Satallax, possibly higher-order set theoryarg.ciirc.cvut.cz/slides/2015-cebrownpragueslides.pdf ·...

Satallax, possiblyhigher-order set

theory

Brown

Before theIntroduction

Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax, possibly higher-order set theory

Chad E. Brown

November 2015


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Outline

Before the Introduction

Introduction

Higher Order Tableau

Higher-Order Example

The Other Levels


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Regarding Formalization of Mathematics

Library (Theorems, Definitions)

Foundation (HO Tarski Grothendieck Set Theory)

Small Proof Checker (ensuring correctness)

Naming by Hashing (ensuring originality)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels






Developments (Documents/Articles/Theories)

Interactive Proof ConstructionImport from the Library

Populate the Library


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels









Automation

Fill Gaps in Interactive Proofs


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels









Automation


Satallax


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels









Automation


Satallax

Egal

Egal


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Outline


Introduction



The Other Levels


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax

◮ Automated theorem prover for

extensional higher-order logic with choice

◮ Instantiation Based

◮ Open Source

Alt version already: Satallax-MaLeS (Kuhlwein)

◮ First and/or Second in THF Division of CASC since2011.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax

◮ Objective Caml

◮ Foreign Function Interface to MiniSat (Een, Sorensson2003) (C++)

◮ Incremental Use of MiniSat

◮ Optional use of first-order prover E


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax: The General Idea

◮ Goal: Show a set of assumptions A is unsatisfiable.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



◮ Initialize: Create a state ΣA.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




◮ Search: ΣA → Σ1 → Σ2 → · · ·


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




◮ Search: ΣA → Σ1 → Σ2 → · · ·

◮ Associate states with sets of propositional clauses:

CΣA ⊆ C

Σ1 ⊆ CΣ2 ⊆ · · ·


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




◮ Search: ΣA → Σ1 → Σ2 → · · ·


CΣA ⊆ C

Σ1 ⊆ CΣ2 ⊆ · · ·

◮ Soundness: CΣn propositionally unsat. ⇒ A is unsat.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




◮ Search: ΣA → Σ1 → Σ2 → · · ·


CΣA ⊆ C

Σ1 ⊆ CΣ2 ⊆ · · ·

◮ Soundness: CΣn propositionally unsat. ⇒ A is unsat.

◮ Completeness: A is unsat. ⇒ a fair search leads tosuch a propositional unsatisisfiable state Σn.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax: Theoretical Basis

◮ Directed, Cut-Free, Ground Tableau System for HOL(Brown, Smolka [LMCS 2010])

◮ Extended to include Choice (Backes, Brown[2010-2011])

◮ Restricted Instantiations

◮ “Satisfiability” = Henkin Satisfiability (Sparse Function

Spaces)

◮ Tableau Refutability Sound and Complete


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Simple Example

Assume ∀x .x = b and prove (¬pb → pa) → pb.

Refute a “branch” with two formulas:

◮ ∀x .x = b

◮ ¬((¬pb → pa) → pb).


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Simple Example: Tableau Refutation

∀x .x = b

¬((¬pb → pa) → pb)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b

a = b

⊥


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Simple Example: Satallax Refutation

◮ 1 ∀x .x = b

◮ 2 (¬pb → pa) → pb

MiniSat Clauses

[1][−2]


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


◮ 1 ∀x .x = b

◮ 2 (¬pb → pa) → pb

◮ 3 ¬pb → pa

◮ 4 pb

MiniSat Clauses

[1][−2][2 ⊔ 3][2 ⊔ −4]


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


◮ 1 ∀x .x = b

◮ 2 (¬pb → pa) → pb

◮ 3 ¬pb → pa

◮ 4 pb

◮ 5 pa

MiniSat Clauses

[1][−2][2 ⊔ 3][2 ⊔ −4]

[−3 ⊔ 4 ⊔ 5]


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


◮ 1 ∀x .x = b

◮ 2 (¬pb → pa) → pb

◮ 3 ¬pb → pa

◮ 4 pb

◮ 5 pa

◮ 6 a = b

MiniSat Clauses

[1][−2][2 ⊔ 3][2 ⊔ −4]

