Normalization in Intuitionistic Set Theoriesbuchholtz/... · Normalization in Intuitionistic Set Theories UlrikBuchholtz Stanford April13,2010 Ulrik Buchholtz (Stanford) Normalization

Normalization in Intuitionistic Set Theories

Ulrik Buchholtz

Stanford

April 13, 2010

Ulrik Buchholtz (Stanford) Normalization of IZF April 13, 2010 1 / 37

Outline

Outline

1 Introduction

2 Propositional logic

3 Set theory


Introduction

Introduction

We summarize the work by Wojciech Moczydłowski, primarily from his2007 PhD dissertation, Investigation on Sets and Types, supervised byRobert Constable and Richard Shore at Cornell.

The thesis was awarded the 2007 Sacks Prize by the ASL.

Any mistakes in the following are almost surely mine.


Propositional logic IPC

IPC

We warm up by studying intuitionistic propositional logic. Formulas:

ϕ ::= p | ⊥ | ϕ∧ϕ | ϕ∨ϕ | ϕ→ ϕ

Rules:

Γ ,ϕ ` ϕΓ ` ⊥Γ ` ϕ

Γ ,ϕ ` ψΓ ` ϕ→ ψ

Γ ` ϕ→ ψ Γ ` ϕΓ ` ψ

Γ ` ϕ Γ ` ψΓ ` ϕ∧ψ

Γ ` ϕ∧ψ

Γ ` ϕΓ ` ϕ∧ψ

Γ ` ψΓ ` ϕ

Γ ` ϕ∨ψΓ ` ψ

Γ ` ϕ∨ψ

Γ ` ϕ∨ψ Γ ,ϕ ` θ Γ ,ψ ` θΓ ` θ


Propositional logic The calculus

The λ→ calculus

Simply-typed lambda calculus, λ→. The types are IPC formulas. The (raw)terms are given by

M ::= x |MN | λx:ϕ,M | inl(M) | inr(M)

| caseM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2

)| 〈M,N〉 | fst(M) | snd(M) | magic(M)

These correspond exactly to the inference rules for IPC, and give notationsfor IPC proofs.


Propositional logic Typing

Typing for λ→

Typing rules for terms of λ→ match the rules of IPC:

Γ , x:ϕ ` x:ϕΓ `M:⊥

Γ ` magic(M):ϕ

Γ , x:ϕ `M:ψx 6∈ dom(Γ)

Γ ` (λx:ϕ,M):ϕ→ ψ

Γ `M:ϕ→ ψ Γ ` N:ϕ

Γ ` (MN):ψ

Γ `M:ϕ Γ ` N:ψ

Γ ` 〈M,N〉:ϕ∧ψ

Γ `M:ϕ∧ψ

Γ ` fst(M):ϕ

Γ `M:ϕ∧ψ

Γ ` snd(M):ψ


Propositional logic Typing

Typing for λ→, II

Typing rules for disjunction:

Γ `M:ϕΓ ` inl(M):ϕ∨ψ

Γ `M:ψ

Γ ` inr(M):ϕ∨ψ

Γ `M:ϕ∨ψ Γ , x:ϕ ` N1:θ Γ ,y:ψ ` N2:θ

Γ ` caseM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2

):θ

Note: If Γ `M:ϕ, then FV(M) ⊂ dom(Γ).


Propositional logic Reductions

Reduction for λ→

For our later application to set theory, it essential to use a deterministicreduction. Reduce terms to values:

V ::= λx:ϕ,M | inl(M) | inr(M) | 〈M,N〉

Non-values have a principal argument:In MN, M is the principal argument.

In caseM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2

), M is the principal argument.

In fst(M), snd(M) and magic(M), M is the principal argument.


Propositional logic Reductions

Reduction for λ→, II

A non-value whose principal argument is value may be reduced:

fst〈M,N〉 −→M

snd〈M,N〉 −→ N

(λx:ϕ,M)N −→M[N/x]

case inlM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2

)−→ N1[M/x]

case inrM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2

)−→ N2[M/x]

If the principal argument is a non-value, then that may be reduced (lazyevaluation).


Propositional logic Properties

Properties for λ→

Lemma (Correspondence)

If Γ ` O:ϕ, then rg(Γ) ` ϕ. If IPC+ Γ ` ϕ, then there is a term M of λ→

so that Γ ′ `M:ϕ.

Lemma (Inversion)

We can determine the final typing judgment in a proof by inspecting theproof term.

Lemma (Subject-reduction)

If Γ `M:ϕ and M −→ N, then Γ ` N:ϕ.

Lemma (Progress)

Non-values can always be reduced in an empty context.


