A nominal syntax for internal parametricity Thorsten Altenkirch, Ambrus Kaposi University of Nottingham TYPES Meeting, Tallinn 18-21 May 2015
Introduction Parametricity and univalence ◮ Parametricity says that terms respect logical relations. ◮ Two functions are related if they map related inputs to related outputs. ◮ Two pairs are related if they are componentwise related. ◮ Two types are related if there is a relation between them. ◮ In homotopy type theory, terms respect equality. ◮ Two functions are equal if they map equal inputs to equal outputs (function extensionality). ◮ Two pairs are equal if they are componentwise equal. ◮ Two types are equal if there is a relation between them and this relation is the graph of an equivalence (univalence). ◮ Our goal is to replace intensional equality in type theory by one which is defined recursively over the type structure. ◮ Inspiration: ◮ Bernardy-Moulin: Internal parametricity, 2012 ◮ Bezem-Coquand-Huber: The cubical sets model of type theory, 2013 ◮ Altenkirch-McBride-Swierstra: Observational type theory, 2007 ◮ Martin-L¨ of: An intuitionistic theory of types, 1972 Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 2 / 16
External parametricity Parametricity ◮ Parametricity says that terms respect logical relations. A : U , u : A , s : A → A ⊢ t : A ρ 0 ≡ ( A �→ N , u �→ zero , s �→ suc) ρ 1 ≡ ( A �→ Bool , u �→ true , s �→ not) ◮ If ρ 0 and ρ 1 are related, then t [ ρ 0 ] is related to t [ ρ 1 ]. ◮ It can’t happen that t [ ρ 0 ] ≡ suc (suc zero) and t [ ρ 1 ] ≡ false. ∼ A : N → Bool → U x ∼ A b : ≡ ( x even) is b ◮ A simpler example: A : U , x : A ⊢ t : A Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 3 / 16
External parametricity Specifying a logical relation ◮ The logical relation for a type A: Γ ⊢ A : U Γ ⊢ 0 Γ , 1 Γ : Γ = ⇒ Γ Γ = ⊢∼ A : A [0] → A [1] → U Γ = ⊢ ◮ The context of related elements: · = ≡ · (Γ , x : A ) = ≡ Γ = , x 0 : A [0] , x 1 : A [1] , x 2 : x 0 ∼ A x 1 ◮ Substitutions 0, 1 project out the corresponding components: ≡ () : · ⇒ ∅ i · i Γ , x : A ≡ ( i Γ , x �→ x i ) : (Γ , x : A ) = ⇒ Γ , x : A Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 4 / 16
External parametricity Defining the logical relation Γ ⊢ A : U Γ = ⊢∼ A : A [0] → A [1] → U f 0 ∼ Π( x : A ) . B f 1 ≡ Π( x 0 : A [0] , x 1 : A [1] , x 2 : x 0 ∼ A x 1 ) . f 0 x 0 ∼ B f 1 x 1 ( a , b ) ∼ Σ( x : A ) . B ( a ′ , b ′ ) ≡ Σ( x 2 : a ∼ A a ′ ) . b ∼ B [ x 0 �→ a , x 1 �→ a ′ ] b ′ A ∼ U B ≡ A → B → U (parametricity) A ∼ U B ≡ A ≃ B (later) Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 5 / 16
External parametricity Parametricity Γ ⊢ t : A Γ = ⊢ t = : t [0] ∼ A t [1] This can be proven by induction on the term t : ≡ f = u [0] u [1] u = ( f u ) = ( λ x . t ) = ≡ λ x 0 , x 1 , x 2 . t = x = ≡ x 2 U = ≡∼ U ( t [ ρ ]) = ≡ t = [ ρ = ] ( ρ = is pointwise) The previous example: ( A : U , u : A , s : A → A ) = ⊢ t = : t [0] ∼ A t [1] Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 6 / 16
Internal parametricity Internalisation ◮ The type of param is not well-formed: · ⊢ param : Π( A : U , t : A ) . t ∼ A t ◮ We need a substitution from ( A : U , t : A ) to ( A : U , t : A ) = . ◮ We define R Γ : Γ ⇒ Γ = : Γ = R · ≡ () R Γ . x : A ≡ (R Γ , x , x , refl x ) R Γ 0 Γ 1 Γ Γ Γ ⊢ a : A Γ ⊢ refl a ≡ ( a = )[R Γ ] : a ∼ A [R Γ ] a ◮ Now we can define param: · ⊢ λ A , t . refl t : Π( A : U , t : A ) . t ∼ A [R A : U ] t ◮ Before adding refl we didn’t extend the theory. Now (refl x ) is a new normal form if x is a variable. Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 7 / 16
Internal parametricity Parametricity of parametricity ◮ Given (refl x ) as a new normal form, we need to say what (refl x ) = is, i.e. refl = x 0 x 1 x 2 . ◮ We could add a new term former refl = : Γ = ⊢ a 0 : A [0] Γ = ⊢ a 1 : A [1] Γ = ⊢ a 2 : a 0 ∼ A a 1 Γ ⊢ A : U Γ = ⊢ refl = a 0 a 1 a 2 : A == [R = Γ ] a 0 a 1 a 2 a 0 a 1 a 2 (refl a 0 ) (refl a 1 ) Γ = ⊢ refl a 2 : A == [R Γ = ] a 0 a 0 (refl a 0 ) a 1 a 1 (refl a 1 ) a 2 a 2 ◮ refl and refl = ways correspond to the two ways of degenerating a line into a square. ◮ We have refl a ≡ a = [R]. ◮ We don’t know how to compute refl = a 0 a 1 a 2 . Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 8 / 16
