redtt cartesian cubical proof assistant favonia university of - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

redtt specialities nbe algorithm cubicaltt adopts a similar one 13

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

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

todo correctness of nbe 14

todo correctness of nbe equality types 14

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

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

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

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