[PPT] - Reynolds Parametricity Patricia Johann Appalachian State University PowerPoint Presentation

SLIDE 1

Reynolds’ Parametricity

Patricia Johann Appalachian State University

cs.appstate.edu/

∼johannp

Based on joint work with Neil Ghani, Fredrik Nordvall Forsberg, Federico Orsanigo, and Tim Revell

OPLSS 2016

SLIDE 2

Course Outline

Topic: Reynolds’ theory of parametric polymorphism for System F Goals: - extract the fibrational essence of Reynolds’ theory

generalize Reynolds’ construction to very general models
Lecture 1: Reynolds’ theory of parametricity for System F
Lecture 2: Introduction to fibrations
Lecture 3: A bifibrational view of parametricity
Lecture 4: Bifibrational parametric models for System F

SLIDE 3

Course Outline

Topic: Reynolds’ theory of parametric polymorphism for System F Goals: - extract the fibrational essence of Reynolds’ theory

generalize Reynolds’ construction to very general models
Lecture 1: Reynolds’ theory of parametricity for System F
Lecture 2: Introduction to fibrations
Lecture 3: A bifibrational view of parametricity
Lecture 4: Bifibrational parametric models for System F

SLIDE 4

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

SLIDE 5

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

Today we will use relations fibrations on categories other than Set

that are both equality preserving arrow fibrations and ∀-fibrations to interpret System F

SLIDE 6

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

Today we will use relations fibrations on categories other than Set

that are both equality preserving arrow fibrations and ∀-fibrations to interpret System F – types as fibred functors

SLIDE 7

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

Today we will use relations fibrations on categories other than Set

that are both equality preserving arrow fibrations and ∀-fibrations to interpret System F – types as fibred functors – terms as fibred natural transformations

SLIDE 8

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

Today we will use relations fibrations on categories other than Set

that are both equality preserving arrow fibrations and ∀-fibrations to interpret System F – types as fibred functors – terms as fibred natural transformations

This gives very general parametric models for System F

SLIDE 9

Where Were We?

Last time we set up all the infrastructure we need to give our bifibra-

tional parametric model of System F

Today we will use relations fibrations on categories other than Set

that are both equality preserving arrow fibrations and ∀-fibrations to interpret System F – types as fibred functors – terms as fibred natural transformations

This gives very general parametric models for System F
Throughout, let Rel(U) be an equality preserving arrow fibration and

∀-fibration

SLIDE 10

Fibrational Semantics of Types

Define fibred functors

[ [∆ ⊢ τ] ] : |Rel(U)||∆| → Rel(U) by

SLIDE 11

Fibrational Semantics of Types

Define fibred functors

[ [∆ ⊢ τ] ] : |Rel(U)||∆| → Rel(U) by – Type variables: [ [∆ ⊢ αi] ]oX = Xi and [ [∆ ⊢ αi] ]rR = Ri

SLIDE 12

Fibrational Semantics of Types

Define fibred functors

[ [∆ ⊢ τ] ] : |Rel(U)||∆| → Rel(U) by – Type variables: [ [∆ ⊢ αi] ]oX = Xi and [ [∆ ⊢ αi] ]rR = Ri – Arrow types: [ [∆ ⊢ τ1 → τ2] ] = [ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]

SLIDE 13

Fibrational Semantics of Types

Define fibred functors

[ [∆ ⊢ τ] ] : |Rel(U)||∆| → Rel(U) by – Type variables: [ [∆ ⊢ αi] ]oX = Xi and [ [∆ ⊢ αi] ]rR = Ri – Arrow types: [ [∆ ⊢ τ1 → τ2] ] = [ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ] – Forall types: [ [∆ ⊢ ∀α.τ] ] = ∀[ [∆, α ⊢ τ] ]

SLIDE 14

Fibrational Semantics of Types

Define fibred functors

[ [∆ ⊢ τ] ] : |Rel(U)||∆| → Rel(U) by – Type variables: [ [∆ ⊢ αi] ]oX = Xi and [ [∆ ⊢ αi] ]rR = Ri – Arrow types: [ [∆ ⊢ τ1 → τ2] ] = [ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ] – Forall types: [ [∆ ⊢ ∀α.τ] ] = ∀[ [∆, α ⊢ τ] ]