Propositional logic Realizability

Realizability for λ→

The terms of the untyped calculus λ→ are obtained from λ→ by erasingthe types:

M ::= x |MN | λx,M | inl(M) | inr(M)

| caseM of(inl x⇒ N1

inr y⇒ N2


We can erase the types of λ→-terms to get λ→-terms.


Propositional logic Realizability

Realizability for λ→, II

We use untyped closed lambda-terms as realizers. This works because thereductions are type-oblivious. We define a realizability relation betweenrealizers and formulas:

M p iff M ↓M ⊥ iff ⊥

M ϕ∧ψ iff M ↓ 〈M1,M2〉∧ (M1 ϕ)∧ (M2 ψ)

M ϕ∨ψ iff (M ↓ inl(M1)∧M1 ϕ)

∨ (M ↓ inr(M2)∧M2 ψ)

M ϕ→ ψ iff (M ↓ λx,M1)∧ ∀N, (N ϕ)→ (M1[N/x] ψ)


Propositional logic Normalization

Normalization for λ→

A realizability environment ρ is partial function from proof variables torealizers. Write ρ � Γ if ρ(x) ψ for (x:ψ) ∈ Γ .

Theorem

If Γ `M:ϕ, then for all ρ � Γ , we have M[ρ] ϕ.

Corollaries:Normalization: If `M:ϕ, then M normalizes.IPC is consistent: There is no M with `M:⊥.The disjunction property for IPC: If ` ϕ∨ψ, then ` ϕ or ` ψ.


Set theory

Set theory

Moczydłowski studies Intuitionistic ZF with Replacement. Terms andformulas are defined by a mutual grammar:

t ::= a | ∅ | {t, t} | ω |⋃t | P(t)

| Sϕ(a,~f)(t,~t) | Rϕ(a,b,~f)(t,~t)

ϕ ::= ⊥ | t ∈ t | t = t | t ∈I t

| ϕ∨ϕ | ϕ∧ϕ | ϕ→ ϕ

| ∀a,ϕ | ∃a,ϕ

Here:

Sϕ(a,~f)(t,~t) ≡ { v ∈ t | ϕ(c,~t) }

Rϕ(a,b,~f)(t,~t) ≡ { c | (∀x ∈ t,∃!y,ϕ(x,y,~t))∧ (∃x ∈ t,ϕ(x, c,~t)) }


Set theory The axioms

Axioms of IZFR

(IN) a ∈ b↔ ∃c, c ∈I b∧ c = a

(EQ) a = b↔ ∀d, (d ∈I a→ d ∈ b)∧ (d ∈I b→ d ∈ a)(EMPTY) c ∈I ∅ ↔ ⊥(PAIR) c ∈I {a,b}↔ c = a∨ c = b

(INF) c ∈I ω↔ c = ∅∨ ∃b ∈ ω, c = S(b)

(SEPϕ(a,~f)) c ∈I Sϕ(a,~f)(a, ~f)↔ c ∈ a∧ϕ(c, ~f)

(UNION) c ∈I⋃a↔ ∃b ∈ a, c ∈ b

(POWER) c ∈I P(a)↔ ∀b ∈ c,b ∈ a

continues . . .


Set theory The axioms

Axioms of IZFR, II

continued:

(REPLϕ(a,b,~f)) c ∈I Rϕ(a,b,~f)(a, ~f)↔

(∀x ∈ a,∃!y,ϕ(x,y, ~f))∧ (∃x ∈ a,ϕ(x, c, ~f))(INDϕ(a,~f)) (∀a, (∀b ∈I a,ϕ(b, ~f))→ ϕ(a, ~f))

→ ∀a,ϕ(a, ~f)


Set theory The calculus

The λZ calculus

We use two sets of variables, proof- and set-variables.Terms:

M ::= x |M t |MN | λa,M | λx:ϕ,M

| inl(M) | inr(M) | caseM of(inl x:ϕ⇒ N1

inr y:ψ⇒ N2


| [t,M] | let [a, x:ϕ] :=M inN

continues . . .



The λZ calculus, II

M ::= . . . | inProp(t,u,M) | inRep(t,u,M)

| eqProp(t,u,M) | eqRep(t,u,M)

| pairProp(t,u1,u2,M) | pairRep(t,u1,u2,M)

| unionProp(t,u,M) | unionRep(t,u,M)

| sepϕ(a,~f)Prop(t,u, ~u,M) | sepϕ(a,~f)Rep(t,u, ~u,M)

| powerProp(t,u,M) | powerRep(t,u,M)

| infProp(t,M) | infRep(t,M)

| replϕ(a,b,~f)Prop(t,u, ~u,M) | replϕ(a,b,~f)Rep(t,u, ~u,M)

| indϕ(a,~f)(M,~t)



The λZ calculus, III

We’ll abbreviate the -Prop and -Rep-axioms as

axProp(t, ~u,M) | axRep(t, ~u,M),

where the length of ~u depends on the axiom.


