---
# System prepended metadata

title: Setlang

---

# Setlang

# Learn Setlang in Y Minutes (Revised: Set-Theoretic Focus)

# In Setlang, everything is a set. Types are sets, and values are sets.

# 1. Primitives as Sets

# Integers are sets containing a single number
x = 5  # x is the set {5}

# Floats are sets containing a single floating-point number
y = 3.14  # y is the set {3.14}

# Strings are sets containing a single string
s = "Hello"  # s is the set {"Hello"}

# Booleans are sets containing True or False
t = True  # t is the set {True}
f = False  # f is the set {False}

# The empty set
empty = (--)

# 2. Types as Sets

# Int is the set of all integers
Int : (- ..., -2, -1, 0, 1, 2, ... -)

# Float is the set of all floating-point numbers
Float : (- ... -)  # Conceptually, all possible floats

# String is the set of all possible strings
String : (- ... -)  # Conceptually, all possible strings

# Bool is the set containing True and False
Bool : (- True, False -)

# 3. Creating Subsets On-the-Fly

# Subset of Int
PositiveInt = (- x in Int where x > 0 -)

# Subset of String
ShortString = (- s in String where len(s) < 10 -)

# Subset of Float
UnitInterval = (- x in Float where 0 <= x and x <= 1 -)

# 4. Functions as Set Operations

# Functions create subsets based on their input and output types
add : Int -> Int -> Int
add = \a, b -> a + b  # Returns a subset of Int

# The result of add is always in Int
result = add 3 4  # result is the set {7}, which is a subset of Int

# 5. Custom Types as Sets

# Point is a set of all possible 2D coordinates
Point : (- [x: Float, y: Float] -)

# Creating a point is selecting a specific element from the Point set
p = Point{x: 3.0, y: 4.0}  # p is a singleton subset of Point

# 6. Subsets in Function Parameters

# Function that works on a subset of Int
doublePositive : PositiveInt -> PositiveInt
doublePositive = \x -> x * 2

# This implicitly creates a subset when called
doubled = doublePositive 5  # doubled is the set {10}

# 7. Set Operations

numbers = (- 1, 2, 3, 4, 5 -)
evens = (- 2, 4 -)
odds = (- 1, 3, 5 -)

union = numbers or (- 4, 5, 6 -)        # (- 1, 2, 3, 4, 5, 6 -)
intersection = numbers & (- 4, 5, 6 -) # (- 4, 5 -)
difference = numbers -- (- 4, 5, 6 -)   # (- 1, 2, 3 -)

# 8. Pattern Matching on Sets

describe = \s ->
    when s is
        (--)            -> "Empty set"
        (- x -)         -> "Singleton set"
        (- x, y -)      -> "Set with two elements"
        (- _ where len(s) > 10 -) -> "Large set"
        _               -> "Set with multiple elements"

# Defining days of the week with known subsets
DayOfWeek : (-
    Weekday,
    Weekend
-)

Weekday : (-
    Monday,
    Tuesday,
    Wednesday,
    Thursday,
    Friday
-)

Weekend : (-
    Saturday,
    Sunday
-)

# Explicitly defining known subsets
Saturday : self & :<Weekend
Sunday : self & :<Weekend

Monday : self & :<Weekday
Tuesday : self & :<Weekday
Wednesday : self & :<Weekday
Thursday : self & :<Weekday
Friday : self & :<Weekday

# Function that benefits from known subsets
describeDayType : DayOfWeek -> String
describeDayType = \day ->
    when day is
        Weekend -> "It's the weekend!"
        Weekday -> "It's a weekday."
    # No need for a catch-all case, as DayOfWeek's subsets are fully known

# More specific function that uses individual days
describeDay : DayOfWeek -> String
describeDay = \day ->
    when day is
        Saturday -> "It's Saturday!"
        Sunday -> "It's Sunday!"
        Monday -> "It's Monday."
        Tuesday -> "It's Tuesday."
        Wednesday -> "It's Wednesday."
        Thursday -> "It's Thursday."
        Friday -> "It's Friday."
    # Again, no catch-all needed due to exhaustiveness

# Function that would require a catch-all without known subsets
unsafeDayDescription : DayOfWeek -> String
unsafeDayDescription = \day ->
    when day is
        Saturday -> "Weekend - Saturday"
        Sunday -> "Weekend - Sunday"
        Monday -> "Weekday - Monday"
        _ -> "Other day"  # Catch-all required if subsets weren't known

# Usage
print(describeDayType(Saturday))  # "It's the weekend!"
print(describeDay(Wednesday))     # "It's Wednesday."
print(unsafeDayDescription(Friday))  # "Other day"

# Demonstrating performance benefit
isWeekend : DayOfWeek -> Bool
isWeekend = \day -> day in Weekend
# This check can be optimized by the compiler due to known subset relationships

# 9. Traits as Set Constraints

Printable : (- T where exists toString : T -> String -)

impl Printable for Point {
    toString = \p -> "(" + toString(p.x) + ", " + toString(p.y) + ")"
}

# 10. Generics with Set Constraints

max : (T : (- x where exists gt : T -> T -> Bool -)) -> T -> T -> T
max = \T, a, b -> if T::gt(a, b) then a else b

# 11. Effect System with Sets

Effect : (- IO, State, Error, Pure -)

Eff : (E : (- e in Effect -)) -> (A : Type) -> (-
    computation : () -> A with effects E
-)

# 12. Infinite Sets and Lazy Evaluation

NaturalNumbers = (- n in Int where n >= 0 -)
EvenNaturals = (- n in NaturalNumbers where n % 2 == 0 -)

# Take first 5 even naturals (lazy evaluation)
first_five_evens = take 5 EvenNaturals  # Results in {0, 2, 4, 6, 8}

# 13. Dependent Types as Parameterized Sets

Vector : (n : NaturalNumbers) -> (- v : [Float] where len(v) == n -)

v3 : Vector 3
v3 = [1.0, 2.0, 3.0]

# 14. Modules as Namespaced Sets

module Math {
    PI = 3.14159  # PI is the set {3.14159}
    
    CircleArea : Float -> Float
    CircleArea = \r -> PI * r * r
    
    export (PI, CircleArea)
}

# 15. Metaprogramming with Set Operations

macro repeat(n, expr) {
    # Expands to a set of n repetitions of expr
    # Implementation details omitted for brevity
}

# This revised guide emphasizes Setlang's fundamental set-theoretic
# nature, showing how all language constructs, including types,
# values, and operations, are fundamentally based on set theory.

# Learn Setlang in Y Minutes: Subsets, Comprehensions, and Assumes

# 1. Basic Set Definitions

# Define a set of days
DayOfWeek : (-
    (-Monday-),
    (-Tuesday-),
    (-Wednesday-),
    (-Thursday-),
    (-Friday-),
    (-Saturday-),
    (-Sunday-)
-)

# That ^ will automatically create subsets of any referenced sets that are not defined. 

Also Every primitive value is a set. so DayOfWeek could also be written: 

DayOfWeek : (-
    Monday,
    Tuesday,
    Wednesday,
    Thursday,
    Friday,
    Saturday,
    Sunday
-)


# 2. Subset Creation

# Define Weekday as a subset of DayOfWeek
Weekday : DayOfWeek where * in (-
    DayOfWeek::Monday,
    DayOfWeek::Tuesday,
    DayOfWeek::Wednesday,
    DayOfWeek::Thursday,
    DayOfWeek::Friday
-)

# Define Weekend using set difference
Weekend : DayOfWeek -- Weekday

# 3. Set Comprehensions

# Even numbers up to 10
Evens : (- x in (- 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 -) where x % 2 == 0 -)

# Squares of numbers
Squares : (- x * x for x in (- 1, 2, 3, 4, 5 -) -)

# Conditional comprehension
PositiveEvens : (- x for x in Int where x > 0 and x % 2 == 0 -)

# 4. The 'assumes' Keyword

# Function that assumes its input is positive
sqrt : (x : Float) -> Float
sqrt = \x ->
    assumes x >= 0
    # Implementation here

# Function with multiple assumptions
divide : (x : Float, y : Float) -> Float
divide = \x, y ->
    assumes y != 0
    assumes is_finite(x) and is_finite(y)
    x / y

# 5. Combining Subsets and Assumes

# A type for positive integers
PositiveInt : Int where * > 0

# Function that works only on PositiveInt
factorial : PositiveInt -> Int
factorial = \n ->
    assumes n <= 20  # To prevent overflow
    if n == 1 then 1 else n * factorial(n - 1)

# 6. Complex Set Definitions

# Define a set of shapes
Shape : (-
    (-Circle { radius: Float }-),
    (-Rectangle { width: Float, height: Float }-),
    (-Triangle { base: Float, height: Float }-)
-)

# Subset of shapes with area greater than 10
LargeShape : Shape where
    when * is
        (-Circle { radius }-) -> pi * radius * radius > 10
        (-Rectangle { width, height }-) -> width * height > 10
        (-Triangle { base, height }-) -> 0.5 * base * height > 10

# 7. Functions with Set Comprehensions

# Function to filter a set based on a predicate
filter : (A : Type) -> (A -> Bool) -> (- A -) -> (- A -)
filter = \A, pred, set -> (- x in set where pred(x) -)

# Usage
EvenSquares = filter(Int, \x -> x % 2 == 0, Squares)

# 8. Assumes in Type Definitions

# A type for non-empty strings
NonEmptyString : String
NonEmptyString = \s ->
    assumes len(s) > 0
    s

# 9. Combining Set Operations

# Intersection of two subsets
WeekendOrMonday : Weekend or (- DayOfWeek::Monday -)

# 10. Recursive Set Definitions

# Binary Tree structure
BinaryTree : (T : Type) -> (-
    (-Leaf-),
    (-Node { value: T, left: BinaryTree T, right: BinaryTree T }-)
-)

# This guide demonstrates Setlang's powerful subset creation,
# set comprehensions, and the 'assumes' keyword for adding
# runtime checks and compile-time constraints.

# Learn Setlang in Y Minutes: Immutability

# 1. Immutable Variables

# Variables are immutable by default
x = 5
# x = 6  # This would cause a compile-time error

# Creating a new binding with the same name is allowed in a new scope
y = {
    x = 10  # This creates a new 'x', doesn't modify the outer one
    x  # Returns 10
}
# x is still 5 here

# 2. Immutable Data Structures

# Lists are immutable
numbers = [1, 2, 3, 4, 5]
# numbers[0] = 10  # This would cause a compile-time error

# Creating a new list with a changed element
new_numbers = [0] ++ numbers[1..]  # [0, 2, 3, 4, 5]

# 3. Immutable Records

Point : { x: Float, y: Float }

p1 = Point{ x: 3.0, y: 4.0 }
# p1.x = 5.0  # This would cause a compile-time error

# Creating a new record with updated fields
p2 = { ...p1, x: 5.0 }  # { x: 5.0, y: 4.0 }

# 4. Working with Immutable Data

# Function to "modify" a list (actually returns a new list)
add_to_list : [Int] -> Int -> [Int]
add_to_list = \list, item -> list ++ [item]

original = [1, 2, 3]
updated = add_to_list(original, 4)  # [1, 2, 3, 4]
# original is still [1, 2, 3]

# 5. Recursion and Immutability

# Recursive function to sum a list
sum : [Int] -> Int
sum = \list ->
    when list is
        [] -> 0
        [x, ...xs] -> x + sum(xs)

total = sum([1, 2, 3, 4, 5])  # 15

# 6. Immutable Set Operations

Set1 = (- 1, 2, 3 -)
Set2 = (- 3, 4, 5 -)

Union = Set1 or Set2  # (- 1, 2, 3, 4, 5 -)
# Set1 and Set2 remain unchanged

# 7. Immutable Transformations

# Map function (returns a new list)
map : (A -> B) -> [A] -> [B]
map = \f, list ->
    when list is
        [] -> []
        [x, ...xs] -> [f(x)] ++ map(f, xs)

doubles = map(\x -> x * 2, [1, 2, 3])  # [2, 4, 6]

# 8. Handling State Changes

# Instead of mutating state, we return new state
type Counter = { count: Int }

increment : Counter -> Counter
increment = \counter -> { count: counter.count + 1 }

c1 = Counter{ count: 0 }
c2 = increment(c1)  # { count: 1 }
c3 = increment(c2)  # { count: 2 }
# c1 is still { count: 0 }

# 9. Immutable Algorithms

# Quicksort implementation showcasing immutability
quicksort : [Int] -> [Int]
quicksort = \list ->
    when list is
        [] -> []
        [pivot, ...rest] ->
            left = quicksort([x for x in rest where x <= pivot])
            right = quicksort([x for x in rest where x > pivot])
            left ++ [pivot] ++ right

sorted = quicksort([3, 1, 4, 1, 5, 9, 2, 6])  # [1, 1, 2, 3, 4, 5, 6, 9]

# 10. Working with Immutable Sets

# Function to add an element to a set (returns a new set)
add_to_set : (T : Type) -> (- T -) -> T -> (- T -)
add_to_set = \T, set, item -> set or (- item -)

Set3 = (- 1, 2, 3 -)
Set4 = add_to_set(Int, Set3, 4)  # (- 1, 2, 3, 4 -)
# Set3 is still (- 1, 2, 3 -)

# This guide demonstrates Setlang's commitment to immutability,
# showing how it affects data structures, functions, and overall
# programming style, promoting safety and predictability.

