\documentclass[11pt]{article} \usepackage[margin=1.6in]{geometry} \usepackage{proof} \usepackage{amsmath,amsthm,amssymb} \usepackage[raiselinks=false,colorlinks=true,citecolor=blue,urlcolor=blue,linkcolor=blue,bookmarksopen=true,dvips]{hyperref} % font{ \usepackage{pxfonts} %% fix sans serif \renewcommand\sfdefault{cmss} \DeclareMathAlphabet{\mathsf}{OT1}{cmss}{m}{n} \SetMathAlphabet{\mathsf}{bold}{OT1}{cmss}{b}{n} % }font \theoremstyle{definition} \newtheorem{task}{Task} \newcommand\pimp{\mathrel{\supset}} \newcommand\pand{\mathrel{\wedge}} \newcommand\por{\mathrel{\vee}} \newcommand\ptrue{\top} \newcommand\pfalse{\bot} \newcommand\pspades{\spadesuit} \newcommand\pforall[3]{\forall #1{:}#2.\, #3} \newcommand\pexists[3]{\exists #1{:}#2.\, #3} \newcommand\qtri\blacktriangledown \newcommand\ptri[3]{\qtri #1{:}#2.\, #3} \newcommand\true{\;\textit{true}} \newcommand\ddd{\raisebox{0.2em}[1.3em]{$\vdots$}} \newcommand\com{\raisebox{0.3em}{$\ ,\ \ $}} \newcommand\hyp[2][]{\infer[#1]{#2}{}} \newcommand\sub[3]{\infer[#2]{#3}{#1}} \newcommand{\lred}{\mathrel{\raisebox{0.5em}{$\Longrightarrow_R$}}} \newcommand{\lexp}{\mathrel{\raisebox{0.5em}{$\Longrightarrow_E$}}} \newcommand{\DD}{\mathcal{D}} \newcommand{\EE}{\mathcal{E}} \newcommand{\FF}{\mathcal{F}} \newcommand{\GG}{\mathcal{G}} % some URLs used below \newcommand\tutchGuideURL{http://www.cs.cmu.edu/~fp/courses/15317-f09/software/tutch/doc/html/tutch_ovr.html} \newcommand\tutchProofTermsURL{http://www.cs.cmu.edu/~fp/courses/15317-f09/software/tutch/doc/html/tutch_4.html\#SEC18} \newcommand\tutchProofTermsRefURL{http://www.cs.cmu.edu/~fp/courses/15317-f09/software/tutch/doc/html/tutch_9.html\#SEC28} \title{Constructive Logic (15-317), Fall 2009 \\ Assignment 2: Quantifiers and Proof Terms} \author{William Lovas \texttt{(wlovas@cs)}} \date{Out: Thursday, September 10, 2009 \\ Due: Thursday, September 17, 2009 (before class)} \begin{document} \maketitle Previously, your work in this course has concerned only the propositional fragment of intuitionistic logic. In this homework, you will delve into the exciting world of first-order logic by solving problems involving the quantifiers $\forall$ and $\exists$. Furthermore, you'll have a chance to explore the Curry-Howard correspondence between logic and computation by writing proof terms representing your deductions. Finally, you'll continue to expand your understanding of harmony in definitions of logics by playing with a new quantifier. The Tutch portion of your work (Section~\ref{sec:tutch}) should be submitted electronically using the command \begin{verbatim} $ /afs/andrew/course/15/317/bin/submit -r hw02 \end{verbatim} from any Andrew server. You may check the status of your submission by running the command \begin{verbatim} $ /afs/andrew/course/15/317/bin/status hw02 \end{verbatim} If you have trouble running either of these commands, email William. The written portion of your work (Section~\ref{sec:tri}) should be submitted at the beginning of class. If you are familiar with \LaTeX, you are encouraged to use this document as a template for typesetting your solutions, but you may alternatively write your solutions \textit{neatly} by hand. \section{Tutch Proofs and Proof Terms (25 points)} \label{sec:tutch} Tutch allows you to give an \textit{annotated proof} for a proposition by declaring it with \texttt{annotated proof}. An annotated proof is just like a regular Tutch proof, but each line \texttt{A} is annotated with the term that justifies it \texttt{M : A}\@. Such an annotated proof is essentially a typing derivation for the proof term at its conclusion. Here's a simple example showing that conjunction is commutative: \begin{verbatim} annotated proof andComm : A & B => B & A = begin [ u : A & B; snd u : B; fst u : A; (snd u, fst u) : B & A ]; fn u => (snd u, fst u) : A & B => B & A end; \end{verbatim} Since a proof term determines the structure of the proof, Tutch also allows you to give just the proof