Typed Lambda Calculus

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Delia KESNER IRIF, CNRS et Universit ´e Paris [email protected] www.irif.fr/˜kesner

Typed Lambda Calculus

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Motivations

Partial specification of programs Avoid meaningless programs (1+true) Avoid memory violation

Avoid programs with undefined semantics

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Simply Typed Lambda Calculus

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Curry-Howard Isomorphism

Logical system ⇔ Language

Propositions ⇔ Types

Proofs ⇔ _Programs

Proof normalisation ⇔ Program Evaluation

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Curry’58, Howard’68

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Adding (simply) types to λ -calculus

Grammar for types:

A,B::= b (base types) |

A→B (functional types)

Example :

int→bool bool→(bool→int) (bool→bool)→int

Remark

→is right-associative,e.g.A1→A2→A3abbreviatesA1→(A2→A3). Every typeAcan be written asA1→. . .→An→b, whereA1, . . . ,An(n≥0)are arbitrary types andbis a base type.

The standard order between types is given by A<A→BandB<A→B.

Thus base types are minimal with respect this order.

In a typed framework substitutions are always well-typed, i.e,t{x\u}means thatx anduhave the same type.

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Typing Environment

Atyping environmentΓis afinitefunction from variables to types, usually written x1:A1, . . . ,xn:An.

Thus for example,x:A,y:Bandy:B,x:Aare two different notations for the same typing environment.

ThedomainofΓ =x1:A1, . . . ,xn:An, writtendom(Γ), is the set{x1, . . . ,xn}. We writeΓ,x:Afor the typing environment extendingΓwith the pairx:A. It is only defined iffx<dom(Γ).

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Typed Lambda Calculus in Church-Style

Γ,x:A`x:A (ax) Γ,x:A`t:B

Γ`λx:A.t:A→B (→i) Γ`t:A→B Γ`u:A Γ`t u:B (→e)

We can also add constants: each constantchas an associated typeTC(c) :A.

Γ`c:TC(c) We can also add the let constructor:

Γ`t:A Γ,x:A`u:B Γ`letx:A=tinu:B

We denote byΓ`_Ct:Athe derivability/typing relation. We say thattistypablein Church-Style iff there isΓ^andAsuch thatΓ`_Ct:A.

Remark:Γ`t:Adenotes a sequent not a derivation!

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Examples

Example of typable term in an empty environment:

y:A→A`y:A→A

`λy:A→A.y: (A→A)→(A→A)

x:A`x:A

`λx:A.x:A→A

`(λy:A→A.y)(λx:A.x) :A→A Example of typable term in a non-empty environment:

ci:int→int`ci:int→int

ci:int→int`ci:int→int ci:int→int`3 :int ciint→int`ci3 :int

ci:int→int`ci(ci3) :int Example of non-typable term:λx.xx.

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Typed Properties

[Unicity]: IfΓ`_Ct:AandΓ`_Ct:B, thenA≡B.

Proof.

By induction ont.

[Weakening and Strengthening]: LetΓ ={x:B|x∈fv(t)}andΓ⊆∆1⊆∆2. Then∆1`_Ct:Aiff∆2`_Ct:A.

Proof.

By induction ont.

[Subject Reduction] IfΓ`_Ct:Aandt→β t⁰, thenΓ`_Ct⁰:A.

Proof.

By induction onΓ`_Ct:A(blackboard).

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Typing Algorithm

Type(Γ,c) =TC(c)

Type(Γ,x) =A ifx:A∈Γ

Type(Γ, λx:A.t) =A→B ifType((Γ,x:A),t)=B

Type(Γ,t u) =B ifType(Γ,t)=A→BandType(Γ,u)=A Type(Γ,letx:A=tinu) =B ifType(Γ,t)=AandType((Γ,x:A),u)=B Type(Γ,t) =error otherwise

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Properties of the Typing Algorithm

[Termination]

For every termtand every environmentΓ, the callType(Γ,t)terminates.

[Soundness]

IfType(Γ,t)=A, thenΓ`_Ct:A. [Completeness]

IfΓ`_Ct:A, thenType(Γ,t)=A. Said differently:

IfType(Γ,t)=erreur, thentis not typable inΓ.