# Setlang's Expose ^ Operator Explanation

1. Basic Usage with Exposable Trait

Exposable : Trait
Exposable = (-
    ["expose", Self -> Exposed Self]
-)

Exposed : (T : Type) -> Type
Exposed = \T -> (-
    e : T where
        exists get : () -> T,
        exists set : T -> ()
-)

# Example type implementing Exposable
Number : { value: Int }

impl Exposable for Number {
    expose = \self -> 
        Exposed Number {
            get = \() -> self.value,
            set = \new_value -> { ...self, value: new_value }
        }
}

2. ^ Operator Behavior

x = Number{ value: 5 }

# Exposing the getter/setter object
x_obj = ^x  # Returns Exposed Number

# Using the exposed object
current_value = x_obj.get()  # 5
x_obj.set(10)

# Chaining operations
squared_value = x_obj.get().square()  # 25

# Reverting to implicit form
x_implicit = x^

3. Property Access with ^

Point : { x: Float, y: Float }

impl Exposable for Point {
    expose = \self -> 
        Exposed Point {
            get = \() -> self,
            set = \new_point -> new_point
        }
}

p = Point{ x: 3.0, y: 4.0 }

# Exposing the getter/setter object of Point, not its properties
point_obj = ^p.x  # Returns Exposed Point, not just x

# Accessing property through the exposed object
x_value = point_obj.get().x  # 3.0

# Reverting property to implicit form (if it was explicit)
p.x^  # This would revert x to implicit form if it was explicit

# Converting the whole object to implicit form
p^.x  # This accesses x on the implicit form of p

4. Effect System Integration

Effect : (-
    ...,  # Other effects
    Expose
-)

expose_number : Number -> Eff (-Expose-) Int
expose_number = \num ->
    exposed = ^num  # This returns Exposed Number
    exposed.get()

5. Chaining with ^

complex_operation = \point ->
    # Expose Point object, get its value, access x property
    x_value = ^point.get().x
    # Set a new x value and return to implicit form
    new_point = point.set({ ...point.get(), x: x_value * 2 })^
    new_point.x  # Implicitly get the new x value

result = complex_operation(^p)

# Key Points:

1. ^obj always returns an Exposed version of the entire object, not just a property.
2. obj.property^ reverts a specific property to implicit form if it was explicit.
3. obj^.property converts the whole object to implicit form before accessing the property.
4. The .get() method on Exposed objects takes no arguments and returns the current value.
5. The .set() method on Exposed objects takes a new value and updates the object.
6. Using ^ triggers the Expose effect, which must be handled or propagated.

This explanation demonstrates the precise behavior of the ^ operator
in Setlang, showing how it interacts with the Exposable trait and
effect system while maintaining clear distinctions between object
and property-level operations.


# Setlang Syntax and Keyword Explainer

1. Comments
   # Single line comment
   /* Multi-line
      comment */

2. Set Notation
   (- ... -)        Set literal
   (--)             Empty set

3. Basic Types
   Int              Integer type
   Float            Floating-point type
   String           String type
   Bool             Boolean type (True, False)

4. Variables and Assignment
   x = 5            Variable assignment
   x : Int = 5      Variable assignment with type annotation

5. Set Operations
   or               Set union
   &                Set intersection
   --               Set difference
   in               Set membership

6. Functions
   \x -> x * 2      Lambda function
   f x = x * 2      Function definition (sugar for \x -> x * 2)
   f(x, y)          Function call with multiple arguments
   f x              Function call with single argument (no parentheses needed)

7. Control Flow
   if ... then ... else ...     Conditional statement
   when ... is ...              Pattern matching

8. Types and Traits
   Type             Keyword for type declarations
   Trait            Keyword for trait declarations
   impl             Keyword for trait implementation

9. Generics and Constraints
   List T           Generic type
   (T : Type where ...) -> ...  Type constraint in function signature

10. Modules
    module ... { ... }     Module definition
    import ...             Import statement
    export ...             Export statement

11. Effect System
    Effect                 Effect type declaration
    with effects ...       Effect annotation

12. Custom Operators
    (op) : A -> B -> C     Custom operator definition

13. Set Comprehension
    (- x in S where ... -)  Set comprehension syntax

14. Arrays
    [1, 2, 3]              Array literal
    arr[0]                 Array indexing

15. Objects
    { x: 5, y: 10 }        Object literal
    obj.x                  Object property access

16. Pattern Matching Keywords
    when                   Start of pattern matching expression
    is                     Separator between pattern and result
    _                      Wildcard pattern

17. Quantifiers
    forall                 Universal quantifier
    exists                 Existential quantifier

18. Type-related Keywords
    where                  Used in type constraints and set comprehensions
    Self                   Refers to the implementing type in traits

19. Dependent Types
    (n : Nat) -> ...       Dependent type syntax

20. Metaprogramming
    macro                  Macro definition keyword

21. Special Set Theory Symbols
    <:                     Subset relation
    <->                    Bijection

22. Built-in Functions
    len                    Length of collection
    toString               Convert to string
    take                   Take first n elements

23. Trait-related
    ::                     Scope resolution for trait methods

24. Lambda Calculus Symbols
    λ                      Alternative to \ for lambda expressions (if supported)

25. Logical Operators
    and                    Logical AND
    or                     Logical OR
    not                    Logical NOT

26. Comparison Operators
    ==                     Equality
    !=                     Inequality
    <, <=, >, >=           Comparison operators

27. Arithmetic Operators
    +, -, *, /             Basic arithmetic
    %                      Modulo
    ^                      Exponentiation (if supported)

28. Bitwise Operators (if supported)
    &, |, ^, ~, <<, >>     Bitwise operations

29. Special Characters
    _                      Underscore (often used as wildcard)
    ...                    Spread operator or variadic arguments

30. Docstrings
    /// Documentation      Documentation comment

This explainer covers the main syntactic elements and keywords
of Setlang, showcasing its unique blend of set theory, functional
programming, and advanced type system features.


# 5. DSL Operations (corrected function calls)

entity : Variable -> Predicate
entity = \v -> "entity" ~> v

has : Variable -> String -> Variable -> Predicate
has = \e, attr, v -> "has" ~> [e, attr, v]

(:-) : Predicate -> Predicate -> Rule
(:-) = \head, body -> { head: head, body: body }

# 6. Query Execution (corrected function calls)

execute : KB -> Query -> [Variable, Entity or Attribute]
execute = \kb, q -> 
    [result where
        forall p in q.where, satisfied(kb, p, result) and
        forall v in q.select, exists [var, _] in result, var == v
    ]

satisfied : KB -> Predicate -> [Variable, Entity or Attribute] -> Bool
satisfied = \kb, pred, bindings ->
    when pred is
        { name: "entity", args: [v] } ->
            exists e in kb.entities,
                [v, e] in bindings
        { name: "has", args: [e, attr, v] } ->
            exists [ent, a, val] in kb.attributes,
                [e, ent] in bindings and
                a == attr and
                [v, val] in bindings
        { name: rel, args: vars } ->
            exists [r, ents] in kb.relations,
                r == rel and
                forall v in vars, exists e in ents, [v, e] in bindings

# 7. Example Usage 


Entity : Type
Attribute : Type
Relation : Type

# 2. Knowledge Base
KB : {
    entities: Entity,
    attributes: [Entity, String, Attribute],
    relations: [String, Entity]
}

# 3. Query Types
Variable : Type
Variable = String

Predicate : Type
Predicate = {
    name: String,
    args: [Variable or Entity or Attribute]
}

Rule : Type
Rule = {
    head: Predicate,
    body: Predicate
}

Query : Type
Query = {
    select: Variable,
    where: Predicate
}

# 4. Custom Operators (unchanged)

(?-) : Variable -> Predicate -> Query
(?-) = \select, where -> { select: select, where: where }

(~>) : String -> Variable -> Predicate
(~>) = \name, vars -> { name: name, args: vars }

# Define a query: Find all entities and their ages
ageQuery = ["X", "Age"] ?- [
    entity "X",
    has("X", "age", "Age")  # Corrected: parentheses added
]

# Execute the query
results = execute(exampleKB, ageQuery)  # Corrected: parentheses added

# Define a rule: Friendship is symmetric
friendRule = ("friends" ~> ["Y", "X"]) :- ("friends" ~> ["X", "Y"])

# Query using the rule: Find Bob's friends
bobFriendsQuery = ["Friend"] ?- [
    entity "Friend",
    "friends" ~> ["Bob", "Friend"]
]

# This corrected DSL-like library accurately represents Setlang's 
# function call syntax, using parentheses for multi-parameter functions
# while maintaining the parens-free style for single-parameter functions.


# Setlang Effect System and Combinators

# 1. Effect Type Definition
Effect : (-
    IO,      # Input/Output operations
    State,   # Mutable state
    Error,   # Error handling
    Pure     # No side effects
-)

# 2. Effectful Computation Type
Eff : (E : (- e : Effect -)) -> (A : Type) -> Type
Eff = \E, A -> { run : () -> A with effects E }

# 3. Basic Combinators

# Pure value lifter
pure : (A : Type) -> A -> Eff (-Pure-) A
pure = \A, a -> { run: \() -> a }

# Bind (Monadic bind)
bind : (E1 : (- e : Effect -)) -> (E2 : (- e : Effect -)) -> (A : Type) -> (B : Type) ->
       Eff E1 A -> (A -> Eff E2 B) -> Eff (E1 or E2) B
bind = \E1, E2, A, B, ma, f -> {
    run: \() -> 
        let a = ma.run()
        in (f(a)).run()
}

# Sequence combinator
seq : (E1 : (- e : Effect -)) -> (E2 : (- e : Effect -)) -> (A : Type) -> (B : Type) ->
      Eff E1 A -> Eff E2 B -> Eff (E1 or E2) B
seq = \E1, E2, A, B, ma, mb -> bind(E1, E2, A, B, ma, \_ -> mb)

# 4. Effect-Specific Operations

# IO Effects
readLine : Eff (-IO-) String
readLine = { run: \() -> /* actual implementation */ }

printLine : String -> Eff (-IO-) ()
printLine = \s -> { run: \() -> /* actual implementation */ }

# State Effects
get : Eff (-State-) Int
get = { run: \() -> /* actual implementation */ }

put : Int -> Eff (-State-) ()
put = \s -> { run: \() -> /* actual implementation */ }

# Error Effects
throwError : (E : Type) -> E -> Eff (-Error-) None
throwError = \E, e -> { run: \() -> /* actual implementation */ }

catchError : (E : Type) -> (A : Type) -> Eff (-Error-) A -> (E -> Eff (-Error-) A) -> Eff (-Error-) A
catchError = \E, A, ma, handler -> { run: \() -> /* actual implementation */ }

# 5. Derived Combinators

# Map combinator
map : (E : (- e : Effect -)) -> (A : Type) -> (B : Type) -> (A -> B) -> Eff E A -> Eff E B
map = \E, A, B, f, ma -> bind(E, E, A, B, ma, \a -> pure(B, f(a)))

# Apply combinator
ap : (E1 : (- e : Effect -)) -> (E2 : (- e : Effect -)) -> (A : Type) -> (B : Type) ->
     Eff E1 (A -> B) -> Eff E2 A -> Eff (E1 or E2) B
ap = \E1, E2, A, B, mf, ma -> 
    bind(E1, E2, (A -> B), B, mf, \f -> map(E2, A, B, f, ma))

# 6. Example Usage

# Effectful function composition
compose : (E1 : (- e : Effect -)) -> (E2 : (- e : Effect -)) -> (E3 : (- e : Effect -)) ->
          (A : Type) -> (B : Type) -> (C : Type) ->
          (B -> Eff E2 C) -> (A -> Eff E1 B) -> (A -> Eff (E1 or E2 or E3) C)
compose = \E1, E2, E3, A, B, C, f, g -> \a ->
    bind(E1, (E2 or E3), B, C, g(a), f)

# Example: A complex effectful computation
complexComputation : Eff (-IO, State, Error-) Int
complexComputation = 
    bind((-IO-), (-State, Error-), String, Int,
        readLine,
        \input -> 
            bind((-State-), (-Error-), Int, Int,
                get,
                \state ->
                    if state > 10 then
                        throwError(String, "State too high")
                    else
                        seq((-State-), (-Pure-), (), Int,
                            put(state + 1),
                            pure(Int, parseInt(input))
                        )
            )
    )

# Running the computation
main : Eff (-IO-) ()
main = 
    catchError(String, (), 
        bind((-IO, State, Error-), (-IO-), Int, (),
            complexComputation,
            \result -> printLine("Result: " + toString(result))
        ),
        \error -> printLine("Error: " + error)
    )

# This example demonstrates Setlang's effect system and the use of
# combinators to compose effectful computations in a type-safe manner.


