mike dupont
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Browse files- #Unimath.txt# +0 -289
- .gitignore +49 -0
- README.md +2 -0
- Unimath-10_1.png +0 -3
- Unimath-10_2.png +0 -3
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#Unimath.txt#
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HLF 2016, Sep. 22, 2016, Heidelberg.
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UniMath
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by Vladimir Voevodsky
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from the Institute for Advanced Study in Princeton, NJ.
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Part 1. Univalent foundations
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2
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Univalent Foundations
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UniMath library
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Today we face a problem that involves two difficult to satisfy conditions.
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On the one hand we have to find a way for computer assisted verification of
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mathematical proofs.
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This is necessary, first of all, because we have to stop the dissolution of the
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concept of proof in mathematics.
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On the other hand we have to preserve the intimate connection between
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mathematics and the world of human intuition.
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This connection is what moves mathematics forward and what we often
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experience as the beauty of mathematics.
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3
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Univalent Foundations
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UniMath library
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The Univalent Foundations (UF) is, a yet imperfect, solution to this problem.
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In their original form, the UF combined three components:
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• the view of mathematics as the study of structures on sets and their higher
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analogs,
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• the idea that the higher analogs of sets are reflected in the set-based
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mathematics as homotopy types,
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• the idea that one can formalize our intuition about structures on these higher
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analogs using the Martin-Lof Type Theory (MLTT) extended with the Law of
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Excluded Middle for propositions (LEM) , the Axiom of Choice for sets (AC),
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the Univalence Axiom (UA) and the Resizing Rules (RR).
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4
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Univalent Foundations
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UniMath library
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The main new concepts that were since added to these are the following:
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• the understanding that a lot of mathematics can be formalized in the MLTT
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without the LEM and the AC and that excluding these two axioms one
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obtains foundations for a new form of constructive mathematics,
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• the understanding that classical mathematics appears as a subset of this new
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constructive mathematics,
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• the understanding that the MLTT extended with the UA is an imperfect
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formalization system for this constructive mathematics and that it should be
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possible to integrate the UA into the MLTT obtaining a new type theory
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with better computational properties.
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5
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Univalent Foundations
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UniMath library
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What does it mean for a formalization system to be constructive?
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Some expressions in type theory are said to be in normal form. Any
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expression can be automatically and deterministically “normalized”, that is, an
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equivalent expression in normal form can be computed.
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In type theory there are type expressions and element expressions. If “T” is a
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type (expression) and “o” is an element (expression) one writes “o:T” if the
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type of “o” is “T”.
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6
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Univalent Foundations
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UniMath library
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In most type systems there is the type of natural numbers. In the UniMath it is
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written as “nat”.
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There is the zero element “O:nat” and the successor function “S” from “nat” to
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“nat” that intuitively corresponds to the function that takes “n” to “1+n”.
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A constructive system satisfies the canonicity property for “nat”, which asserts
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that the normal form of any expression “o:nat” has the form “S(S(….(SO)..))”.
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By counting how many “S” there is in the normal form one obtains an actual
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natural number from any element expression of type “nat”.
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7
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Univalent Foundations
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UniMath library
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This is a tremendously strong property.
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Consider the example: a set “X:hSet” is defined to be finite if there exists an
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isomorphism between it and the standard finite set “stn n”. Here “n” is an
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expression of type “nat”. It is well defined and one obtains a function “fincard”
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from finite sets to “nat” called the cardinality - the number of elements of the
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set.
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Now suppose that I have proved, constructively, that “X” is finite. Then
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“(fincard X):nat”
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is defined. By normalizing “fincard X” I obtain an actual natural number.
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If I had a constructive proof of Faltings’s Theorem, stating that the number of
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rational points on a curve of genus >1 is finite, I could find the actual number
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of points on any curve of genus >1.
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8
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Univalent Foundations
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UniMath library
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We don’t know whether such a proof exists. It is a very interesting and hard
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problem.
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The reason that the MLTT+UA is an imperfect system for constructive
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formalization is that while MLTT itself has the canonicity property MLTT+UA
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does not.
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Therefore, formalizing the proof of Faltings’s Theorem in the UniMath, which is
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based on MLTT+UA, would not immediately give us an algorithm to compute
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the number of rational points on a curve of genus >1.
