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Added Agda code in latex, plus beginning of an introduction

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Mysaa
2023-07-30 19:07:51 +02:00
parent 357e2a087d
commit e9eedbff2d
6 changed files with 311 additions and 42 deletions
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report/M1Report.tex
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@@ -1,4 +1,4 @@
% !TeX spellcheck = fr_FR
% !TeX spellcheck = en_US
\documentclass[10pt,a4paper]{article}
\include{./header.tex}
@@ -21,8 +21,17 @@
\newpage
\section{Introduction}
In the first place, i was supposed to do this internship in Nottingham (UK) with Thorsten Altenkirsh. But, because of administrative issues, i was not able to go there, and Thorsten Altenkirsh contacted Ambrus Kaposi in Budapest, whose university agreed to accept me. Therefore, i went to Budapest and i did the internship under the physical supervision of Ambrus Kaposi, and the remote supervision of Thorsten Altenkirsch.
The original subject of the internship was \enquote{Developping a simplified account of normalization by evaluation using the theory of categories with families}. But there was a lot to do, starting with learning how to use Agda.
\subsection{Introduction to the problem}
What i do call a \enquote{logic} is a set of definitions containing formulæ and a notion of provability of those formulæ, plus a set of operators/equalities to construct or reduce these rules. I have studied the most common logical frameworks, that is, Propositional Logic, First-order logic with infinitary predicates, and Predicate Logic. For each of those logics, one can define a notion of \emph{model}. A model of a certain kind of logic is something that implements all of the logic's definitions, operators and equalities. From all of those models, one can extract the \emph{initial model}, also called \emph{syntax}. It is the smallest of all models, which means that from any model of that logic, we have a morphism from the syntax to that model.
Then, our goal is for each logic to prove the completeness of a specific class of models. Completeness can be stated as such: \enquote{For any formula that is \emph{true} in all models of the specified class, then the formula has a proof in the syntax}. By being true in the model, one can understand that the formula has a proof from the model.
\subsection{Motivation}
\subsection{Structure of this report}
\section{A first account of Completeness and Normalization}
@@ -31,11 +40,158 @@
\subsection{Merging the two proofs}
\section{Predicate Logic}
\subsection{SOGAT Presentation of FFOL}
\begin{tcolorbox}
\[
\begin{array}{lcl}
\Tm & : & \Set^+ \\
\\
\For & : & \Set \\
- \implies - & : & \For \rightarrow \For \rightarrow \For\\
\forall & : & (\Tm \rightarrow \For) \rightarrow \For \\
\R & : & \Tm \rightarrow \Tm \rightarrow \For\\
\\
\Pf & : & \For \rightarrow \Prop^+ \\
\lam & : & (\Pf A \rightarrow \Pf B) \rightarrow \Pf (A \implies B)\\
\app & : & \Pf (A \implies B) \rightarrow (\Pf A \rightarrow \Pf B)\\
\foralli & : & (t : \Tm \rightarrow^+ A\;t) \rightarrow \Pf (\forall A)\\
\foralle & : & \Pf (\forall A) \rightarrow (t : \Tm) \rightarrow \Pf (A\;t)\\
\end{array}
\]
\end{tcolorbox}
\subsection{Turning a SOGAT Presentation into a GAT}
\begin{tcolorbox}
\agda{agda/Con.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Tm.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Tm+.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/For.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Pf.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Pf+.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/R.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Imp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/Forall.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/LamApp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ForallR.tex}
\end{tcolorbox}
\section{Implementing the Syntax}
\subsection{Separated Contexts}
\begin{tcolorbox}
\agda{agda/ICont.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IFor.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubt.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ITm.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubtT.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubTF.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IIdCompT.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IConp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubtC.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IConpTp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IPf.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IRen.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubtP.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubtS.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISubpP.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IIdCompP.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ICon.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ISub.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/IIdComp.tex}
\end{tcolorbox}
\begin{tcolorbox}
\agda{agda/ICExt.tex}
\end{tcolorbox}
\subsection{Transport Hell}
%\section{Completeness and Normalization for FFOL}
\section{Summary}
report/header.tex
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@@ -1,6 +1,5 @@
% Loading packages
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
% Loading packages
\usepackage{autofe}
\usepackage{hyperref}
\hypersetup{
colorlinks=true,
@@ -16,7 +15,6 @@
\usepackage{csquotes}
\usepackage{listings}
\usepackage{lstautogobble}
\usepackage{algorithm2e}
\usepackage{svg}
\usepackage{tikz}
\usepackage{multirow}
@@ -34,6 +32,7 @@
\usepackage{mathtools}
\usepackage[page,header]{appendix}
\usepackage{minitoc}
\usepackage{mathtools}
\usepackage{geometry}
@@ -59,9 +58,6 @@
\def\nDownarrow{\not\mspace{1mu}\Downarrow}
\let\pprec\preccurlyeq
\DeclareUnicodeCharacter{03BB}{$\lambda$}
\DeclareUnicodeCharacter{03B1}{$\alpha$}
\DeclareUnicodeCharacter{03C0}{$\pi$}
% Création des environnement globaux
\newtheorem{theorem}{Théorème}
@@ -95,7 +91,46 @@
% Macros caractères spécifiques au document
\newcommand\Tm{\operatorname{Tm}}
\newcommand\Set{\operatorname{Set}}
\newcommand\For{\operatorname{For}}
\newcommand\Prop{\operatorname{Prop}}
\newcommand\R{\operatorname{R}}
\newcommand\lam{\operatorname{lam}}
\newcommand\app{\operatorname{app}}
\newcommand\foralli{\operatorname{\operatorname{\forall i}}}
\newcommand\foralle{\operatorname{\operatorname{\forall e}}}
\newcommand\Pf{\operatorname{Pf}\;}
% Agda Config
\usepackage{agda}
\usepackage{newunicodechar}
\newunicodechar{∘}{\ensuremath{\mathnormal{\circ}}}
\newunicodechar{≡}{\ensuremath{\mathnormal{\equiv}}}
\newunicodechar{◇}{\ensuremath{\mathnormal{\diamond}}}
\newunicodechar{α}{\ensuremath{\mathnormal{\alpha}}}
\newunicodechar{β}{\ensuremath{\mathnormal{\beta}}}
\newunicodechar{γ}{\ensuremath{\mathnormal{\gamma}}}
\newunicodechar{δ}{\ensuremath{\mathnormal{\delta}}}
\newunicodechar{ε}{\ensuremath{\mathnormal{\varepsilon}}}
\newunicodechar{σ}{\ensuremath{\mathnormal{\sigma}}}
\newunicodechar{π}{\ensuremath{\mathnormal{\pi}}}
\newunicodechar{▹}{\ensuremath{\mathnormal{\triangleright}}}
\newunicodechar{⊢}{\ensuremath{\mathnormal{\vdash}}}
\newunicodechar{⇒}{\ensuremath{\mathnormal{\Rightarrow}}}
\newunicodechar{∀}{\ensuremath{\mathnormal{\forall}}}
\newunicodechar{≈}{\ensuremath{\mathnormal{\approx}}}
\newunicodechar{ₜ}{\ensuremath{{}_t}}
\newunicodechar{ₚ}{\ensuremath{{}_p}}
\newunicodechar{⁰}{\ensuremath{{}^0}}
\newcommand\agda[1]{
\small
\begin{code}
\input{#1}
\end{code}
}
% Création des environnements spécifiques au document