VT-DSL

1. Overview
2. General Syntax
3. Separators
4. Undefined and Skipped Regions
5. Variations
6. Multicolumn Content

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1. Overview

VT-DSL is a domain-specific language for describing variation and sign tables. It is designed to be:

• Human-readable
• AI-friendly for generation and parsing
• Structurally regular while semantically flexible

The DSL describes table structure, not mathematical correctness.

2. General Syntax

header [var] : [domain]
      row [row-label] [/height] : [row-content]
      ...
      

A file must contain exactly one header, followed by zero or more row declarations. All bracketed components are optional.

header — defines the first row; must come before any row.

• var: Defines the variable name.
• domain: Defines the boundaries and values, separated by commas (,).

Values can be text (delimited by ' or ") or math (delimited by $). You can nest math inside text quotes: "variation of $\abs{x}$".

header $x$ : $-\infty$, $0$, $+\infty$
      

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row — defines a horizontal row.

• row-label: label of the row, placed at the first column, e.g., "sign of $f'(x)$".
• /height: Relative height multiplier, e.g., /2.
• row-content: Sequence of cell contents (text, math,signs, ...) and separators.

Signs + and - are native tokens and do not require quotes.

row "sign of $f'(x)$" /1.5 : + -
      

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3. Separators

Separators describe vertical lines between cells. They are optional.

Syntax	Meaning
&	Invisible separator
|&	single bar
||&	double bar
:&	Dashed
::&	Double dashed

row  : ::& + :& - |&
      row  : -& + & - ||&
      

try it Labels can be attached to a separator using &{position:value}. There are 9 positions:

top-left	top-center (alias top)	top-right
center-left	center-center (alias center)	center-right
bottom-left	bottom-center (alias bottom)	bottom-right

row : + |&{top-left:"top-left", bottom-right:"bottom-right"} -
      row : + |&{center:$0$} -
      

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4. Undefined and Skipped Regions

• UNDEF: Renders a hashed region where a function is not defined.
• SKIP: Skips a cell entirely; nothing is drawn, and the next content moves to the next cell.

row : + UNDEF +
      row : UNDEF SKIP UNDEF
      

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5. Variations

Three commands describe function variation:

Command	Meaning
INC	Increasing arrow
DEC	Decreasing arrow
CONST	Constant (horizontal)

row : INC CONST DEC
      

try it Options:

• arrow: Set head to "start", "end", "both", or "none".
• vPos: (For CONST) Set to "top", "center", or "bottom".

row : INC{arrow:"start"} CONST{arrow:"none", vPos:"bottom"} DEC{arrow:"both"}
      

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Extrema and Limits: we can specify extrema and limits by attaching them to a separator

row : &{bottom-right:$-\infty$} INC |&{top:$-1$} DEC &{bottom-left:$-\infty$}
      

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Note : Labels on boundary separators must be explicitly aligned inward: use right-aligned positions at the left edge and left-aligned positions at the right edge.

6. Multicolumn Content

Any content type can span multiple columns using the colSpan option. If multiple options are used, separate them with a comma.

row : "text"{colSpan:2} INC
      row : UNDEF{colSpan:3}
      row : DEC{arrow:"none", colSpan:2} CONST
      

try it