that using Is_Recursive_Cnst this assertion checks not only the syntactic information of the proof goal at hand, but it also extracts an essential part of the semantic information of constants appearing in the goal, by investigating how these constants are defined in the underlying proof context.

As a whole, this assertion reads as following:

there exists a term, named Trm 1, such that there exists an occurrence of Trm 1, named Trm_Occ 1, such that for all induction terms, named Trm 2, there exists an occurrence of Trm 2, named Trm_Occ 2, such that Trm_Occ 1 is defined recursively and Trm_Occ 2 appears as an argument of Trm_Occ 1.

Attentive readers may have noticed that we quantified over induction terms within the quantification over Trm_Occ 1, so that this induction heuristics checks if all induction terms occur as arguments of the same constant.

The LiFtEr interpreter confirms that the model proof is compatible with this heuristic as well: the constant, itrev, is defined recursively and has an occurrence that takes the only induction variable xs as the first argument.

4.4 Example 4. One should apply induction on the nth argument of a function where the nth parameter in the definition of the function always involves a data-constructor.

The previous example checks if all induction terms are arguments of the same occurrence of a recursively defined function. Sometimes we can even estimate on which arguments of such function we should apply induction by inspecting the definitions of the function more carefully.

We introduce three constructs to support such reasoning: Is_Nth_Arg_Of, Is_Nth_Ind, and Pattern. Is_Nth_Arg_Of takes a term occurrence, a number, and another term occurrence, and it checks if the first term occurrence is the th argument of the second term occurrence where counting starts at . Is_Nth_Ind takes a term occurrence and a number and checks if the term is passed to the induct method as the th induction term. Pattern takes a term occurrence, a number, one of three patterns, All_Only_Var, All_Const, and Mixed. Each of such patterns describes how the term is defined.

For example, Pattern (Numb n, Trm_Occ m, All_Only_Var) denotes that the th parameter is always a variable on the left-hand side of the definition of the term that has the term occurrence, Trm_Occ m. Likewise, All_Const denotes the case where the corresponding parameter of the definition of a particular constant always involves a data-constructor, whereas Mixed denotes that the corresponding parameter is a variable in some clauses but involves a data-constructor in other clauses. With these atomic assertions in mind, we write the following LiFtEr assertion:

 Not (Some_Rule (Rule 1, True))
Imply
 Some_Trm (Trm 1,
  Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
    Is_Recursive_Cnst (Trm_Occ 1)
   And
    All_Ind (Trm 2,
     Some_Trm_Occ_Of (Trm_Occ 2, Trm 2,
      Some_Numb (Numb 1,
        Pattern (Numb 1, Trm_Occ 1, All_Constr)
       And
        Is_Nth_Arg_Of (Trm_Occ 2, Numb 1, Trm_Occ 1))))));

This roughly translates to the following English sentence:

if there is no argument in the rule field in the induct method, then there exists a recursively defined constant, Trm 1, with an occurrence, Trm_Occ 1, such that for all induction terms Trm 2, there exists an occurrence, Trm_Occ 2, of Trm 2, such that there exists a number, Numb 1, such that the (Numb 1)th parameter involves a data-constructor in all the clauses of the definition of Trm 1, and Trm_Occ 2 appears as the (Numb 1)th argument of Trm_Occ 1 in the proof goal.

Note that we added Not(Some_Rule(Rule 1,True)) to focus on the case where the induct method does not take any auxiliary lemma in the rule field, since this heuristic is known to be less reliable when there is an auxiliary lemma passed to the induct method. Furthermore, it is important to be aware that 1 in Numb 1 is merely the identifier of this variable, and the value of Numb 1 can be a value that is not .

LiFtEr’s interpreter confirms that Nipkow’s model solution to the lemma about itrev and rev conforms to this heuristic: there exists an occurrence of itrev, such that itrev is recursively defined and for the only induction term, xs, there is an occurrence of xs on the left-hand side of the proof goal, such that itrev’s first parameter involves data-constructor in all clauses of its definition, and this occurrence of xs appears as the first argument of the occurrence of itrev 444Note that in reality the counting starts at internally. Therefore, “the first argument” in this English sentence is processed as the th argument within LiFtEr..

4.5 Example 5. Induction terms should appear as arguments of a function that has a related .induct rule in the rule field.

When the induct method takes an auxiliary lemma in the rule field that Isabelle automatically derives from the definition of a constant, it is often true that we should apply induction on terms that appear as arguments of an occurrence of such constant.

