Cut-elimination for Intuitionistic Provability Logic Iris van der - - PowerPoint PPT Presentation

Introduction to provability logic Cut-elimination

Cut-elimination for Intuitionistic Provability Logic

Iris van der Giessen

Utrecht University i.vandergiessen@uu.nl

April 26, 2019

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

PA ⊢ ¬Prov(⊥) → ¬Prov(¬Prov(⊥)) ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

PA ⊢ ¬Prov(⊥) → ¬Prov(¬Prov(⊥)) ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

PA ⊢ ¬Prov(⊥) → ¬Prov(¬Prov(⊥)) ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

GL ⊢ ¬ ⊥ → ¬ (¬ ⊥ ) ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

GL ⊢ ¬⊥ → ¬(¬⊥) ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

GL ⊢ (¬⊥) → ⊥ ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

GL ⊢ (⊥ → ⊥) → ⊥ ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Provability in Peano Arithmetic PA

◮ What can Peano arithmetic say about its own provability? ◮ Provability predicate Prov(A): ‘There exists a G¨

• del number p coding a correct proof from

the axioms of PA of formula A (coded by G¨

• del number A).’

Example (Formal second incompleteness theorem)

GL ⊢ (⊥ → ⊥) → ⊥ ◮ Normal modal logic GL is given by Hilbert system:

◮ Propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

◮ Solovays completeness theorem

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Intuitionistic G¨

• del-L¨
• b logic

◮ Intuitionistic provability logic is given by Hilbert system GLi:

◮ Intuitionistic propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

Fact

Intuitionistic provability logic is not the logic of the provability theory in Heyting Arithmetic!

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Intuitionistic G¨

• del-L¨
• b logic

◮ Intuitionistic provability logic is given by Hilbert system GLi:

◮ Intuitionistic propositional tautologies ◮ K-axiom: (A → B) → A → B ◮ G¨

• del-L¨
• b’s axiom: (A → A) → A

◮ closed under Modus Ponens, Substitution and Necessitation.

Fact

Intuitionistic provability logic is not the logic of the provability theory in Heyting Arithmetic!

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Gentzen style G3ip+GLR

◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | A ◮ Define ¬A = A → ⊥ and ⊡Γ = Γ ∪ Γ

p, Γ ⇒ p, p atomic (At) Γ, ⊥ ⇒ C (L⊥) Γ ⇒ A Γ ⇒ B R∧ Γ ⇒ A ∧ B Γ, A, B ⇒ C L∧ Γ, A ∧ B ⇒ C Γ ⇒ Ai R∨i Γ ⇒ A1 ∨ A2 Γ, A ⇒ C Γ, B ⇒ C L∨ Γ, A ∨ B ⇒ C Γ, A ⇒ B R → Γ ⇒ A → B Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A տ Π can also contain boxed formulas.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Gentzen style G3ip+GLR

◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | A ◮ Define ¬A = A → ⊥ and ⊡Γ = Γ ∪ Γ

p, Γ ⇒ p, p atomic (At) Γ, ⊥ ⇒ C (L⊥) Γ ⇒ A Γ ⇒ B R∧ Γ ⇒ A ∧ B Γ, A, B ⇒ C L∧ Γ, A ∧ B ⇒ C Γ ⇒ Ai R∨i Γ ⇒ A1 ∨ A2 Γ, A ⇒ C Γ, B ⇒ C L∨ Γ, A ∨ B ⇒ C Γ, A ⇒ B R → Γ ⇒ A → B Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A տ Π can also contain boxed formulas.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Gentzen style G3ip+GLR

◮ A := p | ⊥ | A ∨ A | A ∧ A | A → A | A ◮ Define ¬A = A → ⊥ and ⊡Γ = Γ ∪ Γ

p, Γ ⇒ p, p atomic (At) Γ, ⊥ ⇒ C (L⊥) Γ ⇒ A Γ ⇒ B R∧ Γ ⇒ A ∧ B Γ, A, B ⇒ C L∧ Γ, A ∧ B ⇒ C Γ ⇒ Ai R∨i Γ ⇒ A1 ∨ A2 Γ, A ⇒ C Γ, B ⇒ C L∨ Γ, A ∨ B ⇒ C Γ, A ⇒ B R → Γ ⇒ A → B Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A տ Π can also contain boxed formulas.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Cut-elimination in G3ip+GLR

