paper step1

#Unimath.txt#

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

Unimath.html

@@ -0,0 +1,13 @@
+<!DOCTYPE html>
+<html>
+<head>
+<title>2016_09_22_HLF_Heidelberg.key</title>
+<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
+<meta name="generator" content="pdftohtml 0.36"/>
+<meta name="date" content="2024-02-05T05:31:35+00:00"/>
+</head>
+<frameset cols="100,*">
+<frame name="links" src="Unimath_ind.html"/>
+<frame name="contents" src="Unimaths.html"/>
+</frameset>
+</html>

Unimath.org

@@ -1,311 +1,240 @@
-<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.\\
-UniMath\\
-by Vladimir Voevodsky  \\
-from the Institute for Advanced Study in Princeton, NJ. \\
+<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.  UniMath by Vladimir
+Voevodsky from the Institute for Advanced Study in Princeton, NJ.
 
 --------------
 
-<<2>>Part 1. Univalent foundations\\
-2\\
+<<2>>Part 1. Univalent foundations 2
 
 --------------
 
-<<3>>[[file:Unimath-3_1.png]]\\
-[[file:Unimath-3_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-Today we face a problem that involves two difficult to satisfy conditions. \\
-On the one hand we have to find a way for computer assisted verification of \\
-mathematical proofs.\\
-This is necessary, first of all, because we have to stop the dissolution of the \\
-concept of proof in mathematics.\\
-On the other hand  we have to preserve the intimate connection between \\
-mathematics and the world of human intuition.\\
-This connection is what moves mathematics forward and what we often \\
-experience as the beauty of mathematics. \\
-3\\
+<<3>>[[file:Unimath-3_1.png]] [[file:Unimath-3_2.png]] Univalent Foundations
+UniMath library Today we face a problem that involves two difficult to
+satisfy conditions.  On the one hand we have to find a way for computer
+assisted verification of mathematical proofs.  This is necessary, first
+of all, because we have to stop the dissolution of the concept of
+proof in mathematics.
+#+begin_src interesting
+On the other hand we have to preserve the intimate connection between
+mathematics and the world of human intuition.  This connection is what
+moves mathematics forward and what we often experience as the beauty
+of mathematics.
+#+end_src
+3
 
---------------
-
-<<4>>[[file:Unimath-4_1.png]]\\
-[[file:Unimath-4_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-The Univalent Foundations (UF) is, a yet imperfect, solution to this problem.\\
-In their original form,  the UF combined three components:\\
-• the view of mathematics as the study of structures on sets and their higher \\
-analogs, \\
-• the idea that the higher analogs of sets are reflected in the set-based \\
-mathematics as homotopy types, \\
-• the idea that one can formalize our intuition about structures on these higher \\
-analogs using the Martin-Lof  Type Theory (MLTT) extended with the Law of \\
-Excluded Middle for propositions (LEM) , the Axiom of Choice for sets (AC), \\
-the Univalence Axiom (UA) and the Resizing Rules (RR).\\
-4\\
-
---------------
-
-<<5>>[[file:Unimath-5_1.png]]\\
-[[file:Unimath-5_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-The main new concepts that were since added to these are the following: \\
-• the understanding that a lot of mathematics can be formalized in the MLTT \\
-without the LEM and the AC and that excluding these two axioms one \\
-obtains foundations for a /new form of constructive mathematics/,\\
-• the understanding that classical mathematics appears as a subset of this new \\
-constructive mathematics,\\
-• the understanding that the MLTT extended with the UA is an imperfect \\
-formalization system for this constructive mathematics and that it should be \\
-possible  to integrate the UA into the MLTT obtaining a new type theory \\
-with better computational properties.\\
-5\\
-
---------------
-
-<<6>>[[file:Unimath-6_1.png]]\\
-[[file:Unimath-6_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-What does it mean for a formalization system to be constructive?\\
-Some expressions in type theory are said to be in normal form. Any \\
-expression can be automatically and deterministically “normalized”, that is, an \\
-equivalent expression in normal form can be computed. \\
-In type theory there are type expressions and element expressions. If “T” is a \\
-type (expression) and “o” is an element (expression) one writes “o:T” if the \\
-type of “o” is “T”. \\
-6\\
-
---------------
-
-<<7>>[[file:Unimath-7_1.png]]\\
-[[file:Unimath-7_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-In most type systems there is the type of natural numbers. In the UniMath it is \\
-written as “nat”.\\
-There is the zero element “O:nat” and the successor function “S” from “nat” to \\
-“nat” that intuitively corresponds to the function that takes “n” to “1+n”. \\
-A constructive system satisfies the /canonicity property/ for “nat”, which asserts \\
-that the normal form of any expression “o:nat” has the form “S(S(....(SO)..))”.\\
-By counting how many “S” there is in the normal form one obtains an actual \\
-natural number from any element expression of type “nat”. \\
-7\\
+--------------
+
+<<4>>[[file:Unimath-4_1.png]] [[file:Unimath-4_2.png]] Univalent Foundations
+UniMath library The Univalent Foundations (UF) is, a yet imperfect,
+solution to this problem.  In their original form, the UF combined
+three components: • the view of mathematics as the study of structures
+on sets and their higher analogs, • the idea that the higher analogs
+of sets are reflected in the set-based mathematics as homotopy types, •
+the idea that one can formalize our intuition about structures on
+these higher analogs using the Martin-Lof Type Theory (MLTT) extended
+with the Law of Excluded Middle for propositions (LEM) , the Axiom of
+Choice for sets (AC), the Univalence Axiom (UA) and the Resizing Rules
+(RR).  4
+
+--------------
+
+<<5>>[[file:Unimath-5_1.png]] [[file:Unimath-5_2.png]] Univalent Foundations
+UniMath library The main new concepts that were since added to these
+are the following: • the understanding that a lot of mathematics can
+be formalized in the MLTT without the LEM and the AC and that
+excluding these two axioms one obtains foundations for a /new form of
+constructive mathematics/, • the understanding that classical
+mathematics appears as a subset of this new constructive mathematics,
+• the understanding that the MLTT extended with the UA is an imperfect
+formalization system for this constructive mathematics and that it
+should be possible to integrate the UA into the MLTT obtaining a new
+type theory with better computational properties.  5
 
 --------------
 
-<<8>>[[file:Unimath-8_1.png]]\\
-[[file:Unimath-8_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-This is a tremendously strong property. \\
-Consider the example: a set “X:hSet” is defined to be finite if there exists an \\
-isomorphism between it and the standard finite set “stn n”. Here “n” is an \\
-expression of type “nat”. It is well defined and one obtains a function “fincard”  \\
-from finite sets to “nat” called the cardinality - the number of elements of the \\
-set.\\
-Now suppose that I have proved, constructively, that “X” is finite. Then \\
-“(fincard X):nat” \\
-is defined. By normalizing “fincard X” I obtain an actual natural number.\\
-If I had a constructive proof of /Faltings's Theorem, /stating that the number of \\
-rational points on a curve of genus >1 is finite, I could find the actual number \\
-of points on any curve of genus >1.  \\
-8\\
+<<6>>[[file:Unimath-6_1.png]] [[file:Unimath-6_2.png]] Univalent Foundations
+UniMath library What does it mean for a formalization system to be
+constructive?  Some expressions in type theory are said to be in
+normal form. Any expression can be automatically and deterministically
+“normalized”, that is, an equivalent expression in normal form can be
+computed.  In type theory there are type expressions and element
+expressions. If “T” is a type (expression) and “o” is an element
+(expression) one writes “o:T” if the type of “o” is “T”.  6
 