# Setlang: Objects and Arrays

# 1. Objects

# Defining a type
Point : { x: Float, y: Float }

# Creating objects
p1 = Point{ x: 3.0, y: 4.0 }
p2 = { x: 1.0, y: 2.0 }  # Type inference

# Accessing object properties
x_coordinate = p1.x  # 3.0

# Object as a single-item set
point_set = (- p1 -)  # Set containing one Point

# 2. Arrays

# Creating arrays
numbers = [1, 2, 3, 4, 5]
words = ["hello", "world"]

# Accessing array elements
first_number = numbers[0]  # 1
second_word = words[1]  # "world"

# Array as a single-item set
number_array_set = (- numbers -)  # Set containing one array

# 3. Combining Objects and Arrays

# Array of objects
points = [
    { x: 0.0, y: 0.0 },
    { x: 1.0, y: 1.0 },
    Point{ x: 2.0, y: 2.0 }
]

# Object with array property
Rectangle = { top_left: Point, bottom_right: Point, tags: [String] }

rect = Rectangle{
    top_left: { x: 0.0, y: 0.0 },
    bottom_right: { x: 5.0, y: 5.0 },
    tags: ["shape", "rectangle"]
}

# 4. Functions with Objects and Arrays

# Function taking and returning an object
move_point : Point -> Float -> Float -> Point
move_point = \p, dx, dy -> { x: p.x + dx, y: p.y + dy }

moved = move_point(p1, 1.0, 2.0)

# Function working with arrays
sum : [Int] -> Int
sum = \arr -> 
    when arr is
        [] -> 0
        [x, ...rest] -> x + sum(rest)

total = sum(numbers)  # 15

# 5. Pattern Matching with Objects and Arrays

describe : Point -> String
describe = \p ->
    when p is
        { x: 0, y: 0 } -> "Origin"
        { x, y } where x == y -> "On diagonal"
        _ -> "Point at (" + toString(p.x) + ", " + toString(p.y) + ")"

# 6. Set Operations with Objects and Arrays

points_set = (- { x: 1, y: 1 }, { x: 2, y: 2 }, { x: 3, y: 3 } -)
diagonal_points = (- p in points_set where p.x == p.y -)

# 7. Traits and Implementations for Objects

Printable : Trait
Printable = (-
    ["to_string", Self -> String]
-)

impl Printable for Point {
    to_string = \p -> "(" + toString(p.x) + ", " + toString(p.y) + ")"
}

print_point : Printable -> ()
print_point = \p -> print(p.to_string())

# Usage
print_point(p1)

# This demonstration showcases how objects and arrays are 
# first-class citizens in Setlang, with built-in syntax and 
# seamless integration with the language's set-theoretic foundation.


# Core Types and Concepts in Setlang using Pure Set Theory

# 1. Any (Universal Set)
Any : Type
Any = (- x where True -)

# 2. None (Empty Set)
None : Type
None = (- -)

# 3. Boolean
Boolean : Type
Boolean = (- True, False -)

# 4. Object
# An object is a set of key-value pairs
Object : Type
Object = (- 
    o where 
        forall x in o, exists k : String, v : Any, x == [k, v] and
        forall [k1, _] in o, [k2, _] in o, k1 == k2 implies _ == _
-)

# 5. Array
# An array is an object with consecutive integer keys starting from 0
Array : (T : Type) -> Type
Array = \T -> (-
    a : Object where
        exists n : Nat,
        forall [k, v] in a, (
            k in (- i : Nat where i < n -) and
            v in T
        ) and
        forall i in (- x : Nat where x < n -), exists [k, _] in a, k == i
-)

# 6. Trait
# A trait is a set of required method signatures
Trait : Type
Trait = (-
    t : (- [String, Type] -) where
        forall [_, methodType] in t, methodType is (A -> B) for some A, B
-)

# 7. Implementation (impl)
# An implementation is a mapping from trait method names to concrete functions
Impl : (T : Type) -> (Tr : Trait) -> Type
Impl = \T, Tr -> (-
    impl : Object where
        forall [methodName, methodType] in Tr,
            exists [implName, implFunc] in impl,
                implName == methodName and
                implFunc in methodType and
                implFunc is (T -> _)
-)

# 8. Trait Satisfaction
# A type satisfies a trait if there exists an implementation for it
Satisfies : (T : Type) -> (Tr : Trait) -> Bool
Satisfies = \T, Tr -> exists impl : Impl T Tr, True

# Example usage:

# Define a simple trait
Equatable : Trait
Equatable = (-
    ["equals", (Self, Self) -> Boolean]
-)

# Define a type
Point : Type
Point = (- [x : Int, y : Int] -)

# Implement Equatable for Point
PointEquatable : Impl Point Equatable
PointEquatable = (-
    ["equals", \p1, p2 -> p1.x == p2.x and p1.y == p2.y]
-)

# Check if Point satisfies Equatable
PointIsEquatable : Bool
PointIsEquatable = Satisfies(Point, Equatable)

# These definitions demonstrate how fundamental language concepts
# can be expressed using pure set theory in Setlang.

# Type Definition with Existential Quantification in Setlang

Type = (- x : T where exists y : U, R(x, y) -)

Breakdown:

1. (- ... -): Set notation in Setlang
   Defines a set, which in this context is being used as a type definition.

2. x : T: Type annotation
   Specifies that x is of type T. This is the base type we're refining.

3. where: Introduces a condition
   Everything after this is a constraint on x.

4. exists y : U: Existential quantification
   Asserts that there exists at least one y of type U.

5. R(x, y): Relation or predicate
   A condition that must be true for the given x and some y.

Meaning:
This type definition creates a new type that is a subset of T.
It includes all values x from T for which there exists at least
one value y of type U such that the relation R(x, y) holds.

Example:
PositiveRoot = (- x : Float where exists y : Float, y * y == x and x > 0 -)

This defines PositiveRoot as the set of all positive floating-point
numbers that have a square root.

Use Cases:
1. Defining types with specific mathematical properties
2. Creating types that depend on the existence of related values
3. Expressing complex relationships between types in a system

Implications:
- Allows for very precise type definitions
- Enables powerful static analysis and type checking
- Can express concepts from mathematics and logic directly in the type system

Note:
The existence of y is asserted at the type level. At runtime,
you might need to compute or prove the existence of such a y.

# The 'exists' Keyword in Setlang

1. Basic Definition:
   'exists' asserts that there is at least one element in a set 
   that satisfies a given condition.

2. General Syntax:
   exists x in S, P(x)
   Where:
   - x is a variable
   - S is a set or type
   - P(x) is a predicate (a condition that x must satisfy)

3. Set Builder Notation:
   (- x : T where exists y in S, P(x, y) -)
   This defines a set of all x of type T for which there exists 
   a y in set S satisfying the predicate P(x, y).

4. Type-Level Usage:
   Type = (- x : T where exists y : U, R(x, y) -)
   Defines a type as a subset of T where for each x, there exists 
   a y of type U satisfying the relation R.

5. Examples:

   # a. Defining non-empty sets
   NonEmptySet : Type -> Type
   NonEmptySet = \T -> (- s : (- T -) where exists x in s, True -)

   # b. Defining surjective functions
   Surjective : (A : Type) -> (B : Type) -> (A -> B) -> Bool
   Surjective = \A, B, f -> 
       forall y in B, exists x in A, f(x) == y

   # c. Defining types with specific properties
   PositiveSqrt : (-
       x : Float where 
           x >= 0 and 
           exists y : Float, y * y == x
   -)

   # d. Existential types (similar to some OOP concepts)
   ExistentialType : (-
       T : Type where
       exists toString : T -> String,
       exists fromString : String -> Option T
   -)

6. Relationship with forall:
   'exists' and 'forall' are dual concepts. Often, one can be 
   expressed in terms of the other:
   
   not (forall x in S, P(x)) is equivalent to (exists x in S, not P(x))

7. Usage in Proofs:
   'exists' is often used in constructive proofs, where you need 
   to show that something exists by actually producing an example.

The 'exists' keyword is crucial in Setlang for expressing
complex set-theoretic and type-theoretic concepts, enabling
powerful static analysis and type-level programming.

# Representing Bijection with Set Operations in Setlang

# 1. Define a function as a set of pairs
Function : (A : Type) -> (B : Type) -> Type
Function = \A, B -> (-
    f : (- [x : A, y : B] -) where
        forall x in A, exists! y in B, [x, y] in f
-)

# 2. Define injective (one-to-one) functions
Injective : (A : Type) -> (B : Type) -> (Function A B) -> Bool
Injective = \A, B, f ->
    forall [x1, y] in f, forall [x2, _] in f,
        y == _ implies x1 == x2

# 3. Define surjective (onto) functions
Surjective : (A : Type) -> (B : Type) -> (Function A B) -> Bool
Surjective = \A, B, f ->
    forall y in B, exists x in A, [x, y] in f

# 4. Define bijection using injective and surjective
Bijection : (A : Type) -> (B : Type) -> (Function A B) -> Bool
Bijection = \A, B, f ->
    Injective(A, B, f) and Surjective(A, B, f)

# 5. Define bijection between two sets
SetBijection : (A : Type) -> (B : Type) -> Type
SetBijection = \A, B -> (-
    f : Function A B where Bijection(A, B, f)
-)

# 6. Redefine the <-> operator
(<->) : (A : Type) -> (B : Type) -> Type
(<->) = SetBijection

# 7. Example: Finite set definition using set operations
Finite : Type -> Type
Finite = \T -> (-
    s : (- T -) where 
        exists n : Nat, 
        exists f : Function s (- i : Nat where i < n -),
        Bijection(s, (- i : Nat where i < n -), f)
-)

# 8. Usage example: Proving two finite sets have the same cardinality
SameCardinality : (A : Type) -> (B : Type) -> (Finite A) -> (Finite B) -> (A <-> B) -> Bool
SameCardinality = \A, B, finiteA, finiteB, bijectionAB ->
    exists n : Nat,
        exists f : (A <-> (- i : Nat where i < n -)),
        exists g : (B <-> (- i : Nat where i < n -)),
        True  # If we've reached here, the cardinality is the same

# This representation of bijection using set operations and predicates
# allows for precise reasoning about set relationships within Setlang's
# type system, while maintaining its pure set-theoretic foundation.