◮ Weakening: Γ ⇒ C Γ, B ⇒ C ◮ Contraction: Γ, B, B ⇒ C Γ, B ⇒ C

Theorem (Cut-elimination)

Let Σl Γ ⇒ D Σr D, ∆ ⇒ C cut(D) Γ, ∆ ⇒ C be a topmost cut. This can be transformed into a cut-free derivation with the same end-sequent. ◮ Subformula property ◮ G3ip+GLR equivalent to intuitionistic Hilbert system GLi

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Cut-elimination in G3ip+GLR

◮ Weakening: Γ ⇒ C Γ, B ⇒ C ◮ Contraction: Γ, B, B ⇒ C Γ, B ⇒ C

Theorem (Cut-elimination)

Let Σl Γ ⇒ D Σr D, ∆ ⇒ C cut(D) Γ, ∆ ⇒ C be a topmost cut. This can be transformed into a cut-free derivation with the same end-sequent. ◮ Subformula property ◮ G3ip+GLR equivalent to intuitionistic Hilbert system GLi

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Cut-elimination in G3ip+GLR

◮ Weakening: Γ ⇒ C Γ, B ⇒ C ◮ Contraction: Γ, B, B ⇒ C Γ, B ⇒ C

Theorem (Cut-elimination)

Let Σl Γ ⇒ D Σr D, ∆ ⇒ C cut(D) Γ, ∆ ⇒ C be a topmost cut. This can be transformed into a cut-free derivation with the same end-sequent. ◮ Subformula property ◮ G3ip+GLR equivalent to intuitionistic Hilbert system GLi

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Standard method

◮ Induction on degree of cut-formula d and cut-height h

Σl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Σl ⊡Γ, B ⇒ B Γ ⇒ B Σl ⊡Γ, B ⇒ B cut Γ, ⊡Γ ⇒ B contr ⊡Γ ⇒ B Σl ⊡Γ, B ⇒ B Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C cut B, Γ, ⊡∆, C ⇒ C cut ⊡Γ, Γ, ⊡∆, C ⇒ C contraction ⊡Γ, ⊡∆, C ⇒ C GLR Πl, Πr, Γ, ∆ ⇒ C

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Standard method

◮ Induction on degree of cut-formula d and cut-height h

Σl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Σl ⊡Γ, B ⇒ B Γ ⇒ B Σl ⊡Γ, B ⇒ B cut Γ, ⊡Γ ⇒ B contr ⊡Γ ⇒ B Σl ⊡Γ, B ⇒ B Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C cut B, Γ, ⊡∆, C ⇒ C cut ⊡Γ, Γ, ⊡∆, C ⇒ C contraction ⊡Γ, ⊡∆, C ⇒ C GLR Πl, Πr, Γ, ∆ ⇒ C

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Starting point

◮ Valentini 1983: Syntactic proof for classical sequent calculus for GL based on sets.

→ explicit weakening rules → no contraction rules

◮ Gor´ e and Ramanayake 2008: Syntactic proof for classical sequent calculus for GL based on multisets.

→ explicit weakening and contraction rules.

Idea

Defining an extra induction parameter, called width w. Induction value becomes (d, w, h). ◮ G3ip+GLR: weakening and contraction as admissible rules.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Starting point

◮ Valentini 1983: Syntactic proof for classical sequent calculus for GL based on sets.

→ explicit weakening rules → no contraction rules

◮ Gor´ e and Ramanayake 2008: Syntactic proof for classical sequent calculus for GL based on multisets.

→ explicit weakening and contraction rules.

Idea

Defining an extra induction parameter, called width w. Induction value becomes (d, w, h). ◮ G3ip+GLR: weakening and contraction as admissible rules.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Starting point

◮ Valentini 1983: Syntactic proof for classical sequent calculus for GL based on sets.

→ explicit weakening rules → no contraction rules

◮ Gor´ e and Ramanayake 2008: Syntactic proof for classical sequent calculus for GL based on multisets.

