On automating the extraction of programs from proofs using product types

Introduction • We are interested in programming language with the point of view: Proofs as Programs (Curry-Howard correspondence). Introduction • We are interested in programming language with the point of view: Proofs as Programs (Curry-Howard correspondence). • The specifications are the types and the lambda-terms are the extracted programs (the code). Introduction • We are interested in programming language with the point of view: Proofs as Programs (Curry-Howard correspondence). • The specifications are the types and the lambda-terms are the extracted programs (the code). • The verification of the types (compilation) is a proof of program. Introduction • We are interested in programming language with the point of view: Proofs as Programs (Curry-Howard correspondence). • The specifications are the types and the lambda-terms are the extracted programs (the code). • The verification of the types (compilation) is a proof of program. • The ProPre system was designed as a prototype to show the feasibility of the theory. Motivation • The difficulty is to find formal proofs automatically. Motivation • The difficulty is to find formal proofs automatically. The term quot(x, y, y) computes ⌊ x y ⌋. Motivation • The difficulty is to find formal proofs automatically. The term quot(x, y, y) computes ⌊ x y ⌋. • The proofs are expressed in natural deduction style. Motivation • The difficulty is to find formal proofs automatically. The term quot(x, y, y) computes ⌊ x y ⌋. • The proofs are expressed in natural deduction style. • The automated termination proofs = techniques of rewriting systems. Motivation • Is it possible to go further than the ProPre system but using the same theory ? Motivation • Is it possible to go further than the ProPre system but using the same theory ? • We analyse the proofs made in the system. Motivation • Is it possible to go further than the ProPre system but using the same theory ? • We analyse the proofs made in the system. • We then develop some particular formal proofs using product types. Motivation • Is it possible to go further than the ProPre system but using the same theory ? • We analyse the proofs made in the system. • We then develop some particular formal proofs using product types. • The formal proofs are released from the termination part. Motivation • Is it possible to go further than the ProPre system but using …