Sequent Calculus Cut Elimination

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Sequent Calculus Cut Elimination

Akim Demailleakim@lrde.epita.fr

EPITA — École Pour l’Informatique et les Techniques Avancées

June 10, 2016

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Sequent Calculus Cut Elimination

1 Problems of Natural Deduction

2 Sequent Calculus

3 Natural Deduction in Sequent Calculus

A. Demaille Sequent Calculus, Cut Elimination 2 / 40

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Preamble

The following slides are implicitly dedicated to classicallogic.

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Problems of Natural Deduction

2 Sequent Calculus

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Normalization

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Proofs hard to find

Some elimination rules used formulas coming out of the blue.

··

· A∨B

[A]·

·· C

[B]·

·· C ∨E C

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Negation is awkward

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Sequent Calculus

2 Sequent Calculus Syntax

LK — Classical Sequent Calculus Cut Elimination

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Gerhard Karl Erich Gentzen (1909–1945)

German logician and mathematician.

SA (1933)

An assistant of David Hilbert in Göttingen (1935–1939).

Joined the Nazi Party (1937).

Worked on the V2.

Died of malnutrition in a prison camp (August 4, 1945).

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Syntax

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Sequents

Sequent

Asequentis an expression Γ`∆, whereΓ,∆are (finite) sequences of formulas.

Variants:

Sets Γ,∆can be taken as (finite) sets.

Simplifies the structural group

Prevents close examination of. . . the structural rules Single Sided Sequents can be forced to be `Γ

Half the number of rules is needed

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Sequents

Sequent

Variants:

Half the number of rules is needed Not for intuitionistic systems

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Sequents

Sequent

Variants:

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Sequents

Sequent

Variants:

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Sequents

Sequent

Variants:

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Sequents

Sequent

Variants:

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Sequents

Sequent

Variants:

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Sequents

Sequent

Variants:

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Reading a Sequent

Γ`∆ If all the formulas of Γare true, then one of the formulas of∆is true.

Commas on the left hand side stand for “and”

Turnstile,`, stands for “implies”

Commas on the right hand side stand for “or”

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Reading a Sequent

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Reading a Sequent

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Reading a Sequent

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Some Special Sequents

Γ`A Ais true under the hypotheses Γ

`A Ais true

Γ` Γis in contradiction A` ¬A

` contradiction

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Some Special Sequents

`A Ais true

` contradiction

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Some Special Sequents

`A Ais true

Γ` Γis in contradiction

A` ¬A

` contradiction

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Some Special Sequents

`A Ais true

` contradiction

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Some Special Sequents

`A Ais true

` contradiction

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LK — Classical Sequent Calculus

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LK — Classical Sequent Calculus

LK — Gentzen 1934

logistischerklassischer Kalkül.

There are several possible exposures Different sets of inference rules We follow [Girard, 2011]

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Identity Group

Id A`A

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

Gentzen’s Hauptsatz

Thecut rule is redundant, i.e., any sequent provable with cuts is provable without.

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Identity Group

Id A`A

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

Gentzen’s Hauptsatz

Thecut rule is redundant, i.e., any sequent provable with cuts is provable without.

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Structural Group

Γ`∆

`X Γ`τ(∆)

Γ`∆ X` σ(Γ)`∆

Γ`∆

`W Γ`A,∆

Γ`∆ W` Γ,A`∆ Γ`A,A,∆

`C Γ`A,∆

Γ,A,A`∆ C` Γ,A`∆

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Structural Group

Γ`∆

`X Γ`τ(∆)

Γ`∆ X` σ(Γ)`∆ Γ`∆

`W Γ`A,∆

Γ`∆ W`

Γ,A`∆

Γ`A,A,∆

`C Γ`A,∆

Γ,A,A`∆ C` Γ,A`∆

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Structural Group

Γ`∆

`X Γ`τ(∆)

Γ`∆ X` σ(Γ)`∆ Γ`∆

`W Γ`A,∆

Γ`∆ W`

Γ,A`∆ Γ`A,A,∆

`C Γ`A,∆

Γ,A,A`∆ C` Γ,A`∆

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Logical Group: Negation

Γ,A`∆

` ¬ Γ` ¬A,∆

Γ`A,∆

¬ ` Γ,¬A`∆

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B `∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,Γ⁰`A∧B,∆,∆⁰