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Typed Lambda Calculus in Curry-Style

Γ,x:A`x:A (ax) Γ,x:A`t:B

Γ`λx.t:A→B (→i) Γ`t:A→B Γ`u:A Γ`t u:B (→e)

We can also add constants: each constantchas an associated typeTC(c) :A.

Γ`c:TC(c) We can also add lets:

Γ`t:A Γ,x:A`u:B Γ`letx=tinu:B

We denote byΓ`_λt:Athe derivability/typing relation. We say thattistypablein Curry-Style iff there isΓ^andAsuch thatΓ`_λt:A.

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Properties

Unicitydoes not hold anymore:

`_λλx.x:int→int `_λλx.x:bool→bool The identity function behaves in the same way forintandbool:

Polymorphism

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Difficulties for a Typing Algorithm

Type(Γ, λx.t)=A→B i f there existsAs.t.Type((Γ,x:A),T)=B Type(Γ,letx=tinu)=B i f there existsAs.t.Type(Γ,t)=Aand

Type((Γ,x:A),u)=B

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Monomorphic Type Inference

Thetype inference problemfor a termt:∃Γ∃Asuch thatΓ`_λt:A?

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Towards a Monomorphic Type Inference Algorithm

General Technical Tools:

We consider a table oftypes for constants, calledT C. E.g.T C(3)=intand T C(ci)=int→int.

Lettbe a term. For each sub-termuoftwe introduce a type variableαu. We associate to every termta set of equationsS E(t)={α1u1, . . . , αmum}. The solution to the type inference problem for the termtwill be given by themost general unifier(mgu) of the corresponding set of equations.

t S E(t)

x {αtαx}

c {αtT C(c)}

u v {αuαv→αt} ∪S E(u)∪S E(v) λx.u {αtαx→αu} ∪S E(u)

letx=uinv {αtαv;αxαu} ∪S E(u)∪S E(v)

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Examples

t0:=λf.λg.f g

S E(t0)={αt₀αf →αλg.f g, αλg.f gαg→αf g, αf αg→αf g, αf αf, αgαg} The mgu ofS E(t0)is the substitutionσt₀such thatσt₀(αt₀)=(αg→αf g)→(αg→αf g).

t0is typable withinstances ofσ_t₀(α_t₀)which are types of the form(A→B)→(A→B), for any arbitrary typesAandB.

t1:=let f=λx.xin f(f z)

S E(t1)={αt₁αf(f z), αf αλx.x, αλx.xαx→αx, αxαx, αf αf z→αf(f z), αf αf, αf αz→αf z, αzαz}

The mgu ofS E(t1)is the substitutionσt₁such thatσt₁(αt₁)=αf(f z). t1is typable withinstances ofσt₁(αt₁)which are arbitrary typesA.

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Properties of the Type Inference Algorithm

Theorem

(Soundness)Ifσis a solution ofS E(t), thentis (Curry) typable. Moreover,

∆`_λt:σ⁰(αt), where∆ ={x:σ⁰(αx)|x∈fv(t)}andσ⁰is an instance ofσ.

Theorem

(Completeness)Iftis (Curry) typable, thenS E(t)admits a solution.

Theorem

(Principality)Iftis (Curry) typable, i.e.∆`_λt:A, thenAis an instance of the principal type (i.e.A=σ⁰(σ(αt))), whereσis the mgu of systemS E(t)andσ⁰is a substitution.

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Polymorphic Lambda Calculus

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Motivations

General behaviour of an operator, i.e. without looking at the particular nature (or type) of its parameters.

More clear and concise programs.

Reutilisation of operators.

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Different Forms of Polymorphism

Ad-hoc polymorphisme (overloaded constructors), originally described by Strachey.

Subtyping polymorphism, introduced by Wegner and Cardelli: SiΓ`t:A→B, Γ`u:A⁰etA⁰≤Aalorstu:B.

Parametric polymorphism, introduced by Reynolds and Girard.