 --------------
 
-<<9>>[[file:Unimath-9_1.png]]\\
-[[file:Unimath-9_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-We don't know whether such a proof exists. It is a very interesting and hard \\
-problem. \\
-The reason that the MLTT+UA is an imperfect system for constructive  \\
-formalization is that while MLTT itself has the canonicity property MLTT+UA \\
-does not.\\
-Therefore, formalizing the proof of Faltings's Theorem in the UniMath, which is \\
-based on MLTT+UA, would not immediately give us an algorithm to compute \\
-the number of rational points on a curve of genus >1.\\
-This is where a new type theory that integrates the UA into the MLTT in such \\
-a way as to preserve the canonicity would help. \\
-9\\
+<<7>>[[file:Unimath-7_1.png]] [[file:Unimath-7_2.png]] Univalent Foundations
+UniMath library In most type systems there is the type of natural
+numbers. In the UniMath it is written as “nat”.  There is the zero
+element “O:nat” and the successor function “S” from “nat” to “nat”
+that intuitively corresponds to the function that takes “n” to “1+n”.
+A constructive system satisfies the /canonicity property/ for “nat”,
+which asserts that the normal form of any expression “o:nat” has the
+form “S(S(....(SO)..))”.  By counting how many “S” there is in the
+normal form one obtains an actual natural number from any element
+expression of type “nat”.  7
 
 --------------
 
-<<10>>[[file:Unimath-10_1.png]]\\
-[[file:Unimath-10_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-The search for such a type theory became one of the main driving forces in \\
-the development of the UF.\\
-Today several groups are working on the construction and  implementation in \\
-a proof assistant of candidate type theories.  \\
-The /cubical type theory/ and the prototype proof assistant /cubicaltt/ created by \\
-the group of Thierry Coquand with the help of many researchers from \\
-different parts of the world is at the most advanced stage of development \\
-today. \\
-A proof in the UniMath easily translates into a proof in the cubilatt.\\
-10\\
+<<8>>[[file:Unimath-8_1.png]] [[file:Unimath-8_2.png]] Univalent Foundations
+UniMath library This is a tremendously strong property.  Consider the
+example: a set “X:hSet” is defined to be finite if there exists an
+isomorphism between it and the standard finite set “stn n”. Here “n” is
+an expression of type “nat”. It is well defined and one obtains a
+function “fincard” from finite sets to “nat” called the cardinality -
+the number of elements of the set.  Now suppose that I have proved,
+constructively, that “X” is finite. Then “(fincard X):nat” is defined. By
+normalizing “fincard X” I obtain an actual natural number.  If I had a
+constructive proof of /Faltings's Theorem, /stating that the number of
+rational points on a curve of genus >1 is finite, I could find the
+actual number of points on any curve of genus >1.  8
 
 --------------
 
-<<11>>[[file:Unimath-11_1.png]]\\
-[[file:Unimath-11_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-The new form of the UF that emerges can be seen as combining the following \\
-components:\\
-• the view of mathematics as the study of structures on sets and their higher \\
-analogs, \\
-• the view of mathematics as constructive with the classical mathematics being \\
-a subset consisting of the results that require LEM and/or AC among their \\
-assumptions,\\
-• the idea that the higher analogs of sets are reflected in the set-based \\
-mathematics as constructive homotopy types - objects of the new \\
-constructive homotopy theory that can so far be formulated only in terms of \\
-cubical sets,\\
-• the idea that one can formalize our intuition about structures on these higher \\
-analogs using Cubical Type Theory (CTT).\\
-11\\
+<<9>>[[file:Unimath-9_1.png]] [[file:Unimath-9_2.png]] Univalent Foundations
+UniMath library We don't know whether such a proof exists. It is a
+very interesting and hard problem.  The reason that the MLTT+UA is an
+imperfect system for constructive formalization is that while MLTT
+itself has the canonicity property MLTT+UA does not.  Therefore,
+formalizing the proof of Faltings's Theorem in the UniMath, which is
+based on MLTT+UA, would not immediately give us an algorithm to
+compute the number of rational points on a curve of genus >1.  This is
+where a new type theory that integrates the UA into the MLTT in such a
+way as to preserve the canonicity would help.  9
 
 --------------
 
-<<12>>[[file:Unimath-12_1.png]]\\
-[[file:Unimath-12_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-In addition to the understanding that to obtain a formal system for the new \\
-constructive mathematics the UA needs to be integrated into the MLTT \\
-constructively, several more things are felt as lacking in the MLTT+UA:\\
-• higher inductive types, \\
-• resizing rules,\\
-• a possible strict extensional equality combined with the “fibrancy discipline”,\\
-• as yet unknown mechanism to construct the types of structures that involve \\
-infinite hierarchies of coherence conditions.  \\
-Surprisingly,  it might be easier to add these features to the CTT than to the \\
-MLTT. The work in these directions is ongoing. \\
-12\\
+<<10>>[[file:Unimath-10_1.png]] [[file:Unimath-10_2.png]] Univalent
+Foundations UniMath library The search for such a type theory became
+one of the main driving forces in the development of the UF.  Today
+several groups are working on the construction and implementation in a
+proof assistant of candidate type theories.  The /cubical type theory/
+and the prototype proof assistant /cubicaltt/ created by the group of
+Thierry Coquand with the help of many researchers from different parts
+of the world is at the most advanced stage of development today.  A
+proof in the UniMath easily translates into a proof in the cubilatt.
+10
+
+--------------
+
+<<11>>[[file:Unimath-11_1.png]] [[file:Unimath-11_2.png]] Univalent
+Foundations UniMath library The new form of the UF that emerges can be
+seen as combining the following components: • the view of mathematics
+as the study of structures on sets and their higher analogs, • the
+view of mathematics as constructive with the classical mathematics
+being a subset consisting of the results that require LEM and/or AC
+among their assumptions, • the idea that the higher analogs of sets
+are reflected in the set-based mathematics as constructive homotopy
+types - objects of the new constructive homotopy theory that can so
+far be formulated only in terms of cubical sets, • the idea that one
+can formalize our intuition about structures on these higher analogs
+using Cubical Type Theory (CTT).  11
+
+--------------
+
+<<12>>[[file:Unimath-12_1.png]] [[file:Unimath-12_2.png]] Univalent
+Foundations UniMath library In addition to the understanding that to
+obtain a formal system for the new constructive mathematics the UA
+needs to be integrated into the MLTT constructively, several more
+things are felt as lacking in the MLTT+UA: • higher inductive types, •
+resizing rules, • a possible strict extensional equality combined with
+the “fibrancy discipline”, • as yet unknown mechanism to construct the
+types of structures that involve infinite hierarchies of coherence
+conditions.  Surprisingly, it might be easier to add these features to
+the CTT than to the MLTT. The work in these directions is ongoing.  12
 
 --------------
 
-<<13>>Part 2. The UniMath library\\
-13\\
+<<13>>Part 2. The UniMath library 13
 
 --------------
 
-<<14>>[[file:Unimath-14_1.png]]\\
-[[file:Unimath-14_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-In the development of the UniMath library we attempt to do something that \\
-might be compared with the effort by the Bourbaki group to write a \\
-systematic exposition of mathematics based on the set theory and the view of \\
-mathematics as studying structures on sets.\\
-The effort by Bourbaki stalled at some point around the middle of the 20th \\
-century, in part, because it was very complicated to describe the emerging \\
-category-theoretic constructions in set-theoretic terms.\\
-14\\
+<<14>>[[file:Unimath-14_1.png]] [[file:Unimath-14_2.png]] Univalent
+Foundations UniMath library In the development of the UniMath library
+we attempt to do something that might be compared with the effort by
+the Bourbaki group to write a systematic exposition of mathematics
+based on the set theory and the view of mathematics as studying
+structures on sets.  The effort by Bourbaki stalled at some point
+around the middle of the 20th century, in part, because it was very
+complicated to describe the emerging category-theoretic constructions
+in set-theoretic terms.  14
 
