Sync with unshared/manuscripts/2020.exteqaxiom.

2020.Extensional equality preservation and verified generic programming/About.lidr

@@ -24,7 +24,7 @@ import LexerFriendlyReasoning   -- not yet working as intended
 
 This paper is about \emph{extensional equality preservation} in
 dependently typed languages like Idris~\citep{idrisbook}, % idristutorial,
-Agda~\citep{norell2007thesis} and Coq~\citep{CoqProofAssistant_8_11} that
+Agda~\citep{norell2007thesis} and Coq~\citep{CoqProofAssistant} that
 implement %some version of
 Martin-Löf's intensional type
 theory \citep{martinlof1984}. % , nordstrom1990programming
@@ -161,8 +161,8 @@ but not the converse, normally referred to as \emph{function extensionality}:
 When working with functions, extensional equality is often the notion of
 interest and libraries of formalized mathematics typically provide
 definitions like |<=>| and basic results like |IEqImplEE|, see for
-example |homot| and |eqtohomot| in Part A of the UniMath \citep{UniMath}
-foundations or |eqfun| in Coq \citep{CoqProofAssistant_8_11}.
+example |homot| and |eqtohomot| in Part A of the UniMath library \citep{UniMath}
+or |eqfun| in Coq \citep{CoqProofAssistant}.
 
 In reasoning about generic programs in the style of the Algebra of
 Programming \citep{DBLP:books/daglib/0096998,mu2009algebra} and, more
@@ -179,10 +179,15 @@ Higher order functions are a distinguished trait of functional
 programming languages \citep{bird2014thinking} and many well known
 function combinators can be shown to preserve extensional equality.
 %
-For example the arrow-function |map| for |Identity|, |List|, |Maybe|
-and for many other polymorphic data types preserve extensional
-equality.
-%
+%For example the arrow-function |map| for |Identity|, |List|, |Maybe|
+%and for many other polymorphic data types preserve extensional
+%equality.
+% Tim: is this the place to hint to Coq and Agda standard libs?
+In particular, the arrow-function |map| for |Identity|, |List|, |Maybe|
+and for many other polymorphic data types preserves extensional
+equality. The standard libraries of Agda and Coq provide several
+instances\footnote{e.g. Agda for |Maybe|: \url{https://agda.github.io/agda-stdlib/Data.Maybe.Properties.html}}
+\footnote{e.g. Coq for |List|: \url{https://coq.inria.fr/distrib/current/stdlib/Coq.Lists.List.html}}.
 
 Similarly, if |h| takes two function arguments it preserves
 extensional equality (in two arguments) if |f1 <=> g1| and |f2 <=> g2|
@@ -205,7 +210,7 @@ two-argument version of extensional equality preservation:
 The right hand side is a chain of equal expressions connected by the
 |={| proofs |}=| of the individual steps within special braces and
 ending in |QED|, see ``Preorder reasoning'' in
-\citep{idrisreference,idristutorial}.
+\citep{idrisdocs}.
 %
 The steps with |Refl| are just for human readability, they could be
 omitted as far as Idris is concerned.
@@ -302,17 +307,16 @@ computational version of function extensionality.
 % programming.
 
 This paper is a contribution towards \emph{pragmatic} verified generic
-programming. In this sense it might become obsolete when fully computational notions
-of function extensionality will become available in mainstream
-programming.
-%
-However, from a more abstract mathematical perspective, there are good reasons
-to not use axioms that are stronger than necessary.
-%
-For example, there are interesting models of type theory that refute
-function extensionality \citep{streicher1993investigations,
-  vanGlehn_2015, 10.1145/3018610.3018620}, and results that do not
-rely on it can be interpreted in these models.
+programming. From this perspective, it might become obsolete when fully
+computational notions of function extensionality will become available
+in mainstream programming.
+%
+From a more mathematical perspective, there are good reasons not to rely
+on axioms that are stronger than necessary: there are interesting models
+of type theory that refute function extensionality
+\citep{streicher1993investigations, vanGlehn_2015,
+  10.1145/3018610.3018620}, and our results can be interpreted in these
+models.
 %
 
 The paper has been generated from literate Idris files.

