From 8021868ae945157b0d7b948c0fd0c147a981cc40 Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 20 Jul 2026 10:35:35 +0100 Subject: [PATCH 01/19] not liftable --- draft/draft.tex | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 29debd5..15050de 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -292,6 +292,7 @@ \newcommand{\spto}{\mathrel{\tikz{\draw[-{Stealth}] (0.0,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);}}} \newcommand{\rto}{\mathrel{\tikz{\draw[-{Stealth}] (0.04,0) -- (0.4,0); \draw (0.17,0.07) -- (0.17,-0.07);\draw (0.04,0) -- (0,-0.07);\draw (0.04,0) -- (0,0.07);}}} \newcommand{\powf}{\mathcal{P}} +\newcommand{\powfi}{\mathcal{P}_{\mathsf{inj}}} \newcommand{\sappr}{\sqsupseteq} \newcommand{\Dom}{\mathsf{Dom}} @@ -433,7 +434,7 @@ We want to say that a natural order structure entails an order over a functor de \begin{proof} Using~\autoref{lem:set-ord-str} we only prove the case for every $h\in\Hom(1,\powf FZ)$, $k\in\Hom(1,\powf FY)$. Additionally, for every $g\c Y\to Z$, such that $h\subseteq\powf Fg(k)$, we define $k'\in\powf FY$ that that $k'=\{y\mid Fg(y)\in h\}$. We show that $k'\subseteq k$ and $\powf Fg(k')=h$. - Assuming $y\in k'$ we have $Fg(y)\in h$. Since $h\subseteq \powf Fg(k)$ we have $Fg(y)\in\powf Fh(h)$, so we have $y\in k$ that means $k'\subseteq k$. + Assuming $y\in k'$ we have $Fg(y)\in h$. Since $h\subseteq \powf Fg(k)$ we have $Fg(y)\in\powf Fg(k)$, so we have $y\in k$ that means $k'\subseteq k$. Assuming $z\in\powf Fg(k')$, then there exists $y'\in k'$ that $z=Fg(y')$, so $z\in h$ and $\powf Fg(k')\subseteq h$. @@ -2163,7 +2164,7 @@ We recall that in the above diagram $\sigma_3$ is a bisimulation, and the rest a %\end{proof} \begin{example} And another counter-example!!! -Assuming that $F$ is the powerset endofunctor over the category of sets and injective maps. Then for every $X$, we define the order on $FX$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder.\\ +Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. We show the functor with $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined as below: \begin{gather*} \alpha(x)= @@ -2186,7 +2187,10 @@ $\sigma$ is defined as below: $R$ is symmetric, and $\sigma$ is a witness for $R$ to be an AM-simulation, but $R$ is not a bisimulation in the traditional sense because $(2,1)\in R$, and $2\to 3$, but $(3,1)$ or $(3,2)$ are not in $R$. It is easy to see that it is not an AM-bisimulation as well because we can not define a function that can serve as an evidence for it as $3$ does not appear in any pair in $R$, while it exists in $\alpha(2)$. This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr;(FR)^\dagger;\appr=\powf X\times \powf X$, so $R\subseteq \appr;(FR)^\dagger;\appr$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. So, unlike their definition, in the context of the relator-based definitions that only work in $\Set$, this counter-example does not live. + +The mentioned ordering is not liftable. Assuming $h\in\Hom(\nats,\powfi \nats)$, $g\c \nats\to \nats$, and $k\in\Hom(\nats,\powfi \nats)$, and they are defined for every $n$ in $\nats$ as $h(n)=\{2\times n\}$, $g(n)=3\times n$, and $k(n)=\{n\}$, then $|h(n)|=|\powfi g(k(n))|=1$ that means $h\appr\powfi g\comp k$ is satisfied, but there is no $k'$ that $h=\powfi g\comp k$ because we can never have $h(1)=\powfi g\comp k'(1)$, as assuming $k'(1)=\{n\}$, and $n$ must be a natural number, then we should have $2=3\times n$ that is impossible. \end{example} + \subsection{The concrete proof} %\begin{lemma}\label{lem:sim-opsim-inc1}\ppnote{Actually, this lemma holds for every functor in an arbitrary category.} % Assuming that $\sigma\c R\to\powf R$ is witness for a symmetric relation $R$ to be an AM simulation on $\powf$-coalgebra $(X,\alpha)$, then for all $(x_1,x_2)\in R$ we have: From cc4999dcb9de4267e8c31369b3b666dbf0a707c8 Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 20 Jul 2026 12:18:32 +0100 Subject: [PATCH 02/19] minor --- draft/draft.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/draft/draft.tex b/draft/draft.tex index 15050de..a87cf02 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2186,7 +2186,7 @@ $\sigma$ is defined as below: \end{gather*} $R$ is symmetric, and $\sigma$ is a witness for $R$ to be an AM-simulation, but $R$ is not a bisimulation in the traditional sense because $(2,1)\in R$, and $2\to 3$, but $(3,1)$ or $(3,2)$ are not in $R$. It is easy to see that it is not an AM-bisimulation as well because we can not define a function that can serve as an evidence for it as $3$ does not appear in any pair in $R$, while it exists in $\alpha(2)$. -This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr;(FR)^\dagger;\appr=\powf X\times \powf X$, so $R\subseteq \appr;(FR)^\dagger;\appr$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. So, unlike their definition, in the context of the relator-based definitions that only work in $\Set$, this counter-example does not live. +This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr\comp(FR)^\dagger\comp\appr=\powf X\times \powf X$, so $R\subseteq( \appr\comp(FR)^\dagger\comp\appr)$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. So, unlike their definition, in the context of the relator-based definitions that only work in $\Set$, this counter-example does not live. The mentioned ordering is not liftable. Assuming $h\in\Hom(\nats,\powfi \nats)$, $g\c \nats\to \nats$, and $k\in\Hom(\nats,\powfi \nats)$, and they are defined for every $n$ in $\nats$ as $h(n)=\{2\times n\}$, $g(n)=3\times n$, and $k(n)=\{n\}$, then $|h(n)|=|\powfi g(k(n))|=1$ that means $h\appr\powfi g\comp k$ is satisfied, but there is no $k'$ that $h=\powfi g\comp k$ because we can never have $h(1)=\powfi g\comp k'(1)$, as assuming $k'(1)=\{n\}$, and $n$ must be a natural number, then we should have $2=3\times n$ that is impossible. \end{example} From 3d36dc32935237bebe6ee7652d1a5e8cd34af3e3 Mon Sep 17 00:00:00 2001 From: Sergey Goncharov Date: Mon, 20 Jul 2026 14:05:33 +0100 Subject: [PATCH 03/19] pc sync --- draft/draft.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/draft/draft.tex b/draft/draft.tex index 003d259..19894dd 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -3077,7 +3077,7 @@ The following example justifies why the $g$ in~\autoref{def:coliftable-ord} shou \end{align*}\qed \end{proof} \begin{cor} - Since the symmetrization of left-lax Barr relator that is laxed with a lifatble order structure is natural, and normal, it is a normal relational connector. So, it is a sound and complete relator. + Since the symmetrization of left-lax Barr relator with a lifatble order structure is natural, and normal, it is a normal relational connector. So, it is a sound and complete relator. \end{cor} \begin{cor} By~\autoref{prop:all-rel-compa}.(1), if $\appr$ is liftable, then the mid-lax Barr relator is a normal relation connector, and thus a sound and complete relator as well. From 9da15b278cd6b7cf8499d546610b247c9734a061 Mon Sep 17 00:00:00 2001 From: Sergey Goncharov Date: Mon, 20 Jul 2026 15:17:54 +0100 Subject: [PATCH 04/19] pc sync --- draft/draft.tex | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 9f03924..8367017 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2163,8 +2163,7 @@ We recall that in the above diagram $\sigma_3$ is a bisimulation, and the rest a % We have $(Fp_1)^\dagger\comp\sigma$. %\end{proof} \begin{example} -And another counter-example!!! -Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. We show the functor with $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ +Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. Let us call this functor $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined as below: \begin{gather*} \alpha(x)= From 61fed4f74a856edd8b029f235f0f1c1d22864569 Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 20 Jul 2026 19:01:30 +0100 Subject: [PATCH 05/19] Barr! --- draft/draft.tex | 45 +++++++++++++++++++++++++++++++++++++++++++-- 1 file changed, 43 insertions(+), 2 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index a87cf02..c095486 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2172,7 +2172,24 @@ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined \{1,2\} & x=1 \\ \{2,3\} & x=2\\ \{3\} & x=3 - \end{cases} + \end{cases}\qquad +\begin{tikzpicture}[scale=0.1] + \tikzstyle{every node}+=[inner sep=0pt] + \draw [black] (23.8,-25.2) circle (3); + \draw (23.8,-25.2) node {$1$}; + \draw [black] (42.4,-25.2) circle (3); + \draw (42.4,-25.2) node {$2$}; + \draw [black] (33,-34.6) circle (3); + \draw (33,-34.6) node {$3$}; + \draw [black] (22.477,-22.52) arc (234:-54:2.25); + \fill [black] (25.12,-22.52) -- (26,-22.17) -- (25.19,-21.58); + \draw [black] (41.077,-22.52) arc (234:-54:2.25); + \fill [black] (43.72,-22.52) -- (44.6,-22.17) -- (43.79,-21.58); + \draw [black] (26.8,-25.2) -- (39.4,-25.2); + \fill [black] (39.4,-25.2) -- (38.6,-24.7) -- (38.6,-25.7); + \draw [black] (40.28,-27.32) -- (35.12,-32.48); + \fill [black] (35.12,-32.48) -- (36.04,-32.27) -- (35.33,-31.56); +\end{tikzpicture} \end{gather*} $\sigma$ is defined as below: \begin{gather*} @@ -3138,6 +3155,15 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{proof} They all follow in an obvious way from~\autoref{lem:liftable} and~\autoref{lem:coliftable}. The last one needs $\appr\comp\appr=\appr$ that comes from transitivity of $\appr$. \qed \end{proof} +\begin{prop}\label{prop:lax-relator-full-comm} + Assuming the axiom of choice, for a functor $F$ with a liftable order we have: + \begin{gather*} + \bar{F}r\comp\appr=\appr\comp\bar{F} + \end{gather*} +\end{prop} +\begin{proof} + \todo{Write it down. You have it in your notes.} +\end{proof} %\begin{example} % In the category of sets, subset over the powerset functor is an example of a liftable order. Using~\autoref{lem:set-ord-str} we only prove the case for every $h\in\Hom(1,\powf Z)$, $k\in\Hom(1,\powf Y)$. Additionally, for every $g\c Y\to Z$, such that $h\subseteq\powf g(k)$, we define $k'\in\powf Y$ that that $k'=\{y\mid g(y)\in h\}$. We show that $\powf g(k')=h$. % @@ -3259,7 +3285,22 @@ Perhaps if we can relax the definition of liftable by allowing $g$ to be a relat \begin{remark} With a similar argument we can prove that the symmetrization of a right-lax Barr-relator of $F$ is a Barr-relator. \end{remark} -\todo{Try $FX=\powf(X^2)$ to see if the symmetrization of its lax Barr relator is a Barr relator. The order is just the set inclusion.} +\begin{lemma}\label{lem:lax-relator-str} + For a relation $r$, for functions $k_s\c F\pi_1(\bar{F}r)\to\powf FX$ and $k_b\c F\pi_2(\bar{F}r)\to\powf FX$, defined as + \begin{gather*} + k_s(x)=\{(x',y)\mid(x,y)\in\bar{F}r,x'\appr x, (x',y)\notin\bar{F}r\},\\ + k_b(x)=\{(x',y)\mid(x,y)\in\bar{F}r,x\appr x', (x',y)\notin\bar{F}r\}, + \end{gather*} + we have + \begin{gather*} + (\bar{F}r)\comp\appr=\bar{F}r\cup(\bigcup_{x\in F\pi_1(\bar{F}r)} k_s(x)),\\ + (\bar{F}r)\comp\sappr=\bar{F}r\cup(\bigcup_{x\in F\pi_1(\bar{F}r)} k_b(x)). + \end{gather*} +\end{lemma} +\begin{proof} + \todo{Write it down. You have it in your notes.} +\end{proof} +\todo{Try $FX=\powf(X^2)$ to see if the symmetrization of its lax Barr relator is a Barr relator. The order is just the set inclusion. See if~\autoref{prop:lax-relator-full-comm} or~\autoref{lem:lax-relator-str} can help!} \end{document} From 5eef8a7955e773d465b0511ee66b679531578d2e Mon Sep 17 00:00:00 2001 From: partowp Date: Wed, 22 Jul 2026 13:04:41 +0100 Subject: [PATCH 06/19] presentation --- ACV 2026/Feedback | 3 + ACV 2026/README.md | 93 ++++ ACV 2026/beamercolorthemefocus.sty | 71 ++++ ACV 2026/beamerfontthemefocus.sty | 47 +++ ACV 2026/beamerinnerthemefocus.sty | 110 +++++ ACV 2026/beamerouterthemefocus.sty | 255 +++++++++++ ACV 2026/beamerthemefocus.sty | 60 +++ ACV 2026/references.bib | 110 +++++ ACV 2026/res/HO-GSOS(1).png | Bin 0 -> 32843 bytes ACV 2026/res/HO-GSOS.png | Bin 0 -> 52492 bytes ACV 2026/res/Haskell-Logo.png | Bin 0 -> 12223 bytes ACV 2026/res/arxiv.png | Bin 0 -> 52662 bytes ACV 2026/res/haskell.png | Bin 0 -> 52690 bytes ACV 2026/res/ut-logo.png | Bin 0 -> 277105 bytes ACV 2026/slides.pdf | Bin 0 -> 527697 bytes ACV 2026/slides.tex | 655 +++++++++++++++++++++++++++++ ACV 2026/speech.tex | 33 ++ ACV 2026/طرح.odt | Bin 0 -> 25165 bytes 18 files changed, 1437 insertions(+) create mode 100644 ACV 2026/Feedback create mode 100644 ACV 2026/README.md create mode 100644 ACV 2026/beamercolorthemefocus.sty create mode 100644 ACV 2026/beamerfontthemefocus.sty create mode 100644 ACV 2026/beamerinnerthemefocus.sty create mode 100644 ACV 2026/beamerouterthemefocus.sty create mode 100644 ACV 2026/beamerthemefocus.sty create mode 100644 ACV 2026/references.bib create mode 100644 ACV 2026/res/HO-GSOS(1).png create mode 100644 ACV 2026/res/HO-GSOS.png create mode 100644 ACV 2026/res/Haskell-Logo.png create mode 100644 ACV 2026/res/arxiv.png create mode 100644 ACV 2026/res/haskell.png create mode 100644 ACV 2026/res/ut-logo.png create mode 100644 ACV 2026/slides.pdf create mode 100644 ACV 2026/slides.tex create mode 100644 ACV 2026/speech.tex create mode 100644 ACV 2026/طرح.odt diff --git a/ACV 2026/Feedback b/ACV 2026/Feedback new file mode 100644 index 0000000..4fff444 --- /dev/null +++ b/ACV 2026/Feedback @@ -0,0 +1,3 @@ +- You should first make a list of the things that you want to talk about before making the slides. +- If you have a proposition about strong separated HO-GSOS, you should not present it as a proposition about separated HO-GSOS, but with the same result! You were thinking that you can do this because you will get simplicity, but no! +- You panicked after Filipo said you only have 5 minutes! :) diff --git a/ACV 2026/README.md b/ACV 2026/README.md new file mode 100644 index 0000000..a1406d3 --- /dev/null +++ b/ACV 2026/README.md @@ -0,0 +1,93 @@ +# Asraneh and Oberseminar + + + +## Getting started + +To make it easy for you to get started with GitLab, here's a list of recommended next steps. + +Already a pro? Just edit this README.md and make it your own. Want to make it easy? [Use the template at the bottom](#editing-this-readme)! + +## Add your files + +- [ ] [Create](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#create-a-file) or [upload](https://docs.gitlab.com/ee/user/project/repository/web_editor.html#upload-a-file) files +- [ ] [Add files using the command line](https://docs.gitlab.com/ee/gitlab-basics/add-file.html#add-a-file-using-the-command-line) or push an existing Git repository with the following command: + +``` +cd existing_repo +git remote add origin https://git8.cs.fau.de/le79nide/asraneh-and-oberseminar.git +git branch -M main +git push -uf origin main +``` + +## Integrate with your tools + +- [ ] [Set up project integrations](https://git8.cs.fau.de/le79nide/asraneh-and-oberseminar/-/settings/integrations) + +## Collaborate with your team + +- [ ] [Invite team members and collaborators](https://docs.gitlab.com/ee/user/project/members/) +- [ ] [Create a new merge request](https://docs.gitlab.com/ee/user/project/merge_requests/creating_merge_requests.html) +- [ ] [Automatically close issues from merge requests](https://docs.gitlab.com/ee/user/project/issues/managing_issues.html#closing-issues-automatically) +- [ ] [Enable merge request