No definition for [

[∆ ⊢ τ] ] on morphisms is needed because the domain

f [

[∆ ⊢ τ] ] is discrete

SLIDE 15

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

SLIDE 16

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

Proof: By induction on the structure of τ

SLIDE 17

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

Proof: By induction on the structure of τ
If τ = ∀α.τ ′, then [

[∆ ⊢ τ] ] = ∀[ [∆, α ⊢ τ ′] ] is an equality preserving fibred functor whenever [ [∆, α ⊢ τ ′] ] is, just by the definition of ∀ : (|Rel(U)|n+1 →Eq Rel(U)) → (|Rel(U)|n →Eq Rel(U))

SLIDE 18

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

Proof: By induction on the structure of τ
If τ = ∀α.τ ′, then [

[∆ ⊢ τ] ] = ∀[ [∆, α ⊢ τ ′] ] is an equality preserving fibred functor whenever [ [∆, α ⊢ τ ′] ] is, just by the definition of ∀ : (|Rel(U)|n+1 →Eq Rel(U)) → (|Rel(U)|n →Eq Rel(U))

Indeed, the very existence of ∀ in a ∀-fibration requires that if F is

equality preserving then so is ∀F

SLIDE 19

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

Proof: By induction on the structure of τ
If τ = ∀α.τ ′, then [

[∆ ⊢ τ] ] = ∀[ [∆, α ⊢ τ ′] ] is an equality preserving fibred functor whenever [ [∆, α ⊢ τ ′] ] is, just by the definition of ∀ : (|Rel(U)|n+1 →Eq Rel(U)) → (|Rel(U)|n →Eq Rel(U))

Indeed, the very existence of ∀ in a ∀-fibration requires that if F is

equality preserving then so is ∀F

In our model, the Identity Extension Lemma is “baked into” the inter-

pretation of types, rather than something to be proved post facto

SLIDE 20

Type Interpretations are Equality Preserving

Proposition The interpretation of every System F type is an equality

preserving fibred functor

Proof: By induction on the structure of τ.
If τ = ∀α.τ ′, then [

[∆ ⊢ τ] ] = ∀[ [∆, α ⊢ τ ′] ] is an equality preserving fibred functor whenever [ [∆, α ⊢ τ ′] ] is, just by the definition of ∀ : (|Rel(U)|n+1 →Eq Rel(U)) → (|Rel(U)|n →Eq Rel(U))

