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Overview

Comment: | Penultimate draft of the report. After spell checking and such, this will become the final draft. |
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Downloads: | Tarball | ZIP archive | SQL archive |

Timelines: | family | ancestors | descendants | both | trunk |

Files: | files | file ages | folders |

SHA1: | 98924328add2f2586fae104b095ae6bdbe806d8f |

User & Date: | andy 2015-04-18 19:48:55 |

Context

2015-04-18
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21:14 | Final draft of report. (Unless I think of something else to add...) check-in: 04f1cd0696 user: andy tags: trunk | |

19:48 | Penultimate draft of the report. After spell checking and such, this will become the final draft. check-in: 98924328ad user: andy tags: trunk | |

18:47 | Added a TODO comment describing the problems around applying the substitution to the body of an expanded clause, instead of appending them as unification goals. check-in: b5c5523821 user: andy tags: trunk | |

Changes

Changes to report/arend-report.bib.

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author={Page, Rex and Eastlund, Carl and Felleisen, Matthias}, booktitle={Proceedings of the 2008 international workshop on Functional and declarative programming in education}, pages={21--30}, year={2008}, organization={ACM} } @inproceedings{reichelt2010types, title={Treating Sets as Types in a Proof Assistant for Ordinary Mathematics}, author={Reichelt, Sebastian}, date={September 4, 2010}, organization={Institute of Informatics, University of Warsaw, Warszawa, Poland}, url={http://hlm.sourceforge.net/types.pdf}, |
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author={Page, Rex and Eastlund, Carl and Felleisen, Matthias}, booktitle={Proceedings of the 2008 international workshop on Functional and declarative programming in education}, pages={21--30}, year={2008}, organization={ACM} } @inproceedings{Ford:2004:PEG:964001.964011, author = {Ford, Bryan}, title = {Parsing Expression Grammars: A Recognition-based Syntactic Foundation}, booktitle = {Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages}, series = {POPL '04}, year = {2004}, isbn = {1-58113-729-X}, location = {Venice, Italy}, pages = {111--122}, numpages = {12}, url = {http://doi.acm.org/10.1145/964001.964011}, doi = {10.1145/964001.964011}, acmid = {964011}, publisher = {ACM}, address = {New York, NY, USA}, keywords = {BNF, GTDPL, TDPL, context-free grammars, lexical analysis, packrat parsing, parsing expression grammars, regular expressions, scannerless parsing, syntactic predicates, unified grammars}, } @inproceedings{reichelt2010types, title={Treating Sets as Types in a Proof Assistant for Ordinary Mathematics}, author={Reichelt, Sebastian}, date={September 4, 2010}, organization={Institute of Informatics, University of Warsaw, Warszawa, Poland}, url={http://hlm.sourceforge.net/types.pdf}, |

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% All premises/conclusions in inference rules should be in math mode \renewcommand{\predicate}[1]{$ #1 $} % Setup fonts. \setmainfont[Mapping=tex-text,Numbers=OldStyle]{Junicode} \setsansfont[Mapping=tex-text,Numbers=OldStyle,Scale=MatchLowercase]{Gill Sans MT} \setmonofont[Mapping=tex-text,Scale=MatchLowercase]{Fantasque Sans Mono} % Hyperlinks. We might want these inline, or in the sidebar. \newcommand{\link}[2]{#2\footnote{\url{#1}}} % Citations. Both in the text and in the sidebar. The NoHyper makes the actual % citation link to the bibliography at the end, rather than to the bibentry % in the margin. It also has the unfortunate side effect of making any URLs ................................................................................ proof-trees, as the traditional paragraph-proof notation tends to obscure the fact that a proof is composed of subproofs. \item Because proofs are hierarchical, their construction is very similar to that of other kinds of programming. A proof is iteratively broken down into smaller parts, lemmas can be used, like functions, to abstract out commonly-used subproofs, or to break down proofs that would otherwise be too large and unweildy to understand. \item The lack of traditional negation eliminates some of the more-confusing elements of traditional logic programming. In particular, Arend has nothing corresponding to ``negation as failure''; it is not possible in Arend to know, in a computational sense, when a proposition $P$ fails. Thus reasoning in Arend is entirely about things that are known to be true, never about things that are assumed to be false. ................................................................................ \item \emph{Proof assistants} aim to aid in the development and checking of formal proofs. While some systems only check proofs for completeness, more modern systems will typically also aid the user in the development of a proof. Recent examples include Coq \mcitep{Coq:manual}, Twelf \mcitep{pfenning1999system}, and Abella \mcitep{gacek08ijcar} Several element of Arend's design were influenced by Abella, most notably the use of a constructive two-level logic for describing and reasoning about systems. \item More recently, several systems have emerged which try to bridge the gap between traditional functional programming and theorem proving. Systems such as Agda \mcitep{norell:thesis}, Idris \mcitep{brady2011idris}, and Beluga \mcitep{pientka2010beluga} are based on the Curry-Howard isomorphism (see below) and thus represent propositions as types, and proofs as functional programs. These systems offer varying ................................................................................ support for visualizing proof structure, its consistency, and in the degree of ``assistance'' it gives to the user. \item \mcitet{suppes1981future} offers an interesting perspective, that of an instructor of mathematics, not computer science, at Stanford University, and furthermore, one who had, at the time of this publication, been using interactive theorem proving in undergraduate coursework for almost twenty years. Suppes raises several points of concern which are still revelant, most notably the conflict between providing a system which is ``intelligent'' and, hence, easy to use, and one which meets the pedagogical goals of teaching \emph{introductory} material (the sort of material which a more intelligent proof assistant would tend to elide). Thus, although Arend is actually capable of automatically proving a reasonable large set of propositions, during proof construction it intentionally refrains from doing so, thus forcing the user to carry out --- and, hopefully, to learn --- the elementary steps of proof construction. ................................................................................ \inference[Assume-A] {} {A} \] However, this assumption only applies within the ``scope'' of the subproof of $A -> B$. In order to express this, we introduce the \emph{hypothetical judgements} using the symbol $|-$. $A_1, A_2, \ldots A_n |- C$ should be read ``$C$ is true \emph{assuming} $A_1$, etc.'' We call the $A_i$ the \emph{antecedants} of the judgement, and $C$ the \emph{consequent}. (Although not strictly correct, we will sometimes refer to the antecedants and consequent of a \emph{rule}, since a rule can have at most one hypothetical judgement as its conclusion.) \footnote{Note that $|-$ is \emph{not} a logical connective; in particular, it is not valid to say, e.g., $A \land (B |- C)$.} We use $\Gamma$ to signify any collection of $P_i$. To enable the use of $|-$ in our derivations we add the following axiom, known as the \emph{assumption rule}: \[ \inference[Assume] {} ................................................................................ derivations produced by queries are a restricted form of the derivations constructed as proofs. Thus, queries against the specification can serve as an introduction to the creation of simple proofs. Since Arend's specification logic is a restricted form of the Horn-clause classical logic used in traditional Prolog, a resolution-style proof search procedure is sufficient to execute queries against a specification. The operational semantics of this procedure are presented, in an abbreviated form, in \hyperref[spec-execute]{Appendix~1}. Note that in keeping with its intuitionistic foundation, Arend has no negation operator. (Intuitionistically, $\neg P$ can be defined as $P \rightarrow \bot$, but Arend's specification logic lacks the rightward $\rightarrow$ implication operator. This operator \emph{is} present in the reasoning logic, which thus requires a more sophisticated handling.) \subsection{Reasoning logic} \newthought{Arend's reasoning logic} is closer to a full expression of first-order intuitionistic logic, with extensions to support proofs by single-induction. The inference rules defining the reasoning logic are presented in Figure \ref{reasoning-rules}. It supports rightward implication ($\rightarrow$), with the restriction that the premise cannot contain nested implications, only conjunctions, disjunctions, and atomic goals.\footnote{This restriction subsumes the \emph{stratification} restriction in Abella; stratification in some form is required to ensure monotonicity of the logic.} This implies that antecedants cannot contain implications, since they cannot be added by $\rightarrow_R$, and are not allowed in definitions in the specification logic. %%% ------------------- Reasoning logic rules -------------------- \begin{figure}[!p] % Assumption rule ................................................................................ % Implication rule \[ \inference[${->}_R$] {\Gamma,P |- Q} {\Gamma |- P -> Q} \qquad\qquad \text{\small\parbox{5cm}{(There is no ${->}_L$ rule, because implications are not allowed in antecedants.)}} \] \vspace{1em} % Forall rules \[ \inference[$\forall_R$] ................................................................................ derivation (because this query has no free variables, no substitution is displayed). The rules of the specification (the $\mathbb{N}$ specification, given in full in \hyperref[sec:nat-spec]{appendix II}) are displayed below the input line. \item The interactive \emph{proof assistant} allows the user to construct proofs for given propositions, in the context of a particular specification. \end{itemize} \begin{figure*}[ht!] \begin{centering} ................................................................................ \end{centering} \caption{The browser-based run-eval-print-loop interface} \label{fig:repl} \end{figure*} \begin{figure*}[ht!] \begin{centering} \includegraphics[width=6.5in]{proof-start-sample.png} \end{centering} \caption{The proof assistant interface, with an incomplete inductive proof} \label{fig:passist1} \end{figure*} \clearpage \section{Implementation} \newthought{Arend currently exists} in two semi-overlapping implementations: \begin{itemize} \item A \emph{client-based} implementation, written in JavaScript and running largely in the browser. (The core JavaScript proof-checker can also run offline, via a JavaScript engine such as \link{https://nodejs.org/}{node.js} or \link{https://iojs.org/en/index.html}{io.js}.) This implementation is currently incomplete. \item A \emph{server-based} implementation written in Prolog, running on top of the \link{http://swi-prolog.org}{SWI-Prolog} HTTP server. \end{itemize} Both implementations share some front-end code, mostly related to the representation of terms as JavaScript objects, and to the rendering of rules and proofs as HTML. Details... \subsection{Client-based implementation (JavaScript)} Lines of code (UI vs. checker), number of test cases, portability layer for browser/offline compatibility, etc. Libraries used: \begin{itemize} \item jQuery \item Lo-Dash No-dash, our Lo-dash wrapper \item jsCheck \item PEG.js \item qunit \end{itemize} Module structure: \begin{itemize} \item Browser/Node compatibiltiy layer \item terms \item unify \item parser + spec grammar \item term rendering \end{itemize} \subsection{Server-based implementation (SWI-Prolog)} Lines of code, for the HTTP server component and the checker component, separately. Hopefully the LoC for the checker will be significantly less than that for the JS implementation! If we get any tests running, number of test cases. (Development details? Num. of files, SCM used, total hours, etc.) It should be noted that the Prolog implementation of Arend is mostly portable, relying only on ISO-Prolog predicates, commonly-available libraries, and a few SWI-Prolog-specific extensions (mostly dealing with limiting the depth of search trees). It is interesting to directly compare the two implementations of unification (one of the few modules which is semantically similar in both implementations). Both systems implement the Robinson unification algorithm \mcitep{robinson1965machine}. The Javascript implementation consists of 136 lines of non-comment code and is quite difficult to follow; the Prolog implementation requires only 69 lines, an almost 50\% reduction! This despite the fact that the Prolog implementation does \emph{not} use the built-in unification to simplify the algorithm at all; it would be largely the same if only traditional assignment were used. Our future goal is for these two implementations to be built off the same underlying codebase, for example, by moving the client implementation towards a full implementation of the Warren Abstract Machine \mcitep{warren1983abstract}. This would allow the Prolog code currently used in the server implementation to transparently run in a JavaScript environment, whether the browser or one of the aforementioned offline engines. Module structure: \begin{itemize} \item explicit substitutions \item specification execution \item proof checking and elaboration \item server, Pengines, etc. \end{itemize} \subsection{Proof representation} Informally, the content of a derivation is relatively simple: a tree of \textsc{-Left} and/or \textsc{-Right} rule applications, drawn from the rules of the reasoning logic in figure \ref{reasoning-rules}. In practice, a more detailed representation is required, one which, in particular, necessates the use of \emph{explicit substitution}, rather than the implicit substitution that would result from naive use of (for example) the system unification in Prolog. For derivations purely derived from the specifcation language, explicit substitution is not required. This is because for any valid derivation of a proposition expressed in the specification language, the substitution is \emph{consistent} throughout the derivation tree; it is not possible for different ``branches'' to have different substitutions. (Disjunction may, of course, result in the generation of multiple distinct derivations, each with its own unifying substitution, but within any single derivation there is always a single consistent substitution.) However, because the reasoning language posesses rightward implication, a new element is introduced to derivations: the list of \emph{antecedants}. Case analysis on a disjunction antecedant is superficially similar applying the $\land$-R rule, with an important distinction: the branches produced each have \emph{independent} bindings. Consider, for example, case analysis on $\mathsf{nat}(X)$ in the course of a proof: \[ \inference[] {X = z, \mathsf{nat}(z) |- \ldots & X = s(X'), \mathsf{nat}(X') |- \ldots} ................................................................................ inconsistent, and actually impossible to express, because Horn clauses forbid such conjunctions as conclusions. Conversely, given the conclusion $X = z \lor X = s(x')$ we have the choice of which unification to use, and thus there is no inconsistency.