6,802 research outputs found
Constructions, inductive types and strong normalization
This thesis contains an investigation of Coquand's Calculus of Constructions, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalization proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalization argument can be extended to non-algebraic inductive types. We emphasize that our interpretation is sound for large eliminations, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalization argument for Girard's System F. This formal proof has been developed and checked using the..
On Strong Normalization of the Calculus of Constructions with Type-Based Termination
The original publication is available at www.springerlink.comInternational audienceTermination of recursive functions is an important property in proof assistants based on dependent type theories; it implies consistency and decidability of type checking. Type-based termination is a mechanism for ensuring termination that uses types annotated with size information to check that recursive calls are performed on smaller arguments. Our long-term goal is to extend the Calculus of Inductive Constructions with a type-based termination mechanism and prove its logical consistency. In this paper, we present an extension of the Calculus of Constructions (including universes and impredicativity) with sized natural numbers, and prove strong normalization and logical consistency. Moreover, the proof can be easily adapted to include other inductive types
Definitions by Rewriting in the Calculus of Constructions
The main novelty of this paper is to consider an extension of the Calculus of
Constructions where predicates can be defined with a general form of rewrite
rules. We prove the strong normalization of the reduction relation generated by
the beta-rule and the user-defined rules under some general syntactic
conditions including confluence. As examples, we show that two important
systems satisfy these conditions: a sub-system of the Calculus of Inductive
Constructions which is the basis of the proof assistant Coq, and the Natural
Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award
Inductive types in the Calculus of Algebraic Constructions
In a previous work, we proved that an important part of the Calculus of
Inductive Constructions (CIC), the basis of the Coq proof assistant, can be
seen as a Calculus of Algebraic Constructions (CAC), an extension of the
Calculus of Constructions with functions and predicates defined by higher-order
rewrite rules. In this paper, we prove that almost all CIC can be seen as a
CAC, and that it can be further extended with non-strictly positive types and
inductive-recursive types together with non-free constructors and
pattern-matching on defined symbols.Comment: Journal version of TLCA'0
Inductive-data-type Systems
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last
two authors presented a combined language made of a (strongly normalizing)
algebraic rewrite system and a typed lambda-calculus enriched by
pattern-matching definitions following a certain format, called the "General
Schema", which generalizes the usual recursor definitions for natural numbers
and similar "basic inductive types". This combined language was shown to be
strongly normalizing. The purpose of this paper is to reformulate and extend
the General Schema in order to make it easily extensible, to capture a more
general class of inductive types, called "strictly positive", and to ease the
strong normalization proof of the resulting system. This result provides a
computation model for the combination of an algebraic specification language
based on abstract data types and of a strongly typed functional language with
strictly positive inductive types.Comment: Theoretical Computer Science (2002
Higher-Order Termination: from Kruskal to Computability
Termination is a major question in both logic and computer science. In logic,
termination is at the heart of proof theory where it is usually called strong
normalization (of cut elimination). In computer science, termination has always
been an important issue for showing programs correct. In the early days of
logic, strong normalization was usually shown by assigning ordinals to
expressions in such a way that eliminating a cut would yield an expression with
a smaller ordinal. In the early days of verification, computer scientists used
similar ideas, interpreting the arguments of a program call by a natural
number, such as their size. Showing the size of the arguments to decrease for
each recursive call gives a termination proof of the program, which is however
rather weak since it can only yield quite small ordinals. In the sixties, Tait
invented a new method for showing cut elimination of natural deduction, based
on a predicate over the set of terms, such that the membership of an expression
to the predicate implied the strong normalization property for that expression.
The predicate being defined by induction on types, or even as a fixpoint, this
method could yield much larger ordinals. Later generalized by Girard under the
name of reducibility or computability candidates, it showed very effective in
proving the strong normalization property of typed lambda-calculi..
Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach
We present an extension of the second-order logic AF2 with iso-style
inductive and coinductive definitions specifically designed to extract programs
from proofs a la Krivine-Parigot by means of primitive (co)recursion
principles. Our logic includes primitive constructors of least and greatest
fixed points of predicate transformers, but contrary to the common approach, we
do not restrict ourselves to positive operators to ensure monotonicity, instead
we use the Mendler-style, motivated here by the concept of monotonization of an
arbitrary operator on a complete lattice. We prove an adequacy theorem with
respect to a realizability semantics based on saturated sets and
saturated-valued functions and as a consequence we obtain the strong
normalization property for the proof-term reduction, an important feature which
is absent in previous related work.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Building Decision Procedures in the Calculus of Inductive Constructions
It is commonly agreed that the success of future proof assistants will rely
on their ability to incorporate computations within deduction in order to mimic
the mathematician when replacing the proof of a proposition P by the proof of
an equivalent proposition P' obtained from P thanks to possibly complex
calculations. In this paper, we investigate a new version of the calculus of
inductive constructions which incorporates arbitrary decision procedures into
deduction via the conversion rule of the calculus. The novelty of the problem
in the context of the calculus of inductive constructions lies in the fact that
the computation mechanism varies along proof-checking: goals are sent to the
decision procedure together with the set of user hypotheses available from the
current context. Our main result shows that this extension of the calculus of
constructions does not compromise its main properties: confluence, subject
reduction, strong normalization and consistency are all preserved
- …