Sequent Calculus Cut Elimination
Akim Demaille [email protected]
EPITA — École Pour l’Informatique et les Techniques Avancées
June 10, 2016 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 Preamble
The following slides are implicitly dedicated to classical logic.
A. Demaille Sequent Calculus, Cut Elimination 3 / 40 Problems of Natural Deduction
1 Problems of Natural Deduction
2 Sequent Calculus
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 4 / 40 Normalization
A. Demaille Sequent Calculus, Cut Elimination 5 / 40 Proofs hard to find
Some elimination rules used formulas coming out of the blue.
[A] [B] · · · · · · · · · A ∨ B C C ∨E C
A. Demaille Sequent Calculus, Cut Elimination 6 / 40 Negation is awkward
A. Demaille Sequent Calculus, Cut Elimination 7 / 40 Sequent Calculus
1 Problems of Natural Deduction
2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 8 / 40 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).
A. Demaille Sequent Calculus, Cut Elimination 9 / 40 Syntax
1 Problems of Natural Deduction
2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 10 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents
Sequent A sequent is an expression Γ ` ∆, where Γ, ∆ 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 Not for intuitionistic systems
A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Reading a Sequent
If all the formulas of Γ are true, Γ ` ∆ 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”
A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent
If all the formulas of Γ are true, Γ ` ∆ 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”
A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent
If all the formulas of Γ are true, Γ ` ∆ 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”
A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent
If all the formulas of Γ are true, Γ ` ∆ 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”
A. Demaille Sequent Calculus, Cut Elimination 12 / 40 ` AA is true
Γ ` Γ is in contradiction
A ` ¬A
` contradiction
Some Special Sequents
Γ ` AA is true under the hypotheses Γ
A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Γ ` Γ is in contradiction
A ` ¬A
` contradiction
Some Special Sequents
Γ ` AA is true under the hypotheses Γ
` AA is true
A. Demaille Sequent Calculus, Cut Elimination 13 / 40 A ` ¬A
` contradiction
Some Special Sequents
Γ ` AA is true under the hypotheses Γ
` AA is true
Γ ` Γ is in contradiction
A. Demaille Sequent Calculus, Cut Elimination 13 / 40 ` contradiction
Some Special Sequents
Γ ` AA is true under the hypotheses Γ
` AA is true
Γ ` Γ is in contradiction
A ` ¬A
A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Some Special Sequents
Γ ` AA is true under the hypotheses Γ
` AA is true
Γ ` Γ is in contradiction
A ` ¬A
` contradiction
A. Demaille Sequent Calculus, Cut Elimination 13 / 40 LK — Classical Sequent Calculus
1 Problems of Natural Deduction
2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 14 / 40 LK — Classical Sequent Calculus
LK — Gentzen 1934 logistischerklassischer Kalkül.
There are several possible exposures Different sets of inference rules We follow [Girard, 2011]
A. Demaille Sequent Calculus, Cut Elimination 15 / 40 Identity Group
Γ ` A, ∆ Γ0, A ` ∆0 Id Cut A ` A Γ, Γ0 ` ∆, ∆0
Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without.
A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Identity Group
Γ ` A, ∆ Γ0, A ` ∆0 Id Cut A ` A Γ, Γ0 ` ∆, ∆0
Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without.