See, for example, our alternative proof, alt_prf, for our ongoing example theorem. When Nipkow et al. defined the itrev function with the fun keyword, Isabelle automatically derived the auxiliary lemma itrev.induct, and the occurrence of itrev on the left-hand side of the equation takes xs and ys as its arguments. Furthermore, the alternative proof passes xs and ys to the rule field in the same order they appear as the arguments of the occurrence of itrev in the proof goal.

We introduce Is_Rule_Of to relate a term occurrence with an auxiliary lemma passed to the rule field. Is_Rule_Of takes a term occurrence and an auxiliary lemma in the rule field of the induct method, and it checks if the rule was derived by Isabelle at the time of defining the term. Moreover, we introduce Is_Nth_Ind, which let us specify the order of induction terms passed to the induct method. Using these constructs, we can encode the aforementioned heuristic as following:

 Some_Rule (Rule 1, True)
Imply
 Some_Rule (Rule 1,
  Some_Trm (Trm 1,
   Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
     (Rule 1 Is_Rule_Of Trm_Occ 1)
    And
     (All_Ind (Trm 2,
      (Some_Trm_Occ_Of (Trm_Occ 2, Trm 2,
        Some_Numb (Numb 1,
          Is_Nth_Arg_Of (Trm_Occ 2, Numb 1, Trm_Occ 1)
         And
          (Trm 2 Is_Nth_Ind Numb 1)))))))));

As a whole this LiFtEr assertion checks if the following holds:

if there exists a rule, Rule 1, in the rule field of the induct method, then there exists a term Trm 1 with an occurrence Trm_Occ 1, such that Rule 1 is derived by Isabelle when defining Trm 1, and for all induction terms Trm 2, there exists an occurrence Trm_Occ 2 of Trm 2 such that, there exists a number Numb 1, such that Trm_Occ 2 is the (Numb 1)th argument of Trm_Occ 1 and that Trm 2 is the (Numb 1)th induction terms passed to the induct method.

Our alternative proof is compatible with this heuristic: there is an argument, itrev.induct, in the rule field, and the occurrence of its related term, itrev, in the proof goal takes all the induction terms, xs and ys, as its arguments in the same order.

4.6 Example 6. Generalize variables in induction terms.

Isabelle’s induct method offers the arbitrary field, so that users can specify which terms to be generalized in induction steps; however, it is known to be a hard problem to decide which terms to generalize.

Of course LiFtEr cannot not provide you with a decision procedure to determine which terms to generalize, but it let you describe heuristics to identify variables that are likely to be generalized by experienced Isabelle users. For example, experienced users know that it is usually a bad idea to pass induction terms themselves to the arbitrary field. We also know that it is often a good idea to generalize variables appearing within induction terms if induction terms are compound terms.

We can encode the former heuristic using Are_Same_Trm, which checks if two terms are the same term or not. For instance, we can write the following assertion:

All_Arb (Trm 1,
 Not (Some_Ind (Trm 2,
  Are_Same_Trm (Trm 1, Trm 2))));

By now, it should be easy to see that this assertion checks if the following holds:

for all terms in the arbitrary field, there is no induction term of the same term in the induct method.

The latter heuristic involves the description of the term structure constituting the proof goal. For this purpose we use Is_In_Trm_Loc to check if a term occurrence resides within another term occurrence. With this construct, we can encode the latter heuristic as following:

 Some_Ind (Trm 1,
  Some_Trm_Occ_Of (Trm_Occ 1, Trm 1,
   (All_Trm (Trm 2,
     Some_Trm_Occ_Of (Trm_Occ 2, Trm 2,
       ((Trm_Occ 2 Is_In_Trm_Loc Trm_Occ 1)
        And
         Is_Free (Trm_Occ 2))
      Imply
       Some_Arb (Trm 3,
        Are_Same_Trm (Trm 2, Trm 3)))))));

Again, we used Imply to avoid applying this generalization heuristics to the cases without compound induction terms.

5 Induction Heuristics Across Problem Domains

In Section 4 we wrote six example assertions in LiFtEr. When writing these six assertions, we emphasized that none of them is specific to the data structure list or the function itrev appearing the proof goal. I this section we demonstrate that the LiFtEr assertions written in Section 4 are applicable across domains, taking an inductive problem from a completely different domain as an example. The following code is the formalization of a simple stack machine from Concrete Semantics [17]:

type_synonym vname = string
type_synonym val   = int
type_synonym state = "vname => val"
datatype instr     = LOADI val | LOAD vname | ADD
type_synonym stack = "val list"

fun exec1 :: "instr => state => stack => stack" where
  "exec1 (LOADI n) _ stk        =  n       # stk"
| "exec1 (LOAD x)  s stk        =  s(x)    # stk"
| "exec1  ADD      _ (j#i#stk)  =  (i + j) # stk"

fun exec :: "instr list => state => stack => stack" where
  "exec []     _ stk = stk"
| "exec (i#is) s stk = exec is s (exec1 i s stk)"

exec1 defines how the stack machine in a certain state transforms a given stack into a new one by executing one instruction, whereas exec specifies how the machine executes a series of instructions one by one. Nipkow et al. proved the following lemma using structural induction.

