# Proofs, Theorems and Algorithms

All proof directives can be included using the prf:kind pattern, where the proof directives are shown in Table 1. The directive is enumerated by default and can take in an optional title argument which is shown in brackets after the proof.

Same as Sphinx Proof 🎉

The implementation and documentation for proofs, theorems, etc. is based on Sphinx Proof, the syntax can be used interchangeably. We have reused the examples in that extension here to show off the various parts of the MyST extension.

Changes to the original extension include being able to click on the proof label (e.g. "Theorem 1"), and having a link to that proof anchor. We have also updated the styles from both Sphinx and JupyterBook to be more distinct from admonitions.

You can also reference proofs with any cross-reference syntax (including the {prf:ref} role). We recommend the markdown link syntax.

The following options for proof directives are supported:

• label: text

A unique identifier for your theorem that you can use to reference it with a Markdown link or the {prf:ref} role. Cannot contain spaces or special characters.

• class: text

Value of the theorem’s class attribute which can be used to add custom CSS or JavaScript. This can also be the optional dropdown class to initially hide the proof.

• nonumber: flag (empty)

Turns off auto numbering.

## Referencing Proofs¶

You can refer to a proof using the standard link syntax:

Compatibility with Sphinx Proof

You may also use the the {prf:ref} role like: {prf:ref}my-theorem, which will replace the reference with the theorem number like so: Theorem 2. When an explicit text is provided, this caption will serve as the title of the reference. For example, {prf:ref}Orthogonal-Projection-Theorem <my-theorem>  will produce: Orthogonal-Projection-Theorem.

## Hiding Proof Content¶

To hide the directive, simply add :class: dropdown as a directive option.

Example

Theorem 1

This is an example of how to hide the content of a directive.

MyST Syntax:

{prf:theorem}
:class: dropdown

This is an example of how to hide the content of a directive.


## Proof Examples¶

### Proofs¶

MyST Syntax

{prf:proof}
:label: full-proof

We'll omit the full proof.

But we will prove sufficiency of the asserted conditions.

To this end, let $y \in \mathbb R^n$ and let $S$ be a linear subspace of $\mathbb R^n$.

Let $\hat y$ be a vector in $\mathbb R^n$ such that $\hat y \in S$ and $y - \hat y \perp S$.

Let $z$ be any other point in $S$ and use the fact that $S$ is a linear subspace to deduce

{math}
\| y - z \|^2
= \| (y - \hat y) + (\hat y - z) \|^2
= \| y - \hat y \|^2  + \| \hat y - z  \|^2


Hence $\| y - z \| \geq \| y - \hat y \|$, which completes the proof.


Source: QuantEcon

### Theorems¶

MyST Syntax

{prf:theorem} Orthogonal-Projection-Theorem
:label: my-theorem

Given $y \in \mathbb R^n$ and linear subspace $S \subset \mathbb R^n$,
there exists a unique solution to the minimization problem

{math}
\hat y := \argmin_{z \in S} \|y - z\|


The minimizer $\hat y$ is the unique vector in $\mathbb R^n$ that satisfies

* $\hat y \in S$

* $y - \hat y \perp S$

The vector $\hat y$ is called the **orthogonal projection** of $y$ onto $S$.


Source: QuantEcon

### Axioms¶

MyST Syntax

{prf:axiom} Completeness of $\mathbb{R}$
:label: my-axiom

Every Cauchy sequence on the real line is convergent.


Source: Stachurski, 2009

### Lemmas¶

MyST Syntax

{prf:lemma}
:label: my-lemma

If $\hat P$ is the fixed point of the map $\mathcal B \circ \mathcal D$ and $\hat F$ is the robust policy as given in [(7)](https://python-advanced.quantecon.org/robustness.html#equation-rb-oc-ih), then

{math}
:label: rb_kft

K(\hat F, \theta) = (\theta I - C'\hat P C)^{-1} C' \hat P  (A - B \hat F)



Source: QuantEcon

### Definitions¶

MyST Syntax

{prf:definition}
:label: my-definition

The *economical expansion problem* (EEP) for
$(A,B)$ is to find a semi-positive $n$-vector $p>0$
and a number $\beta\in\mathbb{R}$, such that

$$&\min_{\beta} \hspace{2mm} \beta \\ &\text{s.t. }\hspace{2mm}Bp \leq \beta Ap$$


Source: QuantEcon

### Criteria¶

MyST Syntax

{prf:criterion} Weyl's criterion
:label: weyls-criterion

Weyl's criterion states that the sequence $a_n$ is equidistributed modulo $1$ if
and only if for all non-zero integers $m$,

{math}
\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{j=1}^{n} \exp^{2 \pi i m a_j} = 0



Source: Wikipedia

### Remarks¶

MyST Syntax

{prf:remark}
:label: my-remark

More generally there is a class of density functions
that possesses this feature, i.e.

$$\exists g: \mathbb{R}_+ \mapsto \mathbb{R}_+ \ \ \text{ and } \ \ c \geq 0, \ \ \text{s.t. the density } \ \ f \ \ \text{of} \ \ Z \ \ \text{ has the form } \quad f(z) = c g(z\cdot z)$$

This property is called **spherical symmetry** (see p 81. in Leamer
(1978))


Source: QuantEcon

### Conjectures¶

MyST Syntax

{prf:conjecture} Fake $\gamma$ conjecture
:label: my-conjecture

This is a dummy conjecture to illustrate that one can use math in titles.


### Corollaries¶

MyST Syntax

{prf:corollary}
:label: my-corollary

If $A$ is a convergent matrix, then there exists a matrix norm such
that $\vert \vert A \vert \vert < 1$.


Source: QuantEcon

### Algorithms¶

MyST Syntax

{prf:algorithm} Ford–Fulkerson
:label: my-algorithm

**Inputs** Given a Network $G=(V,E)$ with flow capacity $c$, a source node $s$, and a sink node $t$

**Output** Compute a flow $f$ from $s$ to $t$ of maximum value

1. $f(u, v) \leftarrow 0$ for all edges $(u,v)$
2. While there is a path $p$ from $s$ to $t$ in $G_{f}$ such that $c_{f}(u,v)>0$
for all edges $(u,v) \in p$:

1. Find $c_{f}(p)= \min \{c_{f}(u,v):(u,v)\in p\}$
2. For each edge $(u,v) \in p$

1. $f(u,v) \leftarrow f(u,v) + c_{f}(p)$ *(Send flow along the path)*
2. $f(u,v) \leftarrow f(u,v) - c_{f}(p)$ *(The flow might be "returned" later)*


Source: Wikipedia

### Examples¶

MyST Syntax

{prf:example}
:label: my-example

Next, we shut down randomness in demand and assume that the demand shock
$\nu_t$ follows a deterministic path:

{math}
\nu_t = \alpha + \rho \nu_{t-1}


Again, we’ll compute and display outcomes in some figures

python
ex2 = SmoothingExample(C2=[, ])

x0 = [0, 1, 0]
ex2.simulate(x0)



Source: QuantEcon

### Properties¶

MyST Syntax

{prf:property}
:label: my-property

This is a dummy property to illustrate the directive.


### Observations¶

MyST Syntax

{prf:observation}
:label: my-observation

This is a dummy observation directive.


### Propositions¶

MyST Syntax

{prf:proposition}
:label: my-proposition

This is a dummy proposition directive.


### Assumptions¶

MyST Syntax

{prf:assumption}
:label: my-assumption

This is a dummy assumption directive.

References
1. Stachurski, J. (2009). Economic Dynamics: Theory and Computation (Vol. 1). The MIT Press.