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Oct 07, 2022 •404 likes •673 views

redtt cartesian cubical proof assistant favonia university of minnesota oslo, 2018/8/28 joint work with Carlo Angiuli, Evan Cavallo, Robert Harper, Anders Mrtberg and Jonathan Sterling 1 type theory A type A = B type M

  1. redtt cartesian cubical proof assistant favonia university of minnesota oslo, 2018/8/28 joint work with Carlo Angiuli, Evan Cavallo, Robert Harper, Anders Mörtberg and Jonathan Sterling 1

  2. type theory Γ ⊦ A type Γ ⊦ A = B type Γ ⊦ M : A Γ ⊦ M = N : A 2

  3. cubical type theory formal intervals 𝕁 x: 𝕁 ∈ Γ Γ ⊦ 0: 𝕁 Γ ⊦ 1: 𝕁 Γ ⊦ x: 𝕁 Γ ⊦ r: 𝕁 Γ ⊦ r: 𝕁 Γ ⊦ s: 𝕁 Γ ⊦ r: 𝕁 Γ ⊦ s: 𝕁 Γ ⊦ ¬r: 𝕁 Γ ⊦ r ∧ s: 𝕁 Γ ⊦ r ∨ s: 𝕁 3

  4. cubical type theory formal intervals 𝕁 x 1 : 𝕁 , x 2 : 𝕁 , ..., x n : 𝕁 ⊦ M : A ⬄ M is an n-cube in A y x M ⟨ 0/x ⟩ M ⟨ 1/x ⟩ M ⟨ y/x ⟩ 4

  5. cubical type theory formal intervals 𝕁 ordinary typing rules hold uniformly Γ , a:A ⊦ M : B Γ ⊦ λ a.M : (a:A) → B with any number of 𝕁 in the Γ 5

  6. cubical type theory formal intervals 𝕁 ordinary typing rules hold uniformly Γ , a:A ⊦ M : B Γ ⊦ λ a.M : (a:A) → B with any number of 𝕁 in the Γ function extensionality due to dimensions commuting with function application 5

  7. cubical type theory formal intervals 𝕁 canonicity any closed term of ℕ is equal to some numeral type-theory tango: internalization of judgmental structure, harmony 6

  8. cubical type theories base category structural rules + operators {0,1, ∧ , ∨ , ¬,...} most developed: cartesian, de morgan 7

  9. cubical type theories base category structural rules + operators {0,1, ∧ , ∨ ,¬,...} most developed: cartesian, de morgan kan structure co fi brations, fi berwise fi brant replacement 7

  10. cubical type theories base category structural rules + operators {0,1, ∧ , ∨ ,¬,...} most developed: cartesian, de morgan kan structure co fi brations, fi berwise fi brant replacement mythos proofs or realizers? 7

  11. Agda cubicaltt yacctt redtt RedPRL de morgan cartesian 0 ⇝ 1, i=0/1 r ⇝ s, r=s proofs realizers 8

  12. Agda cubicaltt yacctt redtt RedPRL de morgan cartesian 0 ⇝ 1, i=0/1 r ⇝ s, r=s proofs realizers chalmers cmu gothenburg fancy spartan Agda cubicaltt yacctt redtt RedPRL 8

  13. redtt specialities higher inductive types two-level type theory nbe-like algorithm (conjectured correct) extension types judgmental re fi nements holes, tactics, uni fi cation 9

  14. redtt specialities higher inductive types two-level type theory nbe-like algorithm (conjectured correct) extension types see judgmental re fi nements demo holes, tactics, uni fi cation 9

  15. redtt specialities higher inductive types a general schema; indexed ones on the way see chtt part 4 [Cavallo & Harper] 10

  16. redtt specialities two-level type theory (no equality types yet) type pretype 11

  17. redtt specialities todo: many-level type theory discrete: paths equal to equality consistent with (strict) UIP discrete type pretype 12

  18. redtt specialities nbe algorithm cubicaltt adopts a similar one 13

  19. redtt specialities nbe algorithm cubicaltt adopts a similar one di ffi culty 1: value re-evaluation: loop x [0/x] di ffi culty 2: constraints: r=s 13

  20. redtt specialities nbe algorithm cubicaltt adopts a similar one di ffi culty 1: value re-evaluation: loop x [0/x] di ffi culty 2: constraints: r=s decidable: Φ ⊧ r = s 13

  21. todo correctness of nbe 14

  22. todo correctness of nbe equality types 14

  23. todo correctness of nbe equality types user-de fi ned tactic, pattern matching, etc 14

  24. todo correctness of nbe equality types user-de fi ned tactic, pattern matching, etc improved kan operations of universes 14

  25. todo correctness of nbe equality types user-de fi ned tactic, pattern matching, etc improved kan operations of universes synthetic homotopy theory (!) 14

  26. synthetic homotopy theory todo standard redtt homotopy theory "obvious" RedPRL 15

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