 --------------
 
-<<15>>[[file:Unimath-15_1.png]]\\
-[[file:Unimath-15_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-One may however ask, is there any mathematical innovation in what we are \\
-doing? Is there a discovery of the unknown in the work on the UniMath?\\
-We have already seen how well-known problems in fields such as arithmetic \\
-algebraic geometry can be related to the search for a new foundation of \\
-constructive mathematics and for building proofs in the UniMath.\\
-Here is a different example.\\
-15\\
+<<15>>[[file:Unimath-15_1.png]] [[file:Unimath-15_2.png]] Univalent
+Foundations UniMath library One may however ask, is there any
+mathematical innovation in what we are doing? Is there a discovery of
+the unknown in the work on the UniMath?  We have already seen how
+well-known problems in fields such as arithmetic algebraic geometry can
+be related to the search for a new foundation of constructive
+mathematics and for building proofs in the UniMath.  Here is a
+different example.  15
 
 --------------
 
-<<16>>[[file:Unimath-16_1.png]]\\
-[[file:Unimath-16_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-Some years ago, at the IAS, I had a conversation at lunch with Armand Borel. I \\
-mentioned how I like Bourbaki “Algebra” and how it helped me to become a \\
-mathematician. \\
-I then mentioned that some places there were really dense. For example, said I, \\
-the description of the tensor product was hard to follow. \\
-Of course, said Borel, /we have invented tensor product to get a systematic \\
-exposition of multi-linear maps/. \\
-It was new research, this is why it was not very smoothly written. \\
-16\\
+<<16>>[[file:Unimath-16_1.png]] [[file:Unimath-16_2.png]] Univalent
+Foundations UniMath library Some years ago, at the IAS, I had a
+conversation at lunch with Armand Borel. I mentioned how I like
+Bourbaki “Algebra” and how it helped me to become a mathematician.  I
+then mentioned that some places there were really dense. For example,
+said I, the description of the tensor product was hard to follow.  Of
+course, said Borel, /we have invented tensor product to get a
+systematic exposition of multi-linear maps/.  It was new research,
+this is why it was not very smoothly written.  16
 
 --------------
 
-<<17>>[[file:Unimath-17_1.png]]\\
-[[file:Unimath-17_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-I was amazed.\\
-It is hard to imagine today's mathematics without the concept of the tensor \\
-product. It would never occurred to me that it was invented by Bourbaki with \\
-the only purpose to obtain a more systematic exposition of multi-linear maps \\
-of vector spaces!\\
-This example shows how a major innovation can emerge from the work on \\
-systematization of knowledge. \\
-17\\
+<<17>>[[file:Unimath-17_1.png]] [[file:Unimath-17_2.png]] Univalent
+Foundations UniMath library I was amazed.  It is hard to imagine
+today's mathematics without the concept of the tensor product. It
+would never occurred to me that it was invented by Bourbaki with the
+only purpose to obtain a more systematic exposition of multi-linear
+maps of vector spaces!  This example shows how a major innovation can
+emerge from the work on systematization of knowledge.  17
 
 --------------
 
-<<18>>[[file:Unimath-18_1.png]]\\
-[[file:Unimath-18_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-Finally, a few words to those mathematicians who will decide to understand \\
-UniMath and maybe to contribute to it. \\
-The UniMath library is being created using the proof assistant Coq. It is freely \\
-available on GitHub.\\
-The language of Coq is a very substantial extension of the MLTT and UniMath \\
-uses a very small subset of the full Coq language that approximately \\
-corresponds to the original MLTT.\\
-18\\
+<<18>>[[file:Unimath-18_1.png]] [[file:Unimath-18_2.png]] Univalent
+Foundations UniMath library Finally, a few words to those
+mathematicians who will decide to understand UniMath and maybe to
+contribute to it.  The UniMath library is being created using the
+proof assistant Coq. It is freely available on GitHub.  The language
+of Coq is a very substantial extension of the MLTT and UniMath uses a
+very small subset of the full Coq language that approximately
+corresponds to the original MLTT.  18
 
 --------------
 
-<<19>>[[file:Unimath-19_1.png]]\\
-[[file:Unimath-19_2.png]]\\
-[[file:Unimath-19_3.png]]\\
-Univalent Foundations\\
-UniMath library\\
-The first file in the UniMath after the /Basics/preamble.v/ is /Basics/PartA/.v.\\
-The first line in /Basics/PartA.v/ after the preamble section is as follows:\\
-    \\
-It should be understood as a declaration of intent to define a constant called \\
-/fromempty /whose type is described by the expression that is written to the \\
-right of the colon. \\
-Following this line there is a paragraph that starts with /Proof./ and ends with \\
-/Defined. /where the constant is actually defined using the little sub-programs of \\
-Coq called tactics which help to build complex expressions of the underlying \\
-type theory language in simple steps.  \\
-19\\
+<<19>>[[file:Unimath-19_1.png]] [[file:Unimath-19_2.png]]
+[[file:Unimath-19_3.png]] Univalent Foundations UniMath library The
+first file in the UniMath after the /Basics/preamble.v/ is
+/Basics/PartA/.v.  The first line in /Basics/PartA.v/ after the
+preamble section is as follows:
+    
+It should be understood as a declaration of intent to define a constant
+called /fromempty /whose type is described by the expression that is
+written to the right of the colon.  Following this line there is a
+paragraph that starts with /Proof./ and ends with /Defined. /where the
+constant is actually defined using the little sub-programs of Coq
+called tactics which help to build complex expressions of the
+underlying type theory language in simple steps.  19
 
 --------------
 
-<<20>>[[file:Unimath-20_1.png]]\\
-[[file:Unimath-20_2.png]]\\
-Univalent Foundations\\
-UniMath library\\
-A mathematician who wants to understand UniMath should expect a very \\
-non-linear learning curve:\\
-• In the lectures that I gave in Oxford and in the similar lectures in the Hebrew \\
-University it took me the whole first lecture to explain what that first line \\
-and the following it paragraph really mean.\\
-• In the next lecture I was able to explain the next few hundred lines of PartA.\\
-• By the fourth lecture in Oxford, the video of which can be found on my \\
-website, I was explaining the invariant formalization of fibration sequences.\\
-20\\
+<<20>>[[file:Unimath-20_1.png]] [[file:Unimath-20_2.png]] Univalent
+Foundations UniMath library A mathematician who wants to understand
+UniMath should expect a very non-linear learning curve: • In the
+lectures that I gave in Oxford and in the similar lectures in the
+Hebrew University it took me the whole first lecture to explain what
+that first line and the following it paragraph really mean.  • In the
+next lecture I was able to explain the next few hundred lines of
+PartA.  • By the fourth lecture in Oxford, the video of which can be
+found on my website, I was explaining the invariant formalization of
+fibration sequences.  20
 
 --------------
 
-<<21>>I hope that was able to show how important Univalent Foundations are and \\
-how important is the work on libraries such as UniMath.\\
-Thank you!\\
-21\\
+<<21>>I hope that was able to show how important Univalent Foundations
+are and how important is the work on libraries such as UniMath.  Thank
+you!  21
 
