Hyades

Render Hyades source to Unicode/ASCII text art. Hyades uses LaTeX-like syntax but outputs multi-line plain text instead of PDF. It supports math (fractions, integrals, matrices, Greek letters), tables with box-drawing frames, lists, flex-like layout (hbox/vbox), user-defined macros, and computation.

IMPORTANT: Always display the rendered output to the user directly in your response. The output is pre-formatted multi-line text art that depends on exact character alignment. You MUST use a fenced code block (triple backticks) or equivalent monospace/preformatted element to preserve spacing. Never display the output as regular prose text — column alignment, box-drawing characters, and fraction bars will break without a monospace font.

Input is a complete Hyades document. Plain text outside math delimiters is rendered as prose paragraphs. Use $...$ for inline math and $$...$$ for display math (centered on its own line).

Math

Supports ~95% of LaTeX math syntax.

Inline vs Display

Inline math: $a^2 + b^2 = c^2$ flows with text.

Display math gets its own block:

$$ a^2 + b^2 = c^2 $$

Use $...$ for inline, $$...$$ for display. Display math is centered.

Variables and Operators

$$ a + b - c $$
$$ a \times b \div c $$
$$ a \cdot b $$
$$ \pm x \quad \mp y $$

Superscripts and Subscripts

$$ x^2 \quad x_i \quad x_i^2 \quad x^{n+1} \quad x_{i,j} \quad a^{b^c} $$

Use braces for multi-character exponents/subscripts: x^{n+1}, x_{i,j}.

Bracing rules: Use braces for multi-token expressions: x^{n+1}, x_{i,j}. Commands that take arguments (like \mathbf{v}) work without extra braces: x_\mathbf{v} is equivalent to x_{\mathbf{v}}, matching TeX behavior.

Fractions

$$ \frac{a}{b} $$
$$ \frac{x^2 + 1}{\sqrt{x^2 - 1}} $$
$$ \frac{\partial f}{\partial x} $$

Fractions nest to infinite depth. \dfrac and \tfrac are accepted as aliases for \frac.

Roots

$$ \sqrt{x} $$
$$ \sqrt[3]{x} $$
$$ \sqrt[n]{x^2 + y^2} $$

Optional argument gives the nth root: \sqrt[3]{x}.

Parentheses and Grouping

$$ (a + b) \quad [a + b] \quad \{a + b\} \quad |x| \quad \Vert x \Vert $$

Curly braces need escaping: \{ and \}.

Auto-Scaling Delimiters (\left/\right)

\left and \right make delimiters grow to match their contents:

$$ \left( \frac{a}{b} \right) \quad \left[ \frac{a}{b} \right] \quad \left| \frac{a}{b} \right| $$

Works with all delimiter types: ( ) [ ] \{ \} | \| \lfloor \rfloor \lceil \rceil \langle \rangle.

Use \left. or \right. for invisible (null) delimiters — essential for evaluation bars:

$$ \left.\frac{df}{dx}\right|_{x=0} $$

Use \middle inside \left...\right for scaled interior delimiters:

$$ \left( \frac{a}{b} \middle| \frac{c}{d} \right) $$

Explicit Delimiter Sizing (\big through \Bigg)

\big, \Big, \bigg, \Bigg set delimiter sizes explicitly.

\bigl/\bigr, \Bigl/\Bigr, \biggl/\biggr, \Biggl/\Biggr produce standalone one-sided delimiters:

$$ f(x)\bigr|_{x=0} $$

Floor and Ceiling

$$ \lfloor x \rfloor $$
$$ \lceil x \rceil $$
$$ \left\lfloor \frac{x}{2} \right\rfloor $$

Greek Letters

Lowercase: \alpha \beta \gamma \delta \epsilon \zeta \eta \theta \lambda \mu \nu \xi \pi \rho \sigma \tau \phi \varphi \chi \psi \omega

Uppercase (only where different from Latin): \Gamma \Delta \Theta \Lambda \Xi \Pi \Sigma \Phi \Psi \Omega

Relations

$$ a \neq b \quad a \leq b \quad a \geq b $$
$$ a \ll b \quad a \gg b $$
$$ a \approx b \quad a \equiv b \quad a \sim b \quad a \propto b $$
$$ a \prec b \quad a \succ b \quad a \preceq b \quad a \succeq b $$

