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Theorems, Lemmas, and Proofs ​

MaTiSSe provides numbered theorem-like environments for scientific presentations — theorems, lemmas, corollaries, definitions, examples, exercises, remarks, and propositions — together with an unnumbered proof environment.

All environments use the same fenced-div syntax and integrate with the cross-reference system.

Theorem-like environments ​

Syntax ​

markdown
::: {#PREFIX-id}
## Optional title
Body content (full Markdown, including math).
:::

PREFIX determines the environment type and its display label. id is a unique identifier used for cross-referencing (see below).

Supported prefixes ​

PrefixDisplay labelExample heading
thmTheoremTheorem 1
lemLemmaLemma 1
corCorollaryCorollary 1
prpPropositionProposition 1
defDefinitionDefinition 1
exmExampleExample 1
exrExerciseExercise 1
remRemarkRemark 1

Counters are per-prefix and sequential within the presentation.

Examples ​

Theorem without a title:

markdown
::: {#thm-cauchy}
$$|\langle u, v\rangle|^2 \leq \|u\|^2 \|v\|^2$$
:::

Renders as: Theorem 1 followed by the equation.

Definition with a title:

markdown
::: {#def-lipschitz}
## Lipschitz continuity
A function $f : \mathbb{R}^n \to \mathbb{R}^m$ is $L$-Lipschitz if for all
$x, y$:
$$\|f(x) - f(y)\| \leq L\,\|x - y\|$$
:::

Renders as: Definition 1 (Lipschitz continuity).

Example with inline math:

markdown
::: {#exm-gradient-descent}
## Gradient descent step
Starting from $x_0$, iterate:
$$x_{k+1} = x_k - \alpha \nabla f(x_k)$$
with step size $\alpha > 0$.
:::

Proof environment ​

Proofs use the .proof class instead of a #PREFIX-id identifier. They are unnumbered and always end with the QED symbol ∎.

markdown
::: {.proof}
Let $\varepsilon > 0$.  By compactness of the unit sphere there exists a
finite $\varepsilon$-net …

The bound follows by taking the supremum over all unit vectors.
:::

If the body already contains ∎ or \square, the auto-appended symbol is suppressed to avoid duplication.

A proof can follow immediately after a theorem block:

markdown
::: {#thm-fundamental}
## Fundamental theorem of calculus
If $f$ is continuous on $[a,b]$ then
$$\int_a^b f(x)\,dx = F(b) - F(a)$$
where $F' = f$.
:::

::: {.proof}
Define $F(x) = \int_a^x f(t)\,dt$ and differentiate under the integral sign…
:::

Cross-references ​

Labelling ​

Every theorem-like block carries an implicit label derived from its #PREFIX-id. The label is also registered in the global label registry for use as a cross-reference target:

markdown
::: {#thm-main-result}
## Main result
…
:::

HTML id: thm-main-result.

Referencing ​

Use @PREFIX-id anywhere in slide content (including inside other environments) to insert a hyperlinked cross-reference:

markdown
As shown in @thm-main-result, the bound is tight.

Renders as a link: Theorem 1 → <a href="#thm-main-result">Theorem 1</a>.

If the target label is not found a ?? warning link is emitted and a message is printed at build time.

All supported reference prefixes ​

PrefixDisplay label
figFigure
tblTable
eqEquation
secSection
diaDiagram
thmTheorem
lemLemma
corCorollary
prpProposition
defDefinition
exmExample
exrExercise
remRemark

Complete example ​

markdown
#### Key mathematical results

::: {#def-convex}
## Convex function
$f$ is convex if for all $x, y$ and $\lambda \in [0,1]$:
$$f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y)$$
:::

::: {#thm-jensen}
## Jensen's inequality
For a convex function $f$ and random variable $X$:
$$f(\mathbb{E}[X]) \leq \mathbb{E}[f(X)]$$
:::

::: {.proof}
Apply @def-convex repeatedly to the finite expectation sum…
:::

By @thm-jensen the empirical risk is an upper bound on the expected risk.

HTML output and CSS customisation ​

html
<!-- theorem -->
<div class="theorem theorem-thm" id="thm-cauchy">
  <div class="theorem-header"><strong>Theorem 1 (Cauchy–Schwarz)</strong></div>
  <div class="theorem-body">…</div>
</div>

<!-- proof -->
<div class="theorem theorem-proof" id="proof-1">
  <div class="theorem-header"><em>Proof.</em></div>
  <div class="theorem-body">
    …
    <span class="qed">∎</span>
  </div>
</div>

Useful CSS selectors for custom styling:

css
.theorem              { border-left: 3px solid #4a90d9; padding: 0.5em 1em; }
.theorem-thm          { background: rgba(74,144,217,0.05); }
.theorem-def          { border-left-color: #2e8b57; }
.theorem-proof        { border-left-color: #888; font-style: italic; }
.theorem-header       { font-size: 110%; margin-bottom: 0.3em; }
.qed                  { float: right; }
.cross-ref            { color: #4a90d9; text-decoration: none; }

Apply via css_overtheme in your metadata block.

See also ​