approvals](https://docs.gitlab.com/ee/user/project/merge_requests/approvals/) +- [ ] [Set auto-merge](https://docs.gitlab.com/ee/user/project/merge_requests/merge_when_pipeline_succeeds.html) + +## Test and Deploy + +Use the built-in continuous integration in GitLab. + +- [ ] [Get started with GitLab CI/CD](https://docs.gitlab.com/ee/ci/quick_start/index.html) +- [ ] [Analyze your code for known vulnerabilities with Static Application Security Testing (SAST)](https://docs.gitlab.com/ee/user/application_security/sast/) +- [ ] [Deploy to Kubernetes, Amazon EC2, or Amazon ECS using Auto Deploy](https://docs.gitlab.com/ee/topics/autodevops/requirements.html) +- [ ] [Use pull-based deployments for improved Kubernetes management](https://docs.gitlab.com/ee/user/clusters/agent/) +- [ ] [Set up protected environments](https://docs.gitlab.com/ee/ci/environments/protected_environments.html) + +*** + +# Editing this README + +When you're ready to make this README your own, just edit this file and use the handy template below (or feel free to structure it however you want - this is just a starting point!). Thanks to [makeareadme.com](https://www.makeareadme.com/) for this template. + +## Suggestions for a good README + +Every project is different, so consider which of these sections apply to yours. The sections used in the template are suggestions for most open source projects. Also keep in mind that while a README can be too long and detailed, too long is better than too short. If you think your README is too long, consider utilizing another form of documentation rather than cutting out information. + +## Name +Choose a self-explaining name for your project. + +## Description +Let people know what your project can do specifically. Provide context and add a link to any reference visitors might be unfamiliar with. A list of Features or a Background subsection can also be added here. If there are alternatives to your project, this is a good place to list differentiating factors. + +## Badges +On some READMEs, you may see small images that convey metadata, such as whether or not all the tests are passing for the project. You can use Shields to add some to your README. Many services also have instructions for adding a badge. + +## Visuals +Depending on what you are making, it can be a good idea to include screenshots or even a video (you'll frequently see GIFs rather than actual videos). Tools like ttygif can help, but check out Asciinema for a more sophisticated method. + +## Installation +Within a particular ecosystem, there may be a common way of installing things, such as using Yarn, NuGet, or Homebrew. However, consider the possibility that whoever is reading your README is a novice and would like more guidance. Listing specific steps helps remove ambiguity and gets people to using your project as quickly as possible. If it only runs in a specific context like a particular programming language version or operating system or has dependencies that have to be installed manually, also add a Requirements subsection. + +## Usage +Use examples liberally, and show the expected output if you can. It's helpful to have inline the smallest example of usage that you can demonstrate, while providing links to more sophisticated examples if they are too long to reasonably include in the README. + +## Support +Tell people where they can go to for help. It can be any combination of an issue tracker, a chat room, an email address, etc. + +## Roadmap +If you have ideas for releases in the future, it is a good idea to list them in the README. + +## Contributing +State if you are open to contributions and what your requirements are for accepting them. + +For people who want to make changes to your project, it's helpful to have some documentation on how to get started. Perhaps there is a script that they should run or some environment variables that they need to set. Make these steps explicit. These instructions could also be useful to your future self. + +You can also document commands to lint the code or run tests. These steps help to ensure high code quality and reduce the likelihood that the changes inadvertently break something. Having instructions for running tests is especially helpful if it requires external setup, such as starting a Selenium server for testing in a browser. + +## Authors and acknowledgment +Show your appreciation to those who have contributed to the project. + +## License +For open source projects, say how it is licensed. + +## Project status +If you have run out of energy or time for your project, put a note at the top of the README saying that development has slowed down or stopped completely. Someone may choose to fork your project or volunteer to step in as a maintainer or owner, allowing your project to keep going. You can also make an explicit request for maintainers. diff --git a/ACV 2026/beamercolorthemefocus.sty b/ACV 2026/beamercolorthemefocus.sty new file mode 100644 index 0000000..ffbf9ff --- /dev/null +++ b/ACV 2026/beamercolorthemefocus.sty @@ -0,0 +1,71 @@ +% Copyright (C) 2018 Pasquale Claudio Africa. +% 2018 Sebastian Friedl. +% +% This file is part of beamerthemefocus. +% +% beamerthemefocus is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% beamerthemefocus is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with beamerthemefocus. If not, see . + +\mode + + +% DEFINE COLORS. --------------------------------------------------------------- +\definecolor{main}{RGB}{64, 64, 64} +\definecolor{background}{RGB}{239, 239, 239} + +\definecolor{alert}{RGB}{180, 0, 0} +\definecolor{example}{RGB}{0, 110, 0} + + +% SET COLORS. ------------------------------------------------------------------ +\setbeamercolor{normal text}{fg=text, bg=background} +\setbeamercolor{alerted text}{fg=alert} +\setbeamercolor{example text}{fg=example} + +\setbeamercolor{titlelike}{fg=background, bg=main} +\setbeamercolor{frametitle}{parent={titlelike}} + +\setbeamercolor{footline}{fg=background, bg=main} + +\setbeamercolor{block title}{bg=main!80!background, fg=background} +\setbeamercolor{block body}{bg=main!10!background, fg=text} + +\setbeamercolor{block title alerted}{bg=alert, fg=background} +\setbeamercolor{block body alerted}{bg=alert!10!background, fg=text} + +\setbeamercolor{block title example}{bg=example, fg=background} +\setbeamercolor{block body example}{bg=example!10!background, fg=text} + +\setbeamercolor{itemize item}{fg=text} +\setbeamercolor{itemize subitem}{fg=text} + +\setbeamercolor{enumerate item}{fg=main!70!black} +\setbeamercolor{enumerate subitem}{fg=main!70!black} + +\setbeamercolor{description item}{fg=main!70!black} +\setbeamercolor{description subitem}{fg=main!70!black} + +\setbeamercolor{caption name}{fg=main} + +\setbeamercolor{section in toc}{fg=main} +\setbeamercolor{subsection in toc}{fg=main} +\setbeamercolor{section number projected}{bg=main} +\setbeamercolor{subsection number projected}{bg=main} + +\setbeamercolor{bibliography item}{fg=main} +\setbeamercolor{bibliography entry author}{fg=main!70!black} +\setbeamercolor{bibliography entry title}{fg=main} +\setbeamercolor{bibliography entry location}{fg=main} +\setbeamercolor{bibliography entry note}{fg=main} + +\mode diff --git a/ACV 2026/beamerfontthemefocus.sty b/ACV 2026/beamerfontthemefocus.sty new file mode 100644 index 0000000..26b5885 --- /dev/null +++ b/ACV 2026/beamerfontthemefocus.sty @@ -0,0 +1,47 @@ +% Copyright (C) 2018 Pasquale Claudio Africa. +% 2018 Sebastian Friedl. +% +% This file is part of beamerthemefocus. +% +% beamerthemefocus is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% beamerthemefocus is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with beamerthemefocus. If not, see . + +\mode + + +% SET FONTS. ------------------------------------------------------------------- +\setbeamerfont{title}{size=\huge, shape=\scshape\bfseries} +\setbeamerfont{subtitle}{size=\Large, shape=\scshape, parent=structure} +\setbeamerfont{author}{size=\large, shape=\upshape} + +\setbeamerfont{institute}{size=\normalsize, shape=\scshape} +\setbeamerfont{date}{size=\small, shape=\scshape} + +\setbeamerfont{sectiontitle}{size=\huge, series=\scshape\bfseries} +\setbeamerfont{frametitle}{size=\Large, shape=\scshape} + +\setbeamerfont{footline}{size=\scriptsize} + +\setbeamerfont{focusframe}{size=\huge, shape=\scshape} + +\setbeamerfont{description item}{shape=\bfseries} + +\setbeamerfont{caption name}{shape=\bfseries} + +\setbeamerfont{bibliography item}{size=\small, shape=\scshape} +\setbeamerfont{bibliography entry author}{size=\small, shape=\scshape} +\setbeamerfont{bibliography entry title}{size=\small, series=\scshape\bfseries} +\setbeamerfont{bibliography entry location}{size=\small, shape=\scshape\normalfont} +\setbeamerfont{bibliography entry note}{size=\small, shape=\scshape\normalfont} + +\mode diff --git a/ACV 2026/beamerinnerthemefocus.sty b/ACV 2026/beamerinnerthemefocus.sty new file mode 100644 index 0000000..c990d22 --- /dev/null +++ b/ACV 2026/beamerinnerthemefocus.sty @@ -0,0 +1,110 @@ +% Copyright (C) 2018 Pasquale Claudio Africa. +% 2018 Sebastian Friedl. +% +% This file is part of beamerthemefocus. +% +% beamerthemefocus is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% beamerthemefocus is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with beamerthemefocus. If not, see . + +\mode + +\RequirePackage{tikz} + + +% CUSTOMIZE STRUCTURE ELEMENTS. ------------------------------------------------ +\setbeamertemplate{blocks}[default] + +\setbeamertemplate{section in toc}[square] +\setbeamertemplate{subsection in toc}[square] + +\setbeamertemplate{itemize items}[square] +\setbeamertemplate{itemize subitem}[triangle] + + +% STRUCTURE FRAME TEMPLATE DEFINITIONS. ---------------------------------------- +% Title page. +\defbeamertemplate*{title page}{focus}{% + {\usebeamercolor{frametitle}\colorlet{focus@@temp}{bg}% + \begin{tikzpicture}[overlay, remember picture] + \fill[color=focus@@temp] (current page.north west) rectangle ([shift = {(0, -0.45\paperheight)}] current page.north east); + \end{tikzpicture}} + + \vspace{-1.65\baselineskip} + \begin{minipage}[b][0.35\paperheight]{\textwidth} + \vspace{\baselineskip} + \usebeamerfont{title} + \usebeamercolor[fg]{frametitle} + \inserttitle + \end{minipage} + + \begin{minipage}[t][0.1\paperheight]{\textwidth} + \usebeamerfont{subtitle} + \usebeamercolor[fg]{frametitle} + \insertsubtitle + \end{minipage} + + % Set the title graphic in a zero-height box, so that + % the position of other elements is not affected. + {\vfuzz=9999pt\vbox to 0pt { + \raggedleft + \inserttitlegraphic + }} + + \begin{minipage}[t]{\textwidth} + \usebeamerfont{author} + \insertauthor + \end{minipage} + + \vspace*{\baselineskip} + \begin{minipage}[t]{\textwidth} + \usebeamerfont{institute} + \insertinstitute + \end{minipage} + + \vspace*{\baselineskip} + \begin{minipage}[t]{\textwidth} + \usebeamerfont{date}{\insertdate} + \end{minipage} + + \vspace*{5\baselineskip} + + \addtocounter{framenumber}{-1} +} + +% Section page. +\defbeamertemplate*{section page}{focus}{% + {% + \usebeamercolor{frametitle}\colorlet{focus@@temp}{bg}% + \begin{tikzpicture}[overlay, remember picture] + \fill[color=focus@@temp] (current page.north west) rectangle ([shift = {(0, -0.45\paperheight)}] current page.north east); + \end{tikzpicture}% + } + + \vspace{-2\baselineskip} + \begin{minipage}[b][0.45\paperheight]{\textwidth} + \usebeamerfont{sectiontitle} + \usebeamercolor[fg]{frametitle} + \let\hyperlink\@secondoftwo\insertsection + \end{minipage} + + \begin{minipage}[t][0.55\paperheight]{\textwidth} + \end{minipage} +} + +\AtBeginSection{% + \begin{frame}[plain, noframenumbering]{} + \sectionpage + \end{frame}% +} + +\mode diff --git a/ACV 2026/beamerouterthemefocus.sty b/ACV 2026/beamerouterthemefocus.sty new file mode 100644 index 0000000..3f05f33 --- /dev/null +++ b/ACV 2026/beamerouterthemefocus.sty @@ -0,0 +1,255 @@ +% Copyright (C) 2018 Pasquale Claudio Africa. +% 2018 Sebastian Friedl. +% +% This file is part of beamerthemefocus. +% +% beamerthemefocus is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% beamerthemefocus is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with beamerthemefocus. If not, see . + +\mode + +\RequirePackage{appendixnumberbeamer}% Don't number appendix frames. +\RequirePackage{etoolbox}% \BeforeBeginEnvironment +\RequirePackage{tikz} + + +% FRAMETITLE TEMPLATES. -------------------------------------------------------- +\defbeamertemplate*{frametitle}{focus}{% + % If not title page. + \ifnum\value{framenumber}>0% + \vspace{-1pt}% + \begin{beamercolorbox}[wd=\paperwidth,leftskip=0.55cm,rightskip=0.55cm,sep=0.2cm]{frametitle}% + \strut\insertframetitle\strut% + \end{beamercolorbox}% + \fi% +} + +% Plain header. +\defbeamertemplate{frametitle}{plain}{% + % If not title page. + \ifnum\value{framenumber}>0% + \vspace{-1pt}% + \begin{beamercolorbox}[wd=\paperwidth,leftskip=0.55cm,rightskip=0.55cm,sep=0.2cm,ignorebg]{frametitle}% + \strut% + \end{beamercolorbox}% + \fi% +} + + +% FOOTLINE TEMPLATES. ---------------------------------------------------------- +% Lenghts for the progress bar footline. +\newlength{\focus@pbar@height}% Progress bar height. +\newlength{\focus@pbar@leftoffset} +\newlength{\focus@pbar@rightoffset} + +\defbeamertemplate*{footline}{progressbar}{% + % If not appendix. + \ifnum\mainend<0% From package appendixnumberbeamer. + % + \settowidth{\focus@pbar@leftoffset}{1}% + \addtolength{\focus@pbar@leftoffset}{1.5em}% + % + \settowidth{\focus@pbar@rightoffset}{\inserttotalframenumber}% + \addtolength{\focus@pbar@rightoffset}{1.5em}% + % + % If not title page. + \ifnum\c@framenumber>0% + \ifnum\c@framenumber<\inserttotalframenumber% + \begin{tikzpicture}[inner xsep=0.5em, inner ysep=0.5ex]\usebeamerfont{footline} + \pgfmathsetmacro{\focus@pbar@progress}% + {(\paperwidth-\focus@pbar@leftoffset-\focus@pbar@rightoffset)*(\insertframenumber/\inserttotalframenumber)} + + \clip (0,0) rectangle ++(\paperwidth,\the\focus@pbar@height); + \fill[footline.bg] (0,0) rectangle ++(\the\focus@pbar@leftoffset,\the\focus@pbar@height); + + \fill[footline.bg] (\the\focus@pbar@leftoffset,0) rectangle ++(\focus@pbar@progress pt,\the\focus@pbar@height) + ++(0,{-0.5*\the\focus@pbar@height}) node[anchor=east, text=footline.fg] {\strut\insertframenumber}; + + \fill[footline.bg] (\paperwidth,0) rectangle ++(-\the\focus@pbar@rightoffset,\the\focus@pbar@height) + ++(0,{-0.5*\the\focus@pbar@height}) node[anchor=west, text=footline.fg] {\strut\inserttotalframenumber}; + \end{tikzpicture}% + \else% + \begin{tikzpicture}[inner xsep=0.5em, inner ysep=0.5ex] + \clip (0,0) rectangle ++(\paperwidth,\the\focus@pbar@height); + \fill[footline.bg] (0,0) rectangle ++(\paperwidth,\the\focus@pbar@height); + + \node[anchor=east, footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut\insertframenumber}; + \node[footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut/}; + \node[anchor=west, footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut\inserttotalframenumber}; + \end{tikzpicture}% + \fi% + \fi% + \fi% +} + +% Full bar footline. +\defbeamertemplate{footline}{fullbar}{% + % If not appendix. + \ifnum\mainend<0% From package