Γ,A,B `∆

∧ ` Γ,A∧B`∆

Multiplicative

⊗

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B`∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,Γ⁰`A∧B,∆,∆⁰

Γ,A,B `∆

∧ ` Γ,A∧B`∆

Multiplicative

⊗

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B`∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,Γ⁰`A∧B,∆,∆⁰

Γ,A,B `∆

∧ ` Γ,A∧B`∆

Multiplicative

⊗

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B`∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,A,B `∆

∧ `

Multiplicative

⊗

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B`∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,Γ⁰`A∧B,∆,∆⁰

Γ,A,B `∆

∧ ` Γ,A∧B`∆

Multiplicative

⊗

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Logical Group: Conjunction

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ ` Γ,A∧B`∆

Γ,B `∆ r∧ ` Γ,A∧B `∆

Additive

˘

Γ`A,∆ Γ⁰`B,∆⁰

` ∧ Γ,A,B `∆

∧ ` Multiplicative

⊗

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨ Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ`A∨B,∆

Γ,A`∆ Γ⁰,B `∆⁰

∨ ` Γ,Γ⁰,A∨B`∆,∆⁰

Multiplicative

˙

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨

Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ`A∨B,∆

Γ,A`∆ Γ⁰,B `∆⁰

∨ ` Γ,Γ⁰,A∨B`∆,∆⁰

Multiplicative

˙

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨

Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ`A∨B,∆

Γ,A`∆ Γ⁰,B `∆⁰

∨ ` Γ,Γ⁰,A∨B`∆,∆⁰

Multiplicative

˙

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨

Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ,A`∆ Γ⁰,B `∆⁰

∨ `

Multiplicative

˙

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨

Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ`A∨B,∆

Γ,A`∆ Γ⁰,B `∆⁰

∨ ` Γ,Γ⁰,A∨B`∆,∆⁰

Multiplicative

˙

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Logical Group: Disjunction

Γ`A,∆

`l∨ Γ`A∨B,∆

Γ`B,∆

`r∨

Γ`A∨B,∆

Γ,A`∆ Γ,B`∆

∨ ` Γ,A∨B `∆

Additive

⊕

Γ`A,B,∆

` ∨ Γ,A`∆ Γ⁰,B `∆⁰

∨ `

Multiplicative

˙

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Logical Group: Implication

Γ`∆,A Γ⁰,B `∆⁰

⇒`

Γ,Γ⁰,A⇒B `∆,∆⁰

Γ,A`B,∆

`⇒ Γ`A⇒B,∆

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Logical Group: Implication

Γ`∆,A Γ⁰,B `∆⁰

⇒`

Γ,Γ⁰,A⇒B `∆,∆⁰

Γ,A`B,∆

`⇒

Γ`A⇒B,∆

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Prove A ∧ B ` A ∧ B

A`A l∧ ` A∧B `A

B `B

r∧ ` A∧B `B

` ∧ A∧B `A∧B

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Prove A ∧ B ` A ∧ B

A`A l∧ ` A∧B `A

B `B

r∧ ` A∧B `B

` ∧ A∧B `A∧B

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Prove A ∧ B ` A ∨ B

A`A l∧ ` A∧B `A

`r∨ A∧B `A∨B

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Prove A ∧ B ` A ∨ B

A`A l∧ ` A∧B `A

`r∨ A∧B `A∨B

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Prove A ∨ B ` A ∨ B

A`A

`l∨

A`A∨B

B `B

`r∨ B `A∨B

∨ ` A∨B `A∨B

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Prove A ∨ B ` A ∨ B

A`A

`l∨

A`A∨B

B `B

`r∨ B `A∨B

∨ ` A∨B `A∨B

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Prove the equivalence of the two ∧ rules

