change notation for anamorphisms

thesis/main.tex

@@ -99,6 +99,8 @@
 
 
 \usepackage{noto-mono}
+\usepackage{mathpazo}
+\usepackage{unicode-math}
 \usepackage{mathrsfs}
 \usepackage[colorinlistoftodos,prependcaption,textsize=tiny]{todonotes}
 \usepackage{xargs}
@@ -106,6 +108,43 @@
 %\usepackage{fontspec}
 %\setmonofont{Noto Sans Mono}
 
+\usepackage{pict2e}
+
+% \newcommand{\lbparen}{%
+%   \mathopen{%
+%     \sbox0{$()$}%
+%     \setlength{\unitlength}{\dimexpr\ht0+\dp0}%
+%     \raisebox{-\dp0}{%
+%       \begin{picture}(.32,1)
+%       \linethickness{\fontdimen8\textfont3}
+%       \roundcap
+%       \put(0,0){\raisebox{\depth}{$($}}
+%       \polyline(0.32,0)(0.1,0)(0.1,1)(0.32,1)
+%       \end{picture}%
+%     }%
+%   }%
+% }
+
+% \newcommand{\rbparen}{%
+%   \mathclose{%
+%     \sbox0{$()$}%
+%     \setlength{\unitlength}{\dimexpr\ht0+\dp0}%
+%     \raisebox{-\dp0}{%
+%       \begin{picture}(.32,1)
+%       \linethickness{\fontdimen8\textfont3}
+%       \roundcap
+%       \put(0.02,0){\raisebox{\depth}{$)$}}
+%       \polyline(0.1,0)(0.32,0)(0.32,1)(0.1,1)
+%       \end{picture}%
+%     }%
+%   }%
+% }
+
+\newcommand{\lbparen}{〖}
+% https://unicodeplus.com/U+3016
+
+\newcommand{\rbparen}{〗}
+
 % category C
 \newcommand*{\C}{\ensuremath{\mathscr{C}}}
 \newcommand*{\D}{\ensuremath{\mathscr{D}}}
@@ -142,7 +181,7 @@
   }
 }
 % terminal coalgebra
-\newcommand*{\coalg}[1]{\ensuremath{\llbracket #1 \rrbracket}}
+\newcommand*{\coalg}[1]{\ensuremath{\lbparen#1\rbparen}}
 
 \begin{document}
 \pagestyle{plain}
@@ -168,6 +207,7 @@ Erlangen, \today{} \rule{7cm}{1pt}\\
 \newcommandx{\info}[2][1=]{\todo[inline,linecolor=OliveGreen,backgroundcolor=OliveGreen!25,bordercolor=OliveGreen,#1]{#2}}
 \newcommandx{\improvement}[2][1=]{\todo[inline,linecolor=Plum,backgroundcolor=Plum!25,bordercolor=Plum,#1]{#2}}
 
+
 % for creating custom labels like (Fixpoint)
 \makeatletter
 \newcommand{\customlabel}[2]{%

thesis/src/03_partiality-monads.tex

@@ -357,11 +357,12 @@ Finality of the coalgebras $(DX, out : DX \rightarrow X + DX)_{X \in \obj{\C}}$
       \arrow["out", from=3-1, to=3-5]
       \arrow["{id \times out}", from=1-1, to=1-2]
       \arrow["dstl", from=1-2, to=1-3]
-      \arrow["{\coalg{-}}", dashed, from=1-1, to=3-1]
+      \arrow["{\coalg{\text{-}}}", dashed, from=1-1, to=3-1]
       \arrow["{f \times g + id}", from=1-3, to=1-5]
-      \arrow["{id + \coalg{-}}", dashed, from=1-5, to=3-5]
+      \arrow["{id + \coalg{\text{-}}}", dashed, from=1-5, to=3-5]
     \end{tikzcd}\]
-  We write $\coalg{-}$ to abbreviate the used coalgebra, i.e.\ in this diagram $\coalg{-} = \coalg{(f\times g + id) \circ dstl \circ (id \times out)}$
+  We write $\coalg{\text{-}}$ to abbreviate the used coalgebra, i.e.\ in the previous diagram
+  \[\coalg{\text{-}} = \coalg{(f\times g + id) \circ dstl \circ (id \times out)}.\]
 