appendixnumberbeamer. + % + \settowidth{\focus@pbar@leftoffset}{1}% + \addtolength{\focus@pbar@leftoffset}{1.5em}% + % + \settowidth{\focus@pbar@rightoffset}{\inserttotalframenumber}% + \addtolength{\focus@pbar@rightoffset}{1.5em}% + % + % If not title page. + \ifnum\c@framenumber>0% + \begin{tikzpicture}[inner xsep=0.5em, inner ysep=0.5ex] + \clip (0,0) rectangle ++(\paperwidth,\the\focus@pbar@height); + \fill[footline.bg] (0,0) rectangle ++(\paperwidth,\the\focus@pbar@height); + + \node[anchor=east, footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut\insertframenumber}; + \node[footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut/}; + \node[anchor=west, footline.fg] at ({\paperwidth-\the\focus@pbar@rightoffset},{0.5*\focus@pbar@height}) {\strut\inserttotalframenumber}; + \end{tikzpicture}% + \fi% + \fi% +} + +% Empty footline. +\defbeamertemplate{footline}{none}{} + +\DeclareOptionBeamer{numbering}{\def\beamer@focus@numbering{#1}} +\ExecuteOptionsBeamer{numbering=progressbar} +\ProcessOptionsBeamer + +\def\beamer@focus@numberingprogressbar{progressbar} +\def\beamer@focus@numberingfullbar{fullbar} +\def\beamer@focus@numberingnone{none} + + +% BACKGROUND CANVAS TEMPLATES. ------------------------------------------------- +\defbeamertemplate*{background canvas}{focus}{% + \begin{tikzpicture} + \clip (0,0) rectangle ++(\paperwidth,\paperheight); + \fill[normal text.bg] (0,0) rectangle ++(\paperwidth,\paperheight); + \end{tikzpicture}% +} + +\defbeamertemplate{background canvas}{focusplain}{% + \begin{tikzpicture} + \clip (0,0) rectangle ++(\paperwidth,\paperheight); + \fill[normal text.bg] (0,0) rectangle ++(\paperwidth,\paperheight); + \end{tikzpicture}% +} + +\defbeamertemplate{background canvas}{focusframe}{% + \begin{tikzpicture} + \clip (0,0) rectangle ++(\paperwidth,\paperheight); + \fill[frametitle.bg] (0,0) rectangle ++(\paperwidth,\paperheight); + \end{tikzpicture}% +} + + +% HOOKS FOR CREATING FRAMES. --------------------------------------------------- +\BeforeBeginEnvironment{frame}{% + \setbeamertemplate{background canvas}[focus]% + \setbeamertemplate{frametitle}[focus]% + % + % Reset footline height and determine it for the current slide. + \setlength{\focus@pbar@height}{0cm}% + \focus@calculatefootheight% + % + % If not appendix. + \ifnum\mainend<0 % From package appendixnumberbeamer. + \settoheight{\focus@pbar@height}{\usebeamerfont{footline}1234567890/}% + \addtolength{\focus@pbar@height}{6pt}% + % + \ifx\beamer@focus@numbering\beamer@focus@numberingprogressbar% + \setbeamertemplate{footline}[progressbar]% + \else% + \ifx\beamer@focus@numbering\beamer@focus@numberingfullbar% + \setbeamertemplate{footline}[fullbar]% + \fi% + \fi% + % + \focus@calculatefootheight% + \fi% +} + +% Enable noframenumbering option. +\define@key{beamerframe}{noframenumbering}[true]{% + \setbeamertemplate{footline}[none]% + \setlength{\focus@pbar@height}{0cm}% + \focus@calculatefootheight% + % + \addtocounter{framenumber}{-1}% +} + + +% Enable plain option. +\define@key{beamerframe}{plain}[true]{% + \setbeamertemplate{background canvas}[focusplain]% + \setbeamertemplate{frametitle}[plain]% + % + \setbeamertemplate{footline}[none]% +} + + +% Full vertical centering +% (from https://tex.stackexchange.com/questions/247826/beamer-full-vertical-centering). +\define@key{beamerframe}{c}[true]{% + \beamer@frametopskip=0pt plus 1fill\relax% + \beamer@framebottomskip=0pt plus 1fill\relax% + \beamer@frametopskipautobreak=0pt plus 0.4\paperheight\relax% + \beamer@framebottomskipautobreak=0pt plus 0.6\paperheight\relax% + \def\beamer@initfirstlineunskip{}% +} + + +% Enable focus option. +\providebool{focus@standout} +\define@key{beamerframe}{focus}[true]{% + \booltrue{focus@standout}% + \begingroup% + \setkeys{beamerframe}{noframenumbering}% + \setbeamertemplate{background canvas}[focusframe]% + \setbeamertemplate{frametitle}[plain]% + % + \setkeys{beamerframe}{c}% + \centering% + \usebeamerfont{focusframe}% + \usebeamercolor[fg]{frametitle}% +} + +\apptocmd{\beamer@reseteecodes} +{% + \ifbool{focus@standout}% + {% + \endgroup% + \boolfalse{focus@standout}% + }{}% +}{}{} + + +% Recalculate the footline's size and refresh other parameters. +% Partially copied from the definition of \beamer@calculateheadfoot. +\def\focus@calculatefootheight{% + \footheight=\focus@pbar@height% + \advance\footheight by 4pt% + \sidebarheight=\paperheight% + \advance\sidebarheight by-\headheight% + \advance\sidebarheight by\headdp% + \advance\sidebarheight by-\footheight% + \advance\sidebarheight by 4pt% + \footskip=\footheight% + \textheight=\paperheight% + \advance\textheight by-\footheight% + \advance\textheight by-\headheight% + \@colht\textheight% + \@colroom\textheight% + \vsize\textheight% +} + +\mode diff --git a/ACV 2026/beamerthemefocus.sty b/ACV 2026/beamerthemefocus.sty new file mode 100644 index 0000000..964cf74 --- /dev/null +++ b/ACV 2026/beamerthemefocus.sty @@ -0,0 +1,60 @@ +% Copyright (C) 2018 Pasquale Claudio Africa. +% 2018 Sebastian Friedl. +% +% This file is part of beamerthemefocus. +% +% beamerthemefocus is free software: you can redistribute it and/or modify +% it under the terms of the GNU General Public License as published by +% the Free Software Foundation, either version 3 of the License, or +% (at your option) any later version. +% +% beamerthemefocus is distributed in the hope that it will be useful, +% but WITHOUT ANY WARRANTY; without even the implied warranty of +% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the +% GNU General Public License for more details. +% +% You should have received a copy of the GNU General Public License +% along with beamerthemefocus. If not, see . + +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{beamerthemefocus}[2018/07/15 v2.0 Focus Beamer theme] + +\mode + + +% THEME OPTIONS. --------------------------------------------------------------- +\DeclareOptionBeamer{numbering}{% + \PassOptionsToPackage{numbering=#1}{beamerouterthemefocus} +} + +\newif\if@focus@loadfirafonts +\@focus@loadfirafontstrue + +\DeclareOptionBeamer{nofirafonts}{\@focus@loadfirafontsfalse} +\ProcessOptionsBeamer + + +% LOAD EXTERNAL PACKAGES. ------------------------------------------------------ +\if@focus@loadfirafonts + \RequirePackage[T1]{fontenc} + + \PassOptionsToPackage{type1}{FiraSans} + \PassOptionsToPackage{type1}{FiraMono} + + \RequirePackage{FiraSans} + \RequirePackage{FiraMono} +\fi + +\usecolortheme{focus} +\usefonttheme{focus} +\useinnertheme{focus} +\useoutertheme{focus} + +\setbeamertemplate{navigation symbols}{} + + +% SET MARGINS. ----------------------------------------------------------------- +\setbeamersize{text margin left=0.75cm, text margin right=0.75cm} +\setlength{\leftmargini}{0.75cm} + +\mode diff --git a/ACV 2026/references.bib b/ACV 2026/references.bib new file mode 100644 index 0000000..27b39f7 --- /dev/null +++ b/ACV 2026/references.bib @@ -0,0 +1,110 @@ +@article{amir, + title = {Implication via Spacetime}, + author = {Tabatabai, Amirhossein Akbar}, + journal = {Mathematics, Logic, and Their Philosophies: Essays in Honour of Mohammad Ardeshir}, + volume = {49}, + pages = {161}, + year = {2021}, + publisher = {Springer Nature} +} + +@phdthesis{ardeshir1995aspects, + title = {Aspects of basic logic}, + author = {Ardeshir, Mohammad}, + year = {1995}, + school = {Marquette University} +} + +@article{celani2001closer, + title = {A closer look at some subintuitionistic logics}, + author = {Celani, Sergio and Jansana, Ramon}, + journal = {Notre Dame Journal of Formal Logic}, + volume = {42}, + number = {4}, + pages = {225--255}, + year = {2001}, + publisher = {Duke University Press} +} + +@incollection{DBLP:books/ox/07/Feferman07, + author = {Solomon Feferman}, + editor = {Stewart Shapiro}, + title = {Predicativity}, + booktitle = {The Oxford Handbook of Philosophy of Mathematics and Logic}, + series = {Oxford handbooks in philosophy}, + pages = {590--624}, + publisher = {Oxford University Press}, + year = {2007}, + timestamp = {Wed, 06 May 2020 15:26:14 +0200}, + biburl = {https://dblp.org/rec/books/ox/07/Feferman07.bib}, + bibsource = {dblp computer science bibliography, https://dblp.org} +} + +@incollection{dovsen1993modal, + title = {Modal translations in K and D}, + author = {Do{\v{s}}en, Kosta}, + booktitle = {Diamonds and defaults}, + pages = {103--127}, + year = {1993}, + publisher = {Springer} +} + +@article{feferman2007predicativity, + title = {Predicativity}, + author = {Feferman, Solomon}, + year = {2007}, + publisher = {Citeseer} +} + +@incollection{Kennedy2009-KENOGL, + editor = {Dov Gabbay}, + title = {On {G}\"odel's Logic}, + year = {2009}, + author = {Juliette Kennedy and Mark van Atten}, + booktitle = {The Handbook of the History of Logic}, + publisher = {Elsevier} +} + +@article{mckinsey1944algebra, + title = {The algebra of topology}, + author = {McKinsey, John Charles Chenoweth and Tarski, Alfred}, + journal = {Annals of mathematics}, + pages = {141--191}, + year = {1944}, + publisher = {JSTOR} +} + +@article{tabatabai2017geometric, + title = {Geometric modality and weak exponentials}, + author = {Tabatabai, Amirhossein Akbar}, + journal = {arXiv preprint arXiv:1711.01736}, + year = {2017} +} + +@article{van2001, + title = {Intuitionistic Logic}, + author = {Dirk van Dalen}, + journal = {The Blackwell Guide to Philosophica Logic}, + editor = {L. Gobble. Blackwell}, + publisher = {Blackwell}, + year = {2001}, + pages = {224–257} +} + +@article{van2017predicativity, + title = {Predicativity and parametric polymorphism of Brouwerian implication}, + author = {van Atten, Mark}, + journal = {arXiv preprint arXiv:1710.07704}, + year = {2017} +} + +@article{russell1908mathematical, + title={Mathematical logic as based on the theory of types}, + author={Russell, Bertrand}, + journal={American journal of mathematics}, + volume={30}, + number={3}, + pages={222--262}, + year={1908}, + publisher={JSTOR} +} diff --git a/ACV 2026/res/HO-GSOS(1).png b/ACV 2026/res/HO-GSOS(1).png new file mode 100644 index 0000000000000000000000000000000000000000..009c45c72f058cdc0feb6dfb9e0b087dca03db6f GIT binary patch literal 32843 zcmeFYbyQrzvNwt(xCaOXhu{nv+}+(>2X}Wuu;4Zjg1cLA*93yQySux;Avx#VbKduT z>%F_yd;eWv7JIs?tGa$&Rb9Pz?@$Ffab$!K2oMku$dVEwN)QlGAP5LZS2!4O%V-H4 z75Lw8PZbSkB||r2dq+D{3u_>;vxhy980c zzIzrfWSU%;;xPBQT!K??M7&WJO)7T2DVc`j=TGy7T{N6iw+FnOjH~OB9n)?D&LpMyGIabOFBrseY*e z98WASA-jG0wmq>>veGRAOBTRVEe6e_4&c#AVsX5P{JpRBh^I577UBPxG7)+XV{of^ z2^X}c3^p*xLReTqQdsz(Hi0ck^@-(`=oY~LrLX)8l}7ZCct0i!;ft=y&mU@8a9C1h zqs0MzdHSQdB$5sem^z3kzI18?I;k!$_E>K^QPJEwCC5ai9`%*X(Kk9B_?MTw7F(eM zJ--*3ieoCU4oLhmHlk>zac#|peS$N>*}!1AYBXVn9657ACDK13=)4`ZnxuJI{o*b>P3_bfoRE9gYhhA_tLqMRgSb&{OLso{%*v^LD z(8SINNbhcA5B5F?2p)cSdqZO@pfj-%(9FV?m-M8$os`(ZgqKvERhCiKUKnU@A>ru= zRQ8lpG4`}F<}@MY=R@Fe=K>3`0XiEJyW3dXI&r!4lKzp)1-^c5W*{a0!{ThkOR6EO zKrC$M2qgYQ|B0RvAnI=6%1p|KK+NN4V#=i?BK8jv;BUO7=FZOcTnr3uZf^8$EcAAc zW(-W6oSY1d%nZ!T05Au@$-~y!&>djwMD{A;uM82Old+?Py|aa#E%B>NLnAvEXI@fL zFrE0H}Z0LFvC-O!$aiJpFZ#*Ozey;Ot}7BG8!?lnX(%*0azG6nF2n2GBN>hFar$%Z0v?6 ztj6pthAc*=|0YV(*2&q>))@FI3M`!70xXA}gOSCEnTZkH(G&>y#LUDBFfuVU0hj_! 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z?fke+!Q2oDxET#d*D1h`a7qI*$5XF0N0_cx>AsFxh0R$J{{Ieq~!HFnPyFF`XGnF88bydGCARXphzG+cQq43HvS<5UoNg?xZ=%l_C4o5X)x3aV# zDtukG?_$eCi>UfJ_NMMV(K;2=ChhocpvKxWjn`^pt6jtFN2x{1 zPx-W75NI+Vr7A7w+Rvu#Zy{N_?Xwi>HAsz=5b$|nxOm}yTNR~x6rX>+?(kX9N%gci z%z4)LBYf@!=DIa6hU)Szcs5GGInQMYneBLP-W7*u(pw0B&14M0-pVQ|&F{%+8c_fgmAA6S zyV|C@Y>VBmq_Y|gX;BDUSyGctuoZ=stealth, thick] + \node[circle, draw, minimum size=1cm] (brewing tea) at (0,1.8) {$\mathsf{brewing\;tea}$}; + \node[circle, draw, minimum size=1cm] (idle) at (4,1.8) {$\mathsf{idle}$}; + \node[circle, draw, minimum size=1cm] (active) at (4,0) {$\mathsf{active}$}; + \node[circle, draw, minimum size=1cm] (brewing coffee) at (8,1.8) {$\mathsf{brewing\;coffee}$}; + + \draw[->] (idle) -- node[left] {insert coin} (active); + \draw[->] (active) to[bend right=20] node[below] {select coffee} (brewing coffee); + \draw[->] (active) to[bend left=20] node[below] {select tea} (brewing tea); + + \draw[->] (brewing coffee) -- node[above] {take coffee} (idle); + \draw[->] (brewing tea) -- node[above] {take tea} (idle); + \end{tikzpicture} + \end{center} +% \end{columns} + \end{alertblock} +\end{frame} + +\begin{frame}{Simulation and Bisimulation} + + \begin{columns}[T] + + \column{0.47\textwidth} + + \begin{alertblock}{Similar to} + \centering + \begin{tikzpicture}[>=stealth, thick, scale=1, every node/.style={transform shape}] + \node[circle, draw, minimum size=.9cm] (tea) at (0,1.8) {\scriptsize brewing tea}; + \node[circle, draw, minimum size=.9cm] (idle) at (4,1.8) {\scriptsize idle}; + \node[circle, draw, minimum size=.9cm] (active) at (4,0) {\scriptsize active}; + + \draw[->] (idle) -- node[left,font=\tiny] {insert coin} (active); + \draw[->] (active) to[bend left=20] + node[below,font=\tiny] {select tea} (tea); + + \draw[->] (tea) -- node[above,font=\tiny] {take tea} (idle); + \end{tikzpicture} + \end{alertblock} + + \column{0.47\textwidth} + + \begin{alertblock}{Bisimilar with} + + \centering + + \begin{tikzpicture}[>=stealth, thick, scale=.9, every node/.style={transform shape}] + \node[circle, draw, minimum size=1cm] (tea) at (0,1.8) {\scriptsize brewing tea/coffee}; + \node[circle, draw, minimum size=1cm] (idle) at (4,1.8) {\scriptsize idle}; + \node[circle, draw, minimum size=1cm] (active) at (4,0) {\scriptsize active}; + + \draw[->] (idle) -- node[left,font=\tiny] {insert coin} (active); + \draw[->] (active) to[bend left=20] + node[below,font=\tiny] {select tea} (tea); + \draw[->] (active) to[bend left=10] node[above,font=\tiny,yshift=2mm] {select coffee} (tea); + + \draw[->] (tea) -- node[above,font=\tiny] {take tea} (idle); + \end{tikzpicture} + + \end{alertblock} + + \end{columns} +% +% \vspace{-2mm} +% + \begin{block}{} + \begin{center} + \begin{tikzpicture}[>=stealth, thick, scale=.7, every node/.style={transform shape}] + \node[circle, draw, minimum size=1cm] (brewing tea) at (0,1.8) {$\mathsf{brewing\;tea}$}; + \node[circle, draw, minimum size=1cm] (idle) at (4,1.8) {$\mathsf{idle}$}; + \node[circle, draw, minimum size=1cm] (active) at (4,0) {$\mathsf{active}$}; + \node[circle, draw, minimum size=1cm] (brewing coffee) at (8,1.8) {$\mathsf{brewing\;coffee}$}; + + \draw[->] (idle) -- node[left] {insert coin} (active); + \draw[->] (active) to[bend right=20] node[below] {select coffee} (brewing coffee); + \draw[->] (active) to[bend left=20] node[below] {select tea} (brewing tea); + + \draw[->] (brewing coffee) -- node[above] {take coffee} (idle); + \draw[->] (brewing tea) -- node[above] {take tea} (idle); + \end{tikzpicture} + \end{center} + % \end{columns} + \end{block} + +\end{frame} + +\begin{frame}{Mathematical Definitions} + + \footnotesize + + \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.48\textwidth} + + \begin{alertblock}{Labelled Transition Systems} + A \emph{labelled transition system} is a triple $(X,A,T)$, where $T\;\subseteq\;X\times A\times X$. + We write $x\xrightarrow{a}x'$ + for $(x,a,x')\in T$. + \end{alertblock} + + \vspace{2mm} + + \begin{block}{Simulation and Bisimulation} + + A relation $R\subseteq X\times Y$ is a simulation from $(X,A,T)$ to $(Y,A,S)$ whenever + \[ + x\mathrel{R}y,\; + x\xrightarrow{a}x' + \Rightarrow + \exists\,y'.