 --------------

Unimath.org~

@@ -0,0 +1,311 @@
+<<1>>HLF 2016, Sep. 22, 2016, Heidelberg.\\
+UniMath\\
+by Vladimir Voevodsky  \\
+from the Institute for Advanced Study in Princeton, NJ. \\
+
+--------------
+
+<<2>>Part 1. Univalent foundations\\
+2\\
+
+--------------
+
+<<3>>[[file:Unimath-3_1.png]]\\
+[[file:Unimath-3_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Today we face a problem that involves two difficult to satisfy conditions. \\
+On the one hand we have to find a way for computer assisted verification of \\
+mathematical proofs.\\
+This is necessary, first of all, because we have to stop the dissolution of the \\
+concept of proof in mathematics.\\
+On the other hand  we have to preserve the intimate connection between \\
+mathematics and the world of human intuition.\\
+This connection is what moves mathematics forward and what we often \\
+experience as the beauty of mathematics. \\
+3\\
+
+--------------
+
+<<4>>[[file:Unimath-4_1.png]]\\
+[[file:Unimath-4_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The Univalent Foundations (UF) is, a yet imperfect, solution to this problem.\\
+In their original form,  the UF combined three components:\\
+• the view of mathematics as the study of structures on sets and their higher \\
+analogs, \\
+• the idea that the higher analogs of sets are reflected in the set-based \\
+mathematics as homotopy types, \\
+• the idea that one can formalize our intuition about structures on these higher \\
+analogs using the Martin-Lof  Type Theory (MLTT) extended with the Law of \\
+Excluded Middle for propositions (LEM) , the Axiom of Choice for sets (AC), \\
+the Univalence Axiom (UA) and the Resizing Rules (RR).\\
+4\\
+
+--------------
+
+<<5>>[[file:Unimath-5_1.png]]\\
+[[file:Unimath-5_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The main new concepts that were since added to these are the following: \\
+• the understanding that a lot of mathematics can be formalized in the MLTT \\
+without the LEM and the AC and that excluding these two axioms one \\
+obtains foundations for a /new form of constructive mathematics/,\\
+• the understanding that classical mathematics appears as a subset of this new \\
+constructive mathematics,\\
+• the understanding that the MLTT extended with the UA is an imperfect \\
+formalization system for this constructive mathematics and that it should be \\
+possible  to integrate the UA into the MLTT obtaining a new type theory \\
+with better computational properties.\\
+5\\
+
+--------------
+
+<<6>>[[file:Unimath-6_1.png]]\\
+[[file:Unimath-6_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+What does it mean for a formalization system to be constructive?\\
+Some expressions in type theory are said to be in normal form. Any \\
+expression can be automatically and deterministically “normalized”, that is, an \\
+equivalent expression in normal form can be computed. \\
+In type theory there are type expressions and element expressions. If “T” is a \\
+type (expression) and “o” is an element (expression) one writes “o:T” if the \\
+type of “o” is “T”. \\
+6\\
+
+--------------
+
+<<7>>[[file:Unimath-7_1.png]]\\
+[[file:Unimath-7_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In most type systems there is the type of natural numbers. In the UniMath it is \\
+written as “nat”.\\
+There is the zero element “O:nat” and the successor function “S” from “nat” to \\
+“nat” that intuitively corresponds to the function that takes “n” to “1+n”. \\
+A constructive system satisfies the /canonicity property/ for “nat”, which asserts \\
+that the normal form of any expression “o:nat” has the form “S(S(....(SO)..))”.\\
+By counting how many “S” there is in the normal form one obtains an actual \\
+natural number from any element expression of type “nat”. \\
+7\\
+
+--------------
+
+<<8>>[[file:Unimath-8_1.png]]\\
+[[file:Unimath-8_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+This is a tremendously strong property. \\
+Consider the example: a set “X:hSet” is defined to be finite if there exists an \\
+isomorphism between it and the standard finite set “stn n”. Here “n” is an \\
+expression of type “nat”. It is well defined and one obtains a function “fincard”  \\
+from finite sets to “nat” called the cardinality - the number of elements of the \\
+set.\\
+Now suppose that I have proved, constructively, that “X” is finite. Then \\
+“(fincard X):nat” \\
+is defined. By normalizing “fincard X��� I obtain an actual natural number.\\
+If I had a constructive proof of /Faltings's Theorem, /stating that the number of \\
+rational points on a curve of genus >1 is finite, I could find the actual number \\
+of points on any curve of genus >1.  \\
+8\\
+
+--------------
+
+<<9>>[[file:Unimath-9_1.png]]\\
+[[file:Unimath-9_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+We don't know whether such a proof exists. It is a very interesting and hard \\
+problem. \\
+The reason that the MLTT+UA is an imperfect system for constructive  \\
+formalization is that while MLTT itself has the canonicity property MLTT+UA \\
+does not.\\
+Therefore, formalizing the proof of Faltings's Theorem in the UniMath, which is \\
+based on MLTT+UA, would not immediately give us an algorithm to compute \\
+the number of rational points on a curve of genus >1.\\
+This is where a new type theory that integrates the UA into the MLTT in such \\
+a way as to preserve the canonicity would help. \\
+9\\
+
+--------------
+
+<<10>>[[file:Unimath-10_1.png]]\\
+[[file:Unimath-10_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The search for such a type theory became one of the main driving forces in \\
+the development of the UF.\\
+Today several groups are working on the construction and  implementation in \\
+a proof assistant of candidate type theories.  \\
+The /cubical type theory/ and the prototype proof assistant /cubicaltt/ created by \\
+the group of Thierry Coquand with the help of many researchers from \\
+different parts of the world is at the most advanced stage of development \\
+today. \\
+A proof in the UniMath easily translates into a proof in the cubilatt.\\
+10\\
+
+--------------
+
+<<11>>[[file:Unimath-11_1.png]]\\
+[[file:Unimath-11_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The new form of the UF that emerges can be seen as combining the following \\
+components:\\
+• the view of mathematics as the study of structures on sets and their higher \\
+analogs, \\
+• the view of mathematics as constructive with the classical mathematics being \\
+a subset consisting of the results that require LEM and/or AC among their \\
+assumptions,\\
+• the idea that the higher analogs of sets are reflected in the set-based \\
+mathematics as constructive homotopy types - objects of the new \\
+constructive homotopy theory that can so far be formulated only in terms of \\
+cubical sets,\\
+• the idea that one can formalize our intuition about structures on these higher \\
+analogs using Cubical Type Theory (CTT).\\
+11\\
+
+--------------
+
+<<12>>[[file:Unimath-12_1.png]]\\
+[[file:Unimath-12_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In addition to the understanding that to obtain a formal system for the new \\
+constructive mathematics the UA needs to be integrated into the MLTT \\
+constructively, several more things are felt as lacking in the MLTT+UA:\\
+• higher inductive types, \\
+• resizing rules,\\
+• a possible strict extensional equality combined with the “fibrancy discipline”,\\
+• as yet unknown mechanism to construct the types of structures that involve \\
+infinite hierarchies of coherence conditions.  \\
+Surprisingly,  it might be easier to add these features to the CTT than to the \\
+MLTT. The work in these directions is ongoing. \\
+12\\
+
+--------------
+
+<<13>>Part 2. The UniMath library\\
+13\\
+
+--------------
+
+<<14>>[[file:Unimath-14_1.png]]\\
+[[file:Unimath-14_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+In the development of the UniMath library we attempt to do something that \\
+might be compared with the effort by the Bourbaki group to write a \\
+systematic exposition of mathematics based on the set theory and the view of \\
+mathematics as studying structures on sets.\\
+The effort by Bourbaki stalled at some point around the middle of the 20th \\
+century, in part, because it was very complicated to describe the emerging \\
+category-theoretic constructions in set-theoretic terms.\\
+14\\
+
+--------------
+
+<<15>>[[file:Unimath-15_1.png]]\\
+[[file:Unimath-15_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+One may however ask, is there any mathematical innovation in what we are \\
+doing? Is there a discovery of the unknown in the work on the UniMath?\\
+We have already seen how well-known problems in fields such as arithmetic \\
+algebraic geometry can be related to the search for a new foundation of \\
+constructive mathematics and for building proofs in the UniMath.\\
+Here is a different example.\\
+15\\
+
+--------------
+
+<<16>>[[file:Unimath-16_1.png]]\\
+[[file:Unimath-16_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Some years ago, at the IAS, I had a conversation at lunch with Armand Borel. I \\
+mentioned how I like Bourbaki “Algebra” and how it helped me to become a \\
+mathematician. \\
+I then mentioned that some places there were really dense. For example, said I, \\
+the description of the tensor product was hard to follow. \\
+Of course, said Borel, /we have invented tensor product to get a systematic \\
+exposition of multi-linear maps/. \\
+It was new research, this is why it was not very smoothly written. \\
+16\\
+
+--------------
+
+<<17>>[[file:Unimath-17_1.png]]\\
+[[file:Unimath-17_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+I was amazed.\\
+It is hard to imagine today's mathematics without the concept of the tensor \\
+product. It would never occurred to me that it was invented by Bourbaki with \\
+the only purpose to obtain a more systematic exposition of multi-linear maps \\
+of vector spaces!\\
+This example shows how a major innovation can emerge from the work on \\
+systematization of knowledge. \\
+17\\
+
+--------------
+
+<<18>>[[file:Unimath-18_1.png]]\\
+[[file:Unimath-18_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+Finally, a few words to those mathematicians who will decide to understand \\
+UniMath and maybe to contribute to it. \\
+The UniMath library is being created using the proof assistant Coq. It is freely \\
+available on GitHub.\\
+The language of Coq is a very substantial extension of the MLTT and UniMath \\
+uses a very small subset of the full Coq language that approximately \\
+corresponds to the original MLTT.\\
+18\\
+
+--------------
+
+<<19>>[[file:Unimath-19_1.png]]\\
+[[file:Unimath-19_2.png]]\\
+[[file:Unimath-19_3.png]]\\
+Univalent Foundations\\
+UniMath library\\
+The first file in the UniMath after the /Basics/preamble.v/ is /Basics/PartA/.v.\\
+The first line in /Basics/PartA.v/ after the preamble section is as follows:\\
+    \\
+It should be understood as a declaration of intent to define a constant called \\
+/fromempty /whose type is described by the expression that is written to the \\
+right of the colon. \\
+Following this line there is a paragraph that starts with /Proof./ and ends with \\
+/Defined. /where the constant is actually defined using the little sub-programs of \\
+Coq called tactics which help to build complex expressions of the underlying \\
+type theory language in simple steps.  \\
+19\\
+
+--------------
+
+<<20>>[[file:Unimath-20_1.png]]\\
+[[file:Unimath-20_2.png]]\\
+Univalent Foundations\\
+UniMath library\\
+A mathematician who wants to understand UniMath should expect a very \\
+non-linear learning curve:\\
+• In the lectures that I gave in Oxford and in the similar lectures in the Hebrew \\
+University it took me the whole first lecture to explain what that first line \\
+and the following it paragraph really mean.\\
+• In the next lecture I was able to explain the next few hundred lines of PartA.\\
+• By the fourth lecture in Oxford, the video of which can be found on my \\
+website, I was explaining the invariant formalization of fibration sequences.\\
+20\\
+
+--------------
+
+<<21>>I hope that was able to show how important Univalent Foundations are and \\
+how important is the work on libraries such as UniMath.\\
+Thank you!\\
+21\\
+
+--------------