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Parametric Polymorphism

Motivation:

Many indentity functions:

Idint :int→int,Idbool:bool→bool,Idint→int: (int→int)→(int→int), . . . Many functions to add an element to a list:

A jint:int→list(int)→list(int),A jbool:bool→list(bool)→list(bool), . . . Idea:

Id:∀α.α→α and thus:

Idint =Id[int]

Idbool=Id[bool]

Idint→int=Id[int→int]

RemarkThe key idea is theinstantiationrelation between the general typeα, and the particular used typesint,bool,int→int.

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Girard-Reynolds Polymorphism

Types:A::=b|α|A→A|∀α.A Notation :∀α.∀β.A=∀(α, β).A. Expressions:

t::= x|c|t t|

λx:A.t|letx:A=tint| t[A]|Λαt

Reduction Rules:

(λx:A.t)u → t{x\u}

letx:A=uint → t{x\u}

(Λαt)[A] → t{α\A} Reduction Rules:

LetId= Λαλx:α.x

Id[int] → λx:int.x (Id[int]) 3 → (λx:int.x) 3→ 3

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Type Free Variables

By induction on types:

tfv(b) = ∅

tfv(α) = {α}

tfv(∀α.A) = tfv(A)\ {α}

tfv(A→B) = tfv(A)∪tfv(B) A typeAisclosedifftfv(A)=∅.

Example :tfv(β→(∀α.α→γ))={β, γ}and∀α.∀β.(α→β)is closed.

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Polymorphic Lambda Calculus in Church-Style (Girard)

The same rules used for monomorphic Church-style plus:

Γ`t:A α<tfv(Γ) Γ`Λαt:∀α.A

Γ`t:∀α.A Γ`t[B] :A{α\B}

wheretfv(Γ) :=S

x∈Γtfv(Γ(x))is the set oftype free variablesofΓ^. We denote byΓ`_Gt:Athe derivability/typing relation.

Example :

x:α`x:α

`λx:α.x:α→α

`Λαλx:α.x:∀α.(α→α)

`(Λαλx:α.x)[int] :int→int `3 :int

`((Λαλx:α.x)[int]) 3 :int

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Polymorphic Type Inference

Theorem

(Schubert)The inference problem in Girard system isundecidable.

Idea to become decidable:to restrict the grammar of types.

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ML Polymorphism

Type matrix:M::=b|α|M→M ML Type:S::=∀α1. . . .∀αn.M

A type matrix is a particular case of type.

Type for constants:

T C(+) : int→int→int T C(fst) : ∀α∀β.(α→β)→α T C(snd) : ∀α∀β.(α→β)→β T C(ifthenelse) : ∀α.(bool→α→α)→α T C(fix) : ∀α.(α→α)→α

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Type Instance

Definition

An ML typeAis an instance of an ML type∀α1. . . .∀αn.B, writtenA≤ ∀α1. . . .∀αn.Biff there existC1, . . . ,Cns.t.A=B{α1, . . . , αn\C1, . . . ,Cn}.

In particular, forn=0, we haveA≤BiffA≡B. Example :

int→int ≤ ∀α.α→α

bool→int ≤ ∀α.∀β.α→β

bool→int ∀α.α→α

∀β.int→β ≤ ∀α.∀β.α→β

∀β.(β→β)→ ∀β.(β→β) ∀α.α→α

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Type Generalization

Definition

TheGenoperator is given byGen(A,Γ)=∀α1. . . .∀αn.A, where each variableαiis free in Abut not inΓ, ı.e. foralli=1. . .n, we haveαi∈tfv(A)\tfv(Γ).

Example :LetA=α→βandΓ=x:β,y:∀α.α. ThenGen(A,Γ)=∀α.α→β.

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Polymorphic Lambda Calculus in Curry-Style

A≤Γ(x) Γ`x:A

A≤T C(c) Γ`c:A Γ`t:A→B Γ`u:A

Γ`t u:B

Γ,x:A`t:B Γ`λx.t:A→B Γ`u:A Γ,x:Gen(A,Γ)`t:B

Γ`letx=uint:B

We denote byΓ`_MLt:Athe derivability/typing relation.