term, by declaring it with \texttt{term}: \begin{verbatim} term andComm : A & B => B & A = fn u => (snd u, fst u); \end{verbatim} For more examples, see \href{\tutchProofTermsURL}{Chapter~4} of the \href{\tutchGuideURL}{\textit{Tutch User's Guide}}. The proof terms are very similar to the ones given in lecture and are summarized in \href{\tutchProofTermsRefURL}{Section~A.2.1} of the \textit{Guide}. \begin{task}[6 pts] Prove the theorem $(A \por C) \pand (B \pimp C) \pimp (A \pimp B) \pimp C$ using Tutch. Give a proof, an annotated proof, and a proof term. \begin{verbatim} proof implOr : (A | C) & (B => C) => (A => B) => C annotated proof implOr : (A | C) & (B => C) => (A => B) => C term implOr : (A | C) & (B => C) => (A => B) => C \end{verbatim} \end{task} \begin{task}[13 pts] Prove the following theorems using Tutch, and provide proof terms. \begin{verbatim} proof curry : (A & B => C) => (A => B => C) proof qcurry : ((?x:t. B(x)) => C) => (!x:t. B(x) => C) term curry : (A & B => C) => (A => B => C) term qcurry : ((?x:t. B(x)) => C) => (!x:t. B(x) => C) proof compose : (!x:t. A(x) => B(x)) => (!x:t. B(x) => C(x)) => !x:t. A(x) => C(x) term compose : (!x:t. A(x) => B(x)) => (!x:t. B(x) => C(x)) => !x:t. A(x) => C(x) \end{verbatim} \end{task} \pagebreak[2] \begin{task}[6 pts] Prove the following theorems using Tutch. \begin{verbatim} proof distribAllAnd : (!x:t. A(x) & B(x)) <=> (!x:t. A(x)) & (!x:t. B(x)) proof distribExAnd1 : (?x:t. A(x) & B(x)) => (?x:t. A(x)) & (?x:t. B(x)) \end{verbatim} \end{task} On Andrew machines, you can check your progress against the requirements file \texttt{/afs/andrew/course/15/317/req/hw02.req} by running the command \begin{verbatim} $ /afs/andrew/course/15/317/bin/tutch -r hw02 \end{verbatim} \section{A Mixed-Up Quantifier (15 points)} \label{sec:tri} In recitation, we saw that we could not prove $(\pforall x \tau {A(x)}) \pimp \pexists x \tau {A(x)} \true$---our universal quantifier permits the domain of quantification to be empty! We were able to hack around this by proving a different proposition, but suppose we wanted to directly define a universal quantifier $\ptri x \tau {A(x)}$ that did \textit{not} permit vacuous quantification. What would such a quantifier look like? Its introduction rule ${\qtri}I^a$ is similar to ${\forall}I^a$: we must prove $A(a)$ for a new parameter $a : \tau$. However, to ensure the domain of quantification is non-empty, we must also supply an element $t : \tau$. \[ \infer[{\qtri}I^a] {\ptri x \tau {A(x)} \true} {t : \tau & \deduce[\ddd]{A(a) \true}{\hyp{a : \tau}}} \] Then we can give two elimination rules, one with an existential character and one with a universal character. \[ \infer[{\qtri}E_{\exists}^a] {C \true} {\ptri x \tau {A(x)} \true & \deduce[\ddd]{C \true}{\hyp{a : \tau}}} \quad\qquad \infer[{\qtri}E_{\forall}] {A(t) \true} {\ptri x \tau {A(x)} \true & t : \tau} \] As usual, rules that introduce a parameter restrict its scope to the premise in which it is introduced. In particular, in the elimination rule ${\qtri}E_\exists^a$, the parameter $a$ may not appear in the conclusion of the rule, $C \true$. % Take care in your deductions to respect the scopes of all parameters! \begin{task}[2 pts] Show that this captures our intuition about what non-vacuous universal quantification should mean by giving a deduction of $(\ptri x \tau {A(x)}) \pimp \pexists x \tau {A(x)} \true$. \end{task} \begin{task}[4 pts] Are these elimination rules locally sound? If so, give a local reduction for each elimination rule; if one does not exist, explain why. \end{task} \begin{task}[4 pts] Are these elimination rules locally complete? If so, give a local expansion for for an arbitrary deduction of $\ptri x \tau {A(x)}$; if one does not exist, explain why. \end{task} \begin{task}[5 pts] Make up a proof term for each of the rules, and express the local reductions and local expansions you found above using proof terms. \end{task} \end{document}