### From Trinidad W.I.¶

In [4]:

```
uli2 "West Indies" "https://upload.wikimedia.org/wikipedia/commons/9/98/Caribbean_general_map.png"
```

Out[4]:

- West Indies

In [31]:

```
uli2 "Indians came to Trinidad in the 19th century as indentured labourers" "https://thepeopleafterslavery.files.wordpress.com/2014/03/indentured-labourers-2.png"
```

Out[31]:

- Indians came to Trinidad in the 19th century as indentured labourers

In [6]:

```
uli2 "Today Trinidad is multi-ethnic and multi-cultural" "http://currentriggers.com/wp-content/uploads/2016/11/4-17.jpg"
```

Out[6]:

- Today Trinidad is multi-ethnic and multi-cultural

In [25]:

```
uli2 "Divali and other religious festivals celebrated by all" "https://www.guardian.co.tt/image-3.1974716.c274fa76ed?size=512"
```

Out[25]:

- Divali and other religious festivals celebrated by all

In [9]:

```
uli2 "Cricket!" "https://akm-img-a-in.tosshub.com/indiatoday/images/story/201810/AP18297590913026.jpeg?P8W81HcX8oQiGA9xATv_s0lOWQKR3LH9"
```

Out[9]:

- Cricket!

- An F# DSL for scientific computing focused on safety, expressiveness, and interoperability
- Take the best bits of typed functional programming and apply it to scientific and mathematical computing
- Use F# features to create a more advanced type system for scientific computing

- Dominated by C/C++, Fortran, Java, Matlab, Python, R, Julia,
- Older languages are statically typed and procedural
- Newer languages are dynamically typed and have object-oriented features
- All are imperative languages that depend on mutable state and variables
- Newer languages like Python and Julia rely heavily on meta-programming

- Functional first:
- Everything is an expression
- Functions as first-class citizens
- Higher-order functions
- Immutable variables
- Avoid side-effects
- Pattern matching
- User-defined operators

- Best of both worlds:
- Functional + object-oriented
- Immutable + mutable variables
- Static + dynamic types using the .NET DLR

- Classes, interfaces, inheritance, polymorphism
- Type extensibility with object expressions, extension methods
- Powerful meta-programming capabilities: type providers, computation expressions
- Interoperabilty - Can interoperate with any library or language with C FFI
- C++, Python, Java
- NumPy, Keras, TensorFlow, Torch

In [10]:

```
let titanic = new CsvFile("https://raw.githubusercontent.com/datasciencedojo/datasets/master/titanic.csv")
titanic.["Pclass"].First().Type <- typeof<int>
let dt = Frame(titanic)
query {
for r in dt do
groupBy r?Pclass into g
sortBy g.Key
select (
let survived = (g.Where(fun p -> p?Survived = "1").Count()) |> float
let died = (g.Where(fun p -> p?Survived = "0").Count()) |> float
let ctotal = survived + died
let psurvived = round(100.0 * survived / ctotal)
let pdied = round(100.0 * died / ctotal)
(g.Key, pdied, psurvived)
)} |> Util.Table
```

Out[10]:

Item1 | Item2 | Item3 |
---|---|---|

1 | 37 | 63 |

2 | 53 | 47 |

3 | 76 | 24 |

- Built on .NET runtime and SDK
- Comprehensive open-source tooling and environments - .NET Core / VS Code / Jupyter,...et.al
- Huge open-source community providing libraries, resources, support
- Microsoft fully committed to open source .NET and regularly releases libraries like ML.NET

In [11]:

```
let v1 = Vec<1000>.Rand //Create a vector of length 1000
let v2 = Vec<500>.Rand //Create a vector of length 500
v1 + v2 //Can we do this?
```

- E.g. A vector or array type that depends on its length
- More advanced program verification than regular types
- More errors can be caught before the program is run

- Types can depend on most
*runtime*values - Idris, Agda, ATS, F*....
- Programmers write proof-carrying code

The Future of Programming is Dependent Types... https://medium.com/background-thread/the-future-of-programming-is-dependent-types-programming-word-of-the-day-fcd5f2634878

Idris, a language that will change the way you think about programming...https://crufter.com/idris-a-language-that-will-change-the-way-you-think-about-programming

...but

- Still a niche area in programming
- Languages like Idris and ATS have a relatively small user community
- Lack of tooling, libraries, resources, commercial support

- http://okmij.org/ftp/Computation/lightweight-dependent-typing.html
- Attempt to bring safety benefits of dependent typing to mainstream languages
- Began with Dependent ML - precursor to ATS

- Values must be statically known before evaluation

- Type-level arithmetic on natural number dimensions
- Addition, subtraction, multiplication, equal to, greater than, et..al

- Define function constraints which accept only objects of certain dimensions e.g only 3x3 real matrices
- Define function constraints which accept objects with dimensions in a certain range e.g matrix functions that only accept square matrices
- Sylvester can express type-level constraints and conditions simply without elaborate logical apparatus

- Many languages like C++ and D support type-level arithmetic
- C++ can use static numeric values as template parameters
- Both languages can use static checks and compiler asserts to do type level static checks

- Types can vary with values and not simply fail statjc checks
- Do not rely on compiler asserts
- Types are part of the problem domain e.g. arithmetic, linear algebra
- Rich set of type operators and constraints and checks e.g arithmetic comparison operators

In [14]:

```
// Create typed instance of some natural numbers
let a = new N<1000>()
let b = new N<900>()
a +< b
```

Out[14]:

Sylvester.Arithmetic.Bool+False

In [15]:

```
a +> b
```

Out[15]:

Sylvester.Arithmetic.Bool+True

In [16]:

```
a +!= b
```

Out[16]:

Sylvester.Arithmetic.Bool+True

In [17]:

```
a +== b
```

Out[17]:

Sylvester.Arithmetic.Bool+False

In [18]:

```
check((b * a) +== zero) //Causes type error
```

This expression was expected to have type 'Success' but here has type 'Failure'

In [19]:

```
check ((a - b) +> zero) // But this is ok
```

In [21]:

```
zero - a //This results in another type, not a compiler error
```

Out[21]:

N10Underflow

In [ ]:

```
```