From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS

---
 source/know/concept/dynkins-formula/index.md | 23 ++++++++---------------
 1 file changed, 8 insertions(+), 15 deletions(-)

(limited to 'source/know/concept/dynkins-formula')

diff --git a/source/know/concept/dynkins-formula/index.md b/source/know/concept/dynkins-formula/index.md
index c0d20c5..307f098 100644
--- a/source/know/concept/dynkins-formula/index.md
+++ b/source/know/concept/dynkins-formula/index.md
@@ -39,11 +39,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-kolmogorov"/>
-<label for="proof-kolmogorov">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-kolmogorov">Proof.</label>
+
+{% include proof/start.html id="proof-kolmogorov" -%}
 We define a new process $$Y_t \equiv h(X_t)$$, and then apply Itō's lemma, leading to:
 
 $$\begin{aligned}
@@ -84,9 +81,8 @@ $$\begin{aligned}
     \hat{L}\{h(X_0)\}
     \approx \frac{1}{t} \mathbf{E}[Y_t - Y_0| X_0]
 \end{aligned}$$
+{% include proof/end.html id="proof-kolmogorov" %}
 
-</div>
-</div>
 
 The general definition of resembles that of a classical derivative,
 and indeed, the generator $$\hat{A}$$ can be thought of as a differential operator.
@@ -104,11 +100,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-dynkin"/>
-<label for="proof-dynkin">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-dynkin">Proof.</label>
+
+{% include proof/start.html id="proof-dynkin" -%}
 The proof is similar to the one above.
 Define $$Y_t = h(X_t)$$ and use Itō’s lemma:
 
@@ -136,9 +129,9 @@ $$\begin{aligned}
     = \mathbf{E}\bigg[ Y_\tau - Y_0 - \int_0^\tau \hat{L}\{h(X_t)\} \dd{t} \bigg| X_0 \bigg]
 \end{aligned}$$
 
-Isolating this equation for $$\mathbf{E}[Y_\tau | X_0]$$ then gives Dynkin's formula.
-</div>
-</div>
+Isolating this equation for $$\mathbf{E}[Y_\tau \!\mid\! X_0]$$ then gives Dynkin's formula.
+{% include proof/end.html id="proof-dynkin" %}
+
 
 A common application of Dynkin's formula is predicting
 when the stopping time $$\tau$$ occurs, and in what state $$X_\tau$$ this happens.
-- 
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