Indeed, the very existence of ∀ in a ∀-fibration requires that if F is

equality preserving then so is ∀F

In our model, the Identity Extension Lemma is “baked into” the inter-

pretation of types, rather than something to be proved post facto

If U is faithful, then the ∀-fibration requirement can be reformulated

in terms of more basic concepts using opfibrational structure of U

SLIDE 21

Fibrational Semantics of Terms - The Set Up

In a CCC, for all X and Y , there is an object X ⇒ Y and a isomor-

phism λ : Hom(W × X, Y ) ∼ = Hom(W, X ⇒ Y ) that is natural in W

SLIDE 22

Fibrational Semantics of Terms - The Set Up

In a CCC, for all X and Y , there is an object X ⇒ Y and a isomor-

phism λ : Hom(W × X, Y ) ∼ = Hom(W, X ⇒ Y ) that is natural in W

The unit of this adjunction is the evaluation map

evX,Y = λ−1(id X⇒Y ) : (X ⇒ Y ) × X → Y

SLIDE 23

Fibrational Semantics of Terms - The Set Up

In a CCC, for all X and Y , there is an object X ⇒ Y and a isomor-

phism λ : Hom(W × X, Y ) ∼ = Hom(W, X ⇒ Y ) that is natural in W

The unit of this adjunction is the evaluation map

evX,Y = λ−1(id X⇒Y ) : (X ⇒ Y ) × X → Y

In a ∀-fibration, for every F and G, there is are isomorphisms

ϕn : Hom(F ◦ πn, G) ∼ = Hom(F, ∀nG) that are natural in n

SLIDE 24

Fibrational Semantics of Terms - term variables

Define fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ] by

SLIDE 25

Fibrational Semantics of Terms - term variables

Define fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ] by

If

∆ ⊢ τi xi : τi ∈ Γ ∆; Γ ⊢ xi : τi then [ [∆; Γ ⊢ xi : τi] ] = πi

SLIDE 26

Fibrational Semantics of Terms - term variables

Define fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ] by

If

∆ ⊢ τi xi : τi ∈ Γ ∆; Γ ⊢ xi : τi then [ [∆; Γ ⊢ xi : τi] ] = πi

πi is the ith projection on both B and E

SLIDE 27

Fibrational Semantics of Terms - term variables

Define fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ] by

If

∆ ⊢ τi xi : τi ∈ Γ ∆; Γ ⊢ xi : τi then [ [∆; Γ ⊢ xi : τi] ] = πi

πi is the ith projection on both B and E
This specializes to our Set interpretation of variables

SLIDE 28

Fibrational Semantics of Terms - term abstractions

If

∆; Γ, x : τ1 ⊢ t : τ2 ∆; Γ ⊢ λx.t : τ1 → τ2 then [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] = λ[ [∆; Γ, x : τ1 ⊢ t : τ2] ]

SLIDE 29

Fibrational Semantics of Terms - term abstractions

If

∆; Γ, x : τ1 ⊢ t : τ2 ∆; Γ ⊢ λx.t : τ1 → τ2 then [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] = λ[ [∆; Γ, x : τ1 ⊢ t : τ2] ]

This is sensible because

[ [∆; Γ, x : τ1 ⊢ t : τ2] ] : [ [∆ ⊢ Γ] ] × [ [∆ ⊢ τ1] ] → [ [∆ ⊢ τ2] ]

SLIDE 30

Fibrational Semantics of Terms - term abstractions

If

∆; Γ, x : τ1 ⊢ t : τ2 ∆; Γ ⊢ λx.t : τ1 → τ2 then [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] = λ[ [∆; Γ, x : τ1 ⊢ t : τ2] ]

This is sensible because

[ [∆; Γ, x : τ1 ⊢ t : τ2] ] : [ [∆ ⊢ Γ] ] × [ [∆ ⊢ τ1] ] → [ [∆ ⊢ τ2] ]

λ is the right adjoint to × in both B and E

SLIDE 31

Fibrational Semantics of Terms - term abstractions

If

∆; Γ, x : τ1 ⊢ t : τ2 ∆; Γ ⊢ λx.t : τ1 → τ2 then [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ λx.t : τ1 → τ2] ] = λ[ [∆; Γ, x : τ1 ⊢ t : τ2] ]

This is sensible because

[ [∆; Γ, x : τ1 ⊢ t : τ2] ] : [ [∆ ⊢ Γ] ] × [ [∆ ⊢ τ1] ] → [ [∆ ⊢ τ2] ]

λ is the right adjoint to × in both B and E
This specializes to our Set interpretation of term abstractions

SLIDE 32

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

SLIDE 33

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

This is sensible because

[ [∆; Γ ⊢ t1 : τ1] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ1] ]

SLIDE 34

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

This is sensible because

[ [∆; Γ ⊢ t1 : τ1] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ1] ] [ [∆; Γ ⊢ t2 : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ])

SLIDE 35

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

This is sensible because

[ [∆; Γ ⊢ t1 : τ1] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ1] ] [ [∆; Γ ⊢ t2 : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ] : [ [Γ] ] → ([ [τ1] ] ⇒ [ [τ2] ]) × [ [τ1] ]

SLIDE 36

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

This is sensible because

[ [∆; Γ ⊢ t1 : τ1] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ1] ] [ [∆; Γ ⊢ t2 : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ] : [ [Γ] ] → ([ [τ1] ] ⇒ [ [τ2] ]) × [ [τ1] ]

−, − : (X → Y )×(X → W ) → X → (Y ×W ) is f, gX = fX ×gX

SLIDE 37

Fibrational Semantics of Terms - term applications

If

∆; Γ ⊢ t1 : τ1 ∆; Γ ⊢ t2 : τ1 → τ2 ∆; Γ ⊢ t2t1 : τ2 then [ [∆; Γ ⊢ t2t1 : τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2] ] [ [∆; Γ ⊢ t2t1 : τ2] ] = evτ1,τ2 ◦ [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ]