[−3 ⊔ 4 ⊔ 5][4 ⊔ −5 ⊔ −6]


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


◮ 1 ∀x .x = b

◮ 2 (¬pb → pa) → pb

◮ 3 ¬pb → pa

◮ 4 pb

◮ 5 pa

◮ 6 a = b

◮ Unsatisfiable

MiniSat Clauses

[1][−2][2 ⊔ 3][2 ⊔ −4]

[−3 ⊔ 4 ⊔ 5][4 ⊔ −5 ⊔ −6]

[−1 ⊔ 6]


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Outline


Introduction



The Other Levels


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Branches

We write [s] for the βη-normal form of s.A branch is a finite set of normal formulas (conjunctive)

B = {s1, . . . , sn}

◮ A branch B is closed if ⊥ ∈ B .

◮ A branch B is open if it is not closed.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


Brown, Smolka [LMCS 2010]Tableau calculus inductively defines the set of refutablebranches.Complete for Henkin models without Choice


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


Brown, Smolka [LMCS 2010]A few unsurprising rules...

T¬s, ¬s

⊥T 6=

s 6=ι s

⊥T→

s → t

¬s | t

T¬→

¬(s → t)

s,¬t

T∀∀αs

[st]t : α T¬∀

¬∀αs

¬[sx ]x ∈ Vα fresh


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


Brown, Smolka [LMCS 2010]Mating and decomposition...

Tmatδs , ¬δt

s 6= tTdec

δs 6=ι δt

s 6= t

δ a variable (also, for arity > 1)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


Brown, Smolka [LMCS 2010]...and more rules...extensionality, equality

Tcons =ι t , u 6=ι v

s 6= u , t 6= u | s 6= v , t 6= vTbe

s 6=o t

s , ¬t | ¬s , t

Tbqs =o t

s , t | ¬s , ¬t

Tfes 6=αβ t

¬[∀x .sx = tx ]x /∈ Vs ∪ Vt

Tfqs =αβ t

[∀x .sx = tx ]x /∈ Vs ∪ Vt


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Restrictions on Instantiations

T∀∀αs

[st]t ∈ U

α

Restrict instantiations in the T∀ rule based to Uα on thetype:


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


T∀∀αs

[st]t ∈ U

α


Uι Only terms s occurring as s 6= t or t 6= s on branch.If there are none, use some arbitrary “default” term oftype ιFinitely many!


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


T∀∀αs

[st]t ∈ U

α



Uo Only ⊥ and ¬⊥ (false and true)Finitely many!


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


T∀∀αs

[st]t ∈ U

α



Uo Only ⊥ and ¬⊥ (false and true)Finitely many!

Uαβ Only normal terms using variables free on the branchInfinitely many, of course.

If only quantifiers at o and ι, the procedure sometimesterminates.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Simple Example Revisited

∀x .x = b

¬((¬pb → pa) → pb)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

Rule used:

T¬→

¬(s → t)

s,¬t


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pbpa

Rule used:

T→s → t

¬s | t


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

Left branch is closed.

T¬s, ¬s

⊥


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

Rule used:

Tmatδs , ¬δt

s 6= t


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b

Rule used:

Tmatδs , ¬δt

s 6= t


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b

Note a, b ∈ Uι since a 6= b is on the branch.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b

a = b

Note a, b ∈ Uι since a 6= b is on the branch.Rule used:

T∀∀αs

[st]t ∈ U

α


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


∀x .x = b

¬((¬pb → pa) → pb)¬pb → pa

¬pb

¬¬pb⊥

pa

a 6= b

a = b

⊥

Right branch is closed.

T¬s, ¬s

⊥

Refutation complete


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Adding Choice Operators

New Logical Constants:

◮ For each type α, εα has type (αo)α.

◮ εαp is an element such that p(εαp) if such an elementexists.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Higher Order Tableau with ChoiceExtended to include Choice (Backes, Brown [2010-2011])

◮ Extend Mating and Decomposition to allow for ε.

Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t

(also for arity > 1)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t

(also for arity > 1)◮ Add (restricted) choice rule.

