Type-Safe Proof-Carrying Code with TypeScript

This challenge focuses on creating a type system in TypeScript that can encode not just data and functions, but also the proofs that certain properties about that code hold. This is a foundational concept for verifiable computation, where a verifier can trust that a piece of code has been executed correctly without needing to re-execute it themselves.

Problem Description

Your task is to design and implement a TypeScript type system that allows for the creation and verification of "proof-carrying code" (PCC). This means:

1. Representing Code with Proofs: You need to define types that can encapsulate a piece of code (represented as a TypeScript function or a value) along with a corresponding "proof" that a specific property about that code is true.
2. Defining Properties: The system should allow for the definition of verifiable properties. For this challenge, we'll focus on a simplified notion of properties, such as "this function always returns a positive number" or "this value is within a specific range."
3. Verification Mechanism: You need to implement a mechanism to verify if the provided proof indeed guarantees the stated property for the given code. This verification should be compile-time (using TypeScript's type system) or, for a more advanced version, runtime verifiable.

Key Requirements:

• Type Safety: The entire system must be type-safe within TypeScript.
• Proof Representation: A clear and structured way to represent proofs alongside code.
• Property Representation: A way to define the properties to be proven.
• Verification Logic: Implement the logic to check if a proof is valid for a given code and property.

Expected Behavior:

The system should allow you to:

• Declare a function and a proof that it satisfies a certain property.
• Attempt to use such a function. If the proof is invalid, a type error should occur (ideally at compile-time).
• Attempt to use a function without a valid proof (or with an invalid proof) and have it rejected by the type system.

Important Edge Cases:

• False Proofs: What happens if you try to associate a proof that doesn't actually satisfy the property? The type system should prevent this.
• Complex Properties: While we'll start simple, consider how the system might scale to more complex properties.
• Type Inference: How well does your system integrate with TypeScript's natural type inference?

Examples

Example 1: Verifying a Positive Return Value

Concept: We want to assert that a function add always returns a positive number.

Input:

Imagine a scenario where you define a function add and want to ensure its output is positive.

// Assume we have a way to define a 'Proof' and a 'Property'
type Positive = { __brand: 'Positive' };
type IsPositive<T> = T extends number ? (T > 0 ? Positive : never) : never;

// A function that we *claim* always returns a positive number
function add(a: number, b: number): number {
  return a + b;
}

// We need a way to wrap 'add' with its proof of positivity
// Let's imagine a type like:
// CodeWithProof<typeof add, IsPositive<ReturnType<typeof add>>>

Expected Output (Conceptual):

A type system that allows us to wrap add and its proof, and then infer that any result obtained from this wrapped function is guaranteed to be Positive.

// Type-level representation of 'add' having a proof of being Positive
type SafeAdd = Prove<typeof add, IsPositive<ReturnType<typeof add>>>;

// Now, when we use SafeAdd:
declare function useSafeAdd(): SafeAdd;
const result = useSafeAdd(); // Type of 'result' should be 'Positive & number'

Explanation:

The Prove type would internally check if add's return type satisfies the IsPositive property. If it does, SafeAdd would be a type representing add with the added guarantee of its output being positive.

Example 2: Verifying a Range Bound

Concept: We want to assert that a function getThreshold always returns a number within the range [0, 100].

Input:

// Property: number is between 0 and 100 inclusive
type Range0to100 = { __brand: 'Range0to100' };
type IsInRange0to100<T> = T extends number
  ? (T >= 0 && T <= 100 ? Range0to100 : never)
  : never;

function getThreshold(): number {
  // This function is *intended* to return a value between 0 and 100
  return Math.random() * 100;
}

// We want to wrap getThreshold with its proof of being in range

Expected Output (Conceptual):

type SafeThresholdGetter = Prove<typeof getThreshold, IsInRange0to100<ReturnType<typeof getThreshold>>>;

declare function useSafeThreshold(): SafeThresholdGetter;
const threshold = useSafeThreshold(); // Type of 'threshold' should be 'Range0to100 & number'

Explanation:

Similar to Example 1, Prove would verify the property. If getThreshold were to return a value outside [0, 100] (e.g., 150), the Prove mechanism would ideally produce a type error.

Example 3: Attempting to Prove Falsehood (Compile-time Error)

Concept: Show how the system prevents incorrect proofs.

Input:

// Property: always returns null
type IsNull = { __brand: 'IsNull' };
type IsNotNull<T> = T extends null ? never : IsNull; // Property is NOT null

function alwaysPositive(x: number): number {
  return Math.abs(x) + 1; // Always positive, never null
}

// Attempt to prove that 'alwaysPositive' is 'IsNull'
// This should fail to compile or produce a type error

Expected Output (Conceptual):

A TypeScript compile-time error indicating that the provided proof is invalid for the given function.

// This line should cause a type error:
type InvalidProofAttempt = Prove<typeof alwaysPositive, IsNull>;

Explanation:

The Prove mechanism, when encountering alwaysPositive and the IsNull property, would detect that alwaysPositive cannot satisfy the IsNull property (as it always returns a number). This mismatch should be flagged by the type checker.

Constraints

• TypeScript Version: Assume a recent version of TypeScript (e.g., 4.0 or later).
• Type System Focus: The primary goal is to leverage TypeScript's type system for verification, not necessarily complex runtime code generation or execution.
• Proof Complexity: For this challenge, proofs can be represented as simple branded types or union types that signify a successful check. You don't need to implement complex logical satisfiability solvers. The "proof" itself is more of a declaration that the property holds, and the system validates this declaration.
• Function Signatures: Assume functions operate on basic primitive types or simple object structures.

Notes

• Branded Types: Branded types are a powerful tool in TypeScript for creating distinct types that are not structurally compatible. They are excellent for representing the "guarantee" of a property.
• Conditional Types: Conditional types will be crucial for defining properties and implementing verification logic within the type system.
• Advanced Verification: A more advanced version might involve runtime verification, where the "proof" is an actual executable piece of code or data structure that can be checked at runtime. However, for this challenge, focus on compile-time type-level verification.
• Success Criteria: Success is defined by your ability to create a set of types that:
  • Can represent functions and their claimed properties.
  • Prevent the association of invalid proofs with code.
  • Allow for the inference of guaranteed properties when valid proofs are used.