→ explicit weakening and contraction rules.

Idea

Defining an extra induction parameter, called width w. Induction value becomes (d, w, h). ◮ G3ip+GLR: weakening and contraction as admissible rules.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Starting point

◮ Valentini 1983: Syntactic proof for classical sequent calculus for GL based on sets.

→ explicit weakening rules → no contraction rules

◮ Gor´ e and Ramanayake 2008: Syntactic proof for classical sequent calculus for GL based on multisets.

→ explicit weakening and contraction rules.

Idea

Defining an extra induction parameter, called width w. Induction value becomes (d, w, h). ◮ G3ip+GLR: weakening and contraction as admissible rules.

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Definitions

◮ 2-ary GLR rules ρ over a sequent Γ ⇒ C Σ =

· · · ρ · · · · · · · · · ρ · · · · · · · · · ⊡Γ′′, C · · · ⇒ D ρ Π, Γ′′, · · · ⇒ D · · · ⊡Γ′ · · · C ⇒ C GLR Γ′ · · · ⇒ C Γ ⇒ C

◮ GLR(2, Σ) is the number of 2-ary GLR rules ρ over end-sequent Γ ⇒ C of Σ, such that

• 1. C is the diagonal formula of the 1-ary GLR rule below ρ
• 2. C is not introduced by weakening at ρ

⊡Γ, A ⇒ A GLR Π, Γ ⇒ A

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Definitions

◮ 2-ary GLR rules ρ over a sequent Γ ⇒ C Σ =

· · · ρ · · · · · · · · · ρ · · · · · · · · · ⊡Γ′′, C · · · ⇒ D ρ Π, Γ′′, · · · ⇒ D · · · ⊡Γ′ · · · C ⇒ C GLR Γ′ · · · ⇒ C Γ ⇒ C

◮ GLR(2, Σ) is the number of 2-ary GLR rules ρ over end-sequent Γ ⇒ C of Σ, such that

• 1. C is the diagonal formula of the 1-ary GLR rule below ρ
• 2. C is not introduced by weakening at ρ

⊡Γ, A ⇒ A GLR Π, Γ ⇒ A

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Definitions

◮ 2-ary GLR rules ρ over a sequent Γ ⇒ C Σ =

· · · ρ · · · · · · · · · ρ · · · · · · · · · ⊡Γ′′, ⊡C ⇒ D ρ Π, Γ′′, C ⇒ D · · · ⊡Γ′, C ⇒ C GLR Γ′ ⇒ C Γ ⇒ C

◮ GLR(2, Σ) is the number of 2-ary GLR rules ρ over end-sequent Γ ⇒ C of Σ, such that

• 1. C is the diagonal formula of the 1-ary GLR rule below ρ
• 2. C is not introduced by weakening at ρ

⊡Γ, A ⇒ A GLR Π, Γ ⇒ A

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Definitions

◮ 2-ary GLR rules ρ over a sequent Γ ⇒ C Σ =

· · · ρ · · · · · · · · · ρ · · · · · · · · · ⊡Γ′′, ⊡C ⇒ D ρ Π, Γ′′, C ⇒ D · · · ⊡Γ′, C ⇒ C GLR Γ′ ⇒ C Γ ⇒ C

◮ GLR(2, Σ) is the number of 2-ary GLR rules ρ over end-sequent Γ ⇒ C of Σ, such that

• 1. C is the diagonal formula of the 1-ary GLR rule below ρ
• 2. C is not introduced by weakening at ρ

Definition (Width)

The width of a cut is defined as w(cut) = GLR(2, ‘left premise’).

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Definitions

◮ 2-ary GLR rules ρ over a sequent Γ ⇒ C

· · · ρ · · · · · · · · · ρ · · · · · · · · · ⊡Γ′′, ⊡C ⇒ D ρ Π, Γ′′, C ⇒ D · · · ⊡Γ′, C ⇒ C GLR Γ′ ⇒ C Γ ⇒ C Σr C ⇒ A cut Γ ⇒ A

◮ GLR(2, Σ) is the number of 2-ary GLR rules ρ over end-sequent Γ ⇒ C of Σ, such that

• 1. C is the diagonal formula of the 1-ary GLR rule below ρ
• 2. C is not introduced by weakening at ρ

Definition (Width)

The width of a cut is defined as w(cut) = GLR(2, ‘left premise’).