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Example

LetA=∀α.α→α

y:α`y:α

`λy.y:α→α

int→int≤A

f:A` f:int→int f :A`1 :int f :∀α.α→α` f1 :int

`let f=λy.yin f1 :int

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Properties

[Substitution]:

IfΓ,x:∀α1. . . .∀αn.B`_MLt:AandΓ`_MLu:B, thenΓ`_MLt{x\u}:A.

[Subject Reduction]:

IfΓ`_MLt:Aandt→ t⁰, thenΓ`_MLt⁰:A.

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Polymorphic Type Inference

Lett:=letx=λy.yinx. Let consider the set of equations:

{αtαx, αxGen(αλy.y,∅), αλy.yαy→αy}

If we treat the second equation we obtainαx≡ ∀αλy.y.αλy.y, which isincorrect sinceαx

should be a functional type (an arrow).

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Towards a Type Inference Algorithm: Notations

LetΓ =x1:A1, . . . ,xn:Anbe an environment and letσbe a type substitution. then σ(Γ)=x1:σ(A1), . . . ,xn:σ(An).

We noteinst(∀α1. . . .∀αn.A)the typeA{α1, . . . , αn\β1, . . . , βn}, whereβ1, . . . , βnarefresh variables.

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Damas-Milner-Tofte Algorithm

Input: an environmentΓ^{and a term}ts.t.fv(t)⊆Γ^.

Output: an ML typeAand a type substitutionσs.t.σ(Γ)`_MLt:A(idis the empty type substitution).

W((∆,x:A),x) = (inst(A),id)

W(∆,c) = (inst(T C(c)),id)

W(∆,λx.u) = (ρB(αx)→B, ρB)

whereW((∆,x:αx),u)=(B, ρB)andαxis a fresh variable

W(∆,u v) = (µ(α), µ◦ρC◦ρB)

whereW(∆,u)=(B, ρB),W(ρB(∆),v)=(C, ρC), αis a fresh variable andµ=mgu{ρC(B)C→α} W(∆,letx=uinv) = (C, ρC◦ρB)

whereW(∆,u)=(B, ρB)andW((ρB(∆),x:Gen(B, ρB(∆))),v)=(C, ρC)

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Premier exemple

W(∅, λx.∗4x)=(int→int,{α/int→int, β/int, αx/int})

W(x:αx,∗4x)=(int,{α/int→int, β/int, αx/int}), computesmgu{int→intαx→β} W(x:αx,∗4)=(int→int,{α/int→int}), computesmgu{int→int→intint→α}

W(x:αx,∗)=(int→int→int,id) W(x:αx,4)=(int,id)

W(x:αx,x)=(αx,id)

Then∅ `_MLλx.∗4x:int→int.

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Second Example

W(∅,let f =λx.xinf2)=(int,{β/int, γ/int}) W(∅, λx.x)=(αx→αx,id)

W(x:αx,x)=(αx,id)

W(f:∀αx.αx→αx,f2)=(int,{β/int, γ/int}), wheremgu{β→βint→γ}={β/int, γ/int}

W(f:∀αx.αx→αx,f)=(β→β,id) W(f:∀αx.αx→αx,2)=(int,id)

Then∅ `_MLlet f=λx.xin f2 :int.

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Properties of the Type Inference Algorithm

Theorem

(Soundness)IfW(∆,t)=(A, σ), thenσ(∆)`_MLt:A.

Theorem

(Completeness)Let∆ =x1:α!, . . . ,xn:αnwherefv(t)⊆ {x1, . . . ,xn}. Letτbe a type substitution, Ifτ(∆)`_MLt:B, thenW(∆,t)=(A, σ), whereBis an instance ofAandτis an instance ofσ.

Corollary :∅ `_MLt:AiffW(∅,t)=(B, σ)andAis an instance ofB.

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Strong Normalization of Simply Typed

Lambda Calculus

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Typed Properties

[Strong Normalization] Everysimply typedterm is normalising:

ifΓ`_λt:A, thent∈S Nβ.

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Defining Strongly Normalizing Terms

t∈S Nβ

iff there is no infiniteβ-reduction sequence starting att. t∈S Nβ

iff everyβ-reduction sequence starting attis finite.

