I A T I O N Z A N M I I I I I I I I 2 1 4 8 1 1 1 2 - - PowerPoint PPT Presentation

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Aug 31, 2022 187 likes •359 views

O L R I A T I O N Z A N M I I I I I I I I 2 1 4 8 1 1 1 2 4 8 = 1 4 2 1 8 1 1 2 4 8 2 1 4 8 1 4 2 8 1 1 1 4 2 1 8 1 8 1 4 2 8 1 1 2 4 1 1 2 4 8 1 4 8 1 2 normalized 8 2 4 1 1 4 1 2 1

SLIDE 1

N O I L A M R Z A T I O N

SLIDE 2

I I I I I I I I

SLIDE 3

2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1

=

SLIDE 4

2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 2 1 4 8 1 normalized

SLIDE 5

Γ ⊢ M ≡ N : A

Consistency Type-checking Unification Completeness

O

Normalization

(…many others)

SLIDE 6

⊤ ◇ ⊥ abort(M) A × B ⟨M, N⟩ fst(M) snd(M) λx.M A → B M(N) x A + B inl(M) inr(M) case(x.M; y.N; O)

SLIDE 7

(λx.M) N ≡ M[N/x] fst ⟨M, N⟩ ≡ M snd ⟨M, N⟩ ≡ N case(x.M; y.N; inl(O)) ≡ M[O/x] case(x.M; y.N; inr(O)) ≡ N[O/y]

SLIDE 8

(λx.M) N ↦ M[N/x] fst ⟨M, N⟩ ↦ M snd ⟨M, N⟩ ↦ N case(x.M; y.N; inl(O)) ↦ M[O/x] case(x.M; y.N; inr(O)) ↦ N[O/y]

M M'

β rules

λx.fst⟨x,x⟩ ≡ λx.x : A → A

SLIDE 9

◇ inl(◇) λx.x inl(λx.◇) fst(x) x(y)(z) x case(…; x)

SLIDE 10

F ≡ λx.F(x) : A → B

both sides are stuck!

P ≡ ⟨fst(P),snd(P)⟩ : A × B M ≡ ◇ : ⊤

η-expansion

SLIDE 11

all terms stuck terms

normal forms

β η

eval reify

Normalization By Evaluation

SLIDE 12

(λx.M)N (λx.M')N (λx.M')N'

steps 1 step steps

(λx.M)N {x.M}N {x.M}N'

1 step 1 step

M'[N'/x]

steps

M[N'/x]

steps … … steps

SLIDE 13

all terms

normal forms

β η

e v a l r e i f y

special domain

λx.M

{x.M}

SLIDE 14

all terms

normal forms

domain

e v a l reify

x:A, y:B, z:C ⊢ λw.w(y)

1 2 3

x:A, y:B, z:C ⊢ λw.w(y) A, B, C ⊢ λ 0(2)

2 De Bruijn indices: from the right

A, B, C ⊢ λ 3(1)

De Bruijn levels: from the le

Trick: use a different scheme in the domain

SLIDE 15

…, p:Path_.A(M;N) ⊢ p@0 ≡ M : A …, p:Path_.A(M;N) ⊢ p@1 ≡ N : A …, p:Path_.A(M;N), i=0 ⊢ p@i ≡ M : A …, p:Path_.A(M;N) ⊢ p@i : A …, p:Path_.A(M;N), i=1 ⊢ p@i ≡ N : A …, i=0 ⊢ loopi ≡ base : S1 … ⊢ loopi : S1 …, i=1 ⊢ loopi ≡ base : S1

CUBICAL

SLIDE 16

Normalization by Evaluation Hereditary Substitution

β-reduction + η-expansion

TODO: Cubical Type Theory

alternative method