Tε[∀x .¬(sx)] | [s(εs)]

εs accessible, x /∈ Vs

When is εαs accessible? Depends on α:


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t


Tε[∀x .¬(sx)] | [s(εs)]



ι: εs 6=ι t or t 6=ι εs on the branch


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t


Tε[∀x .¬(sx)] | [s(εs)]




α1 · · ·αnι: (εs)u1 · · · un 6=ι t on the branch


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t


Tε[∀x .¬(sx)] | [s(εs)]





o: εs or ¬εs on the branch


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels



Tmatεs , ¬εt

s 6= tTdec

εs 6=ι εt

s 6= t


Tε[∀x .¬(sx)] | [s(εs)]





o: εs or ¬εs on the branch

α1 · · ·αno: (εs)u1 · · · un or its negation on the branch


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Outline


Introduction



The Other Levels


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Epsilon Induction

Axiom of ∈-Induction (Foundation):∀P .(∀x .(∀y .y ∈ x → Py) → Px) → ∀x .PxTheorem: ∀x .x 6∈ x

Proof:

Use λx .x 6∈ x for P in ∈-Induction.

Prove ∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x .Assume IH: ∀y .y ∈ x → y 6∈ y . Prove x 6∈ x .

Assume x ∈ x .

Use x for y : x ∈ x → x 6∈ x .


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Epsilon Induction

Axiom of ∈-Induction (Foundation):∀P .(∀x .(∀y .y ∈ x → Py) → Px) → ∀x .PxTheorem: ∀x .x 6∈ x

Proof:

Use λx .x 6∈ x for P in ∈-Induction.

Prove ∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x .Assume IH: ∀y .y ∈ x → y 6∈ y . Prove x 6∈ x .

Assume x ∈ x .

Use x for y : x ∈ x → x 6∈ x . Contradiction.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Tableau Refutation of Example

¬∀x .x 6∈ x

∀P .(∀x .(∀y .y ∈ x → Py) → Px) → ∀x .PxP := λx .x 6∈ x

(∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x) → ∀x .x 6∈ x

¬(∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x)¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)

∀y .y ∈ a → y 6∈ y

a ∈ a

y := a

a ∈ a → a 6∈ a

a 6∈ a a 6∈ a

∀x .x 6∈ x

⊥


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Tableau Refutation of Example

¬∀x .x 6∈ x

∀P .(∀x .(∀y .y ∈ x → Py) → Px) → ∀x .PxP := λx .x 6∈ x

(∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x) → ∀x .x 6∈ x

¬(∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x)¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)

∀y .y ∈ a → y 6∈ y

a ∈ a

y := a

a ∈ a → a 6∈ a

a 6∈ a

⊥a 6∈ a

⊥

∀x .x 6∈ x

⊥


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax Refutation of Example

Idea: Generate

◮ Formulas,

◮ Instantiation Terms, and

◮ Propositional Clauses

Success when Propositional Clauses are Unsatisfiable


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Satallax Refutation of Termination Example

MiniSat Clauses

[1][−2]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5][5 ⊔ 6][5 ⊔ 7]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)◮ 6 ∀y .y ∈ a → y 6∈ y◮ 7 a ∈ a


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5][5 ⊔ 6][5 ⊔ 7]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)◮ 6 ∀y .y ∈ a → y 6∈ y◮ 7 a ∈ a◮ Instantiation a. Use in 6


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5][5 ⊔ 6][5 ⊔ 7][−6 ⊔ 8]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)◮ 6 ∀y .y ∈ a → y 6∈ y◮ 7 a ∈ a◮ Instantiation a. Use in 6◮ 8 a ∈ a → a 6∈ a


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5][5 ⊔ 6][5 ⊔ 7][−6 ⊔ 8]

[−8 ⊔ −7 ⊔ −7]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)◮ 6 ∀y .y ∈ a → y 6∈ y◮ 7 a ∈ a◮ Instantiation a. Use in 6◮ 8 a ∈ a → a 6∈ a◮ -7 a 6∈ a


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels


MiniSat Clauses

[1][−2]

[−1 ⊔ 3][−3 ⊔ −4 ⊔ 2]

[4 ⊔ −5][5 ⊔ 6][5 ⊔ 7][−6 ⊔ 8]

[−8 ⊔ −7 ⊔ −7]

