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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/251230477 Generalized Developments in -calculus (SLIDES) Article February 2009 CITATIONS READS 0 121 2 authors: Yiorgos

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Generalized Developments in λ-calculus (SLIDES)

Article · February 2009

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Generalized Developments in λ-calculus

Yiorgos Stavrinos

(joint work with George Koletsos) February 6, 2009

Yiorgos Stavrinos: Generalized Developments in λ-calculus 1/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Outline

  • Generalized developments
  • λ-calculi with types
  • Embedding
  • Finiteness of gen. developments
  • Conclusion

Yiorgos Stavrinos: Generalized Developments in λ-calculus 2/17

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SLIDE 4

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • V = {x, y, z, . . .} infinite set of variables

Λ: set of λ-terms M ::= x | λx.P | PQ β-reduction: (λx.P)Q

  • redex

− →β P[Q/x]

contractum

  • Generalized development

generalized β-redex (g-redex): (λx1 . . . λxn.P)Q1 . . . Qn F a set of g-redex occurrences in M M

∆

− →β M1

∆1

− →β M2

∆2

− →β . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 3/17

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SLIDE 5

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • V = {x, y, z, . . .} infinite set of variables

Λ: set of λ-terms M ::= x | λx.P | PQ β-reduction: (λx.P)Q

  • redex

− →β P[Q/x]

contractum

  • Generalized development

generalized β-redex (g-redex): (λx1 . . . λxn.P)Q1 . . . Qn F a set of g-redex occurrences in M (M, F)

∆∈F

− →β M1

∆1

− →β M2

∆2

− →β . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 3/17

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SLIDE 6

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • V = {x, y, z, . . .} infinite set of variables

Λ: set of λ-terms M ::= x | λx.P | PQ β-reduction: (λx.P)Q

  • redex

− →β P[Q/x]

contractum

  • Generalized development

generalized β-redex (g-redex): (λx1 . . . λxn.P)Q1 . . . Qn F a set of g-redex occurrences in M (M, F)

∆∈F

− →β (M1, F1)

∆1∈F1

− →β M2

∆2

− →β . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 3/17

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SLIDE 7

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • V = {x, y, z, . . .} infinite set of variables

Λ: set of λ-terms M ::= x | λx.P | PQ β-reduction: (λx.P)Q

  • redex

− →β P[Q/x]

contractum

  • Generalized development

generalized β-redex (g-redex): (λx1 . . . λxn.P)Q1 . . . Qn F a set of g-redex occurrences in M (M, F)

∆∈F

− →β (M1, F1)

∆1∈F1

− →β (M2, F2)

∆2∈F2

− →β . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 3/17

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SLIDE 8

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • V = {x, y, z, . . .} infinite set of variables

Λ: set of λ-terms M ::= x | λx.P | PQ β-reduction: (λx.P)Q

  • redex

− →β P[Q/x]

contractum

  • Generalized development

generalized β-redex (g-redex): (λx1 . . . λxn.P)Q1 . . . Qn F a set of g-redex occurrences in M (M, F)

∆∈F

− →β (M1, F1)

∆1∈F1

− →β (M2, F2)

∆2∈F2

− →β . . .

  • Λ′: set of indexed λ-terms

M ::= x | λx.P | PQ | (λ0x1 . . . λ0xn.P)Q1 . . . Qn

  • generalized indexed redex

β0-reduction: (λ0x1 . . . λ0xn.P)Q1 . . . Qn − →β0 (λ0x2 . . . λ0xn.P[Q1/x])Q2 . . . Qn

Yiorgos Stavrinos: Generalized Developments in λ-calculus 3/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Example

β-reductions: ((λx.λy.x)λx.x)((λx.xx)(λx.xx)) − →β ((λx.λy.x)λx.x)((λx.xx)(λx.xx)) − →β . . . ((λx.λy.x)λx.x)((λx.xx)(λx.xx)) − →β (λy.λx.x)((λx.xx)(λx.xx)) − →β λx.x

  • gen. developments:

((λ0x.λy.x)λx.x)((λ0x.xx)(λx.xx))

