proof assistants

Type checker may be wrong – Lean and the Curry-Howard correspondence

2 months ago

Type checkers underpin proof assistants like Lean and Rocq, enabling verification of mathematical proofs through the Curry-Howard correspondence.
The Curry-Howard correspondence maps logical operations to type constructs: conjunction to product types, implication to function types, etc.
Proof assistants validate proofs by checking that a program returns the correct type, but this requires the program to terminate.
The halting problem (undecidable) means type checkers cannot always determine if a program terminates, leading to incompleteness.
Formal languages restrict recursion to ensure termination, but this makes them unable to prove or disprove all statements (Gödel's first incompleteness theorem).
The halting problem implies Gödel's incompleteness theorem via the Curry-Howard correspondence, showing type checkers are provably imperfect.
Consequently, type checkers may reject correct code (though never accept incorrect code), so they can be 'wrong' in the sense of incompleteness.