Definition symbols: \coloneqq (≔), \eqqcolon (≕), and := (parsed as a single relation ≔):

$$ \Phi := S $$
$$ f \coloneqq x^2 + 1 $$

Colon alone is treated as punctuation (no space before, space after), useful in set-builder notation:

$$ \{x : x > 0\} $$

The \not prefix negates a relation: \not= → ≠, \not\in → ∉, \not\leq → ≰, \not\equiv → ≢, \not\subset → ⊄, \not\exists → ∄

Set Theory

$$ x \in A \quad x \notin A $$
$$ A \subset B \quad A \subseteq B \quad A \cup B \quad A \cap B \quad \emptyset $$
$$ A \setminus B $$

Logic

$$ \forall x \in \mathbb{R} \quad \exists x $$
$$ p \implies q \quad p \iff q $$
$$ \neg p \quad p \land q \quad p \lor q $$
$$ \therefore \quad \because \quad \top \quad \bot $$

Arrows

\rightarrow, \Rightarrow, \mapsto, \longmapsto, \hookrightarrow, \hookleftarrow, \leftarrow, \leftrightarrow, \Leftarrow, \Leftrightarrow, \uparrow, \downarrow, \updownarrow, \nearrow, \searrow, \nwarrow, \swarrow.

Extensible arrows with text labels:

$$ A \xrightarrow{f} B \xleftarrow{g} C $$

Standard Functions

Standard functions are set in upright type:

$$ \sin x \quad \cos(x) \quad \tan x $$
$$ \log x \quad \log_2 x \quad \ln x \quad \exp(x) $$
$$ \sin^2 x + \cos^2 x = 1 $$

Also: \arcsin, \arccos, \arctan, \sinh, \cosh, \tanh, \det, \dim, \ker, \deg, \gcd, \sup, \inf, \max, \min, \argmax, \argmin, \limsup, \liminf.

Custom Function Names (\fn / \operatorname)

\fn{name} or \operatorname{name} produces an upright function name for functions Hyades doesn't know:

$$ \fn{softmax}\left(x_i\right) = \frac{e^{x_i}}{\sum_j e^{x_j}} $$
$$ \operatorname{Tr}(A) $$

Text in Math (\text)

\text{words} inserts upright text inside math mode:

$$ x = 0 \text{ if } y > 0 $$

Accents and Decorations

Single-character: \hat{x}, \bar{x}, \tilde{x}, \vec{x}, \dot{x}, \ddot{x}, \acute{x}, \grave{x}, \breve{x}, \check{x}

Wide accents: \overline{AB}, \underline{text}, \overrightarrow{AB}, \overleftarrow{AB}, \widehat{xyz}, \widetilde{xyz}

Overset and Underset

$$ \overset{n}{=} $$
$$ \underset{x}{y} $$
$$ \stackrel{\text{def}}{=} $$

\overset{top}{base} places annotation above base; \underset{bot}{base} below. \stackrel is an alias for \overset.

Overbrace and Underbrace

$$ \overbrace{a + b + c}^{3 \text{ terms}} $$
$$ \underbrace{x + y + z}_{n \text{ terms}} $$

Primes

$$ f'(x) \quad f''(x) \quad f'''(x) \quad f^{(n)}(x) $$

Math Fonts

\mathbf{v} → bold, \mathbb{R} → blackboard bold (ℕℤℚℝℂ), \mathcal{L} → calligraphic, \mathfrak{R} → Fraktur (𝔄𝔅ℭ), \mathsf{A} → sans-serif (𝖠𝖡𝖢), \mathscr{A} → script (same as \mathcal), \boldsymbol{\alpha} → bold (works with Greek: 𝛂𝛃, and symbols: \boldsymbol{\nabla} → 𝛁, \boldsymbol{\partial} → 𝛛)

Dots and Ellipses

\ldots (low dots, for commas), \cdots (centered, for operators), \vdots (vertical), \ddots (diagonal, for matrices)

Summation

$$ \sum_{i=1}^{n} x_i $$

\Sum and \SUM give progressively larger variants.

Products

$$ \prod_{i=1}^{n} x_i $$

\Prod and \PROD give larger variants.