∗

− →β0 (λy.λx.x)((λx.xx)(λx.xx)) ((λ0x.λ0y.x)λx.x)((λ0x.xx)(λx.xx))

∗

− →β0 (λ0y.λx.x)((λx.xx)(λx.xx)) − →β0 λx.x

Yiorgos Stavrinos: Generalized Developments in λ-calculus 4/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • Gen. developments are simulated by β0-reduction

To any term M ∈ Λ and a set F of g-redex occurrences in M we associate a term M′ ∈ Λ′ where all g-redex occurrences in F are indexed, i.e. if M ≡ . . . (λ x.P) Q . . . where (λ x.P) Q ∈ F then M′ ≡ . . . (λ0 x.P) Q . . .

Proposition

  • Gen. development :

(M, F)

∗

− →β (Mn, Fn) Λ

  • β0-reduction :

M′

∗

− →β0 M′

n

Λ′

Theorem (finiteness of gen. developments)

All gen. developments of M ∈ Λ are finite and end with the same term. Equivalently:

∗

− →β0 is strongly normalizable and has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 5/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • Gen. developments are simulated by β0-reduction

To any term M ∈ Λ and a set F of g-redex occurrences in M we associate a term M′ ∈ Λ′ where all g-redex occurrences in F are indexed, i.e. if M ≡ . . . (λ x.P) Q . . . where (λ x.P) Q ∈ F then M′ ≡ . . . (λ0 x.P) Q . . .

Proposition

  • Gen. development :

(M, F)

∗

− →β (Mn, Fn) Λ

  • β0-reduction :

M′

∗

− →β0 M′

n

Λ′

Theorem (finiteness of gen. developments)

All gen. developments of M ∈ Λ are finite and end with the same term. Equivalently:

∗

− →β0 is strongly normalizable and has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 5/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Simply typed λ-calculus, λ→

Types σ ::= α | σ → σ, where α ∈ {α1, α2, . . .} Contexts Γ = {x1 : σ1, . . . , xk : σk}

(Ax.)

Γ, x : σ ⊢ x : σ Γ, x : σ ⊢ M : τ

(→I)

Γ ⊢ λx.M : σ → τ Γ ⊢ M : σ → τ Γ ⊢ N : σ

(→E)

Γ ⊢ MN : τ Subject reduction property: Γ ⊢ M : σ M

∗

− →β N Γ ⊢ N : σ

Theorem

If Γ ⊢ M : σ, then M is strongly normalizable and has the Church-Rosser property.

Proof: By the method of reducibility: strong normalization [Tait 1967, Girard 1971], C-R property [Koletsos 1985]

Yiorgos Stavrinos: Generalized Developments in λ-calculus 6/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Intersection types, system D

Types σ ::= α | σ → σ | σ ∩ σ, where α ∈ {α1, α2, . . .} D := λ→ + (∩I), (∩E) Γ ⊢ M : σ Γ ⊢ M : τ

(∩I)

Γ ⊢ M : σ ∩ τ Γ ⊢ M : σ1 ∩ σ2

(∩E) i = 1, 2

Γ ⊢ M : σi

Theorem

  • 1. Γ ⊢ M : σ if and only if M is strongly normalizable.
  • 2. If Γ ⊢ M : σ, then M has the Church-Rosser property.

Proof: Characterizing the strongly normalizable terms [Pottinger 1980], C-R property [Koletsos-Stavrinos 1997]

Yiorgos Stavrinos: Generalized Developments in λ-calculus 7/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

The sets ΛC, ΛC!

Add to the variables of Λ a set C = {c1, c2, . . . , ci, . . .} of new variables. So now the set of variables is VC = {x, y, z, . . .} ∪ C. ΛC : M ::= x | λx.P | ciPQ | (λ x.P) Q, where x, x ∈ VC \ C, ci ∈ C ΛC!: the set of λ-terms M ∈ ΛC s.t. the ci’s occur at most once in M

Example

ci, λx.yx / ∈ ΛC, but λx.ciyx ∈ ΛC (λx.λy.x)y, (λx.λy.x)yz, ci((λx.λy.x)y)z ∈ ΛC (λx.c1yx)(λx.c1yx) / ∈ ΛC!, but (λx.c3yx)(λx.c1yx) ∈ ΛC!