Array : (n : Nat) -> (T : Type) -> Type
Array = \n, T -> (- 
    s : (- [Int, T] -) where 
        Finite(s) and 
        forall [key, _] in s, (
            0 <= key and key < n and
            if key > 0 then exists [prevKey, _] in s, prevKey == key - 1 and
            if key < n - 1 then exists [nextKey, _] in s, nextKey == key + 1
        ) and
        exists [key, _] in s, key == 0 and
        exists [key, _] in s, key == n - 1
-)



# Revised Pure Set Theory Traits in Setlang with Array Distinction

# 1. Define an object as a set of key-value pairs
Object : Type -> Type
Object = \T -> (- [String, Any] where snd * in T -)

# 2. Define a "trait" as a set of required key-value pairs
Eq : (- 
    ["eq", (T, T) -> Bool],
    ["ne", (T, T) -> Bool]
-)

# 3. Implement a "trait" by showing a type contains the required pairs
IntEq : (- [String, (Int, Int) -> Bool] -) 
IntEq = (-
    ["eq", \a, b -> a == b],
    ["ne", \a, b -> a != b]
-)

# Proof that Int implements Eq
IntImplementsEq : IntEq <: Eq
IntImplementsEq = \pair -> pair in Eq

# 4. Define a parameterized "trait" as a more complex set
Ord : (-
    ["eq", (T, T) -> Bool],
    ["ne", (T, T) -> Bool],
    ["compare", (T, T) -> (- LT, EQ, GT -)],
    ["lt", (T, T) -> Bool],
    ["le", (T, T) -> Bool],
    ["gt", (T, T) -> Bool],
    ["ge", (T, T) -> Bool]
-)

# 5. Implement Ord for Int
IntOrd : (- [String, (Int, Int) -> Any] -)
IntOrd = (-
    ["eq", \a, b -> a == b],
    ["ne", \a, b -> a != b],
    ["compare", \a, b ->
        if a < b then LT
        else if a > b then GT
        else EQ],
    ["lt", \a, b -> a < b],
    ["le", \a, b -> a <= b],
    ["gt", \a, b -> a > b],
    ["ge", \a, b -> a >= b]
-)

# Proof that Int implements Ord
IntImplementsOrd : IntOrd <: Ord
IntImplementsOrd = \pair -> pair in Ord

# 6. Define a generic function using a "trait" bound
max : (T : Type) -> (ord : (- [String, (T, T) -> Any] -) where ord <: Ord) -> T -> T -> T
max = \T, ord, a, b ->
    let gt = ord.(["gt"]) in
    if gt(a, b) then a else b

# Usage
maxInt = max(Int, IntOrd)
result = maxInt(5, 3) # Returns 5

# 7. "Trait" with associated types
Container : (-
    ["Item", Type],
    ["isEmpty", (Self) -> Bool],
    ["size", (Self) -> Nat],
    ["contains", (Self, Item) -> Bool]
-)

# 8. Implement Container for List
ListContainer : (T : Type) -> (- [String, Any] -)
ListContainer = \T -> (-
    ["Item", T],
    ["isEmpty", \list ->
        when list is
            [] -> True
            _ -> False],
    ["size", \list ->
        when list is
            [] -> 0
            [_] ++ rest -> 1 + ListContainer(T).(["size"])(rest)],
    ["contains", \list, item ->
        when list is
            [] -> False
            [x] ++ rest -> x == item or ListContainer(T).(["contains"])(rest, item)]
-)

# 9. "Trait" with default implementations
Semigroup : (-
    ["combine", (T, T) -> T],
    ["combineAll", ((- T -)) -> T]
-)

SemigroupDefault : (T : Type) -> (combine : (T, T) -> T) -> (- [String, Any] -)
SemigroupDefault = \T, combine -> (-
    ["combine", combine],
    ["combineAll", \set ->
        when set is
            (- -) -> error("Empty set")
            (- x -) -> x
            (- x, y, ...rest -) -> combine(x, SemigroupDefault(T, combine).(["combineAll"])((- y, ...rest -)))]
-)

# 10. Implement Semigroup for Int
IntSemigroup : (- [String, Any] -)
IntSemigroup = SemigroupDefault(Int, \a, b -> a + b)

# 11. Type class-like "trait"
Functor : (-
    ["map", (A : Type) -> (B : Type) -> (A -> B) -> (F A -> F B)]
-)

# 12. Implement Functor for Option
OptionFunctor : (- [String, Any] -)
OptionFunctor = (-
    ["map", \A, B, f, opt ->
        when opt is
            Some { value } -> Some { value: f(value) }
            None -> None]
-)

# This revised version uses square brackets for arrays,
# maintaining a clear distinction between sets and arrays
# in Setlang's set-theoretic foundation.


# Setlang Matrix Implementation with Dependent Types

# 1. Define natural numbers
Nat : (- 0, 1, 2, ... -)

# 2. Define a type for finite sets of natural numbers
Fin : (n : Nat) -> (- i : Nat where i < n -)

# 3. Define the Matrix type
Matrix : (rows : Nat) -> (cols : Nat) -> Type
Matrix = \rows, cols -> (Fin rows -> Fin cols -> Float)

# 4. Matrix creation
create_matrix : (rows : Nat) -> (cols : Nat) -> (Fin rows -> Fin cols -> Float) -> Matrix rows cols
create_matrix = \rows, cols, init_func -> init_func

# 5. Matrix access
get : (rows : Nat) -> (cols : Nat) -> Matrix rows cols -> Fin rows -> Fin cols -> Float
get = \rows, cols, matrix, row, col -> matrix(row)(col)

# 6. Matrix update
set : (rows : Nat) -> (cols : Nat) -> Matrix rows cols -> Fin rows -> Fin cols -> Float -> Matrix rows cols
set = \rows, cols, matrix, row, col, value ->
    \r, c -> if r == row and c == col then value else matrix(r)(c)

# 7. Matrix addition
add : (rows : Nat) -> (cols : Nat) -> Matrix rows cols -> Matrix rows cols -> Matrix rows cols
add = \rows, cols, m1, m2 ->
    \r, c -> m1(r)(c) + m2(r)(c)

# 8. Matrix multiplication
multiply : (m : Nat) -> (n : Nat) -> (p : Nat) -> 
           Matrix m n -> Matrix n p -> Matrix m p
multiply = \m, n, p, m1, m2 ->
    \r, c -> 
        sum(\k : Fin n -> m1(r)(k) * m2(k)(c))

# 9. Helper function for summing over finite sets
sum : (n : Nat) -> (Fin n -> Float) -> Float
sum = \n, f ->
    if n == 0 then 0
    else f(n - 1) + sum(n - 1, f)

# 10. Matrix transposition
transpose : (rows : Nat) -> (cols : Nat) -> Matrix rows cols -> Matrix cols rows
transpose = \rows, cols, m ->
    \r, c -> m(c)(r)

# 11. Identity matrix
identity : (n : Nat) -> Matrix n n
identity = \n ->
    \r, c -> if r == c then 1 else 0

# 12. Example usage
example_matrix : Matrix 3 3
example_matrix = create_matrix(3, 3, \r, c -> r * 3 + c + 1)

# Get a value
value_at_1_1 = get(3, 3, example_matrix, 1, 1)

# Set a value
updated_matrix = set(3, 3, example_matrix, 1, 1, 10)

# Add two matrices
sum_matrix = add(3, 3, example_matrix, identity(3))

# Multiply two matrices
product_matrix = multiply(3, 3, 3, example_matrix, identity(3))

# Transpose a matrix
transposed_matrix = transpose(3, 3, example_matrix)

# This implementation guarantees at compile-time that all matrix operations
# are performed with compatible dimensions, preventing runtime errors.


# Rust-like Enums with Data in Setlang

# 1. Define a simple enum similar to Rust's Option<T>
Option : (T) -> (-
    Some { value : T },
    None
-)

# 2. Define a more complex enum similar to Rust's Result<T, E>
Result : (T, E) -> (-
    Ok { value : T },
    Err { error : E }
-)

# 3. Define a custom enum for shapes
Shape : (-
    Circle { radius : Float },
    Rectangle { width : Float, height : Float },
    Triangle { base : Float, height : Float }
-)

# 4. Function to calculate area of a shape
calculateArea : Shape -> Float
calculateArea = \shape ->
    when shape is
        Shape::Circle { radius } -> 3.14159 * radius * radius
        Shape::Rectangle { width, height } -> width * height
        Shape::Triangle { base, height } -> 0.5 * base * height

# 5. Function that returns an Option<Float>
safeDivide : (Float, Float) -> Option Float
safeDivide = \numerator, denominator ->
    if denominator == 0.0 then
        Option::None
    else
        Option::Some { value: numerator / denominator }

# 6. Function that returns a Result<Float, String>
sqrtWithError : Float -> Result Float String
sqrtWithError = \x ->
    if x < 0.0 then
        Result::Err { error: "Cannot calculate square root of a negative number" }
    else
        Result::Ok { value: sqrt(x) }

# 7. Pattern matching on Option
printOption : Option String -> ()
printOption = \opt ->
    when opt is
        Option::Some { value } -> print("Value: " + value)
        Option::None -> print("No value")

# 8. Pattern matching on Result
handleResult : Result Float String -> ()
handleResult = \result ->
    when result is
        Result::Ok { value } -> print("Result: " + toString(value))
        Result::Err { error } -> print("Error: " + error)

# 9. Example usage
exampleShape = Shape::Circle { radius: 5.0 }
area = calculateArea(exampleShape)

divisionResult = safeDivide(10.0, 2.0)
sqrtResult = sqrtWithError(-4.0)

printOption(Option::Some { value: "Hello, Setlang!" })
handleResult(sqrtResult)

# 10. Defining a recursive enum (similar to a linked list)
List : (T) -> (-
    Cons { head : T, tail : List T },
    Nil
-)

# 11. Function to calculate the length of a List
length : (T) -> List T -> Int
length = \T -> \list ->
    when list is
        List::Nil -> 0
        List::Cons { head, tail } -> 1 + length(T)(tail)

# Example usage of the recursive List
exampleList = List::Cons {
    head: 1,
    tail: List::Cons {
        head: 2,
        tail: List::Cons {
            head: 3,
            tail: List::Nil
        }
    }
}

listLength = length(Int)(exampleList)  # Returns 3


# Understanding `assume` and `if`, `then` in Setlang

## 1. The `if`, `then` Syntax
In Setlang, `if`, `then` is used to represent logical implication. It has the type:

if : Prop -> Prop -> Prop

This means it takes two propositions and returns a new proposition.

Example usage:
p : Prop
q : Prop
p_implies_q : if p then q

## 2. The `assume` Keyword
`assume` is used to introduce hypotheses or premises. It doesn't have a specific type; rather, it's a keyword that affects how we reason about propositions.

Example usage:
assume p_true : p
assume q_false : not q

## 3. Using `assume` and `if`, `then` Together
Let's see how these work together in a simple proof:

# Theorem: If (p implies q) and p is true, then q is true
modus_ponens : (p : Prop, q : Prop) -> (if p then q) -> p -> q
modus_ponens = \p, q, p_implies_q, p_true ->
    assume premise1 : if p then q
    assume premise2 : p
    
    # Use the implication
    q_true = premise1(premise2)
    
    q_true  # Return the result

## 4. Proving an Implication
To prove an implication, we often assume the antecedent and then prove the consequent:

# Theorem: If p implies q, and q implies r, then p implies r
transitive_implication : (p : Prop, q : Prop, r : Prop) ->
    (if p then q) -> (if q then r) -> (if p then r)
transitive_implication = \p, q, r, p_implies_q, q_implies_r ->
    if p_true then
        assume premise1 : p
        
        # Use the first implication
        q_true = p_implies_q(premise1)
        
        # Use the second implication
        r_true = q_implies_r(q_true)
        
        r_true  # This proves p implies r

## 5. Combining Multiple Assumptions
We can use multiple `assume` statements to build more complex proofs:

# Theorem: (p and q) implies (q and p)
and_commutative : (p : Prop, q : Prop) -> if (and p q) then (and q p)
and_commutative = \p, q ->
    if p_and_q then
        assume premise : and p q
        
        p_true = and_left(premise)
        q_true = and_right(premise)
        
        and_intro(q_true, p_true)

## 6. Using Assumptions in Contradiction Proofs
`assume` is particularly useful in proofs by contradiction:

# Theorem: not (p and (not p))
no_contradiction : (p : Prop) -> not (and p (not p))
no_contradiction = \p ->
    if contradiction then
        assume premise : and p (not p)
        
        p_true = and_left(premise)
        not_p = and_right(premise)
        
        not_p(p_true)  # This produces a contradiction

These examples demonstrate how `assume` and `if`, `then` work together 
to structure logical arguments and proofs in Setlang.