Set theory Typing

Typing for the λZ calculus

Same rules as for IPC plus first-order rules:

Γ `M:ϕ a 6∈ FVs(Γ)Γ ` (λa,M):∀a,ϕ

Γ `M:∀a,ϕΓ `M t:ϕ[t/a]

Γ `M:ϕ[t/a]

Γ ` [t,M]:∃a,ϕ

Γ `M:∃a,ϕ Γ , x:ϕ ` N:ψa 6∈ FVs(Γ ,ψ)

Γ ` (let [a, x:ϕ] :=M inN):ψ

continues . . .


Set theory Typing

Typing for the λZ calculus, II

Plus rules for the axioms, first equality (EQ):

Γ `M:∀d, (d ∈I t→ d ∈ u)∧ (d ∈I u→ d ∈ t)Γ ` eqRep(t,u,M):t = u

Γ `M:t = uΓ ` eqProp(t,u,M):∀d, (d ∈I t→ d ∈ u)∧ (d ∈I u→ d ∈ t)

continues . . .


Set theory Typing

Typing for the λZ calculus, III

Then, membership (IN):

Γ `M:∃c, c ∈I u∧ t = c

Γ ` inRep(t,u,M):t ∈ u

Γ `M:t ∈ uΓ ` inProp(t,u,M):∃c, c ∈I u∧ t = c

continues . . .


Set theory Typing

Typing for the λZ calculus, IV

The other axioms all follow the same pattern:

Γ `M:ϕA(t, ~u)Γ ` axRep(t, ~u,M):t ∈I tA(~u)

Γ `M:t ∈I tA(~u)

Γ ` axProp(t, ~u,M):ϕA(t, ~u)

continues . . .


Set theory Typing

Typing for the λZ calculus, V

Except ∈I-induction (IND):

Γ `M:∀c, (∀b,b ∈I c→ ϕ(b,~t))→ ϕ(c,~t)

Γ ` indϕ(a,~f)(M,~t):∀a,ϕ(a,~t)


Set theory Reductions

Reduction for the λZ calculus

Same reductions as for IPC plus:

(λa,M) t −→M[t/a]

let [a, x:ϕ] := [t,M] inN −→ N[t/a][M/x]

axProp(t, ~u, axRep(t, ~u,M)) −→M

indϕ(a,~f)(M,~t) −→ λc,M c (λb, λx:(b ∈I c),

indϕ(a,~f)(M,~t)b)

Values:

V ::= λa,M | λx:ϕ,M | inr(M) | inr(M)

| [t,M] | 〈M,N〉 | axRep(t, ~u,M)


Set theory Realizability

Realizability for the λZ calculus

Realizability for IZF was first defined by David McCarty in his 1984PhD-thesis. Moczydłowski builds on this work to prove normalization.To get realizers, we erase types and sets from λZ-terms to get λZ-terms(all sets disappear or become ∅):

M ::= x |M ∅ |MN | λa,M | λx,M

| inl(M) | inr(M) | caseM of(inl x⇒ N1

inr y⇒ N2


| [∅,M] | let [a, x] :=M inN| axProp(M) | axRep(M) | ind(M)



Lambda names

The idea is to have a version of the cumulative hierarchy that for each setincludes realizers as evidence for the placement in the hiearchy.

Definition

A λ-name is a set of pairs (v,B) where v ∈ λZvc and B is a λ-name.The class of λ-names is denoted Vλ.We have

Vλ =⋃α∈Ord

Vλα, Vλα =⋃β<α

P(λZvc × Vλβ),

and for a λ-name A we let λrk(A) denote the smallest ordinal α withA ∈ Vλα.



Prerealizability

Define M A ∈I B, M A ∈ B and M A = B for M ∈ λZc, andA,B ∈ Vλ:

M A ∈I B ≡M ↓ v∧ (v,A) ∈ BM A ∈ B ≡M ↓ inRep(N)∧N ↓ [∅,O]

∧ ∃C ∈ Vλ,O ↓ 〈O1,O2〉∧O1 C ∈I B∧O2 A = C

M A = B ≡M ↓ eqRep(M0)∧M0 ↓ λa,M1

∀D ∈ Vλ,M1[∅/a] ↓ 〈O,P〉∧O ↓ λx,O1 ∧ (∀N D ∈I A,O1[N/x] D ∈ B)∧ P ↓ λx,P1 ∧ (∀N D ∈I A,P1[N/x] D ∈ A)



Enriched language

Definition

For C ∈ Vλ, let C+ ≡ {(M,A) |M A ∈ C}.

