clean

#Unimath.txt#

@@ -1,289 +0,0 @@
-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
-
-

.gitignore

@@ -0,0 +1,49 @@
+# -*- mode: gitignore; -*-
+*~
+\#*\#
+/.emacs.desktop
+/.emacs.desktop.lock
+*.elc
+auto-save-list
+tramp
+.\#*
+
+# Org-mode
+.org-id-locations
+*_archive
+
+# flymake-mode
+*_flymake.*
+
+# eshell files
+/eshell/history
+/eshell/lastdir
+
+# elpa packages
+/elpa/
+
+# reftex files
+*.rel
+
+# AUCTeX auto folder
+/auto/
+
+# cask packages
+.cask/
+dist/
+
+# Flycheck
+flycheck_*.el
+
+# server auth directory
+/server/
+
+# projectiles files
+.projectile
+
+# directory configuration
+.dir-locals.el
+
+# network security
+/network-security.data
+

README.md

@@ -1,3 +1,5 @@
 ---
 license: creativeml-openrail-m
 ---
+
+This contains papers and different forms

Unimath.org~

@@ -1,311 +0,0 @@
-<<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\\
-
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-
-<<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\\
-
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-
-<<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\\
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-
-<<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\\
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