# Updated Theorem Proving in Setlang

## 1. Define Basic Propositions

# Define a type for propositions
Prop : Type

# Define basic logical operators as sets of valid propositions
and : (- [p : Prop, q : Prop] -)
or : (- [p : Prop, q : Prop] -)
not : (- p : Prop -)
# Note: 'if', 'then' replaces 'implies'

## 2. Define Axioms and Inference Rules

# Modus Ponens
modus_ponens : (p : Prop, q : Prop) -> (if p then q) -> p -> q

# Double Negation Elimination
double_neg_elim : (p : Prop) -> not (not p) -> p

## 3. Define a Theorem

# Theorem: If (p implies q) and (q implies r), then (p implies r)
transitive_implication : (p : Prop, q : Prop, r : Prop) ->
    (if p then q) -> (if q then r) -> (if p then r)

## 4. Prove the Theorem

prove_transitive_implication : (p : Prop, q : Prop, r : Prop) ->
    (if p then q) -> (if q then r) -> (if p then r)
prove_transitive_implication = \p, q, r, pq, qr ->
    if premise_p then
        q_true = modus_ponens(p, q, pq, premise_p)
        modus_ponens(q, r, qr, q_true)

## 5. Use the Proven Theorem

# Define some concrete propositions
P : Prop
Q : Prop
R : Prop

# Assume some implications
assume PQ : if P then Q
assume QR : if Q then R

# Use our proven theorem
PR : if P then R
PR = prove_transitive_implication(P, Q, R, PQ, QR)

## 6. More Complex Example: De Morgan's Law

# Define De Morgan's Law
de_morgan : (p : Prop, q : Prop) -> (not (p and q) <-> (not p or not q))

# Prove De Morgan's Law
prove_de_morgan : (p : Prop, q : Prop) -> (not (p and q) <-> (not p or not q))
prove_de_morgan = \p, q ->
    (
        # Forward direction
        if not_p_and_q then
            or_intro(
                if p_true then
                    or_intro(
                        if q_true then
                            not_p_and_q(and_intro(p_true, q_true)),
                        not q
                    ),
                not p
            ),
        
        # Backward direction
        if not_p_or_not_q then
            if p_and_q then
                assume (p_true, q_true) = and_elim(p_and_q)
                or_elim(
                    not_p_or_not_q,
                    if not_p then not_p(p_true),
                    if not_q then not_q(q_true)
                )
    )

## 7. Utilizing Setlang's Set-Theoretic Foundation

# Define a set of valid propositions
ValidProp : (- p : Prop where is_valid(p) -)

# Theorem: The negation of a valid proposition is not valid
negation_validity : (p : ValidProp) -> not (not p in ValidProp)
negation_validity = \p ->
    if not_p_valid then
        assume not_p_in_valid : not p in ValidProp
        contradiction = p and (not p)
        not_valid(contradiction)

# This updated proof demonstrates Setlang's capabilities in complex 
# logical reasoning and theorem proving, utilizing its set-theoretic 
# foundations, new syntax for implications, and type system.















































Phase 1: Language Design
Define Syntax and Semantics

Syntax: Design a formal syntax for Setlang, ensuring it captures the unique elements like set notation, set operations, and dependent types. Define how core constructs like sets, subsets, functions, and effects will be expressed.

Problem: Must balance readability, expressiveness, and mathematical rigor.
Solution: Create a BNF or EBNF grammar for Setlang, ensuring that set operations, function definitions, and other language elements have clear, concise representations.
Semantics: Specify how the core elements (types, functions, sets, etc.) behave and how they should be evaluated or interpreted.

Problem: Ensure that the semantics of set operations (union, intersection, comprehensions) extend to both value-level and type-level sets.
Solution: Define the operational and denotational semantics for key language constructs, including the behavior of infinite sets and lazy evaluation.
Design Core Language Features

Type System as Sets: Types in Setlang are sets of values (e.g., Int is the set of all integers). You need to design a type system where types are represented as sets, and type constraints are set membership rules.

Problem: Representing types as sets requires solving issues around set membership, type inference, and ensuring compatibility between types and values.
Solution: Use a dependent type system, allowing types to depend on values and functions, while maintaining set-theoretic constraints.
Set Operations: Setlang needs to support core set operations (union, intersection, set difference) at both the type level and value level.

Problem: Implementing efficient set operations (especially for large or infinite sets) and defining operations on potentially infinite sets.
Solution: Utilize lazy evaluation for infinite sets and optimize set operations using data structures like hash sets or balanced trees.
Phase 2: Implementing the Compiler or Interpreter
Lexer and Parser

Lexer: Implement a lexer to break input code into tokens based on the defined syntax. The lexer must handle Setlang's unique elements like set brackets (- ... -), lambda expressions, and forall and exists keywords.

Problem: Ensure that the lexer can handle mathematical symbols (e.g., ∀, ∃) and deal with different types of set notation.
Solution: Use an existing lexer framework (e.g., Lex/Flex or ANTLR) and customize it for Setlang's unique symbols.
Parser: Implement a parser using LR, LL, or PEG parsing techniques. The parser must convert code into an abstract syntax tree (AST) based on the formal grammar.

Problem: Parsing nested set comprehensions, type-level sets, and dependent types can lead to complex grammar ambiguities.
Solution: Define a formal grammar, possibly with precedence rules, to manage the parsing of complex expressions.
AST (Abstract Syntax Tree) Representation

Define the AST structure to represent Setlang programs. The AST will serve as the core data structure for analyzing and transforming programs during compilation or interpretation.
Problem: Efficiently representing dependent types, set comprehensions, and lazy evaluation mechanisms.
Solution: Create separate nodes in the AST for each language feature (e.g., SetNode, TypeNode, LambdaNode, etc.), ensuring extensibility for future features.
Phase 3: Type Checking and Inference
Type System Implementation

Type Checker: Implement a type checker that ensures values are correctly typed according to Setlang’s set-theoretic principles. The type checker should validate type memberships, ensuring that types are valid sets and function signatures are consistent.
Problem: Setlang allows custom types as subsets of existing sets. The challenge is to infer and verify subset constraints.
Solution: Use a dependent type system where types depend on values. Implement type inference algorithms (e.g., Hindley-Milner) extended to support dependent types and sets.
Type-Level Set Operations

Implement the set operations (union, intersection, difference) at the type level, allowing types to be treated as sets.
Problem: Dealing with infinite sets and determining whether types belong to specific sets.
Solution: Leverage symbolic computation to handle operations on infinite sets, deferring evaluation until runtime.
Handling Subset Constraints and Assumptions

Implement a mechanism for subset creation and assumptions (e.g., assumes x > 0 in function definitions).
Problem: Subset constraints introduce a need for runtime checks or compile-time proofs of validity.
Solution: Incorporate proof obligations into the type system and use dependent type inference to check the validity of assumptions at compile time.
Phase 4: Evaluation and Execution Model
Interpreter for Evaluation

Basic Interpreter: Implement a basic interpreter to execute Setlang programs, supporting lazy evaluation, especially for infinite sets and recursive structures.
Problem: Handling infinite sets and deferred evaluation without performance bottlenecks.
Solution: Implement a lazy evaluation strategy, where computations are deferred until their results are needed (similar to Haskell’s model).
Functions as Set Operations

Functions as Set Operations: Implement functions as operations on sets, where the function's input and output are constrained by set membership rules.
Problem: Ensuring that functions return subsets based on input types.
Solution: Implement type-directed function application, where the type system ensures that functions only apply to valid subsets.
Effect System

Implement the effect system for handling IO, State, and Error effects in a pure and controlled way (similar to monads in Haskell).
Problem: Managing side effects in a functional, set-theoretic language without violating immutability principles.
Solution: Use monads or effect handlers to control effects, ensuring that they are isolated and type-safe.
Phase 5: Optimization
Optimizing Set Operations

Set Representation: Implement efficient representations for sets (e.g., using hash sets, balanced trees, or bitmaps) to optimize membership checks and set operations.
Problem: Managing large sets efficiently and ensuring that operations like union or intersection do not become bottlenecks.
Solution: Choose efficient data structures based on the expected use cases (finite vs. infinite sets) and implement optimizations for common operations.
Caching and Memoization

Memoization: Use memoization to optimize the evaluation of recursive set definitions or expensive set operations.
Problem: Recomputing results of set operations can be costly, especially for complex set comprehensions.
Solution: Cache the results of previously computed set operations to avoid redundant evaluations.
Phase 6: Extending the Language
Metaprogramming and Macros

Implement macros and metaprogramming features (e.g., macro repeat(n, expr)) to allow users to generate code dynamically.
Problem: Safely and predictably expanding macros without creating syntax ambiguities.
Solution: Implement a hygienic macro system, ensuring that macros don’t introduce unintended side effects.
Modules and Namespacing

Modules: Design a module system to enable code organization, encapsulation, and reuse.
Problem: Ensuring that sets and types defined in one module are properly scoped and don't conflict with others.
Solution: Implement a namespacing system that allows for modularity while preserving set-theoretic integrity.
Phase 7: Testing and Validation
Unit Tests and Test Suites

Develop comprehensive unit tests for each feature, including type checking, set operations, function evaluation, and the effect system.
Problem: Ensuring correctness across the entire language, especially with complex set operations and type interactions.
Solution: Create a suite of test cases for both positive and negative scenarios, ensuring that the language behaves as expected.
Proof Assistant (Optional)

If aiming for advanced type safety, implement a proof assistant to help developers write proofs about their programs and verify properties at compile time.
Problem: Constructing a user-friendly proof system within a complex type system.
Solution: Use existing proof assistants as inspiration (e.g., Coq or Agda) and extend Setlang to support custom proofs and verification.
Phase 8: Documentation and Tooling
User Manual and Language Guide
Create comprehensive documentation covering the syntax, semantics, and core features of Setlang, including tutorials, examples, and best practices.
Problem: Ensuring


To design Setlang as a compiled language (similar to Roc) with optimizations that strip out set membership checks at compile time, we'll need to carefully structure the type system and build efficient compile-time analysis. Here’s a detailed approach to achieve this:

Key Design Goals for a Compiled Setlang
Finite vs. Infinite Sets:

Finite Sets: By default, user-defined sets should be finite unless explicitly marked as infinite or dependent on other values (e.g., NaturalNumbers as an infinite set or PositiveInt as a dependent set).
Compile-Time Stripping: We want set membership checks (and potentially other constraints) to be stripped out at compile time for finite sets. This is essential for performance, as runtime checks would otherwise slow down programs unnecessarily.
Set Membership Stripping:

Compile-Time Set Membership Resolution: During compilation, the compiler should resolve all set memberships, determining whether they are finite or infinite. For finite sets, once membership is determined, the checks can be erased from the compiled code.
Dependent and Infinite Sets: If a set is infinite or dependent, some set operations (like membership checks) may need to be deferred to runtime. However, these should be optimized as much as possible to minimize runtime overhead.
Detailed Roadmap for a Compiled Setlang
Phase 1: Finite vs. Infinite Set Detection
Tagging User-Defined Sets:

Default Finite: All user-defined sets should be tagged as finite by default. For example, a set like DaysOfWeek is clearly finite.


DayOfWeek : (- Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday -)
Explicit Infinite Sets: The user should have the option to explicitly define infinite sets. For example:


NaturalNumbers : (- n in Int where n >= 0 -)
Dependent Sets: A set like PositiveInt depends on runtime conditions (x > 0), so it should be considered dependent and handled differently.