Lemma

If M ∈ ΛC and M

∗

− →β N, then N ∈ ΛC.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 8/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Embedding Λ′ in ΛC

Define a function | | : ΛC → Λ′

  • |x| = x
  • |λx.M| = λx.|M|
  • |ciMN| = |M| |N|
  • |(λx1 . . . λxn.M)N1 . . . Nn| = (λ0x1 . . . λ0xn.|M|)|N1| . . . |Nn|

| | surjective, not injective, e.g. |(λx.c1yx)y| = |(λx.c2yx)y| = (λ0x.yx)y Representatives of M′ ∈ Λ′: [M′]

def

= = |M′|−1 = {M ∈ ΛC | M′ = |M|} Define an equivalence relation ∼ s.t. ciMN ∼ cjMN. Then [M′] ∈ ΛC/

∼.

Proposition

[ ] : Λ′ → ΛC/

∼

M′ → [M′]

  • ne-to-one correspondence (an embedding of Λ′ in ΛC)

Yiorgos Stavrinos: Generalized Developments in λ-calculus 9/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Embedding Λ′ in ΛC

Define a function | | : ΛC → Λ′

  • |x| = x
  • |λx.M| = λx.|M|
  • |ciMN| = |M| |N|
  • |(λx1 . . . λxn.M)N1 . . . Nn| = (λ0x1 . . . λ0xn.|M|)|N1| . . . |Nn|

| | surjective, not injective, e.g. |(λx.c1yx)y| = |(λx.c2yx)y| = (λ0x.yx)y Representatives of M′ ∈ Λ′: [M′]

def

= = |M′|−1 = {M ∈ ΛC | M′ = |M|} Define an equivalence relation ∼ s.t. ciMN ∼ cjMN. Then [M′] ∈ ΛC/

∼.

Proposition

[ ] : Λ′ → ΛC/

∼

M′ → [M′]

  • ne-to-one correspondence (an embedding of Λ′ in ΛC)

Yiorgos Stavrinos: Generalized Developments in λ-calculus 9/17

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SLIDE 17

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Embedding Λ′ in ΛC

Define a function | | : ΛC → Λ′

  • |x| = x
  • |λx.M| = λx.|M|
  • |ciMN| = |M| |N|
  • |(λx1 . . . λxn.M)N1 . . . Nn| = (λ0x1 . . . λ0xn.|M|)|N1| . . . |Nn|

| | surjective, not injective, e.g. |(λx.c1yx)y| = |(λx.c2yx)y| = (λ0x.yx)y Representatives of M′ ∈ Λ′: [M′]

def

= = |M′|−1 = {M ∈ ΛC | M′ = |M|} Define an equivalence relation ∼ s.t. ciMN ∼ cjMN. Then [M′] ∈ ΛC/

∼.

Proposition

[ ] : Λ′ → ΛC/

∼

M′ → [M′]

  • ne-to-one correspondence (an embedding of Λ′ in ΛC)

Yiorgos Stavrinos: Generalized Developments in λ-calculus 9/17

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SLIDE 18

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Embedding Λ′ in ΛC

Define a function | | : ΛC → Λ′

  • |x| = x
  • |λx.M| = λx.|M|
  • |ciMN| = |M| |N|
  • |(λx1 . . . λxn.M)N1 . . . Nn| = (λ0x1 . . . λ0xn.|M|)|N1| . . . |Nn|

| | surjective, not injective, e.g. |(λx.c1yx)y| = |(λx.c2yx)y| = (λ0x.yx)y Representatives of M′ ∈ Λ′: [M′]

def

= = |M′|−1 = {M ∈ ΛC | M′ = |M|} Define an equivalence relation ∼ s.t. ciMN ∼ cjMN. Then [M′] ∈ ΛC/

∼.