A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Γ ` ∆ Γ ` ∆ `W W` Γ ` A, ∆ Γ, A ` ∆
Γ ` A, A, ∆ Γ, A, A ` ∆ `C C` Γ ` A, ∆ Γ, A ` ∆
Structural Group
Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆
A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Γ ` A, A, ∆ Γ, A, A ` ∆ `C C` Γ ` A, ∆ Γ, A ` ∆
Structural Group
Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆
Γ ` ∆ Γ ` ∆ `W W` Γ ` A, ∆ Γ, A ` ∆
A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Structural Group
Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆
Γ ` ∆ Γ ` ∆ `W W` Γ ` A, ∆ Γ, A ` ∆
Γ ` A, A, ∆ Γ, A, A ` ∆ `C C` Γ ` A, ∆ Γ, A ` ∆
A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Logical Group: Negation
Γ, A ` ∆ Γ ` A, ∆ `¬ ¬` Γ ` ¬A, ∆ Γ, ¬A ` ∆
A. Demaille Sequent Calculus, Cut Elimination 18 / 40 Additive ˘
Multiplicative ⊗
Γ, A ` ∆ l∧` Γ, A ∧ B ` ∆
Γ, B ` ∆ r∧` Γ, A ∧ B ` ∆
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆
Logical Group: Conjunction
Γ ` A, ∆ Γ ` B, ∆ `∧ Γ ` A ∧ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘
Multiplicative ⊗
Γ, B ` ∆ r∧` Γ, A ∧ B ` ∆
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆
Logical Group: Conjunction
Γ, A ` ∆ l∧` Γ ` A, ∆ Γ ` B, ∆ Γ, A ∧ B ` ∆ `∧ Γ ` A ∧ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Multiplicative ⊗
Additive ˘
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆
Logical Group: Conjunction
Γ, A ` ∆ l∧` Γ ` A, ∆ Γ ` B, ∆ Γ, A ∧ B ` ∆ `∧ Γ ` A ∧ B, ∆ Γ, B ` ∆ r∧` Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘
Multiplicative ⊗
Logical Group: Conjunction
Γ, A ` ∆ l∧` Γ ` A, ∆ Γ ` B, ∆ Γ, A ∧ B ` ∆ `∧ Γ ` A ∧ B, ∆ Γ, B ` ∆ r∧` Γ, A ∧ B ` ∆
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 ˘
⊗
Logical Group: Conjunction
Γ, A ` ∆ l∧` Γ ` A, ∆ Γ ` B, ∆ Γ, A ∧ B ` ∆ Additive `∧ Γ ` A ∧ B, ∆ Γ, B ` ∆ r∧` Γ, A ∧ B ` ∆
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ Multiplicative `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Logical Group: Conjunction
Γ, A ` ∆ l∧` Γ ` A, ∆ Γ ` B, ∆ Γ, A ∧ B ` ∆ Additive `∧ Γ ` A ∧ B, ∆ Γ, B ` ∆ r∧` ˘ Γ, A ∧ B ` ∆
Γ ` A, ∆ Γ0 ` B, ∆0 Γ, A, B ` ∆ Multiplicative `∧ ∧` Γ, Γ0 ` A ∧ B, ∆, ∆0 Γ, A ∧ B ` ∆ ⊗
A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ⊕
Multiplicative ˙
Γ, A ` ∆ Γ, B ` ∆ ∨` Γ ` B, ∆ Γ, A ∨ B ` ∆ `r∨ Γ ` A ∨ B, ∆
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0
Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 Additive ⊕
Multiplicative ˙
Γ, A ` ∆ Γ, B ` ∆ ∨` Γ, A ∨ B ` ∆
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0
Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆
Γ ` B, ∆ `r∨ Γ ` A ∨ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 Multiplicative ˙
Additive ⊕
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0
Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆ Γ, A ` ∆ Γ, B ` ∆ ∨` Γ ` B, ∆ Γ, A ∨ B ` ∆ `r∨ Γ ` A ∨ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 Additive ⊕
Multiplicative ˙
Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆ Γ, A ` ∆ Γ, B ` ∆ ∨` Γ ` B, ∆ Γ, A ∨ B ` ∆ `r∨ Γ ` A ∨ B, ∆
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 ⊕
˙
Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆ Γ, A ` ∆ Γ, B ` ∆ Additive ∨` Γ ` B, ∆ Γ, A ∨ B ` ∆ `r∨ Γ ` A ∨ B, ∆
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 Multiplicative `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 Logical Group: Disjunction
Γ ` A, ∆ `l∨ Γ ` A ∨ B, ∆ Γ, A ` ∆ Γ, B ` ∆ Additive ∨` Γ ` B, ∆ Γ, A ∨ B ` ∆ ⊕ `r∨ Γ ` A ∨ B, ∆
Γ ` A, B, ∆ Γ, A ` ∆ Γ0, B ` ∆0 Multiplicative `∨ ∨` Γ ` A ∨ B, ∆ Γ, Γ0, A ∨ B ` ∆, ∆0 ˙
A. Demaille Sequent Calculus, Cut Elimination 20 / 40 Γ, A ` B, ∆ `⇒ Γ ` A ⇒ B, ∆
Logical Group: Implication
Γ ` ∆, A Γ0, B ` ∆0 ⇒` Γ, Γ0, A ⇒ B ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 21 / 40 Logical Group: Implication
Γ ` ∆, A Γ0, B ` ∆0 Γ, A ` B, ∆ ⇒` `⇒ Γ, Γ0, A ⇒ B ` ∆, ∆0 Γ ` A ⇒ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 21 / 40 Prove A ∧ B ` A ∧ B
A ` A B ` B l∧` r∧` A ∧ B ` A A ∧ B ` B `∧ A ∧ B ` A ∧ B
A. Demaille Sequent Calculus, Cut Elimination 22 / 40 Prove A ∧ B ` A ∧ B
A ` A B ` B l∧` r∧` A ∧ B ` A A ∧ B ` B `∧ A ∧ B ` A ∧ B
A. Demaille Sequent Calculus, Cut Elimination 22 / 40 Prove A ∧ B ` A ∨ B
A ` A l∧` A ∧ B ` A `r∨ A ∧ B ` A ∨ B
A. Demaille Sequent Calculus, Cut Elimination 23 / 40 Prove A ∧ B ` A ∨ B
A ` A l∧` A ∧ B ` A `r∨ A ∧ B ` A ∨ B
A. Demaille Sequent Calculus, Cut Elimination 23 / 40 Prove A ∨ B ` A ∨ B
A ` A B ` B `l∨ `r∨ A ` A ∨ B B ` A ∨ B ∨` A ∨ B ` A ∨ B
A. Demaille Sequent Calculus, Cut Elimination 24 / 40 Prove A ∨ B ` A ∨ B
A ` A B ` B `l∨ `r∨ A ` A ∨ B B ` A ∨ B ∨` A ∨ B ` A ∨ B
A. Demaille Sequent Calculus, Cut Elimination 24 / 40 Γ ` A, ∆ Γ ` B, ∆ Γ ` A, ∆ Γ0 ` B, ∆0 ` ∧× ======W ` ======W ` Γ, Γ ` A ∧ B, ∆, ∆ Γ, Γ0 ` A, ∆ Γ, Γ0 ` B, ∆0 ======` C ======` W ======` W Γ, Γ ` A ∧ B, ∆ Γ, Γ0 ` A, ∆, ∆0 Γ, Γ0 ` B, ∆, ∆0 ======C ` ` ∧+ Γ ` A ∧ B, ∆ Γ, Γ0 ` A ∧ B, ∆, ∆0
Prove the equivalence of the two ∧ rules
Additive Multiplicative
Γ ` A, ∆ Γ ` B, ∆ Γ ` A, ∆ Γ0 ` B, ∆0 ` ∧+ ≡ ` ∧× Γ ` A ∧ B, ∆ Γ, Γ0 ` A ∧ B, ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 25 / 40 Γ ` A, ∆ Γ0 ` B, ∆0 ======W ` ======W ` Γ, Γ0 ` A, ∆ Γ, Γ0 ` B, ∆0 ======` W ======` W Γ, Γ0 ` A, ∆, ∆0 Γ, Γ0 ` B, ∆, ∆0 ` ∧+ Γ, Γ0 ` A ∧ B, ∆, ∆0