2020.Extensional equality preservation and verified generic programming/Appendix.lidr

-% -*-Latex-*-
-
-%if False
-
-> module Appendix
-
-> import Syntax.PreorderReasoning
-> import Data.List
-
-> import About
-> import Functors
-> import Monads
-> import Applications
-
-> %default total
-> %auto_implicits off
-> %access public export
-
-%endif
-
-\appendix 
-\label{section:appendix}
-

2020.Extensional equality preservation and verified generic programming/Applications.lidr

@@ -50,12 +50,12 @@ monadic systems
 
 where |M| is an \emph{uncertainty} monad: |List|, |Maybe|, |Dist|
 \citep{10.1017/S0956796805005721}, |SimpleProb|
-\citep{2017_Botta_Jansson_Ionescu}, etc.
+\citep{ionescu2009, 2017_Botta_Jansson_Ionescu}, etc.
 %
 For monadic systems, one can derive a number of general results.
 %
 One is that every deterministic system can be embedded in a monadic
-systems:
+system:
 
 
 > embed : {X : Type} -> {M : Type -> Type} -> Fat.VeriMonadFat M => DetSys X -> MonSys M X

2020.Extensional equality preservation and verified generic programming/Conclusions.lidr

@@ -12,9 +12,9 @@
 \label{section:conclusions}
 
 In dependently typed programming in the context of Martin-Löf type
-theories \citep{nordstrom1990programming}, the problem of how to specify
-abstract data types for verified generic programming is still not well
-understood.
+theories \citep{nordstrom1990programming, martinlof1984}, the problem of
+how to specify abstract data types for verified generic programming is
+still not well understood.
 %
 In this work, we have shown that requiring functors to preserve
 extensional equality of arrows yields abstract data types that are

2020.Extensional equality preservation and verified generic programming/Example.lidr

@@ -42,10 +42,10 @@ In dynamical systems theory \citep{Kuznetsov:1998:EAB:289919,
   thomas2012catastrophe}, a prominent notion is that of the flow (or
 iteration) of a system.
 %
-A \emph{deterministic} dynamical system on a set |X| is an endofunction on |X|.
-The set |X| is often called the \emph{state space} of the system.
-%
-After explaining this simpler, deterministic, case we will get to a more general, monadic, case.
+We start by discussing deterministic systems and then generalize to the
+monadic case. A \emph{deterministic} dynamical system on a set |X| is an
+endofunction on |X|.  The set |X| is often called the \emph{state space}
+of the system.
 
 Given a deterministic system |f : X -> X|, its $n$-th iterate or flow,
 is typically denoted by |pow f n : X -> X| and is defined by induction
@@ -147,26 +147,20 @@ fuzzy systems?
 Can we extend our results to the general case of \emph{monadic}
 dynamical systems?
 %
-Monadic dynamical systems \citep{ionescu2009, 2017_Botta_Jansson_Ionescu}
+Monadic dynamical systems \citep{ionescu2009}
 on a set |X| are functions of type |X -> M X| where |M| is a monad.
 %
-When |M| is equal to the identity monad, one recovers the
-deterministic case.
-%
-For |M| equal to |List| one has non-deterministic systems and |M| equal
-to the finite probability distribution monads |Dist|
-\citep{10.1017/S0956796805005721} or |SimpleProb|
-\citep{2017_Botta_Jansson_Ionescu} formalizes the notion of stochastic
-systems.
-%%Tim, Nuria: perhaps avoid "equal" altogether?
-%%When |M| is the identity monad, one recovers the
-%%deterministic case.
-%%%
-%%When |M| is |List| one has non-deterministic systems. Taking for |M|
-%%a finite probability monad (e.g. |Dist|
+When |M| is the identity monad, one recovers the deterministic case.
+%
+%%For |M| equal to |List| one has non-deterministic systems and |M| equal
+%%to the finite probability distribution monads |Dist|
 %%\citep{10.1017/S0956796805005721} or |SimpleProb|
-%%\citep{2017_Botta_Jansson_Ionescu}) captures the notion of stochastic
+%%\citep{2017_Botta_Jansson_Ionescu} formalizes the notion of stochastic
 %%systems.
+When |M| is |List| one has non-deterministic systems, and finite
+probability monads (e.g. |Dist| \citep{10.1017/S0956796805005721} or
+|SimpleProb| \citep{ionescu2009, 2017_Botta_Jansson_Ionescu}) capture the notion of
+stochastic systems.
 %
 Other monads encode other notions of uncertainty, see
 \citep{ionescu2009,giry1981}.
@@ -187,8 +181,8 @@ with |pure| and function composition with Kleisli composition (|>=>|):
 > flowMonR f     Z     =  pure
 > flowMonR f  (  S n)  =  f >=> flowMonR f n
 
