[PPT] - On Continuous Normalization Klaus Aehlig Felix Joachimski PowerPoint Presentation

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 1

✤ ✣ ✜ ✢

On Continuous Normalization

Klaus Aehlig Felix Joachimski Mathematisches Institut LMU M¨ unchen

{aehlig|joachski}@mathematik.uni-muenchen.de

CSL 2002

http://www.mathematik.uni-muenchen.de/˜{aehlig|joachski}

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 2

Some history

Cut-elimination: a cut ¬A ∨ ¬B

A ∧ B ⊥ is replaced by cuts on A and B.

This process is defined by induction on (semi-formal) derivations.
Want to separate the operational definition from the proof-theoretical analysis

(well-foundedness of the derivation involves ordinal notation systems and strong means).

In order to compute the normalized derivation in a primitive recursive way
Mints 1978 introduced repetition rule∗ Γ ⊢ A

Γ ⊢ A to compute the last rule of the normalized derivation (“Please wait; your proof will soon be computed (hopefully)”)

This procedure even applies to non-wellfounded derivations (it is continuous).

∗ which has the subformula property!

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 3

Goal

Transfer ideas to λ-calculus (as Schwichtenberg 1998 and others did).

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 3

Goal

Transfer ideas to λ-calculus (as Schwichtenberg 1998 and others did).
Interesting, because some λ-terms (like Y ) have infinite “normal forms”, or are

even diverging (like Ω).

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 3

Goal

Transfer ideas to λ-calculus (as Schwichtenberg 1998 and others did).
Interesting, because some λ-terms (like Y ) have infinite “normal forms”, or are

even diverging (like Ω).

Need a coinductive λ-calculus to write out the normal forms!

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 3

Goal

Transfer ideas to λ-calculus (as Schwichtenberg 1998 and others did).
Interesting, because some λ-terms (like Y ) have infinite “normal forms”, or are

even diverging (like Ω).

Need a coinductive λ-calculus to write out the normal forms!

✬ ✫ ✩ ✪

Thus: define and analyze a primitive recursive normalization function ( )β : Λ(co) → Λco

R

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 3

Goal

Transfer ideas to λ-calculus (as Schwichtenberg 1998 and others did).
Interesting, because some λ-terms (like Y ) have infinite “normal forms”, or are

even diverging (like Ω).

Need a coinductive λ-calculus to write out the normal forms!

✬ ✫ ✩ ✪

Thus: define and analyze a primitive recursive normalization function ( )β : Λ(co) → Λco

R

In particular, explain why and how many repetition rules are needed.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr Examples. Θ := tt with t = λλ.0.110

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr Examples. Θ := tt with t = λλ.0.110 = “λtλf(f(ttf))”

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr Examples. Θ := tt with t = λλ.0.110 = “λtλf(f(ttf))” Y := λ.(λ.1.00)(λ.1.00) = “λf.(λx(f(xx)))(λx(f(xx)))”

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr Examples. Θ := tt with t = λλ.0.110 = “λtλf(f(ttf))” Y := λ.(λ.1.00)(λ.1.00) = “λf.(λx(f(xx)))(λx(f(xx)))” Y co

r

:= rY co

r

= r(r(r . . .

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 4

Terms

Terms. Inductive and coinductive λ-calculus in de Bruijn-notation (x ∈ N).

Λ ∋ r, s ::= x | rs | λr Λco ∋ r, s ::=co x | rs | λr ΛR ∋ r, s ::= x | rs | λr | Rr | βr Λco

R

∋ r, s ::=co x | rs | λr | Rr | βr Examples. Θ := tt with t = λλ.0.110 = “λtλf(f(ttf))” Y := λ.(λ.1.00)(λ.1.00) = “λf.(λx(f(xx)))(λx(f(xx)))” Y co

r

:= rY co

r

= r(r(r . . . Observational equality. r ≃k s iff the outermost k constructors are identical. E.g., 1λλ0 ≃2 1λ(0 2).

Equality. r = s iff r ≃k s for all k.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 5

Continuous normalization

Substitution. r[s] is well-defined

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 5

Continuous normalization

Substitution. r[s] is well-defined and continuous with identity as modulus of

continuity, i.e., r ≃k r′ ∧ s ≃k s′ ⇒ r[s] ≃k r′[s′].

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 5

Continuous normalization

Substitution. r[s] is well-defined and continuous with identity as modulus of

continuity, i.e., r ≃k r′ ∧ s ≃k s′ ⇒ r[s] ≃k r′[s′].