Prove the equivalence of the two ∧ rules
Additive Multiplicative
Γ ` A, ∆ Γ ` B, ∆ Γ ` A, ∆ Γ0 ` B, ∆0 ` ∧+ ≡ ` ∧× Γ ` A ∧ B, ∆ Γ, Γ0 ` A ∧ B, ∆, ∆0
Γ ` A, ∆ Γ ` B, ∆ ` ∧× Γ, Γ ` A ∧ B, ∆, ∆ ======` C Γ, Γ ` A ∧ B, ∆ ======C ` Γ ` A ∧ B, ∆
A. Demaille Sequent Calculus, Cut Elimination 25 / 40 Prove the equivalence of the two ∧ rules
Additive Multiplicative
Γ ` A, ∆ Γ ` B, ∆ Γ ` A, ∆ Γ0 ` B, ∆0 ` ∧+ ≡ ` ∧× Γ ` A ∧ B, ∆ Γ, Γ0 ` A ∧ B, ∆, ∆0
Γ ` A, ∆ Γ ` B, ∆ Γ ` A, ∆ Γ0 ` B, ∆0 ` ∧× ======W ` ======W ` Γ, Γ ` A ∧ B, ∆, ∆ Γ, Γ0 ` A, ∆ Γ, Γ0 ` B, ∆0 ======` C ======` W ======` W Γ, Γ ` A ∧ B, ∆ Γ, Γ0 ` A, ∆, ∆0 Γ, Γ0 ` B, ∆, ∆0 ======C ` ` ∧+ Γ ` A ∧ B, ∆ Γ, Γ0 ` A ∧ B, ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 25 / 40 Γ ` A[t/x], ∆ Γ, A ` ∆ ` ∃ ∃ ` Γ ` ∃x · A, ∆ Γ, ∃x · A ` ∆ In ` ∀ and ∃ `, x 6∈ FV(Γ, ∆).
Logical Group: Quantifiers
Γ ` A, ∆ Γ, A[t/x] ` ∆ ` ∀ ∀ ` Γ ` ∀x · A, ∆ Γ, ∀x · A ` ∆
A. Demaille Sequent Calculus, Cut Elimination 26 / 40 Logical Group: Quantifiers
Γ ` A, ∆ Γ, A[t/x] ` ∆ ` ∀ ∀ ` Γ ` ∀x · A, ∆ Γ, ∀x · A ` ∆
Γ ` A[t/x], ∆ Γ, A ` ∆ ` ∃ ∃ ` Γ ` ∃x · A, ∆ Γ, ∃x · A ` ∆ In ` ∀ and ∃ `, x 6∈ FV(Γ, ∆).
A. Demaille Sequent Calculus, Cut Elimination 26 / 40 Single Sided Defining the Negation
Alternatively, one can define the negation as a notation instead of defining it by inference rules.
¬(¬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, ∆
A. Demaille Sequent Calculus, Cut Elimination 27 / 40 Single Sided Defining the Negation
Alternatively, one can define the negation as a notation instead of defining it by inference rules.
¬(¬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, ∆
A. Demaille Sequent Calculus, Cut Elimination 27 / 40 Single Sided Defining the Negation
Alternatively, one can define the negation as a notation instead of defining it by inference rules.
¬(¬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, ∆
A. Demaille Sequent Calculus, Cut Elimination 27 / 40 Single Sided The Full Sequent Calculus
` Γ, A ` ¬A, ∆ Id Cut ` ¬A, A ` Γ, ∆
` Γ ` Γ ` A, A, Γ X W C ` σ(Γ) ` A, Γ ` A, Γ
` A, ∆ ` B, ∆ ` A, ∆ ` B, ∆ l∨ r∨ ∧ ` A ∨ B, ∆ ` A ∨ B, ∆ ` A ∧ B, ∆
` A, ∆ ` A[t/x], ∆ ∀ ∃ ` ∀x · A, ∆ ` ∃x · A, ∆
A. Demaille Sequent Calculus, Cut Elimination 28 / 40 Cut Elimination
1 Problems of Natural Deduction
2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 29 / 40 nothing can be said!
premisses can only use subformulas of the conclusion!
