fix the open problem solved by Hew Wolff (#108)

* fix the open problem solved by Hew Wolff

* wip

* clean everything with clean.sh

* add files with embedded fonts

* fix font size for back cover

* fix back cover size

* fix barcode position

sofp-src/cover/sofp-back-cover-no-bg.tex.src

@@ -1,6 +1,6 @@
 \onecolumn
 \thispagestyle{empty}
-\newgeometry{top=2cm, left=2.5cm, right=3.9cm,bottom=1cm}
+\newgeometry{top=2.5cm, left=2.5cm, right=3.9cm, bottom=1cm}
 
 \begin{wrapfigure}{l}{0.4\columnwidth}
 \includegraphics[width=0.4\columnwidth]{monads_evil_face}
@@ -53,4 +53,5 @@ matrices or simplifying the expressions:
 
 The author received a Ph.D. in theoretical physics. After a career in academic research, he works as a software engineer.
 
-%\vspace{0.2cm}\hspace*{\fill}\includegraphics[scale=1.0]{barcode}
+
+%\vspace{3.0cm}\hspace*{\fill}\colorbox{white}{\includegraphics[scale=1.0,width=50.8mm,height=30.5mm]{barcode}}

sofp-src/cover/sofp-back-cover.tex

@@ -5,6 +5,7 @@
 \usepackage{graphicx}
 \usepackage{pagecolor}
 \usepackage{changepage}
+\usepackage{relsize}
 
 \geometry{
   paperwidth=\CoverWidth in,%

sofp-src/lyx/sofp-applicative.lyx

@@ -28713,21 +28713,11 @@ status open
 
 \begin_layout Plain Layout
 It is unknown how to characterize or enumerate all polynomial functors that
- are monads (see Problems
+ are monads (see Problem
 \begin_inset space ~
 \end_inset
 
 
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "par:Problem-monads-1"
-plural "false"
-caps "false"
-noprefix "false"
-
-\end_inset
-
-–
 \begin_inset CommandInset ref
 LatexCommand ref
 reference "par:Problem-monads"

sofp-src/lyx/sofp-summary.lyx

@@ -3045,7 +3045,7 @@ do not store
 
 .
 \begin_inset Note Note
-status open
+status collapsed
 
 \begin_layout Plain Layout
 By Yoneda we have 
@@ -3181,14 +3181,56 @@ If
 Exercise 
 \begin_inset CommandInset label
 LatexCommand label
-name "par:Exercise-additional-13"
+name "par:Problem-monads-1"
 
 \end_inset
 
 
 \begin_inset CommandInset ref
 LatexCommand ref
-reference "par:Exercise-additional-13"
+reference "par:Problem-monads-1"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+This was an open problem but it was solved by Hew Wolff.
+\end_layout
+
+\end_inset
+
+ Prove that 
+\begin_inset Formula $L^{A}\triangleq\bbnum 1+\underbrace{A\times A\times...\times A}_{n\text{ times}}$
+\end_inset
+
+ cannot be made into a monad if 
+\begin_inset Formula $n\ge2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsubsection
+Exercise 
+\begin_inset CommandInset label
+LatexCommand label
+name "par:Exercise-additional-13-1"
+
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "par:Exercise-additional-13-1"
 plural "false"
 caps "false"
 noprefix "false"
@@ -4541,87 +4583,7 @@ open problems
 \begin_layout Standard
 The author of this book does not know how to answer the following questions
  and also could not find any answers in existing books or papers.
-\end_layout
-
-\begin_layout Subsubsection
-Problem 
-\begin_inset CommandInset label
-LatexCommand label
-name "par:Problem-monads-1"
-
-\end_inset
-
-
-\begin_inset CommandInset ref
-LatexCommand ref
-reference "par:Problem-monads-1"
-plural "false"
-caps "false"
-noprefix "false"
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Do all polynomial functors of the form 
-\begin_inset Formula $P_{n}^{A}\triangleq\bbnum 1+\overbrace{A\times A\times...\times A}^{n\text{ times, }n\ge2}$
-\end_inset
-
- fail to be monads? An example is the functor 
-\begin_inset Formula $P_{2}^{A}\triangleq\bbnum 1+A\times A$
-\end_inset
-
-, which is not a monad because all possible implementations of 
-\begin_inset listings
-inline true
-status open
-
-\begin_layout Plain Layout
-
-pure
-\end_layout
-
-\end_inset
-
- and 
-\begin_inset listings
-inline true
-status open
-
-\begin_layout Plain Layout
-
-flatMap
-\end_layout
-
-\end_inset
-
- methods for 
-\begin_inset Formula $P_{2}$
-\end_inset
-
- fail the monad laws.
-\begin_inset Foot
-status open
-
-\begin_layout Plain Layout
-See discussion here: 
-\family typewriter
-
-\begin_inset CommandInset href
-LatexCommand href
-target "https://stackoverflow.com/questions/49742377"
-literal "false"
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
+
 \end_layout
 
