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

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Oct 07, 2022 404 likes •676 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

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redtt

university of minnesota

favonia

slo, 2018/8/28

joint work with Carlo Angiuli, Evan Cavallo, Robert Harper, Anders Mörtberg and Jonathan Sterling

cartesian cubical proof assistant

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type theory

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

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cubical type theory

formal intervals 𝕁 Γ ⊦ 0:𝕁 Γ ⊦ 1:𝕁 Γ ⊦ x:𝕁 x:𝕁 ∈ Γ Γ ⊦ r∧s:𝕁 Γ ⊦ r:𝕁 Γ ⊦ s:𝕁 Γ ⊦ r∨s:𝕁 Γ ⊦ r:𝕁 Γ ⊦ s:𝕁 Γ ⊦ ¬r:𝕁 Γ ⊦ r:𝕁

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cubical type theory

formal intervals 𝕁

⬄ M is an n-cube in A x1:𝕁, x2:𝕁, ..., xn:𝕁 ⊦ M : A

M⟨0/x⟩ M⟨1/x⟩ M⟨y/x⟩ x y

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cubical type theory

formal intervals 𝕁

rdinary typing rules hold uniformly

Γ ⊦ λa.M : (a:A) → B Γ, a:A ⊦ M : B with any number of 𝕁 in the Γ

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cubical type theory

formal intervals 𝕁

rdinary typing rules hold uniformly

Γ ⊦ λa.M : (a:A) → B Γ, a:A ⊦ M : B with any number of 𝕁 in the Γ function extensionality due to dimensions commuting with function application

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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

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cubical type theories

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

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cubical type theories

base category most developed: cartesian, de morgan structural rules + operators {0,1,∧,∨,¬,...} kan structure cofibrations, fiberwise fibrant replacement

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cubical type theories

base category most developed: cartesian, de morgan structural rules + operators {0,1,∧,∨,¬,...} kan structure cofibrations, fiberwise fibrant replacement mythos proofs or realizers?

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8 RedPRL yacctt redtt Agda cubicaltt

cartesian r⇝s, r=s de morgan 0⇝1, i=0/1 realizers proofs

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8 RedPRL yacctt redtt Agda cubicaltt

cartesian r⇝s, r=s de morgan 0⇝1, i=0/1 cmu chalmers gothenburg realizers proofs spartan fancy

RedPRL yacctt redtt Agda cubicaltt

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redtt specialities

higher inductive types two-level type theory nbe-like algorithm (conjectured correct) extension types judgmental refinements holes, tactics, unification

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redtt specialities

higher inductive types two-level type theory nbe-like algorithm (conjectured correct) extension types judgmental refinements holes, tactics, unification

see demo

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redtt specialities

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

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redtt specialities

two-level type theory

type pretype (no equality types yet)

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redtt specialities

todo: many-level type theory

type pretype discrete discrete: paths equal to equality consistent with (strict) UIP

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redtt specialities

nbe algorithm cubicaltt adopts a similar one

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redtt specialities

nbe algorithm cubicaltt adopts a similar one difficulty 1: value re-evaluation: loopx[0/x] difficulty 2: constraints: r=s

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redtt specialities

nbe algorithm cubicaltt adopts a similar one difficulty 1: value re-evaluation: loopx[0/x] difficulty 2: constraints: r=s decidable: Φ ⊧ r = s

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todo

correctness of nbe

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todo

correctness of nbe equality types

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todo

correctness of nbe equality types user-defined tactic, pattern matching, etc

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todo

correctness of nbe improved kan operations of universes equality types user-defined tactic, pattern matching, etc

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todo

correctness of nbe improved kan operations of universes synthetic homotopy theory (!) equality types user-defined tactic, pattern matching, etc