Proposition

[ ] : Λ′ → ΛC/

∼

M′ → [M′]

  • ne-to-one correspondence (an embedding of Λ′ in ΛC)

Yiorgos Stavrinos: Generalized Developments in λ-calculus 9/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Embedding Λ′ in ΛC

Define a function | | : ΛC → Λ′

  • |x| = x
  • |λx.M| = λx.|M|
  • |ciMN| = |M| |N|
  • |(λx1 . . . λxn.M)N1 . . . Nn| = (λ0x1 . . . λ0xn.|M|)|N1| . . . |Nn|

| | surjective, not injective, e.g. |(λx.c1yx)y| = |(λx.c2yx)y| = (λ0x.yx)y Representatives of M′ ∈ Λ′: [M′]

def

= = |M′|−1 = {M ∈ ΛC | M′ = |M|} Define an equivalence relation ∼ s.t. ciMN ∼ cjMN. Then [M′] ∈ ΛC/

∼.

Proposition

[ ] : Λ′ → ΛC/

∼

M′ → [M′]

  • ne-to-one correspondence (an embedding of Λ′ in ΛC)

Yiorgos Stavrinos: Generalized Developments in λ-calculus 9/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (lifting)

M − →β N ΛC ↑ ↑ M′ − →β0 N′ Λ′

Proposition (projecting)

M − →β N ΛC ↓ ↓ |M| − →β0 |N| Λ′

Yiorgos Stavrinos: Generalized Developments in λ-calculus 10/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (lifting)

M − →β N ΛC ↑ ↑ M′ − →β0 N′ Λ′

Proposition (projecting)

M − →β N ΛC ↓ ↓ |M| − →β0 |N| Λ′

Yiorgos Stavrinos: Generalized Developments in λ-calculus 10/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (lifting)

M ∈ ΛC! − →β N ΛC ↑ ↑ M′ − →β0 N′ Λ′

Proposition (projecting)

M − →β N ΛC ↓ ↓ |M| − →β0 |N| Λ′

Yiorgos Stavrinos: Generalized Developments in λ-calculus 10/17

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SLIDE 23

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (lifting)

M ∈ ΛC! − →β N ΛC ↑ ↑ M′ − →β0 N′ Λ′

Proposition (projecting)

M − →β N ΛC ↓ ↓ |M| − →β0 |N| Λ′

Yiorgos Stavrinos: Generalized Developments in λ-calculus 10/17

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SLIDE 24

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (lifting)

M ∈ ΛC! − →β N ΛC ↑ ↑ M′ − →β0 N′ Λ′

Proposition (projecting)

M − →β N ΛC ↓ ↓ |M| − →β0 |N| Λ′

Yiorgos Stavrinos: Generalized Developments in λ-calculus 10/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

An overall schema

V VC Λ ∪ Λ′

[ ]

− → ΛC (ΛC/

∼) ⊂ D

∪ ∪ Λ ΛC! ⊂ λ→

Yiorgos Stavrinos: Generalized Developments in λ-calculus 11/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • one extra variable c,

intersection types [Parigot-Krivine 1990, Koletsos-Stavrinos 1997] y : σ, c : (σ → (τ → ρ) → υ) ∩ (σ → τ → ρ) ⊢ (λx.cyx)(λx.cyx) : υ

  • two extra variables f , g,

simple types 0, 0 → 0, (0 → 0) → 0, . . . [Ghilezan 1996, Ghilezan-Kunˇ cak 2001] y : 0, f : 0 → 0 → 0, g : (0 → 0) → 0 ⊢ (λx.f yx)(g(λx.f yx)) : 0

  • many extra variables c1, c2, . . .,

simple types [Koletsos-Stavrinos 2008] y : σ, c3 : σ → (τ → ρ) → υ, c1 : σ → τ → ρ ⊢ (λx.c3yx)(λx.c1yx) : υ

Yiorgos Stavrinos: Generalized Developments in λ-calculus 12/17

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SLIDE 27

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • one extra variable c,

intersection types [Parigot-Krivine 1990, Koletsos-Stavrinos 1997] y : σ, c : (σ → (τ → ρ) → υ) ∩ (σ → τ → ρ) ⊢ (λx.cyx)(λx.cyx) : υ