 \begin_layout Subsubsection
@@ -4737,6 +4699,33 @@ not
  obtained by a chain of these constructions?
 \end_layout
 
+\begin_layout Standard
+For example, functors of the form 
+\begin_inset Formula $L^{A}\triangleq\bbnum 1+A\times A$
+\end_inset
+
+, 
+\begin_inset Formula $L^{A}\triangleq\bbnum 1+A\times A\times A$
+\end_inset
+
+, etc., cannot be obtained via these constructions.
+ All those functors are not monads (see Exercise
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "par:Problem-monads-1"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+).
+\end_layout
+
 \begin_layout Subsubsection
 Problem 
 \begin_inset CommandInset label

sofp-src/scripts/clean.sh

@@ -1 +1 @@
-rm -rf build vol1 vol2 vol3 pdf* mdoc sofp-*.pdf
+rm -rf build tex vol1 vol2 vol3 pdf* mdoc sofp-*.pdf sofp.pdf 

sofp-src/scripts/prepare_volume.sh

@@ -9,15 +9,15 @@ function pdfPages {
 }
 
 vol1_ISBN="ISBN (vol.~1): 978-1-4710-4004-7"
-vol1_ISBN_barcode="vol1_isbn_barcode.pdf"
+vol1_ISBN_barcode="vol1_isbn_barcode_embedded_fonts.pdf"
 vol1_url="https://www.lulu.com/shop/sergei-winitzki/the-science-of-functional-programming-part-i/paperback/product-dyyq2zm.html"
 
 vol2_ISBN="ISBN (vol.~2): 978-1-4461-9146-0"
-vol2_ISBN_barcode="vol2_isbn_barcode.pdf"
+vol2_ISBN_barcode="vol2_isbn_barcode_embedded_fonts.pdf"
 vol2_url="https://www.lulu.com/shop/sergei-winitzki/the-science-of-functional-programming-part-ii/paperback/product-655e7wm.html"
 
 vol3_ISBN="ISBN (vol.~3): 978-1-4461-9136-1"
-vol3_ISBN_barcode="vol3_isbn_barcode.pdf"
+vol3_ISBN_barcode="vol3_isbn_barcode_embedded_fonts.pdf"
 vol3_url="https://www.lulu.com/shop/sergei-winitzki/the-science-of-functional-programming-part-ii/paperback/product-p668z4q.html"
 
 v=$1
@@ -124,7 +124,7 @@ esac
   echo "Using volume $v ISBN '$isbn'"
   sed -i.bak -e 's|\({\\footnotesize{}\)ISBN: [^}]*\(}\\\\\)|\1'"$isbn"'\2|;' $dir/$name.tex
   # Add barcode to back cover.
-  sed -i.bak -e 's|%\(.*\){barcode}.*|\1{'"$barcode"'}|' $dir/sofp-back-cover-no-bg.tex
+  sed -i.bak -e 's|%\(.*\){barcode}\(.*\)$|\1{'"$barcode"'}\2|' $dir/sofp-back-cover-no-bg.tex
 
 mv $dir/$name.tex $dir/sofp.tex
 

sofp-src/tex/sofp-applicative.tex

@@ -4711,7 +4711,7 @@ \subsection{Relationship between monads and applicative functors}
 functors with monoidal fixed types. (All our examples of non-applicative
 functors involve non-polynomial functors.) But we have seen that not
 all polynomial functors are monads.\footnote{It is unknown how to characterize or enumerate all polynomial functors
-that are monads (see Problems~\ref{par:Problem-monads-1}\textendash \ref{par:Problem-monads}).} Example~\ref{subsec:Example-applicative-not-monad} shows a simple
+that are monads (see Problem~\ref{par:Problem-monads}).} Example~\ref{subsec:Example-applicative-not-monad} shows a simple
 applicative functor\index{applicative functor!not a monad} ($L^{A}\triangleq\bbnum 1+A\times A$)
 that does \emph{not} have a lawful monad implementation (Exercise~\ref{subsec:Exercise-1-monads-7-not-a-monad}).
 