◮ 1 ∈-Induction◮ -2 ¬∀x .x 6∈ x◮ Generate Instantiation λx .x 6∈ x◮ Use in 1◮ 3 ∈-Induction with P := λx .x 6∈ x .◮ -4 ¬∀x .(∀y .y ∈ x → y 6∈ y) → x 6∈ x◮ 2 ∀x .x 6∈ x◮ Fresh Witness a for -4◮ -5 ¬((∀y .y ∈ a → y 6∈ y) → a 6∈ a)◮ 6 ∀y .y ∈ a → y 6∈ y◮ 7 a ∈ a◮ Instantiation a. Use in 6◮ 8 a ∈ a → a 6∈ a◮ -7 a 6∈ a◮ Propositional Clauses Unsatisfiable


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Outline


Introduction



The Other Levels


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels









Automation


Satallax

Egal

Egal


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Foundation

◮ Higher-Order Tarski Grothendieck Set Theory

◮ (Very) Simple Type Theory with Choice◮ No polymorphism, no product types, etc.◮ The kind of logic Satallax targets◮ Curry-Howard proof terms (simple proof checker)

◮ Set Theory Axioms: Extensionality, ∈-Induction, EmptySet, Unions, Power Sets, Replacement, Universes

◮ The systems with big libraries can translate into it.






theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

Higher-Order Tarski Grothendieck

◮ No schemes (finitely many axioms)

◮ ∈-Induction (Higher-Order):

∀P : ιo.(∀x .(∀y ∈ x → Py) → Px) → ∀x .Px

◮ Implies Regularity: ∀x .x 6= ∅ → ∃y ∈ x .x ∩ y = ∅


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




∀P : ιo.(∀x .(∀y ∈ x → Py) → Px) → ∀x .Px


◮ Replacement (Higher-Order): replAF for {Fx |x ∈ A}

∀F : ιι.∀y .y ∈ {Fx |x ∈ A} ↔ ∃x ∈ A.y = Fx


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




∀P : ιo.(∀x .(∀y ∈ x → Py) → Px) → ∀x .Px



∀F : ιι.∀y .y ∈ {Fx |x ∈ A} ↔ ∃x ∈ A.y = Fx

◮ Grothendieck Universes: every N is in some U where U

is transitive, closed under unions, power sets andreplacement.


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels




∀P : ιo.(∀x .(∀y ∈ x → Py) → Px) → ∀x .Px



∀F : ιι.∀y .y ∈ {Fx |x ∈ A} ↔ ∃x ∈ A.y = Fx

◮ Grothendieck Universes: every N is in some U where U

is transitive, closed under unions, power sets andreplacement.

◮ Claim: Grothendieck Universes imply Tarski’s Axiom A:every N is in some M such that

◮ X ∈ M and Y ⊆ X imply Y ∈ M◮ M is closed under power sets◮ X ⊆ M implies either X ∈ M or X has the same

“potency” as M


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels









Automation



theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels





Developments

Interactive

Prover 1

· · ·Developments

Interactive

Prover n

Automation

Automated

Prover 1

· · ·Automation

Automated

Prover m


theory

Brown


Introduction

Higher OrderTableau

Higher-OrderExample

The Other Levels

References

◮ Chad E. Brown. Reducing Higher-Order Theorem Proving to aSequence of SAT Problems.CADE 23. 2011. (JAR 2013)

◮ Julian Backes and Chad E. Brown. Analytic Tableaux for Higher-OrderLogic with Choice.

◮ Chad E. Brown and Gert Smolka. Analytic Tableaux for Simple TypeTheory and its First-Order Fragment.Logical Methods in Computer Science. Volume 6, Issue 2. 2010 Journalof Automated Reasoning 2011.

◮ Andreas Teucke. Translating a Satallax Refutation to a TableauRefutation Encoded in Coq.

◮ Chad E. Brown and Christine Rizkallah. From Classical ExtensionalHigher-Order Tableau to Intuitionistic Intentional Natural Deduction.Third International Workshop on Proof Exchange for Theorem Proving.June 2013

◮ Niklas Een, Niklas Sorensson. An Extensible SAT-solver.SAT 2003.

◮ Armin Biere. Picosat essentials.JSAT 2008.

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