mirror of
https://git8.cs.fau.de/theses/bsc-leon-vatthauer.git
synced 2024-05-31 07:28:34 +02:00
Finishing touches on partiality chapter
This commit is contained in:
parent
f37458fe97
commit
75d6990bd5
SSH key fingerprint:
SHA256:G4+ddwoZmhLPRB1agvXzZMXIzkVJ36dUYZXf5NxT+u8
2 changed files with 47 additions and 56 deletions
|
|
@ -125,6 +125,7 @@ private
|
|||
where
|
||||
f' = (η _ +₁ idC) ∘ f
|
||||
g' = (η _ +₁ idC) ∘ g
|
||||
w : X × Z ⇒ monadK.F.F₀ (X × K.₀ U + K.₀ Y × Z) + X × Z
|
||||
w = [ i₁ ∘ K.₁ i₁ ∘ τ _ , (K.₁ i₂ ∘ σ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (f' ⁂ idC) ] ∘ distributeˡ⁻¹ ∘ (idC ⁂ g')
|
||||
w-law₁ : f' # ∘ π₁ ≈ extend [ f' # ∘ π₁ , η _ ∘ π₁ ] ∘ w #
|
||||
w-law₁ = sym (begin
|
||||
|
|
|
|||
|
|
@ -21,23 +21,15 @@ To ensure that programs are partial, we recall the following notion by Cockett a
|
|||
\begin{definition}[Restriction Structure]
|
||||
A restriction structure on a category \(\mathcal{C}\) is a mapping \(dom : \mathcal{C}(X, Y) \rightarrow \mathcal{C}(X , X)\) with the following properties:
|
||||
\begin{alignat*}{1}
|
||||
f \circ (\tdom f) & = f\tag*{(R1)}\label{R1} \\
|
||||
(\tdom f) \circ (\tdom g) & = (\tdom g) \circ (\tdom f)\tag*{(R2)}\label{R2} \\
|
||||
\tdom(g \circ (\tdom f)) & = (\tdom g) \circ (\tdom f)\tag*{(R3)}\label{R3} \\
|
||||
(\tdom h) \circ f & = f \circ \tdom (h \circ f)\tag*{(R4)}\label{R4}
|
||||
f \circ (\tdom f) & = f \\
|
||||
(\tdom f) \circ (\tdom g) & = (\tdom g) \circ (\tdom f) \\
|
||||
\tdom(g \circ (\tdom f)) & = (\tdom g) \circ (\tdom f) \\
|
||||
(\tdom h) \circ f & = f \circ \tdom (h \circ f)
|
||||
\end{alignat*}
|
||||
for any \(X, Y, Z \in \vert\mathcal{C}\vert, f : X \rightarrow Y, g : X \rightarrow Z, h: Y \rightarrow Z\).
|
||||
\end{definition}
|
||||
|
||||
\begin{lemma}
|
||||
For any \(f : X \rightarrow Y\) the restriction morphism \(\tdom f\) is idempotent.
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
This follows by~\ref{R3} and~\ref{R1}:
|
||||
\[\tdom f \circ \tdom f = \tdom (f \circ \tdom f) = \tdom f\tag*{\qedhere}\]
|
||||
\end{proof}
|
||||
|
||||
The idempotent morphism \(\tdom f : X \rightarrow X\) represents the domain of definiteness of \(f : X \rightarrow Y\). In the category of partial functions this takes the following form:
|
||||
The morphism \(\tdom f : X \rightarrow X\) represents the domain of definiteness of \(f : X \rightarrow Y\). In the category of partial functions this takes the following form:
|
||||
|
||||
\[
|
||||
(\tdom f)(x) = \begin{cases}
|
||||
|
|
@ -55,7 +47,7 @@ That is, \(\tdom f\) is only defined on values where \(f\) is defined and for th
|
|||
|
||||
For a suitable defined partiality monad \(T\) the Kleisli category \(\C^T\) should be a restriction category.
