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. . . . . . . . . . . . . . A Parametric CPS to Sprinkle CIC with Classical Reasoning Pierre-Marie Pdrot University of Ljubljana LOLA 2017 19th June 2017 Pdrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . .

  1. . . . . . . . . . . . . . . A Parametric CPS to Sprinkle CIC with Classical Reasoning Pierre-Marie Pédrot University of Ljubljana LOLA 2017 19th June 2017 Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 / 33

  2. . . . . . . . . . . . . CIC, I’m loving it . Dependent Type Theory is awesome! The pinacle of the Curry-Howard correspondence: You can program with it “A pure functional programming with crazily precise types.” You can prove with it “A incredibly rich constructive logic with built-in computation.” Everything at the same time! “Prove your programs! Program your proofs!” Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 33

  3. . . . . . . . . . . . . CIC, I’m loving it . Dependent Type Theory is awesome! The pinacle of the Curry-Howard correspondence: You can program with it “A pure functional programming with crazily precise types.” You can prove with it “A incredibly rich constructive logic with built-in computation.” Everything at the same time! “Prove your programs! Program your proofs!” Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 33

  4. . . . . . . . . . . . . CIC, I’m loving it . Dependent Type Theory is awesome! The pinacle of the Curry-Howard correspondence: You can program with it “A pure functional programming with crazily precise types.” You can prove with it “A incredibly rich constructive logic with built-in computation.” Everything at the same time! “Prove your programs! Program your proofs!” Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 / 33

  5. . . . . . . . . . . . . . . . An efgective object That’s just not theoretical ramble. Lots of actual, serious, big developments. CompCert, VST, RustBelt... Four Colour Theorem, Feit-Thompson... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 33

  6. . . . . . . . . . . . . . . . An efgective object That’s just not theoretical ramble. Lots of actual, serious, big developments. CompCert, VST, RustBelt... Four Colour Theorem, Feit-Thompson... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . 3 / 33

  7. . . . . . . . . . . . . A Classical Problem . In practice, many people reason in the dreaded classical logic. Both a theoretical and practical limitation! CIC is deadcore intuitionistic Requires that you write your statements in the right way Most non-logicians don’t care about this fuss (both CS and math...) It would be nice to have a type theory... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 33 em : Π( A : □ ) . A ∨ ¬ A

  8. . . . . . . . . . . . . A Classical Problem . In practice, many people reason in the dreaded classical logic. Both a theoretical and practical limitation! CIC is deadcore intuitionistic Requires that you write your statements in the right way Most non-logicians don’t care about this fuss (both CS and math...) It would be nice to have a type theory... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 33 em : Π( A : □ ) . A ∨ ¬ A

  9. . . . . . . . . . . . . . . A Classical Problem In practice, many people reason in the dreaded classical logic. Both a theoretical and practical limitation! CIC is deadcore intuitionistic Requires that you write your statements in the right way Most non-logicians don’t care about this fuss (both CS and math...) Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 4 / 33 em : Π( A : □ ) . A ∨ ¬ A It would be nice to have a classical type theory...

  10. Classical logic is incompatible with univalence! (Your mileage may vary.) . forall A . . . . . . . . Attempt 1: The Truth is Out There There is a very simple straightforward solution. Axiom classical Type . A A Pro: Simple, local, works in Coq, be my guest. Cons: Axioms are dangerous, you have to show consistency externally Classical logic holds in the well-known Set model, blah-blah... Non-trivial interactions: e.g. classical CIC implies proof-irrelevance. The logic does not compute anymore, axioms block reduction... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 33

  11. Classical logic is incompatible with univalence! (Your mileage may vary.) . . . . . . . . . . . . Attempt 1: The Truth is Out There . There is a very simple straightforward solution. Pro: Simple, local, works in Coq, be my guest. Cons: Axioms are dangerous, you have to show consistency externally Classical logic holds in the well-known Set model, blah-blah... Non-trivial interactions: e.g. classical CIC implies proof-irrelevance. The logic does not compute anymore, axioms block reduction... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 33 Axiom classical : forall ( A : Type ) , A ∨ ¬ A .

  12. . . . . . . . . . . . . Attempt 1: The Truth is Out There . There is a very simple straightforward solution. Pro: Simple, local, works in Coq, be my guest. Cons: Axioms are dangerous, you have to show consistency externally Classical logic holds in the well-known Set model, blah-blah... Non-trivial interactions: e.g. classical CIC implies proof-irrelevance. The logic does not compute anymore, axioms block reduction... Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 / 33 Axiom classical : forall ( A : Type ) , A ∨ ¬ A . Classical logic is incompatible with univalence! (Your mileage may vary.)

  13. . . . . . . . . . . . . . . Since Griffjn, it’s folklore that control operators implement classical logic. Essentially allows to reify context evaluation. E callcc M callcc k E M E k The type of callcc is Peirce’s law, the minimal logic equivalement of EM. Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . . . 6 / 33 Attempt 2: CIC and call / cc are in a boat callcc : (( A → B ) → A ) → A

  14. . . . . . . . . . . . . . . . . Since Griffjn, it’s folklore that control operators implement classical logic. Essentially allows to reify context evaluation. The type of callcc is Peirce’s law, the minimal logic equivalement of EM. Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . . . . . . . . . 6 / 33 Attempt 2: CIC and call / cc are in a boat callcc : (( A → B ) → A ) → A E [ callcc M ] ≡ β callcc ( λ k . E [ M ( E ◦ k )])

  15. . Pro: Computational by construction. . . . . . . . . . . Cons: . Needs a whole new proof assistant implementation. Reminder: Coq is a 33-year old project. Changes the global meaning of logical connectives. What does x A B means? ... and it changes it so much that it also proves False !!! Pro: At least my proofs are going to be easier. Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . . . 7 / 33 . . . . . . . . . . . . Attempt 2: CIC and call / cc are in a boat “Just” throw call / cc into CIC!

  16. . . . . . . . . . . . . . . Pro: Computational by construction. Cons: Needs a whole new proof assistant implementation. Reminder: Coq is a 33-year old project. Changes the global meaning of logical connectives. ... and it changes it so much that it also proves False !!! Pro: At least my proofs are going to be easier. Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . 7 / 33 . . . . . . . . . . . . Attempt 2: CIC and call / cc are in a boat “Just” throw call / cc into CIC! What does Σ x : A . B means?

  17. . . . . . . . . . . . . . . Pro: Computational by construction. Cons: Needs a whole new proof assistant implementation. Reminder: Coq is a 33-year old project. Changes the global meaning of logical connectives. ... and it changes it so much that it also proves False !!! Pro: At least my proofs are going to be easier. Pédrot (U. Ljubljana) A Parametric CPS 19/06/2017 . . . . . . . . . . . . . . 7 / 33 . . . . . . . . . . . . Attempt 2: CIC and call / cc are in a boat “Just” throw call / cc into CIC! What does Σ x : A . B means?

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