-Notice, however, that now the implementation of |flowMonL| and
-|flowMonR| depends on |(>=>)|, which is a monad-specific operation.
+Notice, however, that now the implementations of |flowMonL| and
+|flowMonR| depend on |(>=>)|, which is a monad-specific operation.
 %
 This means that, in proving properties of the flow of monadic systems,
 we can no longer rely on a specific \emph{definition} of |(>=>)|: we

2020.Extensional equality preservation and verified generic programming/Functors.lidr

@@ -57,8 +57,9 @@ When considering ADT specifications of functor (and of natural
 transformation, monad, etc.) in dependently typed languages, one has to
 distinguish between two related but different situations.
 
-One in which the specification is part of an attempt at formalizing
-category theory. In this situation, one has to expect the notion of
+One in which the specification is put forward in the context of
+formalizations of category theory, see for example \citep{UniMath,
+CoqProofAssistant}. In this situation, one has to expect the notion of
 category to be in place and that of functor to be predicated on that of
 its source and target categories. A functor ADT in this situation is an
 answer to the question ``What shall the notion of functor look like in
@@ -152,7 +153,7 @@ generality (supporting a larger class of functors) by parameterising
 over the equalities used, but this leads to quite a bit of book-keeping
 (basically a setoid-based framework).
 %
-We instead stop at this point and hope to show that it is a pragmatic
+We instead stop at this point and show that it is a pragmatic
 compromise between generality and convenient usage.
 
 \paragraph*{Equality preservation examples.}
@@ -171,11 +172,11 @@ proof looks as follows:
 %  {-" below=\the\belowdisplayskip"-}
 
 > mapListPresEE :  {A, B : Type} -> (f, g : A -> B) -> f <=> g -> mapList f <=> mapList g
-> mapListPresEE f g fExtg [] = Refl
-> mapListPresEE f g fExtg (a :: as) =
+> mapListPresEE f g fEEg [] = Refl
+> mapListPresEE f g fEEg (a :: as) =
 >   ( mapList f (a :: as) )          ={ Refl }=
->   ( f a :: mapList f as )          ={ cong {f = (::{-"~"-} mapList f as)} (fExtg a) }=
->   ( g a :: mapList f as )          ={ cong (mapListPresEE f g fExtg as) }=
+>   ( f a :: mapList f as )          ={ cong {f = (::{-"~"-} mapList f as)} (fEEg a) }=
+>   ( g a :: mapList f as )          ={ cong (mapListPresEE f g fEEg as) }=
 >   ( g a :: mapList g as )          ={ Refl }=
 >   ( mapList g (a :: as) )          QED
 
@@ -259,7 +260,7 @@ If we try to implement preservation of extensional equality we end up with
 
 > mapReaderPresEE :  {A, B : Type} -> (f, g : A -> B) ->
 >                    f <=> g -> mapReader f <=> mapReader g
-> mapReaderPresEE f g fExtEqg r  =
+> mapReaderPresEE f g fEEg r  =
 >   ( mapReader f r)        ={ Refl }=
 >   ( f . r )               ={ whatnow1 r }=     -- here we need |f = g| to proceed
 >   ( g . r )               ={ Refl }=
@@ -267,7 +268,8 @@ If we try to implement preservation of extensional equality we end up with
 
 Notice the question mark in front of |whatnow1r|. This introduces an
 unresolved proof step and allows us to ask Idris to help us implementing
-this step, see \citep{idrisguides}. Among others, we can ask about the
+this step, see ``Elaborator Reflection -- Holes'' in \citep{idrisdocs}.
+Among others, we can ask about the
 type of |whatnow1r|. Perhaps not surprisingly, this turns out to be |f
 . r = g . r|.
 