Definition

Let L(Vλ) be the first-order language obtained by enriching the signatureof IZFR with constants for all λ-names.

Definition

A realizabilty environment ρ is a partial function from variables in L(Vλ) tothe class of λ-names, Vλ.



Realizability for IZFR

We define by mutual induction, for ϕ a formula of L(Vλ), a term t ofL(Vλ), and for an environment ρ defined on the free variables in ϕ or t, arealizability relation M ρ ϕ (for M ∈ λZc), and a denotation JtKρ ∈ Vλ.

JaKρ ≡ ρ(a)JAKρ ≡ AJωKρ ≡ ω ′ (a suitable λ-name for ω)

JtA(~u)Kρ ≡ {(axRep(N),B) ∈ λZvc × Vλγ | N ρ ϕA(B, ~JuKρ)}(for a suitable ordinal γ depending on the λ-ranks of the parametersto the axiom)



Realizability for IZFR, II

M ρ ⊥ iff ⊥M ρ t ∈I s iff M JtKρ ∈I JsKρM ρ t ∈ s iff M JtKρ ∈ JsKρM ρ t = s iff M JtKρ = JsKρM ρ ϕ∧ψ iff M ↓ 〈M1,M2〉∧ (M1 ρ ϕ)∧ (M2 ρ ψ)

M ρ ϕ∨ψ iff (M ↓ inl(M1)∧M1 ρ ϕ)

∨ (M ↓ inr(M2)∧M2 ρ ψ)

M ρ ϕ→ ψ iff (M ↓ λx,M1)∧ ∀N ρ ϕ,M1[N/x] ρ ψ

M ρ ∃a,ϕ iff M ↓ [∅,N]∧ ∃A ∈ Vλ,N ρ ϕ[A/a]

M ρ ∀a,ϕ iff M ↓ λa,N∧ ∀A ∈ Vλ,N[∅/a] ρ ϕ[A/a]



Properties of this interpretation

Lemma

If A ∈ Vλα, then there is a β < α, so that if M B ∈ A, then B ∈ Vλβ.If M B = A, then B ∈ Vλα.If M B ∈I A, then λrkB < λrkA.

LemmaFor any intensional axiom we have

(M,C) ∈ JtA(~u)Kρ iff M = axRep(N) and N ρ ϕA(C, ~JuKρ)


Set theory Normalization

Normalization of λZ

We write ρ � Γ `M:ϕ if ρ assigns lambda-names to free first-ordervariables and realizers to context proof variables, so that for (x:ψ) ∈ Γ , wehave ρ(x) ρ ψ.

Theorem

If Γ `M:ϕ, then for all ρ � Γ `M:ϕ, we have M[ρ] ρ ϕ.

CorollaryIf `M:ϕ, then M normalizes.


Set theory Applications

Applications

CorollaryIf IZFR ` ϕ∨ψ, then IZFR ` ϕ or IZFR ` ψ.

Corollary

If IZFR ` ∃x,ϕ(x), then there is a term t so that IZFR ` ϕ(t).

Corollary

If IZFR ` ∃x ∈ ω,ϕ(x), then there is number n so that IZFR ` ϕ(n̄).


Set theory Strong normalization

Failure of strong normalization

An obstacle to strong normalization of intuitionistic set theories is Crabbé’sCounterexample:

Let t = { x ∈ ∅ | x ∈ x→ ⊥ }. There is a term M:(t ∈ ∅ → ⊥) that doesnot normalize if we allow reductions under the binder.


Set theory Extensions

Extensions

Moczydłowski’s approach extends to give normalizing calculi for IZFR withcountably many inaccessibles. He also gives a dependent set theory thatproves collection (which is stronger than replacement, intuitionistically).


Set theory Extensions

References

Wojciech Moczydłowski.Investigations on Sets and Types.PhD thesis, Cornell University, August 2007.Awarded the 2007 Sacks Prize.

Wojciech Moczydłowski.Normalization of IZF with replacement.Log. Methods Comput. Sci., 4(2):2:1, 29, 2008.


Documents

Normalization in Intuitionistic Set Theoriesbuchholtz/... · Normalization in Intuitionistic Set Theories UlrikBuchholtz Stanford April13,2010 Ulrik Buchholtz (Stanford) Normalization