Dilation situation.

src/logic/index.tex

@@ -2,3 +2,6 @@
 \label{part:logic}
 
 This section is plagiarised from 
+
+\input{./tp/index.tex}
+\input{./notation.tex}

src/logic/notation.tex

@@ -0,0 +1,11 @@
+\chapter{Notations}
+\label{chap:logic-notations}
+
+\begin{tabular}{lll}
+    \textbf{Notation} & \textbf{Description} & \textbf{Source} \\
+    \hline
+    % ---- Topological Vector Spaces ----
+    $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ & Language $\mathcal{L}$ with function symbols $\mathcal{F}$, relation symbols $\mathcal{R}$, and constant symbols $\mathcal{C}$. & \autoref{definition:language} \\
+    $\mathcal{M} = (M, f^\cm, R^\cm, c^\cm)$ & $\mathcal{L}$-structure with functions $\bracsn{f^\cm|f \in \mathcal{F}}$, relations $\bracsn{R^\cm|R \in \mathcal{R}}$, and constants $\bracsn{c^\cm|c \in \mathcal{C}}$. & \autoref{definition:language-structure}
+\end{tabular}
+

src/logic/tp/lang.tex

@@ -11,5 +11,46 @@
     \end{enumerate}
 \end{definition}
 
+\begin{definition}[$\mathcal{L}$-Structure]
+\label{definition:language-structure}
+    Let $L = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language, then a \textbf{\mathcal{L}-structure} $\mathcal{M}$ is a family
+    \[
+    \mathcal{M} = (M, f^\cm, R^\cm, c^\cm| f \in \mathcal{F}, R \in \mathcal{R}, c \in \mathcal{C})
+    \]
+
+    where
+
+    \begin{enumerate}
+        \item $M$ is a non-empty set, known as the \textbf{universe}, \textbf{domain}, or \textbf{underlying set} of $\mathcal{M}$. 
+        \item For each $f \in \mathcal{F}$, $f^\cm: M^{n_f} \to M$ is a function in $n$-variables in $M$. 
+        \item For each $R \in \mathcal{R}$, $R^\cm \subset M^{n_R}$ is a $n$-ary relation.
+        \item For each $c \in \mathcal{C}$, $c^\cm \in M$ is a constant.  
+    \end{enumerate}
+    
+    The assignments $f^\cm$, $R^\cm$, and $c^\cm$ are the \textbf{interpretations} of $(\mathcal{F}, \mathcal{R}, \mathcal{C})$. 
+\end{definition}
+
+\begin{definition}[Terms]
+\label{definition:logic-terms}
+    Let $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language, then the set of \textbf{$\mathcal{L}$-terms} is the smallest set of expressions $\mathcal{T}(\mathcal{L})$ such that:
+    \begin{enumerate}
+        \item $\mathcal{C} \subset \mathcal{T}(\mathcal{L})$. 
+        \item For each $n \in \natp$, the variable symbol $v_n \in \mathcal{T}(\mathcal{L})$. 
+        \item For each $f \in \mathcal{F}$ and $\bracsn{t_j}_1^{n_f} \subset \mathcal{T}(\mathcal{L})$, the expression $f(t_1, \cdots, t_{n_f}) \in \mathcal{T}(\mathcal{L})$. 
+    \end{enumerate}
+\end{definition}
+
+\begin{lemma}
+\label{lemma:logic-terms-induction}
+    Let $\mathcal{L} = (\mathcal{F}, \mathcal{R}, \mathcal{C})$ be a language and $\mathcal{T}_0(\mathcal{L}) = \mathcal{C} \cup \seq{v_n}$. For each $n \in \natp$, inductively define
+    \[
+    \mathcal{T}_{n}(\mathcal{L}) = \mathcal{T}_{n-1}(\mathcal{L}) \cup \bracsn{f(t_1, \cdots, t_{n_f})|f \in \mathcal{F}, \bracsn{t_j}_1^{n_f} \in \mathcal{T}_{n-1}(\mathcal{L})}
+    \]
+
+    then $\mathcal{T} = \bigcup_{n \in \natz}\mathcal{T}_n$. For each $n \in \natz$, $\mathcal{T}_n$ is the set of \textbf{$\mathcal{L}$-terms of complexity at most $n$}.
+\end{lemma}
+
+
+