Optimize last images, add proof template, improve CSS

1 files changed, 13 insertions, 21 deletions

diff --git a/source/know/concept/ito-process/index.md b/source/know/concept/ito-process/index.md
index f192e28..2756e33 100644
--- a/source/know/concept/ito-process/index.md
+++ b/source/know/concept/ito-process/index.md
@@ -61,6 +61,7 @@ since only the current value of $$X_t$$ determines the future,
 and $$B_t$$ is also a Markov process.
 
 
+
 ## Itō's lemma
 
 Classically, given $$y \equiv h(x(t), t)$$,
@@ -83,11 +84,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-lemma"/>
-<label for="proof-lemma">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-lemma">Proof.</label>
+
+{% include proof/start.html id="proof-lemma" -%}
 We start by applying the classical chain rule,
 but we go to second order in $$x$$.
 This is also valid classically,
@@ -133,14 +131,15 @@ $$\begin{aligned}
 
 Where $$\chi_1^2(\dd{t})$$ is the generalized chi-squared distribution
 with one term of variance $$\dd{t}$$.
-</div>
-</div>
+{% include proof/end.html id="proof-lemma" %}
+
 
 The most important application of Itō's lemma
 is to perform coordinate transformations,
 to make the solution of a given Itō SDE easier.
 
 
+
 ## Coordinate transformations
 
 The simplest coordinate transformation is a scaling of the time axis.
@@ -208,6 +207,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 
+
 ## Existence and uniqueness
 
 It is worth knowing under what condition a solution to a given SDE exists,
@@ -232,11 +232,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-existence"/>
-<label for="proof-existence">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-existence">Proof.</label>
+
+{% include proof/start.html id="proof-existence" -%}
 If we define $$Y_t \equiv X_t^2$$,
 then Itō's lemma tells us that the following holds:
 
@@ -275,9 +272,8 @@ $$\begin{aligned}
     \\
     &\le (Y_0 + 3 K t) \exp\!\big(3 K t\big)
 \end{aligned}$$
+{% include proof/end.html id="proof-existence" %}
 
-</div>
-</div>
 
 If a solution exists, it is also worth knowing whether it is unique.
 Suppose that $$f$$ and $$g$$ satisfy the following inequalities,
@@ -301,11 +297,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-<div class="accordion">
-<input type="checkbox" id="proof-uniqueness"/>
-<label for="proof-uniqueness">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-uniqueness">Proof.</label>
+
+{% include proof/start.html id="proof-uniqueness" -%}
 We define $$D_t \equiv X_t \!-\! Y_t$$ and $$Z_t \equiv D_t^2 \ge 0$$,
 together with $$F_t \equiv f(X_t) \!-\! f(Y_t)$$ and $$G_t \equiv g(X_t) \!-\! g(Y_t)$$,
 such that Itō's lemma states:
@@ -347,9 +340,8 @@ $$\begin{aligned}
     \\
     &\le Z_0 \exp\!\Big( \big( 2 K \!+\! K^2 \big) t \Big)
 \end{aligned}$$
+{% include proof/end.html id="proof-uniqueness" %}
 
-</div>
-</div>
 
 Using these properties, it can then be shown
 that if all of the above conditions are satisfied,