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This is where a new type theory that integrates the UA into the MLTT in such
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a way as to preserve the canonicity would help.
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9
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Univalent Foundations
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UniMath library
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The search for such a type theory became one of the main driving forces in
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the development of the UF.
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Today several groups are working on the construction and implementation in
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a proof assistant of candidate type theories.
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The cubical type theory and the prototype proof assistant cubicaltt created by
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the group of Thierry Coquand with the help of many researchers from
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different parts of the world is at the most advanced stage of development
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today.
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A proof in the UniMath easily translates into a proof in the cubilatt.
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10
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Univalent Foundations
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UniMath library
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The new form of the UF that emerges can be seen as combining the following
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components:
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• the view of mathematics as the study of structures on sets and their higher
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analogs,
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• the view of mathematics as constructive with the classical mathematics being
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a subset consisting of the results that require LEM and/or AC among their
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assumptions,
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• the idea that the higher analogs of sets are reflected in the set-based
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mathematics as constructive homotopy types - objects of the new
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constructive homotopy theory that can so far be formulated only in terms of
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cubical sets,
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• the idea that one can formalize our intuition about structures on these higher
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analogs using Cubical Type Theory (CTT).
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Univalent Foundations
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UniMath library
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In addition to the understanding that to obtain a formal system for the new
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constructive mathematics the UA needs to be integrated into the MLTT
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constructively, several more things are felt as lacking in the MLTT+UA:
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• higher inductive types,
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• resizing rules,
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• a possible strict extensional equality combined with the “fibrancy discipline”,
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• as yet unknown mechanism to construct the types of structures that involve
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infinite hierarchies of coherence conditions.
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Surprisingly, it might be easier to add these features to the CTT than to the
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MLTT. The work in these directions is ongoing.
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Part 2. The UniMath library
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Univalent Foundations
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UniMath library
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In the development of the UniMath library we attempt to do something that
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might be compared with the effort by the Bourbaki group to write a
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systematic exposition of mathematics based on the set theory and the view of
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mathematics as studying structures on sets.
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The effort by Bourbaki stalled at some point around the middle of the 20th
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century, in part, because it was very complicated to describe the emerging
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category-theoretic constructions in set-theoretic terms.
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Univalent Foundations
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UniMath library
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One may however ask, is there any mathematical innovation in what we are
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doing? Is there a discovery of the unknown in the work on the UniMath?
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We have already seen how well-known problems in fields such as arithmetic
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algebraic geometry can be related to the search for a new foundation of
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constructive mathematics and for building proofs in the UniMath.
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Here is a different example.
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Univalent Foundations
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UniMath library
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Some years ago, at the IAS, I had a conversation at lunch with Armand Borel. I
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mentioned how I like Bourbaki “Algebra” and how it helped me to become a
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mathematician.
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I then mentioned that some places there were really dense. For example, said I,
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the description of the tensor product was hard to follow.
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Of course, said Borel, we have invented tensor product to get a systematic
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exposition of multi-linear maps.
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It was new research, this is why it was not very smoothly written.
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Univalent Foundations
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UniMath library
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I was amazed.
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It is hard to imagine today’s mathematics without the concept of the tensor
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product. It would never occurred to me that it was invented by Bourbaki with
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the only purpose to obtain a more systematic exposition of multi-linear maps
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of vector spaces!
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This example shows how a major innovation can emerge from the work on
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systematization of knowledge.
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Univalent Foundations
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UniMath library
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Finally, a few words to those mathematicians who will decide to understand
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UniMath and maybe to contribute to it.
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The UniMath library is being created using the proof assistant Coq. It is freely
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available on GitHub.
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The language of Coq is a very substantial extension of the MLTT and UniMath
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uses a very small subset of the full Coq language that approximately
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corresponds to the original MLTT.
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Univalent Foundations
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The first file in the UniMath after the Basics/preamble.v is Basics/PartA.v.
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The first line in Basics/PartA.v after the preamble section is as follows:
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It should be understood as a declaration of intent to define a constant called
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fromempty whose type is described by the expression that is written to the
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right of the colon.
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Following this line there is a paragraph that starts with Proof. and ends with
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Defined. where the constant is actually defined using the little sub-programs of
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Coq called tactics which help to build complex expressions of the underlying
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type theory language in simple steps.