Additive Multiplicative Γ`A,∆ Γ`B,∆

` ∧+

Γ`A∧B,∆

≡ Γ`A,∆ Γ⁰ `B,∆⁰

` ∧×

Γ,Γ⁰ `A∧B,∆,∆⁰

Γ`A,∆ Γ`B,∆

` ∧× Γ,Γ`A∧B,∆,∆

==============`C Γ,Γ`A∧B,∆

============C` Γ`A∧B,∆

Γ`A,∆

=========W` Γ,Γ⁰`A,∆

============`W Γ,Γ⁰`A,∆,∆⁰

Γ⁰`B,∆⁰

=========W` Γ,Γ⁰`B,∆⁰

============`W Γ,Γ⁰`B,∆,∆⁰

` ∧+ Γ,Γ⁰`A∧B,∆,∆⁰

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Prove the equivalence of the two ∧ rules

` ∧+

Γ`A∧B,∆

≡ Γ`A,∆ Γ⁰ `B,∆⁰

` ∧×

Γ,Γ⁰ `A∧B,∆,∆⁰

Γ`A,∆ Γ`B,∆

` ∧×

Γ,Γ`A∧B,∆,∆

==============`C Γ,Γ`A∧B,∆

Γ`A,∆

=========W` Γ,Γ⁰`A,∆

============`W Γ,Γ⁰`A,∆,∆⁰

Γ⁰`B,∆⁰

=========W` Γ,Γ⁰`B,∆⁰

============`W Γ,Γ⁰`B,∆,∆⁰

` ∧+ Γ,Γ⁰`A∧B,∆,∆⁰

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Prove the equivalence of the two ∧ rules

` ∧+

Γ`A∧B,∆

≡ Γ`A,∆ Γ⁰ `B,∆⁰

` ∧×

Γ,Γ⁰ `A∧B,∆,∆⁰

Γ`A,∆ Γ`B,∆

` ∧×

Γ,Γ`A∧B,∆,∆

==============`C Γ,Γ`A∧B,∆

============C` Γ`A∧B,∆

Γ`A,∆

=========W` Γ,Γ⁰`A,∆

============`W Γ,Γ⁰`A,∆,∆⁰

Γ⁰`B,∆⁰

=========W` Γ,Γ⁰`B,∆⁰

============`W Γ,Γ⁰`B,∆,∆⁰

` ∧+

Γ,Γ⁰`A∧B,∆,∆⁰

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Logical Group: Quantifiers

Γ`A,∆

` ∀ Γ` ∀x·A,∆

Γ,A[t/x]`∆

∀ ` Γ,∀x·A`∆

Γ`A[t/x],∆

` ∃ Γ` ∃x·A,∆

Γ,A`∆

∃ ` Γ,∃x·A`∆ In ` ∀and∃ `,x 6∈FV(Γ,∆).

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Logical Group: Quantifiers

Γ`A,∆

` ∀ Γ` ∀x·A,∆

Γ,A[t/x]`∆

∀ ` Γ,∀x·A`∆ Γ`A[t/x],∆

` ∃ Γ` ∃x·A,∆

Γ,A`∆

∃ ` Γ,∃x·A`∆ In` ∀ and∃ `,x 6∈FV(Γ,∆).

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Single Sided

Defining the Negation

Alternatively, one can define the negation as a notationinstead of defining it by inference rules.

¬(¬p) := p

¬(A∧B) := ¬A∨ ¬B

¬(A∨B) := ¬A∧ ¬B

¬(∀x·A) := ∃x· ¬A

¬(∃x·A) := ∀x· ¬A

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Single Sided

¬(¬p) := p

¬(A∧B) := ¬A∨ ¬B

¬(A∨B) := ¬A∧ ¬B

¬(∀x·A) := ∃x· ¬A

¬(∃x·A) := ∀x· ¬A

Then define the sequents as`Γ I.e., Γ`∆;` ¬Γ,∆

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Single Sided

¬(¬p) := p

¬(A∧B) := ¬A∨ ¬B

¬(A∨B) := ¬A∧ ¬B

¬(∀x·A) := ∃x· ¬A

¬(∃x·A) := ∀x· ¬A

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Single Sided

The Full Sequent Calculus

` ¬A,AId

`Γ,A ` ¬A,∆ Cut

`Γ,∆

`Γ

`σ(Γ)X

`Γ W

`A,Γ

`A,A,Γ C

`A,Γ

`A,∆

`A∨B,∆l∨

`B,∆

`A∨B,∆r∨

`A,∆ `B,∆

`A∧B,∆ ∧

`A,∆

` ∀x·A,∆∀

`A[t/x],∆

∃

` ∃x·A,∆

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Cut Elimination

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Subformula Property

What can be told from the last rule of a proof?