Unimath_ind.html

@@ -0,0 +1,28 @@
+<!DOCTYPE html><html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
+<head>
+<title></title>
+</head>
+<body>
+<a href="Unimaths.html#1" target="contents" >Page 1</a><br/>
+<a href="Unimaths.html#2" target="contents" >Page 2</a><br/>
+<a href="Unimaths.html#3" target="contents" >Page 3</a><br/>
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+<a name=1></a>HLF 2016, Sep. 22, 2016, Heidelberg.<br/>
+UniMath<br/>
+by Vladimir Voevodsky &#160;<br/>
+from the Institute for Advanced Study in Princeton, NJ.&#160;<br/>
+<hr/>
+<a name=2></a>Part 1. Univalent foundations<br/>
+2<br/>
+<hr/>
+<a name=3></a><img src="Unimath-3_1.png"/><br/>
+<img src="Unimath-3_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+Today&#160;we&#160;face&#160;a&#160;problem&#160;that&#160;involves&#160;two&#160;difficult&#160;to&#160;satisfy&#160;conditions.&#160;<br/>
+On&#160;the&#160;one&#160;hand&#160;we&#160;have&#160;to&#160;find&#160;a&#160;way&#160;for&#160;computer&#160;assisted&#160;verification&#160;of&#160;<br/>mathematical&#160;proofs.<br/>
+This&#160;is&#160;necessary,&#160;first&#160;of&#160;all,&#160;because&#160;we&#160;have&#160;to&#160;stop&#160;the&#160;dissolution&#160;of&#160;the&#160;<br/>concept&#160;of&#160;proof&#160;in&#160;mathematics.<br/>
+On&#160;the&#160;other&#160;hand&#160;&#160;we&#160;have&#160;to&#160;preserve&#160;the&#160;intimate&#160;connection&#160;between&#160;<br/>mathematics&#160;and&#160;the&#160;world&#160;of&#160;human&#160;intuition.<br/>
+This&#160;connection&#160;is&#160;what&#160;moves&#160;mathematics&#160;forward&#160;and&#160;what&#160;we&#160;often&#160;<br/>experience&#160;as&#160;the&#160;beauty&#160;of&#160;mathematics.&#160;<br/>
+3<br/>
+<hr/>
+<a name=4></a><img src="Unimath-4_1.png"/><br/>
+<img src="Unimath-4_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+The&#160;Univalent&#160;Foundations&#160;(UF)&#160;is,&#160;a&#160;yet&#160;imperfect,&#160;solution&#160;to&#160;this&#160;problem.<br/>
+In&#160;their&#160;original&#160;form,&#160;&#160;the&#160;UF&#160;combined&#160;three&#160;components:<br/>
+•&#160;the&#160;view&#160;of&#160;mathematics&#160;as&#160;the&#160;study&#160;of&#160;structures&#160;on&#160;sets&#160;and&#160;their&#160;higher&#160;<br/>
+analogs,&#160;<br/>
+•&#160;the&#160;idea&#160;that&#160;the&#160;higher&#160;analogs&#160;of&#160;sets&#160;are&#160;reflected&#160;in&#160;the&#160;set-based&#160;<br/>
+mathematics&#160;as&#160;homotopy&#160;types,&#160;<br/>
+•&#160;the&#160;idea&#160;that&#160;one&#160;can&#160;formalize&#160;our&#160;intuition&#160;about&#160;structures&#160;on&#160;these&#160;higher&#160;<br/>
+analogs&#160;using&#160;the&#160;Martin-Lof&#160;&#160;Type&#160;Theory&#160;(MLTT)&#160;extended&#160;with&#160;the&#160;Law&#160;of&#160;<br/>Excluded&#160;Middle&#160;for&#160;propositions&#160;(LEM)&#160;,&#160;the&#160;Axiom&#160;of&#160;Choice&#160;for&#160;sets&#160;(AC),&#160;<br/>the&#160;Univalence&#160;Axiom&#160;(UA)&#160;and&#160;the&#160;Resizing&#160;Rules&#160;(RR).<br/>
+4<br/>
+<hr/>
+<a name=5></a><img src="Unimath-5_1.png"/><br/>
+<img src="Unimath-5_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+The&#160;main&#160;new&#160;concepts&#160;that&#160;were&#160;since&#160;added&#160;to&#160;these&#160;are&#160;the&#160;following:&#160;<br/>
+•&#160;the&#160;understanding&#160;that&#160;a&#160;lot&#160;of&#160;mathematics&#160;can&#160;be&#160;formalized&#160;in&#160;the&#160;MLTT&#160;<br/>
+without&#160;the&#160;LEM&#160;and&#160;the&#160;AC&#160;and&#160;that&#160;excluding&#160;these&#160;two&#160;axioms&#160;one&#160;<br/>obtains&#160;foundations&#160;for&#160;a&#160;<i>new&#160;form&#160;of&#160;constructive&#160;mathematics</i>,<br/>
+•&#160;the&#160;understanding&#160;that&#160;classical&#160;mathematics&#160;appears&#160;as&#160;a&#160;subset&#160;of&#160;this&#160;new&#160;<br/>
+constructive&#160;mathematics,<br/>
+•&#160;the&#160;understanding&#160;that&#160;the&#160;MLTT&#160;extended&#160;with&#160;the&#160;UA&#160;is&#160;an&#160;imperfect&#160;<br/>
+formalization&#160;system&#160;for&#160;this&#160;constructive&#160;mathematics&#160;and&#160;that&#160;it&#160;should&#160;be&#160;<br/>possible&#160;&#160;to&#160;integrate&#160;the&#160;UA&#160;into&#160;the&#160;MLTT&#160;obtaining&#160;a&#160;new&#160;type&#160;theory&#160;<br/>with&#160;better&#160;computational&#160;properties.<br/>
+5<br/>
+<hr/>
+<a name=6></a><img src="Unimath-6_1.png"/><br/>
+<img src="Unimath-6_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+What&#160;does&#160;it&#160;mean&#160;for&#160;a&#160;formalization&#160;system&#160;to&#160;be&#160;constructive?<br/>
+Some&#160;expressions&#160;in&#160;type&#160;theory&#160;are&#160;said&#160;to&#160;be&#160;in&#160;normal&#160;form.&#160;Any&#160;<br/>expression&#160;can&#160;be&#160;automatically&#160;and&#160;deterministically&#160;“normalized”,&#160;that&#160;is,&#160;an&#160;<br/>equivalent&#160;expression&#160;in&#160;normal&#160;form&#160;can&#160;be&#160;computed.&#160;<br/>
+In&#160;type&#160;theory&#160;there&#160;are&#160;type&#160;expressions&#160;and&#160;element&#160;expressions.&#160;If&#160;“T”&#160;is&#160;a&#160;<br/>type&#160;(expression)&#160;and&#160;“o”&#160;is&#160;an&#160;element&#160;(expression)&#160;one&#160;writes&#160;“o:T”&#160;if&#160;the&#160;<br/>type&#160;of&#160;“o”&#160;is&#160;“T”.&#160;<br/>