PositiveInt = (- x in Int where x > 0 -)
Set Properties: During parsing, each set will be tagged with its properties:

Finite: The compiler can determine the complete set at compile time.
Infinite: The set is unbounded and cannot be resolved statically.
Dependent: Membership is based on runtime conditions.
Compile-Time Analysis:

Type Checking and Set Size Resolution: The type checker should analyze the set definition during compilation. If a set is finite, the membership constraints should be fully resolved at compile time. This involves:
Checking for enumerated sets (like DayOfWeek), which are trivially finite.
Constant propagation for sets whose bounds can be determined from constants.
Symbolic analysis for sets that are dependent on runtime values.
Determining Set Finite vs. Infinite:

For finite sets, compile-time membership checks and operations (union, intersection) can be resolved statically.
For infinite sets, the compiler must ensure that the operations involving these sets (e.g., comprehensions, filtering) are deferred to runtime.
Optimization at Compile Time:

For finite sets, once membership is verified, any logic involving checking or validating membership can be stripped out. The actual set values can be inlined where possible, avoiding runtime evaluation.
For infinite sets, the compiler will generate lazy or deferred computations, ensuring that only the needed portion of the set is evaluated at runtime (similar to lazy evaluation in Haskell).
Phase 2: Compiler Design and Type System
Finite Set Compilation:

Set Inlining: For sets that are finite, membership checks should be compiled away. If the set is statically known (e.g., DayOfWeek), it can be inlined directly into the generated code.
Example: For a subset function:

Weekday = (- x in DayOfWeek where x != Saturday and x != Sunday -)
The compiler resolves this subset and replaces it with the static set {Monday, Tuesday, Wednesday, Thursday, Friday} in the compiled output. Any membership checks against this set will be stripped out, and the set is directly used in comparisons.
Type-Level Set Operations:

Compile-Time Type Checking: The type checker should infer whether set operations involve finite or infinite sets. For finite sets, all operations (like union, intersection) are resolved during type checking.
Type Constraints: Type constraints related to set membership should be verified during compilation, and if valid, these constraints will not need to exist in the compiled output.
For example, a function that accepts only positive integers can be guaranteed to work on valid inputs at compile time, eliminating the need for runtime membership checks.

doublePositive : PositiveInt -> PositiveInt
doublePositive = \x -> x * 2
Runtime Handling of Infinite/Dependent Sets:

For infinite or dependent sets, runtime checks cannot be entirely stripped out. However, optimizations can still be applied:
Lazy Evaluation: For infinite sets (like NaturalNumbers), set operations (such as comprehensions) should be lazily evaluated.
Predicate Evaluation: For dependent sets, membership constraints should be compiled into efficient runtime checks (e.g., for PositiveInt, a simple x > 0 check).
Phase 3: Set Membership Stripping and Code Generation
Strip Set Membership for Finite Sets:

During the code generation phase, once the type checker confirms that a set is finite, any membership checks against that set can be stripped out. For example:

isWeekday : DayOfWeek -> Bool
isWeekday = \day -> day in Weekday
If Weekday is known at compile time, the day in Weekday check can be replaced by a simple comparison against the inlined finite set {Monday, Tuesday, Wednesday, Thursday, Friday}.
Generated Code Example: For a function like this:


isWeekday : DayOfWeek -> Bool
isWeekday = \day -> day in Weekday
The compiler will resolve Weekday to a finite set and compile the following:


bool isWeekday(DayOfWeek day) {
    switch(day) {
        case Monday:
        case Tuesday:
        case Wednesday:
        case Thursday:
        case Friday:
            return true;
        default:
            return false;
    }
}
Handling Infinite Sets and Lazy Evaluation: For sets that cannot be fully determined at compile time (e.g., infinite sets), the compiler should generate code that evaluates membership lazily:



firstFiveEvens = take 5 EvenNaturals
The compiler will translate this into code that lazily generates and evaluates only the first five even numbers at runtime, rather than computing the entire infinite set of even numbers.

Phase 4: Set Operations and Optimizations
Optimizing Set Operations:

Union, Intersection, Difference: For finite sets, the operations should be resolved at compile time. For example, if two finite sets are being intersected, the compiler can precompute the result of the intersection and replace the operation with the result.
For infinite sets, operations like union or intersection should be deferred to runtime using lazy evaluation or iterative computation.
Compile-Time Errors for Invalid Operations:

If a user tries to apply operations (like comprehensions or filtering) on infinite sets without bounds, the compiler should raise errors or warnings. For example, a comprehension that attempts to compute all elements of an infinite set without constraints should result in a compile-time error.
Phase 5: Testing and Performance Validation
Finite Set Membership Optimization Tests:

Write test cases for functions and set operations to ensure that membership checks are correctly stripped for finite sets.
Ensure that the compiler correctly identifies and optimizes all finite set operations.
Benchmarking:

Performance Testing: Benchmark the performance of Setlang’s compiled code, especially comparing cases with membership checks (infinite sets) versus cases where they have been stripped out (finite sets).
Test set operation optimizations to ensure that union, intersection, and difference are being efficiently handled at compile time.
Additional Notes
Default Set Behavior: By making finite sets the default and using lazy evaluation for infinite sets, the language will ensure efficient runtime behavior, where possible, while maintaining the expressiveness of set-theoretic operations.

User-Defined Constraints: By allowing users to define finite and infinite sets explicitly, Setlang can optimize for cases where users define bounded sets but remain flexible enough to handle complex, infinite set scenarios when needed.

This approach will allow Setlang to be compiled efficiently, with most set membership logic stripped out for finite sets, while still supporting complex and infinite sets when necessary, using a blend of compile-time and runtime techniques.


Here's a quick version of a simple **Setlang** compiler using **Zig** and **MLIR** to demonstrate the basic structure of the language. The idea is to keep the initial version simple and focus on compiling basic set operations and functions.

### **Simple Setlang Features**
1. **Basic Types**: `Int`, `Bool`, `String`
2. **Finite Sets**: Simple enumerations like `{1, 2, 3}`, or `{True, False}`
3. **Basic Functions**: Lambda expressions with set membership
4. **Set Operations**: Union, Intersection, Difference
5. **Simple Compilation**: Compile set operations and checks to **MLIR** via **Zig**

### **Simple Setlang Syntax Example**
```setlang
# Simple set definitions
IntSet = (- 1, 2, 3 -)
BoolSet = (- True, False -)

# Function on sets
doublePositive : Int -> Int
doublePositive = \x -> if x > 0 then x * 2 else 0

# Simple set membership check
checkInSet : Int -> Bool
checkInSet = \x -> x in IntSet
```

### **Step-by-Step Implementation Plan**

#### 1. **Parsing Setlang with Zig**
You can use Zig to create a parser for the **Setlang** syntax. The parser will read the input, tokenize it, and build an **AST** (Abstract Syntax Tree) representing the program.

- **Lexing**: Identify tokens such as `IntSet`, `{`, `}`, `->`, `if`, etc.
- **Parsing**: Build the AST for set definitions, functions, and set membership operations.

**Example Zig Lexer:**
```zig
const std = @import("std");

const TokenType = enum {
    Ident,
    Number,
    LCurly,
    RCurly,
    Arrow,
    Lambda,
    If,
    Then,
    Else,
    In,
    EOF,
};

const Token = struct {
    typ: TokenType,
    value: []const u8,
};

fn lex(input: []const u8) []Token {
    var tokens: []Token = undefined;
    // Simple lexing logic: Iterate through input and create tokens
    return tokens;
}
```

#### 2. **Building an AST in Zig**
The next step is to take the tokens and create an **AST** that represents the code structure of Setlang.

```zig
const SetNode = struct {
    set_name: []const u8,
    elements: []const u8, // for simplicity, assume all elements are strings
};

const FunctionNode = struct {
    name: []const u8,
    param_type: []const u8,
    return_type: []const u8,
    body: []const u8,  // We'll simplify function bodies as strings for now
};
```

#### 3. **Code Generation with MLIR**
After building the AST, you can generate **MLIR** (Multi-Level Intermediate Representation) code using Zig. MLIR will allow us to compile and optimize the Setlang code and eventually lower it to **LLVM IR** for further compilation.

You can start by generating MLIR code for **set definitions** and **simple functions**.

**Example MLIR Code Generation:**
```zig
const mlir = @import("mlir");

fn genSetDefinition(setNode: SetNode) void {
    std.debug.print("mlir set definition: {s}\n", .{ setNode.set_name });
    std.debug.print("mlir elements: {s}\n", .{ setNode.elements });
    // Generate MLIR representation for the set
    // e.g., "IntSet = (- 1, 2, 3 -)"
    // MLIR code would look like:
    // "intset = set { 1, 2, 3 }"
}

fn genFunction(funcNode: FunctionNode) void {
    std.debug.print("mlir function: {s} -> {s}\n", .{ funcNode.name, funcNode.return_type });
    // Generate MLIR for the function
    // For example, "doublePositive = \x -> if x > 0 then x * 2 else 0"
    // would generate a conditional branch in MLIR:
    //
    // %x = block_argument
    // %0 = cmp gt %x, 0
    // cond_br %0, ^bb_true, ^bb_false
    // ^bb_true:
    //     %1 = mul %x, 2
    //     return %1
    // ^bb_false:
    //     return 0
}
```

#### 4. **Compiling and Linking**
Once the **MLIR** is generated, you can use the **MLIR optimizer** to optimize the intermediate code and lower it to **LLVM IR** or directly generate machine code. Zig can handle the integration with MLIR and LLVM.

**Lowering MLIR to LLVM IR:**
```zig
fn lowerToLLVM(mlir_code: []const u8) void {
    // Use MLIR's built-in functionality to lower to LLVM IR
    // and invoke the LLVM backend for final code generation
    // Use mlir-opt and mlir-translate tools here
}
```

#### 5. **Handling Set Membership at Compile Time**
In this simple version, we'll handle finite set membership checks at compile time using the parser and the AST. If a set is known to be finite (like `{1, 2, 3}`), membership checks can be **evaluated and stripped** during compilation.

For example, the check:
```setlang
checkInSet : Int -> Bool
checkInSet = \x -> x in IntSet
```
Would generate:
```mlir
%0 = cmp eq %x, 1
%1 = cmp eq %x, 2
%2 = cmp eq %x, 3
%3 = or %0, %1
%4 = or %3, %2
return %4
```

#### 6. **Building and Running the Compiler**
- Use **Zig** to manage the overall compiler pipeline, handling parsing, AST generation, and MLIR code generation.
- Utilize **MLIR** to optimize and lower the code to LLVM IR.
- Test the compiler with simple Setlang programs, ensuring that set operations and function compilation work as expected.

---

### **Next Steps for Expansion**
- **Add Support for Infinite Sets**: Introduce lazy evaluation for infinite sets.
- **Extend Type System**: Implement dependent types and more advanced type-level operations.
- **Optimizations**: Improve performance by stripping set membership checks for finite sets at compile time.

This quick version of Setlang focuses on basic parsing, set operations, and generating MLIR code for compilation via **Zig**. You can build on this foundation to handle more complex language features over time.


Using **Rust** to build the **Setlang compiler** with **MLIR** (via **Inkdrop** for Rust bindings) is a powerful choice because Rust's safety features and performance align well with compiler development. Below is a rewritten guide to integrate **Rust** with **MLIR** using **Inkdrop** for building your Setlang compiler.

### **Steps to Use Rust + MLIR (Inkdrop)**

#### 1. **Set Up MLIR and Rust Environment**

- Install **MLIR** and **LLVM** as part of the LLVM project:
  ```bash
  git clone https://github.com/llvm/llvm-project.git
  cd llvm-project
  mkdir build && cd build
  cmake -G Ninja ../llvm -DLLVM_ENABLE_PROJECTS="mlir" -DLLVM_BUILD_EXAMPLES=ON -DLLVM_TARGETS_TO_BUILD="X86"
  ninja
  ```

- Install **Rust** via `rustup` if you haven't already:
  ```bash
  curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh
  ```

- Add the **Inkdrop** crate to your project. Inkdrop is a Rust binding for MLIR, making it easier to work with MLIR in Rust. Set up the project with `cargo`:

  ```bash
  cargo new setlang_compiler --bin
  cd setlang_compiler
  ```

  Then, edit your `Cargo.toml` file to add Inkdrop as a dependency:
  ```toml
  [dependencies]
  inkdrop = "0.1.0" # Update with the correct version if available
  ```

#### 2. **Write a Basic Rust Parser for Setlang**

In this step, you’ll create a **Rust lexer and parser** for Setlang to tokenize the input and build an Abstract Syntax Tree (AST).

- **Lexer**: Tokenizes the input code into meaningful elements (e.g., `Ident`, `SetOp`, `Number`, etc.).
- **Parser**: Converts the tokens into an AST.

**Lexer Example in Rust**:
```rust
#[derive(Debug, PartialEq)]
enum TokenType {
    Ident,
    Number,
    LCurly,
    RCurly,
    Arrow,
    Lambda,
    If,
    Then,
    Else,
    In,
    EOF,
}

#[derive(Debug)]
struct Token {
    typ: TokenType,
    value: String,
}

struct Lexer {
    input: String,
    position: usize,
}

impl Lexer {
    fn new(input: String) -> Self {
        Lexer { input, position: 0 }
    }

    fn next_token(&mut self) -> Token {
        // Basic lexing logic to tokenize the input
        Token {
            typ: TokenType::EOF,
            value: String::new(),
        }
    }
}
```

**AST Representation in Rust**:
```rust
enum ASTNode {
    SetNode(SetNode),
    FunctionNode(FunctionNode),
}

struct SetNode {
    set_name: String,
    elements: Vec<String>,
}

struct FunctionNode {
    name: String,
    param_type: String,
    return_type: String,
    body: String, // Simplified body as a string for now
}
```

#### 3. **Integrating MLIR with Inkdrop**

After building the AST, the next step is to generate **MLIR** code using Inkdrop. Inkdrop provides **MLIR bindings** for Rust, allowing you to generate and optimize MLIR code directly from Rust.

