From bef0b76eab8233185695f9160cc5ec627fe31953 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Wed, 16 Sep 2026 13:48:54 -0400
Subject: [PATCH] Cleaned up notations for logic.

---
 src/logic/notation.tex | 4 +++-
 src/logic/tp/lang.tex  | 2 +-
 2 files changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/logic/notation.tex b/src/logic/notation.tex
index 84f89c5..3e6043a 100644
--- a/src/logic/notation.tex
+++ b/src/logic/notation.tex
@@ -6,6 +6,8 @@
     \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}
+    $\mathcal{M} = (M, f^\cm, R^\cm, c^\cm)$ & $\mathcal{L}$-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} \\
+    $\mathcal{T}(\mathcal{L})$ & Set of $\mathcal{L}$-terms. & \autoref{definition:logic-terms} \\
+    $\mathcal{T}_n(\mathcal{L})$ & Set of $\mathcal{L}$-terms of complexity at most $n$. & \autoref{lemma:logic-terms-induction}
 \end{tabular}
 
diff --git a/src/logic/tp/lang.tex b/src/logic/tp/lang.tex
index 38acfb0..153027c 100644
--- a/src/logic/tp/lang.tex
+++ b/src/logic/tp/lang.tex
@@ -47,7 +47,7 @@
     \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 $n$}.
+    then $\mathcal{T}(\mathcal{L}) = \bigcup_{n \in \natz}\mathcal{T}_n(\mathcal{L})$. For each $n \in \natz$, $\mathcal{T}_n$ is the set of \textbf{$\mathcal{L}$-terms of complexity at most $n$}.
 \end{lemma}