LIX janvier
39
pages
Documents
2010
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
39
pages
Documents
2010
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
Niveau: Supérieur
Parsifal? Typical David Baelde University of Minnesota LIX, 14 janvier 2010
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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
Publié par
Publié le
01 janvier 2010
Parsifal
↔
Typical
David Baelde
University of Minnesota
LIX,
14
janvier
2010
Introduction
LINC,µMALL,µ
IctionistiIntui
IedroriF-tsr
LJ,G
ISequent calculus
,tec.
IUseful structures (focusing)
IProof-search : logic programming, model-checking, theorem proving
Coq
IutnIoitinistic
IedrHigher-or
INatural deduction
IMore complex proof structures
IVery little proof-search
Introduction
This talk
ICoq lacks automation
IWe want more expressivity : heterogeneous (co)inductive definitions, recursive definitions
ICoq is a necessary comparison anyway
Outline
Iµ
JL
IEquality
IFixed points
IGeneric quantification
µLJ
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:A4=BandA0θ=.Aθ} Γ,A0`P
Derivations correspond to natural numbers :
`nat0 `nat(s0)
Γ`BθA=4B Γ`Aθ
:
4 =
simplify
Let’s
)
X
>
4 =
`uPΓ
Fixed points
=u
(s
Γθ`Pθ:θ∈,µB~t`G{}),Γ=u`vsc(u.uv=Γ`~tB)(µ`B:ΓesulΓG`t~)Bµ(B,Γt~Bµsdowboily)Itsy∧Nnirgllwoehoftnto(X4=tXnaX=Y.∨∃=0tanXtanλx.x(λN.y.x==0∨∃ta)YYsn∧feµ=andtnat0
X
=
∧
)
=
(X
4 =
Let’s
.
∃Y
∨
0
u=u`
:
simplify
Fixed points
u=Γ,PΓv`u.u()}=v∈θscP`:θ{GθΓ~B`tGΓ,µ)~t`B(µB~tΓ,Bµ`Γt~)Bµ(B`Γ:seulgrinowllfoheotxλ.N0=x..y∃∨ys=xy)∧NboItsdilntowXtYsnatYnatdef=µ(λna
=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(λN.λx.x=0∨ ∃y.x=
def =
Fixed points
Let’sypliifms:
nat X4= (X=0∨ ∃Y.X=s Y∧nat Y)
natd=efµ(λN.λx.x=0∨ ∃y.x=s y∧Ny)
It boils down to the following rules :
Γ`B(µB)~tΓ,B(µB)~t`G Γ`µB~tΓ, µB~t`G
{Γθ`Pθ:θ∈csu(.)} u=v Γ,u=v`P
Γ`u=u
Example
sxnatx
`nat xaxiom =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∨∃nyat.(ssxx)=`ysnat∧nytax`nat xµL∨L =y,nat(s x)`nat y=L
x