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Univalent Foundations
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UniMath library
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A mathematician who wants to understand UniMath should expect a very
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non-linear learning curve:
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• In the lectures that I gave in Oxford and in the similar lectures in the Hebrew
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University it took me the whole first lecture to explain what that first line
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and the following it paragraph really mean.
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• In the next lecture I was able to explain the next few hundred lines of PartA.
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• By the fourth lecture in Oxford, the video of which can be found on my
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website, I was explaining the invariant formalization of fibration sequences.
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I hope that was able to show how important Univalent Foundations are and
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how important is the work on libraries such as UniMath.
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Thank you!
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.gitignore
ADDED
@@ -0,0 +1,49 @@
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README.md
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license: creativeml-openrail-m
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license: creativeml-openrail-m
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This contains papers and different forms
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Unimath.org~
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<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.\\
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UniMath\\
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by Vladimir Voevodsky \\
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from the Institute for Advanced Study in Princeton, NJ. \\
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--------------
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<<2>>Part 1. Univalent foundations\\
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2\\
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--------------
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<<3>>[[file:Unimath-3_1.png]]\\
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[[file:Unimath-3_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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Today we face a problem that involves two difficult to satisfy conditions. \\
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On the one hand we have to find a way for computer assisted verification of \\
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mathematical proofs.\\
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This is necessary, first of all, because we have to stop the dissolution of the \\
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concept of proof in mathematics.\\
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On the other hand we have to preserve the intimate connection between \\
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mathematics and the world of human intuition.\\
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This connection is what moves mathematics forward and what we often \\
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experience as the beauty of mathematics. \\
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3\\
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--------------
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<<4>>[[file:Unimath-4_1.png]]\\
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[[file:Unimath-4_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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The Univalent Foundations (UF) is, a yet imperfect, solution to this problem.\\
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In their original form, the UF combined three components:\\
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• the view of mathematics as the study of structures on sets and their higher \\
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analogs, \\
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• the idea that the higher analogs of sets are reflected in the set-based \\
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mathematics as homotopy types, \\
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• the idea that one can formalize our intuition about structures on these higher \\
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analogs using the Martin-Lof Type Theory (MLTT) extended with the Law of \\
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Excluded Middle for propositions (LEM) , the Axiom of Choice for sets (AC), \\
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the Univalence Axiom (UA) and the Resizing Rules (RR).\\
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4\\
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--------------
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<<5>>[[file:Unimath-5_1.png]]\\
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[[file:Unimath-5_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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The main new concepts that were since added to these are the following: \\
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• the understanding that a lot of mathematics can be formalized in the MLTT \\
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without the LEM and the AC and that excluding these two axioms one \\
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obtains foundations for a /new form of constructive mathematics/,\\
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• the understanding that classical mathematics appears as a subset of this new \\
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constructive mathematics,\\
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• the understanding that the MLTT extended with the UA is an imperfect \\
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formalization system for this constructive mathematics and that it should be \\
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possible to integrate the UA into the MLTT obtaining a new type theory \\
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with better computational properties.\\
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5\\
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--------------
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<<6>>[[file:Unimath-6_1.png]]\\
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[[file:Unimath-6_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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What does it mean for a formalization system to be constructive?\\
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Some expressions in type theory are said to be in normal form. Any \\
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expression can be automatically and deterministically “normalized”, that is, an \\
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equivalent expression in normal form can be computed. \\
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In type theory there are type expressions and element expressions. If “T” is a \\
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type (expression) and “o” is an element (expression) one writes “o:T” if the \\
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type of “o” is “T”. \\
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6\\
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--------------
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<<7>>[[file:Unimath-7_1.png]]\\
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[[file:Unimath-7_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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In most type systems there is the type of natural numbers. In the UniMath it is \\
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written as “nat”.\\
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There is the zero element “O:nat” and the successor function “S” from “nat” to \\
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“nat” that intuitively corresponds to the function that takes “n” to “1+n”. \\
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A constructive system satisfies the /canonicity property/ for “nat”, which asserts \\
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that the normal form of any expression “o:nat” has the form “S(S(....(SO)..))”.\\
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By counting how many “S” there is in the normal form one obtains an actual \\