} Because of this, each branch of a derivation must maintain its own substitution, applied to its goals as they are expanded. \subsection{Capture avoidance} The use of explicit substitution means that the proof-checker must also contend with another troublesome problem which Prolog conceals: avoiding \emph{accidental capture} when expanding a call to a goal. Consider the goal $\mathsf{nat}(X)$. We would expect this goal to succeed twice, first with the solution $X = z$. ................................................................................ $X \mapsto Y$ to $\forall Y, P(X,Y)$; clearly, one of the $Y$'s will need to be renamed. Since the $Y$ in the substitution is most likely bound somewhere else in the derivation, we choose to rename the $Y$ bound within the $\forall$. \subsection{Unification} The Robinson unification algorithm is presented, in natural deduction style, in figure \ref{unification-rules}. Note that the same algorithm is used, with minor modification, in both the Javascript and Prolog implementations (for a direct comparison of these two implementations, see comments below). \begin{figure}[h!] \[ \inference[Wildcard] {} {\_ \equiv t \mid \varepsilon} \] ................................................................................ {} {\mathtt{[]} \equiv \mathtt{[]} \mid \varepsilon} \qquad\qquad \inference[Cons] {t \equiv s \mid \theta_1 & \hat{t} \theta_1 \equiv \hat{s} \theta_1 \mid \theta_2} {(t:\hat{t}) \equiv (s:\hat{s}) \mid \theta_2} \] \label{unification-rules} \end{figure} (This presentation is based in part on that in \mcitet{pfenning2006logic}.) \subsection{Proof Checking} The proof checker is implemented in approximately 450 lines of Prolog code. It essentially implements the rules of the reasoning logic presented in figure \ref{reasoning-rules}, by recursively checking the proof tree. That is, it first checks the root node to ensure that the proof object is consistent with the conclusion (premises and succedant) by ensuring that it has the correct number and type of subproofs. The subproofs are then recursively checked, working through the tree until the axiomatic rules are reached (or, in the case of an incomplete proof, a \texttt{hole} is found, indicating an unproved subtree). There are two complications to a straightforward top-down recursive implementation: ................................................................................ \subsection{Proof Elaboration} The heart of the incremental proof checking algorithm is the predicate \texttt{elaborate} (defined in file \texttt{checker.pl}). \texttt{elaborate} works in conjunction with \texttt{check}, the proof checker itself. The purpose of elaborate is to take an incomplete proof, together with a reference to one of its leaves (i.e., to a $?$), and to ``elaborate'' it with respect to either its succedant, or one of its premises. As described above, the form of the selected element dictates the forms of the subproofs. \texttt{elaborate} ``fills in'' these subproofs, to a single level only (i.e., all the subproofs it constructs are themselves $?$). In a logic which did not include unification or the ability to call defined predicates (atomic goals), we could elaborate any node of the tree, independent of the rest of the tree. ................................................................................ \begin{figure}[h!] All left-rules instantiate the Proof to a term of the form \[ \text{case}(\text{Type},N,\text{Keep},[\text{Proofs}\ldots]) \] In the rules below we leave $N$ (the index in the list of antecedants of the judgment to be cased upon) implicit and omit Keep (a Boolean flag indicating whether the targeted antecedant should remain in the list, or be removed), and write them as \[ \text{Type}(\text{Proofs}\ldots) \] \[ \inference[False-Left] ................................................................................ it has the form $P \land Q$, then the output proof will be elaborated to \[ \proof{P \land Q}{\Gamma}{\text{product}([ \proof{P}{\Gamma}{\text{hole}}, \proof{Q}{\Gamma}{\text{hole}} ])} \] The subproofs of the product will have the correct succedants and premises, but their own proofs will be empty, ready for further elaboration. \subsection{Inductive reasoning} Arend's reasoning logic supports proofs of a universal quantification both generically and by \emph{induction}. Arend's induction is technically on the height of derivations, although from the perspective of the user it supports full structural induction on terms; this subsumes the usual natural number induction. Arend supports only single-induction proofs (i.e., induction on multiple premisses is not allowed) and supports only induction global to a proof (i.e., nested inductions are not allowed, although they can be ``faked'' by using lemmas). These restrictions imply that the induction hypothesis can be regarded as being global to a proof, thus eliminating the need to restrict the scope of difference induction hypotheses to different branches of the proof tree. Internally, induction is implemented by \emph{goal tagging}. When an inductive proof is declared, a particular goal in the antecedants is selected, by the user, to be the target of the induction.\marginnote{For example, in a proof of $\mathsf{nat}(X) |- \mathsf{add}(X,0,X)$ we would induct on $\mathsf{nat}(X)$.} This goal must be a user goal; it cannot be a built-in operator such as conjunction, disjunction, or unification. The functor of the goal is internally flagged as being ``big'' (indicated as $\uparrow$) and the induction hypothesis is defined in terms of the same goal, but flagged as ``small'' ($\downarrow$). For example, to prove that $\mathsf{nat}(X) -> \mathsf{add}(X,0,X)$ our proof would proceed by induction on $\mathsf{nat}(X)$, and thus we would have the inductive hypothesis $\mathsf{nat}^\downarrow(X) -> \mathsf{add}(X,0,X)$. When a $\uparrow$ goal is expanded by case analysis or backchaining, any sub-goals in its expansion are flagged as $\downarrow$, indicating that they are ``smaller'' than the original goal. The induction hypothesis can only be applied to goals which are ``small'', ................................................................................ being a flaw in our system, this surprising result reflects the fact that one computational interpretation of $\bot$ is that of a non-terminating computation. In fact, this result is simply the computational analogue of the well-known \emph{ex falso quodlibet} principle of classical logic. \clearpage \section{Future work} \begin{itemize} \item Arend's specification logic is simple enough that formalization of its properties should not be difficult, nonetheless, for formal correctness and consistency work needs to be done to show that it is both \emph{sound} (everything it proves true is true) and \emph{non-deterministically complete} (when it ................................................................................ conclusion. Since one of the difficulties of the derivation-tree proof format is its often-extravagant use of horizontal space, it should be possible to automatically \emph{extract} a sub-proof as a lemma. This could be done even when the lemma is not generally useful, but simply as a way of ``naming'' a portion of the proof or commenting on its purpose. \item The implementaiton of Arend is primitively minimal, suitable only for small-scale usage. Real-world usage would require integration with existing grading systems and automatic checks of student-submitted proofs. On a higher level we could easily imagine a set of features aimed at creating an ecosystem around student-created specifications, proofs, and lemmas, perhaps with collaboration features for multi-student use. \item Although we believe Arend's derivation-tree presentation of proofs has ................................................................................ \subsection{Equational reasoning} One very large extension to the logic which we would like to investigate would be the incorporation of \emph{equational reasoning}. Arend's current concept of equality is based entirely on unification: two terms are equal if they can be made equal by some substitution. Equational reasoning allows equality to be defined via rules, for example \marginnote{For computational purposes equational rules are often regarded as unidirectional \emph{rewite rules}: $a = b$ becomes $a \mapsto b$.} \[ \inference[$=$-Refl] {} {x = x} \qquad \inference[$=$-Symm] ................................................................................ \inference[$\times$-dist-$+$] {} {a \times (b + c) = a \times b + a \times c} \] Equational reasoning is a particularly powerful mechanism for reasoning about \emph{functional programs} and programming languages; we can state equivalencies between expressions and then prove that two programs, or two classes of programs, are equivalent under those rules. Even more powerfully, if we wish to prove some property of an entire language, it is sufficient to show that the equivalencies preserve that property. Although the additional of equational reasoning to Arend would give the system a marked increase in logical power, it would also correspondingly increase the complexity of its implementation and presentation. Equivalence rules require a different treatment of terms and variables from unification, and it is not entirely clear how the two will interact, logically. Equivalences over programs ................................................................................ logic variables, but can both contain, and be contained by, them. Although logics exist which provide support for reasoning about binding structures \mcitep{miller05tocl} their usage in proofs involves additional complexity that may be at odd with our goal of use in \emph{introductory} curriculum. \clearpage \section{Appendix I: Syntax of the Specification Logic} \label{spec-syntax} \newcommand{\nterm}[1]{\mathop{\langle\text{#1}\rangle}} \newcommand{\sterm}[1]{\mathop{\text{\emph{#1}}}} \newcommand{\term}[1]{\mathop{\text{\texttt{``#1''}}}} \newthought{The following is a simplified presentation} of the grammar of the specification language. In particular, precedence rules for infix operators have been omitted, but are consistent with normal usage. Note that while the grammar includes rules defining infix arithmetic and comparison operators, these operators have no \emph{semantic} significance within the specification logic. They are present under the assumption that specifications may want to use them, and will expect them to have their normal precendence ordering. \begin{align*} \nterm{start} &<- \nterm{definitions}? \\ \\ \nterm{definitions} &<- \nterm{definition} \nterm{definitions}* \\ \nterm{definition} &<- \sterm{Rulename}? \nterm{predicate} \\ &\mid \nterm{infix-decl} \\ ................................................................................ &\mid \term{(} \nterm{term} \term{)} \\ &\mid \sterm{Atom} \\ &\mid \sterm{Variable} \\ \\ \nterm{compound} &<- \sterm{Atom} \term{(} \nterm{termlist} \term{)} \\ \end{align*} \subsection{Operational Semantics}\label{spec-execute} This may or may not get written. If it does, it will be based on the stack-based semantics of \mcitet{pfenning2006logic}. \clearpage \section[Appendix II: Specification for N, lists]{Appendix II: Specification for $\mathbb{N}$, lists} \label{sec:nat-spec} \newthought{The following is an example specification} included with Arend, one which |
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% All premises/conclusions in inference rules should be in math mode \renewcommand{\predicate}[1]{$ #1 $} % Setup fonts. \setmainfont[Mapping=tex-text,Numbers=OldStyle]{Junicode} \setsansfont[Mapping=tex-text,Numbers=OldStyle,Scale=MatchLowercase]{Gill Sans MT} \setmonofont[Mapping=tex-text,Scale=MatchLowercase]{Fantasque Sans Mono} % The \lnum{} command displays numbers normally (not old style). The font % setting above sets them to old-style by default, and you cannot "turn off" % a font feature, only create a new font with the feature turned off. \newfontface\liningnum{Junicode} \newcommand{\lnum}[1]{{\liningnum #1}} % Hyperlinks. We might want these inline, or in the sidebar. \newcommand{\link}[2]{#2\footnote{\url{#1}}} % Citations. Both in the text and in the sidebar. The NoHyper makes the actual % citation link to the bibliography at the end, rather than to the bibentry % in the margin. It also has the unfortunate side effect of making any URLs ................................................................................ proof-trees, as the traditional paragraph-proof notation tends to obscure the fact that a proof is composed of subproofs. \item Because proofs are hierarchical, their construction is very similar to that of other kinds of programming. A proof is iteratively broken down into smaller parts, lemmas can be used, like functions, to abstract out commonly-used subproofs, or to break down proofs that would otherwise be too large and unwieldy to understand. \item The lack of traditional negation eliminates some of the more-confusing elements of traditional logic programming. In particular, Arend has nothing corresponding to ``negation as failure''; it is not possible in Arend to know, in a computational sense, when a proposition $P$ fails. Thus reasoning in Arend is entirely about things that are known to be true, never about things that are assumed to be false. ................................................................................ \item \emph{Proof assistants} aim to aid in the development and checking of formal proofs. While some systems only check proofs for completeness, more modern systems will typically also aid the user in the development of a proof. Recent examples include Coq \mcitep{Coq:manual}, Twelf \mcitep{pfenning1999system}, and Abella \mcitep{gacek08ijcar} Several element of Arend's design were influenced by Abella, most notably the use of a constructive two-level logic for describing and reasoning about systems. For a full history of the development of computer-assisted proof systems, see \mcitet{geuvers2009proof}. \item More recently, several systems have emerged which try to bridge the gap between traditional functional programming and theorem proving. Systems such as Agda \mcitep{norell:thesis}, Idris \mcitep{brady2011idris}, and Beluga \mcitep{pientka2010beluga} are based on the Curry-Howard isomorphism (see below) and thus represent propositions as types, and proofs as functional programs. These systems offer varying ................................................................................ support for visualizing proof structure, its consistency, and in the degree of ``assistance'' it gives to the user. \item \mcitet{suppes1981future} offers an interesting perspective, that of an instructor of mathematics, not computer science, at Stanford University, and furthermore, one who had, at the time of this publication, been using interactive theorem proving in undergraduate coursework for almost twenty years. Suppes raises several points of concern which are still relevant, most notably the conflict between providing a system which is ``intelligent'' and, hence, easy to use, and one which meets the pedagogical goals of teaching \emph{introductory} material (the sort of material which a more intelligent proof assistant would tend to elide). Thus, although Arend is actually capable of automatically proving a reasonable large set of propositions, during proof construction it intentionally refrains from doing so, thus forcing the user to carry out --- and, hopefully, to learn --- the elementary steps of proof construction. ................................................................................ \inference[Assume-A] {} {A} \] However, this assumption only applies within the ``scope'' of the subproof of $A -> B$. In order to express this, we introduce the \emph{hypothetical judgments} using the symbol $|-$. $A_1, A_2, \ldots A_n |- C$ should be read ``$C$ is true \emph{assuming} $A_1$, etc.'' We call the $A_i$ the \emph{antecedents} of the judgment, and $C$ the \emph{consequent}. (Although not strictly correct, we will sometimes refer to the antecedents and consequent of a \emph{rule}, since a rule can have at most one hypothetical judgment as its conclusion.) \footnote{Note that $|-$ is \emph{not} a logical connective; in particular, it is not valid to say, e.g., $A \land (B |- C)$.} We use $\Gamma$ to signify any collection of $P_i$. To enable the use of $|-$ in our derivations we add the following axiom, known as the \emph{assumption rule}: \[ \inference[Assume] {} ................................................................................ derivations produced by queries are a restricted form of the derivations constructed as proofs. Thus, queries against the specification can serve as an introduction to the creation of simple proofs. Since Arend's specification logic is a restricted form of the Horn-clause classical logic used in traditional Prolog, a resolution-style proof search procedure is sufficient to execute queries against a specification. Note that in keeping with its intuitionistic foundation, Arend has no negation operator. (Intuitionistically, $\neg P$ can be defined as $P \rightarrow \bot$, but Arend's specification logic lacks the rightward $\rightarrow$ implication operator. This operator \emph{is} present in the reasoning logic, which thus requires a more sophisticated handling.) % spec-execute -- Mention operational semantics here. \subsection{Reasoning logic} \newthought{Arend's reasoning logic} is closer to a full expression of first-order intuitionistic logic, with extensions to support proofs by single-induction. The inference rules defining the reasoning logic are presented in Figure \ref{reasoning-rules}. It supports rightward implication ($\rightarrow$), with the restriction that the premise cannot contain nested implications, only conjunctions, disjunctions, and atomic goals.\footnote{This restriction subsumes the \emph{stratification} restriction in Abella; stratification in some form is required to ensure monotonicity of the logic.} This implies that antecedents cannot contain implications, since they cannot be added by $\rightarrow_R$, and are not allowed in definitions in the specification logic. %%% ------------------- Reasoning logic rules -------------------- \begin{figure}[!p] % Assumption rule ................................................................................ % Implication rule \[ \inference[${->}_R$] {\Gamma,P |- Q} {\Gamma |- P -> Q} \qquad\qquad \text{\small\parbox{5cm}{(There is no ${->}_L$ rule, because implications are not allowed in antecedents.)}} \] \vspace{1em} % Forall rules \[ \inference[$\forall_R$] ................................................................................ derivation (because this query has no free variables, no substitution is displayed). The rules of the specification (the $\mathbb{N}$ specification, given in full in \hyperref[sec:nat-spec]{appendix II}) are displayed below the input line. \item The interactive \emph{proof assistant} allows the user to construct proofs for given propositions, in the context of a particular specification. Figure~\ref{fig:passist1} illustrates the proof assistant interface: the rules of the current specification are displayed in the pane on the left; since the current proof is inductive, the inductive hypothesis is included. (Lemmas, once proved, are also displayed as rules.) The right pane displays the current proof statement, and the proof, here, in progress. The user can double-click on any antecedent, or any goal, to perform the default elaboration for that element. For example, double-clicking an antecedent $P \land Q$ would apply the $\land$-L rule, producing a subproof with antecedents $P,Q$. For situations in which there is more than one possible action (e.g., the $\lor$-R rules, or when backchaining against the IH), keyboard interaction is used to select the appropriate action. \end{itemize} \begin{figure*}[ht!] \begin{centering} ................................................................................ \end{centering} \caption{The browser-based run-eval-print-loop interface} \label{fig:repl} \end{figure*} \begin{figure*}[t!] \begin{centering} \includegraphics[width=6.5in]{proof-sample.png} \end{centering} \caption{The proof assistant interface, with an incomplete inductive proof} \label{fig:passist1} \end{figure*} \clearpage \section{Implementation} \newthought{Arend is implemented as} a web-based system, with a server component, written in Prolog and running in the \link{http://swi-prolog.org}{SWI-Prolog} and a browser-based frontend. Currently, the implementation of Arend consists of \begin{itemize} \item \lnum{1,401} lines of Prolog \item \lnum{6,198} lines of Javascript (of which \lnum{442} lines are test code) \item \lnum{493} lines of PEG grammar specification \item \lnum{501} lines of HTML \item \lnum{129} lines of CSS \item \lnum{41} source code files in total \end{itemize} The following libraries and applications are used in the development of Arend; these will be described in detail in the following section. \begin{itemize} \item \link{https://nodejs.org/}{Node.js} \item \link{http://www.swi-prolog.org/}{SWI-Prolog} \item \link{http://pengines.swi-prolog.org/docs/index.html}{Pengines} \item \link{https://jquery.com/}{jQuery} --- General utilities for Javascript in a browser environment \item \link{https://lodash.com/}{Lodash} --- Collection utilities for Javascript \item \link{http://pegjs.org/}{PEG.js} --- Parsing Expression Grammar parser generator for Javascript. \item \link{https://qunitjs.com/}{QUnit} --- Javascript test framework \item \link{http://www.jscheck.org/}{JSCheck} --- Randomized testing engine for Javascript. \end{itemize} Arend's development is tracked using the \link{http://fossil-scm.org}{Fossil} source control management system. As of \lnum{April 18, 2015,} the project history consisted of \lnum{294} commits spanning eight months of development. Arend, its source code and project history, can be found on the web at \url{http://fossil.twicetwo.com/arend.pl}. \subsection{Automated testing} Arend's Javascript code is run through a test suite consisting of \lnum{133} automated tests, however, of these, \lnum{12} central tests are randomized, running, by default, \lnum{20} randomly generated tests each. Thus, a total of \lnum{361} individual tests are run. The test suite can be run in the browser, or offline, via Node.js. Testing is handled via the \link{https://qunitjs.com/}{QUnit} test framework; a small compatibility layer was written to allow QUnit tests to run offline. Randomized testing is provided by \link{http://www.jscheck.org/}{JSCheck} a ``port'' of the Haskell QuickCheck framework to Javascript. We have extended JSCheck with support for randomized generation of Arend data types: atoms, variables, and ground and non-ground terms up to a limited depth. QUnit is designed for automated testing in a browser environment. Arend's development, as far as is possible, tries to target both browser and offline environments; this is true even for Javascript code. Thus, we wrote ``QNodeit'', a small compatibility layer that allows the complete suite of QUnit tests to run offline, via Node.js. All non-user-interface tests run successfully in both the browser and offline environments. QNodeit also integrates JSCheck into QUnit, allowing JSCheck's randomized tests to be used naturally within QUnit. \subsection{Client-side implementation} Although Arend's client-side code only serves to provide a user-interface to the backend's proof-checking and manipulation engine, it contains a significant amount of logic itself. Early in Arend's development we felt that the best course would be to implement the entire proof checking engine in Javascript, thus allowing