This is sensible because

[ [∆; Γ ⊢ t1 : τ1] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ1] ] [ [∆; Γ ⊢ t2 : τ1 → τ2] ] : [ [∆ ⊢ Γ] ] → ([ [∆ ⊢ τ1] ] ⇒ [ [∆ ⊢ τ2] ]) [ [∆; Γ ⊢ t2 : τ1 → τ2] ], [ [∆; Γ ⊢ t1 : τ1] ] : [ [Γ] ] → ([ [τ1] ] ⇒ [ [τ2] ]) × [ [τ1] ]

−, − : (X → Y )×(X → W ) → X → (Y ×W ) is f, gX = fX ×gX
This specializes to our Set interpretation of term applications

SLIDE 38

Fibrational Semantics of Terms - type abstractions

If

∆, α; Γ ⊢ t : τ ∆; Γ ⊢ Λα.t : ∀α.τ then [ [∆; Γ ⊢ Λα.t : ∀α.τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ ∀α.τ] ] = [ [∆ ⊢ Γ] ] → ∀[ [∆, α ⊢ τ] ] [ [∆; Γ ⊢ Λα.t : ∀α.τ] ] = ϕ|∆|[ [∆, α; Γ ⊢ t : τ] ]

This is sensible because α is not free in Γ, so

[ [∆, α; Γ ⊢ t : τ] ] : [ [∆, α ⊢ Γ] ] → [ [∆, α ⊢ τ] ] = [ [∆ ⊢ Γ] ] ◦ π|∆| → [ [∆, α ⊢ τ] ]

SLIDE 39

Fibrational Semantics of Terms - type applications

If

∆; Γ ⊢ t : ∀α.τ2 ∆ ⊢ τ1 ∆; Γ ⊢ t τ1 : τ2[α → τ1] then [ [∆; Γ ⊢ t τ1 : τ2[α → τ1]] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ2[α → τ1]] ] [ [∆; Γ ⊢ t τ1 : τ2[α → τ1]] ] = ϕ−1

|∆|[

[∆; Γ ⊢ t : ∀α.τ2] ] ◦ id |∆|, [ [∆ ⊢ τ1] ]

This is sensible because

[ [∆; Γ ⊢ t : ∀α.τ2] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ ∀α.τ2] ] = [ [∆ ⊢ Γ] ] → ∀[ [∆, α ⊢ τ2] ]

SLIDE 40

Validating β- and η-Rules

Our model is sensible by construction

SLIDE 41

Validating β- and η-Rules

Our model is sensible by construction
Reynolds’ model is an instance of ours, assuming a constructive metathe-
ry — e.g., the Calculus of Constructions with impredicative Set

SLIDE 42

Validating β- and η-Rules

Our model is sensible by construction
Reynolds’ model is an instance of ours, assuming a constructive metathe-
ry — e.g., the Calculus of Constructions with impredicative Set
Proposition If ∆ ⊢ τ1 and ∆, α; Γ ⊢ t : τ2
1. [

[∆; Γ ⊢ (Λα.t)τ1 : τ2[α → τ1]] ] = [ [∆; Γ ⊢ t[α → τ1] : τ2[α → τ1]] ]

2. [

[∆; Γ ⊢ t : ∀β.τ] ] = [ [∆; Γ ⊢ Λα.t α : ∀β.τ] ]

SLIDE 43

Validating β- and η-Rules

Our model is sensible by construction
Reynolds’ model is an instance of ours, assuming a constructive metathe-
ry — e.g., the Calculus of Constructions with impredicative Set
Proposition If ∆ ⊢ τ1 and ∆, α; Γ ⊢ t : τ2
1. [

[∆; Γ ⊢ (Λα.t)τ1 : τ2[α → τ1]] ] = [ [∆; Γ ⊢ t[α → τ1] : τ2[α → τ1]] ]

2. [

[∆; Γ ⊢ t : ∀β.τ] ] = [ [∆; Γ ⊢ Λα.t α : ∀β.τ] ]

Proposition If ∆; Γ ⊢ t1 : τ1 and ∆; Γ, x : τ1 ⊢ t2 : τ2
1. [

[∆; Γ ⊢ (λx.t2)t1 : τ2] ] = [ [∆; Γ ⊢ t2[x → t1] : τ2] ]

2. [

[∆; Γ ⊢ t : τ1 → τ2] ] = [ [∆; Γ ⊢ λx.tx : τ1 → τ2] ]