@@ -290,7 +292,7 @@ extify (<=>)| it is easy to show
 
 > mapReaderPresEE2 :  {E, A, B : Type} -> (f, g : A -> B) ->
 >                     f <=> g  ->  mapReader f <==> mapReader g
-> mapReaderPresEE2 f g fExtEqg r x = fExtEqg (r x)
+> mapReaderPresEE2 f g fEEg r x = fEEg (r x)
 
 Thus, |Reader E| is an example of a functor which does not preserve,
 but rather \emph{transforms} the notion of equality.
@@ -332,4 +334,4 @@ the monad ADT design.
 %% We are interested in ADTs that support well verified generic programming
 %% and we discuss the advantages and the disadvantages of different
 %% approaches from an DSL perspective. As in section \ref{section:example},
-%% we discuss appplications in dynamical systems and control theory.
+%% we discuss applications in dynamical systems and control theory.

2020.Extensional equality preservation and verified generic programming/Makefile

@@ -16,7 +16,6 @@ main.tex: main.lhs \
           Applications.lidr \
           RelatedWork.lidr \
           Conclusions.lidr \
-          Appendix.lidr \
           references.bib
 	${LHS2TEX} --poly main.lhs > main.almost_tex
 	./spacefix.bash main.almost_tex > main.tex

2020.Extensional equality preservation and verified generic programming/Monads.lidr

@@ -32,10 +32,9 @@ We discuss the role of extensional equality preservation for deriving
 monad laws and for verifying the equivalence between the different ADT
 formulations.
 %
-Note that there are many possible versions of the axioms and we do not
-claim to have found the ``optimal'' axioms for Monad specifically, but
-we want to explain the trade-offs between different kinds of
-formulations.
+There are many possible way of formulating Monad axioms. Here, we do not
+argue for or against a specific formulation but rather discuss the most
+popular ones and their trade-offs.
 
 \subsection{The traditional view}\label{sec:MonadTradView}
 
@@ -157,7 +156,7 @@ The same approach can be followed for implementing bind, another monad
 combinator similar to Kleisli composition:
 
 >   (>>=)  :  {A, B : Type} -> {M : Type -> Type} -> ThinVeriMonad M => M A -> (A -> M B) -> M B
->   ma >>= f = join (map f ma)
+>             ma >>= f = join (map f ma)
 
 Starting from a \emph{thin} monad ADT as in the example above and
 adding monadic operators that fulfil a specification
@@ -544,11 +543,9 @@ The lifting operation is required to satisfy W1--W3 for any objects
 
 \paragraph*{From tradition to Wadler and back.}
 %
-We here briefly explain how the two monad definitions can be seen as
-views on the same mathematical concept.
-%
-We do this because we would like the corresponding ADT formulations to
-also preserve this relationship.
+The two monad definitions can be seen as views on the same mathematical
+concept and we would like the corresponding ADT formulations to preserve
+this relationship.
 
 It turns out that, if (|M|, |etaN|, |muN|) fulfil the properties T1--T5
 of the traditional view, then the object part of |M|, |etaN|, and the

2020.Extensional equality preservation and verified generic programming/main.lhs

@@ -115,7 +115,6 @@ generic programming.
 %include Applications.lidr
 %include RelatedWork.lidr
 %include Conclusions.lidr
-%%include Appendix.lidr
 
 \section*{Acknowledgments}
 %**PJ[after review]: The authors thank the JFP editors and reviewers, whose comments have

2020.Extensional equality preservation and verified generic programming/references.bib

@@ -22,6 +22,15 @@
   year =	 2020
 }
 
+@misc{idrisdocs,
+  author =	 {{The Idris Community}},
+  title =	 {{D}ocumentation for the {I}dris {L}anguage},
+  howpublished = {\url{http://docs.idris-lang.org/en/latest/}},
+  url =		 {http://docs.idris-lang.org/en/latest/},
+  year =	 2020
+}
+
+
 @book{idrisbook,
   author =	 {Edwin Brady},
   publisher =	 {Manning Publications Co.},
@@ -41,7 +50,7 @@
   author =	 {Nicola Botta},
   howpublished = {\url{https://gitlab.pik-potsdam.de/botta/IdrisLibs}},
   title =	 {{I}dris{L}ibs},
-  year =	 {2016--2020}
+  year =	 {2016-2021}
 }
 
 @inproceedings{ionescujansson:LIPIcs:2013:3899,
@@ -269,7 +278,7 @@
                   (DSLsofMath)},
   url =
                   {https://www.chalmers.se/en/projects/Pages/Domain-Specific-Languages-of-Mathematics-QDSLsofMathQ.aspx},
-  year =	 {2014-2020}
+  year =	 {2014-2021}
 }
 