Integrals

$$ \int_a^b f(x) \, dx $$
$$ \iint f(x,y) \, dx\,dy $$
$$ \oint_C f(z) \, dz $$

Also: \iiint, \oiint. Larger variants: \Int, \INT, etc.

Limits

\lim places its subscript directly below:

$$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$
$$ \max_{x \in [0,1]} f(x) $$
$$ \argmax_{\theta} L(\theta) $$

Calculus Operators

$$ \frac{dy}{dx} \quad \frac{d^2y}{dx^2} \quad \frac{\partial f}{\partial x} $$
$$ \nabla f \quad \nabla \cdot \mathbf{F} \quad \nabla \times \mathbf{F} \quad \nabla^2 f $$

Matrices

Six types with different delimiters. & separates columns, \\ or ; separates rows. LaTeX-style \begin{pmatrix}...\end{pmatrix} also works for all matrix types.

$$ \pmatrix{a & b \\ c & d} $$    % (parentheses)
$$ \bmatrix{a & b \\ c & d} $$    % [brackets]
$$ \Bmatrix{a & b \\ c & d} $$    % {braces}
$$ \vmatrix{a & b \\ c & d} $$    % |bars| (determinants)
$$ \Vmatrix{a & b \\ c & d} $$    % ‖double bars‖ (norms)
$$ \matrix{a & b \\ c & d} $$     % no delimiters

Cases (Piecewise Functions)

$$f(x) = \cases{
    x    & \text{if } x > 0 \\
    0    & \text{if } x = 0 \\
    -x   & \text{if } x < 0
}$$

Aligned Equations

& marks the alignment point:

$$\aligned{
    f(x) &= x^2 + 2x + 1 \\
         &= (x + 1)^2
}$$

Use \intertext{...} inside aligned to insert prose between rows. Intertext is left-aligned at the document margin while equations remain centered:

$$\aligned{
    x &= a \\
    \intertext{where}
    a &= b + c
}$$

\tag works on aligned blocks — the tag is right-justified and vertically centered:

$$\aligned{
    f(x) &= x^2 + 2x + 1 \\
         &= (x + 1)^2
} \tag{7}$$

Boxed

$$ \boxed{E = mc^2} $$

Draws a box-drawing frame around the content.

Phantom

$$ a + \phantom{bbb} + c $$

Invisible placeholder — renders as whitespace with the same dimensions as the content.

Smash

$$ \sqrt{\smash{y^3}} $$

\smash{content} renders content but collapses its height to a single line (the baseline row). Used to prevent tall content from affecting surrounding constructs like roots or delimiters.

Substack

$$ \sum_{\substack{i=1 \\ j>0}} x_{ij} $$

Stacks multiple lines vertically (for multi-line subscripts/superscripts).

Tag

$$ E = mc^2 \tag{1} $$

Equation tag rendered as (text).

Number Theory

$$ a \equiv b \pmod{n} $$
$$ a \bmod n \quad a \mid b $$
$$ \binom{n}{k} $$

Style Commands (No-ops)

\displaystyle, \textstyle, \scriptstyle, \scriptscriptstyle are accepted but transparent (Hyades has no math size concept). \notag and \nonumber are also silently ignored. This allows pasting LaTeX source that uses these commands without errors.

Math Spacing

\! (negative thin), \, (thin), \: (medium), \; (thick), \quad (1em), \qquad (2em)

Atom-Type Overrides (\mathord, \mathbin, \mathrel)

Override the default spacing category of a symbol. Analogous to TeX's {=} idiom for suppressing operator spacing, but explicit and readable:

$a \mathord{=} b$          %% "a=b"  — = treated as ordinary symbol (no spacing)
$a = b$                    %% "a = b" — default relation spacing
$f \mathrel{:} A \to B$    %% "f : A → B" — : treated as relation (symmetric spacing)
$a : b$                    %% "a: b"  — default punctuation spacing
$a \mathbin{\triangle} b$  %% "a △ b" — triangle with binary operator spacing

• \mathord{X}: Suppress spacing — X renders as an ordinary symbol with no surrounding gaps. Use this instead of TeX's {=} or {}= idiom which does not work in Hyades.