Reduction. → is the compatible closure of (λr)s → r[s].

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 5

Continuous normalization

Substitution. r[s] is well-defined and continuous with identity as modulus of

continuity, i.e., r ≃k r′ ∧ s ≃k s′ ⇒ r[s] ≃k r′[s′].

Reduction. → is the compatible closure of (λr)s → r[s].

Normal forms. NF ∋ r, s ::=co x r | λr | Rr | βr.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 5

Continuous normalization

Substitution. r[s] is well-defined and continuous with identity as modulus of

continuity, i.e., r ≃k r′ ∧ s ≃k s′ ⇒ r[s] ≃k r′[s′].

Reduction. → is the compatible closure of (λr)s → r[s].

Normal forms. NF ∋ r, s ::=co x r | λr | Rr | βr. Continuous normalization. rβ := r@ε where for r, s ∈ Λco we define r@ s ∈ NF by

✬ ✫ ✩ ✪

(rs)@ s := R.r@(s, s ) x@ s := x s β (λr)@ε := λrβ (λr)@(s, s ) := β.r[s]@ s

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0)

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . .

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f → (λf.f(Θf))f

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f → (λf.f(Θf))f → f(Θf)

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f → (λf.f(Θf))f → f(Θf) while Yf

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f → (λf.f(Θf))f → f(Θf) while Yf := (λx.f(xx))(λx.f(xx))

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 6

Examples

(SKK)β = RRββλRRββ0

with S := λλλ.2 0 .1 0, K := λλ1. Note that SKK →2 λ.K0(K0) →2 λ0.

For Θ = tt with t = λλ.0.110 and Y = λ.(λ.1.00)(λ.1.00) we get

nf Θ = nf Y = λ. 0( 0( 0(. . . Θβ = Rβλ.R(0(RRββR(0(RRββR(0(. . . Y β = λ.RβR(0( Rβ R(0( Rβ R(0(. . . Note that Θf = (λx, f.f(xxf))(λx, f.f(xxf))f → (λf.f(Θf))f → f(Θf) while Yf := (λx.f(xx))(λx.f(xx)) → fYf.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 7

Properties

Continuity: r ≃k r′ ∧

s ≃k s′ = ⇒ r@ s ≃k r′@ s′. In particular rβ ≃k r′β for r ≃k r′.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 7

Properties

Continuity: r ≃k r′ ∧

s ≃k s′ = ⇒ r@ s ≃k r′@ s′. In particular rβ ≃k r′β for r ≃k r′.

Soundness: r ∈ Λ ∧ rβ ∈ ΛR =

⇒ r →∗ rβ∗ where r∗ is r without β and R (for r ∈ ΛR).

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 7

Properties

Continuity: r ≃k r′ ∧

s ≃k s′ = ⇒ r@ s ≃k r′@ s′. In particular rβ ≃k r′β for r ≃k r′.

Soundness: r ∈ Λ ∧ rβ ∈ ΛR =

⇒ r →∗ rβ∗ where r∗ is r without β and R (for r ∈ ΛR).

Continuous normal forms: ∃s ∈ Λco.sβ = r iff ⊢ r with

s, s ⊢ r

s ⊢ Rr

⊢ r

r ⊢ x

r ⊢ r ⊢ λr

s ⊢ r

s, s ⊢ βr

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 7

Properties

Continuity: r ≃k r′ ∧

s ≃k s′ = ⇒ r@ s ≃k r′@ s′. In particular rβ ≃k r′β for r ≃k r′.

Soundness: r ∈ Λ ∧ rβ ∈ ΛR =

⇒ r →∗ rβ∗ where r∗ is r without β and R (for r ∈ ΛR).