If the last rule is a cut,

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

nothing can be said!

Otherwise. . .

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ `

Γ,A∧B`∆ · · ·

premisses can only usesubformulas of the conclusion!

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Subformula Property

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

Otherwise. . .

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ `

Γ,A∧B`∆ · · ·

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Subformula Property

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

Otherwise. . .

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ `

Γ,A∧B`∆ · · ·

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Subformula Property

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

Otherwise. . .

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ,A`∆ l∧ `

Γ,A∧B`∆ · · ·

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Cut Elimination

replace “complex” cuts by simpler cuts (smaller formulas) until the cut is on the simplest form, the identity

where it is not longer needed!

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Cut Elimination

Logical Rules

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ⁰,A`∆⁰

l∧ ` Γ⁰,A∧B `∆⁰

Cut Γ,Γ⁰ `∆,∆⁰

;

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰`∆,∆⁰

For all the connectives.

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Cut Elimination

Logical Rules

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ⁰,A`∆⁰

l∧ ` Γ⁰,A∧B `∆⁰

Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,∆ Γ⁰,A`∆⁰

Cut Γ,Γ⁰`∆,∆⁰

For all the connectives.

(73)

Cut Elimination

Logical Rules

Γ`A,∆ Γ`B,∆

` ∧ Γ`A∧B,∆

Γ⁰,A`∆⁰

l∧ ` Γ⁰,A∧B `∆⁰

Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,∆ Γ⁰,A`∆⁰

Cut Γ,Γ⁰`∆,∆⁰

(74)

Cut Elimination

Removal of a Cut

Identity A`A

·· Γ,A·`∆

Cut Γ,A`∆

;

··

· Γ,A`∆

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Cut Elimination

Removal of a Cut

Identity A`A

·· Γ,A·`∆

Cut Γ,A`∆

;

··

· Γ,A`∆

(76)

Cut Elimination

Commutations

Γ,B `A,∆ Γ,C `A,∆

∨ `

Γ,B∨C `A,∆ Γ⁰,A`∆⁰ Cut Γ,B∨C,Γ⁰ `∆,∆⁰

;

Γ,B `A,∆ Γ⁰,A`∆⁰ Cut Γ,B,Γ⁰ `∆,∆⁰

Γ,C `A,∆ Γ⁰,A`∆⁰ Cut Γ,C,Γ⁰`∆,∆⁰

∨ ` Γ,B∨C,Γ⁰`∆,∆⁰

Beware of the duplication!

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Cut Elimination

Commutations

Γ,B `A,∆ Γ,C `A,∆

∨ `

Γ,B∨C `A,∆ Γ⁰,A`∆⁰ Cut Γ,B∨C,Γ⁰ `∆,∆⁰

; Γ,B `A,∆ Γ⁰,A`∆⁰

Cut Γ,B,Γ⁰ `∆,∆⁰

Γ,C `A,∆ Γ⁰,A`∆⁰ Cut Γ,C,Γ⁰`∆,∆⁰

∨ `

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Cut Elimination

Commutations

Γ,B `A,∆ Γ,C `A,∆

∨ `

Γ,B∨C `A,∆ Γ⁰,A`∆⁰ Cut Γ,B∨C,Γ⁰ `∆,∆⁰

; Γ,B `A,∆ Γ⁰,A`∆⁰

Cut Γ,B,Γ⁰ `∆,∆⁰

Γ,C `A,∆ Γ⁰,A`∆⁰ Cut Γ,C,Γ⁰`∆,∆⁰

∨ ` Γ,B∨C,Γ⁰`∆,∆⁰

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Cut Elimination

Structural Rules: Weakening

Γ`∆

`W

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰`∆,∆⁰

;