|
||||
|
||||
Lastly we also recall the following notion by Bucalo et al.~\cite{eqlm} which captures what it means for a monad to have no other side effect besides some sort of non-termination:
|
||||
Lastly, we also recall the following notion by Bucalo et al.~\cite{eqlm} which captures what it means for a monad to have no other side effect besides some sort of non-termination:
|
||||
|
||||
\begin{definition}[Equational Lifting Monad]\label{def:eqlm}
|
||||
A commutative monad \(T\) is called an \textit{equational lifting monad} if the following diagram commutes:
|
||||
|
|
@ -110,7 +102,7 @@ The endofunctor \(MX = X + 1\) extends to a monad by taking \(\eta_X = i_1 : X \
|
|||
& (id + !) \circ dstl \circ (id \times (f + id)) \\
|
||||
= \; & (id + !) \circ ((id \times f) + (id \times id)) \circ dstl \\
|
||||
= \; & ((id \times f) + !) \circ dstl \\
|
||||
= \; & ((id \times f) + id) \circ (id + !) \circ dstl
|
||||
= \; & ((id \times f) + id) \circ (id + !) \circ dstl.
|
||||
\end{alignat*}
|
||||
|
||||
The other strength laws and commutativity can be proven by using simple properties of distributive categories, we added these proofs to the formalization for completeness.
|
||||
|
|
@ -131,20 +123,20 @@ The endofunctor \(MX = X + 1\) extends to a monad by taking \(\eta_X = i_1 : X \
|
|||
\end{tikzcd}
|
||||
\]
|
||||
|
||||
This is easily proven by pre-composing with \(i_1\) and \(i_2\):
|
||||
This is easily proven by pre-composing with \(i_1\) and \(i_2\), indeed
|
||||
\begin{alignat*}{1}
|
||||
& (id\;+\;!) \circ dstl \circ \langle i_1 , i_1 \rangle
|
||||
\\=\;&(id\;+\;!) \circ dstl \circ (id \times i_1) \circ \langle i_1 , id \rangle
|
||||
\\=\;&(id\;+\;!) \circ i_1 \circ \langle i_1 , id \rangle
|
||||
\\=\;&i_1 \circ \langle i_1 , id \rangle
|
||||
\\=\;&i_1 \circ \langle i_1 , id \rangle,
|
||||
\end{alignat*}
|
||||
|
||||
and
|
||||
\begin{alignat*}{1}
|
||||
& (id\;+\;!) \circ dstl \circ \langle i_2 , i_2 \rangle
|
||||
\\=\;&(id\;+\;!) \circ dstl \circ (id \times i_2) \circ \langle i_2 , id \rangle
|
||||
\\=\;&(id\;+\;!) \circ i_2 \circ \langle i_2 , id \rangle
|
||||
\\=\;&i_2 \;\circ \; ! \circ \langle i_2 , id \rangle
|
||||
\\=\;&i_2 \;\circ \; !\tag*{\qedhere}
|
||||
\\=\;&i_2 \;\circ \; !.\tag*{\qedhere}
|
||||
\end{alignat*}
|
||||
\end{proof}
|
||||
|
||||
|
|
@ -152,7 +144,7 @@ In the setting of classical mathematics this monad is therefore sufficient for m
|
|||
|
||||
\section{The Delay Monad}
|
||||
Capretta's delay monad~\cite{delay} is a coinductive datatype whose inhabitants can be viewed as suspended computations.
|
||||
This datatype is usually defined by the two coinductive constructors \(now : X \rightarrow D\;X\) and \(later : D\;X \rightarrow D\;X\), where \(now\) is for lifting a value inside a computation and \(later\) intuitively delays a computation by one time unit. See \autoref{chp:setoids} or~\cite{delay}, for a type theoretical study of this monad.
|
||||
This datatype is usually defined by the two coinductive constructors \(now : X \rightarrow DX\) and \(later : DX \rightarrow DX\), where \(now\) is for lifting a value inside a computation and \(later\) intuitively delays a computation by one time unit. See \autoref{chp:setoids} for a type theoretical study of this monad.
|
||||
Categorically we obtain the delay monad by the final F-coalgebras \(DX = \nu A. X + A\), which we assume to exist. In this section we will show that these final F-coalgebras indeed yield a monad that is strong and commutative.
|
||||
|
||||
Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecursion and prove theorems by coinduction. By \autoref{lem:lambek} the coalgebra structure \(out : DX \rightarrow X + DX\) is an isomorphism, whose inverse can be decomposed into the two constructors mentioned before: \(out^{-1} = [ now , later ] : X + DX \rightarrow DX\).
