LIX janvier

David Baelde

Description

Niveau: Supérieur
Parsifal? Typical David Baelde University of Minnesota LIX, 14 janvier 2010

• inductive definitions


• parsifal? typical


• logic programming


• generic quantification


• complex proof


• introduction linc


• very little


• sequent calculus

Sujets

• Philippus Baldaeus
• Logic programming
• Sequent calculus

Informations

• Publié par: profil-nechor-2012
• Publié le: 01 janvier 2010
• Nombre de lectures: 22

Extrait

Parsifal

↔

Typical

David Baelde

University of Minnesota

LIX,

14

janvier

2010

Introduction

LINC, µ MALL, µ

I ctionistiIntui

I edroriF-tsr

LJ,G

I Sequent calculus

,tec.

I Useful structures (focusing)

I Proof-search : logic programming, model-checking, theorem proving

Coq

I utnIoitinistic

I edrHigher-or

I Natural deduction

I More complex proof structures

I Very little proof-search

Introduction

This talk

I Coq lacks automation

I We want more expressivity : heterogeneous (co)inductive definitions, recursive definitions

I Coq is a necessary comparison anyway

Outline

I µ

JL

I Equality

I Fixed points

I Generic quantification

µ LJ

Definitions

Origins inlogic programming:

nat0

4 =>

nat ( s X ) = 4

antX

Focusedderivations correspond to natural numbers :

T `

nat0

axiom

. at ( s0 ) ∀ L ,⊃ L T ` n , . . .

Definitions

Origins inlogic programming:

ant0

= 4 > nat ( s X ) 4 =

antX

Definitionsprovide moreructstrueandssneisevrpsexe

{(Γ, B ` P )θ: A 4 = BandA 0 θ= . A θ} Γ, A 0 ` P

Derivations correspond to natural numbers :

` nat0 ` nat ( s0 )

Γ` B θ A = 4 B Γ` A θ

:

4 =

simplify

Let’s

)

X

>

4 =

` uP Γ

Fixed points

= u

( s

Γθ` P θ:θ∈ ,µ B ~ t ` G {}) ,Γ= u ` v sc ( u . uv = Γ` ~ tB )(µ` B : Γ esul Γ G ` t ~ ) B µ( B ,Γ t ~ B µ sdowboil y ) It sy ∧ N nirgllwoehoftnto ( X 4 = tXnaX = Y .∨∃= 0 tanXtan λ x . x (λ N . y . x == 0 ∨∃ ta ) YYsn ∧ fe µ= an d t nat0

X

=

∧

)

=

( X

4 =

Let’s

.

∃ Y

∨

0

u = u `

:

simplify

Fixed points

u =Γ, P Γ v ` u . u ()}= v ∈θ scP `:θ{ G θΓ ~ B ` t G Γ,µ ) ~ t ` B (µ B ~ t Γ, B µ`Γ t ~ ) B µ( B `Γ :seulgrinowllfoheot x λ. N0 = x .. y ∃∨ ys = xy )∧ N boItsdilntow XtYsnatY nat def =µ(λ na

nirgluse Γ : B I ` t µ b ( o ) i B lsdowntothefollow Γ G ` t ~ B µ, { G ` t P `θΓ Γ` ~ t ~ X t t µ a B n B (Γ,) ~ µ B Γ Pu `= u ` v

= 4 ( X = 0 ∨ ∃ Y . X = s Y

Let’syfiimpls:

s y ∧ Ny )

∧ nat Y )

u =

Fixed points

csu (θ:θ∈)} Γ, u . = v

nat

µ(λ N .λ x . x = 0 ∨ ∃ y . x =

def =

Fixed points

Let’sypliifms:

nat X 4 = ( X = 0 ∨ ∃ Y . X = s Y ∧ nat Y )

nat d = ef µ(λ N .λ x . x = 0 ∨ ∃ y . x = s y ∧ Ny )

It boils down to the following rules :

Γ` B (µ B ) ~ t Γ, B (µ B ) ~ t ` G Γ`µ B ~ t Γ, µ B ~ t ` G

{Γθ` P θ:θ∈ csu ( . )} u = v Γ, u = v ` P

Γ` u = u

Example

sx natx

` nat x axiom = s x , nat x ` nat x = L s x = s y ∧ nat y ` nat x ∧ L ∃ L s x = 0 ` nat x = L ∃ y . s x = s y ∧ nat y ` nat x ( s x ) = 0 ∨∃ n y at . ( s s x x ) = ` ys nat ∧ nyta x ` nat x µ L ∨ L = y , nat ( s x )` nat y = L

x