André Hirschowitz

Higher-Order Abstract Syntax in Coq

The terms of the simply-typed λ-calculus can be used to express the higher-order abstract syntax of objects such as logical formulas, proofs, and programs. Support for the manipulation of such objects is provided in several programming languages (e.g. λProlog, Elf). Such languages also provide embedded implication, a tool which is widely used for expressing hypothetical judgments in natural deduction. In this paper, we show how a restricted form of second-order syntax and embedded implication can be used together with induction in the Coq Proof Development system. We specify typing rules and evaluation for a simple functional language containing only function abstraction and application, and we fully formalize a proof of type soundness in the system. One difficulty we encountered is that expressing the higher-order syntax of an object-language as an inductive type in Coq generates a class of terms that contains more than just those that directly represent objects in the language. We overcome this difficulty by defining a predicate in Coq that holds only for those terms that correspond to programs. We use this predicate to express and prove the adequacy for our syntax.

Higher-Order Abstract Syntax with Induction in Coq

Three important properties of Higher-Order Abstract Syntax are the (higher-order) induction principle, which allows proofs by induction, the (higher-order) injection principle, which asserts that equal terms have equal heads and equal sons, and the extensionality principle, which asserts that functional terms which are pointwise equal are equal. Higher-order abstract syntax is implemented for instance in the Edinburgh Logical Framework and the above principles are satisfied by this implementation. But although they can be proved at the meta level, they cannot be proved at the object level and furthermore, it is not so easy to know how to formulate them in a simple way at the object level. We explain here how Second-Order Abstract Syntax can be implemented in a more powerful type system (Coq) in such a way as to make available or provable (at the object level) the corresponding induction, injection and extensionality principles.

New evidence for Green's conjecture on syzygies of canonical curves

What we call the generic Greens conjecture predicts what are the numbers of syzygies of the generic canonical curve of genus g. Green and Lazarsfeld have observed that curves with nonmaximal Clifford index have extra syzygies and we call specific Greens conjecture the stronger prediction that the curves which have the numbers of syzygies expected for generic curves are precisely those with maximal Clifford index. In this note, we prove that, as stated above, the generic and specific Greens conjectures for canonical curves are equivalent at least when g is odd.

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x1,...,xr be closed points in general position in projective spacePn, then the linear subspaceV ofH0 (⨑n,O(d)) (the space of homogeneous polynomials of degreed on ⨑n) formed by those polynomials which are singular at eachxi, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x1,...,xr. As such, the “expected” value for the dimension ofV is max(0,h0(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.

Fibres de ’t Hooft Speciaux et Applications

On s’interesse aux fibres vectoriels algebriques de rang deux sur un espace projectif de dimension deux ou trois sur un corps de base algebriquement clos.

Nested Abstract Syntax in Coq

We illustrate Nested Abstract Syntax as a high-level alternative representation of languages with binding constructs, based on nested datatypes. Our running example is a partial solution in the Coq proof assistant to the POPLmark Challenge. The resulting formalization is very compact and does not require any extra library or special logical apparatus. Along the way, we propose an original, high-level perspective on environments.

Existence De Faisceaux Réflexifs De Rang Deux SurP 3 À Bonne Cohomologie

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A theory for game theories

Game semantics is a valuable source of fully abstract models of programming languages or proof theories based on categories of so-called games and strategies. However, there are many variants of this technique, whose interrelationships largely remain to be elucidated. This raises the question: what is a category of games and strategies? Our central idea, taken from the first authors PhD thesis [11], is that positions and moves in a game should be morphisms in a base category: playing move m in position f consists in factoring f through m, the new position being the other factor. Accordingly, we provide a general construction which, from a selection of legal moves in an almost arbitrary category, produces a category of games and strategies, together with subcategories of deterministic and winning strategies. As our running example, we instantiate our construction to obtain the standard category of Hyland-Ong games subject to the switching condition. The extension of our framework to games without the switching condition is handled in the first authors PhD thesis [11].

On the Stratification of Nested Hilbert Schemes

Abstract We consider nested Hilbert schemes of the type H N−1, N defined by H N−1, N = {(Z, W) ∈ H N−1 × H N |Z ⊂ W}. We show that H N−1, N is stratified by irreducible subvarieties of the type H Φ, ψ = {(Z, W) ∈ H N−1, N |Z ∈ H Φ, W ∈ H ψ}, where H Φ is the locally closed subscheme of the Hilbert scheme parameterizing finite schemes with Hilbert function Φ, and we compute the dimension of the strata H Φ, ψ. We also show that H N−2, N is irreducible and compute the dimension of certain strata. The results apply to the classification of globally generated and very ample Hilbert functions.

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH1(Z, O) →H1(X, EndE) is surjective.