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Book digitized by Google and uploaded to the Internet Archive by user tpb.

The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics.

Book digitized by Google from the library of Harvard University and uploaded to the Internet Archive by user tpb.

This document looks at specific applications of calculus to engineering. It is noted that for an automobile traveling at constant speed, the mathematical relationship between the directional heading and the angular deflection of the steering wheel can be calculated with respect to time. An analysis is presented which derives the relationship from basic geometrical and kinematic principles. The derivation is seen as an example of calculus use in the mathematical modeling of dynamic systems. This module contains both exercises and a model exam, with answers provided for both at the conclusion of the document. (MP)

This document looks at specific applications of calculus to engineering. It is noted that for an automobile traveling at constant speed, the mathematical relationship between the directional heading and the angular deflection of the steering wheel can be calculated with respect to time. An analysis is presented which derives the relationship from basic geometrical and kinematic principles. The derivation is seen as an example of calculus use in the mathematical modeling of dynamic systems. This module contains both exercises and a model exam, with answers provided for both at the conclusion of the document. (MP)

1Legends of Saints and Sinners

By Douglas Hyde

"I have called the present volume "Legends of Saints and Sinners," which to a certain extent it is; but I mean it for a book of Irish Christian folk-lore. My idea in compiling it has been to give for the first time a collection of genuine Irish folk-lore which might be called "Christian." By this I mean folk-stories and folk-poems which are either entirely founded upon Christian conceptions, or else are so far coloured by them, that they could never have been told—at least in their present shape—had not Christianity established itself in Ireland. Every one of these stories conforms fairly to this standard, except one or two, which I give as necessary corollaries. They are all translations from the Irish. I have found hardly any such stories in English. They were mostly collected by myself from the mouths of native speakers, but three or four of them I have taken from Irish MSS. in my own possession, and a few more were given me by my friends. Not one of these stories was ever translated into English before, with the exception of those which I have taken from my own "Religious Songs of Connacht." - Summary by From The Introduction

This book is a collection of folklore tales collected by Dubhglas de hIde in the Connacht region of Ireland in the 1880s and 1890s. - Summary by Brian Ó Broin