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The HOL Light manual (1.1)
, 2000
"... ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's A ..."
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ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. \x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous functionvalued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with rstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordina...
Uniqueness of Normal Proofs in Implicational Intuitionistic Logic
 Journal of Logic, Language and Information
, 1999
"... . A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has be ..."
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. A minimal theorem in a logic L is an Ltheorem which is not a nontrivial substitution instance of another Ltheorem. Komori (1987) raised the question whether every minimal implicational theorem in intuitionistic logic has a unique normal proof in the natural deduction system NJ. The answer has been known to be partially positive and generally negative. It is shown here that a minimal implicational theorem A in intuitionistic logic has a unique finormal proof in NJ whenever A is provable without nonprime contraction. The nonprime contraction rule in NJ is the implication introduction rule whose cancelled assumption differs from a propositional variable and appears more than once in the proof. Our result improves the known partial positive solutions to Komori's problem. Also, we present another simple example of a minimal implicational theorem in intuitionistic logic which does not have a unique fijnormal proof in NJ. Key words: natural deduction, uniqueness of normal proofs, coh...
Strategic Computation and Deduction
, 2009
"... I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these thi ..."
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I'd like to conclude by emphasizing what a wonderful eld this is to work in. Logical reasoning plays such a fundamental role in the spectrum of intellectual activities that advances in automating logic will inevitably have a profound impact in many intellectual disciplines. Of course, these things take time. We tend to be impatient, but we need some historical perspective. The study of logic has a very long history, going back at least as far as Aristotle. During some of this time not very much progress was made. It's gratifying to realize how much has been accomplished in the less than fty years since serious e orts to mechanize logic began.
The HOL Light manual (1.0)
, 1998
"... ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's A ..."
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Cited by 1 (0 self)
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ion is in a precise sense a converse operation to application. Given 49 50 CHAPTER 5. PRIMITIVE BASIS OF HOL LIGHT a variable x and a term t, which may or may not contain x, one can construct the socalled lambdaabstraction x: t, which means `the function of x that yields t'. (In HOL's ASCII concrete syntax the backslash is used, e.g. "x. t.) For example, x: x + 1 is the function that adds one to its argument. Abstractions are not often seen in informal mathematics, but they have at least two merits. First, they allow one to write anonymous functionvalued expressions without naming them (occasionally one sees x 7! t[x] used for this purpose), and since our logic is avowedly higher order, it's desirable to place functions on an equal footing with firstorder objects in this way. Secondly, they make variable dependencies and binding explicit; by contrast in informal mathematics one often writes f(x) in situations where one really means x: f(x). We should give some idea of how ordinary...
Superdeduction at Work
"... Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that models ..."
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Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that models appropriately superdeduction. We finaly examplify on several examples, including equality and noetherian induction, the usefulness of this approach which is implemented in the lemuridæ system, written in TOM.
GentzenPrawitz Natural Deduction as a Teaching Tool
, 907
"... Abstract. We report a fouryears experiment in teaching reasoning to undergraduate students, ranging from weak to gifted, using GentzenPrawitz’s style natural deduction. We argue that this pedagogical approach is a good alternative to the use of Boolean algebra for teaching reasoning, especially fo ..."
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Abstract. We report a fouryears experiment in teaching reasoning to undergraduate students, ranging from weak to gifted, using GentzenPrawitz’s style natural deduction. We argue that this pedagogical approach is a good alternative to the use of Boolean algebra for teaching reasoning, especially for computer scientists and formal methods practioners. 1
V Implementing FormulasasTypesasObjects Implementing FormulasasTypesasObjects
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unknown title
, 2000
"... Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL "of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the Curry ..."
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Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL &quot;of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the CurryHoward `propositions as types ' analogy, and to the BrouwerHeytingKolmogorov interpretation of intuitionistic logic we can treat MLTT as a framework for the specification and derivation of correct functional programs. However, programming in MLTT has two weaknesses: ffl we are limited in the functions that we can naturally express; ffl the functions that we do write naturally are often inefficient. Programming with continuations allows us partially to address these problems. The continuationpassing programming style is known to offer a number of advantages to the functional programmer. We can also observe a relationship between continuation passing and type lifting in categorial grammar. We present computation rules which allow us to use continuations with inductivelydefined types, and with types not presented inductively. We justify the new elimination rules using the usual prooftheoretic semantics. We show that the new rules preserve the consistency of the theory. We show how to use wellorderings to encode continuationpassing operators for inductively defined types. Acknowledgements An earlier version of some of the material in Chapter 6 appeared as [70]. I would like to thank: ffl Peter Kay of Massey University's Albany campus, and Steve Reeves of Waikato University, for providing invaluable support and guidance; ffl Ross Renner, and the School of Mathematical and Computing Sciences at Victoria University of Wellington, for indulging me with time and money to visit Waikato University to talk with Steve; ffl Mhairi for caring (again) that I should finish a thesis; ffl Keir and Ailidh for not caring about theses at all.