  • two extra variables f , g,

simple types 0, 0 → 0, (0 → 0) → 0, . . . [Ghilezan 1996, Ghilezan-Kunˇ cak 2001] y : 0, f : 0 → 0 → 0, g : (0 → 0) → 0 ⊢ (λx.f yx)(g(λx.f yx)) : 0

  • many extra variables c1, c2, . . .,

simple types [Koletsos-Stavrinos 2008] y : σ, c3 : σ → (τ → ρ) → υ, c1 : σ → τ → ρ ⊢ (λx.c3yx)(λx.c1yx) : υ

Yiorgos Stavrinos: Generalized Developments in λ-calculus 12/17

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SLIDE 28

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

  • one extra variable c,

intersection types [Parigot-Krivine 1990, Koletsos-Stavrinos 1997] y : σ, c : (σ → (τ → ρ) → υ) ∩ (σ → τ → ρ) ⊢ (λx.cyx)(λx.cyx) : υ

  • two extra variables f , g,

simple types 0, 0 → 0, (0 → 0) → 0, . . . [Ghilezan 1996, Ghilezan-Kunˇ cak 2001] y : 0, f : 0 → 0 → 0, g : (0 → 0) → 0 ⊢ (λx.f yx)(g(λx.f yx)) : 0

  • many extra variables c1, c2, . . .,

simple types [Koletsos-Stavrinos 2008] y : σ, c3 : σ → (τ → ρ) → υ, c1 : σ → τ → ρ ⊢ (λx.c3yx)(λx.c1yx) : υ

Yiorgos Stavrinos: Generalized Developments in λ-calculus 12/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proposition (ΛC! ⊂ λ→)

Let M ∈ ΛC! and c1, . . . , cn the extra variables occurring in M. If Γ is an arbitrary context for the free variables of M, except c1, . . . , cn, then there exist simple types σ1, . . . , σn, σ s.t. Γ, c1 : σ1, . . . , cn : σn ⊢ M : σ.

Proposition (ΛC ⊂ D)

Let M ∈ ΛC and c1, . . . , cn the extra variables occurring in M. If Γ is an arbitrary context for the free variables of M, except c1, . . . , cn, then there exist intersection types σ1, . . . , σn, σ s.t. Γ, c1 : σ1, . . . , cn : σn ⊢ M : σ.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 13/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Theorem

If M ∈ ΛC! (ΛC), then M is strongly normalizable and has the Church-Rosser property.

Theorem (finiteness of gen. developments)

1.

∗

− →β0 is strongly normalizable. 2.

∗

− →β0 has the Church-Rosser property.

Proof: 1. M∈ ΛC! − →β M1 − →β M2 − →β . . . contradiction ↑ ↑ ↑ M′ − →β0 M′

1 −

→β0 M′

2 −

→β0 . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 14/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Theorem

If M ∈ ΛC! (ΛC), then M is strongly normalizable and has the Church-Rosser property.

Theorem (finiteness of gen. developments)

1.

∗

− →β0 is strongly normalizable. 2.

∗

− →β0 has the Church-Rosser property.

Proof: 1. M∈ ΛC! − →β M1 − →β M2 − →β . . . contradiction ↑ ↑ ↑ M′ − →β0 M′

1 −

→β0 M′

2 −

→β0 . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 14/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Theorem

If M ∈ ΛC! (ΛC), then M is strongly normalizable and has the Church-Rosser property.

Theorem (finiteness of gen. developments)

1.

∗

− →β0 is strongly normalizable. 2.

∗

− →β0 has the Church-Rosser property.

Proof: 1. M∈ ΛC! − →β M1 − →β M2 − →β . . . contradiction ↑ ↑ ↑ M′ − →β0 M′

1 −

→β0 M′

2 −

→β0 . . .

Yiorgos Stavrinos: Generalized Developments in λ-calculus 14/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proof (cont.) 2. M1

β ∗

← − M∈ ΛC!