Firstinductive alternative:

Iftis aβ-normal form, thent∈S N

If∀t⁰[(t→β t⁰)impliest⁰∈S N], thent∈S N (the first line is a special case of the second one) Secondinductive alternative:

t₁, . . . ,t_n∈S Nimpliesx~t=x t₁ . . .t_n∈S N.

t∈S Nimpliesλx.t∈S N.

t{x\u}~r∈S Nandu∈S Nimplies(λx.t)u~r∈S N.

In both cases one shows that t∈S Nifft∈S Nβ .

Definition (Measuring

S Nβ

-terms)

Givent∈S Nβ, we define themeasureµβ(t)asmax{n∈IN| t→ⁿ_βt⁰}.

Note thatt→β t⁰impliesµβ(t⁰)< µβ(t), so thatt∈S Nβandt→β t⁰impliest⁰∈S Nβ.

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Some General Remarks About S N

_β

-Terms

u∈S Nβiffλy.u∈S Nβ.

u1, . . . ,un∈S Nβiffx u1. . .un∈S Nβ.

In general, ift∈S Nβ, then every subterm oftis alsoS Nβ. but the converse is not true, e.g.(λx..xx)(λx..xx).

This is becauseS Nβis not stable by substitution. Example:x x∈S Nβ,λy.y y∈S Nβ, but(x x){x\λy.y y}= ∆ ∆<S Nβ.

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First Proof of the SN property

This first proof is due to Tait.

Uses thefirstalternative definition ofS Nβ

It is based on a predicateS Cto characterizestrong computableterms.

Definition

Lettbe of typeA=A1→. . .→An→τ. Thent∈S Ciff forallui∈S Cof typeAiwe havet~u=t u1. . .un∈S Nβ.

The previous definition implies

1 S C⊆S Nβ.

2 S Cis closed underβ(i.e.t∈S Candt→_β t⁰impliest⁰∈S C).

3 x∈S Cfor every variablex(using 1).

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Lemma

Ift,u1, . . . ,un(n≥1)∈S Nβandt{x\u1}u2. . .un∈S Nβ, then(λx.t)u1u2. . .un∈S Nβ.

Proof.

By the first alternative definition ofS Nβ, it is sufficient to show thatallthe reducts of (λx.t)u1. . .unare inS Nβ. We reason byinductiononµ(t)+ Σiµ(ui). Case analysis on the reducts:

(λx.t⁰)u1. . .un, wheret→ t⁰. Thenµ(t⁰)< µ(t), we conclude by thei.h.

(λx.t)u1. . .u⁰_i. . .un, whereui→ u⁰_i. Thenµ(u⁰_i)< µ(ui), we conclude by thei.h.

t{x\u1}u2. . .un. We conclude by the hypothesis.

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Lemma

Lettbe a typed term. Letσbe a type preserving substitution mapping all the free variables oftto terms inSC. Thentσ∈S C.

Proof.

We proceed byinductionon the typed termt. Ift=x, thenxσ=σ(x)∈S Cby hypothesis.

Ift=uv, then let~r∈S C. We haveuσandvσinS Cbyi.h.Then (uv)σ~r=uσvσ~r∈S Nβby definition.

Ift=λx.u, then(λx.u)σ=αλx.uσ. Sinceσ∪ {x\x}verifies the hypothesis of the lemma, then by thei.h.u(σ∪ {x\x})=uσ∈S C⊆S Nβ. To showλx.uσ∈S Cwe show(λx.uσ)~r∈S Nβfor~r∈S C⊆S Nβ. This follows from the previous lemma.

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Lemma

Every typed term is inS C.

Proof.

Using the previous lemma with the identity substitution (which verifies the hypothesis of

the current lemma).

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Theorem

Every typed term is inS Nβ.

Proof.

Using the previous lemma and the fact theS C⊆S Nβ.

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Second proof of the SN property

Can be found in Femke van Raamsdonk’s Thesis.

Uses thesecondalternative definition ofS Nβ 1 DefineΛA(terms of typeA) inductively:

Ifxis a variable of typeA, thenx∈ΛA.