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Proof idea

Theorem (Cut-elimination)

Let Σl Γ ⇒ D Σr D, ∆ ⇒ C cut(D) Γ, ∆ ⇒ C be a topmost cut. This can be transformed into a cut-free derivation with the same end-sequent. ◮ Induction on (d, w, h)

Σl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Proof idea: w > 0

Σ′

l

⊡Γ′, ⊡B, ⊡Λ, D ⇒ D ρ Π, Γ, B, Λ ⇒ D Θl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr ⊡B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

. . . ⊡Γ ⇒ B Σl ⊡Γ, B ⇒ B Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C cut B, Γ, ⊡∆, C ⇒ C cut ⊡Γ, Γ, ⊡∆, C ⇒ C contraction ⊡Γ, ⊡∆, C ⇒ C GLR Πl, Πr, Γ, ∆ ⇒ C

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Proof idea: w > 0

Σ′

l

⊡Γ′, ⊡B, ⊡Λ, D ⇒ D ρ Π, Γ, B, Λ ⇒ D Θl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr ⊡B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

. . . ⊡Γ ⇒ B Σl ⊡Γ, B ⇒ B Γ ⇒ B Σr B, B, ⊡∆, C ⇒ C cut B, Γ, ⊡∆, C ⇒ C cut ⊡Γ, Γ, ⊡∆, C ⇒ C contraction ⊡Γ, ⊡∆, C ⇒ C GLR Πl, Πr, Γ, ∆ ⇒ C

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Proof idea: w > 0

Σ′

l

⊡Γ′, ⊡B, ⊡Λ, D ⇒ D ρ Π, Γ, B, Λ ⇒ D Θl ⊡Γ, B ⇒ B GLR Πl, Γ ⇒ B Σr ⊡B, ⊡∆, C ⇒ C GLR Πr, B, ∆ ⇒ C cut(B) Πl, Πr, Γ, ∆ ⇒ C

• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Σ D, ⊡Γ, ⊡Λ ⇒ D ρ′ Π, B, Γ, Λ ⇒ D Θl ⊡Γ, B ⇒ B GLR Γ ⇒ B Σl ⊡Γ, B ⇒ B cut4 Γ, ⊡Γ ⇒ B contraction ⊡Γ ⇒ B

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Towards termination

◮ G3ip+GLR ⊢ ( ⇒ C) if and only if GLi ⊢ C ◮ Terminating calculus G4ip+GLR. Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A is replaced by 5 left implication rules Π does not contain any boxed formulas. ◮ G4ip+GLR ⊢ (Γ ⇒ C) if and only if G3ip+GLR ⊢ (Γ ⇒ C)

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Towards termination

◮ G3ip+GLR ⊢ ( ⇒ C) if and only if GLi ⊢ C ◮ Terminating calculus G4ip+GLR. Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A is replaced by 5 left implication rules Π does not contain any boxed formulas. ◮ G4ip+GLR ⊢ (Γ ⇒ C) if and only if G3ip+GLR ⊢ (Γ ⇒ C)

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Towards termination

◮ G3ip+GLR ⊢ ( ⇒ C) if and only if GLi ⊢ C ◮ Terminating calculus G4ip+GLR. Γ, A → B ⇒ A Γ, B ⇒ C L → Γ, A → B ⇒ C ⊡Γ, A ⇒ A GLR Π, Γ ⇒ A is replaced by 5 left implication rules Π does not contain any boxed formulas. ◮ G4ip+GLR ⊢ (Γ ⇒ C) if and only if G3ip+GLR ⊢ (Γ ⇒ C)

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic

Introduction to provability logic Cut-elimination

Summary

◮ Intuitionistic provability logic ◮ Syntactic proof of cut-elimination for G3ip+GLR.

◮ Introduce third parameter width (Valentini)

◮ Semantic proof of cut-elimination for G4ip+GLR. GLi

G3ip+GLR G4ip+GLR

Iris van der Giessen Cut-elimination for Intuitionistic Provability Logic