\; + y\xrightarrow{a}y' + \;\land\; + x'Ry'. + \] + A simulation relation like $R$ is a bisimulation whenever + \[ + x\mathrel{R}y,\; + y\xrightarrow{a}y' + \Rightarrow + \exists\,x'.\; + x\xrightarrow{a}x' + \;\land\; + x'Ry'. + \] + \end{block} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.48\textwidth} + \begin{alertblock}{Similarity and Bisimilaritiy} + The \emph{similarity relation} $\precsim\subseteq X\times Y$ is the greatest simulation relation. Equivalently + \vspace{-0.3cm} + \[ + x\precsim y + \iff + \exists\,R.\; + xRy, + \] + where $R$ is a simulation.\\ + Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm} + \[ + x\sim y + \iff + \exists\,R.\; + xRy, + \] + where $R$ is a bisimulation. + \end{alertblock} + \vspace{-0.2cm} + Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first. + \end{columns} + +\end{frame} + +\begin{frame}{Coalgebras: A Uniform View of Transition Systems} + + \small + + \begin{columns}[T] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{0.48\textwidth} + \begin{alertblock}{Coalgebras} + \vspace{-2mm} + Fix an endofunctor + \vspace{-2mm} + \[ + F:\mathbf{C}\rightarrow\mathbf{C}. + \] + \vspace{-2mm} + An $F$-coalgebra is a pair $(X,\gamma)$, consisting of an object $X$, and a map + \vspace{-2mm} + \[ + \gamma:X\rightarrow FX. + \] + \vspace{-2mm} + Different choices of $F$ describe + different transition systems. + \vspace{-1mm} + \end{alertblock} + + \vspace{-1mm} + + \begin{block}{Our Goal} + \centering + Develop + \textbf{simulation, bisimulation} + once, for all coalgebras. + \end{block} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{0.48\textwidth} + + + \begin{alertblock}{Different kinds of systems} + \vspace{-2mm} + \begin{itemize} + \setlength{\itemsep}{2pt} + \item Labeled Transition Systems ($FX=\powf(A\times X)$) + \item Kripke frames ($FX=\powf X$) + \item \textcolor{red}{Operational Semantics of Programming Languages (\emph{Abstract HO-GSOS, and Big-step SOS!})} + \item Markov Chains ($FX=(\mathcal{D}X)^A$) + \item Mealy Machines ($FX=(O\times X)^I$) + \item[$\vdots$] + \end{itemize} + \vspace{-1mm} + \end{alertblock} + + \end{columns} + +\end{frame} + +\begin{frame}{Relators, Simulations and Bisimulations} + + \footnotesize + + \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + + \begin{alertblock}{Relator} + Assuming $F:\Set\to\Set$ is a functor, an $F$-relator a monotone map that sends a morphism of $\rel$ that is a relation $X\rto Y$ to a relation $FX\rto FY$. + \end{alertblock} + \vspace{-0.3cm} + \begin{block}{R-Simulation} + For an $F$-relator $\relar$, a relation $r:X\rto Y$ is a $\relar$-simulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if $r\subseteq \beta\comp\relar r\comp\alpha^{-1}$. + \end{block} + \vspace{-0.3cm} + + \begin{alertblock}{Symmetrization of a Relator} + Symmetrization of a relator $\relar$ is defined as: + \begin{gather*} + \hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} + \end{gather*} + \end{alertblock} + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + \begin{alertblock}{Symmetric Relator} + A relator $\relar$ is symmetric if $\relar=\hat{\relar}$ + \end{alertblock} + \vspace{-0.2cm} + \begin{block}{R-Bisimulation} + For an $F$-relator $\relar$, a relation $r:X\rto Y$ is a $\relar$-bisimulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if it is an $R$-simulation, and $R$ is a symmetric relator. + \end{block} + \vspace{-0.2cm} + \begin{block}{R-Similarity} + $\relar$-similarity is the greatest $\relar$-simulation. + \end{block} + \vspace{-0.3cm} + \begin{block}{R-Bisimilarity} + $\relar$-bisimilarity is the greatest $\relar$-bisimulation. + \end{block} + + \end{columns} + +\end{frame} + +\begin{frame}{Some Examples of Relators} + \footnotesize + \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + \begin{alertblock}{Example: Trivial Relator} + For an arbitrary set endofunctor $F$, the relator $\mathbf{Q}$ takes every relation $r:X\rto Y$ to $FX\times FY$. + \end{alertblock} + It is a symmetric relator. + \begin{block}{} + But it is a bad relator because every relation is a $\mathbf{Q}$-bisimulation! + \end{block} + \begin{alertblock}{Example: Egli-Milner Relator} + The relator $\emre$ is a $\powf$-relator defined as: + \begin{gather*} + \emre r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\} + \end{gather*} + \end{alertblock} + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + We give a family of examples: + \begin{alertblock}{Example: Barr Relators} + Given a set endofunctor $F$, and $r$ is a relation that $r=\pi_2\comp\pi_1^{-1}$ ($\pi_1$ and $\pi_2$ are functions) the Barr relator $\bar{F}$ is defined as + $\bar{F}r=\bar{F}\pi_2\comp(\bar{F}\pi_1)^{-1}$. + \end{alertblock} + \begin{block}{} + The symmetrization of the Egli-Milner relator is a Barr relator, i.e., + \begin{gather*} + \hat{\emre}=\bar{\powf} + \end{gather*} + \end{block} + Barr relators are nicely behaved relators.\\ + A trivial relator is a not a Barr relator. +\end{columns} +\end{frame} + +\begin{frame}{Lax Barr relators} + \footnotesize + \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + An ordering of a set endofunctor $F$ is a functor $G$ that commutes in the following diagram: + \begin{equation*} + \begin{tikzcd}[ampersand replacement=\&] + \& \mathbf{PoSet} \& \\ + \Set \&\& \Set + \arrow["U", from=1-2, to=2-3] + \arrow["G", from=2-1, to=1-2] + \arrow["F"', from=2-1, to=2-3] + \end{tikzcd} + \end{equation*} + Meaning that for every $X$, there is a poset $(FX,\appr)$. + \begin{block}{} + For example $(\powf X,\subseteq)$ for every set $X$. + \end{block} + With orderings we define \emph{lax Barr relators}.\\ + Lax Barr relators are different abstractions of the Egli-Milner relator. + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + \begin{block}{} + There are multiple of them: + \begin{itemize} + \item Left-lax: $\appr\comp\bar{F}r$ + \item Right-lax: $\bar{F}\comp\appr$ + \item Bi-lax: $\appr\comp\bar{F}r\comp\appr$ + \item Mid-lax: $F\pi_1\comp\appr\comp(F\pi_2)^{-1}$ + \end{itemize} + \end{block} + \begin{alertblock}{Liftability} + An ordering on a set endofunctor $F$ is called \emph{liftable}, if\vspace{-0.3cm} + \begin{gather*} + \alpha\appr\beta\Rightarrow\alpha\comp f\appr \beta\comp f,\\ + \alpha\appr\beta\Rightarrow Ff'\comp\alpha\appr Ff'\comp \beta,\\ + h\appr Fg\comp k\Rightarrow \exists k', h=Fg\comp k\;\&\;k'\appr k. + \end{gather*} + \end{alertblock} + \end{columns} +\end{frame} + +\begin{frame}{Lax Barr relators (2)} + \begin{alertblock}{Proposition} + Given a set endofunctor $F$, with a liftable ordering + \begin{itemize} + \item mid-lax and left-lax relators of $F$ are equal, + \item Assuming the axiom of choice, righ-lax and left-lax relators of $F$ are equal, + \item the symmetrizaion of the left-lax relator is \emph{sound} and \emph{complete} with respect to \emph{behavioural equivalence}. + \end{itemize} + \end{alertblock} +\end{frame} + +\begin{frame}{Span-based definitions} + \footnotesize + \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + \begin{alertblock}{Aczel-Mendler Bisimulation} + For coalgebras $(X,\alpha)$, and $(Y,\beta)$ a span $(R,X,Y)$ is an \emph{AM-bisimulation}, if there is a \emph{witness} $\gamma$ that commutes in the following: + \begin{equation*} + \begin{tikzcd}[ampersand replacement=\&] + X \& R \& Y \\ + FX \& FR \& FY + \arrow["\alpha"', from=1-1, to=2-1] + \arrow["{{p_1}}"', from=1-2, to=1-1] + \arrow["{{p_2}}", from=1-2, to=1-3] + \arrow["{{\gamma}}", from=1-2, to=2-2] + \arrow["\beta", from=1-3, to=2-3] + \arrow["{Fp_1}", from=2-2, to=2-1] + \arrow["{Fp_2}"', from=2-2, to=2-3] + \end{tikzcd} + \end{equation*} + \end{alertblock}\vspace{-0.2cm} + In a \emph{regular category}, if we change $FR$ with $\bar{F}R$ that shows the image of $$, and the legs of $\bar{F}$ (Barr relator in $\Set$), we have an \emph{Hermida-Jacobs} bisimulation. + \begin{block}{}\vspace{-0.2cm} + Every AM-bisimulation is an HJ-bisimulation. + \end{block} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + \column{.47\textwidth} + \vspace{0.1cm} + Assuming the axiom of choice, every HJ-bisimulation is an AM-bisimulation. + \begin{block}{} + We have \emph{AM-simulation}, if we lax the diagram:\vspace{-0.4cm} + \begin{equation*} + \begin{tikzcd}[ampersand replacement=\&] + X \& R \& Y \\ + FX \& FR \& FY + \arrow["\alpha"', from=1-1, to=2-1] + \arrow["\appr"{marking, allow upside down}, draw=none, from=1-1, to=2-2] + \arrow["{p_1}"', from=1-2, to=1-1] + \arrow["{p_2}", from=1-2, to=1-3] + \arrow["\sigma", from=1-2, to=2-2] + \arrow["\beta", from=1-3, to=2-3] + \arrow["\appr"{marking, allow upside down}, draw=none, from=2-2, to=1-3] + \arrow["{{Fp_1}}", from=2-2, to=2-1] + \arrow["{{Fp_2}}"', from=2-2, to=2-3] + \end{tikzcd} + \end{equation*} + \end{block}\vspace{-0.3cm} + We have found a symmetric AM-simulation that is not a bisimulation! + \begin{alertblock}{Proposition} + Given a symmetric span $(R,X)$, with the symmetry evidence $s:R\to R$, assuming $R$ is an AM-simulation on a coalgebra $(X,\alpha)$ with evidence $\sigma$, then $R$ is an AM-bisimulation iff $s$ is a coalgebra endomorphism on $(R,\sigma)$. + \end{alertblock} + \end{columns} +\end{frame} + +\begin{frame}{Motivation!} + \footnotesize + We have a coalgbera $(\mS,\gamma)$ in an arbitrary category $\mathbb{C}$. + \begin{block}{} + -$\mS$ is representing terms of a programming language,\\ + -and $\gamma$ is giving their big-step reduction. + \end{block} + \begin{alertblock}{Main Goal} + We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm} + \begin{gather*} + t\sim s\Rightarrow C[t]\sim C[s] + \end{gather*} + \end{alertblock}\vspace{-0.5cm} + -In the literature, this is often proved using \emph{Howe's method}. \\ + -The method, applies a closure on the bisimilarity relation on $\mS$.\\ + -Then one should prove that the result is a simulation.\\ + -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ + -We have found out that it is not always the case.\\ + -That is why we want to know when exactly a symmetric simulation is a bisimulation. +\end{frame} + +\bibliographystyle{apalike} +\bibliography{references} +\addtocounter{framenumber}{-1} +\begin{frame}{\quad} +% \vspace{-1cm} + \begin{center} + \textcolor[rgb]{0.384, 0.384, 0}{\Huge Thank you for your attention! :-)} + \end{center} +\end{frame} + +\end{document} diff --git a/ACV 2026/speech.tex b/ACV 2026/speech.tex new file mode 100644 index 0000000..e0374fd --- /dev/null +++ b/ACV 2026/speech.tex @@ -0,0 +1,33 @@ +\documentclass{paper} +\pagestyle{empty} +\begin{document} + +\setcounter{section}{-1} +\section{Greetings} +\section{Agenda} +\begin{itemize} + \item Three parts: + \begin{enumerate} + \item What I know: + \begin{itemize} + \item Impredicativity + \item in logic: BHK + \end{itemize} + \item What has been done: + \begin{itemize} + \item An idea to avoid impredicative implication + \item Construct a logical system with this idea + \item There are well-behaved algebraic models for this system. + \end{itemize} + \item What we're going to do: + \begin{itemize} + \item Analiticity and the problem of $cut$ + \item Introduce a new analytic system + \item what we hope to achive + \end{itemize} + \end{enumerate} +\end{itemize} +\part{Impredicativity} +\section{What is impredicativity?} +\textbullet We know impredicative definitions are +\end{document} \ No newline at end of file diff --git a/ACV 2026/طرح.odt b/ACV 2026/طرح.odt new file mode 100644 index 0000000000000000000000000000000000000000..7318b13a9cf376be61e9744d8c28b07c7d87502c GIT binary patch literal 25165 zcmb5U1CVIT(k|GxZTD{5wr$(J+qP}nwr$(kZQHhHpZnhbA|~d{otTQKh+J7KE32|r zMSb<+}O003w+deYV%VT2(A008)>{#5~3TUeVox!ai-*x6ZG7#TQO*xJy! 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As the mentioned property is equivalent with $\sigma$ being a bisimulation, and bisimulation is unique, then $\sigma_1=\sigma_2$. \end{proof} % +\begin{prop} + Assuming that the ordering on the functor $F$ is natural, and gives posets, given witness $\sigma$ for a symmetric object $R$ to be an HJ-simulation, if $(Fs)^\dagger\comp\sigma=\sigma\comp s$, then $\sigma$ is a witness for $R$ to be an $HJ$-bisimulation. +\end{prop} +\begin{proof} + Since $\sigma$ is a witness that $R$ is an $HJ$-simulation we have + \begin{gather*} + \alpha\comp p_1\appr (Fp_1)^\dagger\comp\sigma,\\ + (Fp_2)^\dagger\comp\sigma\appr\alpha\comp p_2. + \end{gather*} + So, we have: + \begin{align*} + \alpha\comp p_1\appr (Fp_1)^\dagger\comp\sigma&\\ + \Rightarrow&\alpha\comp p_1\comp s\appr (Fp_1)^\dagger\comp\sigma\comp s&\by{naturality of $\appr$}\\ + \Rightarrow&\alpha\comp p_1\comp s\appr (Fp_1)^\dagger\comp (Fs)^\dagger\comp\sigma&\by{\eqref{eq:diag-sym-rel}}\\ + \Rightarrow&\alpha\comp p_2\appr (Fp_1)^\dagger\comp (Fs)^\dagger\comp\sigma&\by{assumption}\\ + \Rightarrow&\alpha\comp p_2\appr (Fp_2)^\dagger\comp\sigma&\by{\eqref{eq:diag-sym-rel}}\\ + \Rightarrow&\alpha\comp p_2= (Fp_2)^\dagger\comp\sigma&\by{anti symmetry of $\appr$} + \end{align*} + Similarly we have: + \begin{align*} + (Fp_2)^\dagger\comp\sigma\appr\alpha\comp p_2&\\ + \Rightarrow&(Fp_2)^\dagger\comp\sigma\comp s\appr\alpha\comp p_2\comp s&\by{naturality of $\appr$}\\ + \Rightarrow&(Fp_2)^\dagger\comp(Fs)^\dagger\comp\sigma\appr\alpha\comp p_2\comp s&\by{\eqref{eq:diag-sym-rel}}\\ + \Rightarrow&(Fp_2)^\dagger\comp(Fs)^\dagger\comp\sigma\appr\alpha\comp p_1&\by{assumption}\\ + \Rightarrow&(Fp_1)^\dagger\comp\sigma\appr\alpha\comp p_1&\by{\eqref{eq:diag-sym-rel}}\\ + \Rightarrow&(Fp_1)^\dagger\comp\sigma=\alpha\comp p_1&\by{anti symmetry of $\appr$} + \end{align*}\qed +\end{proof} +% Now, we give a counter example of a symmetric relation on $\Set$ that is a simulation according to~\autoref{def:sim}, i.e, exists the morphism $\sigma$ that commutes laxly in~\eqref{eq:diag-lax-sim}, but $\sigma$ is not a coalgebraic bisimulation, although the relation that we give is clearly a bisimulation in the classic sense. We set $R=\{(A,B),(B,A),(C_1,C_2),(C_2,C_1),(C'_2,C_2),(C_2,C'_2),(C_2,C_2)\}$, $F=\mathbf{Id}$, $\appr=\Delta\cup\{(C_1,C_2),(C_2,C'_2)\}$, and the coalgebra $\alpha$ is defined with the following set of reductions: \begin{gather*} @@ -2162,7 +2191,7 @@ We recall that in the above diagram $\sigma_3$ is a bisimulation, and the rest a % \end{gather*} % We have $(Fp_1)^\dagger\comp\sigma$. %\end{proof} -\begin{example} +\begin{example}\label{ex:sym-sim-bisim-count} Assuming that the functor is the powerset endofunctor over the category of sets and injective maps. Let us call this functor $\powfi$. For every $X$, we define the order on $\powfi X$ as $A\appr B$ whenever $|A|\leq |B|$, where $|A|$ and $|B|$ are just cardinalities of $|A|$ and $|B|$ respectively. This is a preorder. Then we define the order over every $\Hom(X,\powfi Y)$ pointwise. We have chosen the category of sets with injective maps because we could not define a functor from $\Set$ to $\preord$ with the mentioned ordering.