Ift∈ΛCandxis a variable of typeB, thenλx.t∈ΛB→C. Ift∈ΛB→Aandu∈ΛB, thent u∈ΛA.

2 DefineS NA=S N∩ΛA.

3 DefineX→Y={t| ∀u.(u∈Ximpliestu∈Y)}.

4 ShowΛA→B= ΛA→ΛB.

5 ShowS NA→S NB⊆S NA→B(easy).

6 Ifu∈S NA₁→S NA₂→. . .→S NAmwithAma base type andt∈S NB, then t{x\u} ∈S NB(induction on SN using 5).

7 ShowS NA→B⊆S NA→S NB(using 6).

8 Show thatΛA⊆S NA(by induction using 7).

9 SinceS NA⊆S N=S Nβwe conclude.

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Third Proof of the SN property

Lemma

Iftanduare typed and belong toS Nβ, thent{x\u} ∈S Nβ.

Proof.

Byinductiononhtype(u), µβ(t),size(t)i. The base casehbase type, 0, 1iis trivial.

Caset=λy.vis by thei.h.onv(size( )strictly decreases).

Caset=y~cwithx,yis by thei.h.onci(µβ( )decreases andsize( )strictly decreases.).

Caset=x. We havex{x\u}=u∈S Nβby hypothesis.

Caset=x b~c. By thei.h.B=b{x\u}andCi=ci{x\u}are inS Nβ. We want to show thatu BC~∈S Nβ. It is sufficient (first alternative definition ofS Nβ) to show that all its reducts are inS Nβ. We reason byinductiononµβ(u)+µβ(B)+ Σiµβ(Ci). The reducts are

u⁰BC, where~ u→ u⁰. Apply thei.h.

u B⁰C, where~ B→ B⁰. Apply thei.h.

u⁰B C1. . .C_i⁰. . .Cn, whereCi→ C⁰_i. Apply thei.h.

v{y\B}C, where~ u=λy.v. Butv{y\B}C~=(zC){z\v{y\B}}~ andtype(v{y\B})<type(u). We thus conclude by thei.h.sincezC~andv{y\B}are typed and inS N_βby thei.h.

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Caset=(λz.b)cd~. By thei.h.B=b{x\u}_andC=c{x\u}_andDi=di{x\u}_{are in} S Nβ. Supposet{x\u}=(λz.B)CD~<S Nβ. ThenB{z\C}D~<S Nβ. But

B{z\C}D~=(b{z\c}d){x\u}~ andµβ(b{z\c}d)~ < µβ(t). ThusB{z\C}D~∈S Nβby thei.h.

Contradiction. Thust{x\u}=(λz.B)CD~∈S Nβ.

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Theorem

Iftis typable, thent∈S Nβ.

Proof.

By induction on the typing derivation oft. Caset=xis trivial.

Caset=λy.uholds by the i.h.

For the caset=u vuse the fact thatt=(z v){z\u}and apply previous lemma (verification of the hypothesis is easy).

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Fourth proof of the SN property

See for example Gandy’s proof by Alexandre Miquel.

A combinatorial proof of strong normalisation for the simply typed lambda-calculus.

http://www.pps.univ-paris-diderot.fr/˜miquel/publis/snlam.pdf

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Strong Normalization of System F

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Reducibility Candidates

Neutral Terms:Terms that are not abstractions.

Strongly Normalizing Terms:S NF

Definition

Areducibility candidateof typeAis a setRof terms of typeAsuch that (CR1) Ift∈ R, thent∈S NF

(CR2) Ift∈ Randt→_F t⁰, thent⁰∈ R

(CR3) Iftis neutral and (t→_F t⁰impliest⁰∈ R), thent∈ R.

IfRandSare reducibility candidates of typeAandBrespectively, thenR → Sis a set of terms of typeA→Bdefined by

t∈ R → Siff∀u.(u∈ Rimpliestu∈ S)

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Remarks

A consequence of(CR3): Iftis neutral and normal, thent∈ R. Rof typeAis never empty, it contains at least the variables of typeA. The set{t∈S NFandtof typeA}is a reducibility candidate.

R → Sis a reducibility candidate.