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natural number from any element expression of type “nat”. \\
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7\\
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--------------
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<<8>>[[file:Unimath-8_1.png]]\\
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[[file:Unimath-8_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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This is a tremendously strong property. \\
|
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Consider the example: a set “X:hSet” is defined to be finite if there exists an \\
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isomorphism between it and the standard finite set “stn n”. Here “n” is an \\
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expression of type “nat”. It is well defined and one obtains a function “fincard” \\
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from finite sets to “nat” called the cardinality - the number of elements of the \\
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set.\\
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Now suppose that I have proved, constructively, that “X” is finite. Then \\
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“(fincard X):nat” \\
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is defined. By normalizing “fincard X��� I obtain an actual natural number.\\
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If I had a constructive proof of /Faltings's Theorem, /stating that the number of \\
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rational points on a curve of genus >1 is finite, I could find the actual number \\
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of points on any curve of genus >1. \\
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8\\
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--------------
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<<9>>[[file:Unimath-9_1.png]]\\
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[[file:Unimath-9_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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We don't know whether such a proof exists. It is a very interesting and hard \\
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problem. \\
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The reason that the MLTT+UA is an imperfect system for constructive \\
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formalization is that while MLTT itself has the canonicity property MLTT+UA \\
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does not.\\
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Therefore, formalizing the proof of Faltings's Theorem in the UniMath, which is \\
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based on MLTT+UA, would not immediately give us an algorithm to compute \\
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the number of rational points on a curve of genus >1.\\
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This is where a new type theory that integrates the UA into the MLTT in such \\
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a way as to preserve the canonicity would help. \\
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9\\
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--------------
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<<10>>[[file:Unimath-10_1.png]]\\
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[[file:Unimath-10_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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The search for such a type theory became one of the main driving forces in \\
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the development of the UF.\\
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Today several groups are working on the construction and implementation in \\
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a proof assistant of candidate type theories. \\
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The /cubical type theory/ and the prototype proof assistant /cubicaltt/ created by \\
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the group of Thierry Coquand with the help of many researchers from \\
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different parts of the world is at the most advanced stage of development \\
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today. \\
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A proof in the UniMath easily translates into a proof in the cubilatt.\\
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10\\
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--------------
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<<11>>[[file:Unimath-11_1.png]]\\
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[[file:Unimath-11_2.png]]\\
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Univalent Foundations\\
|
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UniMath library\\
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The new form of the UF that emerges can be seen as combining the following \\
|
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components:\\
|
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• the view of mathematics as the study of structures on sets and their higher \\
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analogs, \\
|
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• the view of mathematics as constructive with the classical mathematics being \\
|
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a subset consisting of the results that require LEM and/or AC among their \\
|
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assumptions,\\
|
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• the idea that the higher analogs of sets are reflected in the set-based \\
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mathematics as constructive homotopy types - objects of the new \\
|
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constructive homotopy theory that can so far be formulated only in terms of \\
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cubical sets,\\
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• the idea that one can formalize our intuition about structures on these higher \\
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analogs using Cubical Type Theory (CTT).\\
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11\\
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--------------
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<<12>>[[file:Unimath-12_1.png]]\\
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[[file:Unimath-12_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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In addition to the understanding that to obtain a formal system for the new \\
|
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constructive mathematics the UA needs to be integrated into the MLTT \\
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constructively, several more things are felt as lacking in the MLTT+UA:\\
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• higher inductive types, \\
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• resizing rules,\\
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• a possible strict extensional equality combined with the “fibrancy discipline”,\\
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• as yet unknown mechanism to construct the types of structures that involve \\
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infinite hierarchies of coherence conditions. \\
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Surprisingly, it might be easier to add these features to the CTT than to the \\
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MLTT. The work in these directions is ongoing. \\
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12\\
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--------------
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<<13>>Part 2. The UniMath library\\
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13\\
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--------------
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<<14>>[[file:Unimath-14_1.png]]\\
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[[file:Unimath-14_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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In the development of the UniMath library we attempt to do something that \\
|
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might be compared with the effort by the Bourbaki group to write a \\
|
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systematic exposition of mathematics based on the set theory and the view of \\
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mathematics as studying structures on sets.\\