• \mathbin{X}: Force binary operator spacing (gap on both sides).

• \mathrel{X}: Force relation spacing (gap on both sides).

Binary Operators

\oplus, \otimes, \odot, \circ, \bullet, \star, \dagger, \ddagger

Geometry

\angle, \triangle, \perp, \parallel

Famous Equations (all valid input)

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$                                                   %% Quadratic formula
$$ e^{i\pi} + 1 = 0 $$                                                                            %% Euler's identity
$$ f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$                        %% Normal distribution
$$ \fn{Attention}(Q, K, V) = \fn{softmax}\left(\frac{QK^\top}{\sqrt{d_k}}\right) V $$            %% Attention mechanism
$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$                                                   %% Taylor series
$$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} $$                                           %% Gaussian integral
$$ P(A|B) = \frac{P(B|A) P(A)}{P(B)} $$                                                           %% Bayes' theorem
$$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$                          %% Maxwell's equation
$$ i\hbar \frac{\partial}{\partial t} \Psi = \hat{H} \Psi $$                                      %% Schrodinger equation
$$ \fn{softmax}(x_i) = \frac{e^{x_i}}{\sum_j e^{x_j}} $$                                         %% Softmax

Text Formatting

\textbf{bold text}
\textit{italic text}
\texttt{monospace text}
\textbf{\textit{bold italic}}

Nesting \textbf and \textit produces bold-italic text.

Inline math $...$ inside \textit or \textbf is rendered as math (not styled as text):

\textit{Let $f(x) = x^2$ be a function}

Verbatim (No Processing)

\verb|raw \commands { } are preserved|

Special Characters

\textbackslash → , \textdollar → $, \textpercent → %, \textampersand → &, \texthash → #

Text Spacing Commands

\ (backslash-space) — explicit inter-word space. Useful after abbreviations: i.e.\ $f$ renders without extra sentence spacing. \@ — marks the next period as sentence-ending (silently consumed). \, — thin space (works in both text and math mode).

Paragraphs and Line Breaking

Text flows into paragraphs automatically. A single newline is a space; a blank line starts a new paragraph.

Line-Breaking Modes

\linebreaker{greedy}       % Default. Fast, good results.
\linebreaker{knuth}        % TeX-style optimal. Minimizes ugliness.
\linebreaker{raggedright}  % Left-aligned, no justification.

Hyphenation

\sethyphenate{true} / \sethyphenate{false}

Tables

Tables use \table with nested \row and \col:

\table[width:40, frame:single, pad:{l:1,r:1}]{
    \row[frame:{b:double}]{
        \col{Name}
        \col{Value}
    }
    \row{
        \col{Alpha}
        \col{0.05}
    }
    \row{
        \col{Beta}
        \col{0.95}
    }
}

Table Properties

Row properties: frame, pad, align, valign, height. Column properties: width, align, valign, frame, pad, span, reset.

Frame styles: none (space), dotted (┄┆), single (─│), rounded (╭╮╰╯ corners), double (═║), bold (━┃)

Compound edges: frame:{t:double, b:single, l:none, r:none}, pad:{l:1, r:1}

Properties cascade: table → column (down rows) → row → cell.

IMPORTANT: When a row overrides a property (e.g. frame:{b:double}), that override is inherited by ALL subsequent rows. To stop it, the next row must explicitly reset it (e.g. frame:{b:single}). The border property on the table level controls the outer edges and cannot be overridden by row or column frame.

Lists

\fancylist{
    - First item
    - Second item with longer text that
      wraps to the next line
    - Third item
      - Nested child
        - Grandchild
}

Three nesting levels supported. Uses ●/○ bullets in Unicode, - in ASCII.

Layout Primitives

Flex-like box model for spatial control.

Box Types

\begin[width]{hbox}              % horizontal (side by side)
    \child[width][align]{content}
\end{hbox}

\begin[top|middle|bottom]{hbox}  % hbox with vertical alignment
    \child[width][align]{content}
\end{hbox}

\begin[width]{vbox}              % vertical (stacked)
    \child[width][align]{content}
\end{vbox}

Child width: fixed number, auto (fill remaining), intrinsic (natural width), or omitted (equal share). When multiple children omit width, they split the remaining space equally (after fixed-width children). Horizontal alignment: left, center, right. Vertical alignment for hbox: set on \begin bracket — top (default), middle, bottom.