lemma exec_append_model_prf[simp]:
  "exec (is1 @ is2) s stk = exec is2 s (exec is1 s stk)"
  apply(induct is1 arbitrary: stk) by auto

This lemma states that executing a concatenation of two lists of instructions in a state to a stack produces the same stack as executing the first list of the instructions first in the same state to the same stack and executing the second list again in the same state again but to the resulting new stack. As in the case with the equivalence of two reverse functions, there is also an alternative proof based on recursion induction:

lemma exec_append_alt_proof:
  "exec (is1 @ is2) s stk = exec is2 s (exec is1 s stk)"
  apply(induct is1 s stk rule:exec.induct) by auto

Now we check if the heuristics from Section 4 correctly recommends these proofs.

Example 1.

Both exec_append_model_prf and exec_append_alt_prf are compatible with this heuristic. For example, is1 is the only induction term in exec_append_model_prf, and it has occurrences in the proof goal, where it occurs as a variable.

Example 2.

exec_append_model_prf complies with the second example: its only induction term, is1 occurs at the bottom of the syntax tree as a variable, which is an atomic term. exec_append_alt_prf also complies with this heuristic: is1, s, and stk as the arguments of the inner exec on the right-hand side of the equation are all atomic terms at the deepest layer of the syntax tree.

Example 3.

Both proof scripts comply with this heuristic. For example, the inner occurrence of exec on the right-hand side of the equation takes all the induction terms of the alternative proof (namely, is1, s, and stk) as its arguments.

Example 4.

This heuristic works for both proof scripts, but it explains the model answer particularly well: it has a recursively defined constant, exec, and the inner occurrence of exec on the right-hand side of the equation has an occurrence that takes the only induction term is1 as its first argument, and the first parameter of exec always involve a data-constructor in the definition of exec.

Example 5.

This heuristic also works for both proof scripts, but it fits particularly well with the alternative answer: the rule exec.induct is derived by Isabelle when defining exec, while exec has an occurrence as part of the third argument of another exec on the right-hand side of the equation, and this inner occurrence of exec takes all the induction terms (is1, s, and stk) in the same order.

Example 6.

None of our proofs involve induction on a compound term, making Example 6-b rather irrelevant, whereas Example 6-a explains the model answer: the only generalized term, stk, does not appear as an induction term.

6 LiFtEr’s Preprocessor

datatype term =
  Const of string         * typ
| Free  of string         * typ
| Var   of (string * int) * typ
| Bound of int
| Abs   of string * typ * term
| $     of term   * term
Program 2 Isabelle’s default term representation.

The previous examples showed that LiFtEr let us encode our induction heuristics following our intuitive understanding of our proof scripts; however, such intuitive understanding is often disparate from the default term representation of Isabelle/HOL. For example, new Isabelle users may expect that the term, itrev xs ys, has two arguments, xs and ys, at the same level even though in reality xs and ys are not located at the same level in the syntax tree of Isabelle’s default term representation.

Program 2 shows the default term structure of Isabelle. In this data-type declaration, typ represents the type of each term. Const represents constants, Abs stands for lambda abstraction. Variables bound by a lambda abstraction are bound variables denoted by Bound, each of which is identified by an integer representing the corresponding de-Bruijn index. Variables that are not bound by a lambda abstraction are called free variables, represented by Free. Var denotes schematic variable, which corresponds to logical variable in Prolog, and users can instantiate them during the proof process.

   (Const ("Induction_Demo.itrev", _)
  $
    Free ("xs", "’a list"))
$
  Free ("ys", "’a list")

$ represents a function application in Isabelle, and this causes a gap between how a proof goal is represented in Isabelle and how some Isabelle users and the induct method see the goal: $ takes a pair of a function and exactly one argument of the function even when we handle multi-arity functions. In our running example, itrev xs ys may appear as one function application of itrev to two arguments, xs and ys; however, this term is represented by two function applications as shown in the above code-snippet. This means that the two arguments of itrev belong to distinct depths in Isabelle’s internal representation even though for the induct method and many human-engineers they should not be discriminated in terms of the depths in the syntax tree when deciding on which variable one should apply induction.