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The effort by Bourbaki stalled at some point around the middle of the 20th \\
|
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century, in part, because it was very complicated to describe the emerging \\
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category-theoretic constructions in set-theoretic terms.\\
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14\\
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--------------
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<<15>>[[file:Unimath-15_1.png]]\\
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[[file:Unimath-15_2.png]]\\
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Univalent Foundations\\
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UniMath library\\
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One may however ask, is there any mathematical innovation in what we are \\
|
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doing? Is there a discovery of the unknown in the work on the UniMath?\\
|
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We have already seen how well-known problems in fields such as arithmetic \\
|
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algebraic geometry can be related to the search for a new foundation of \\
|
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constructive mathematics and for building proofs in the UniMath.\\
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Here is a different example.\\
|
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15\\
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--------------
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<<16>>[[file:Unimath-16_1.png]]\\
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[[file:Unimath-16_2.png]]\\
|
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Univalent Foundations\\
|
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UniMath library\\
|
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Some years ago, at the IAS, I had a conversation at lunch with Armand Borel. I \\
|
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mentioned how I like Bourbaki “Algebra” and how it helped me to become a \\
|
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mathematician. \\
|
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I then mentioned that some places there were really dense. For example, said I, \\
|
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the description of the tensor product was hard to follow. \\
|
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Of course, said Borel, /we have invented tensor product to get a systematic \\
|
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exposition of multi-linear maps/. \\
|
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It was new research, this is why it was not very smoothly written. \\
|
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16\\
|
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--------------
|
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<<17>>[[file:Unimath-17_1.png]]\\
|
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[[file:Unimath-17_2.png]]\\
|
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Univalent Foundations\\
|
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UniMath library\\
|
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I was amazed.\\
|
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It is hard to imagine today's mathematics without the concept of the tensor \\
|
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product. It would never occurred to me that it was invented by Bourbaki with \\
|
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the only purpose to obtain a more systematic exposition of multi-linear maps \\
|
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of vector spaces!\\
|
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This example shows how a major innovation can emerge from the work on \\
|
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systematization of knowledge. \\
|
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17\\
|
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--------------
|
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<<18>>[[file:Unimath-18_1.png]]\\
|
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[[file:Unimath-18_2.png]]\\
|
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Univalent Foundations\\
|
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UniMath library\\
|
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Finally, a few words to those mathematicians who will decide to understand \\
|
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UniMath and maybe to contribute to it. \\
|
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The UniMath library is being created using the proof assistant Coq. It is freely \\
|
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available on GitHub.\\
|
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The language of Coq is a very substantial extension of the MLTT and UniMath \\
|
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uses a very small subset of the full Coq language that approximately \\
|
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corresponds to the original MLTT.\\
|
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18\\
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--------------
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<<19>>[[file:Unimath-19_1.png]]\\
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[[file:Unimath-19_2.png]]\\
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[[file:Unimath-19_3.png]]\\
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Univalent Foundations\\
|
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UniMath library\\
|
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The first file in the UniMath after the /Basics/preamble.v/ is /Basics/PartA/.v.\\
|
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The first line in /Basics/PartA.v/ after the preamble section is as follows:\\
|
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\\
|
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It should be understood as a declaration of intent to define a constant called \\
|
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/fromempty /whose type is described by the expression that is written to the \\
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right of the colon. \\
|
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Following this line there is a paragraph that starts with /Proof./ and ends with \\
|
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/Defined. /where the constant is actually defined using the little sub-programs of \\
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Coq called tactics which help to build complex expressions of the underlying \\
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type theory language in simple steps. \\
|
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19\\
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--------------
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<<20>>[[file:Unimath-20_1.png]]\\
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[[file:Unimath-20_2.png]]\\
|
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Univalent Foundations\\
|
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UniMath library\\
|
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A mathematician who wants to understand UniMath should expect a very \\
|
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non-linear learning curve:\\
|
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• In the lectures that I gave in Oxford and in the similar lectures in the Hebrew \\
|
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University it took me the whole first lecture to explain what that first line \\
|
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and the following it paragraph really mean.\\
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• In the next lecture I was able to explain the next few hundred lines of PartA.\\
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• By the fourth lecture in Oxford, the video of which can be found on my \\
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website, I was explaining the invariant formalization of fibration sequences.\\
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20\\
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--------------
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<<21>>I hope that was able to show how important Univalent Foundations are and \\
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how important is the work on libraries such as UniMath.\\
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Thank you!\\
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21\\
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--------------
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