Homotopy Type-Theoretic Interpretations of CZF

Transcription

1 Homotopy Type-Theoretic Interpretations of CZF Cesare Gallozzi School of Mathematics University of Leeds HoTT/UF Workshop Oxford, September 9th, 2017

2 Outline 1. Type-theoretic interpretations of CZF 2. New interpretations,2 and 1,2 3. Proof-theoretic characterisation of the formulas valid in,2

3 Part I: Type-theoretic interpretations of CZF

4 Constructive Set Theory (CZF) The underlying logic is intuitionistic rather than classical. Axioms: Extensionality Pairing Union Emptyset Infinity -Induction Bounded Separation a x y(y x y a φ(y)) for φ bounded Strong Collection x a yφ(x, y) b( x a)( y b)φ(x, y) ( y b)( x a)φ(x, y) Subset Collection

5 Aczel s Interpretation 1. Interpretation of sets: V := (W A : U)A The introduction rule is: A : U f : A V sup(a, f ) : V 2. Interpretation of equality. Given A, B : U and f : A V and g : B V we define sup(a, f ). = sup(b, g) as: (Πx : A)(Σy : B) f (x). = g(y) (Πy : B)(Σx : A) f (x). = g(y)

6 Aczel s Interpretation 3. The interpretation of the other formulas follows the propositions-as-types correspondence: φ ψ := φ ψ φ ψ := φ ψ φ ψ := φ + ψ ( x sup(a, f ))φ(x) := (Πx : A) φ(f (x)) ( x sup(a, f ))φ(x) := (Σx : A) φ(f (x)) x φ(x) := (Πα : V) φ(α) x φ(x) := (Σα : V) φ(α) α β := ( y β)(y =. α)

7 Other Interpretations HoTT book interpretation, the type V is a higher inductive type, Id V interprets equality and is defined as a bisimulation relation V is a hset, however V itself is constructed with arbitrary small types formulas are interpreted as hpropositions Remark: Univalence is invoked in the proof of a lemma that gives a description the terms of V. An inspection of the proofs shows that univalence is not needed overall.

8 Other Interpretations Gylterud s PhD thesis, interpretation of a multiset theory in type theory, sets are special multisets multisets are interpreted as small types formulas are interpreted as hprops the identity type interprets equality

9 Part II: New interpretations

10 New Interpretation,2 : sets-as-hsets 1. Idea: sets are interpreted as hsets. Define hset U := (ΣA : U)ishSet(A). Then: V 2 := (W x : hset U)π 1(x) We work in the type theory that uses just the rules for V 2, namely: ML i 1V 2 + FE. 2. Interpretation of equality. As in Aczel s interpretation define sup(a, f ). = sup(b, g),2 as: (Πx : A)(Σy : B) f (x). = g(y),2 (Πy : B)(Σx : A) f (x). = g(y),2

11 New Interpretation,2 : sets-as-hsets 3. Interpretation of the other formulas, as in Aczel s interpretation: φ ψ,2 := φ,2 ψ,2 φ ψ,2 := φ,2 ψ,2 φ ψ,2 := φ,2 + ψ,2 ( x sup(a, f ))φ(x),2 := (Πx : A) φ(f (x)),2 ( x sup(a, f ))φ(x),2 := (Σx : A) φ(f (x)),2 x φ(x),2 := (Πα : V 2) φ(α),2 x φ(x),2 := (Σα : V 2) φ(α),2 α β,2 := ( y β)(y. = α),2 Theorem All axioms of CZF are valid in the interpretation,2.

12 New Interpretation 1,2 : sets-as-hsets, formulas-as-hprops 1. Interpretation of sets as hsets: V 2 := (W x : hset U)π 1(x) 2. Interpretation of equality, we define sup(a, f ). = sup(b, g) 1,2 as: (Πx : A) (Σy : B) f (x). = g(y) 1,2 (Πy : B) (Σx : A) f (x). = g(y) 1,2 3. Interpretation of the other formulas: φ ψ 1,2 := φ 1,2 + ψ 1,2 ( x sup(a, f ))φ(x) 1,2 := (Σx : A) φ(f (x)) 1,2 x φ(x) 1,2 := (Σα : V 2) φ(α) 1,2 Theorem The following axioms are valid in the interpretation 1,2: Extensionality, Pairing, Union, Emptyset, Infinity, -Induction, Bounded Separation.

13 Part III: Proof-theoretic characterisation of,k

14 Proof-theoretic Aspects of,k Definition A CC formula is one in which no unbounded quantifier occurs in the antecedent of an implication. CC formulas include all formulas used in standard practice. Definition A set is a base iff it satisfies the axiom of choice. A set is ΠΣ-generated iff it is generated from finite sets and ω using the operations Π the set of dependent functions and Σ the disjoint union. ΠΣ-AC: every ΠΣ-generated set is a base. Theorem (Rathjen & Tupailo) Let φ be a CC sentence. Then CZF + ΠΣ-AC φ if and only if ML e 1V t : φ for some term t.

15 Proof-theoretic Aspects of,k Lemma ΠΣ-AC is valid in,k. Theorem Given a CC sentence φ, if ML i 1V k + FE s : φ,k for some term s, then CZF + ΠΣ-AC φ. Proof. It follows from the existence of an interpretation of ML i 1V k + FE into ML e 1V such that,k is mapped to.

16 Proof-theoretic Aspects of,k Corollary Given a CC sentence φ of CZF and two numbers h, k N {+ } with 2 h, k. We have: ML i 1V k + FE t : φ,k ML i 1V h + FE s : φ,h for some term t for some term s

17 Future Work We have different interpretations of CZF in type theory. They interpret sets, formulas and equality in different ways. However, they share a core structure. Idea Provide an analysis of these type-theoretic interpretations using a logic enriched type theory.

18 References P. Aczel, The Type Theoretic Interpretation of Constructive Set Theory, in Logic Colloquium 77, North Holland, N. Gambino, P. Aczel, The Generalised Type-Theoretic Interpretation of Constructive Set Theory, in Journal of Symbolic Logic, H. R. Gylterud, From Multisets to Sets in Homotopy Type Theory, preprint available at M. Rathjen, S. Tupailo, Characterizing the Interpretation of Set Theory in Martin-Löf Type Theory. Annals of Pure and Applied Logic 141, no. 3, The Univalent Foundations Program, Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, 2013.