∗

− →β M2 ΛC [lifting] ↑ ↑ ↑ M′

1 β0 ∗

← − M′

∗

− →β0 M′

2

Λ′ M1

∗

− →β M3

β ∗

← − M2 ΛC [M has C-R] ↓ ↓ ↓ M′

1 ∗

− →β0 M′

3 β0 ∗

← − M′

2

Λ′ [projecting]

Corollary (Church-Rosser)

Every term in Λ has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 15/17

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Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proof (cont.) 2. M1

β ∗

← − M∈ ΛC!

∗

− →β M2 ΛC [lifting] ↑ ↑ ↑ M′

1 β0 ∗

← − M′

∗

− →β0 M′

2

Λ′ M1

∗

− →β M3

β ∗

← − M2 ΛC [M has C-R] ↓ ↓ ↓ M′

1 ∗

− →β0 M′

3 β0 ∗

← − M′

2

Λ′ [projecting]

Corollary (Church-Rosser)

Every term in Λ has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 15/17

slide-35
SLIDE 35

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proof (cont.) 2. M1

β ∗

← − M∈ ΛC!

∗

− →β M2 ΛC [lifting] ↑ ↑ ↑ M′

1 β0 ∗

← − M′

∗

− →β0 M′

2

Λ′ M1

∗

− →β M3

β ∗

← − M2 ΛC [M has C-R] ↓ ↓ ↓ M′

1 ∗

− →β0 M′

3 β0 ∗

← − M′

2

Λ′ [projecting]

Corollary (Church-Rosser)

Every term in Λ has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 15/17

slide-36
SLIDE 36

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proof (cont.) 2. M1

β ∗

← − M∈ ΛC!

∗

− →β M2 ΛC [lifting] ↑ ↑ ↑ M′

1 β0 ∗

← − M′

∗

− →β0 M′

2

Λ′ M1

∗

− →β M3

β ∗

← − M2 ΛC [M has C-R] ↓ ↓ ↓ M′

1 ∗

− →β0 M′

3 β0 ∗

← − M′

2

Λ′ [projecting]

Corollary (Church-Rosser)

Every term in Λ has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 15/17

slide-37
SLIDE 37

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Proof (cont.) 2. M1

β ∗

← − M∈ ΛC!

∗

− →β M2 ΛC [lifting] ↑ ↑ ↑ M′

1 β0 ∗

← − M′

∗

− →β0 M′

2

Λ′ M1

∗

− →β M3

β ∗

← − M2 ΛC [M has C-R] ↓ ↓ ↓ M′

1 ∗

− →β0 M′

3 β0 ∗

← − M′

2

Λ′ [projecting]

Corollary (Church-Rosser)

Every term in Λ has the Church-Rosser property.

Yiorgos Stavrinos: Generalized Developments in λ-calculus 15/17

slide-38
SLIDE 38

Outline Generalized developments λ-calculi with types Embedding Finiteness of gen. developments Conclusion

Conclusion “the typed theory is the more general one, and the prior one” Dana Scott, 1980

Yiorgos Stavrinos: Generalized Developments in λ-calculus 16/17

slide-39
SLIDE 39

References

Koletsos G. and Stavrinos Y., Embedding developments into simply typed λ-calculus, “Logical and Semantic Frameworks and Applications” (2008). Koletsos G. and Stavrinos G., Church-Rosser property and intersection types, The Australasian Journal of Logic 6 (2008). Ghilezan S. and Kunˇ cak V., Confluence of Untyped Lambda Calculus via Simple Types, “ICTCS 2001”, Restivo A., Ronchi Della Rocca S., and Roversi L. (eds.), LNCS 2202 (2001). Koletsos G. and Stavrinos G., Church-Rosser theorem for conjunctive type systems, “1st Panhellenic Symposium on Logic”, Kakas A. and Sinachopoulos A. (eds.), University of Cyprus (1997). Ghilezan S., Generalized finiteness of developments in typed lambda calculi, Journal of Automata, Languages and Combinatorics 1 (1996). Krivine J.-L., “Lambda-calcul, types et mod` eles”, Masson (1990). Parigot M., Internal labellings in lambda-calculus, “Mathematical Foundations of Computer Science”, Rovan B. (ed.), LNCS 452 (1990).

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