|
||||
|
|
@ -225,14 +217,13 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
\\=\;&(id + \coalg{\alpha} \circ i_2) \circ out
|
||||
\end{alignat*}
|
||||
Let us verify that \(f^*\) indeed satisfies the requisite property:
|
||||
|
||||
\begin{alignat*}{1}
|
||||
& out \circ \coalg{\alpha} \circ i_1
|
||||
\\=\;&(id + \coalg{\alpha}) \circ \alpha \circ i_1
|
||||
\\=\;&(id + \coalg{\alpha}) \circ [ [ i_1 , i_2 \circ i_2 ] \circ out \circ f , i_2 \circ i_1 ] \circ out
|
||||
\\=\;&[ [ (id + \coalg{\alpha}) \circ i_1 , (id + \coalg{\alpha}) \circ i_2 \circ i_2 ] \circ out \circ f , (id + \coalg{\alpha}) \circ i_2 \circ i_1 ] \circ out
|
||||
\\=\;&[ [ i_1 , i_2 \circ \coalg{\alpha} \circ i_2 ] \circ out \circ f , i_2 \circ \coalg{\alpha} \circ i_1 ] \circ out
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ out
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ out.
|
||||
\end{alignat*}
|
||||
|
||||
We are left to check uniqueness of \(f^*\). Let \(g : DX \rightarrow DY\) and \(out \circ g = [ out \circ f , i_2 \circ g ] \circ out\).
|
||||
|
|
@ -245,39 +236,39 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
\\=\;&[ [ [ i_1 , i_2 \circ [g , id] \circ i_2 ] \circ out \circ f , i_2 \circ [g , id] \circ i_1 ] \circ out , (id + [g , id] \circ i_2) \circ out ]
|
||||
\\=\;&[ [ [ (id + [g , id]) \circ i_1 , (id + [g , id]) \circ i_2 \circ i_2 ] \circ out \circ f , (id + [g , id]) \circ i_2 \circ i_1 ] \circ out
|
||||
\\ \;&, (id + [g , id]) \circ (id + i_2) \circ out ]
|
||||
\\=\;&(id + [g , id]) \circ [ [ [ i_1 , i_2 \circ i_2 ] \circ out \circ f , i_2 \circ i_1 ] \circ out , (id + i_2) \circ out ]
|
||||
\\=\;&(id + [g , id]) \circ [ [ [ i_1 , i_2 \circ i_2 ] \circ out \circ f , i_2 \circ i_1 ] \circ out , (id + i_2) \circ out ].
|
||||
\end{alignat*}
|
||||
|
||||
Thus, \([ g , id ] = \coalg{\alpha}\) by uniqueness of \(\coalg{\alpha}\).
|
||||
It follows: \[g = [ g , id ] \circ i_1 =\;\coalg{\alpha} \circ i_1 = f^*\]
|
||||
\item[\ref{D3}] This follows immediately since \(\tau \) is the unique coalgebra morphism from \((X \times DY, dstl \circ (id \times out))\) into the terminal coalgebra \((D(X \times Y) , out)\).
|
||||
It follows that indeed \[g = [ g , id ] \circ i_1 =\;\coalg{\alpha} \circ i_1 = f^*.\]
|
||||
\item[\ref{D3}] Take \(\tau := \coalg{dstl \circ (id \times out)} : X \times DY \rightarrow D(X \times Y)\), the requisite property then follows by definition.