sofp-src/tex/sofp-back-cover-no-bg.tex

@@ -1,6 +1,6 @@
 \onecolumn
 \thispagestyle{empty}
-\newgeometry{top=2cm, left=2.5cm, right=3.9cm,bottom=1cm}
+\newgeometry{top=2.5cm, left=2.5cm, right=3.9cm, bottom=1cm}
 
 \begin{wrapfigure}{l}{0.4\columnwidth}
 \includegraphics[width=0.4\columnwidth]{monads_evil_face}
@@ -28,7 +28,7 @@
 developed in excruciating detail through 1906
 Scala code snippets, 192 statements with step-by-step
 derivations, 104 diagrams, 223 examples
-with tested Scala code, and 309 exercises. Discussions
+with tested Scala code, and 310 exercises. Discussions
 build upon each chapter\textsf{'}s material further.
 
 Beginners in FP will find tutorials about the \texttt{map}/\texttt{reduce}
@@ -53,4 +53,5 @@
 
 The author received a Ph.D. in theoretical physics. After a career in academic research, he works as a software engineer.
 
-%\vspace{0.2cm}\hspace*{\fill}\includegraphics[scale=1.0]{barcode}
+
+%\vspace{3.0cm}\hspace*{\fill}\colorbox{white}{\includegraphics[scale=1.0,width=50.8mm,height=30.5mm]{barcode}}

sofp-src/tex/sofp-filterable.tex

@@ -3628,7 +3628,7 @@ \subsubsection{Statement \label{subsec:Statement-recursive-filterable-contrafunc
 }
 \end{lstlisting}
 \[
-\xymatrix{S^{B,F^{B}}\ar[r]\sp(0.525){\text{liftOpt}_{S}(f^{:A\rightarrow\bbnum 1+B})}\ar[rd]\sb(0.45){\text{liftOpt}_{F}(f)\triangleq~~~} & S^{A,F^{B}}\ar[d]\sp(0.45){\big(\overline{\text{liftOpt}_{F}}(f)\big)^{\uparrow S^{A,\bullet}}}\\
+\xymatrix{S^{B,F^{B}}\ar[r]\sp(0.525){\text{liftOpt}_{S}(f^{:A\rightarrow\bbnum 1+B})}\ar[rd]\sb(0.45){\text{liftOpt}_{F}(f)\triangleq~~~} & S^{A,F^{B}}\ar[d]\sp(0.4){\big(\overline{\text{liftOpt}_{F}}(f)\big)^{\uparrow S^{A,\bullet}}}\\
 \xyScaleY{1.8pc}\xyScaleX{6pc} & S^{A,F^{A}}
 }
 \]
@@ -3674,7 +3674,7 @@ \subsubsection{Statement \label{subsec:Statement-recursive-filterable-contrafunc
 with respect to lifting in the type parameter $A$ of $S^{A,R}$.
 That law is:
 \[
-\xymatrix{S^{B,R}\ar[r]\sp(0.5){\text{liftOpt}_{S}(f^{:A\rightarrow\bbnum 1+B})}\ar[d]\sb(0.55){(h^{:R\rightarrow R^{\prime}})^{\uparrow S^{B,\bullet}}} & S^{A,R}\ar[d]\sp(0.55){h^{\uparrow S^{A,\bullet}}}\\
+\xymatrix{S^{B,R}\ar[r]\sp(0.5){\text{liftOpt}_{S}(f^{:A\rightarrow\bbnum 1+B})}\ar[d]\sb(0.5){(h^{:R\rightarrow R^{\prime}})^{\uparrow S^{B,\bullet}}} & S^{A,R}\ar[d]\sp(0.4){h^{\uparrow S^{A,\bullet}}}\\
 \xyScaleY{1.8pc}\xyScaleX{6.0pc}S^{B,R^{\prime}}\ar[r]\sp(0.45){\text{liftOpt}_{S}(f)} & S^{A,R^{\prime}}
 }
 \]
@@ -4332,7 +4332,7 @@ \subsection{Naturality laws and natural transformations\label{subsec:Naturality-
 that requirement as an equation called the \textbf{naturality law}\index{naturality law!of headOption@of \texttt{headOption}}
 of \lstinline!headOption!:
 \[
-\xymatrix{\text{List}^{A}\ar[r]\sp(0.55){\text{headOpt}^{A}}\ar[d]\sb(0.5){(f^{:A\rightarrow B})^{\uparrow\text{List}}} & \text{Opt}^{A}\ar[d]\sb(0.4){f^{\uparrow\text{Opt}}}\\
+\xymatrix{\text{List}^{A}\ar[r]\sp(0.55){\text{headOpt}^{A}}\ar[d]\sb(0.5){(f^{:A\rightarrow B})^{\uparrow\text{List}}} & \text{Opt}^{A}\ar[d]\sp(0.45){f^{\uparrow\text{Opt}}}\\
 \xyScaleY{1.6pc}\xyScaleX{4.5pc}\text{List}^{B}\ar[r]\sp(0.55){\text{headOpt}^{B}} & \text{Opt}^{B}
 }
 \]

