leonv/bsc-leon-vatthauer
1
0
Fork 0
mirror of https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git synced 2024-05-31 07:28:34 +02:00
Code Issues Projects Releases Packages Wiki Activity

Finish proof that K is commutative

Browse source
This commit is contained in:
Leon Vatthauer 2024-03-08 07:33:21 +01:00
parent 945601470b
commit 75a6789d18
Signed by: leonv
SSH key fingerprint: SHA256:G4+ddwoZmhLPRB1agvXzZMXIzkVJ36dUYZXf5NxT+u8
2 changed files with 97 additions and 15 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

5
thesis/.vscode/ltex.hiddenFalsePositives.en-US.txt vendored
View file

@ -20,3 +20,8 @@
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\QA (unguarded) Elgot Algebra \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q consists of: An object X for every \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the iteration \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, satisfying: law:fixpoint Fixpoint: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:uniformity Uniformity: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:folding Folding: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q test: \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\QA (unguarded) Elgot Algebra \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q consists of: An object X for every \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the iteration \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, satisfying: law:fixpoint Fixpoint: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:uniformity Uniformity: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:folding Folding: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\Qlaw:diamond Diamond\\E$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\QA (unguarded) Elgot Algebra \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q consists of: An object X for every \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the iteration \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, satisfying: law:fixpoint Fixpoint: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:uniformity Uniformity: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:folding Folding: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\Qlaw:stutter Stutter: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:diamond Diamond: \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\Qlaw:stutter Stutter: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:diamond Diamond: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:compositionality Compositionality: \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\QA (unguarded) Elgot Algebra \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q consists of: An object A for every \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the iteration \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, satisfying: law:fixpoint Fixpoint: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:uniformity Uniformity: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:folding Folding: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}
{"rule":"ENGLISH_WORD_REPEAT_RULE","sentence":"^\\Qlaw:stutter Stutter: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:diamond Diamond: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q law:compositionality Compositionality: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}