|
||||
\qedhere
|
||||
\end{itemize}
|
||||
\end{proof}
|
||||
|
||||
\begin{lemma}
|
||||
The following properties follow from \autoref{lem:delay}:
|
||||
The following properties of \(\mathbf{D}\) hold:
|
||||
\begin{enumerate}
|
||||
\item \(out \circ Df = (f + Df) \circ out\)
|
||||
\item \(f^* = [ f , {(later \circ f)}^* ] \circ out\)
|
||||
\item \(later \circ f^* = {(later \circ f)}^* = f^* \circ later\)
|
||||
\end{enumerate}
|
||||
\end{lemma}
|
||||
\begin{proof}
|
||||
\begin{proof} These identities follow by use of \autoref{lem:delay}:
|
||||
\begin{itemize}
|
||||
\item[1.] Note that definitionally: \(Df = {(now \circ f)}^*\) for any \(f : X \rightarrow TY\). The statement is then simply a consequence of~\ref{D1} and~\ref{D2}:
|
||||
\begin{alignat*}{1}
|
||||
& out \circ Df
|
||||
\\=\;&out \circ {(now \circ f)}^*
|
||||
\\=\;&[ out \circ now \circ f , i_2 \circ {(now \circ f)}^* ] \circ out\tag*{\ref{D2}}
|
||||
\\=\;&(f + Df) \circ out\tag*{\ref{D1}}
|
||||
\\=\;&(f + Df) \circ out.\tag*{\ref{D1}}
|
||||
\end{alignat*}
|
||||
\item[2.] By uniqueness of \(f^*\) it suffices to show:
|
||||
\begin{alignat*}{1}
|
||||
& out \circ [ f , {(later \circ f)}^* ] \circ out
|
||||
\\=\;&[ out \circ f , out \circ {(later \circ f)}^* ] \circ out
|
||||
\\=\;&[out \circ f , [ out \circ later \circ f , i_2 \circ {(later \circ f)}^* ] \circ out ] \circ out\tag*{\ref{D2}}
|
||||
\\=\;&[out \circ f , i_2 \circ [ f , {(later \circ f)}^* ] \circ out ] \circ out\tag*{\ref{D1}}
|
||||
\\=\;&[out \circ f , i_2 \circ [ f , {(later \circ f)}^* ] \circ out ] \circ out.\tag*{\ref{D1}}
|
||||
\end{alignat*}
|
||||
\item[3.]
|
||||
These equations follow by monicity of \(out\):
|
||||
|
|
@ -290,7 +281,7 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
\\=\;&i_2 \circ f^*\tag*{\ref{D1}}
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ i_2
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ out \circ later\tag*{\ref{D1}}
|
||||
\\=\;&out \circ f^* \circ later \tag*{\ref{D2}}
|
||||
\\=\;&out \circ f^* \circ later. \tag*{\ref{D2}}
|
||||
\end{alignat*}
|
||||
\end{itemize}
|
||||
Thus, the postulated properties have been proven.
|
||||
|
|
@ -300,7 +291,7 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
\(\mathbf{D} = (D, now, {(-)}^*)\) is a Kleisli triple.
|
||||
\end{theorem}
|
||||
\begin{proof}
|
||||
We will now use the properties proven in Corollary~\ref{lem:delay} to prove the Kleisli triple laws:
|
||||
We will now use the properties proven in \autoref{lem:delay} to prove the Kleisli triple laws:
|
||||
\begin{itemize}
|
||||
\item[\ref{K1}]
|
||||
By uniqueness of \(now^*\) it suffices to show that \(out \circ id = [ out \circ now , i_2 \circ id ] \circ out\) which instantly follows by~\ref{D1}.
|
||||
|
|
@ -309,7 +300,7 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
& out \circ f^* \circ now
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ out \circ now\tag*{\ref{D2}}
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ i_1\tag*{\ref{D1}}
|
||||
\\=\;&out \circ f
|
||||
\\=\;&out \circ f.
|
||||
\end{alignat*}
|
||||
\item[\ref{K3}] Let \(f : Y \rightarrow Z, g : X \rightarrow Z\) to show \(f^* \circ g^* = {(f^* \circ g)}^*\) by uniqueness of \({(f^* \circ g)}^*\) it suffices to show:
|
||||
\begin{alignat*}{1}
|
||||
|
|
@ -317,7 +308,7 @@ Since \(DX\) is defined as a final coalgebra, we can define morphisms via corecu
|
|||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ out \circ g^*\tag*{\ref{D2}}
|
||||
\\=\;&[ out \circ f , i_2 \circ f^* ] \circ [ out \circ g , i_2 \circ g^* ] \circ out\tag*{\ref{D2}}
|
||||
\\=\;&[ [ out \circ f , i_2 \circ f^* ] \circ out \circ g , i_2 \circ f^* \circ g^* ] \circ out
|
||||
\\=\;&[ out \circ f^* \circ g , i_2 \circ f^* \circ g^* ] \circ out\tag*{\ref{D2}}
|
||||
\\=\;&[ out \circ f^* \circ g , i_2 \circ f^* \circ g^* ] \circ out.\tag*{\ref{D2}}
|
||||
\end{alignat*}
|
||||
\end{itemize}
|
||||
This concludes the proof.
