Type-Level Theorem Prover for Propositional Logic

This challenge asks you to build a type-level theorem prover in TypeScript. The goal is to represent propositional logic statements as TypeScript types and then use TypeScript's type system to prove whether a given conclusion logically follows from a set of premises. This is a fascinating exercise in leveraging TypeScript's powerful type inference and manipulation capabilities for formal reasoning.

Problem Description

You need to design and implement a system that can:

1. Represent Propositional Logic: Define a set of TypeScript types that can represent basic propositional logic concepts such as:

   • Propositions: Individual atomic statements (e.g., P, Q, R).
   • Logical Connectives:
     • Negation (NOT<T>)
     • Conjunction (AND<T1, T2>)
     • Disjunction (OR<T1, T2>)
     • Implication (IMPLIES<T1, T2>)
     • Equivalence (EQUIV<T1, T2>)
   • Logical Truth (TRUE) and Falsity (FALSE)

2. Define Inference Rules: Implement type-level representations of fundamental inference rules from propositional logic. These rules should allow you to derive new logical statements from existing ones. Examples of rules to consider:

   • Modus Ponens: From IMPLIES<A, B> and A, infer B.
   • Introduction of AND: From A and B, infer AND<A, B>.
   • Elimination of AND: From AND<A, B>, infer A and infer B.
   • Introduction of OR (left): From A, infer OR<A, B>.
   • Elimination of OR: From OR<A, B>, IMPLIES<A, C>, and IMPLIES<B, C>, infer C.
   • Double Negation Elimination: From NOT<NOT<A>>, infer A.
   • Proof by Contradiction (Reductio ad Absurdum): If assuming NOT<A> leads to FALSE, then infer A.

3. Implement a Theorem Prover: Create a mechanism (likely a higher-order type or a series of type aliases) that takes a set of premises (as types) and a conclusion (as a type) and determines if the conclusion can be logically derived from the premises using the defined inference rules. The prover should return TRUE if the conclusion is provable, and FALSE otherwise.

Key Requirements:

• Type Safety: All logical operations and deductions must be enforced by the TypeScript type system.
• Expressiveness: The system should be able to represent a reasonable subset of propositional logic.
• Modularity: Inference rules should be implemented as reusable type constructs.

Expected Behavior:

The prover should correctly identify valid logical deductions. For example, if premises are P and IMPLIES<P, Q>, the prover should successfully derive Q. If the conclusion is not derivable, it should correctly indicate that.

Edge Cases:

• Handling complex nested logical structures.
• Proving tautologies (statements that are always true).
• Proving contradictions (statements that are always false).
• Inconsistent premises.

Examples

Example 1: Modus Ponens

// Premises: P, IMPLIES<P, Q>
// Conclusion: Q

type P = 'P';
type Q = 'Q';

type Premises1 = [P, IMPLIES<typeof P, typeof Q>];
type Conclusion1 = Q;

// Expected: TRUE
// How to represent this as a type check?
// You might create a type alias like 'Prove<Premises, Conclusion>'
// and it should resolve to TRUE or FALSE.

Example 2: AND Introduction and Elimination

// Premises: P, Q
// Conclusion: AND<P, Q>

type P = 'P';
type Q = 'Q';

type Premises2 = [P, Q];
type Conclusion2 = AND<typeof P, typeof Q>;

// Expected: TRUE

Example 3: Unprovable Conclusion

// Premises: P
// Conclusion: Q

type P = 'P';
type Q = 'Q';

type Premises3 = [P];
type Conclusion3 = Q;

// Expected: FALSE

Example 4: Proof by Contradiction (Conceptual)

This is more complex and might require a sophisticated type structure. Imagine a type that tries to derive FALSE from NOT<A> given a set of premises. If it succeeds, it proves A.

// Premises: IMPLIES<NOT<P>, FALSE>
// Conclusion: P

type P = 'P';

// The premise IMPLIES<NOT<P>, FALSE> is itself a proof by contradiction for P.
// How to represent this?
// If 'AssumeNotP' leads to 'FALSE' (under premises), then we can assert 'P'.

// A simplified premise for illustration:
type Premise4 = IMPLIES<NOT<typeof P>, FALSE>;
type Premises4 = [Premise4];
type Conclusion4 = typeof P;

// Expected: TRUE

Constraints

• Propositional Logic Scope: Focus on propositional logic, not predicate logic.
• Type Representation: All logical operators and statements must be represented as TypeScript types (interfaces, type aliases, etc.).
• Inference Rules: Implement at least Modus Ponens, AND Introduction/Elimination, and OR Introduction/Elimination. Proof by Contradiction is a stretch goal.
• Prover Mechanism: The core prover logic should be expressible within TypeScript's type system, meaning it should be a type that resolves to TRUE or FALSE.
• Performance: While type-level computation is inherently slower than runtime, avoid overly complex recursive types that could lead to excessive compile times or stack overflows for moderately complex proofs. Aim for a balance between expressiveness and feasibility.

Notes

• Type Aliases vs. Interfaces: Consider how type aliases and interface declarations can be used to model logical structures.
• Tuple Types for Premises: Use tuple types ([T1, T2, ...]) to represent collections of premises.
• Union Types and Conditional Types: These are powerful tools in TypeScript for pattern matching and conditional logic at the type level, which will be crucial for implementing inference rules.
• Recursive Type Definitions: Some inference rules might naturally lead to recursive type definitions. Be mindful of potential infinite recursion and how to handle it.
• Type Inference as Deduction: The act of TypeScript inferring a type or resolving a complex type alias can be seen as performing a deduction.
• Success Looks Like: A set of type definitions and a Prove<Premises, Conclusion> type that accurately resolves to TRUE for valid deductions and FALSE for invalid ones, demonstrating a working type-level theorem prover.