SLIDE 44

Reynolds’ Abstraction Theorem, Generalized

Our model actually gives rise to a λ2-fibration

SLIDE 45

Reynolds’ Abstraction Theorem, Generalized

Our model actually gives rise to a λ2-fibration
Seely showed that λ2-fibrations always soundly interpret System F

SLIDE 46

Reynolds’ Abstraction Theorem, Generalized

Our model actually gives rise to a λ2-fibration
Seely showed that λ2-fibrations always soundly interpret System F
Theorem If Rel(U) is an equality preserving arrow fibration and a

∀-fibration, then there is a λ2-fibration in which types ∆ ⊢ τ are interpreted as equality preserving fibred functors [ [∆ ⊢ τ] ] : |Rel(U)||∆| →Eq Rel(U) and terms ∆; Γ ⊢ t : τ are interpreted as fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ]

SLIDE 47

Reynolds’ Abstraction Theorem, Generalized

Our model actually gives rise to a λ2-fibration
Seely showed that λ2-fibrations always soundly interpret System F
Theorem If Rel(U) is an equality preserving arrow fibration and a

∀-fibration, then there is a λ2-fibration in which types ∆ ⊢ τ are interpreted as equality preserving fibred functors [ [∆ ⊢ τ] ] : |Rel(U)||∆| →Eq Rel(U) and terms ∆; Γ ⊢ t : τ are interpreted as fibred natural transformations [ [∆; Γ ⊢ t : τ] ] : [ [∆ ⊢ Γ] ] → [ [∆ ⊢ τ] ] |Rel(E)||∆|

[ [Γ] ]r

[

[τ] ]r

[

[t] ]r |Rel(U)||∆|

Rel(E)

U

|B||∆| × |B||∆|

[ [Γ] ]o×[ [Γ] ]o

[

[τ] ]o×[ [τ] ]o

[

[t] ]o×[ [t] ]o B × B

SLIDE 48

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

SLIDE 49

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

1. An object interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]o : |B||∆| → B

SLIDE 50

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

1. An object interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]o : |B||∆| → B

2. A relational interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]r : |Rel(E)||∆| → Rel(E)

SLIDE 51

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

1. An object interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]o : |B||∆| → B

2. A relational interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]r : |Rel(E)||∆| → Rel(E)

3. A proof of the Identity Extension Lemma as in the previous lecture,

i.e., a proof that [ [∆ ⊢ τ] ] is equality preserving

SLIDE 52

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

1. An object interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]o : |B||∆| → B

2. A relational interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]r : |Rel(E)||∆| → Rel(E)

3. A proof of the Identity Extension Lemma as in the previous lecture,

i.e., a proof that [ [∆ ⊢ τ] ] is equality preserving

4. An object interpretation of ∆; Γ ⊢ t : τ as a natural transformation

[ [∆; Γ ⊢ t : τ] ]o : [ [∆ ⊢ Γ] ]o → [ [∆ ⊢ τ] ]o

SLIDE 53

Unwinding the Theorem

In particular, for every fibration U : E → B whose relations fibration is an equality preserving arrow fibration and a forall fibration, for every System F type ∆ ⊢ τ and term ∆; Γ ⊢ t : τ, we get:

1. An object interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]o : |B||∆| → B

2. A relational interpretation of ∆ ⊢ τ as a functor [

[∆ ⊢ τ] ]r : |Rel(E)||∆| → Rel(E)

3. A proof of the Identity Extension Lemma as in the previous lecture,

i.e., a proof that [ [∆ ⊢ τ] ] is equality preserving

4. An object interpretation of ∆; Γ ⊢ t : τ as a natural transformation

[ [∆; Γ ⊢ t : τ] ]o : [ [∆ ⊢ Γ] ]o → [ [∆ ⊢ τ] ]o

5. A proof of the Abstraction Theorem as in the previous lecture, i.e.,

a proof that ∆; Γ ⊢ t : τ has a relational interpretation as a natural transformation [ [∆; Γ ⊢ t : τ] ]r : [ [∆ ⊢ Γ] ]r → [ [∆ ⊢ τ] ]r over [ [∆; Γ ⊢ t : τ] ]o × [ [∆; Γ ⊢ t : τ] ]o.

SLIDE 54