|
||||
|
|
@ -386,19 +377,18 @@ Finality of the coalgebras \({(DX, out : DX \rightarrow X + DX)}_{X \in \obj{\C}
|
|||
\arrow["{\pi_2 + id}", from=1-3, to=1-4]
|
||||
\end{tikzcd}
|
||||
\]
|
||||
\item[\ref{S2}] We don't need coinduction to show \(\tau \circ (id \times now) = now\), but we need a small helper lemma:
|
||||
\item[\ref{S2}] We don't need coinduction to show \(\tau \circ (id \times now) = now\), but we will first need to establish
|
||||
\begin{equation*}
|
||||
\tau \circ (id \times out^{-1}) = out^{-1} \circ (id + \tau) \circ dstl \tag*{(\(\ast \))}\label{helper}
|
||||
\tau \circ (id \times out^{-1}) = out^{-1} \circ (id + \tau) \circ dstl, \tag*{(\(\ast \))}\label{helper}
|
||||
\end{equation*}
|
||||
which is a direct consequence of~\ref{D3}.
|
||||
With this the proof is straightforward:
|
||||
With this we are done by
|
||||
\begin{alignat*}{1}
|
||||
& \tau \circ (id \times now) \\
|
||||
=\; & \tau \circ (id \times out^{-1}) \circ (id \times i_1) \\
|
||||
=\; & out^{-1} \circ (id + \tau) \circ dstl \circ (id \times i_1)\tag*{\ref*{helper}} \\
|
||||
=\; & now
|
||||
=\; & now.
|
||||
\end{alignat*}
|
||||
where the last step follows by \(dstl \circ (id \times i_1) = i_1\).
|
||||
\item[\ref{S3}] We need to check \(\tau^* \circ \tau = \tau \circ (id \times id^*)\), the coalgebra for coinduction is:
|
||||
% https://q.uiver.app/#q=WzAsNixbMCwwLCJYIFxcdGltZXMgRERZIl0sWzAsMiwiRChYXFx0aW1lcyBZKSJdLFsxLDAsIlggXFx0aW1lcyAoRFkgKyBERFkpIl0sWzIsMCwiWCBcXHRpbWVzIERZICsgWCBcXHRpbWVzIEREWSJdLFsyLDIsIlggXFx0aW1lcyBZICsgRChYIFxcdGltZXMgWSkiXSxbMiwxLCJYIFxcdGltZXMgWSArIFggXFx0aW1lcyBERFkiXSxbMCwxLCJcXGNvYWxney19IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzAsMiwiaWQgXFx0aW1lcyBvdXQiXSxbMiwzLCJkc3RsIl0sWzEsNCwib3V0Il0sWzUsNCwiaWQgKyBcXGNvYWxney19IiwyXSxbMyw1LCJbIChpZCArIChpZCBcXHRpbWVzIG5vdykpIFxcY2lyYyBkc3RsIFxcY2lyYyAoaWQgXFx0aW1lcyBvdXQpICwgaV8yIF0iLDJdXQ==
|
||||
\[
|
||||
|
|
@ -456,8 +446,8 @@ To prove that \(\mathbf{D}\) is commutative we will use another proof principle
|
|||
\end{corollary}
|
||||
\begin{proof}
|
||||
Let \(g : X \rightarrow D(Y + X)\) be guarded by \(h : X \rightarrow Y + D(Y+X)\) and \(f, i : X \rightarrow DY\) be solutions of g.
|
||||
It suffices to show \({[ now , f ]}^* = {[ now , i ]}^*\), because then follows:
|
||||
\[f = {[ now , f ]}^* \circ g = {[ now , i ]}^* \circ g = i\]
|
||||
It suffices to show \({[ now , f ]}^* = {[ now , i ]}^*\), because then follows that
|
||||
\[f = {[ now , f ]}^* \circ g = {[ now , i ]}^* \circ g = i.\]
|
||||
This is proven by coinduction using
|
||||
% https://q.uiver.app/#q=WzAsNSxbMCwxLCJEWSJdLFszLDEsIlkgKyBEWSJdLFswLDAsIkQoWSArIFgpIl0sWzEsMCwiKFkgKyBYKSArIEQoWStYKSJdLFszLDAsIlkgKyBEKFkrWCkiXSxbMCwxLCJvdXQiXSxbMiwzLCJvdXQiXSxbMyw0LCJbIFsgaV8xICwgaCBdICwgaV8yIF0iXSxbNCwxLCJpZCArIFxcY29hbGd7LX0iXSxbMiwwLCJcXGNvYWxney19IiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d
|
||||
\[
|
||||
|
|
@ -489,7 +479,7 @@ Let us record some facts that we will use to prove commutativity of \(\mathbf{D}
|
|||
\begin{proof}
|
||||
\begin{itemize}
|
||||
\item[\ref{tau1}] This is just~\ref{D3} restated.