107
thesis/src/04_iteration.tex
View file

@ -5,24 +5,43 @@ In this chapter we will draw on the inherent connection between partiality and i
\improvement{Add some text}
\unsure[inline]{Check types etc.}
\begin{definition}[Unguarded Elgot Algebra]
\todo[inline]{Definition unguarded Elgot algebra}
\end{definition}
\begin{definition}[Elgot Algebra]
A (unguarded) Elgot Algebra~\cite{uniformelgot} consists of:
\begin{itemize}
\item An object X
\item for every $f : S \rightarrow X + S$ the iteration $f^\# : S \rightarrow X$, satisfying:
\item An object A
\item for every $f : X \rightarrow A + X$ the iteration $f^\# : X \rightarrow A$, satisfying:
\begin{itemize}
\item \customlabel{law:fixpoint}{\textbf{Fixpoint}}: $f^\# = [ id , f ^\# ] \circ f$
\\ for $f : X \rightarrow A + X$
\item \customlabel{law:uniformity}{\textbf{Uniformity}}: $(id + h) \circ f = g \circ h \Rightarrow f ^\# = g^\# \circ h$
\\ for $f : S \rightarrow X + S,\; g : R \rightarrow A + R,\; h : S \rightarrow R$
\\ for $f : X \rightarrow A + X,\; g : Y \rightarrow A + Y,\; h : X \rightarrow Y$
\item \customlabel{law:folding}{\textbf{Folding}}: $((f^\# + id) \circ h)^\# = [ (id + inl) \circ f , inr \circ h ] ^\#$
\\ for $f : S \rightarrow A + S,\; h : R \rightarrow S + R$
\\ for $f : X \rightarrow A + X,\; h : Y \rightarrow X + Y$
\end{itemize}
\end{itemize}
\end{definition}
\todo[inline]{Diamond and stutter laws}
\customlabel{law:diamond}{\textbf{Diamond}}
\begin{lemma}
Every Elgot algebra $(A , (-)^\#)$ satisfies the following additional laws
\begin{itemize}
\item \customlabel{law:stutter}{\textbf{Stutter}}: $(([ h , h ] + id) \circ f)^\# = (i_1 \circ h , [ h + i_1 , i_2 \circ i_2 ] )^\# \circ inr$
\\ for $f : X \rightarrow (Y + Y) + X, h : Y \rightarrow A$
\item \customlabel{law:diamond}{\textbf{Diamond}}: $((id + \Delta) \circ f)^\# = ([ i_1 , ((id + \Delta) \circ f)^\# + id] \circ f)^\#$
\\ for $f : X \rightarrow A + (X + X)$
\item \customlabel{law:compositionality}{\textbf{Compositionality}}: $((f^\# + id) \circ h)^\# = ([ (id + i_1) \circ f , i_2 \circ i_2 ] \circ [ i_1 , h ])^\# \circ i_2$
\\ for $f : X \rightarrow A + X, h : Y \rightarrow X + Y$
\end{itemize}
\end{lemma}
\begin{proof}
\todo[inline]{Prove diamond and stutter laws}
\end{proof}
\todo[inline]{Mention here that previous lemma implies that unguarded Elgot are id guarded Elgot and diamond gives alternate way of classifying Elgot algebras citing Sergey}
Morphisms between Elgot algebras that are coherent with respect to the iteration operator are of special interest to us, the following definition classifies them.
@ -707,7 +726,7 @@ The following Lemma is central to the proof of commutativity.
\\=\;&(\pi_1 + (\pi_1 \circ \pi_1 \circ \langle id , g \circ \pi_2 \rangle)) \circ dstr \circ (\hat{f} \times id)
\\=\;&(\pi_1 + \pi_1) \circ dstr \circ (\hat{f} \times id)
\\=\;&\pi_1 \circ (\hat{f} \times id)
\\=\;&\hat{f} \circ \pi_1
\\=\;&\hat{f} \circ \pi_1 .
\end{alignat*}
This concludes the proof.
@ -716,8 +735,8 @@ The following Lemma is central to the proof of commutativity.
\begin{theorem}
$\mathbf{K}$ is a commutative monad.
\end{theorem}
\begin{proof}
We use remark~\ref{rem:proofbystability} again:
\begin{proof} We need to show that $\tau^* \circ \sigma = \sigma^* \circ \tau : KX \times KY \rightarrow K(X \times Y)$.
Let us proceed by right stability, consider the following diagram.
% https://q.uiver.app/#q=WzAsNSxbMSwwLCJLWCBcXHRpbWVzIEtZIl0sWzMsMCwiSyhLWCBcXHRpbWVzIFkpIl0sWzEsMiwiSyhYIFxcdGltZXMgS1kpIl0sWzMsMiwiSyhYIFxcdGltZXMgWSkiXSxbMCwzLCJLWCBcXHRpbWVzIFkiXSxbMCwxLCJcXHRhdSJdLFswLDIsIlxcaGF0e1xcdGF1fSIsMl0sWzEsMywiXFxoYXR7XFx0YXV9XioiXSxbMiwzLCJcXHRhdV4qIiwyXSxbNCwwLCJpZCBcXHRpbWVzIFxcZXRhIiwwLHsiY3VydmUiOi0yfV0sWzQsMywiXFxoYXR7XFx0YXV9IiwyLHsiY3VydmUiOjJ9XV0=
\[\begin{tikzcd}
& {KX \times KY} && {K(KX \times Y)} \\
@ -731,11 +750,16 @@ The following Lemma is central to the proof of commutativity.
\arrow["{id \times \eta}", curve={height=-12pt}, from=4-1, to=1-2]
\arrow["{\sigma}"', curve={height=12pt}, from=4-1, to=3-4]
\end{tikzcd}\]
\todo[inline]{Maybe don't omit proofs, give them point by point?}
The proofs for \ref{sharpr1} and the proof that $\sigma^* \circ \tau$ is right iteration preserving are straightforward and can be looked up in the formalization.
The proof that $\tau^* \circ \sigma$ is right iteration preserving is non-trivial, so we will look at it in more detail:
Let $Z \in \obj{\C}, h : Z \rightarrow KY + Z$ and let us introduce a definition for brevity: $\psi = \tau^* \circ \sigma$. We now use remark~\ref{rem:proofbyleftstability} to show that $\psi$ is right iteration preserving:
The diagram commutes since
\[\sigma^* \circ \tau \circ (id \times \eta) = \sigma^* \circ \eta = \sigma\]
and
\[\tau^* \circ \sigma \circ (id \times \eta) = \tau^* \circ K(id \times \eta) \circ \sigma = (\tau \circ (id \times \eta))^* \circ \sigma = \sigma.\]
We are left to show that both $\sigma^* \circ \tau$ and $\tau^* \circ \sigma$ are right iteration preserving. Let $h : Z \rightarrow KY + Z$, indeed