Internal parametricity Higher dimensions ( x : A ) == can be viewed as a context of squares: ( x : A ) == . x 2 : x 0 ∼ A x 1 ) = ≡ ( x 0 : A [0] . x 1 : A [1] ≡ x 00 : A [00] . x 01 : A [01] . x 02 : x 00 ∼ A [0] x 01 . x 12 : x 10 ∼ A [1] x 11 . x 10 : A [10] . x 11 : A [11] . x 20 : x 00 ∼ A [0] x 10 . x 21 : x 01 ∼ A [1] x 11 . x 22 : x 20 ∼ x 0 ∼ A x 1 x 21 The result of R ( x : A ) = and (R x : A ) = : This can be drawn as: refl x 1 x 21 x 2 x 01 x 11 x 1 x 1 x 0 x 1 refl = x 0 x 1 x 2 x 02 x 22 x 12 x 2 x 2 refl x 2 refl x 0 refl x 1 x 20 refl x 0 x 2 x 00 x 10 x 0 x 0 x 0 x 1 Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 9 / 16
Internal parametricity Adding dimension names = we First we add more information to the contexts: for each usage of − will use a new dimension name, so ( x : A ) == will become ( x : A ) ij : (Γ , x : A ) i ≡ Γ i , x i 0 : A [0 i ] , x i 1 : A [1 i ] , x i 2 : A i x i 0 x i 1 This refines the types of R ( x : A ) = and R = x : A : : ( x : A ) j ⇒ ( x : A ) ji R i ( x : A ) j (R i ( x : A ) ) j : ( x : A ) j ⇒ ( x : A ) ij Their targets are different contexts, with dimensions swapped. Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 10 / 16
Internal parametricity A definitional quotient We would like to equate ( x : A ) ij and ( x : A ) ji . These have different variable names, but they contain the same information: eg. x i 1 j 2 corresponds to x j 2 i 1 . We add the following quotients: ρ : ∆ ⇒ Γ Γ ⊢ t : A Γ ij ≡ Γ ji Γ ij ⊢ t ij ≡ t ji : A ij { x �→ t } ij ρ ij ≡ ρ ji : ∆ ij ⇒ Γ ij { x �→ t } ij is ( x �→ t ) ij omitting the last element. Every term former needs to be symmetric, eg. Π types need to know which argument corresponds to which index, because we need (Π( x : A ) . B ) ij ≡ (Π( x : A ) . B ) ji . i.e. Π( x i 0 j 0 : A [0 i 0 j ] , x i 0 j 1 : A [0 i 1 j ] , x i 0 j 1 : A [0 i ] j x i 0 j 0 x i 0 j 1 , . . . ) . B ij . . . ≡ Π( x i 0 j 0 : A [0 i 0 j ] , x i 0 j 1 : A [0 i 1 j ] , . . . , x j 1 i 0 : ( A j [0 i ] x j 0 i 0 x j 1 i 0 ) . B ji . . . . Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 11 / 16
Internal parametricity New rules for Π types New rules for full Π types and relations ( I is a set of dimension names): ξ : Γ ⇒ ( X : U) I Γ . ( x : X ) I [ ξ ] ⊢ B : U ξ : Γ ⇒ { X : U } I Γ ⊢ Π( x : X ) I [ ξ ] . B : U Γ ⊢ Π { x : X } I [ ξ ] . U : U Γ + + ( { x : X } ) I [ ξ ] ⊢ t : B Γ ⊢ λ ( { x : X } { x : X } ) I [ ξ ] . t : Π( ) I [ ξ ] . B Γ ⊢ f : Π( { x : X } ) I [ ξ ] . B Γ ⊢ ω : ( { y : X } ) I [ ξ ] Γ ⊢ app I ( f , ω ) : B [id + + ω ] Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 12 / 16
Internal parametricity The computation rule of refl = We generalise the rule for parametricity from adding one dimension to adding a set of dimensions at a time. The parametricity rule becomes: Γ ⊢ t : A Γ I ⊢ t I : app I ( A I , { x �→ t } I ) The order of dimensions does not matter. Lifting of the universe becomes: U I ≡ λ { X : U } I . Π { x : X } I . U Now we have (refl i a ) j ≡ ( a i [R i Γ ]) j ≡ a ij [R i Γ j ] ≡ a ji [R i Γ j ] ≡ refl i a j , so the problem with the computation rule of refl = disappears. Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 13 / 16
Internal parametricity Operational semantics To rigorously define the theory, we need telescope contexts and substitutions. We defined a call-by-name operational semantics for this theory where the weak-head normal forms are: t , A ::= . . . terms ::= () | ( ν, x �→ g ) | ( ν, x �→ t [ ν ]) ν environments ::= U | (Π( { x : X } ) I [ ρ ] . A )[ ν ] | ( λ ( { x } ) I . t )[ ν ] | n values (whnfs) v n ::= g | app I ( n , ν ) neutral values ::= x | refl i g generic values g Ambrus Kaposi (University of Nottingham) A nominal syntax for internal parametricity May 22, 2015 14 / 16
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