#### Tutorial: (English)

# Sylvester: computer-based math via F# meta-programming

Sylvester is an F#-based DSL that tries to capture the conceptual, logical, symbolic, computational, and visual properties of mathematical language without the need for external parsers or interpreters or interactive environments. Mathematical and logical statements in Sylvester are expressed as conceptual or logical or symbolic relationships between formulae using F# types, operators and functions, and a formula is defined as any F# expression for which an F# code quotation and full expression tree is available. Sylvester can therefore use the full range of F# language features, IDEs, and environments like Jupyter notebooks for applications like theorem proving, numerical computing and machine learning, and visualization via interop with JavaScript libraries like JSXGraph.

One of the biggest barriers to widespread adoption of theorem provers by math students is the relative paucity of feature-rich programming environments for these specialized languages, when compared to mainstream languages like Python or C++ and the sophisticated IDEs and environments available that are designed to assist the user as much as possible.

This situation isn’t helped by the fragmentation of languages and tools that attempt to capture the different aspects of mathematical statements. Typically there are tools like Lean for interactive theorem proving, NumPy and SciPy for numeric programming, Maxima and its siblings for symbolic computing, JSXGraph and D3 and similar libraries for visualization, and these tools generally feature little interoperability with each other.

Sylvester embeds an internal DSL into a mainstream functional language that is expressive enough to capture different aspects of mathematical expressions. As a hybrid functional-OOP language with extensive meta-programming features F# is an ideal choice for building a DSL that can naturally capture the relationships between mathematical expressions, and as a .NET based language can easily interoperate with libraries in other languages like C or target different runtime environments like Web pages or Jupyter notebooks.

This 90-min tutorial will go through using Sylvester in several applications and scenarios within a Jupyter Notebook and VS Code, both from a developer’s POV as in how a certain language feature is implemented, and from a user’s POV as in how different statements and problems for a given computer-based math application can be expressed and solved using a single language and environment. The main topics are:

- Using F# code-quotations for mathematical expressions
- F# type-level arithmetic and arithmetic constraints
- Theorem-proving
- Symbolic manipulation
- Type-safe abstract algebra
- Type-safe number-parameterized linear algebra
- Compiling F# quotations to JavaScript and using visualization libraries like JSXGraph
- Type-safe number-parameterized machine learning using F# type providers and TensorFlow

### Allister Beharry

I’ve been programming computers both as a hobby and professionally for more than 20 years. I like both Windows and Linux. My current focus is computer security, machine learning, conversational user interfaces and .NET scientific programming.