   Next we check the strength laws:
   \begin{itemize}
@@ -371,8 +372,8 @@ Finality of the coalgebras $(DX, out : DX \rightarrow X + DX)_{X \in \obj{\C}}$
               {1 \times DX} \& {1 \times X + DX} \& {1 \times X + 1 \times DX} \& {X + 1 \times DX} \\
               DX \&\&\& {X + DX}
               \arrow["out", from=2-1, to=2-4]
-              \arrow["{\coalg{-}}", dashed, from=1-1, to=2-1]
-              \arrow["{id + \coalg{-}}", from=1-4, to=2-4]
+              \arrow["{\coalg{\text{-}}}", dashed, from=1-1, to=2-1]
+              \arrow["{id + \coalg{\text{-}}}", from=1-4, to=2-4]
               \arrow["{id \times out}", from=1-1, to=1-2]
               \arrow["dstl", from=1-2, to=1-3]
               \arrow["{\pi_2 + id}", from=1-3, to=1-4]
@@ -396,11 +397,11 @@ Finality of the coalgebras $(DX, out : DX \rightarrow X + DX)_{X \in \obj{\C}}$
               {X \times DDY} \& {X \times (DY + DDY)} \& {X \times DY + X \times DDY} \\
               \&\& {X \times Y + X \times DDY} \\
               {D(X\times Y)} \&\& {X \times Y + D(X \times Y)}
-              \arrow["{\coalg{-}}", dashed, from=1-1, to=3-1]
+              \arrow["{\coalg{\text{-}}}", dashed, from=1-1, to=3-1]
               \arrow["{id \times out}", from=1-1, to=1-2]
               \arrow["dstl", from=1-2, to=1-3]
               \arrow["out", from=3-1, to=3-3]
-              \arrow["{id + \coalg{-}}"', from=2-3, to=3-3]
+              \arrow["{id + \coalg{\text{-}}}"', from=2-3, to=3-3]
               \arrow["{[ (id + (id \times now)) \circ dstl \circ (id \times out) , i_2 ]}"', from=1-3, to=2-3]
             \end{tikzcd}\]
     \item[\ref{S4}] To show $D\alpha \circ \tau = \tau \circ (id \times \tau) \circ \alpha$ by coinduction we take the coalgebra:
@@ -409,9 +410,9 @@ Finality of the coalgebras $(DX, out : DX \rightarrow X + DX)_{X \in \obj{\C}}$
               {(X \times Y) \times DZ} \& {(X \times Y) \times (Z+ DZ)} \& {(X\times Y) \times Z + (X \times Y) \times DZ} \\
               \&\& {X \times Y  \times Z + (X \times Y) \times DZ} \\
               {D(X \times Y \times Z)} \&\& {X \times Y \times Z + D(X \times Y \times Z)}
-              \arrow["{\coalg{-}}", dashed, from=1-1, to=3-1]
+              \arrow["{\coalg{\text{-}}}", dashed, from=1-1, to=3-1]
               \arrow["out", from=3-1, to=3-3]
-              \arrow["{id +\coalg{-}}"', from=2-3, to=3-3]
+              \arrow["{id +\coalg{\text{-}}}"', from=2-3, to=3-3]
               \arrow["{id \times out}", from=1-1, to=1-2]
               \arrow["dstl", from=1-2, to=1-3]
               \arrow["{\langle \pi_1 \circ \pi_1 , \langle \pi_2 \circ \pi_1 , \pi_2 \rangle \rangle + id}"', from=1-3, to=2-3]
@@ -451,8 +452,8 @@ To prove that $\mathbf{D}$ is commutative we will use another proof principle pr
       \arrow["out", from=2-1, to=2-4]
       \arrow["out", from=1-1, to=1-2]
       \arrow["{[ [ i_1 , h ] , i_2 ]}", from=1-2, to=1-4]
-      \arrow["{id + \coalg{-}}", from=1-4, to=2-4]
-      \arrow["{\coalg{-}}", dashed, from=1-1, to=2-1]
+      \arrow["{id + \coalg{\text{-}}}", from=1-4, to=2-4]
+      \arrow["{\coalg{\text{-}}}", dashed, from=1-1, to=2-1]
     \end{tikzcd}\]
 \end{proof}