Continuous normal forms: ∃s ∈ Λco.sβ = r iff ⊢ r with

s, s ⊢ r

s ⊢ Rr

⊢ r

r ⊢ x

r ⊢ r ⊢ λr

s ⊢ r

s, s ⊢ βr

Analysis: A R is justified by a β or an application:

⊢ s = ⇒ Rζs ≥ βζs + Aζs with = for complete paths ζ

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 8

Counting reductions

in the leftmost-outermost reduction strategy

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 8

Counting reductions

in the leftmost-outermost reduction strategy Standardization.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 8

Counting reductions

in the leftmost-outermost reduction strategy

Standardization. If r →∗ s ∈ NF ∩ Λ then r ❀ s, given by
r ❀

n

s x r ❀Σ

n x

s r ❀n s λr ❀n λs r[s] s ❀n t (λr)s s ❀n+1 t

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 8

Counting reductions

in the leftmost-outermost reduction strategy

Standardization. If r →∗ s ∈ NF ∩ Λ then r ❀ s, given by
r ❀

n

s x r ❀Σ

n x

s r ❀n s λr ❀n λs r[s] s ❀n t (λr)s s ❀n+1 t

Lemma. r ❀n s =

⇒ rβ ✄n

n sβ

where ✄k

n removes n occurrences of β and k occurences of R.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 8

Counting reductions

in the leftmost-outermost reduction strategy

Standardization. If r →∗ s ∈ NF ∩ Λ then r ❀ s, given by
r ❀

n

s x r ❀Σ

n x

s r ❀n s λr ❀n λs r[s] s ❀n t (λr)s s ❀n+1 t

Lemma. r ❀n s =

⇒ rβ ✄n

n sβ

where ✄k

n removes n occurrences of β and k occurences of R.

Theorem. r ❀n s ∈ NF ∩ Λ =

⇒ rβ ✄n+|s|

n

s where |s| is the number of applications in s.

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 9

Conclusions and further work

Defined and analyzed a primitive recursive normalization function in the

coinductive λ-calculus with repetition rule R.

Explained the use of R, using auxiliary constructor β. This gives information
n the normalization process.
Can be extended to calculi with infinitary branching rules, such as (ω).
Straightforward implementation in Haskell:

beta :: Term -> Term beta r = app r [] app :: Term -> [Term] -> Term app (Lam r) (s:l) = Bet (app (subst r s 0) l) app (Lam r) [] = Lam (beta r) app (Var k) l = foldl App (Var k) (map beta l) app (App r s) l = Rep (app r (s:l))

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Examples in Haskell

s = Lam (Lam (Lam (Var 2 ‘App‘ (Var 0) ‘App‘ (Var 1 ‘App‘ (Var 0))))) k = Lam (Lam (Var 1)) y = Lam (Lam (Var 1 ‘App‘ (Var 0 ‘App‘ (Var 0))) ‘App‘ (Lam (Var 1 ‘App‘ (Var 0 ‘App‘ (Var 0))))) yco r = r ‘App‘ (yco r) theta = Lam (Lam (Var 0 ‘App‘ (Var 1 ‘App‘ (Var 1) ‘App‘ (Var 0)))) ‘App‘ Lam (Lam (Var 0 ‘App‘ (Var 1 ‘App‘ (Var 1) ‘App‘ (Var 0)))) church n = Lam (Lam (iterate (App (Var 1)) (Var 0) !! n))

Test runs:

Main> beta (s ‘App‘ k ‘App‘ k) Rep (Rep (Bet (Bet (Lam (Rep (Rep (Bet (Bet (Var 0))))))))) Main> beta (yco k) Rep (Bet (Lam (Rep (Bet (Lam (Rep (Bet (Lam (Rep (Bet (Lam (Rep (Bet (Lam (Rep (Bet (Lam (Rep {Interrupted!} Main> beta (y ‘App‘ k) Rep (Bet (Rep (Bet (Rep (Bet (Lam (Rep (Bet (Rep (Bet (Lam (Rep (Bet (Rep (Bet (Lam (Rep (Bet (Rep {Interrupted!} Main> beta (theta ‘App‘ k) Rep (Rep (Bet (Bet (Rep (Bet (Lam (Rep (Rep (Bet (Bet (Rep (Bet (Lam (Rep (Rep (Bet (Bet (Rep (Bet (Lam {Interrupted!}

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History – Goal – Terms – Normalization – Examples – Properties – Counting – Conclusions On Continuous Normalization CSL 2002 11

More test runs

Main> beta (church 2 ‘App‘ (church 3)) Rep (Bet (Lam (Rep (Bet (Lam (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (App (Var 1) (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (App (Var 1) (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (App (Var 1) (Var 0))) [..]) Main> beta (church 3 ‘App‘ (church 2)) Rep (Bet (Lam (Rep (Bet (Lam (Rep (Rep (Bet (Bet (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (Rep (Bet (Bet (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Rep (Rep (Bet (Bet (Rep (App (Var 1) (Rep (App (Var 1) (Var 0))) [..] )