\\ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined as below: \begin{gather*} @@ -2192,19 +2221,26 @@ We take $R=\{(1,1),(1,2),(2,1),(2,2)\}$, and $X=\{1,2,3\}$. $\alpha$ is defined \end{gather*} $\sigma$ is defined as below: \begin{gather*} - \sigma(w)= - \begin{cases} - \{(1,2),(2,2)\} & w=(1,2) \\ - \{(2,1),(2,2)\} & w=(2,1) \\ - \{(1,2),(2,1)\} & w=(1,1) \\ - \{(2,2),(1,1)\} & w=(2,2) - \end{cases} + \sigma(w)=\{(1,2),(2,1)\} \end{gather*} $R$ is symmetric, and $\sigma$ is a witness for $R$ to be an AM-simulation, but $R$ is not a bisimulation in the traditional sense because $(2,1)\in R$, and $2\to 3$, but $(3,1)$ or $(3,2)$ are not in $R$. It is easy to see that it is not an AM-bisimulation as well because we can not define a function that can serve as an evidence for it as $3$ does not appear in any pair in $R$, while it exists in $\alpha(2)$. This counter-example also works as a counter-example for Hughes-Jacobs definition of simulation. Actually, $\appr\comp(FR)^\dagger\comp\appr=\powf X\times \powf X$, so $R\subseteq( \appr\comp(FR)^\dagger\comp\appr)$ that means that $R$ is a simulation. Worth noting that they claim that their setting works for an arbitrary category. So, unlike their definition, in the context of the relator-based definitions that only work in $\Set$, this counter-example does not live. The mentioned ordering is not liftable. Assuming $h\in\Hom(\nats,\powfi \nats)$, $g\c \nats\to \nats$, and $k\in\Hom(\nats,\powfi \nats)$, and they are defined for every $n$ in $\nats$ as $h(n)=\{2\times n\}$, $g(n)=3\times n$, and $k(n)=\{n\}$, then $|h(n)|=|\powfi g(k(n))|=1$ that means $h\appr\powfi g\comp k$ is satisfied, but there is no $k'$ that $h=\powfi g\comp k$ because we can never have $h(1)=\powfi g\comp k'(1)$, as assuming $k'(1)=\{n\}$, and $n$ must be a natural number, then we should have $2=3\times n$ that is impossible. + +It is the same for HJ-simulation and HJ-bisimulation. We define $\sigma^\dagger\c R\to (\powf R)^\dagger$ as +\begin{gather*} + \sigma^\dagger(w)=\{(\{1,2\},\{1,2\})\}. +\end{gather*} +Indeed, $\sigma^\dagger$ is a witness for $R$ to be an HJ-simulation, but it is not a witness for $R$ to be an HJ-bisimulation. Similar to the case for AM-bisimulation, we can not have a witness for $R$ to be an HJ-bisimulation. +\end{example} + +\begin{example} + In~\autoref{ex:sym-sim-bisim-count} we gave a counter-example with an ordering that gives preorders. Now, we slightly change the ordering so that it gives posets. We define $\appr$ as follows: + \begin{gather*} + A\appr B \iff + \end{gather*} \end{example} \subsection{The concrete proof} From 748b1b582ddc6347b7581920fd1ba2fdf1e5f99b Mon Sep 17 00:00:00 2001 From: partowp Date: Fri, 24 Jul 2026 10:32:29 +0100 Subject: [PATCH 08/19] some editting --- ACV 2026/slides.tex | 120 +++++++++++++++++++++++++------------------- 1 file changed, 68 insertions(+), 52 deletions(-) diff --git a/ACV 2026/slides.tex b/ACV 2026/slides.tex index 7acb946..538056a 100644 --- a/ACV 2026/slides.tex +++ b/ACV 2026/slides.tex @@ -166,9 +166,9 @@ \begin{itemize} \item Intuitive introduction to coalgebra and (bi)simulation \item Reviewing basic definitions of coalgebra and (bi)simulation + \item Our motivation: To ease proving program equivalence \item Relator-based notions \item Span-based notions - \item Motivation! Howe's method for categorical operational semantics \end{itemize} \end{frame} @@ -200,10 +200,9 @@ %\end{frame} \begin{frame}{From ``System'' to Mathematics} - The word ``system'' needs to be made precise. % \vspace{-0.6cm} \begin{block}{} - We model systems as mathematical notions like \textbf{labeled transition systems}: a set of states, together with a transition relation between them. + We model systems as mathematical notions like \textbf{labeled transition systems}: a set of states, together with a labeled transition relation between them. \end{block} \begin{alertblock}{Example: a vending machine} % \begin{columns} @@ -266,7 +265,8 @@ node[below,font=\tiny] {select tea} (tea); \draw[->] (active) to[bend left=10] node[above,font=\tiny,yshift=2mm] {select coffee} (tea); - \draw[->] (tea) -- node[above,font=\tiny] {take tea} (idle); + \draw[->] (tea) to[bend right=10] node[below,font=\tiny] {take tea} (idle); + \draw[->] (tea) to[bend left=10] node[above,font=\tiny] {take coffee} (idle); \end{tikzpicture} \end{alertblock} @@ -317,28 +317,43 @@ A relation $R\subseteq X\times Y$ is a simulation from $(X,A,T)$ to $(Y,A,S)$ whenever \[ - x\mathrel{R}y,\; - x\xrightarrow{a}x' - \Rightarrow - \exists\,y'.\; - y\xrightarrow{a}y' - \;\land\; - x'Ry'. - \] - A simulation relation like $R$ is a bisimulation whenever - \[ - x\mathrel{R}y,\; - y\xrightarrow{a}y' - \Rightarrow - \exists\,x'.\; - x\xrightarrow{a}x' - \;\land\; - x'Ry'. + \begin{tikzcd}[ampersand replacement=\&] + x \& y \\ + {x'} \& {y'} + \arrow["R"{description}, no head, from=1-1, to=1-2] + \arrow["a"', from=1-1, to=2-1] + \arrow["a", dashed, from=1-2, to=2-2] + \arrow["R"{description}, dashed, no head, from=2-1, to=2-2] + \end{tikzcd} \] +% A simulation relation like $R$ is a bisimulation whenever +% \[ +% x\mathrel{R}y,\; +% y\xrightarrow{a}y' +% \Rightarrow +% \exists\,x'.\; +% x\xrightarrow{a}x' +% \;\land\; +% x'Ry'. +% \] \end{block} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.48\textwidth} + \begin{block}{Bisimulation} + A simulation relation like $R$ is a bisimulation whenever:\vspace{-0.5cm} + \[ + \begin{tikzcd}[ampersand replacement=\&] + x \& y \\ + {x'} \& {y'} + \arrow["R"{description}, no head, from=1-1, to=1-2] + \arrow["a", dashed, from=1-1, to=2-1] + \arrow["a"', from=1-2, to=2-2] + \arrow["R"{description}, dashed, no head, from=2-1, to=2-2] + \end{tikzcd} + \] + \end{block} + \vspace{-0.2cm} \begin{alertblock}{Similarity and Bisimilaritiy} The \emph{similarity relation} $\precsim\subseteq X\times Y$ is the greatest simulation relation. Equivalently \vspace{-0.3cm} @@ -349,17 +364,18 @@ xRy, \] where $R$ is a simulation.\\ - Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm} - \[ - x\sim y - \iff - \exists\,R.\; - xRy, - \] - where $R$ is a bisimulation. + Similarity is defined similarly, where $R$ is a simulation, and it is shown with $\precsim$. +% Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm} +% \[ +% x\sim y +% \iff +% \exists\,R.\; +% xRy, +% \] +% where $R$ is a bisimulation. \end{alertblock} - \vspace{-0.2cm} - Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first. +% \vspace{-0.2cm} +% Ultimately, people care about similarity and bisimilarity, but simulation and bisimulation should be defined first. \end{columns} \end{frame} @@ -421,6 +437,27 @@ \end{frame} +\begin{frame}{Towards Program Equivalence} + \footnotesize + We have developed a coalgbera $(T,\gamma)$ that gives the big-step semantics for an arbitrary programming language in a specified family of them, called \emph{abstract HO-GSOS}. +% \begin{block}{} +% -$\mS$ is representing terms of a programming language,\\ +% -and $\gamma$ is giving their big-step reduction. +% \end{block} + \begin{alertblock}{Main Goal} + We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm} + \begin{gather*} + t\sim s\Rightarrow C[t]\sim C[s] + \end{gather*} + \end{alertblock}\vspace{-0.5cm} + -In the literature, this is often proved using \emph{Howe's method}. \\ + -The method, applies a closure on the bisimilarity relation on $\mS$.\\ + -Then one should prove that the result is a simulation.\\ + -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ + -We have found out that it is not always the case.\\ + -That is why we want to know when exactly a symmetric simulation is a bisimulation. +\end{frame} + \begin{frame}{Relators, Simulations and Bisimulations} \footnotesize @@ -621,27 +658,6 @@ \end{columns} \end{frame} -\begin{frame}{Motivation!} - \footnotesize - We have a coalgbera $(\mS,\gamma)$ in an arbitrary category $\mathbb{C}$. - \begin{block}{} - -$\mS$ is representing terms of a programming language,\\ - -and $\gamma$ is giving their big-step reduction. - \end{block} - \begin{alertblock}{Main Goal} - We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm} - \begin{gather*} - t\sim s\Rightarrow C[t]\sim C[s] - \end{gather*} - \end{alertblock}\vspace{-0.5cm} - -In the literature, this is often proved using \emph{Howe's method}. \\ - -The method, applies a closure on the bisimilarity relation on $\mS$.\\ - -Then one should prove that the result is a simulation.\\ - -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ - -We have found out that it is not always the case.\\ - -That is why we want to know when exactly a symmetric simulation is a bisimulation. -\end{frame} - \bibliographystyle{apalike} \bibliography{references} \addtocounter{framenumber}{-1} From 539b5fec3d6cdd2ca4c6927c221ce6e10841c6f2 Mon Sep 17 00:00:00 2001 From: partowp Date: Fri, 24 Jul 2026 11:44:36 +0100 Subject: [PATCH 09/19] fixing the example --- ACV 2026/slides.tex | 15 +++++++-------- 1 file changed, 7 insertions(+), 8 deletions(-) diff --git a/ACV 2026/slides.tex b/ACV 2026/slides.tex index 538056a..158db8b 100644 --- a/ACV 2026/slides.tex +++ b/ACV 2026/slides.tex @@ -220,8 +220,8 @@ \draw[->] (active) to[bend right=20] node[below] {select coffee} (brewing coffee); \draw[->] (active) to[bend left=20] node[below] {select tea} (brewing tea); - \draw[->] (brewing coffee) -- node[above] {take coffee} (idle); - \draw[->] (brewing tea) -- node[above] {take tea} (idle); + \draw[->] (brewing coffee) -- node[above] {take drink} (idle); + \draw[->] (brewing tea) -- node[above] {take drink} (idle); \end{tikzpicture} \end{center} % \end{columns} @@ -245,7 +245,7 @@ \draw[->] (active) to[bend left=20] node[below,font=\tiny] {select tea} (tea); - \draw[->] (tea) -- node[above,font=\tiny] {take tea} (idle); + \draw[->] (tea) -- node[above,font=\tiny] {take drink} (idle); \end{tikzpicture} \end{alertblock} @@ -265,8 +265,7 @@ node[below,font=\tiny] {select tea} (tea); \draw[->] (active) to[bend left=10] node[above,font=\tiny,yshift=2mm] {select coffee} (tea); - \draw[->] (tea) to[bend right=10] node[below,font=\tiny] {take tea} (idle); - \draw[->] (tea) to[bend left=10] node[above,font=\tiny] {take coffee} (idle); + \draw[->] (tea) -- node[above,font=\tiny] {take drink} (idle); \end{tikzpicture} \end{alertblock} @@ -287,8 +286,8 @@ \draw[->] (active) to[bend right=20] node[below] {select coffee} (brewing coffee); \draw[->] (active) to[bend left=20] node[below] {select tea} (brewing tea); - \draw[->] (brewing coffee) -- node[above] {take coffee} (idle); - \draw[->] (brewing tea) -- node[above] {take tea} (idle); + \draw[->] (brewing coffee) -- node[above] {take drink} (idle); + \draw[->] (brewing tea) -- node[above] {take drink} (idle); \end{tikzpicture} \end{center} % \end{columns} @@ -444,7 +443,7 @@ % -$\mS$ is representing terms of a programming language,\\ % -and $\gamma$ is giving their big-step reduction. % \end{block} - \begin{alertblock}{Main Goal} + \begin{alertblock}{Main Motivation} We want to prove that for terms $t$ and $s$, and a context $C$:\vspace{-0.3cm} \begin{gather*} t\sim s\Rightarrow C[t]\sim C[s] From 4f9fc5a687b67354470e1c887038f819ddfb9ddc Mon Sep 17 00:00:00 2001 From: partowp Date: Fri, 24 Jul 2026 16:42:07 +0100 Subject: [PATCH 10/19] more! --- ACV 2026/slides.tex | 219 ++++++++++++++++++++++++++++++-------------- 1 file changed, 152 insertions(+), 67 deletions(-) diff --git a/ACV 2026/slides.tex b/ACV 2026/slides.tex index 158db8b..3057f6c 100644 --- a/ACV 2026/slides.tex +++ b/ACV 2026/slides.tex @@ -312,9 +312,9 @@ \vspace{2mm} - \begin{block}{Simulation and Bisimulation} + \begin{block}{Simulation} - A relation $R\subseteq X\times Y$ is a simulation from $(X,A,T)$ to $(Y,A,S)$ whenever + A relation $R\subseteq X\times Y$ is a simulation from $(X,A,T)$ to $(Y,A,S)$ whenever: \[ \begin{tikzcd}[ampersand replacement=\&] x \& y \\ @@ -363,7 +363,7 @@ xRy, \] where $R$ is a simulation.\\ - Similarity is defined similarly, where $R$ is a simulation, and it is shown with $\precsim$. + Bisimilarity is defined similarly, where $R$ is a bisimulation, and it is shown with $\sim$. % Similarly, the \emph{bisimilarity relation} $\sim\subseteq X\times Y$ is the greatest bisimulation relation. Equivalently\vspace{-0.3cm} % \[ % x\sim y @@ -450,14 +450,14 @@ \end{gather*} \end{alertblock}\vspace{-0.5cm} -In the literature, this is often proved using \emph{Howe's method}. \\ - -The method, applies a closure on the bisimilarity relation on $\mS$.\\ + -The method, applies a closure on the bisimilarity relation on $T$.\\ -Then one should prove that the result is a simulation.\\ -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ -We have found out that it is not always the case.\\ -That is why we want to know when exactly a symmetric simulation is a bisimulation. \end{frame} -\begin{frame}{Relators, Simulations and Bisimulations} +\begin{frame}{Relators and Simulations} \footnotesize @@ -465,7 +465,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.47\textwidth} - + To define coalgebraic simulation, relators are invented. \begin{alertblock}{Relator} Assuming $F:\Set\to\Set$ is a functor, an $F$-relator a monotone map that sends a morphism of $\rel$ that is a relation $X\rto Y$ to a relation $FX\rto FY$. \end{alertblock} @@ -473,62 +473,101 @@ \begin{block}{R-Simulation} For an $F$-relator $\relar$, a relation $r:X\rto Y$ is a $\relar$-simulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if $r\subseteq \beta\comp\relar r\comp\alpha^{-1}$. \end{block} - \vspace{-0.3cm} - - \begin{alertblock}{Symmetrization of a Relator} - Symmetrization of a relator $\relar$ is defined as: - \begin{gather*} - \hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} - \end{gather*} - \end{alertblock} - - +% \vspace{-0.3cm} +% +% \begin{alertblock}{Symmetrization of a Relator} +% Symmetrization of a relator $\relar$ is defined as: +% \begin{gather*} +% \hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} +% \end{gather*} +% \end{alertblock} +\begin{block}{R-Similarity} + $\relar$-similarity is the greatest $\relar$-simulation. +\end{block} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.47\textwidth} - \begin{alertblock}{Symmetric Relator} - A relator $\relar$ is symmetric if $\relar=\hat{\relar}$ - \end{alertblock} - \vspace{-0.2cm} - \begin{block}{R-Bisimulation} - For an $F$-relator $\relar$, a relation $r:X\rto Y$ is a $\relar$-bisimulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if it is an $R$-simulation, and $R$ is a symmetric relator. + \begin{alertblock}{Example: Trivial Relator} + For an arbitrary set endofunctor $F$, the relator $\mathbf{K}$ takes every relation $r:X\rto Y$ to $FX\times FY$. + \end{alertblock}\vspace{-0.2cm} +% \begin{alertblock}{Symmetric Relator} +% A relator $\relar$ is symmetric if $\relar=\hat{\relar}$ +% \end{alertblock} +% \vspace{-0.2cm} +% \begin{block}{R-Bisimulation} +% For an $F$-relator $\relar$, a relation $r:X\rto Y$ is a $\relar$-bisimulation from a coalgebra $(X,\alpha)$ to a coalgebra $(Y,\beta)$ if it is an $R$-simulation, and $R$ is a symmetric relator. +% \end{block} +% \vspace{-0.2cm} +% \begin{block}{R-Similarity} +% $\relar$-similarity is the greatest $\relar$-simulation. +% \end{block} +% \vspace{-0.3cm} +% \begin{block}{R-Bisimilarity} +% $\relar$-bisimilarity is the greatest $\relar$-bisimulation. +% \end{block} + \begin{block}{} + But it is a bad relator because every relation is a $\mathbf{K}$-bisimulation! \end{block} - \vspace{-0.2cm} - \begin{block}{R-Similarity} - $\relar$-similarity is the greatest $\relar$-simulation. + \begin{alertblock}{Example: Egli-Milner Relator} + The relator $\emre$ is a $\powf$-relator defined as:\vspace{-0.2cm} + \begin{gather*} + \emre r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\} + \end{gather*} + \end{alertblock}\vspace{-0.2cm} + \begin{block}{} + It gives us the traditional definition for simulation on Kripke frames. \end{block} - \vspace{-0.3cm} - \begin{block}{R-Bisimilarity} - $\relar$-bisimilarity is the greatest $\relar$-bisimulation. - \end{block} - \end{columns} \end{frame} -\begin{frame}{Some Examples of Relators} - \footnotesize - \begin{columns}[T,onlytextwidth] +\begin{frame}{Symmetric Relators} +% \footnotesize + +% \begin{columns}[T,onlytextwidth] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \column{.47\textwidth} - \begin{alertblock}{Example: Trivial Relator} - For an arbitrary set endofunctor $F$, the relator $\mathbf{Q}$ takes every relation $r:X\rto Y$ to $FX\times FY$. - \end{alertblock} - It is a symmetric relator. - \begin{block}{} - But it is a bad relator because every relation is a $\mathbf{Q}$-bisimulation! +% \column{.47\textwidth} + \begin{block}{$R$-bisimulation ($R$-bisimularity)} + An $\relar$-simulation (similarity) is a bisimulation (bisimilarity), if $\relar$ is \emph{symmetric}. \end{block} - \begin{alertblock}{Example: Egli-Milner Relator} - The relator $\emre$ is a $\powf$-relator defined as: - \begin{gather*} - \emre r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\} - \end{gather*} - \end{alertblock} + \begin{alertblock}{Symmetrization of a Relator} + Symmetrization of a relator $\relar$ is defined as: + \begin{gather*} + \hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} + \end{gather*} + \end{alertblock} + \begin{block}{Symmetric Relators} + A relator $\relar$ is symmetric if $\relar=\hat{\relar}$ + \end{block}\vspace{-0.5cm} + Egli-Milner is not symmetric. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% \column{.47\textwidth} +% \end{columns} +\end{frame} + +\begin{frame}{Barr Relators} +% \footnotesize +% \begin{columns}[T,onlytextwidth] + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% \column{.47\textwidth} +% \begin{alertblock}{Example: Trivial Relator} +% For an arbitrary set endofunctor $F$, the relator $\mathbf{Q}$ takes every relation $r:X\rto Y$ to $FX\times FY$. +% \end{alertblock} +% It is a symmetric relator. +% \begin{block}{} +% But it is a bad relator because every relation is a $\mathbf{Q}$-bisimulation! +% \end{block} +% \begin{alertblock}{Example: Egli-Milner Relator} +% The relator $\emre$ is a $\powf$-relator defined as: +% \begin{gather*} +% \emre r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\} +% \end{gather*} +% \end{alertblock} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \column{.47\textwidth} - We give a family of examples: +% \column{.47\textwidth} \begin{alertblock}{Example: Barr Relators} Given a set endofunctor $F$, and $r$ is a relation that $r=\pi_2\comp\pi_1^{-1}$ ($\pi_1$ and $\pi_2$ are functions) the Barr relator $\bar{F}$ is defined as $\bar{F}r=\bar{F}\pi_2\comp(\bar{F}\pi_1)^{-1}$. @@ -538,10 +577,13 @@ \begin{gather*} \hat{\emre}=\bar{\powf} \end{gather*} + \end{block}\vspace{-0.5cm} + Barr relators are nicely behaved relators. +% A trivial relator is a not a Barr relator. + \begin{block}{} + So, we always have bisimulations with Barr relators. \end{block} - Barr relators are nicely behaved relators.\\ - A trivial relator is a not a Barr relator. -\end{columns} +%\end{columns} \end{frame} \begin{frame}{Lax Barr relators} @@ -571,34 +613,55 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.47\textwidth} \begin{block}{} - There are multiple of them: \begin{itemize} \item Left-lax: $\appr\comp\bar{F}r$ \item Right-lax: $\bar{F}\comp\appr$ \item Bi-lax: $\appr\comp\bar{F}r\comp\appr$ \item Mid-lax: $F\pi_1\comp\appr\comp(F\pi_2)^{-1}$ \end{itemize} - \end{block} + \end{block}\vspace{-0.2cm} \begin{alertblock}{Liftability} An ordering on a set endofunctor $F$ is called \emph{liftable}, if\vspace{-0.3cm} \begin{gather*} - \alpha\appr\beta\Rightarrow\alpha\comp f\appr \beta\comp f,\\ - \alpha\appr\beta\Rightarrow Ff'\comp\alpha\appr Ff'\comp \beta,\\ h\appr Fg\comp k\Rightarrow \exists k', h=Fg\comp k\;\&\;k'\appr k. \end{gather*} - \end{alertblock} + \end{alertblock}\vspace{-0.2cm} + An ordering and liftablity can be defined for an arbitrary category as well. \end{columns} \end{frame} -\begin{frame}{Lax Barr relators (2)} +\begin{frame}{Symmetrization of Lax Barr relators} + \footnotesize + Having liftability allows us for an arbitrary function $g$ to have:\vspace{-0.2cm} +\begin{equation*} + \begin{tikzcd}[ampersand replacement=\&] + FX \& FX \\ + FY \& FY + \arrow["{\appr_X}"{inner sep=.8ex}, "\shortmid"{marking}, tail, from=1-1, to=1-2] + \arrow["Fg"', from=1-1, to=2-1] + \arrow["Fg", from=1-2, to=2-2] + \arrow["{\appr_Y}"'{inner sep=.8ex}, "\shortmid"{marking}, tail, from=2-1, to=2-2] + \end{tikzcd} +\end{equation*}\vspace{-0.3cm} +% \begin{gather*} +% Fg\comp\appr=\appr\comp Fg\\ +% (Fg)^{-1}\comp\sappr=\sappr\comp(Fg)^{-1} +% \end{gather*} \begin{alertblock}{Proposition} Given a set endofunctor $F$, with a liftable ordering - \begin{itemize} + \begin{enumerate} \item mid-lax and left-lax relators of $F$ are equal, \item Assuming the axiom of choice, righ-lax and left-lax relators of $F$ are equal, \item the symmetrizaion of the left-lax relator is \emph{sound} and \emph{complete} with respect to \emph{behavioural equivalence}. - \end{itemize} + \end{enumerate} \end{alertblock} + \vspace{-0.1cm} + \begin{block}{} + \begin{itemize} + \item Although, not every sound and complete relator is a Barr relator.\\ + \item We do not know when the symmetrizaion of a Barr relator is a Barr relator. + \end{itemize} + \end{block} \end{frame} \begin{frame}{Span-based definitions} @@ -630,9 +693,8 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.47\textwidth} - \vspace{0.1cm} - Assuming the axiom of choice, every HJ-bisimulation is an AM-bisimulation. - \begin{block}{} + \vspace{-0.2cm} + \begin{alertblock}{AM-Simulation} We have \emph{AM-simulation}, if we lax the diagram:\vspace{-0.4cm} \begin{equation*} \begin{tikzcd}[ampersand replacement=\&] @@ -649,14 +711,37 @@ \arrow["{{Fp_2}}"', from=2-2, to=2-3] \end{tikzcd} \end{equation*} - \end{block}\vspace{-0.3cm} - We have found a symmetric AM-simulation that is not a bisimulation! - \begin{alertblock}{Proposition} - Given a symmetric span $(R,X)$, with the symmetry evidence $s:R\to R$, assuming $R$ is an AM-simulation on a coalgebra $(X,\alpha)$ with evidence $\sigma$, then $R$ is an AM-bisimulation iff $s$ is a coalgebra endomorphism on $(R,\sigma)$. - \end{alertblock} + \end{alertblock}\vspace{-0.2cm} + \begin{block}{} + \begin{itemize} + \item Every AM-(bi)simulation is an HJ-(bi)simulation. + \item Assuming AoC every HJ-bisimulation is an AM-bisimulation. + \item Every HJ-bisimulation for a functor of form $\powf F$ in a topos is an AM-bisimulation. + \end{itemize} + \end{block} \end{columns} \end{frame} +\begin{frame}{Symmetric Simulation for Span-based Definitions} + \footnotesize + So far we have: + \begin{block}{} + \begin{itemize} + \item An example of an AM(HJ)-simulation that is not an AM(HJ)-bisimulation. + \item An example of an AM(HJ)-simulation that the witness for AM(HJ)-simulation does not serve as a witness for AM(HJ)-bisimulation for a $\powf$-coalgebra in $\Set$. + \end{itemize} + \end{block} + \begin{alertblock}{Proposition} + Given a symmetric span $(R,X)$, with the symmetry witness $s:R\to R$, assuming $R$ is an HJ-simulation on a coalgebra $(X,\alpha)$ with witness $\sigma$, then $R$ is an AM-bisimulation iff $s$ is a coalgebra endomorphism on $(R,\sigma)$. + \end{alertblock} + \begin{block}{} + \begin{itemize} + \item We have a proof that symmetric AM-simulations for coalgebras of $\powf$ and the maybe fucntor on $\Set$ are AM-bisimulation. + \item We want to know when exactly this proposition holds (perhaps liftability matters). + \end{itemize} + \end{block} +\end{frame} + \bibliographystyle{apalike} \bibliography{references} \addtocounter{framenumber}{-1} From d94b780ca320e32bea843fea5a9a9f90906c9267 Mon Sep 17 00:00:00 2001 From: partowp Date: Sat, 25 Jul 2026 17:19:22 +0100 Subject: [PATCH 11/19] slides --- ACV 2026/slides.tex | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/ACV 2026/slides.tex b/ACV 2026/slides.tex index 3057f6c..50d3f02 100644 --- a/ACV 2026/slides.tex +++ b/ACV 2026/slides.tex @@ -165,7 +165,7 @@ The content of the talk: \begin{itemize} \item Intuitive introduction to coalgebra and (bi)simulation - \item Reviewing basic definitions of coalgebra and (bi)simulation + \item Reviewing definitions of coalgebra and (bi)simulation \item Our motivation: To ease proving program equivalence \item Relator-based notions \item Span-based notions @@ -177,14 +177,14 @@ When can we say that two systems \emph{behave the same}? \end{alertblock} \vspace{-0.4cm} - This question is everywhere, often silently assumed:\vspace{-0.4cm} + This question is everywhere, for example:\vspace{-0.4cm} \begin{itemize} \setlength{\itemsep}{6pt} - \item[\faServer] \textbf{Hardware redesign.} A chip redesigned for cheaper production must still compute exactly what the old one did. + \item[\faServer] \textbf{Hardware redesign.} A chip redesigned for a cheaper price must still compute exactly what the old one did. \item[\faCode] \textbf{Software upgrades.} A bank replaces its backend --- from the customer's view (balances, transactions), nothing should change. \end{itemize} \vspace{0.2cm} - \begin{block}{The catch} + \begin{block}{The point} ``Behaving the same'' is intuitive, but making it \emph{precise} and \emph{checkable} is surprisingly subtle. \end{block} \end{frame} @@ -453,8 +453,8 @@ -The method, applies a closure on the bisimilarity relation on $T$.\\ -Then one should prove that the result is a simulation.\\ -The closure preserves symmetry, and symmetric simulation is a bisimulation in traditional definitions.\\ - -We have found out that it is not always the case.\\ - -That is why we want to know when exactly a symmetric simulation is a bisimulation. + -We need to know how to generally derive bisimulation from simulation.\\ + -And what is the impact of choosing specific notion of simulation (span-based, relation-based)? \end{frame} \begin{frame}{Relators and Simulations} @@ -465,7 +465,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \column{.47\textwidth} - To define coalgebraic simulation, relators are invented. + Relators unify simulation and bisimulation. \begin{alertblock}{Relator} Assuming $F:\Set\to\Set$ is a functor, an $F$-relator a monotone map that sends a morphism of $\rel$ that is a relation $X\rto Y$ to a relation $FX\rto FY$. \end{alertblock} @@ -533,7 +533,7 @@ \begin{alertblock}{Symmetrization of a Relator} Symmetrization of a relator $\relar$ is defined as: \begin{gather*} - \hat{\relar}=\relar r\cap (\relar r^{-1})^{-1} + \hat{\relar}r=\relar r\cap (\relar r^{-1})^{-1} \end{gather*} \end{alertblock} \begin{block}{Symmetric Relators} @@ -658,8 +658,8 @@ \vspace{-0.1cm} \begin{block}{} \begin{itemize} - \item Although, not every sound and complete relator is a Barr relator.\\ - \item We do not know when the symmetrizaion of a Barr relator is a Barr relator. + \item Although, not every sound and complete relator is a Barr relator!\\ + \item We do not know when Clause 3 does not produce a Barr relator. \end{itemize} \end{block} \end{frame} From 1123f4281058992cd5b80f5b4600e1dfdf064328 Mon Sep 17 00:00:00 2001 From: partowp Date: Sat, 25 Jul 2026 17:24:45 +0100 Subject: [PATCH 12/19] prop --- draft/draft.tex | 3 +++ 1 file changed, 3 insertions(+) diff --git a/draft/draft.tex b/draft/draft.tex index 7ace02f..8dee0bc 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -3336,6 +3336,9 @@ Perhaps if we can relax the definition of liftable by allowing $g$ to be a relat \todo{Write it down. You have it in your notes.} \end{proof} \todo{Try $FX=\powf(X^2)$ to see if the symmetrization of its lax Barr relator is a Barr relator. The order is just the set inclusion. See if~\autoref{prop:lax-relator-full-comm} or~\autoref{lem:lax-relator-str} can help!} +\begin{prop} +Assuming that $r$ is a symmetric relation, and it is an $\relar$-simulation on a coalgebra $(X,\alpha)$, then $r$ is a $\hat{\relar}$-bisimulation. +\end{prop} \end{document} From 89aeaf7461fd20b9d916c8a5b60b67800d16188c Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 28 Jul 2026 12:22:02 +0100 Subject: [PATCH 13/19] the prop --- draft/draft.tex | 13 ++++++++++++- 1 file changed, 12 insertions(+), 1 deletion(-) diff --git a/draft/draft.tex b/draft/draft.tex index 8dee0bc..6540bab 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -3337,8 +3337,19 @@ Perhaps if we can relax the definition of liftable by allowing $g$ to be a relat \end{proof} \todo{Try $FX=\powf(X^2)$ to see if the symmetrization of its lax Barr relator is a Barr relator. The order is just the set inclusion. See if~\autoref{prop:lax-relator-full-comm} or~\autoref{lem:lax-relator-str} can help!} \begin{prop} -Assuming that $r$ is a symmetric relation, and it is an $\relar$-simulation on a coalgebra $(X,\alpha)$, then $r$ is a $\hat{\relar}$-bisimulation. +Assuming that $r$ is a symmetric relation, and it is an $\relar$-simulation on a coalgebra $(X,\alpha)$, then $r$ is an $\hat{\relar}$-bisimulation. \end{prop} +\begin{proof} + $r$ being an $\relar$-simulation means that $r\leq \alpha^\op\comp\relar r\comp\alpha$, and $r$ being symmetric means that $r^\op=r$. So, we need to prove that $r\leq\alpha^\op\comp\hat{\relar} r\comp\alpha$. We have: + \begin{align*} + x \mathrel{(\alpha^\op\comp\hat{\relar} r\comp\alpha)} y&\\ + \iff &x\mathrel{(\alpha^\op\comp(\relar r\cap (\relar r)^\op)\comp\alpha)} y\\ + \iff &\exists x',y',\quad x\mathrel{\alpha}x'\quad\&\quad y\mathrel{\alpha}y'\quad\&\quad x'\mathrel{(\relar r\cap (\relar r)^\op)}y'\\ + \iff &\exists x',y',\quad x\mathrel{\alpha}x'\quad\&\quad y\mathrel{\alpha}y'\quad\&\quad x'\mathrel{\relar r}y'\quad\&\quad x'\mathrel{(\relar r)^\op}y'\\ + \iff &\exists x',y',\quad x\mathrel{\alpha}x'\quad\&\quad y\mathrel{\alpha}y'\quad\&\quad x'\mathrel{\relar r}y'\quad\&\quad y'\mathrel{\relar r}x' + \end{align*} + Now, assuming $x\mathrel{r}y$ gives us $x\mathrel{(\alpha^\op\comp\relar r\comp \alpha)} y$ that is equivalent with saying that exist $x'$ and $y'$ such that $x\mathrel{\alpha}x'$, $y\mathrel{\alpha}y'$, and $x'\mathrel{\relar r}y'$. Since $r$ is symmetric, we have $y\mathrel{r} x$ that means that exist $x''$ and $y''$ such that $x\mathrel{\alpha}x''$, $y\mathrel{\alpha}y''$, and $y''\mathrel{\relar r}x''$. On the other hand since $\alpha$ is a function, we have $x''=x'$ and $y''=y'$, so we have $y'\mathrel{\relar r}x'$ that ultimately gives $x\mathrel{(\alpha^\op\comp\hat{\relar}r\comp\alpha)}y$. So, $r$ is an $\hat{\relar}$-bisimulation as well.\qed +\end{proof} \end{document} From e4ac0c6797b1c9e0a407f12fdf15b0ba9cd82c55 Mon Sep 17 00:00:00 2001 From: partowp Date: Tue, 28 Jul 2026 15:27:41 +0100 Subject: [PATCH 14/19] a prop --- draft/draft.