Another problem with regards to the depth of sub-terms occur when a proof goal contains multiple occurrences of the meta-implication or meta-conjunction. For example, if your proof goal take the form of "P x ==> Q x ==> R x", Q x appears at a deeper level than P x does because the meta implication, ==>, associates to the right even though such difference in depth does not make a meaningful difference from the view point of the induct method because the induct method employs its own preprocessing step.

We circumvented this problem by transforming a given proof goal in Isabelle’s default term representation into our custom data-type that is closer to both human intuition and the way the induct method perceives the proof goal.

First, we replaced the function $ application with a new data-constructor for the function application with possibly multiple arguments. Second, we replaced both the meta-implication and meta-conjunction with a new multi-arity meta-implication and a new multi-arity meta-conjuction. Lastly, we tagged each node in the new syntax tree with the path from the root to that node, so that the LiFtEr interpreter is able to look up appropriate nodes quickly when processing LiFtEr quantifiers that have many corresponding sub-terms.

7 Related Work

The recent development in proof automation for higher-order logic takes the meta-tool approach. Gauthier et al., for example, developed an automated tactic prover, TacticToe, on top of the HOL4 [4]. TacticToe leans how human engineers used tactics and applies the knowledge to execute a tactic based Monte Carlo tree search. To automate proofs in Coq [19], Komendantskaya et al. developed ML4PG [9]. ML4PG uses recurrent clustering to mine a proof database and attempts to find a tactic-based proof for a given proof goal. Both of them try to identify useful lemmas or hypotheses as arguments of a tactic; however, they do not identify promising terms as arguments of a tactic even though identify such terms is crucial to apply induction effectively.

The most well-known approach for ITP, called waterfall [11], was invented for a first-order logic, which cannot handle induction without jeopardizing the soundness by introducing axioms. Jiang et al. followed this approach and ran multiple waterfalls [6] to automate ITP in HOL light [5]. However, when deciding induction variables, they naively picked the first free variable with recursive type and left the selection of appropriate induction variables as future work.

To determine induction variables automatically, Nagashima et al. developed a proof strategy language PSL and its default proof strategy, try_hard for Isabelle/HOL [14]. PSL tries to identify useful arguments for the induct method by conducting a depth-first search. Sometimes it is not enough to pass arguments to the induct method, but users have to specify necessary auxiliary lemmas before applying induction. To automate such labor-intensive work, PGT [15], a new extension to PSL, produces many lemmas by transforming the given proof goal while trying to identify a useful one in a goal-oriented manner.

The drawback of PSL and PGT is that they cannot produce recommendations if they fail to complete a proof search: when the search space becomes enormous, neither PSL and PGT gives any advice to Isabelle users.

PaMpeR [13]

, on the other hand, recommends which proof method is likely to be useful to a given proof goal, using a supervised learning applied to the Archive of Formal Proofs

[8]. The key of PaMpeR was its feature extractor: PaMpeR

first applies 108 assertions to each invocation of proof methods and converts each pair of a proof goal with its context and the name of proof method applied to that goal into an array of boolean values of length 108 because this simpler format is amenable for machine learning algorithms to analyze. The limitation of

PaMpeR is, unlike PSL, it cannot recommend which arguments in the induct method to tackle a given proof goal.

Taking the same approach as PaMpeR, Nagashima attempted to build a recommendation tool, MeLoId [12], to automatically suggest promising arguments for the induct method without completing a proof: they wrote many assertions in Isabelle/ML. Unfortunately, encoding induction heuristics as assertions directly in Isabelle/ML caused an immense amount of code-clutter, and they could not encode even the notion of depth in syntax tree due to the problem discussed in Section 6. Therefore, we developed LiFtEr, expecting that LiFtEr serves as a language for feature extraction.

8 Conclusion and Future Work

ITP has been considered as a very challenging task. To address this issue, we presented LiFtEr. LiFtEr is a domain-specific language in the sense that we developed LiFtEr to encode induction heuristics; however, heuristics written in LiFtEr are often not specific to any problem domains. To the best of our knowledge, LiFtEr is the first programming language developed to capture induction heuristics across problem domains, and its interpreter is the first system that executes meta-reasoning on interactive inductive theorem proving.

The novelty of LiFtEr and its interpreter make its syntax and behaviour rather esoteric. Therefore, we explained how to write induction heuristics in LiFtEr and how its interpreter behaves for a given heuristic and invocation of the induct method using six small self-contained examples.

We hope that when combined into the supervised learning framework of MeLoId, assertions written in LiFtEr extract the essence of induction in Isabelle/HOL in a cross-domain style and produce a useful database for machine learning algorithms, so that new Isabelle users can have the recommendation of promising arguments for the induct method in a fully automatic way.

References