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
Otherwise. . . Γ ` A, ∆ Γ ` B, ∆ Γ, A ` ∆ ` ∧ l∧ ` ··· Γ ` A ∧ B, ∆ Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 30 / 40 premisses can only use subformulas of the conclusion!
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 ` ∆ ` ∧ l∧ ` ··· Γ ` A ∧ B, ∆ Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 30 / 40 premisses can only use subformulas of the conclusion!
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 ` ∆ ` ∧ l∧ ` ··· Γ ` A ∧ B, ∆ Γ, A ∧ B ` ∆
A. Demaille Sequent Calculus, Cut Elimination 30 / 40 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 ` ∆ ` ∧ l∧ ` ··· Γ ` A ∧ B, ∆ Γ, A ∧ B ` ∆
premisses can only use subformulas of the conclusion!
A. Demaille Sequent Calculus, Cut Elimination 30 / 40 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!
A. Demaille Sequent Calculus, Cut Elimination 31 / 40 For all the connectives.
Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
Cut Elimination Logical Rules
Γ ` A, ∆ Γ ` B, ∆ Γ0, A ` ∆0 ` ∧ l∧ ` Γ ` A ∧ B, ∆ Γ0, A ∧ B ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
A. Demaille Sequent Calculus, Cut Elimination 32 / 40 For all the connectives.
Cut Elimination Logical Rules
Γ ` A, ∆ Γ ` B, ∆ Γ0, A ` ∆0 ` ∧ l∧ ` Γ ` A ∧ B, ∆ Γ0, A ∧ B ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 32 / 40 Cut Elimination Logical Rules
Γ ` A, ∆ Γ ` B, ∆ Γ0, A ` ∆0 ` ∧ l∧ ` Γ ` A ∧ B, ∆ Γ0, A ∧ B ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0 For all the connectives.
A. Demaille Sequent Calculus, Cut Elimination 32 / 40 · · · Γ, A ` ∆
Cut Elimination Removal of a Cut
· · Identity · A ` A Γ, A ` ∆ Cut Γ, A ` ∆
;
A. Demaille Sequent Calculus, Cut Elimination 33 / 40 Cut Elimination Removal of a Cut
· · Identity · A ` A Γ, A ` ∆ Cut Γ, A ` ∆
; · · · Γ, A ` ∆
A. Demaille Sequent Calculus, Cut Elimination 33 / 40 Beware of the duplication!
Γ, B ` A, ∆ Γ0, A ` ∆0 Γ, C ` A, ∆ Γ0, A ` ∆0 Cut Cut Γ, B, Γ0 ` ∆, ∆0 Γ, C, Γ0 ` ∆, ∆0 ∨ ` Γ, B ∨ C, Γ0 ` ∆, ∆0
Cut Elimination Commutations
Γ, B ` A, ∆ Γ, C ` A, ∆ ∨ ` Γ, B ∨ C ` A, ∆ Γ0, A ` ∆0 Cut Γ, B ∨ C, Γ0 ` ∆, ∆0
;
A. Demaille Sequent Calculus, Cut Elimination 34 / 40 Beware of the duplication!
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 Γ, C ` A, ∆ Γ0, A ` ∆0 Cut Cut Γ, B, Γ0 ` ∆, ∆0 Γ, C, Γ0 ` ∆, ∆0 ∨ ` Γ, B ∨ C, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 34 / 40 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 Γ, C ` A, ∆ Γ0, A ` ∆0 Cut Cut Γ, B, Γ0 ` ∆, ∆0 Γ, C, Γ0 ` ∆, ∆0 ∨ ` Γ, B ∨ C, Γ0 ` ∆, ∆0 Beware of the duplication!
A. Demaille Sequent Calculus, Cut Elimination 34 / 40 Γ ` ∆ ======W` Γ, Γ0 ` ∆ ======`W Γ, Γ0 ` ∆, ∆0
Cut Elimination Structural Rules: Weakening
Γ ` ∆ `W Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
A. Demaille Sequent Calculus, Cut Elimination 35 / 40 Cut Elimination Structural Rules: Weakening
Γ ` ∆ `W Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
Γ ` ∆ ======W` Γ, Γ0 ` ∆ ======`W Γ, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 35 / 40 but wrong
might loop for ever
Nice!