- Install **Inkdrop** and set it up with **Cargo**.
  
- **Generate MLIR Code for Set Definitions**:
  
  **Rust Code Example**:
  ```rust
  use inkdrop::*;

  fn generate_mlir_for_set(set_node: &SetNode) -> String {
      let mut context = Context::new();
      context.append_dialect::<mlir::standard::StandardDialect>();

      let module = Module::new(&context);
      let mut builder = Builder::new(&context);

      // Define the MLIR set operation (simplified)
      let mlir_code = format!("set {} {{ {} }}", set_node.set_name, set_node.elements.join(", "));
      println!("{}", mlir_code);

      mlir_code
  }
  ```

- **Generate MLIR for Functions**:

  **Rust Code Example**:
  ```rust
  fn generate_mlir_for_function(function_node: &FunctionNode) -> String {
      let mut context = Context::new();
      context.append_dialect::<mlir::standard::StandardDialect>();

      let module = Module::new(&context);
      let mut builder = Builder::new(&context);

      let function_type = builder.get_function_type(&[], &[]);
      let function = builder.build_function(
          &function_node.name,
          function_type,
          |func| {
              let entry_block = func.append_entry_block();
              builder.set_insertion_point_to_start(&entry_block);

              // Function body logic (simplified for this example)
              println!("Generating MLIR for function {}", function_node.name);
          }
      );

      module.to_string()
  }
  ```

#### 4. **Lowering MLIR to LLVM IR**

Once you have generated the **MLIR**, you can use Inkdrop to further optimize it and lower it to **LLVM IR**.

**Lowering MLIR to LLVM IR with Inkdrop**:
```rust
fn lower_to_llvm(mlir_code: String) {
    // Use Inkdrop's functionality to lower MLIR to LLVM IR
    // Implement LLVM IR lowering using the built-in Inkdrop functions
    println!("Lowering to LLVM IR: {}", mlir_code);
    // You can use MLIR passes for optimization, lowering, etc.
}
```

#### 5. **Handling Set Membership Optimization**

For **finite sets**, the Rust compiler should strip out set membership checks when possible. The logic for this optimization would be written as part of the MLIR generation phase. When the compiler detects a finite set, it replaces set membership checks with direct comparisons.

Example:
```rust
fn optimize_set_membership(set_node: &SetNode) -> String {
    let mlir_code = generate_mlir_for_set(set_node);

    // Check if the set is finite, and replace membership checks with direct comparisons
    if set_node.elements.len() > 0 {
        let optimized_code = format!(
            "if %x == {}",
            set_node.elements.join(" || %x == ")
        );
        optimized_code
    } else {
        mlir_code
    }
}
```

For example:
```setlang
isWeekday : DayOfWeek -> Bool
isWeekday = \day -> day in Weekday
```

Would generate the following Rust-based MLIR code:
```mlir
%0 = cmpi eq, %day, 1 : i32
%1 = cmpi eq, %day, 2 : i32
%2 = cmpi eq, %day, 3 : i32
%3 = cmpi eq, %day, 4 : i32
%4 = cmpi eq, %day, 5 : i32
%5 = or %0, %1
%6 = or %5, %2
%7 = or %6, %3
%8 = or %7, %4
return %8 : i1
```

#### 6. **Building and Running the Compiler**

To compile a Setlang program, you will:

1. Parse the code using **Rust** lexer and parser.
2. Generate the **AST**.
3. Convert the AST into **MLIR** using **Inkdrop**.
4. Lower the **MLIR** to **LLVM IR**.
5. Optimize set operations and function compilation.

**Running the Compiler in Rust**:
```rust
fn main() {
    let input_code = "
    doublePositive : Int -> Int
    doublePositive = \\x -> if x > 0 then x * 2 else 0
    ";

    let mut lexer = Lexer::new(input_code.to_string());
    let tokens = lexer.next_token();

    // Parser would generate AST from tokens
    let ast = parse(tokens);

    // Generate MLIR from AST
    for node in ast {
        match node {
            ASTNode::SetNode(set_node) => {
                generate_mlir_for_set(&set_node);
            },
            ASTNode::FunctionNode(function_node) => {
                generate_mlir_for_function(&function_node);
            },
        }
    }

    // Lower to LLVM IR
    lower_to_llvm(mlir_code);
}
```

#### **Next Steps and Expansions**

- **Add Dependent Sets**: Implement dependent set operations and membership checks that rely on runtime conditions.
- **Type System**: Extend the type system for more complex types (e.g., dependent types, recursive types).
- **Optimizations**: Strip out unnecessary set checks and function calls using MLIR passes and optimizations.

This Rust-based version uses **Inkdrop** for MLIR integration, giving you a powerful way to optimize and compile Setlang to LLVM IR efficiently.


Absolutely! Let's combine the arena allocation effect system with custom allocators using traits. This powerful combination would allow for even more flexible and efficient memory management in Setlang. Here's how it might look:

Title: The Future of Memory Management in Setlang: Custom Allocators and Arena Effects

Imagine a future version of Setlang that not only supports arena allocations through the effect system but also allows developers to define custom allocators using traits. This combination would provide unprecedented control over memory management while maintaining Setlang's commitment to safety and expressiveness.

Custom Allocator Trait

First, let's define a trait for custom allocators:

```setlang
Allocator : Trait
Allocator = (-
    ["allocate", (Size) -> Ptr],
    ["deallocate", (Ptr) -> ()],
    ["reallocate", (Ptr, Size) -> Ptr]
-)

# A type representing memory size
Size : (- Bytes Int, Kilobytes Int, Megabytes Int -)

# A simple pointer type
Ptr : Type
```

Now, let's implement this trait for a custom arena allocator:

```setlang
ArenaAllocator : {
    memory: Ptr,
    size: Size,
    used: Size
}

impl Allocator for ArenaAllocator {
    allocate = \size ->
        if self.used + size <= self.size then
            ptr = self.memory + self.used
            self.used = self.used + size
            ptr
        else
            error("Out of memory in arena")

    deallocate = \_ ->
        # No-op for arena allocator

    reallocate = \ptr, new_size ->
        # Simplified: always allocate new memory
        new_ptr = self.allocate(new_size)
        copy_memory(ptr, new_ptr, min(new_size, self.used))
        new_ptr
}
```

Combining Custom Allocators with the Effect System

Now, let's enhance our effect system to work with custom allocators:

```setlang
Effect : (-
    ...,  # Existing effects
    Alloc
-)

# Define an allocator effect
AllocEffect : Effect
AllocEffect = Alloc { allocator: Allocator }
```

Using Custom Allocators with Arena Effects

Here's how we might use this combined system:

```setlang
# Function that uses a custom allocator
processWithCustomAlloc : [Data] -> Eff (AllocEffect { allocator: ArenaAllocator }) Result
processWithCustomAlloc = \dataset ->
    # All allocations within this function will use the ArenaAllocator
    result = dataset
        |> map(processItem)
        |> reduce(combineResults)
    
    pure(result)

# Main function that sets up and uses the custom arena allocator
main : () -> Eff (IO) ()
main = \() ->
    largeDataset = loadLargeDataset()
    
    # Create a custom arena allocator
    arenaAllocator = ArenaAllocator {
        memory: system_allocate(Megabytes 100),
        size: Megabytes 100,
        used: Bytes 0
    }
    
    # Run the computation with the custom allocator effect
    result = run AllocEffect { allocator: arenaAllocator } in
        processWithCustomAlloc(largeDataset)
    
    printResult(result)
    
    # Manually free the arena memory
    system_deallocate(arenaAllocator.memory)
```

Advanced Features: Allocator Sets and Automatic Selection

To make this system even more powerful, we could introduce the concept of "allocator sets" - collections of allocators that Setlang can automatically choose from based on allocation patterns:

```setlang
AllocatorSet : Trait
AllocatorSet = (-
    ["get_allocator", (Size) -> Allocator]
-)

# An implementation that chooses between different allocators based on size
SmartAllocSet : {
    small_alloc: Allocator,
    medium_alloc: Allocator,
    large_alloc: Allocator
}

impl AllocatorSet for SmartAllocSet {
    get_allocator = \size ->
        when size is
            Bytes n if n < 1024 -> self.small_alloc
            Kilobytes n if n < 1024 -> self.medium_alloc
            _ -> self.large_alloc
}

# Using the AllocatorSet in our effect system
SmartAllocEffect : Effect
SmartAllocEffect = Alloc { allocator_set: AllocatorSet }

# A function that benefits from smart allocation
complexProcessing : [Data] -> Eff (SmartAllocEffect { allocator_set: SmartAllocSet }) Result
complexProcessing = \dataset ->
    # Setlang automatically chooses the right allocator for each allocation
    ...
```

Compile-Time Optimization and Analysis

Setlang's compiler could perform sophisticated analysis to optimize allocator usage:

1. Allocation Pattern Analysis: Suggest optimal allocator configurations based on how a function allocates memory.
2. Escape Analysis: Ensure that memory allocated with special allocators doesn't escape its intended scope.
3. Automatic Allocator Insertion: Automatically wrap code blocks with appropriate allocators based on their allocation patterns.

```setlang
# Compiler might suggest:
# @SuggestedAllocator(SmartAllocSet { 
#     small_alloc: PoolAllocator { size: Kilobytes 64 },
#     medium_alloc: ArenaAllocator { size: Megabytes 10 },
#     large_alloc: SystemAllocator
# })
complexFunction : [Data] -> Result
complexFunction = \data ->
    # Function body with various allocation patterns
    ...
```

Conclusion: A New Frontier in Memory Management

By combining custom allocators, allocator sets, and the effect system, Setlang opens up a new frontier in memory management:

1. Fine-grained Control: Developers can define custom allocation strategies tailored to their specific needs.
2. Safety: The effect system ensures that custom allocation strategies are used correctly and don't leak outside their intended scope.
3. Performance: By allowing different allocators for different scenarios, programs can achieve optimal performance across various use cases.
4. Ergonomics: The compiler's ability to suggest and automatically insert appropriate allocators reduces the burden on developers while still providing the benefits of custom memory management.
5. Flexibility: The ability to compose different allocators and easily switch between them allows for great adaptability as program requirements change.

This vision for Setlang's future demonstrates how the language could continue to push the boundaries of what's possible in systems programming. By providing powerful abstractions for memory management that are both safe and efficient, Setlang would enable developers to write high-performance code with confidence, knowing that the language and compiler are working together to ensure optimal memory usage.

As we look to the future of systems programming, this combination of custom allocators, effect systems, and intelligent compiler analysis in Setlang represents a exciting step forward in the eternal quest to balance performance, safety, and developer productivity.

Excellent question! The Expose trait plays a crucial role in Setlang's memory management and safety. Let's dive deep into how Expose integrates with these aspects:

1. Controlled Mutability

At its core, Expose provides a mechanism for controlled mutability in an otherwise immutable-by-default language. This is fundamental to memory safety because it limits where and when mutations can occur.

```setlang
Counter : {
    value: Int
}

impl Expose for Counter {
    expose = \self -> 
        Exposed Counter {
            get = \() -> self.value,
            set = \new_value -> { ...self, value: new_value }
        }
}
```

2. Temporary Access

Expose allows temporary, controlled access to mutable state. This temporality is key to preventing long-lived mutable references that could lead to data races or use-after-free errors.

```setlang
incrementCounter : Counter -> Counter
incrementCounter = \counter ->
    let exposed = ^counter  # Temporary exposure
    let current = exposed.get()
    exposed.set(current + 1)
    counter^  # Return to immutable state
```

3. Compile-Time Tracking

The compiler can track when and where Expose is used, ensuring that exposed (mutable) states don't outlive their intended scope. This is similar to Rust's borrow checker but with a focus on exposed state rather than borrowing.

