Floris van Doorn's research while affiliated with French National Centre for Scientific Research and other places
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Publications (22)
We will discuss our experiences and design decisions obtained from building a formal library for the convolution of two functions. Convolution is a fundamental concept with applications throughout mathematics. We will focus on the design decisions we made to make the convolution general and easy to use, and the incorporation of this development in...
In differential topology and geometry, the h-principle is a property enjoyed by certain construction problems. Roughly speaking, it states that the only obstructions to the existence of a solution come from algebraic topology. We describe a formalisation in Lean of the local h-principle for first-order, open, ample partial differential relations. T...
We describe a formal proof of the independence of the continuum hypothesis ($\mathsf{CH}$) in the Lean theorem prover. We use Boolean-valued models to give forcing arguments for both directions, using Cohen forcing for the consistency of $\neg \mathsf{CH}$ and a $\sigma$-closed forcing for the consistency of $\mathsf{CH}$.
We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library \textsf{mathlib}, and discuss...
We consider a perimeter surveillance problem introduced by Kingston, Beard, and Holt in 2008 and studied by Davis, Humphrey, and Kingston in 2019. In this problem, $n$ drones surveil a finite interval, moving at uniform speed and exchanging information only when they meet another drone. Kingston, Beard, and Holt described a particular online algori...
The Lean mathematical library mathlib is developed by a community of users with very different backgrounds and levels of experience. To lower the barrier of entry for contributors and to lessen the burden of reviewing contributions, we have developed a number of tools for the library which check proof developments for subtle mistakes in the code an...
The Lean mathematical library mathlib is developed by a community of users with very different backgrounds and levels of experience. To lower the barrier of entry for contributors and to lessen the burden of reviewing contributions, we have developed a number of tools for the library which check proof developments for subtle mistakes in the code an...
We describe a formalization of forcing using Boolean-valued models in the Lean 3 theorem prover, including the fundamental theorem of forcing and a deep embedding of first-order logic with a Boolean-valued soundness theorem. As an application of our framework, we specialize our construction to the Boolean algebra of regular opens of the Cantor spac...
The goal of this dissertation is to present synthetic homotopy theory in the setting of homotopy type theory. We will present various results in this framework, most notably the construction of the Atiyah-Hirzebruch and Serre spectral sequences for cohomology, which have been fully formalized in the Lean proof assistant.
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-Löf type theory. We investigate ordinary groups from this...
We present a development of the theory of higher groups, including infinity groups and connective spectra, in homotopy type theory. An infinity group is simply the loops in a pointed, connected type, where the group structure comes from the structure inherent in the identity types of Martin-L\"of type theory. We investigate ordinary groups from thi...
We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.
In homotopy type theory, we construct the propositional truncation as a colimit, using only non-recursive higher inductive types (HITs). This is a first step towards reducing recursive HITs to non-recursive HITs. This construction gives a characterization of functions from the propositional truncation to an arbitrary type, extending the universal p...
In homotopy type theory, we construct the propositional truncation as a
colimit, using only non-recursive higher inductive types (HITs). This is a
first step towards reducing recursive HITs to non-recursive HITs. This
construction gives a characterization of functions from the propositional
truncation to an arbitrary type, extending the universal p...
Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the constru...
I formalize important theorems about classical propositional logic in the
proof assistant Coq. The main theorems I prove are (1) the soundness and
completeness of natural deduction calculus, (2) the equivalence between natural
deduction calculus, Hilbert systems and sequent calculus and (3) cut
elimination for sequent calculus.
We investigate possible extensions of arbitrary given Pure Type Systems with additional sorts and rules which preserve the normalization property. In particular we identify the following interesting extensions: the disjoint union \(\mathcal{P}+\mathcal{Q}\) of two PTSs \(\mathcal{P}\) and \(\mathcal{Q}\), the PTS \(\forall\mathcal{P}.\mathcal{Q}\)...
We define type theory with explicit conversions. When type checking a term in normal type theory, the system searches for convertibility paths between types. The results of these searches are not stored in the term, and need to be reconstructed every time again. In our system, this information is also represented in the term. The system we define h...
Citations
... For each such ball B ij , its image has measure bounded by (|det(A i )| + ε) Leb(B ij ). Adding these estimates and letting δ tend to 0, we get (2). For the converse inequality ...
... Our second axiom is that this adjunction between synthetic and analytic spectra is an equivalence. As an application, we show that this axiom is is strong enough to fix the homotopy groups (appropriately defined) of the synthetic sphere spectrum to be the stable homotopy groups of the ordinary higher inductive spheres, using some recent work on sequential colimits [SDR20]. ...
... Moreover, there are several partial mechanisations [6,34,38], and Popescu and Traytel [31] investigate the abstract preconditions of the incompleteness theorems using Isabelle/HOL. With the independence of the continuum hypothesis, Han and van Doorn [17] mechanise a specific instance of incompleteness for in Lean. None of these mechanisations approach incompleteness via undecidability. ...
... Under the pertinent identifications (69) this are nothing but the BG-dependent data types (cf. [BvDR18, §4.2]): ...
... The equivalence in Theorem 4.4 is given by mapping a pointed, (n − 1)-connected, n-truncated type X to π n (X), and the inverse is given by mapping a group G to the Eilenberg-Mac Lane space K(G, n), as constructed in [8]. Moreover, [4] shows that the inverse equivalence maps short exact sequences of groups to fiber sequences of types. Note that the n > 1 case of Corollary 4.5 also follows immediately from Proposition 4.3. ...
Reference: Localization in Homotopy Type Theory
... The formalisation was done for the third major version of Lean, which supports reasoning in HoTT by enforcing that its strict, impredicative universe of propositions is avoided in all definitions. It relies on a port of the Lean HoTT library [26] which was developed for Lean 2. The Lean code is available online. 2 The formalisation approximately follows the structure of the paper. Many arguments are directly translated from the informal mathematical style of the paper into Lean code. ...
Reference: Coherence via Wellfoundedness
... With the pushouts (and sequential colimits which are defined similarly), and given that there are more intricate constructions giving propositional truncations [van Doorn, 2016, Kraus, 2016 and higher truncations [Rijke, 2017], we have presented ways to encode all the higher inductive types which we have seen so far as coequalizers. ...
... More recently, however, equality reflection has fallen into disrepute among computer scientists and computationally minded mathematicians. It causes the loss of useful meta-theoretic properties such as strong normalization of terms and decidability of type checking [14], the cornerstones of modern proof assistants like Coq [7], Agda [19] and Lean [10]. Even the property "if an application of a lambda abstraction to an argument is well typed, then its β-reduct is well typed" may not hold if the user assumes nonstandard type equalities. ...
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... [18] develop semantics for several HITs (sphere, torus, suspensions, truncations, pushouts) in certain presheaf toposes, and extend the syntax of cubical type theory [17] with these HITs. Kraus [33] and Van Doorn [19] construct propositional truncation as a sequential colimit. The schemes mentioned so far do not support inductive-inductive types. ...
... The equivalence of Natural Deduction, Sequent Calculus and Hilbert calculus for classical propositional logic, has been formalised in the theorem prover Coq, by Doorn (2015). A major difference between my formalisation and that of Doorn is that they used lists for their contexts in both N and G, whereas I have used sets and multisets respectively. ...