proof-handling with no server at all. Although this did not prove feasible, a significant amount of logic-handling code in Javascript was written, and, so far from being a redundancy, this has proved to be an asset. Arend's client is not ``dumb'', but in fact understands a great deal of the structure of the proofs it is presenting. This allows for richer user-interface possibilities, and more flexible coordination between front- and back-end. The \link{https://lodash.com/}{Lodash} Javascript library provides a set of general utilities, mostly aimed at manipulation of collections (arrays and objects) and enabling a functional style of programming; Lodash is used extensively throughout Arend. Lodash makes its facilities available as methods of a global \texttt{\_} object. Thus, filtering out the odd elements of an array in ``stock'' Lodash would take the form \begin{verbatim} _.filter([1,2,3,4], function(e) { return e % 2 == 0; }); \end{verbatim} We have created a wrapper library around Lodash called ``Nodash'' which integrates Lodash's utility methods into the global prototypes of the datatypes which they operate on. Thus, for example, the \texttt{filter} method, which can be applied to any ``collection'' -- array, object, or string --- would be installed on the global \texttt{Array.prototype}, \texttt{Object.prototype}, and \texttt{String.prototype} objects, so that it is accessible simply as \begin{verbatim} [1,2,3,4].filter(function(e) { return e % 2 == 0; }); \end{verbatim} \marginnote[-1in]{Historically, ``monkey-patching'' the global prototypes was considered highly unsafe, as new properties would be ``enumerable'' and would thus become visible in, for example, \texttt{for-in} loops over the properties of \emph{any} object. However, all modern Javascript implementations support the \texttt{defineProperty} method, which allows the creation of \emph{non-enumerate} properties on objects which do not have this problem. Nodash uses \texttt{defineProperty} to safely add Lodash's methods to the global prototypes.} Note that in Nodash the global \texttt{\_} object is still available, for those methods which do not fit with any of the global prototypes. The client-side implementation includes a complete parser for the specification language. This allows user input to be fully syntax-checked before being sent to the server, and allows for faster feedback to the user when syntax errors occur. The parser is written using \link{http://pegjs.org/}{PEG.js}, a parsing expression grammar (PEG) \mcitep{Ford:2004:PEG:964001.964011} parser-generator. The specification grammar in PEG form consists of \lnum{28} non-terminals and \lnum{53} terminal tokens. Of note is the fact that the grammar uses the standard Unicode character classes in its definition of ``identifier'' and ``operator''; an operator, in particular, is defined as any sequence of characters from the symbolic or punctuation classes\footnote{Classes Sc, Sm, Sk, So, Pc, Pd, and Po.}. This allows for traditional mathematical operators such as $\in$ or $\leq$ to be used directly in specifications. Other modules of note in the client-side implementation include: \begin{itemize} \item \texttt{core} --- contains a small amount of compatibility code that allows the other non-user-interface modules to operate transparently in either the browser or Node. In particular, it provides a browser implementation of the \texttt{require()} function in Node, used to load modules. In the browser, these modules must have already been loaded via standard \texttt{<script>} tags (i.e., dynamic loading is not provided), but, once loaded, they are installed into a global module repository; \texttt{require} then simply returns a reference to the appropriate module. This allows Javascript code to transparently access other modules, without knowing whether they are being dynamically loaded within Node, or have already been loaded in the browser. \item \texttt{terms} --- contains a complete representation of \emph{terms}, the fundamental datatype of logic programming. The \texttt{terms} module supports walking the structure of terms, converting Prolog-style term-lists to Javascript Arrays and vice versa, rendering terms to either specification or Prolog-compatible strings, and enumerating the variables in a term. \item \texttt{unify} --- contains a fully-functional implementation of the Robinson unification algorithm \mcitep{robinson1965machine}. Although this module is exhaustively tested, it is currently minimally used. In the future, we hope to build a pattern-matching utility library on it, to allow for more natural examination and manipulation of term-structures on the client-side. \item \texttt{term\_render} --- contains the rendering engine for converting terms, rules, and derivations to HTML structures. Note that derivation rendering is extensible via a number of ``hook'' functions, which allow client code to extend or replace the rendering of the various proof components: antecedents, consequents, disjunctive consequents, and consequents which could be the target of the inductive hypothesis. \end{itemize} \subsection{Server-side implementation} The core of Arend's proof-checking engine is written in Prolog, and is made available to the client via a HTTP interface. SWI-Prolog provides both a standard HTTP server module, as well as a specialized ``app engine'' module called \textsc{Pengines}, both of which are used in Arend. HTML, CSS, and Javascript files for the client-side interface are served via the standard HTTP server, while two PEngine ``applications'', \texttt{repl} and \texttt{passist} provide the interface to the proof-checker itself. PEngines provides a transparent interface between Javascript code running in the browser, and Prolog code running on the server. With it, our client-side code can directly execute queries against the exported predicates of the two aforementioned applications. The results (success, failure, and substitution) of those queries can then be enumerated. Terms are encoded as JSON objects (our \texttt{terms} module can decode JSON terms into its own types). It should be noted that the Prolog implementation of Arend is mostly portable, relying only on ISO-Prolog predicates, commonly-available libraries, and a few SWI-Prolog-specific extensions and modules (most notably the HTTP server module). It would not be difficult to port the proof-checker itself to another Prolog system. The Prolog core proof-checker implementation consists of the following modules, the functionality of which will be described in detail in the following sections: \begin{itemize} \item \texttt{subst} --- provides support for working with \emph{explicit substitutions}. As described below, Arend cannot use Prolog's own substitution, as the substitutions that are applied to a proof object may differ in subtrees; Prolog applies a substitution globally. Thus, we re-implement both variables and substitutions for our own use. \item \texttt{program} --- provides support for expanding atomic goals in the specification language, and for executing goals in the specification. The \texttt{run/3} predicate produces proof objects compatible with those produced by the full proof checker, but is an independent implementation. \item \texttt{checker} --- the heart of the proof checking system, provides routines for dealing with proof objects, elaborating incomplete proofs, and checking and generating proof objects corresponding to particular statements. \end{itemize} \subsection{Proof representation} Informally, the content of a derivation is relatively simple: a tree of \textsc{-Left} and/or \textsc{-Right} rule applications, drawn from the rules of the reasoning logic in figure \ref{reasoning-rules}. In practice, a more detailed representation is required, one which, in particular, necessitates the use of \emph{explicit substitution}, rather than the implicit substitution that would result from naive use of (for example) the system unification in Prolog. For derivations purely derived from the specification language, explicit substitution is not required. This is because for any valid derivation of a proposition expressed in the specification language, the substitution is \emph{consistent} throughout the derivation tree; it is not possible for different ``branches'' to have different substitutions. (Disjunction may, of course, result in the generation of multiple distinct derivations, each with its own unifying substitution, but within any single derivation there is always a single consistent substitution.) However, because the reasoning language possesses rightward implication, a new element is introduced to derivations: the list of \emph{antecedents}. Case analysis on a disjunction antecedent is superficially similar applying the $\land$-R rule, with an important distinction: the branches produced each have \emph{independent} bindings. Consider, for example, case analysis on $\mathsf{nat}(X)$ in the course of a proof: \[ \inference[] {X = z, \mathsf{nat}(z) |- \ldots & X = s(X'), \mathsf{nat}(X') |- \ldots} ................................................................................ inconsistent, and actually impossible to express, because Horn clauses forbid such conjunctions as conclusions. Conversely, given the conclusion $X = z \lor X = s(x')$ we have the choice of which unification to use, and thus there is no inconsistency.} Because of this, each branch of a derivation must maintain its own substitution, applied to its goals as they are expanded. Proofs are represented as a term of the form \begin{verbatim} proof(Goal,Ctx,Subst,Proof) \end{verbatim} where \texttt{Goal} is the goal to be proved, \texttt{Ctx} is the list of antecedents (initially empty for most goals), \texttt{Subst} is the substitution that results if the goal is proved, and \texttt{Proof} is a proof term for this particular goal. The proof terms vary, depending on the structure of the goal and the contents of the context. Non-axiomatic proof terms include one or more sub-proofs. The possible proof terms are \begin{itemize} \item \texttt{induction(N,Proof)} --- proves a $\forall$ inductively, on the N-th element of the context. \item \texttt{ih(Proof)} --- Proves an atomic goal by backchaining it against the inductive hypothesis. \item \texttt{generic(Proof)} --- Proves a $\forall$ generically (i.e., by substituting a unique constant for the quantified variable). \item \texttt{instan(V,Proof)} --- Proves a $\exists$ by giving a value for the quantified variable. \item \texttt{product([Proofs...])} --- Proves a conjunction by providing a subproof for each conjunct. \item \texttt{choice(N,Proof)} --- Proves the N-th branch of a disjunction. \item \texttt{implies(Proof)} --- Proves an implication, by adding the left-hand-side to the context as an assumption and then proving the right-hand-side. \item \texttt{ctxmem(N)} --- Implements the ``assumption rule''; i.e., proves an atomic goal by unifying it with the N-th element of the context. \item \texttt{backchain(Proof)} --- Proves an atomic goal by backchaining it; i.e., by expanding it into its definition, the disjunction of its clauses. \item \texttt{unify(A,B)} --- Proves a goal of the form \texttt{A = B} (this has the side-effect of adding the substitution produced by $A = B$ to the \texttt{Subst} for this proof subtree). \item \texttt{case(T,N,Keep,[Proofs...])} --- Performs case-analysis on an element of the context. The type \texttt{T} corresponds to the various -Left rules of the reasoning logic. \item \texttt{trivial} --- Proves $\top$. \item \texttt{hole} --- Proves any goal, but represents an incomplete proof. \end{itemize} \subsection{Capture avoidance} The use of explicit substitution means that the proof-checker must also contend with another troublesome problem which Prolog conceals: avoiding \emph{accidental capture} when expanding a call to a goal. Consider the goal $\mathsf{nat}(X)$. We would expect this goal to succeed twice, first with the solution $X = z$. ................................................................................ $X \mapsto Y$ to $\forall Y, P(X,Y)$; clearly, one of the $Y$'s will need to be renamed. Since the $Y$ in the substitution is most likely bound somewhere else in the derivation, we choose to rename the $Y$ bound within the $\forall$. \subsection{Unification} The Robinson unification algorithm is presented, in natural deduction style, in figure~\ref{unification-rules}. Note that the same algorithm is used, with minor modification, in both the Javascript and Prolog implementations (for a direct comparison of these two implementations, see our comments below). \begin{figure}[h!] \[ \inference[Wildcard] {} {\_ \equiv t \mid \varepsilon} \] ................................................................................ {} {\mathtt{[]} \equiv \mathtt{[]} \mid \varepsilon} \qquad\qquad \inference[Cons] {t \equiv s \mid \theta_1 & \hat{t} \theta_1 \equiv \hat{s} \theta_1 \mid \theta_2} {(t:\hat{t}) \equiv (s:\hat{s}) \mid \theta_2} \] \caption{Rules for the Robinson unification algorithm} \label{unification-rules} \end{figure} (This presentation is based in part on that in \mcitet{pfenning2006logic}.) It is interesting to directly compare the two implementations of unification (one of the few modules which is semantically similar in both implementations). Both systems implement the same algorithm. The Javascript implementation consists of 136 lines of non-comment code and is quite difficult to follow; the Prolog implementation requires only 69 lines, an almost 50\% reduction! This despite the fact that the Prolog implementation does \emph{not} use the built-in unification to simplify the algorithm at all; it would be largely the same if only traditional assignment were used. \subsection{Proof Checking} The proof checker is implemented in approximately 450 lines of Prolog code. It essentially implements the rules of the reasoning logic presented in figure \ref{reasoning-rules}, by recursively checking the proof tree. That is, it first checks the root node to ensure that the proof object is consistent with the conclusion (antecedents and consequent) by ensuring that it has the correct number and type of subproofs. The subproofs are then recursively checked, working through the tree until the axiomatic rules are reached (or, in the case of an incomplete proof, a \texttt{hole} is found, indicating an unproved subtree). There are two complications to a straightforward top-down recursive implementation: ................................................................................ \subsection{Proof Elaboration} The heart of the incremental proof checking algorithm is the predicate \texttt{elaborate} (defined in file \texttt{checker.pl}). \texttt{elaborate} works in conjunction with \texttt{check}, the proof checker itself. The purpose of elaborate is to take an incomplete proof, together with a reference to one of its leaves (i.e., to a $?$), and to ``elaborate'' it with respect to either its consequent, or one of its antecedents. As described above, the form of the selected element dictates the forms of the subproofs. \texttt{elaborate} ``fills in'' these subproofs, to a single level only (i.e., all the subproofs it constructs are themselves $?$). In a logic which did not include unification or the ability to call defined predicates (atomic goals), we could elaborate any node of the tree, independent of the rest of the tree. ................................................................................ \begin{figure}[h!] All left-rules instantiate the Proof to a term of the form \[ \text{case}(\text{Type},N,\text{Keep},[\text{Proofs}\ldots]) \] In the rules below we leave $N$ (the index in the list of antecedents of the judgment to be cased upon) implicit and omit Keep (a Boolean flag indicating whether the targeted antecedent should remain in the list, or be removed), and write them as \[ \text{Type}(\text{Proofs}\ldots) \] \[ \inference[False-Left] ................................................................................ it has the form $P \land Q$, then the output proof will be elaborated to \[ \proof{P \land Q}{\Gamma}{\text{product}([ \proof{P}{\Gamma}{\text{hole}}, \proof{Q}{\Gamma}{\text{hole}} ])} \] The subproofs of the product will have the correct consequents and antecedents, but their own proofs will be empty, ready for further elaboration. \subsection{Inductive reasoning} Arend's reasoning logic supports proofs of a universal quantification both generically and by \emph{induction}. Arend's induction is technically on the height of derivations, although from the perspective of the user it supports full structural induction on terms; this subsumes the usual natural number induction. Arend supports only single-induction proofs (i.e., induction on multiple antecedents is not allowed) and supports only induction global to a proof (i.e., nested inductions are not allowed, although they can be ``faked'' by using lemmas). These restrictions imply that the induction hypothesis can be regarded as being global to a proof, thus eliminating the need to restrict the scope of difference induction hypotheses to different branches of the proof tree. Internally, induction is implemented by \emph{goal tagging}. When an inductive proof is declared, a particular goal in the antecedents is selected, by the user, to be the target of the induction.\marginnote{For example, in a proof of $\mathsf{nat}(X) |- \mathsf{add}(X,0,X)$ we would induct on $\mathsf{nat}(X)$.} This goal must be a user goal; it cannot be a built-in operator such as conjunction, disjunction, or unification. The functor of the goal is internally flagged as being ``big'' (indicated as $\uparrow$) and the induction hypothesis is defined in terms of the same goal, but flagged as ``small'' ($\downarrow$). For example, to prove that $\mathsf{nat}(X) -> \mathsf{add}(X,0,X)$ our proof would proceed by induction on $\mathsf{nat}(X)$, and thus we would have the inductive hypothesis $\mathsf{nat}^\downarrow(X) -> \mathsf{add}(X,0,X)$. When a $\uparrow$ goal is expanded by case analysis or backchaining, any sub-goals in its expansion are flagged as $\downarrow$, indicating that they are ``smaller'' than the original goal. The induction hypothesis can only be applied to goals which are ``small'', ................................................................................ being a flaw in our system, this surprising result reflects the fact that one computational interpretation of $\bot$ is that of a non-terminating computation. In fact, this result is simply the computational analogue of the well-known \emph{ex falso quodlibet} principle of classical logic. \clearpage \section{Future work} There are many areas in which Arend could be enhanced and extended. We examine ten possibilities here, including one, equational reasoning, in some depth. \begin{itemize} \item Arend's specification logic is simple enough that formalization of its properties should not be difficult, nonetheless, for formal correctness and consistency work needs to be done to show that it is both \emph{sound} (everything it proves true is true) and \emph{non-deterministically complete} (when it ................................................................................ conclusion. Since one of the difficulties of the derivation-tree proof format is its often-extravagant use of horizontal space, it should be possible to automatically \emph{extract} a sub-proof as a lemma. This could be done even when the lemma is not generally useful, but simply as a way of ``naming'' a portion of the proof or commenting on its purpose. \item The implementation of Arend is primitively minimal, suitable only for small-scale usage. Real-world usage would require integration with existing grading systems and automatic checks of student-submitted proofs. On a higher level we could easily imagine a set of features aimed at creating an ecosystem around student-created specifications, proofs, and lemmas, perhaps with collaboration features for multi-student use. \item Although we believe Arend's derivation-tree presentation of proofs has ................................................................................ \subsection{Equational reasoning} One very large extension to the logic which we would like to investigate would be the incorporation of \emph{equational reasoning}. Arend's current concept of equality is based entirely on unification: two terms are equal if they can be made equal by some substitution. Equational reasoning allows equality to be defined via rules, for example \marginnote{For computational purposes equational rules are often regarded as unidirectional \emph{rewrite rules}: $a = b$ becomes $a \mapsto b$.} \[ \inference[$=$-Refl] {} {x = x} \qquad \inference[$=$-Symm] ................................................................................ \inference[$\times$-dist-$+$] {} {a \times (b + c) = a \times b + a \times c} \] Equational reasoning is a particularly powerful mechanism for reasoning about \emph{functional programs} and programming languages; we can state equivalences between expressions and then prove that two programs, or two classes of programs, are equivalent under those rules. Even more powerfully, if we wish to prove some property of an entire language, it is sufficient to show that the equivalences preserve that property. Although the additional of equational reasoning to Arend would give the system a marked increase in logical power, it would also correspondingly increase the complexity of its implementation and presentation. Equivalence rules require a different treatment of terms and variables from unification, and it is not entirely clear how the two will interact, logically. Equivalences over programs ................................................................................ logic variables, but can both contain, and be contained by, them. Although logics exist which provide support for reasoning about binding structures \mcitep{miller05tocl} their usage in proofs involves additional complexity that may be at odd with our goal of use in \emph{introductory} curriculum. \clearpage \section{Appendix I: Syntax of the Specification Logic} \label{sec:spec-syntax} \newcommand{\nterm}[1]{\mathop{\langle\text{#1}\rangle}} \newcommand{\sterm}[1]{\mathop{\text{\emph{#1}}}} \newcommand{\term}[1]{\mathop{\text{\texttt{``#1''}}}} \newthought{The following is a simplified presentation} of the grammar of the specification language. In particular, precedence rules for infix operators have been omitted, but are consistent with normal usage. Note that while the grammar includes rules defining infix arithmetic and comparison operators, these operators have no \emph{semantic} significance within the specification logic. They are present under the assumption that specifications may want to use them, and will expect them to have their normal precedence ordering. \begin{align*} \nterm{start} &<- \nterm{definitions}? \\ \\ \nterm{definitions} &<- \nterm{definition} \nterm{definitions}* \\ \nterm{definition} &<- \sterm{Rulename}? \nterm{predicate} \\ &\mid \nterm{infix-decl} \\ ................................................................................ &\mid \term{(} \nterm{term} \term{)} \\ &\mid \sterm{Atom} \\ &\mid \sterm{Variable} \\ \\ \nterm{compound} &<- \sterm{Atom} \term{(} \nterm{termlist} \term{)} \\ \end{align*} %\subsection{Operational Semantics}\label{spec-execute} %This may or may not get written. If it does, it will be based on the %stack-based semantics of \mcitet{pfenning2006logic}. \clearpage \section[Appendix II: Specification for N, lists]{Appendix II: Specification for $\mathbb{N}$, lists} \label{sec:nat-spec} \newthought{The following is an example specification} included with Arend, one which |