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 6540bab..94090c5 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -3191,13 +3191,13 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i They all follow in an obvious way from~\autoref{lem:liftable} and~\autoref{lem:coliftable}. The last one needs $\appr\comp\appr=\appr$ that comes from transitivity of $\appr$. \qed \end{proof} \begin{prop}\label{prop:lax-relator-full-comm} - Assuming the axiom of choice, for a functor $F$ with a liftable order we have: + For a functor $F$ with a liftable order we have: \begin{gather*} - \bar{F}r\comp\appr=\appr\comp\bar{F} + (\bar{F}r\comp\appr)\subseteq(\appr\comp\bar{F}r) \end{gather*} \end{prop} \begin{proof} - \todo{Write it down. You have it in your notes.} + Assuming $x \mathrel{(\bar{F}r\comp\appr)} y$ and $\bar{F}r=F\pi_2\comp(F\pi_1)^\op$, then there exist $x'$ and $p$ such that $x\appr x'$, $p\mathrel{F\pi_1}x'$, and $p\mathrel{F\pi_2}y$. So, we have $x\appr F\pi_1(p)$ then by liftability there exists $p'$ such that $p'\appr p$, and $F\pi_1(p')=x$. Then from $p'\appr p$ we get $F\pi_2(p')\appr y$ that is equivalent with $p' \mathrel{(\appr\comp F\pi_2)} y$, and then we have $x \mathrel{(\appr\comp F\pi_2\comp(F\pi_1)^\op)} y$.\qed \end{proof} %\begin{example} % In the category of sets, subset over the powerset functor is an example of a liftable order. Using~\autoref{lem:set-ord-str} we only prove the case for every $h\in\Hom(1,\powf Z)$, $k\in\Hom(1,\powf Y)$. Additionally, for every $g\c Y\to Z$, such that $h\subseteq\powf g(k)$, we define $k'\in\powf Y$ that that $k'=\{y\mid g(y)\in h\}$. We show that $\powf g(k')=h$. From 3090215662d07dd84fc263f2a272035fd2c633a8 Mon Sep 17 00:00:00 2001 From: partowp Date: Fri, 31 Jul 2026 15:57:22 +0100 Subject: [PATCH 15/19] list --- list | 7 +++++++ 1 file changed, 7 insertions(+) create mode 100644 list diff --git a/list b/list new file mode 100644 index 0000000..580c908 --- /dev/null +++ b/list @@ -0,0 +1,7 @@ +Introduce HJ-simulation in relator-form, show uniqness of wittnesses. +Prove a general theorem that symmetric simulation for symmetrized relator yields similarity that is sound and complete for behavioral equivalence (use "Relators and Notions of Simulation Revisited" soundness and compteleness criterion). +Elaborate this for powerset +When is symmetrized relator the Barr relator (definition is in "Relators and Notions of Simulation Revisited")? +At least, try to prove it for the Jacobs-Hughes relator +Introduce notions of simulation, diverging from HJ-simulation. Do we need two of them (with normalization and without)? How are all of them related? +Can we separate Hughes-Jacobs relator from one-sided lax relator? From f7df0d21ea3ff26cb8974e227f20a81395b41c87 Mon Sep 17 00:00:00 2001 From: partowp Date: Fri, 31 Jul 2026 15:58:02 +0100 Subject: [PATCH 16/19] liftablity --- draft/draft.tex | 24 ++++++++++++++++++++++++ 1 file changed, 24 insertions(+) diff --git a/draft/draft.tex b/draft/draft.tex index 94090c5..de54c94 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -304,6 +304,7 @@ }% }% } +\newcommand{\sub}{\mathcal{S}} \newcommand{\bba}{ @@ -474,6 +475,19 @@ We want to say that a natural order structure entails an order over a functor de We could not prove a general statement like~\autoref{prop:lift-gen-func} for coliftability even if the surgectivity of $g$ remains in the definition. Perhaps more conditions on $F$ in~\autoref{prop:lift-gen-func} are needed. Conditions like preserving surjections and unions that make the statement extremely limited, and meaningless. \end{remark} +\begin{example} + The subdistribution functor $\sub\c\Set\to\Set$ is defined as $\sub X=\{\mu\c X\to[0,1]\mid \sum_{x\in X}\mu(x)\}$ on objects, and for $\sub f\c \sub X\to\sub Y$, we have + \begin{gather*} + \sub f(\mu)=y\mapsto\sum_{x\in f^{\mone}(y)}\mu(x), + \end{gather*} + on morphisms. We define the ordering on $\sub X$ for $\mu_1,\mu_2\in \sub X$ as + \begin{gather*} + \mu_1\appr \mu_2\iff \forall x\in X, \mu_1(x)\leq\mu_2(x). + \end{gather*} + The definition derives the ordering on morphisms of each hom-set $\hom(X,\sub Y)$ as usual in $\Set$.\\ + Assuming $\mu\in\sub X$, $g\c X\to Y$, $\nu\in\sub Y$, and $\nu\appr \sub g(\mu)$, we need to prove that there exists $\mu'\in \sub X$, that $\mu'\appr\mu$ and $\nu=\sub g(\mu')$. +\end{example} + \begin{lemma}\label{lem:liftable} Assuming that a a functor $F$ has an order structure $\appr$ that is liftable, then for every $f\in\Hom(X,FY)$ we have: \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] @@ -3037,6 +3051,12 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i $\hat{L}$ is a Barr relator. \end{prop} \begin{proof} + We have + \begin{gather*} + \hat{\emre}r=\emre r\cap (\emre r^\op)^\op,\\ + \hat{\emre}r=\{(S,T)\mid x\in S\Rightarrow \exists y\in T, x\;r\;y\}\cap\{(T,S)\mid x\in S\Rightarrow \exists y\in T, y\;r\;x\}. + \end{gather*} + Assuming that $r=\pi_2\comp(\pi_1)^\op$ we have to prove that $\hat{L}r=\powf\pi_2\comp(\powf\pi_1)^\op$. \todo{Finish.} \end{proof} @@ -3199,6 +3219,9 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \begin{proof} Assuming $x \mathrel{(\bar{F}r\comp\appr)} y$ and $\bar{F}r=F\pi_2\comp(F\pi_1)^\op$, then there exist $x'$ and $p$ such that $x\appr x'$, $p\mathrel{F\pi_1}x'$, and $p\mathrel{F\pi_2}y$. So, we have $x\appr F\pi_1(p)$ then by liftability there exists $p'$ such that $p'\appr p$, and $F\pi_1(p')=x$. Then from $p'\appr p$ we get $F\pi_2(p')\appr y$ that is equivalent with $p' \mathrel{(\appr\comp F\pi_2)} y$, and then we have $x \mathrel{(\appr\comp F\pi_2\comp(F\pi_1)^\op)} y$.\qed \end{proof} +\begin{cor} + Assuming that $r$ is an $(\bar{F}r\comp\appr)$-simulation, then it is an $(\appr\comp\bar{F}r)$-simulation as well. +\end{cor} %\begin{example} % In the category of sets, subset over the powerset functor is an example of a liftable order. Using~\autoref{lem:set-ord-str} we only prove the case for every $h\in\Hom(1,\powf Z)$, $k\in\Hom(1,\powf Y)$. Additionally, for every $g\c Y\to Z$, such that $h\subseteq\powf g(k)$, we define $k'\in\powf Y$ that that $k'=\{y\mid g(y)\in h\}$. We show that $\powf g(k')=h$. % @@ -3336,6 +3359,7 @@ Perhaps if we can relax the definition of liftable by allowing $g$ to be a relat \todo{Write it down. You have it in your notes.} \end{proof} \todo{Try $FX=\powf(X^2)$ to see if the symmetrization of its lax Barr relator is a Barr relator. The order is just the set inclusion. See if~\autoref{prop:lax-relator-full-comm} or~\autoref{lem:lax-relator-str} can help!} +\subsection{Symmetric relation} \begin{prop} Assuming that $r$ is a symmetric relation, and it is an $\relar$-simulation on a coalgebra $(X,\alpha)$, then $r$ is an $\hat{\relar}$-bisimulation. \end{prop} From 99d6862e1271988a08f95c34afc8075c913f9942 Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 3 Aug 2026 20:16:39 +0100 Subject: [PATCH 17/19] liftable --- draft/draft.tex | 216 ++++++++++++++++++++++++++++++------------------ 1 file changed, 134 insertions(+), 82 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index de54c94..861be01 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -406,12 +406,12 @@ We want to say that a natural order structure entails an order over a functor de \end{proof} \begin{definition}[Liftable Order Structure]\label{def:liftable-ord} - A \emph{liftable order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and if $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $h\appr Fg\comp k$ in $\Hom(X,FZ)$, then there is $k'\c X\to FY$ such that $k'\appr k$ in $\Hom(X,FY)$ and $h=Fg\comp k'$. + A \emph{liftable order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ if $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $h\appr Fg\comp k$ in $\Hom(X,FZ)$, and $g$ is surjective, then there is $k'\c X\to FY$ such that $k'\appr k$ in $\Hom(X,FY)$ and $h=Fg\comp k'$.\ppnote{I added the surjection condition on $g$.} \end{definition} \begin{definition}[Coliftable Order Structure]\label{def:coliftable-ord} - A \emph{coliftable order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ that is a natural order structure, and if $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $Fg\comp k\appr h $ in $\Hom(X,FZ)$, and $g$ is surjective, then there is $k'\c X\to FY$ such that $k\appr k'$ in $\Hom(X,FY)$ and $h=Fg\comp k'$. + A \emph{coliftable order structure} on a functor $F$ is a preorder $\appr$ on each Hom-set of the form $\Hom(X,FY)$ if $h\c X\to FZ$, $k\c X\to FY$, $g\c Y\to Z$, $Fg\comp k\appr h $ in $\Hom(X,FZ)$, and $g$ is surjective, then there is $k'\c X\to FY$ such that $k\appr k'$ in $\Hom(X,FY)$ and $h=Fg\comp k'$. \end{definition} \begin{lemma}\label{lem:set-ord-str} @@ -429,6 +429,47 @@ We want to say that a natural order structure entails an order over a functor de The exact same argument in~\autoref{lem:set-ord-str} is true for coliftable order structures. \end{remark} +\begin{lemma}\label{lem:liftable} + Assuming that a a functor $F$ has an order structure $\appr$ that is liftable, then for every surjective $f\in\Hom(X,FY)$ we have: + \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] + \item $Ff\comp\sappr\quad=\quad\sappr\comp Ff$ + \item $(Ff)^\op\comp\appr\quad=\quad\appr\comp (Ff)^\op$ + \end{enumerate} +\end{lemma} +\begin{proof} + $(I)$ Assuming $t\mathrel{(Ff\comp\sappr)} x$, there exists $s$ such that $t\sappr s$ and $Ff(s)=x$. Since $\appr$ is liftable, and thus natural, by~\autoref{def:nat-ord}, from $s\appr t$ we get $x\appr Ff(t)$ that is $t\mathrel{(\sappr\comp Ff)} x$. + + Assuming $t\mathrel{(\sappr\comp Ff)} x$, there exists $y$ such that $Ff(t)=y$ and $y\sappr x$. By~\autoref{def:liftable-ord} since $Ff(t)\sappr x$ there exists $s$ that $t\sappr s$ and $Ff(s)=x$ that is $t\mathrel{(Ff\comp\appr)}s$. + + $(II)$ Basically, by definition of $\op$ and relation composition we have + \begin{gather*} + (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ + (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + \end{gather*} + So it follows directly from applying $\op$ on both sides of $(I)$.\qed +\end{proof} +\begin{lemma}\label{lem:coliftable} + Assuming that a a functor $F$ has an order structure $\appr$ that is coliftable, then for every surjective $f\in\Hom(X,FY)$ we have: + \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] + \item $Ff\comp\appr\quad=\quad\appr\comp Ff$ + \item $(Ff)^\op\comp\sappr\quad=\quad\sappr\comp (Ff)^\op$ + \end{enumerate} +\end{lemma} +\begin{proof} + $(I)$ Assuming $t\mathrel{(Ff\comp\appr)} x$, there exists $s$ such that $t\appr s$ and $Ff(s)=x$. Since $\appr$ is coliftable, and thus natural, by~\autoref{def:nat-ord}, from $t\appr s$ we get $Ff(t)\appr x$ that is $t\mathrel{(\appr\comp Ff)} x$. + + Assuming $t\mathrel{(\appr\comp Ff)} x$, there exists $y$ such that $Ff(t)=y$ and $y\appr x$. By~\autoref{def:coliftable-ord} since $Ff(t)\appr x$ there exists $s$ that $t\appr s$ and $Ff(s)=x$ that is $t\mathrel{(Ff\comp\appr)}s$. + + $(II)$ Basically, by definition of $\op$ and relation composition we have + \begin{gather*} + (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ + (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + \end{gather*} + So it follows directly from applying $\op$ on both sides of $(I)$.\qed +\end{proof} + +\subsection{Powerset Functor} +In this section we discuss set inclusion as an ordering over the powerset functor. \begin{prop}\label{prop:lift-gen-func} For a functor $F\c\Set\to\Set$, a functor of the form $\powf F$, where $\powf$ is the powerset functor, the set inclusion is a liftable order structure. \end{prop} @@ -475,56 +516,98 @@ We want to say that a natural order structure entails an order over a functor de We could not prove a general statement like~\autoref{prop:lift-gen-func} for coliftability even if the surgectivity of $g$ remains in the definition. Perhaps more conditions on $F$ in~\autoref{prop:lift-gen-func} are needed. Conditions like preserving surjections and unions that make the statement extremely limited, and meaningless. \end{remark} -\begin{example} - The subdistribution functor $\sub\c\Set\to\Set$ is defined as $\sub X=\{\mu\c X\to[0,1]\mid \sum_{x\in X}\mu(x)\}$ on objects, and for $\sub f\c \sub X\to\sub Y$, we have - \begin{gather*} - \sub f(\mu)=y\mapsto\sum_{x\in f^{\mone}(y)}\mu(x), - \end{gather*} - on morphisms. We define the ordering on $\sub X$ for $\mu_1,\mu_2\in \sub X$ as - \begin{gather*} - \mu_1\appr \mu_2\iff \forall x\in X, \mu_1(x)\leq\mu_2(x). - \end{gather*} - The definition derives the ordering on morphisms of each hom-set $\hom(X,\sub Y)$ as usual in $\Set$.\\ - Assuming $\mu\in\sub X$, $g\c X\to Y$, $\nu\in\sub Y$, and $\nu\appr \sub g(\mu)$, we need to prove that there exists $\mu'\in \sub X$, that $\mu'\appr\mu$ and $\nu=\sub g(\mu')$. -\end{example} - -\begin{lemma}\label{lem:liftable} - Assuming that a a functor $F$ has an order structure $\appr$ that is liftable, then for every $f\in\Hom(X,FY)$ we have: - \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] - \item $Ff\comp\sappr\quad=\quad\sappr\comp Ff$ - \item $(Ff)^\op\comp\appr\quad=\quad\appr\comp (Ff)^\op$ - \end{enumerate} -\end{lemma} +\subsection{Maybe Functor} +The order structure that we can define for this functor is that for sets $X$ and $Y$, and functions $f,g\c X\to Y+1$ we have $f\appr g$ whenever $\Dom(f)\subseteq\Dom(g)$, and for every $x\in\Dom(f)$ we have $f(x)=g(x)$ ($\Dom(f)$ is the domain of a function $f$). +\begin{prop}\label{prop:maybe-lif} + The order structure on the set-functor $FX=X+1$ is a liftable order. +\end{prop} +% By~\autoref{lem:set-ord-str}, assuming $h\in\Hom(1,Y+1)$, $k\in\Hom(1,X+1)$, $g\c X\to Y$, and $h\appr Fg(k)$, we need to prove that exists $k'\in\Hom(1,X+1)$ such that $k'\appr k$ and $Fg(k')=h$. Since $h\in\Hom(1,Y+1)$ we have two cases: +% \begin{itemize} + % \item $h=\bot$: In this case we take $k'=\bot$, so we have $k'\appr k$, and then we have $Fg(k')=\bot=h$. + % \item $h\in Y$: Since $h\appr Fg(k)$ and $h\neq\bot$, we have $h=Fg(k)$. In this case we take $k'=k$, so $k'\appr k$ and $Fg(k')=h$.\qed + % \end{itemize} \begin{proof} - $(I)$ Assuming $t\mathrel{(Ff\comp\sappr)} x$, there exists $s$ such that $t\sappr s$ and $Ff(s)=x$. Since $\appr$ is liftable, and thus natural, by~\autoref{def:nat-ord}, from $s\appr t$ we get $x\appr Ff(t)$ that is $t\mathrel{(\sappr\comp Ff)} x$. - - Assuming $t\mathrel{(\sappr\comp Ff)} x$, there exists $y$ such that $Ff(t)=y$ and $y\sappr x$. By~\autoref{def:liftable-ord} since $Ff(t)\sappr x$ there exists $s$ that $t\sappr s$ and $Ff(s)=x$ that is $t\mathrel{(Ff\comp\appr)}s$. - - $(II)$ Basically, by definition of $\op$ and relation composition we have + Assuming that $h\in\Hom(X,Z+1)$, $g\c Y\to Z$, $k\in\Hom(X,Y+1)$, such that $h\appr (g+1)\comp k$. We define $k'\in\Hom(X,Y+1)$ as follows: \begin{gather*} - (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ - (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + k'(x)= + \begin{cases} + \bot&x\notin\Dom(h)\\ + k(x)&x\in\Dom(h) + \end{cases} \end{gather*} - So it follows directly from applying $\op$ on both sides of $(I)$.