 @book{Kuznetsov:1998:EAB:289919,
@@ -473,17 +482,15 @@
   year =	 1981
 }
 
-@misc{CoqProofAssistant_8_11,
-  doi =		 {10.5281/ZENODO.1003420},
-  url =		 {https://zenodo.org/record/1003420},
-  author =	 "{The Coq Dev.\ Team}",
-  LONGauthor =	 "{The Coq Development Team}",
-  keywords =	 {proof assistant, mathematical software, formal
-                  proofs},
-  language =	 {en},
-  title =	 {The Coq Proof Assistant, version 8.11.0},
-  publisher =	 {Zenodo},
-  year =	 2020
+@misc{CoqProofAssistant,
+  author       = {{The Coq Development Team}},
+  title        = {The Coq Proof Assistant (Version 8.13.0)},
+  month        = jan,
+  year         = 2021,
+  publisher    = {Zenodo},
+  version      = {8.13},
+  doi          = {10.5281/zenodo.4501022},
+  howpublished = {\url{https://doi.org/10.5281/zenodo.4501022}}
 }
 
 @book{nordstrom1990programming,
@@ -497,9 +504,9 @@
 
 @misc{brede2020,
   author =	 {Brede, Nuria and Botta, Nicola},
-  title =	 {Semantic verification of dynamic programming},
-  howpublished = {In submission to J. Funct. Program.},
-  year =	 2020
+  title =	 {On the correctness of monadic backward induction},
+  howpublished = {Submitted to J. Funct. Program., \url{https://arxiv.org/abs/2008.02143}},
+  year =	 2021
 }
 
 @article{gnesi1981dynamic,
@@ -840,22 +847,48 @@ isbn="978-3-540-71067-7"
 @misc{CoRN_library,
   author =	 {Spitters, Bas and Semeria, Vincent (Maintainers)},
   howpublished = {\url{https://github.com/coq-community/corn}},
-  title =	 {Coq Repository at Nijmegen},
-  year =	 {2000-2020}
+  title =	 {Coq Repository at Nijmegen (Version 1.2.0)},
+  year =	 {2017}
+}
+
+@InProceedings{10.1007/978-3-540-27818-4_7,
+author="Cruz-Filipe, Lu{\'i}s
+and Geuvers, Herman
+and Wiedijk, Freek",
+editor="Asperti, Andrea
+and Bancerek, Grzegorz
+and Trybulec, Andrzej",
+title="C-CoRN, the Constructive Coq Repository at Nijmegen",
+booktitle="Mathematical Knowledge Management",
+year="2004",
+publisher="Springer Berlin Heidelberg",
+address="Berlin, Heidelberg",
+pages="88--103",
+isbn="978-3-540-27818-4"
 }
 
 @misc{CoLoR_library,
   author =	 {Frederic Blanqui and others},
   howpublished = {\url{https://github.com/fblanqui/color}},
-  title =	 {CoLoR: a Coq Library on Rewriting and termination},
-  year =	 {2005-2020}
+  title =	 {CoLoR: a Coq Library on Rewriting and termination (Version 1.8.0)},
+  year =	 {2020}
 }
 
+@article{blanqui_koprowski_2011,
+  title={CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates}, volume={21},
+  DOI={10.1017/S0960129511000120},
+  number={4},
+  journal={Mathematical Structures in Computer Science},
+  publisher={Cambridge University Press},
+  author={Blanqui, Fred{\'e}d{\'e}ric and Koprowski, Adam},
+  year={2011},
+  pages={827--859}}
+
 @misc{arend_prover,
   author =	 {{JetBrains Research}},
   howpublished = {\url{https://arend-lang.github.io/}},
-  title =	 {Arend Theorem Prover},
-  year =	 {2020}
+  title =	 {Arend Theorem Prover (Version 1.6.0)},
+  year =	 {2021}
 }
 
 @Misc{UniMath,
@@ -865,8 +898,7 @@ isbn="978-3-540-71067-7"
                   univalent mathematics},
   url =		 {https://github.com/UniMath/UniMath},
   howpublished = {Available at
-                  \url{https://github.com/UniMath/UniMath}},
-  year =	 2017
+                  \url{https://github.com/UniMath/UniMath}}
 }
 
 @book{pierce_basic_1991,
@@ -884,16 +916,16 @@ isbn="978-3-540-71067-7"