+6<br/>
+<hr/>
+<a name=7></a><img src="Unimath-7_1.png"/><br/>
+<img src="Unimath-7_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+In&#160;most&#160;type&#160;systems&#160;there&#160;is&#160;the&#160;type&#160;of&#160;natural&#160;numbers.&#160;In&#160;the&#160;UniMath&#160;it&#160;is&#160;<br/>written&#160;as&#160;“nat”.<br/>
+There&#160;is&#160;the&#160;zero&#160;element&#160;“O:nat”&#160;and&#160;the&#160;successor&#160;function&#160;“S”&#160;from&#160;“nat”&#160;to&#160;<br/>“nat”&#160;that&#160;intuitively&#160;corresponds&#160;to&#160;the&#160;function&#160;that&#160;takes&#160;“n”&#160;to&#160;“1+n”.&#160;<br/>
+A&#160;constructive&#160;system&#160;satisfies&#160;the&#160;<i>canonicity&#160;property</i>&#160;for&#160;“nat”,&#160;which&#160;asserts&#160;<br/>that&#160;the&#160;normal&#160;form&#160;of&#160;any&#160;expression&#160;“o:nat”&#160;has&#160;the&#160;form&#160;“S(S(….(SO)..))”.<br/>
+By&#160;counting&#160;how&#160;many&#160;“S”&#160;there&#160;is&#160;in&#160;the&#160;normal&#160;form&#160;one&#160;obtains&#160;an&#160;actual&#160;<br/>natural&#160;number&#160;from&#160;any&#160;element&#160;expression&#160;of&#160;type&#160;“nat”.&#160;<br/>
+7<br/>
+<hr/>
+<a name=8></a><img src="Unimath-8_1.png"/><br/>
+<img src="Unimath-8_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+This&#160;is&#160;a&#160;tremendously&#160;strong&#160;property.&#160;<br/>
+Consider&#160;the&#160;example:&#160;a&#160;set&#160;“X:hSet”&#160;is&#160;defined&#160;to&#160;be&#160;finite&#160;if&#160;there&#160;exists&#160;an&#160;<br/>isomorphism&#160;between&#160;it&#160;and&#160;the&#160;standard&#160;finite&#160;set&#160;“stn&#160;n”.&#160;Here&#160;“n”&#160;is&#160;an&#160;<br/>expression&#160;of&#160;type&#160;“nat”.&#160;It&#160;is&#160;well&#160;defined&#160;and&#160;one&#160;obtains&#160;a&#160;function&#160;“fincard”&#160;&#160;<br/>from&#160;finite&#160;sets&#160;to&#160;“nat”&#160;called&#160;the&#160;cardinality&#160;-&#160;the&#160;number&#160;of&#160;elements&#160;of&#160;the&#160;<br/>set.<br/>
+Now&#160;suppose&#160;that&#160;I&#160;have&#160;proved,&#160;constructively,&#160;that&#160;“X”&#160;is&#160;finite.&#160;Then&#160;<br/>
+“(fincard&#160;X):nat”&#160;<br/>
+is&#160;defined.&#160;By&#160;normalizing&#160;“fincard&#160;X”&#160;I&#160;obtain&#160;an&#160;actual&#160;natural&#160;number.<br/>
+If&#160;I&#160;had&#160;a&#160;constructive&#160;proof&#160;of&#160;<i>Faltings’s&#160;Theorem,&#160;</i>stating&#160;that&#160;the&#160;number&#160;of&#160;<br/>rational&#160;points&#160;on&#160;a&#160;curve&#160;of&#160;genus&#160;&gt;1&#160;is&#160;finite,&#160;I&#160;could&#160;find&#160;the&#160;actual&#160;number&#160;<br/>of&#160;points&#160;on&#160;any&#160;curve&#160;of&#160;genus&#160;&gt;1.&#160;&#160;<br/>
+8<br/>
+<hr/>
+<a name=9></a><img src="Unimath-9_1.png"/><br/>
+<img src="Unimath-9_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+We&#160;don’t&#160;know&#160;whether&#160;such&#160;a&#160;proof&#160;exists.&#160;It&#160;is&#160;a&#160;very&#160;interesting&#160;and&#160;hard&#160;<br/>problem.&#160;<br/>
+The&#160;reason&#160;that&#160;the&#160;MLTT+UA&#160;is&#160;an&#160;imperfect&#160;system&#160;for&#160;constructive&#160;&#160;<br/>formalization&#160;is&#160;that&#160;while&#160;MLTT&#160;itself&#160;has&#160;the&#160;canonicity&#160;property&#160;MLTT+UA&#160;<br/>does&#160;not.<br/>
+Therefore,&#160;formalizing&#160;the&#160;proof&#160;of&#160;Faltings’s&#160;Theorem&#160;in&#160;the&#160;UniMath,&#160;which&#160;is&#160;<br/>based&#160;on&#160;MLTT+UA,&#160;would&#160;not&#160;immediately&#160;give&#160;us&#160;an&#160;algorithm&#160;to&#160;compute&#160;<br/>the&#160;number&#160;of&#160;rational&#160;points&#160;on&#160;a&#160;curve&#160;of&#160;genus&#160;&gt;1.<br/>
+This&#160;is&#160;where&#160;a&#160;new&#160;type&#160;theory&#160;that&#160;integrates&#160;the&#160;UA&#160;into&#160;the&#160;MLTT&#160;in&#160;such&#160;<br/>a&#160;way&#160;as&#160;to&#160;preserve&#160;the&#160;canonicity&#160;would&#160;help.&#160;<br/>
+9<br/>
+<hr/>
+<a name=10></a><img src="Unimath-10_1.png"/><br/>
+<img src="Unimath-10_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+The&#160;search&#160;for&#160;such&#160;a&#160;type&#160;theory&#160;became&#160;one&#160;of&#160;the&#160;main&#160;driving&#160;forces&#160;in&#160;<br/>the&#160;development&#160;of&#160;the&#160;UF.<br/>
+Today&#160;several&#160;groups&#160;are&#160;working&#160;on&#160;the&#160;construction&#160;and&#160;&#160;implementation&#160;in&#160;<br/>a&#160;proof&#160;assistant&#160;of&#160;candidate&#160;type&#160;theories.&#160;&#160;<br/>
+The&#160;<i>cubical&#160;type&#160;theory</i>&#160;and&#160;the&#160;prototype&#160;proof&#160;assistant&#160;<i>cubicaltt</i>&#160;created&#160;by&#160;<br/>the&#160;group&#160;of&#160;Thierry&#160;Coquand&#160;with&#160;the&#160;help&#160;of&#160;many&#160;researchers&#160;from&#160;<br/>different&#160;parts&#160;of&#160;the&#160;world&#160;is&#160;at&#160;the&#160;most&#160;advanced&#160;stage&#160;of&#160;development&#160;<br/>today.&#160;<br/>
