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Some lambda calculus and type theory formalized
 Journal of Automated Reasoning
, 1999
"... Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention ..."
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Abstract. We survey a substantial body of knowledge about lambda calculus and Pure Type Systems, formally developed in a constructive type theory using the LEGO proof system. On lambda calculus, we work up to an abstract, simplified, proof of standardization for beta reduction, that does not mention redex positions or residuals. Then we outline the meta theory of Pure Type Systems, leading to the strengthening lemma. One novelty is our use of named variables for the formalization. Along the way we point out what we feel has been learned about general issues of formalizing mathematics, emphasizing the search for formal definitions that are convenient for formal proof and convincingly represent the intended informal concepts.
A realizability interpretation of MartinLöf's type theory
"... In this paper we present a simple argument for normalization of the fragment of MartinLöf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types a ..."
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Cited by 7 (1 self)
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In this paper we present a simple argument for normalization of the fragment of MartinLöf's type theory that contains the natural numbers, dependent function types and the first universe. We do this by building a realizability model of this theory which directly reflects that terms and types are generated simultaneously.
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:
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A formalization of Γ ∞ in Coq
, 2010
"... In this paper we present a formalization of the type systems Γ ∞ in the proof assistant Coq. The family of type systems Γ∞, described in a recent article by Geuvers, McKinna and Wiedijk [9], presents type theory without the need for explicit contexts. A typing judgment in Γ ∞ is of the shape A: ∞ B ..."
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In this paper we present a formalization of the type systems Γ ∞ in the proof assistant Coq. The family of type systems Γ∞, described in a recent article by Geuvers, McKinna and Wiedijk [9], presents type theory without the need for explicit contexts. A typing judgment in Γ ∞ is of the shape A: ∞ B while an ordinary judgment is of the shape Γ ` A: B. This approach of Geuvers et al. makes a bridge between traditional logic and type theory. In the former free variables are really free and contexts are nonexplicit, as in Γ∞. Furthermore Γ ∞ could make it possible to create a stateless version of an LCF style prover. The important part of [9] is a theorem that states that there is a natural correspondence between judgments in Γ ∞ and ordinary Pure Type Systems. Their paper contains an informal proof of this theorem. We have formalized many of their definitions and lemmas which result in a proof of one direction of this correspondence theorem. 1
Russell’s Metatheoretic Study Notes
, 2006
"... We are working on the formalization of Russell’s type theory in the Coq proof assistant [3]. The type system of Russell is based on the Calculus of Constructions with Σtypes (dependent sums), extended by an equivalence on types which subsumes βconversion. The extension permits to identify types an ..."
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We are working on the formalization of Russell’s type theory in the Coq proof assistant [3]. The type system of Russell is based on the Calculus of Constructions with Σtypes (dependent sums), extended by an equivalence on types which subsumes βconversion. The extension permits to identify types and subsets based on them in a manner similar to the Predicate Subtyping feature of PVS. We are aiming at a complete proof of Russell’s metatheoretic properties (structural properties, Subject Reduction, maybe Strong Normalization), the refining steps which led us to the algorithmic system and the corresponding typing algorithm and also the correctness of an interpretation from Russell to the Calculus of Inductive Constructions with metavariables. We started the development using the formalization of the Calculus of Constructions by Bruno Barras [2]. We kept the standard de Bruijn encoding for variable bindings and defined our judgements using dependent inductive predicates. This alone causes some problems for the faithful formalization of the paper results. The proofs offer several other technical difficulties including:
This document in subdirectoryRS/97/51/ Some Lambda Calculus and Type Theory Formalized ∗†
, 909
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
This document in subdirectoryRS/97/18/ How to Believe a MachineChecked Proof 1
, 909
"... Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS ..."
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Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:
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λTypes on the λCalculus with Abbreviations
, 2007
"... In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable te ..."
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In this paper the author presents λδ, a λtyped λcalculus with a single λ binder and abbreviations. This calculus pursues the reuse of the term constructions both at the level of types and at the level of contexts as the main goal. Up to conversion λδ shares with Church λ → the subset of typable terms but in the “propositions as types ” perspective it can encode the implicative fragment of predicative logic without quantifiers because dependent types are allowed. λδ enjoys the properties of Church λ → (mainly subject conversion, strong normalization and decidability of type inference) and, in addition, it satisfies the correctness of types and the uniqueness of types up to conversion. We stress that λδ differs from the Automathrelated λcalculi in that they do not provide for an abbreviation construction at the level of terms. Moreover, unlike many λcalculi, λδ features a type hierarchy with an infinite number of levels both above and below any reference point.