|
||||
\item[\ref{sigma1}]
|
||||
\item[\ref{sigma1}] Indeed, by use of~\ref{tau1}
|
||||
\begin{alignat*}{1}
|
||||
& out \circ \sigma
|
||||
\\=\;&out \circ Dswap \circ \tau \circ swap
|
||||
|
|
@ -497,19 +487,19 @@ Let us record some facts that we will use to prove commutativity of \(\mathbf{D}
|
|||
\\=\;&(swap + Dswap) \circ (id + \tau) \circ dstl \circ (id \times out) \circ swap \tag*{\ref{tau1}}
|
||||
\\=\;&(swap + Dswap) \circ (id + \tau) \circ dstl \circ swap \circ (out \times id)
|
||||
\\=\;&(swap + Dswap) \circ (id + \tau) \circ (swap + swap) \circ dstr \circ (out \times id)
|
||||
\\=\;&(id + \sigma) \circ dstr \circ (out \times id)
|
||||
\\=\;&(id + \sigma) \circ dstr \circ (out \times id).
|
||||
\end{alignat*}
|
||||
\item[\ref{tau2}] By monicity of \(out\):
|
||||
\begin{alignat*}{1}
|
||||
& out \circ \tau \circ (id \times out^{-1})
|
||||
\\=\;&(id + \tau) \circ dstl \circ (id \times out) \circ (id \times out^{-1})\tag*{\ref{tau1}}
|
||||
\\=\;&(id + \tau) \circ dstl
|
||||
\\=\;&(id + \tau) \circ dstl.
|
||||
\end{alignat*}
|
||||
\item[\ref{sigma2}] By monicity of \(out\):
|
||||
\item[\ref{sigma2}] Again, by monicity of \(out\):
|
||||
\begin{alignat*}{1}
|
||||
& out \circ \sigma \circ (out^{-1} \times id)
|
||||
\\=\;&(id + \sigma) \circ dstr \circ (out \times id) \circ (out^{-1} \times id)\tag*{\ref{sigma1}}
|
||||
\\=\;&(id + \sigma) \circ dstr\tag*{\qedhere}
|
||||
\\=\;&(id + \sigma) \circ dstr.\tag*{\qedhere}
|
||||
\end{alignat*}
|
||||
\end{itemize}
|
||||
\end{proof}
|
||||
|
|
@ -521,27 +511,28 @@ Let us record some facts that we will use to prove commutativity of \(\mathbf{D}
|
|||
Using \autoref{cor:solution} it suffices to show that both \(\tau^* \circ \sigma \) and \(\sigma^* \circ \tau \) are solutions of some guarded morphism \(g\).
|
||||
|
||||
Let \(w = (dstr + dstr) \circ dstl \circ (out \times out)\) and take
|
||||
\[g = out^{-1} \circ [ i_1 + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\]
|
||||
\[g = out^{-1} \circ [ i_1 + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w.\]
|
||||
\(g\) is trivially guarded by \([ id + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\).
