Building Recursive Data Structures: Anamorphism Challenge

In functional programming, anamorphisms (often called "unfolds") are a powerful concept for constructing recursive data structures. They allow you to generate a data structure from a seed value by repeatedly applying a function until a termination condition is met. This challenge asks you to implement an anamorphism for a custom recursive type in TypeScript, demonstrating your understanding of type-level programming and recursive structures.

Problem Description

Your task is to create a generic anamorphism function in TypeScript. This function should take a seed value and a "step" function, and produce a recursive data structure. The "step" function will dictate how the structure is built and when it terminates.

Key Requirements:

1. Generic Function: The anamorphism function should be generic to work with various seed types and recursive data structure types.
2. Recursive Data Type: You will need to define a representative recursive data type that the anamorphism will construct. A common example is a list, but you can choose another suitable recursive type (e.g., a tree).
3. Step Function: The step function will receive the current "state" (initially the seed) and return either a value to add to the structure and the next state, or a signal to terminate the construction.
4. Type Safety: The entire implementation must be type-safe within TypeScript.

Expected Behavior:

The anamorphism function should repeatedly apply the step function to the current state. Each application of the step function should produce a "step" – either a new element for the structure and the next state, or an indication to stop. The process continues until the step function signals termination.

Edge Cases:

• Immediate Termination: The step function might immediately signal termination, resulting in an empty structure.
• Infinite Structures (Conceptual): While TypeScript doesn't directly support infinite types at runtime, consider how your anamorphism could conceptually handle infinite generation if the step function never terminated (though you'll want to avoid actual infinite loops in your test cases).

Examples

Let's consider building a List type.

// Define a recursive List type
type List<T> = {
  kind: "Nil";
} | {
  kind: "Cons";
  head: T;
  tail: List<T>;
};

// A helper to create Nil
const Nil = <T>(): List<T> => ({ kind: "Nil" });

// A helper to create Cons
const Cons = <T>(head: T, tail: List<T>): List<T> => ({
  kind: "Cons",
  head,
  tail,
});

// The step function signature for List anamorphism
// It takes the current state (number in this case)
// and returns either:
// - { done: true } to terminate
// - { done: false, value: T, nextState: S } to add an element and continue
type ListStepFn<T, S> = (
  state: S
) =>
  | { done: true }
  | { done: false; value: T; nextState: S };

Example 1: Generating a List of numbers from a count

Input:
  seed: 5 (starting number)
  stepFn: (n: number) => n > 0 ? { done: false, value: n, nextState: n - 1 } : { done: true }

Output:
  {
    kind: "Cons",
    head: 5,
    tail: {
      kind: "Cons",
      head: 4,
      tail: {
        kind: "Cons",
        head: 3,
        tail: {
          kind: "Cons",
          head: 2,
          tail: {
            kind: "Cons",
            head: 1,
            tail: { kind: "Nil" }
          }
        }
      }
    }
  }

Explanation:
The step function is applied to 5. It returns { done: false, value: 5, nextState: 4 }.
A Cons cell with 5 and the result of the next step is created.
This process repeats with 4, 3, 2, and 1.
When the state becomes 0, the step function returns { done: true }, terminating the recursion and creating Nil.

Example 2: Generating an empty list

Input:
  seed: 0
  stepFn: (n: number) => n > 0 ? { done: false, value: n, nextState: n - 1 } : { done: true }

Output:
  { kind: "Nil" }

Explanation:
The step function is immediately called with 0. Since 0 is not greater than 0, it returns { done: true }, terminating the recursion and resulting in an empty Nil list.

Example 3: Generating a list of squares up to a limit

Input:
  seed: 3 (max number to consider)
  stepFn: (max: number) => {
    if (max <= 0) return { done: true };
    const value = max * max; // Calculate the square
    return { done: false, value: value, nextState: max - 1 };
  }

Output:
  {
    kind: "Cons",
    head: 9, // 3*3
    tail: {
      kind: "Cons",
      head: 4, // 2*2
      tail: {
        kind: "Cons",
        head: 1, // 1*1
        tail: { kind: "Nil" }
      }
    }
  }

Explanation:
The step function is applied to 3. It returns { done: false, value: 9, nextState: 2 }.
The list is constructed with 9 and the result of the next step.
This continues for 2 (value 4) and 1 (value 1).
When the state becomes 0, termination occurs.

Constraints

• The anamorphism function must be implemented purely in TypeScript.
• The recursive data type used in your examples should be clearly defined.
• The anamorphism function should handle arbitrary seed types S and element types T.
• The step function's return type should clearly distinguish between termination and continuation.
• Avoid external libraries.

Notes

• Anamorphisms are the dual of catamorphisms (folds). They are about building up structure from a seed.
• Consider how you will represent the "done" state in your step function's return type.
• Think about the recursive nature of building the data structure. How will you chain the results of successive step function calls?
• You'll need to carefully define the types for your recursive structure and the step function.