+A&#160;proof&#160;in&#160;the&#160;UniMath&#160;easily&#160;translates&#160;into&#160;a&#160;proof&#160;in&#160;the&#160;cubilatt.<br/>
+10<br/>
+<hr/>
+<a name=11></a><img src="Unimath-11_1.png"/><br/>
+<img src="Unimath-11_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+The&#160;new&#160;form&#160;of&#160;the&#160;UF&#160;that&#160;emerges&#160;can&#160;be&#160;seen&#160;as&#160;combining&#160;the&#160;following&#160;<br/>components:<br/>
+•&#160;the&#160;view&#160;of&#160;mathematics&#160;as&#160;the&#160;study&#160;of&#160;structures&#160;on&#160;sets&#160;and&#160;their&#160;higher&#160;<br/>
+analogs,&#160;<br/>
+•&#160;the&#160;view&#160;of&#160;mathematics&#160;as&#160;constructive&#160;with&#160;the&#160;classical&#160;mathematics&#160;being&#160;<br/>
+a&#160;subset&#160;consisting&#160;of&#160;the&#160;results&#160;that&#160;require&#160;LEM&#160;and/or&#160;AC&#160;among&#160;their&#160;<br/>assumptions,<br/>
+•&#160;the&#160;idea&#160;that&#160;the&#160;higher&#160;analogs&#160;of&#160;sets&#160;are&#160;reflected&#160;in&#160;the&#160;set-based&#160;<br/>
+mathematics&#160;as&#160;constructive&#160;homotopy&#160;types&#160;-&#160;objects&#160;of&#160;the&#160;new&#160;<br/>constructive&#160;homotopy&#160;theory&#160;that&#160;can&#160;so&#160;far&#160;be&#160;formulated&#160;only&#160;in&#160;terms&#160;of&#160;<br/>cubical&#160;sets,<br/>
+•&#160;the&#160;idea&#160;that&#160;one&#160;can&#160;formalize&#160;our&#160;intuition&#160;about&#160;structures&#160;on&#160;these&#160;higher&#160;<br/>
+analogs&#160;using&#160;Cubical&#160;Type&#160;Theory&#160;(CTT).<br/>
+11<br/>
+<hr/>
+<a name=12></a><img src="Unimath-12_1.png"/><br/>
+<img src="Unimath-12_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+In&#160;addition&#160;to&#160;the&#160;understanding&#160;that&#160;to&#160;obtain&#160;a&#160;formal&#160;system&#160;for&#160;the&#160;new&#160;<br/>constructive&#160;mathematics&#160;the&#160;UA&#160;needs&#160;to&#160;be&#160;integrated&#160;into&#160;the&#160;MLTT&#160;<br/>constructively,&#160;several&#160;more&#160;things&#160;are&#160;felt&#160;as&#160;lacking&#160;in&#160;the&#160;MLTT+UA:<br/>
+•&#160;higher&#160;inductive&#160;types,&#160;<br/>
+•&#160;resizing&#160;rules,<br/>
+•&#160;a&#160;possible&#160;strict&#160;extensional&#160;equality&#160;combined&#160;with&#160;the&#160;“fibrancy&#160;discipline”,<br/>
+•&#160;as&#160;yet&#160;unknown&#160;mechanism&#160;to&#160;construct&#160;the&#160;types&#160;of&#160;structures&#160;that&#160;involve&#160;<br/>
+infinite&#160;hierarchies&#160;of&#160;coherence&#160;conditions.&#160;&#160;<br/>
+Surprisingly,&#160;&#160;it&#160;might&#160;be&#160;easier&#160;to&#160;add&#160;these&#160;features&#160;to&#160;the&#160;CTT&#160;than&#160;to&#160;the&#160;<br/>MLTT.&#160;The&#160;work&#160;in&#160;these&#160;directions&#160;is&#160;ongoing.&#160;<br/>
+12<br/>
+<hr/>
+<a name=13></a>Part 2. The UniMath library<br/>
+13<br/>
+<hr/>
+<a name=14></a><img src="Unimath-14_1.png"/><br/>
+<img src="Unimath-14_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+In&#160;the&#160;development&#160;of&#160;the&#160;UniMath&#160;library&#160;we&#160;attempt&#160;to&#160;do&#160;something&#160;that&#160;<br/>might&#160;be&#160;compared&#160;with&#160;the&#160;effort&#160;by&#160;the&#160;Bourbaki&#160;group&#160;to&#160;write&#160;a&#160;<br/>systematic&#160;exposition&#160;of&#160;mathematics&#160;based&#160;on&#160;the&#160;set&#160;theory&#160;and&#160;the&#160;view&#160;of&#160;<br/>mathematics&#160;as&#160;studying&#160;structures&#160;on&#160;sets.<br/>
+The&#160;effort&#160;by&#160;Bourbaki&#160;stalled&#160;at&#160;some&#160;point&#160;around&#160;the&#160;middle&#160;of&#160;the&#160;20th&#160;<br/>century,&#160;in&#160;part,&#160;because&#160;it&#160;was&#160;very&#160;complicated&#160;to&#160;describe&#160;the&#160;emerging&#160;<br/>category-theoretic&#160;constructions&#160;in&#160;set-theoretic&#160;terms.<br/>
+14<br/>
+<hr/>
+<a name=15></a><img src="Unimath-15_1.png"/><br/>
+<img src="Unimath-15_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+One&#160;may&#160;however&#160;ask,&#160;is&#160;there&#160;any&#160;mathematical&#160;innovation&#160;in&#160;what&#160;we&#160;are&#160;<br/>doing?&#160;Is&#160;there&#160;a&#160;discovery&#160;of&#160;the&#160;unknown&#160;in&#160;the&#160;work&#160;on&#160;the&#160;UniMath?<br/>
+We&#160;have&#160;already&#160;seen&#160;how&#160;well-known&#160;problems&#160;in&#160;fields&#160;such&#160;as&#160;arithmetic&#160;<br/>algebraic&#160;geometry&#160;can&#160;be&#160;related&#160;to&#160;the&#160;search&#160;for&#160;a&#160;new&#160;foundation&#160;of&#160;<br/>constructive&#160;mathematics&#160;and&#160;for&#160;building&#160;proofs&#160;in&#160;the&#160;UniMath.<br/>
+Here&#160;is&#160;a&#160;different&#160;example.<br/>
+15<br/>
+<hr/>
+<a name=16></a><img src="Unimath-16_1.png"/><br/>
+<img src="Unimath-16_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+Some&#160;years&#160;ago,&#160;at&#160;the&#160;IAS,&#160;I&#160;had&#160;a&#160;conversation&#160;at&#160;lunch&#160;with&#160;Armand&#160;Borel.&#160;I&#160;<br/>mentioned&#160;how&#160;I&#160;like&#160;Bourbaki&#160;“Algebra”&#160;and&#160;how&#160;it&#160;helped&#160;me&#160;to&#160;become&#160;a&#160;<br/>mathematician.&#160;<br/>