4. Thread Safety

By controlling when mutation can occur, Expose helps prevent data races in concurrent code. The compiler can ensure that data is not exposed in a mutable way across thread boundaries.

```setlang
threadSafeIncrement : Counter -> Eff (Concurrent) Counter
threadSafeIncrement = \counter ->
    atomicOperation(\() ->
        let exposed = ^counter
        let current = exposed.get()
        exposed.set(current + 1)
        counter^
    )
```

5. Integration with Automatic Reference Counting (ARC)

Expose works hand-in-hand with ARC to ensure memory safety:

- When a value is exposed, the ARC system ensures it's not deallocated while in use.
- The temporary nature of exposure prevents creating multiple mutable references that could confuse the reference counting system.

6. Copy-on-Write Optimization

Expose can be implemented efficiently using copy-on-write semantics. This means that when you expose a value, it's not immediately copied. Instead, a copy is made only when a mutation occurs, preserving performance for read-only operations.

```setlang
impl Expose for LargeDataStructure {
    expose = \self -> 
        Exposed LargeDataStructure {
            get = \() -> self,  # No copy needed for reads
            set = \new_value -> 
                if new_value != self 
                    then new_value  # Copy only on write
                    else self
        }
}
```

7. Granular Control

Expose allows for fine-grained control over which parts of a data structure can be mutated. This granularity enhances safety by limiting the attack surface for potential bugs.

```setlang
ComplexStructure : {
    immutablePart: Int,
    mutablePart: Counter
}

impl Expose for ComplexStructure {
    expose = \self -> 
        Exposed ComplexStructure {
            get = \() -> self,
            set = \new_value -> { 
                ...self, 
                mutablePart: new_value.mutablePart  # Only mutablePart can change
            }
        }
}
```

8. Safe Interoperation with Effect System

Expose can integrate with Setlang's effect system to provide additional safety guarantees:

```setlang
Effect : (-
    ...,
    Mutable
-)

safeUpdate : Counter -> Eff (Mutable) Counter
safeUpdate = \counter ->
    let exposed = ^counter
    let current = exposed.get()
    exposed.set(current + 1)
    pure(counter^)  # Effect system ensures proper handling of mutable state
```

9. Compile-Time Optimization

The compiler can use information about Expose usage to optimize memory layout and access patterns, potentially eliminating runtime checks in cases where safety can be guaranteed at compile-time.

10. Protection Against Use-After-Move

By controlling when a value can be modified, Expose helps prevent use-after-move errors. Once a value has been moved (which would typically happen through Expose), the type system can prevent further use of the original value.

```setlang
moveValue : Counter -> (Counter, Counter)
moveValue = \counter ->
    let exposed = ^counter
    let newCounter = exposed.get()
    (Counter { value: newCounter }, counter^)  # Original counter can't be used after this
```

11. Memory Leak Prevention

The controlled nature of Expose helps prevent memory leaks by ensuring that all exposed (and potentially modified) values are properly handled and returned to an immutable state.

Conclusion:

The Expose trait is a cornerstone of Setlang's approach to memory safety. By providing a controlled mechanism for mutability within an otherwise immutable paradigm, it allows for efficient updates while maintaining strong safety guarantees. Its integration with the type system, effect system, and ARC creates a comprehensive approach to memory management that is both safe and flexible.

This design allows Setlang to offer the performance benefits of mutable operations when needed, while still defaulting to the safety of immutability. It's a sophisticated approach that balances safety, performance, and ergonomics in a unique way, fitting well with Setlang's set-theoretic foundations.


Certainly! An actor system in Setlang would be an excellent way to showcase the language's capabilities for concurrent and distributed programming. Let's design an actor system that leverages Setlang's unique features, including its set-theoretic foundation, effect system, and memory safety guarantees.

Here's what an actor system might look like in Setlang:

```setlang
# Define the Actor trait
Actor : Trait
Actor = (-
    ["receive", (Message) -> Eff (ActorEffect) ()]
-)

# Define the ActorRef type
ActorRef : Type
ActorRef = { id: UUID, address: Address }

# Define the Address type
Address : (- Local UUID, Remote String -)

# Define the Message type
Message : Type
Message = {
    sender: ActorRef,
    content: Any
}

# Define the ActorSystem type
ActorSystem : Type
ActorSystem = {
    actors: (- ActorRef -),  # Set of all actors in the system
    dispatcher: (Message) -> Eff (ActorEffect) ()
}

# Define the ActorEffect
Effect : (-
    ...,  # Other effects
    ActorEffect
-)

ActorEffect : Effect
ActorEffect = (-
    Send { to: ActorRef, message: Message },
    Spawn { actor: Actor },
    Stop { actor: ActorRef }
-)

# Implementation of a simple actor
Counter : Type
Counter = { count: Int }

impl Actor for Counter {
    receive = \message ->
        when message.content is
            "Increment" -> 
                let exposed = ^self
                exposed.set({ count: exposed.get().count + 1 })
                pure(())
            "Get" ->
                send(message.sender, self.count)
            _ -> 
                pure(())  # Ignore unknown messages
}

# Function to create a new actor
spawnActor : (ActorSystem, Actor) -> Eff (ActorEffect) ActorRef
spawnActor = \system, actor ->
    actorRef = ActorRef { 
        id: generateUUID(), 
        address: Local(generateUUID()) 
    }
    Spawn { actor: actor }
    pure(actorRef)

# Function to send a message to an actor
send : (ActorRef, Any) -> Eff (ActorEffect) ()
send = \to, content ->
    Send { 
        to: to, 
        message: Message { 
            sender: getSelfRef(), 
            content: content 
        } 
    }

# Function to stop an actor
stopActor : (ActorSystem, ActorRef) -> Eff (ActorEffect) ()
stopActor = \system, actorRef ->
    Stop { actor: actorRef }

# Example usage
main : () -> Eff (IO, ActorEffect) ()
main = \() ->
    # Create an actor system
    system = createActorSystem()

    # Spawn a counter actor
    counterRef = spawnActor(system, Counter { count: 0 })

    # Send messages to the counter
    send(counterRef, "Increment")
    send(counterRef, "Increment")
    send(counterRef, "Get")

    # Wait for and print the result
    result = receive()
    print("Counter value: " + toString(result))

    # Stop the actor
    stopActor(system, counterRef)
```

Let's break down the key components and features of this actor system:

1. Actor Trait: Defines the interface for all actors, with a `receive` method that handles incoming messages.

2. ActorRef: A reference to an actor, which can be either local or remote.

3. Message: Represents messages sent between actors, including the sender's reference.

4. ActorSystem: Manages the set of all actors and dispatches messages.

5. ActorEffect: An effect that encapsulates actor-related operations like sending messages, spawning new actors, and stopping actors.

6. Set-Theoretic Foundation: The `actors` field in `ActorSystem` is defined as a set, leveraging Setlang's set-theoretic nature.

7. Expose Trait Usage: The `Counter` actor uses the Expose trait to safely mutate its state.

8. Effect System Integration: Actor operations are wrapped in the `ActorEffect`, allowing the compiler to track and manage side effects related to the actor system.

9. Pattern Matching: The `receive` method uses pattern matching to handle different message types.

10. Type Safety: The system leverages Setlang's strong type system to ensure type-safe message passing.

11. Concurrency: The actor model naturally supports concurrent operations, with each actor processing messages independently.

Additional Features We Could Add:

1. Supervision Strategies:
```setlang
SupervisorStrategy : (-
    OneForOne { maxRestarts: Int, withinDuration: Duration },
    AllForOne { maxRestarts: Int, withinDuration: Duration },
    EscalateFailure
-)

impl Actor for Supervisor {
    receive = \message ->
        when message is
            ChildFailure { child, error } ->
                handleFailure(self.strategy, child, error)
}
```

2. Actor Lifecycles:
```setlang
Lifecycle : (-
    PreStart,
    PostStop,
    PreRestart { reason: Error },
    PostRestart { reason: Error }
-)

impl Actor for LifecycleAwareActor {
    receive = \message ->
        when message is
            Lifecycle::PreStart -> self.onPreStart()
            Lifecycle::PostStop -> self.onPostStop()
            # ... other lifecycle handlers
}
```

3. Distributed Actor References:
```setlang
RemoteActorRef : ActorRef
RemoteActorRef = { 
    id: UUID, 
    address: Remote String,
    serializer: (Message) -> Bytes,
    deserializer: (Bytes) -> Message
}
```

4. Actor Persistence:
```setlang
PersistentActor : Trait
PersistentActor = (-
    ["persist", (Event) -> Eff (PersistenceEffect) ()],
    ["recover", () -> Eff (PersistenceEffect) ()]
-)

impl PersistentActor for PersistentCounter {
    persist = \event ->
        PersistenceEffect::Write { event: event }
    recover = \() ->
        events = PersistenceEffect::Read { actorId: self.id }
        applyEvents(events)
}
```

This actor system design in Setlang showcases how the language's unique features can be leveraged to create a safe, efficient, and expressive concurrent programming model. The combination of set-theoretic foundations, strong typing, effect system, and controlled mutability through the Expose trait makes Setlang an excellent fit for implementing complex distributed systems like actor models.



Certainly! In Setlang, we can explicitly define that one set is a subset of another using set-theoretic notation and type constraints. This feature is powerful for creating precise type hierarchies and ensuring type safety. Let's explore how this could be implemented and used in Setlang.

Here's an example showcasing explicit subset relationships:

```setlang
# Define a base set
Number : (- Int, Float -)

# Define subsets
PositiveNumber : (- n in Number where n > 0 -)
NegativeNumber : (- n in Number where n < 0 -)

# More specific subsets
EvenNumber : (- n in Int where n % 2 == 0 -)
OddNumber : (- n in Int where n % 2 != 0 -)

# Explicit subset relationship
PositiveEvenNumber : (- n in PositiveNumber where n in EvenNumber -)

# Function that works on any Number
printNumber : Number -> ()
printNumber = \n -> print "Number: " + toString n

# Function that only accepts PositiveNumber
doublePositive : PositiveNumber -> PositiveNumber
doublePositive = \n -> n * 2

# Function that works on EvenNumber
halveEven : EvenNumber -> Int
halveEven = \n -> n / 2

# Usage
x : PositiveEvenNumber = 6
printNumber x  # Works because PositiveEvenNumber is a subset of Number
y = doublePositive x  # Works because PositiveEvenNumber is a subset of PositiveNumber
z = halveEven x  # Works because PositiveEvenNumber is a subset of EvenNumber

# This would cause a compile-time error:
# negativeNumber : NegativeNumber = 5  # Error: 5 is not in NegativeNumber

# Subset checking
<: : (A : Type) -> (B : Type) -> Bool
<: = \A, B -> forall x in A, x in B

# Usage of subset checking
PositiveEvenNumber <: PositiveNumber  # Returns true
PositiveEvenNumber <: NegativeNumber  # Returns false

# More complex subset relationships
PrimeNumber : (- n in PositiveNumber where isPrime n -)
OddPrimeNumber : (- n in PrimeNumber where n in OddNumber -)

# Function that works on the intersection of two sets
processSpecialNumber : (n in (PositiveNumber & OddNumber)) -> Int
processSpecialNumber = \n -> n * n

# Usage
specialResult = processSpecialNumber 3  # Works
# This would cause a compile-time error:
# invalidResult = processSpecialNumber 4  # Error: 4 is not in (PositiveNumber & OddNumber)

# Parametric subsets
LessThan : (max : Int) -> (- n in Int where n < max -)
GreaterThan : (min : Int) -> (- n in Int where n > min -)

# Usage of parametric subsets
SmallNumber : LessThan 10
LargeNumber : GreaterThan 1000

# Function using parametric subset
printSmallNumber : SmallNumber -> ()
printSmallNumber = \n -> print "Small number: " + toString n

printSmallNumber 7  # Works
# This would cause a compile-time error:
# printSmallNumber 15  # Error: 15 is not in SmallNumber

# Combining subsets
BoundedNumber : (min : Int, max : Int) -> (- n in GreaterThan min where n in LessThan max -)

# Usage of combined subsets
MediumNumber : BoundedNumber 10 100

isMedium : MediumNumber -> Bool
isMedium = \_ -> true  # Always true because of type constraints

isMedium 50  # Returns true
# This would cause a compile-time error:
# isMedium 150  # Error: 150 is not in MediumNumber
```

Key po