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Reducibility with Parameters

LetTbe a type wheretfv(T)⊆~α. We writeT{~α\A}~ for the simultaneous substitution of

~αbyA~. Given~Ra sequence of reducibility candidates, we define a setRED_T(~α, ~R)of terms of typeT{~α\A~}.

IfT=αi, thenRED_T(~α, ~R)=R_i

IfT=A→B, thenRED_A→B(~α, ~R)=RED_A(~α, ~R)→RED_B(~α, ~R)

IfT =∀γ.B, thenRED_∀γ.B(~α, ~R)is the set of termstof typeT{~α\A}~ such that for every typeCand reducibility candidateSof this type, thent[C]∈RED_B(~αγ, ~RS)

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Properties

Lemma

RED_T(~α, ~R)is a reducibility candidate of typeT{~α\A~}

Lemma

RED_T{γ\B}(~α, ~R)=RED_T(~αγ, ~RRED_B(~α, ~R))

Lemma

If for every typeBand candidateS_,t{γ\B} ∈RED_A(~αγ, ~RS), thenΛγt∈RED∀γ.A(~α, ~R)

Lemma

Ift∈RED∀γ.A(~α, ~R),t[B]∈REDA{γ\B}(~α, ~R)for evert typeB.

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Reducible Terms

A termtof typeAisreducibleift∈RED_A(~α, ~S N)where~α=α1. . . αnare the free type variables ofA, andS N~ isS N1. . .S Nn, whereS Niis the set of terms ofS NFof typeαi.

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Final Theorem

Theorem

All terms of systemFare reducible.

Corollary (by CR1)

Corollary

All terms of systemFare inS NF.

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Key Lemma

Lemma

Lettbe a term of typeA. Supposefvt⊆ {x1, . . . ,xn}andxiis of typeBi. Suppose tfv(A,B1, . . . ,Bn)⊆ {α1, . . . , αm}. If{R₁, . . . ,R_m}are reducibility candidates of types {C1, . . . ,Cm}andv1, . . . ,vnare terms of typesB1{~α\C}, . . . ,~ Bn{~α\C}~ which are in RED_B₁(~α, ~~R), . . . ,RED_B_n(~α, ~~R), thent{~α\C}{~~ x\~v} ∈RED_A(~α, ~R)

Typed Lambda Calculus

Typed Lambda Calculus

Motivations

Contents

Simply Typed Lambda Calculus

Curry-Howard Isomorphism

Curry’58, Howard’68

Adding (simply) types to λ -calculus

Typing Environment

Typed Lambda Calculus in Church-Style

Examples

Typed Properties

Proof.

Proof.

Proof.

Typing Algorithm

Properties of the Typing Algorithm

Typed Lambda Calculus in Curry-Style

Properties

Polymorphism

Difficulties for a Typing Algorithm

Monomorphic Type Inference

Towards a Monomorphic Type Inference Algorithm

Examples

Properties of the Type Inference Algorithm

Theorem

Theorem

Theorem

Polymorphic Lambda Calculus

Motivations

Different Forms of Polymorphism

Parametric Polymorphism

Girard-Reynolds Polymorphism

Type Free Variables

Polymorphic Lambda Calculus in Church-Style (Girard)

Polymorphic Type Inference

Theorem

ML Polymorphism

Type Instance

Definition

Type Generalization

Definition

Polymorphic Lambda Calculus in Curry-Style

Example

Properties

Polymorphic Type Inference

Towards a Type Inference Algorithm: Notations

Damas-Milner-Tofte Algorithm

Premier exemple

Second Example

Properties of the Type Inference Algorithm

Theorem

Theorem

Strong Normalization of Simply Typed

Lambda Calculus

Typed Properties

Defining Strongly Normalizing Terms

Definition (Measuring

-terms)

Some General Remarks About S N

-Terms

First Proof of the SN property

Definition

Lemma

Proof.

Lemma

Proof.

Lemma

Proof.

Theorem

Proof.

Second proof of the SN property

Third Proof of the SN property

Lemma

Proof.

Theorem

Proof.

Fourth proof of the SN property

Strong Normalization of System F

Reducibility Candidates