+I&#160;then&#160;mentioned&#160;that&#160;some&#160;places&#160;there&#160;were&#160;really&#160;dense.&#160;For&#160;example,&#160;said&#160;I,&#160;<br/>the&#160;description&#160;of&#160;the&#160;tensor&#160;product&#160;was&#160;hard&#160;to&#160;follow.&#160;<br/>
+Of&#160;course,&#160;said&#160;Borel,&#160;<i>we&#160;have&#160;invented&#160;tensor&#160;product&#160;to&#160;get&#160;a&#160;systematic&#160;<br/>exposition&#160;of&#160;multi-linear&#160;maps</i>.&#160;<br/>
+It&#160;was&#160;new&#160;research,&#160;this&#160;is&#160;why&#160;it&#160;was&#160;not&#160;very&#160;smoothly&#160;written.&#160;<br/>
+16<br/>
+<hr/>
+<a name=17></a><img src="Unimath-17_1.png"/><br/>
+<img src="Unimath-17_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+I&#160;was&#160;amazed.<br/>
+It&#160;is&#160;hard&#160;to&#160;imagine&#160;today’s&#160;mathematics&#160;without&#160;the&#160;concept&#160;of&#160;the&#160;tensor&#160;<br/>product.&#160;It&#160;would&#160;never&#160;occurred&#160;to&#160;me&#160;that&#160;it&#160;was&#160;invented&#160;by&#160;Bourbaki&#160;with&#160;<br/>the&#160;only&#160;purpose&#160;to&#160;obtain&#160;a&#160;more&#160;systematic&#160;exposition&#160;of&#160;multi-linear&#160;maps&#160;<br/>of&#160;vector&#160;spaces!<br/>
+This&#160;example&#160;shows&#160;how&#160;a&#160;major&#160;innovation&#160;can&#160;emerge&#160;from&#160;the&#160;work&#160;on&#160;<br/>systematization&#160;of&#160;knowledge.&#160;<br/>
+17<br/>
+<hr/>
+<a name=18></a><img src="Unimath-18_1.png"/><br/>
+<img src="Unimath-18_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+Finally,&#160;a&#160;few&#160;words&#160;to&#160;those&#160;mathematicians&#160;who&#160;will&#160;decide&#160;to&#160;understand&#160;<br/>UniMath&#160;and&#160;maybe&#160;to&#160;contribute&#160;to&#160;it.&#160;<br/>
+The&#160;UniMath&#160;library&#160;is&#160;being&#160;created&#160;using&#160;the&#160;proof&#160;assistant&#160;Coq.&#160;It&#160;is&#160;freely&#160;<br/>available&#160;on&#160;GitHub.<br/>
+The&#160;language&#160;of&#160;Coq&#160;is&#160;a&#160;very&#160;substantial&#160;extension&#160;of&#160;the&#160;MLTT&#160;and&#160;UniMath&#160;<br/>uses&#160;a&#160;very&#160;small&#160;subset&#160;of&#160;the&#160;full&#160;Coq&#160;language&#160;that&#160;approximately&#160;<br/>corresponds&#160;to&#160;the&#160;original&#160;MLTT.<br/>
+18<br/>
+<hr/>
+<a name=19></a><img src="Unimath-19_1.png"/><br/>
+<img src="Unimath-19_2.png"/><br/>
+<img src="Unimath-19_3.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+The&#160;first&#160;file&#160;in&#160;the&#160;UniMath&#160;after&#160;the&#160;<i>Basics/preamble.v</i>&#160;is&#160;<i>Basics/PartA</i>.v.<br/>
+The&#160;first&#160;line&#160;in&#160;<i>Basics/PartA.v</i>&#160;after&#160;the&#160;preamble&#160;section&#160;is&#160;as&#160;follows:<br/>
+&#160; &#160;&#160;<br/>
+It&#160;should&#160;be&#160;understood&#160;as&#160;a&#160;declaration&#160;of&#160;intent&#160;to&#160;define&#160;a&#160;constant&#160;called&#160;<br/><i>fromempty&#160;</i>whose&#160;type&#160;is&#160;described&#160;by&#160;the&#160;expression&#160;that&#160;is&#160;written&#160;to&#160;the&#160;<br/>right&#160;of&#160;the&#160;colon.&#160;<br/>
+Following&#160;this&#160;line&#160;there&#160;is&#160;a&#160;paragraph&#160;that&#160;starts&#160;with&#160;<i>Proof.</i>&#160;and&#160;ends&#160;with&#160;<br/><i>Defined.&#160;</i>where&#160;the&#160;constant&#160;is&#160;actually&#160;defined&#160;using&#160;the&#160;little&#160;sub-programs&#160;of&#160;<br/>Coq&#160;called&#160;tactics&#160;which&#160;help&#160;to&#160;build&#160;complex&#160;expressions&#160;of&#160;the&#160;underlying&#160;<br/>type&#160;theory&#160;language&#160;in&#160;simple&#160;steps.&#160;&#160;<br/>
+19<br/>
+<hr/>
+<a name=20></a><img src="Unimath-20_1.png"/><br/>
+<img src="Unimath-20_2.png"/><br/>
+Univalent Foundations<br/>
+UniMath library<br/>
+A&#160;mathematician&#160;who&#160;wants&#160;to&#160;understand&#160;UniMath&#160;should&#160;expect&#160;a&#160;very&#160;<br/>non-linear&#160;learning&#160;curve:<br/>
+•&#160;In&#160;the&#160;lectures&#160;that&#160;I&#160;gave&#160;in&#160;Oxford&#160;and&#160;in&#160;the&#160;similar&#160;lectures&#160;in&#160;the&#160;Hebrew&#160;<br/>
+University&#160;it&#160;took&#160;me&#160;the&#160;whole&#160;first&#160;lecture&#160;to&#160;explain&#160;what&#160;that&#160;first&#160;line&#160;<br/>and&#160;the&#160;following&#160;it&#160;paragraph&#160;really&#160;mean.<br/>
+•&#160;In&#160;the&#160;next&#160;lecture&#160;I&#160;was&#160;able&#160;to&#160;explain&#160;the&#160;next&#160;few&#160;hundred&#160;lines&#160;of&#160;PartA.<br/>
+•&#160;By&#160;the&#160;fourth&#160;lecture&#160;in&#160;Oxford,&#160;the&#160;video&#160;of&#160;which&#160;can&#160;be&#160;found&#160;on&#160;my&#160;<br/>
+website,&#160;I&#160;was&#160;explaining&#160;the&#160;invariant&#160;formalization&#160;of&#160;fibration&#160;sequences.<br/>
+20<br/>
+<hr/>
+<a name=21></a>I&#160;hope&#160;that&#160;was&#160;able&#160;to&#160;show&#160;how&#160;important&#160;Univalent&#160;Foundations&#160;are&#160;and&#160;<br/>
+how&#160;important&#160;is&#160;the&#160;work&#160;on&#160;libraries&#160;such&#160;as&#160;UniMath.<br/>
+Thank&#160;you!<br/>
+21<br/>
+<hr/>
+</body>
+</html>