Γ`∆

=======W` Γ,Γ⁰ `∆

=========`W Γ,Γ⁰ `∆,∆⁰

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Cut Elimination

Structural Rules: Weakening

Γ`∆

`W

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰`∆,∆⁰

; Γ`∆

=======W` Γ,Γ⁰ `∆

=========`W Γ,Γ⁰ `∆,∆⁰

(81)

Cut Elimination

Structural Rules: Contraction

Γ`A,A,∆

`C

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

;

Γ`A,A,∆ Γ⁰,A`∆⁰ Cut

Γ,Γ⁰ `A,∆,∆⁰ Γ⁰,A`∆⁰ Cut Γ,Γ⁰,Γ⁰`∆,∆⁰,∆⁰

============== C Γ,Γ⁰ `∆,∆⁰

Nice!

but wrong might loop for ever

(82)

Cut Elimination

Γ`A,A,∆

`C

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,A,∆ Γ⁰,A`∆⁰

Cut

Γ,Γ⁰ `A,∆,∆⁰ Γ⁰,A`∆⁰ Cut Γ,Γ⁰,Γ⁰`∆,∆⁰,∆⁰

============== C Γ,Γ⁰ `∆,∆⁰

Nice!

(83)

Cut Elimination

Γ`A,A,∆

`C

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,A,∆ Γ⁰,A`∆⁰

Cut

Γ,Γ⁰ `A,∆,∆⁰ Γ⁰,A`∆⁰ Cut Γ,Γ⁰,Γ⁰`∆,∆⁰,∆⁰

Nice!

(84)

Cut Elimination

Γ`A,A,∆

`C

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,A,∆ Γ⁰,A`∆⁰

Cut

Γ,Γ⁰ `A,∆,∆⁰ Γ⁰,A`∆⁰ Cut Γ,Γ⁰,Γ⁰`∆,∆⁰,∆⁰

============== C Γ,Γ⁰ `∆,∆⁰

Nice!

but wrong

might loop for ever

(85)

Cut Elimination

Γ`A,A,∆

`C

Γ`A,∆ Γ⁰,A`∆⁰ Cut Γ,Γ⁰ `∆,∆⁰

; Γ`A,A,∆ Γ⁰,A`∆⁰

Cut

Γ,Γ⁰ `A,∆,∆⁰ Γ⁰,A`∆⁰ Cut Γ,Γ⁰,Γ⁰`∆,∆⁰,∆⁰

Nice!

(86)

Cut Elimination

Structural Rules: Complications with Contractions

Γ`A,A,∆

`C Γ`A,∆

Γ⁰,A,A`∆⁰ C` Γ⁰,A`∆⁰

Cut Γ,Γ⁰ `∆,∆⁰

;

Γ`A,A,∆

Γ⁰,A,A`∆⁰ C` Γ⁰,A`∆⁰

Cut Γ,Γ⁰ `A,∆,∆⁰

Γ⁰,A,A`∆⁰ C` Γ⁰,A`∆⁰

Cut Γ,Γ⁰,Γ⁰ `∆,∆⁰,∆⁰

==============C Γ,Γ⁰ `∆,∆⁰

(87)

Cut Elimination

Structural Rules: Complications with Contractions

Γ`A,A,∆

`C Γ`A,∆

Γ⁰,A,A`∆⁰ C` Γ⁰,A`∆⁰

Cut Γ,Γ⁰ `∆,∆⁰

;

Γ`A,A,∆

Γ⁰,A,A`∆⁰ C` Γ⁰,A`∆⁰

Cut Γ,Γ⁰ `A,∆,∆⁰

Γ⁰,A,A`∆⁰ C`

Γ⁰,A`∆⁰

(88)

Natural Deduction in Sequent Calculus

2 Sequent Calculus

(89)

Bibliography I

Girard, J.-Y. (2011).

The Blind Spot: Lectures on Logic.

European Mathematical Society.

https://books.google.fr/books?id=eVZZ0wtazo8C.

Girard, J.-Y., Lafont, Y., and Taylor, P. (1989).

Proofs and Types.

Cambridge University Press.

http://www.cs.man.ac.uk/~pt/stable/Proofs+Types.html.