Width inheritance: boxes without an explicit width inherit their parent's width. \setwidth{N} sets the document-level width.

Rules

\hrule[width]{left_cap}{fill_char}{right_cap}
\vrule[height]{top}{fill}{bottom}
\intersect_rules{...}          % fix junction characters at rule crossings

Width/height can be a number or auto. An auto-width \hrule in a vbox stretches to the box width. An auto-height \vrule in an hbox stretches to match the tallest sibling.

\intersect_rules scans all cells, checks neighboring line arms, and replaces with correct junction glyphs (┌ ┬ ┐ ├ ┼ ┤ └ ┴ ┘). Handles single, double, bold, and mixed styles.

Measuring Content

\measure<content_name,width_var,height_var>{content to measure}

Renders content invisibly, stores it under content_name (retrieve with \recall<content_name>), and stores the measured width and height into integer variables.

Spacing

\vskip{N} (N blank lines), \hskip{N} (N spaces)

Common Layout Patterns

Two-column with gutter:

\begin{hbox}
    \child{Left column}
    \child[2]{}
    \child{Right column}
\end{hbox}

Centering via auto spacers:

\begin{hbox}
    \child[auto]{}
    \child[intrinsic]{Centered content}
    \child[auto]{}
\end{hbox}

Indentation:

\begin{hbox}
    \child[4]{}
    \child[auto]{Indented content}
\end{hbox}

User-Defined Macros

\macro<\greet{name}>{Hello, ${name}!}
\greet{World}       % → Hello, World!

\macro<\fullname{first}{last}>{${first} ${last}}
\fullname{John}{Doe}  % → John Doe

% Optional parameters with defaults
\macro<\heading[char=-]{title}>{
${title}
\hrule[20]{}{${char}}{}
}
\heading{Default}           % uses -
\heading[char:=]{Custom}    % uses =

Document Settings

\setwidth{80}              % Output width in columns (default: 80)
\setunicode{true}          % Use Unicode symbols (default: true)
\setmathitalic{true}       % Italicize math variables (default: true)
\linebreaker{greedy}       % Line-breaking algorithm
\sethyphenate{true}        % Enable hyphenation
\setparskip{1}             % Blank lines between paragraphs
\setmathabove{1}           % Blank lines above display math (default: 0)
\setmathbelow{1}           % Blank lines below display math (default: 0)

Computation

Variables, loops, and conditionals are available.

Variables and Arithmetic

\let<x>{10}
\let<y>{20}
\let<sum>{\add{\valueof<x>,\valueof<y>}}
Result: \valueof<sum>       % → Result: 30

Conditionals

\let<n>{42}
\if{\gt{\valueof<n>,0}}{positive}\else{non-positive}

Loops

\let<i>{1}
\begin{loop}
    \exit_when{\gt{\valueof<i>,5}}
    \valueof<i>\hskip{1}
    \inc<i>
\end{loop}
% Output: 1 2 3 4 5

Content Storage

\assign<name>{content} stores text, \recall<name> retrieves it. \let/\valueof is for integers; \assign/\recall is for text.

Lambdas

\lambda<double>[x]{\mul{\recall<x>,2}}
\recall<double>[5]             % → 10

Arrays

\let<nums[]>{[10, 20, 30]}
\valueof<nums>[0]              % → 10
\len<nums>                     % → 3
\push<nums>{40}                % append

Quick Reference

Math Mode

Text Commands

Table Syntax

\table[width:W, frame:STYLE, pad:{l:L,r:R}, border:STYLE, align:A]{
    \row[frame:{b:STYLE}]{
        \col[width:N, align:A]{content}
    }
}

Layout Syntax

\begin[width]{vbox}
    \child[width][h_align]{content}
\end{vbox}

\begin[top|middle|bottom]{hbox}
    \child[width][h_align]{content}
\end{hbox}

Settings

\setwidth{N}           \setunicode{true|false}
\setmathitalic{true}   \linebreaker{greedy|knuth|raggedright}
\sethyphenate{true}    \setparskip{N}
\setmathabove{N}       \setmathbelow{N}