Γ ` A, A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` A, ∆, ∆0 Γ0, A ` ∆0 Cut Γ, Γ0, Γ0 ` ∆, ∆0, ∆0 ======C Γ, Γ0 ` ∆, ∆0
Cut Elimination Structural Rules: Contraction
Γ ` A, A, ∆ `C Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
A. Demaille Sequent Calculus, Cut Elimination 36 / 40 but wrong
might loop for ever
Nice!
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
A. Demaille Sequent Calculus, Cut Elimination 36 / 40 but wrong
might loop for ever
Cut Elimination Structural Rules: Contraction
Γ ` A, A, ∆ `C Γ ` A, ∆ Γ0, A ` ∆0 Cut Nice! 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
A. Demaille Sequent Calculus, Cut Elimination 36 / 40 might loop for ever
Cut Elimination Structural Rules: Contraction
Γ ` A, A, ∆ `C Γ ` A, ∆ Γ0, A ` ∆0 Cut Nice! Γ, Γ0 ` ∆, ∆0 but ; wrong Γ ` A, A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` A, ∆, ∆0 Γ0, A ` ∆0 Cut Γ, Γ0, Γ0 ` ∆, ∆0, ∆0 ======C Γ, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 36 / 40 Cut Elimination Structural Rules: Contraction
Γ ` A, A, ∆ `C Γ ` A, ∆ Γ0, A ` ∆0 Cut Nice! Γ, Γ0 ` ∆, ∆0 but ; wrong
0 0 Γ ` A, A, ∆ Γ , A ` ∆ might Cut Γ, Γ0 ` A, ∆, ∆0 Γ0, A ` ∆0 loop Cut for ever Γ, Γ0, Γ0 ` ∆, ∆0, ∆0 ======C Γ, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 36 / 40 Γ0, A, A ` ∆0 C` Γ ` A, A, ∆ Γ0, A ` ∆0 Γ0, A, A ` ∆0 Cut C` Γ, Γ0 ` A, ∆, ∆0 Γ0, A ` ∆0 Cut Γ, Γ0, Γ0 ` ∆, ∆0, ∆0 ======C Γ, Γ0 ` ∆, ∆0
Cut Elimination Structural Rules: Complications with Contractions
Γ ` A, A, ∆ Γ0, A, A ` ∆0 `C C` Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
A. Demaille Sequent Calculus, Cut Elimination 37 / 40 Cut Elimination Structural Rules: Complications with Contractions
Γ ` A, A, ∆ Γ0, A, A ` ∆0 `C C` Γ ` A, ∆ Γ0, A ` ∆0 Cut Γ, Γ0 ` ∆, ∆0
;
Γ0, A, A ` ∆0 C` Γ ` A, A, ∆ Γ0, A ` ∆0 Γ0, A, A ` ∆0 Cut C` Γ, Γ0 ` A, ∆, ∆0 Γ0, A ` ∆0 Cut Γ, Γ0, Γ0 ` ∆, ∆0, ∆0 ======C Γ, Γ0 ` ∆, ∆0
A. Demaille Sequent Calculus, Cut Elimination 37 / 40 Natural Deduction in Sequent Calculus
1 Problems of Natural Deduction
2 Sequent Calculus
3 Natural Deduction in Sequent Calculus
A. Demaille Sequent Calculus, Cut Elimination 38 / 40 Recommended readings
[Girard et al., 1989], Chapters 5 & 13 A short (160p.) book addressing all the concerns of this course, and more (especially Linear Logic). Easy and pleasant to read. Available for free.
A. Demaille Sequent Calculus, Cut Elimination 39 / 40 BibliographyI
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.
A. Demaille Sequent Calculus, Cut Elimination 40 / 40