If you have never heard about lambda calculus, here is a quick definition found on Wikipedia:

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

Basically, this is the syntax:

`<function> ::= λ <variable-list> . <expression>`

For example, here is the identity function from mathematics:

Lambda Calculus

` ````
identity = λx.x
```

A more concrete example is its usage to represent types. In the following example:

JavaScript

` ````
function upper(str) {
return str.toUpperCase()
}
```

has the type:

Lambda Calculus

` ````
t' = λ t . t
// We can reduce it to infer the return type
t' = λ string . string
t' = string
// The result type is `string`
```

Lambda Calculus

` ````
true = λx.λy.x
false = λx.λy.y
```

Lambda Calculus

` ````
if = λb.λt.λf.b t f
```

Lambda Calculus

` ````
and = λbb'. if b b' false
or = λbb'. if b true b'
not = λb.if b false true
```

I decided to use OCaml for this example because it has partial application and a nice REPL.

Since the conditional and logical operators are already in the language, I will prepend them with an underscore.

OCaml

` ````
let _true x y = x;;
(* - : 'a -> 'b -> 'a =
``` *)
let _false x y = y;;
(* - : 'a -> 'b -> 'b = *)

Note that it's similar to how an identity function works.

OCaml

` ````
let _if b t f = b t f;;
(* - : ('a -> 'b -> 'c) -> 'a -> 'b -> 'c =
``` *)

For the sake of simplicity, I won't declare all the operators for now. We can already evaluate some simple conditions.

OCaml

` ````
(* This is my condition, for now it's a constant `true` *)
let myCond = _true;;
_if myCond 1 0;;
(* - : int = 1 *)
(* and if my condition is `false` *)
let myCond2 = _false;;
_if myCond2 1 0;;
(* - : int = 0 *)
```

To be honest, I don't know why you would use this in real life code. There is no need to redefine the built-in condition mechanism.

Thanks to Danny Willems for introducing me to this.

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N° SIRET : 821 797 891 00016, RCS Strasbourg TI 821 797 891