\[\sigma^* \circ \tau \circ (id \times h^\#) = \sigma^* ((\tau + id) \circ dstl \circ (id \times h))^\# = (((\sigma^* \circ \tau) + id) \circ dstl \circ (id \times h))^\#. \]
Let $\psi := \tau^* \circ \sigma$ and let us proceed by left stability to show that $\psi$ is right iteration preserving, consider the following diagram
% https://q.uiver.app/#q=WzAsNCxbNCwwLCJLWCBcXHRpbWVzIEtZIl0sWzQsMSwiSyhYIFxcdGltZXMgWSkiXSxbMCwxLCJLWCBcXHRpbWVzIFoiXSxbMCwzLCJYIFxcdGltZXMgWiJdLFswLDEsIlxccHNpIl0sWzIsMCwiaWQgXFx0aW1lcyBoXlxcIyJdLFsyLDEsIigoXFxwc2kgKyBpZCkgXFxjaXJjIGRzdGwgXFxjaXJjIChpZCBcXHRpbWVzIGgpKV5cXCMiLDJdLFszLDIsIlxcZXRhIFxcdGltZXMgaWQiXSxbMywxLCJcXHRhdSBcXGNpcmMgKGlkIFxcdGltZXMgaF5cXCMpIiwyXV0=
\[\begin{tikzcd}
&&&& {KX \times KY} \\
@ -748,8 +772,61 @@ The following Lemma is central to the proof of commutativity.
\arrow["{\eta \times id}", from=4-1, to=2-1]
\arrow["{\tau \circ (id \times h^\#)}"', from=4-1, to=2-5]
\end{tikzcd}\]
\change[inline]{Be more specific}
\todo[inline]{Introduce Diamond and Stutter laws}
which commutes, since
\begin{alignat*}{1}
&\psi \circ (id \times h^\#) \circ (\eta \times id)
\\=\;&\psi \circ (\eta \times id) \circ (id \times h^\#)
\\=\;&\tau^* \circ \eta \circ (id \times h^\#)
\\=\;&\tau \circ (id \times h^\#)
\\=\;&((\tau + id) \circ dstl \circ (id \times h))^\#
\\=\;&((\psi + id) \circ dstl \circ (id \times h))^\# \circ (\eta \times id).\tag{\ref{law:uniformity}}
\end{alignat*}
We are left to show that both $\psi \circ (id \times h^\#)$ and $((\psi + id) \circ dstl \circ (id \times h))^\#$ are left iteration preserving. Let $g : A \rightarrow KX + A$, then $\psi \circ (id \times h^\#)$ is left iteration preserving, since
\begin{alignat*}{1}
&\psi \circ (id \times h^\#) \circ (g^\# \times id)
\\=\;&\psi \circ (g^\# \times id) \circ (id \times h^\#)
\\=\;&\tau^* \circ ((\sigma + id) \circ dstr \circ (g \times id))^\# \circ (id \times h^\#)
\\=\;&((\psi + id) \circ dstr \circ (g \times id))^\# \circ (id \times h^\#)
\\=\;&(((\psi \circ (id \times h^\#)) + id) \circ dstr \circ (g \times id))^\#.\tag{\ref{law:uniformity}}
\end{alignat*}
Lastly, we need to show that
\[((\psi + id) \circ dstl \circ (id \times h))^\# \circ (g^\# \times id) = ((((\psi + id) \circ dstl \circ (id \times h))^\# + id) \circ dstr \circ (g \times id))^\#.\]
Note that by \ref{law:uniformity} the left-hand side can be rewritten as
\[((((\psi + id) \circ dstr \circ (g \times id))^\# + id) \circ dstl \circ (id \times h))^\#.\]
Consider now, that
\begin{alignat*}{1}
&((((\psi + id) \circ dstl \circ (id \times h))^\# + id) \circ dstr \circ (g \times id))^\#
\\=\;&(((((\psi + id) \circ dstl \circ (id \times h))^\#)^* + id) \circ (\eta + id) \circ dstr \circ (g \times id))^\#
\\=\;&(((\psi + id) \circ dstl \circ (id \times h))^\#)^* \circ ((\eta + id) \circ dstr \circ (g \times id))^\#
\\=\;&(((\psi + id) \circ dstl \circ (id \times h))^\#)^* \circ (((\sigma \circ (\eta \times id)) + id) \circ dstr \circ (g \times id))^\#
\\=\;&(((\psi + id) \circ dstl \circ (id \times h))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&(((\psi^* + id) \circ (\eta + id) \circ dstl \circ (id \times h))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ (((\eta + id) \circ dstl \circ (id \times h))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ (((\eta + id) \circ dstl \circ (id \times h))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ ((((\tau \circ (id \times \eta)) + id) \circ dstl \circ (id \times h))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ (((\tau + id) \circ dstl \circ (id \times ((\eta + id) \circ h)))^\#)^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ (\tau \circ (id \times ((\eta + id) \circ h)^\#))^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ \tau^* \circ K(id \times ((\eta + id) \circ h)^\#) \circ \sigma \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ \tau^* \circ \sigma \circ (id \times ((\eta + id) \circ h)^\#) \circ (((\eta + id) \circ g)^\# \times id)
\\=\;&\psi^* \circ \tau^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times ((\eta + id) \circ h)^\#),
\end{alignat*}
and by a symmetric argument
\begin{alignat*}{1}
&((((\psi + id) \circ dstr \circ (g \times id))^\# + id) \circ dstl \circ (id \times h))^\#
\\=\;&\psi^* \circ \sigma^* \circ \tau \circ (((\eta + id) \circ g)^\# \times ((\eta + id) \circ h)^\#).
\end{alignat*}
We are thus done by
\begin{alignat*}{1}
&((\psi + id) \circ dstl \circ (id \times h))^\# \circ (g^\# \times id)
\\=\;&((((\psi + id) \circ dstr \circ (g \times id))^\# + id) \circ dstl \circ (id \times h))^\#\tag{\ref{law:uniformity}}
\\=\;&\psi^* \circ \sigma^* \circ \tau \circ (((\eta + id) \circ g)^\# \times ((\eta + id) \circ h)^\#)
\\=\;&\psi^* \circ \tau^* \circ \sigma \circ (((\eta + id) \circ g)^\# \times ((\eta + id) \circ h)^\#)\tag{\ref{lem:KCommKey}}
\\=\;&((((\psi + id) \circ dstl \circ (id \times h))^\# + id) \circ dstr \circ (g \times id))^\#.
\end{alignat*}
\end{proof}
\begin{theorem}\label{thm:Klifting}