|
||||
|
||||
We are left to show that both \(\tau^* \circ \sigma \) and \(\sigma^* \circ \tau \) are solutions of \(g\). Consider:
|
||||
We are left to show that both \(\tau^* \circ \sigma \) and \(\sigma^* \circ \tau \) are solutions of \(g\). Consider
|
||||
|
||||
\[\tau^* \circ \sigma = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \tau^* \circ \sigma ] ] \circ w = {[ now , \tau^* \circ \sigma]}^* \circ g \]
|
||||
\[\sigma^* \circ \tau = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \sigma^* \circ \tau ] ] \circ w = {[ now , \sigma^* \circ \tau]}^* \circ g \]
|
||||
\[\tau^* \circ \sigma = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \tau^* \circ \sigma ] ] \circ w = {[ now , \tau^* \circ \sigma]}^* \circ g, \]
|
||||
and
|
||||
\[\sigma^* \circ \tau = out^{-1} \circ [ id + \sigma , i_2 \circ [ \tau , later \circ \sigma^* \circ \tau ] ] \circ w = {[ now , \sigma^* \circ \tau]}^* \circ g. \]
|
||||
|
||||
The last step in both equations can be proven generally for any \(f : DX \times DY \rightarrow D(X \times Y)\) using monicity of \(out\):
|
||||
The last step in both equations can be proven generally for any \(f : DX \times DY \rightarrow D(X \times Y)\) using monicity of \(out\)
|
||||
\begin{alignat*}{1}
|
||||
& out \circ {[ now , f ]}^* \circ out^{-1} \circ [ i_1 + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w
|
||||
\\=\; & [ out \circ [ now , f ] , i_2 \circ {[ now , f ]}^* ] \circ [ i_1 + D i_1 \circ \sigma , i_2 \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\tag*{\ref{D2}}
|
||||
\\=\; & [ id + \sigma , i_2 \circ {[ now , f]}^* \circ [ D i_1 \circ \tau , later \circ now \circ i_2 ] ] \circ w\tag*{\ref{D1}}
|
||||
\\=\; & [ id + \sigma , i_2 \circ [ \tau , {[ now , f]}^* \circ later \circ now \circ i_2 ] ] \circ w
|
||||
\\=\; & [ id + \sigma , i_2 \circ [ \tau , {[ later \circ now , later \circ f]}^* \circ now \circ i_2 ] ] \circ w
|
||||
\\=\; & [ id + \sigma , i_2 \circ [ \tau , later \circ f ] ] \circ w
|
||||
\\=\; & [ id + \sigma , i_2 \circ [ \tau , later \circ f ] ] \circ w.
|
||||
\end{alignat*}
|
||||
|
||||
Let us now check the first step in the equation for \(\sigma^* \circ \tau \), the same step for \(\tau^* \circ \sigma \) is then symmetric.
|
||||
|
||||
Again we proceed by monicity of \(out\):
|
||||
Again, we proceed by monicity of \(out\), which yields
|
||||
\begin{alignat*}{1}
|
||||
& out \circ \sigma^* \circ \tau
|
||||
\\=\;&[ out \circ \sigma , i_2 \circ \sigma^* ] \circ out \circ \tau\tag*{\ref{D2}}
|
||||
|
|
@ -557,18 +548,17 @@ Let us record some facts that we will use to prove commutativity of \(\mathbf{D}
|
|||
\\=\;&[ (id + \sigma), i_2 \circ {(out^{-1} \circ (id + \sigma))}^* \circ [D i_1 \circ \tau , D i_2 \circ \tau]] \circ (dstr + dstr) \circ dstl \circ (out \times out)
|
||||
\\=\;&[ (id + \sigma), i_2 \circ [{(out^{-1} \circ i_1)}^* \circ \tau , {(out^{-1} \circ i_2 \circ \sigma)}^* \circ \tau]] \circ w
|
||||
\\=\;&[ (id + \sigma), i_2 \circ [ \tau , {(later \circ \sigma)}^* \circ \tau]] \circ w \tag*{\ref{K1}}
|
||||
\\=\;&[ (id + \sigma), i_2 \circ [ \tau , later \circ \sigma^* \circ \tau]] \circ w
|
||||
\\=\;&[ (id + \sigma), i_2 \circ [ \tau , later \circ \sigma^* \circ \tau]] \circ w,
|
||||
\end{alignat*}
|
||||
|
||||
where
|
||||
\[Ddstr \circ \tau = [ Di_1 \circ \tau , Di_2 \circ \tau ] \circ dstr \tag*{(*)}\label{Dcomm-helper}\]
|
||||
is proven using epicness of \(dstr^{-1}\):
|
||||
indeed follows by epicness of \(dstr^{-1}\):
|
||||
|
||||
\begin{alignat*}{1}
|
||||
& Ddstr \circ \tau \circ dstr^{-1}
|
||||
\\=\;&[ Ddstr \circ \tau \circ (i_1 \times id) , Ddstr \circ \tau \circ (i_2 \times id) ]
|
||||
\\=\;&[ Ddstr \circ D(i_1 \times id) \circ \tau , Ddstr \circ D(i_2 \times id) \circ \tau ]
|
||||
\\=\;&[ Di_1 \circ \tau , Di_2 \circ \tau ]\tag*{\qedhere}
|
||||
\\=\;&[ Di_1 \circ \tau , Di_2 \circ \tau ].\tag*{\qedhere}
|
||||
\end{alignat*}
|
||||
\end{proof}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in a new issue