\qed + We have $\Dom((g+1)\comp k')=\Dom(k')$ and $\Dom(k')=\Dom(h)$, so we have $\Dom((g+1)\comp k')=\Dom(h)$. If $x\in\Dom(h)$, then $h(x)=(g+1)\comp k(x)$ and $(g+1)\comp k(x)=(g+1)\comp k'(x)$, so we have $h(x)=(g+1)\comp k'(x)$. So, we have $h=(g+1)\comp k'$. Additionally, $\Dom(k')\subseteq\Dom(k)$, and for every $x\in \Dom(k')$, we have $k'(x)=k(x)$, so we have $k'\appr k$. \qed \end{proof} -\begin{lemma}\label{lem:coliftable} - Assuming that a a functor $F$ has an order structure $\appr$ that is coliftable, then for every surjective $f\in\Hom(X,FY)$ we have: - \begin{enumerate}[label=(\Roman*), ref=(\Roman*)] - \item $Ff\comp\appr\quad=\quad\appr\comp Ff$ - \item $(Ff)^\op\comp\sappr\quad=\quad\sappr\comp (Ff)^\op$ - \end{enumerate} -\end{lemma} +\begin{prop}\label{prop:maybe-colif} + The order structure on the set-functor $FX=X+1$ is a coliftable order. +\end{prop} \begin{proof} - $(I)$ Assuming $t\mathrel{(Ff\comp\appr)} x$, there exists $s$ such that $t\appr s$ and $Ff(s)=x$. Since $\appr$ is coliftable, and thus natural, by~\autoref{def:nat-ord}, from $t\appr s$ we get $Ff(t)\appr x$ that is $t\mathrel{(\appr\comp Ff)} x$. - - Assuming $t\mathrel{(\appr\comp Ff)} x$, there exists $y$ such that $Ff(t)=y$ and $y\appr x$. By~\autoref{def:coliftable-ord} since $Ff(t)\appr x$ there exists $s$ that $t\appr s$ and $Ff(s)=x$ that is $t\mathrel{(Ff\comp\appr)}s$. - - $(II)$ Basically, by definition of $\op$ and relation composition we have + By~\autoref{lem:set-ord-str}, assuming $h\in\Hom(1,Y+1)$, $k\in\Hom(1,X+1)$, $g\c X\to Y$, and $Fg(k)\appr h$, we need to prove that exists $k'\in\Hom(1,X+1)$ such that $k\appr k'$ and $Fg(k')=h$. Since $h\in\Hom(1,Y+1)$ we have two cases: + \begin{itemize} + \item $Fg(k)=h$: In this case we take $k'=k$, and then we have $Fg(k')=h$. + + \item $Fg(k)=\bot$: It entails that $k=\bot$. We either have $h=\bot$ or $h\in Y$. If $h=\bot$ then we take $k'=k$, and we are done. If $h\in Y$, then by the surjectivity of $g$, there exists $k'$ such that $g(k')=h$ that entails $Fg(k')=h$ as well.\qed + \end{itemize} +\end{proof} + +\subsection{Subdistribution Functor} +The subdistribution functor $\sub\c\Set\to\Set$ is defined as $\sub X=\{\mu\c X\to[0,1]\mid \sum_{x\in X}\mu(x)\}$ on objects, and for $\sub f\c \sub X\to\sub Y$, we have +\begin{gather*} + \sub f(\mu)=y\mapsto\sum_{x\in f^{\mone}(y)}\mu(x), +\end{gather*} +on morphisms. We define the ordering on $\sub X$ for $\mu_1,\mu_2\in \sub X$ as +\begin{gather*} + \mu_1\appr \mu_2\iff \forall x\in X, \mu_1(x)\leq\mu_2(x). +\end{gather*} +The definition derives the ordering on morphisms of each hom-set $\Hom(X,\sub Y)$ as usual in $\Set$. +\begin{prop} + The pointwise ordering on $\sub$ is a liftable ordering. +\end{prop} +\begin{proof} + Assuming that $\mu\in SX$, $g\c X\to Y$ is surjective, $\nu\in SY$, and $\nu\appr \sub g(\mu)$, then there exists $\mu'\in \sub X$ such that $\mu'\appr \mu$ and $\sub g(\mu')=h$.\\ + The assumption $\nu \appr Sg(\mu)$ means that for every $y \in Y$, \begin{gather*} - (Ff\comp\appr)^\op=\sappr\comp(Ff)^\op,\\ - (\appr\comp Ff)^\op=(Ff)^\op\comp\sappr. + \nu(y) \leq Sg(\mu)(y) = \sum_{x \in g^{\mone}(y)} \mu(x). \end{gather*} - So it follows directly from applying $\op$ on both sides of $(I)$.\qed + We construct $\mu'$ as follows: + \begin{gather*} + \mu'(x)= + \begin{cases} + \frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)&Sg(\mu)(g(x))\neq 0\\ + 0&Sg(k)(g(x))= 0 + \end{cases} + \end{gather*} + First, we need to prove that $\mu'$ is a subdistribution. + For every $x \in X$, $\mu'(x) \ge 0$. The total mass of $\mu'$ is + \begin{align*} + \sum_{x \in X} \mu'(x)&\\ + =& \sum_{x \in X} \frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)\\ + =& \sum_{y \in Y} \sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)\\ + =& \sum_{y \in Y} \nu(y)\\ + \leq& 1 + \end{align*} + Thus $\mu'$ is a subdistribution. Now, we prove $Sg(\mu') = \nu$. + For any $y \in Y$, we have: + \begin{align*} + Sg(\mu')(y)&\\ + = &\sum_{x \in g^{\mone}(y)} \mu'(x)\\ + = &\sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)\;\;(Sg(\mu)(y)\neq 0)\\ + = &\frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)\;\;(Sg(\mu)(y)\neq 0)\\ + = &\frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)\;\;(Sg(\mu)(y)\neq 0)\\ + = &\mu(y) + \end{align*} + (If $Sg(\mu)(y) = 0$, then $\nu(y) = 0$, and the sum is $0$ as well, so the equality still holds.) \\ + Now, we are left to prove that $\mu'\appr\mu$. + For every $x\in X$, have the following cases: + \begin{itemize} + \item $Sg(\mu)(g(x))= 0$: We have $\mu'(x)=0$, and obviously $\mu'\appr\mu$. + \item $Sg(\mu)(g(x))\neq 0$: We have $\frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)$, and since $\nu\appr Sg(\mu)$ then we have: + \begin{align*} + &\quad\frac{\nu(g(x))}{Sg(\mu)(g(x))}\leq 1\\ + \Rightarrow&\quad\frac{\nu(g(x))}{Sg(\mu)(g(x))}\comp \mu(x)\leq \mu(x) + \end{align*}\qed + \end{itemize} \end{proof} @@ -2400,39 +2483,8 @@ Now, we prove our main statement. \end{cor} Now, we make the proof more abstract. We prove the statement for set-functors of the form $\powf F$, where $F$ is an arbitrary set-functor, and $\powf$ is the powerset functor. \subsection{Maybe Functor} -We prove that symmetric simulation is a bisimulation for the case that $FX=X+1$. First, we prove it for $\Set$. The order structure that we can define for this functor is that for sets $X$ and $Y$, and functions $f,g\c X\to Y+1$ we have $f\appr g$ whenever $\Dom(f)\subseteq\Dom(g)$, and for every $x\in\Dom(f)$ we have $f(x)=g(x)$ ($\Dom(f)$ is the domain of a function $f$). -Now, we prove that the given order on the maybe functor is liftable and coliftable. -\begin{lemma}\label{lem:maybe-lif} - The order structure on the set-functor $FX=X+1$ is a liftable order. -\end{lemma} -\begin{proof} -% By~\autoref{lem:set-ord-str}, assuming $h\in\Hom(1,Y+1)$, $k\in\Hom(1,X+1)$, $g\c X\to Y$, and $h\appr Fg(k)$, we need to prove that exists $k'\in\Hom(1,X+1)$ such that $k'\appr k$ and $Fg(k')=h$. Since $h\in\Hom(1,Y+1)$ we have two cases: -% \begin{itemize} -% \item $h=\bot$: In this case we take $k'=\bot$, so we have $k'\appr k$, and then we have $Fg(k')=\bot=h$. -% \item $h\in Y$: Since $h\appr Fg(k)$ and $h\neq\bot$, we have $h=Fg(k)$. In this case we take $k'=k$, so $k'\appr k$ and $Fg(k')=h$.\qed -% \end{itemize} -Assuming that $h\in\Hom(X,Z+1)$, $g\c Y\to Z$, $k\in\Hom(X,Y+1)$, such that $h\appr (g+1)\comp k$. We define $k'\in\Hom(X,Y+1)$ as follows: -\begin{gather*} - k'(x)= - \begin{cases} - \bot&x\notin\Dom(h)\\ - k(x)&x\in\Dom(h) - \end{cases} -\end{gather*} -We have $\Dom((g+1)\comp k')=\Dom(k')$ and $\Dom(k')=\Dom(h)$, so we have $\Dom((g+1)\comp k')=\Dom(h)$. If $x\in\Dom(h)$, then $h(x)=(g+1)\comp k(x)$ and $(g+1)\comp k(x)=(g+1)\comp k'(x)$, so we have $h(x)=(g+1)\comp k'(x)$. So, we have $h=(g+1)\comp k'$. Additionally, $\Dom(k')\subseteq\Dom(k)$, and for every $x\in \Dom(k')$, we have $k'(x)=k(x)$, so we have $k'\appr k$. \qed -\end{proof} -\begin{lemma}\label{lem:maybe-colif} - The order structure on the set-functor $FX=X+1$ is a coliftable order. -\end{lemma} -\begin{proof} - By~\autoref{lem:set-ord-str}, assuming $h\in\Hom(1,Y+1)$, $k\in\Hom(1,X+1)$, $g\c X\to Y$, and $Fg(k)\appr h$, we need to prove that exists $k'\in\Hom(1,X+1)$ such that $k\appr k'$ and $Fg(k')=h$. Since $h\in\Hom(1,Y+1)$ we have two cases: - \begin{itemize} - \item $Fg(k)=h$: In this case we take $k'=k$, and then we have $Fg(k')=h$. - - \item $Fg(k)=\bot$: It entails that $k=\bot$. We either have $h=\bot$ or $h\in Y$. If $h=\bot$ then we take $k'=k$, and we are done. If $h\in Y$, then by the surjectivity of $g$, there exists $k'$ such that $g(k')=h$ that entails $Fg(k')=h$ as well.\qed - \end{itemize} -\end{proof} -So, proven by Dubut, for every AM-simulation relation over a coalgebra $(X,\alpha)$ of a functor with a liftable order, we have a witness $\sigma\c R\to R+1$ such that $\alpha\comp p_1=Fp_1\comp\sigma$. +We prove that symmetric simulation is a bisimulation for the case that $FX=X+1$. First, we prove it for $\Set$.\\ +Proven by Dubut, for every AM-simulation relation over a coalgebra $(X,\alpha)$ of a functor with a liftable order, we have a witness $\sigma\c R\to R+1$ such that $\alpha\comp p_1=Fp_1\comp\sigma$. We have shown that the ordering on the maybe functor is liftable and coliftalbe in~\autoref{prop:maybe-lif} and~\autoref{prop:maybe-colif} %\begin{lemma}\label{lem:maybe-func-set} % Assuming that $R$ is a symmetric AM-simulation over an $F$-coalgebra $(X,\alpha)$ that $FX=X+1$, then for every $(x_1,x_2)\in R$ % \begin{gather*} @@ -2441,7 +2493,7 @@ So, proven by Dubut, for every AM-simulation relation over a coalgebra $(X,\alph % \end{gather*} %\end{lemma} %\begin{proof} -% By~\autoref{prop:alph-prod-dubut} and~\autoref{lem:maybe-lif} there exists $\sigma\c R\to R+1$ that is a witness for $R$ to be an AM-simulation, and $Fp_1\comp\sigma=\alpha\comp p_1$. %Since $R$ is symmetric, for every $(x_1,x_2)\in R$ we have the following\sgnote{Consider noting which facts come from the pair $(x_1,x_2)$ (namely \eqref{eq:maybe-func-set-1} and \eqref{eq:maybe-func-set-4}) and which from $(x_2,x_1)$ (namely \eqref{eq:maybe-func-set-2} and \eqref{eq:maybe-func-set-3}); it saves the reader from reconstructing it.}: +% By~\autoref{prop:alph-prod-dubut} and~\autoref{prop:maybe-lif} there exists $\sigma\c R\to R+1$ that is a witness for $R$ to be an AM-simulation, and $Fp_1\comp\sigma=\alpha\comp p_1$. %Since $R$ is symmetric, for every $(x_1,x_2)\in R$ we have the following\sgnote{Consider noting which facts come from the pair $(x_1,x_2)$ (namely \eqref{eq:maybe-func-set-1} and \eqref{eq:maybe-func-set-4}) and which from $(x_2,x_1)$ (namely \eqref{eq:maybe-func-set-2} and \eqref{eq:maybe-func-set-3}); it saves the reader from reconstructing it.}: %% \begin{enumerate} %% \item $\alpha(x_1)=Fp_1\comp\sigma(x_1,x_2)$\label{eq:maybe-func-set-1} %% \item $\alpha(x_2)=Fp_1\comp\sigma(x_2,x_1)$\label{eq:maybe-func-set-2} @@ -2585,7 +2637,7 @@ Assuming $f,g\in\Hom(X,Y+1)$, and that $f'\c X_f\rightarrowtail X$ and $g'\c X_g \arrow["{f'}"', tail, from=2-1, to=1-2] \end{tikzcd} \end{equation*} -\begin{lemma}\label{lem:maybe-lif-abs} +\begin{lemma}\label{prop:maybe-lif-abs} The order structure on the functor $F\c\BC\to\BC$ defined as $FX=X+1$ is a liftable order. \end{lemma} \begin{proof} @@ -3061,7 +3113,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i \end{proof} \begin{definition}[Mid-lax Barr relator] - Given a relation $r$, and take a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$. Assuming that $\appr$ is a partial order over a functor $F$, then the relator over $F$ and shown with $\overrightarrow{F}$ is a \emph{mid-lax Barr relator} if we have: + Given a relation $r$, and take a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$, and $\pi_1$ and $\pi_2$ are surjective, assuming that $\appr$ is a partial order over a functor $F$, then the relator over $F$ and shown with $\overrightarrow{F}$ is a \emph{mid-lax Barr relator} if we have: % A relator over a functor $F$ is a one-sided Barr relator, shown by $\overrightarrow{F}$, iff for a partial order $\appr$ over $F$, a relation $r\c X\rto Y$, and a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $r=\pi_2\comp\pi_1^\op$ we have: \begin{gather*} \overrightarrow{F}r=F\pi_2\comp\appr\comp(F\pi_1)^\op @@ -3194,7 +3246,7 @@ Barr relator is a generalization of the Egli-Milner relator, where the functor i % Since $\appr$ is a coliftable order structure by~\autoref{def:coliftable-ord} there exists a $w$ such that $z\appr w$ and $F\pi_j(w)=y$. So, we have $w \mathrel{(F\pi_j)} y$, $z\mathrel{\appr} w$, and $z\mathrel{(F\pi_i)} x$ that gives $x \mathrel{F\pi_j\comp\appr\comp(F\pi_i)^\op} y$.\qed %\end{proof} \begin{prop}\label{prop:all-rel-compa} - For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$, assuming that a set-functor $F$ has an order structure $\appr$, the following propositions hold: + For a span $(\pi_1\c A\to X,\pi_2\c A\to Y)$ that $\pi_1$ and $\pi_2$ are surjective, assuming that a set-functor $F$ has an order structure $\appr$, the following propositions hold: \begin{enumerate} \item If $\appr$ is liftable, we have $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad=\quad F\pi_2\comp(F\pi_1)^\op\comp\appr$ \item If $\appr$ is coliftable, we have $F\pi_2\comp\appr\comp(F\pi_1)^\op\quad=\quad \appr\comp F\pi_2\comp(F\pi_1)^\op$ From 51b64cc427c1b877e34387057feb6a0fe25fafdc Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 3 Aug 2026 20:29:50 +0100 Subject: [PATCH 18/19] minor --- draft/draft.tex | 18 +++++++++--------- 1 file changed, 9 insertions(+), 9 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index 861be01..e1f0014 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -577,24 +577,24 @@ The definition derives the ordering on morphisms of each hom-set $\Hom(X,\sub Y) \end{cases} \end{gather*} First, we need to prove that $\mu'$ is a subdistribution. - For every $x \in X$, $\mu'(x) \ge 0$. The total mass of $\mu'$ is + For every $x \in X$, $\mu'(x) \ge 0$. We have: \begin{align*} \sum_{x \in X} \mu'(x)&\\ - =& \sum_{x \in X} \frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)\\ - =& \sum_{y \in Y} \sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)\\ - =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)\\ - =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)\\ + =& \sum_{x \in X} \frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ + =& \sum_{y \in Y} \sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)&(Sg(\mu)(g(x))\neq 0)\\ =& \sum_{y \in Y} \nu(y)\\ \leq& 1 \end{align*} - Thus $\mu'$ is a subdistribution. Now, we prove $Sg(\mu') = \nu$. + As for every $x\in X$ that $Sg(\mu)(g(x))=0$, $\mu'(x)=0$, it does not have effect the inequality. Thus $\mu'$ is a subdistribution. Now, we prove $Sg(\mu') = \nu$. For any $y \in Y$, we have: \begin{align*} Sg(\mu')(y)&\\ = &\sum_{x \in g^{\mone}(y)} \mu'(x)\\ - = &\sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)\;\;(Sg(\mu)(y)\neq 0)\\ - = &\frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)\;\;(Sg(\mu)(y)\neq 0)\\ - = &\frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)\;\;(Sg(\mu)(y)\neq 0)\\ + = &\sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)&(Sg(\mu)(y)\neq 0)\\ + = &\frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)&(Sg(\mu)(y)\neq 0)\\ + = &\frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)&(Sg(\mu)(y)\neq 0)\\ = &\mu(y) \end{align*} (If $Sg(\mu)(y) = 0$, then $\nu(y) = 0$, and the sum is $0$ as well, so the equality still holds.) \\ From 8c1054256a4ef67f109cc6d30f3d3bd050e936cb Mon Sep 17 00:00:00 2001 From: partowp Date: Mon, 3 Aug 2026 20:34:08 +0100 Subject: [PATCH 19/19] minor --- draft/draft.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/draft/draft.tex b/draft/draft.tex index e1f0014..996180d 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -581,9 +581,9 @@ The definition derives the ordering on morphisms of each hom-set $\Hom(X,\sub Y) \begin{align*} \sum_{x \in X} \mu'(x)&\\ =& \sum_{x \in X} \frac{\nu(g(x))}{Sg(\mu)(g(x))} \comp \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ - =& \sum_{y \in Y} \sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ - =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)&(Sg(\mu)(g(x))\neq 0)\\ - =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)&(Sg(\mu)(g(x))\neq 0)\\ + =& \sum_{y \in Y} \sum_{x \in g^{\mone}(y)} \frac{\nu(y)}{Sg(\mu)(y)} \comp \mu(x)&(Sg(\mu)(y)\neq 0)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp \sum_{x \in g^{\mone}(y)} \mu(x)&(Sg(\mu)(y)\neq 0)\\ + =& \sum_{y \in Y} \frac{\nu(y)}{Sg(\mu)(y)} \comp Sg(\mu)(y)&(Sg(\mu)(y)\neq 0)\\ =& \sum_{y \in Y} \nu(y)\\ \leq& 1 \end{align*}