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Component-based Proof Assistant

Component-based Proof Assistant. Frenkel Smeijers. Outline. Proof assistant FoolProof. Proof assistant. Interactive construction of proofs Based on typed lambda-calculi Based on propositions as types, proofs as terms principle Formal basis of e.g. Coq, Cocktail. Propositions as Types.

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Component-based Proof Assistant

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  1. Component-based Proof Assistant Frenkel Smeijers

  2. Outline • Proof assistant • FoolProof

  3. Proof assistant • Interactive construction of proofs • Based on typed lambda-calculi • Based on propositions as types, proofs as terms principle • Formal basis of e.g. Coq, Cocktail

  4. Propositions as Types • Unifying approach to lambda calculi and logic • Based on “isomorphism” between types and propositions • Judgment ├ M: τhas two interpretations: • In context , term M has type τ • In context , term M encodes proof of proposition τ

  5. ( I) , x: σ├ M: τ ├ (λx: σ. M): σ  τ ( E)  ├ M: στ ├ N: σ ├ MN: τ (var) ├ M: σ if (M: σ)   Similar correspondences between  and  + and   and   and  ( I) , P├ Q ├ P  Q ( E)  ├ P  Q  ├ P ├ Q (var) ├ P if P   Type deduction vs. logical deduction

  6. x : P y : Q x : P (λy: Q. x) : Q P (λx: P. (λy: Q . x) : P  (Q  P) P Q P Q  P P  (Q  P) Propositions as types example

  7. Interactive construction of terms λ : P(QP) x: P λ : QP y: Q x: P • This is an example of structure editing with bound variables

  8. FoolProof • Component library • Delphi • Components for: • Lexical scanning • Syntax highlighting • Parsing • Tree building • Structure editing • Textual views • Structural views • Context management

  9. FoolProof: distinguishing facilities • “Small and simple” components that fit into a general framework (e.g. Delphi, JBuilder, Visual Studio) • Manipulation of binding structures • Copying, substitution, unification etc. • Context management • Context: structured collection of names and their properties • Management: manipulating contexts in a well-formedness preserving way

  10. FoolProof Test Environment

  11. My proof assistant • Based on abstract descriptions by Marco Brassé • Research usability of FoolProof components • Not an automatic prover • Intended to become FoolProof component itself

  12. Subtasks • Recall typed lambda calculi (as far as necessary) • Study existing proof assistants (Coq, Cocktail, …), Brassé’s notes, FoolProof • Abstract and concrete design of new proof assistant • Implement and integrate

  13. Questions?

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