sofp-src/tex/sofp-summary.tex

@@ -457,7 +457,13 @@ \subsubsection{Exercise \label{par:Exercise-additional-12}\ref{par:Exercise-addi
 If $M$ is a commutative monad then $M\circ M$ is also a lawful commutative
 monad.
 
-\subsubsection{Exercise \label{par:Exercise-additional-13}\ref{par:Exercise-additional-13}}
+\subsubsection{Exercise \label{par:Problem-monads-1}\ref{par:Problem-monads-1}}
+
+\footnote{This was an open problem but it was solved by Hew Wolff.}
+Prove that $L^{A}\triangleq\bbnum 1+\underbrace{A\times A\times...\times A}_{n\text{ times}}$
+cannot be made into a monad if $n\ge2$.
+
+\subsubsection{Exercise \label{par:Exercise-additional-13-1}\ref{par:Exercise-additional-13-1}}
 
 If $M$ is a commutative monad and $W$ is a commutative monoid then
 the monoid $M^{W}$ is commutative.
@@ -632,14 +638,7 @@ \subsection{Open problems\index{open problems}}
 
 The author of this book does not know how to answer the following
 questions and also could not find any answers in existing books or
-papers.
-
-\subsubsection{Problem \label{par:Problem-monads-1}\ref{par:Problem-monads-1}}
-
-Do all polynomial functors of the form $P_{n}^{A}\triangleq\bbnum 1+\overbrace{A\times A\times...\times A}^{n\text{ times, }n\ge2}$
-fail to be monads? An example is the functor $P_{2}^{A}\triangleq\bbnum 1+A\times A$,
-which is not a monad because all possible implementations of \lstinline!pure!
-and \lstinline!flatMap! methods for $P_{2}$ fail the monad laws.\footnote{See discussion here: \texttt{\href{https://stackoverflow.com/questions/49742377}{https://stackoverflow.com/questions/49742377}}}
+papers. 
 
 \subsubsection{Problem \label{par:Problem-monads}\ref{par:Problem-monads}}
 
@@ -660,6 +659,10 @@ \subsubsection{Problem \label{par:Problem-monads}\ref{par:Problem-monads}}
 there any polynomial monads \emph{not} obtained by a chain of these
 constructions?
 
+For example, functors of the form $L^{A}\triangleq\bbnum 1+A\times A$,
+$L^{A}\triangleq\bbnum 1+A\times A\times A$, etc., cannot be obtained
+via these constructions. All those functors are not monads (see Exercise~\ref{par:Problem-monads-1}).
+
 \subsubsection{Problem \label{subsec:Problem-co-pointed-applicative}\ref{subsec:Problem-co-pointed-applicative}}
 
 By Statement~\ref{subsec:Statement-co-pointed-applicative-example},

sofp-src/tex/sofp-typeclasses.tex

@@ -2889,7 +2889,7 @@ \subsection{Pointed functors: motivation and laws\label{subsec:Pointed-functors-
 Since both sides of the law are functions applied to an arbitrary
 value $x^{:A}$, we can omit $x$ and rewrite the law as:
 \begin{equation}
-\text{for any }f^{:A\rightarrow B}:\quad\quad\text{pu}_{F}\bef f^{\uparrow F}=f\bef\text{pu}_{F}\quad.\label{eq:naturality-law-of-pure}
+\text{for any }f^{:A\rightarrow B}:\quad\text{pu}_{F}\bef f^{\uparrow F}=f\bef\text{pu}_{F}\quad.\label{eq:naturality-law-of-pure}
 \end{equation}
 \[
 \xymatrix{\xyScaleY{2.0pc}\xyScaleX{4.0pc}A\ar[r]\sp(0.5){\text{pu}_{F}}\ar[d]\sb(0.45){f} & F^{A}\ar[d]\sp(0.45){f^{\uparrow F}}\\