diff --git a/content/.smart-env/multi/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング_md.ajson b/content/.smart-env/multi/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング_md.ajson
index d874248e..84ac44fd 100644
--- a/content/.smart-env/multi/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング_md.ajson
+++ b/content/.smart-env/multi/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング_md.ajson
@@ -66,3 +66,17 @@
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diff --git a/content/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング.md b/content/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング.md
index 22d8cd78..fe2dd658 100644
--- a/content/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング.md
+++ b/content/多段階計算と増分関数型リアクティブプログラミングのによる信号処理のライブコーディング.md
@@ -59,19 +59,47 @@ mimiumはコードを専用のVMバイトコードへコンパイルし実行す
 
 ## 多段階計算
 
-多段階計算は、型付きラムダ計算に対して、計算のステージを複数段階に分割する明示的なシンタックスを導入するものである。Lisp系言語のquote/splice機能[@lisp]のように、部分的に計算したコード片を埋め込むようなものを想定しているが、不正な値が埋め込まれないことを型システムとして保証することが特徴である。
-
-実用的な例では、Scala 3でのマクロや、関数型組版処理エンジンSaTysFi[@suwaMLStyleModuleSystem2024]のように、言語内DSLを型安全にライブラリとして実装することを想定しているものがある。
+多段階計算は、型付きラムダ計算に対して、計算のステージを複数段階に分割する明示的なシンタックスを導入するものである。Lisp系言語のquote/splice機能[@lisp]のように、部分的に計算したコード片を埋め込むようなものを想定しているが、不正な値が埋め込まれないことを型システムとして保証することが特徴である（逆に、通常マクロに期待される型システムの範囲を超えたメタ操作を行うことは許されない）。実用的な例では、Scala 3でのマクロや、関数型組版処理エンジンSaTysFi[@suwaMLStyleModuleSystem2024]のように、言語内DSLを型安全にライブラリとして実装することを想定しているものがある。mimiumにおいては、多段階計算はコンパイル時に行う計算（＝シグナルグラフのルーティングの決定）と、ランタイム時に行う計算（＝実際のオーディオ処理）を区別するのに用いている。
 
 
-### シンタックス
+### シンタックスの拡張
 
 図nにmimiumの内部表現Lambda-mmmに多段階計算の体系を加えた新しい内部表現 $\lambda_{Mmmm}$(Multi-stage version of $\lambda_{mmm}$)のシンタックスを定義する。
 
+型にはCode Type $\langle t\rangle$を加える
+$$
+\begin{align}
+\tau ::=
+		&\quad R \quad &\\
+      |&\quad I_n \quad &n \in \mathbb{N} \\
+      |&\quad \tau → \tau &\\
+      |&\quad \langle \tau \rangle
+\end{align}
 $$
 
+式にはクオートとスプライスを加える。mem(e)はシングルサンプルのディレイだが、これは遅延時間のデータを保存する必要がないため通常のディレイと区別して使用される。
+$$
+\begin{align}
+e ::=
+	  &\quad R & R \in \mathbb{R}  [number]&\\
+      |&\quad e \ e \quad& [app]&\\
+      |&\quad \lambda x.e& [abs]&\\
+      |&\quad let\; x\; =\; e\; in\; e& [let]&\\
+      |&\quad delay(x,e_{time},v_{bound})&[delay]&\\
+      |&\quad mem(e) &[mem]&\\
+      |&\quad feed\ x.e &[feed]&\\
+      |&\quad `(e) &[quote]&\\
+      |&\quad $(e) &[splice]&
+\end{align}
+$$
 
-
+$$
+\begin{align}
+v ::=
+	  |&\quad R & R \in \mathbb{R} [number]&\\
+      |&\quad cls(e,E) &[closure]&\\
+      |&\quad \langle v \rangle &[code]&\\
+\end{align}
 $$
 
 ### シンタックスシュガー