proofcloud white @ in- ria+cnrs proofcloud · proofcloud robert white @ in-ria+cnrs background...

ProofCloud

RobertWhite @ IN-RIA+CNRS

BackgroundHigher OrderLogic andTheoremProversProofCollaboration

ProofTranslationand ProofChecking

ProofRetrievalEngine:ProofCloud

Conclusionand FutureWork

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ProofCloudA Proof Retrieval Engine for Verified Proofs

Robert White @ INRIA+CNRS

Supervisor: Prof. Gilles Dowek @ INRIA

UITP’16

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Outline

1 BackgroundHigher Order Logic and Theorem ProversProof Collaboration

2 Proof Translation and Proof Checking

3 Proof Retrieval Engine: ProofCloud

4 Conclusion and Future Work

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Motivation and Introduction

1 the first proof retrievel engine.2 1700+ pages of proofs and proof packages with analysis.3 the first complete set of proof checking results about

OpenTheory.4 where are the proofs?5 which proofs are constructive?6 what axioms are used in the proof?7 how long does it take to verify these proofs?

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Higher Order Logic

Simple type theory (STT) is also known as Higher orderlogic (HOL).HOL = simply typed λ-Calculus + boolean types +axioms + inference rules.Most mathematical objects/theories can be expressed inHOL.Interactive and automatic theorem provers & proofcheckers for formal mathematics.HOL Light, ProofPower, HOL4, HOL Zero . . . [HOLfamily].

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Kernel

HOL syntax:

type variables α, βtype operators ptypes A,B ::= α | p(A1, . . . ,An)term variables x , yterm constants cterms M,N ::= x |λx : A.M |MN | c

Polymorphic Type:

= : α→ α→ o

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Primitive Inference Rules

Structural ASSUME{A} ` A

λ Calculus

Γ ` A = BABS

Γ ` λx.A = λx.BBETA

(λx.A)x = A

Instantiation

Γ[x1, . . . , xn ] ` A[x1, . . . , xn ]INST

Γ[t1, . . . , tn ] ` A[t1, . . . , tn ]Γ[α1, . . . , αn ] ` A[α1, . . . , αn ]

INST_TYPEΓ[γ1, . . . , γn ] ` A[γ1, . . . , γn ]

Bi-implication

Γ ` A = B ∆ ` AEQ_MP

Γ ∪ ∆ ` BΓ ` A ∆ ` B

DEDUCTANTISYMRULE(Γ \ {B}) ∪ ∆ \ {A}) ` A = B

Equality

REFL` A = AΓ ` A = B ∆ ` C = D

MK_COMBΓ ∪ ∆ ` A(C) = B(D)

Γ ` A = B ∆ ` B = CTRANS

Γ ∪ ∆ ` A = C

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Dependency

=

>

∧

⇒∀

⊥

∨ ∃

¬

Figure: Constants and Connectives Dependency Analysis

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OpenTheory

HOL family: HOL Light, ProofPower, HOL4, Isabelle . . .Need a platform to reuse proofs from different systems.OpenTheory has a standard format of proofs (*.art).Export proofs and import proofs (in article files).OpenTheory HOL Light:a modified version of HOL Light which allows import andexport of proofs.http://www.gilith.com/research/opentheory/

http://www.gilith.com/research/opentheory/

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Packages and Dependency

base (the standard library)

stream natural-divides

natural-prime

natural-fibonacci modular

probability

gfp

natural-list

Figure: Dependency of Packages of OpenTheory

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OpenTheory Kernel

Figure: Primitive Inference Rules of OpenTheory 5

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OpenTheory 5 & 6

Commands Updated/NewproveHyp newtrans newsym newdefineConstList newhdTl newdefineTypeOp updatedpragma new

Table: Differences between commands of version 5 and version 6

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OpenTheory 5 & 6

Figure: Additional Primitive Inference Rules in OpenTheory 6

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News from OpenTheory

OpenTheory is updating constantly, so will Holide and Dedukti.

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Holide and Dedukti

OpenTheory has a repository of proof packages (articles).Holide translates proofs from OpenTheory articles toDedukti.Dedukti is a proof checker (for proof checking).

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Workflow of OpenTheory, Holide and Dedukti

Figure: Work Flow of OpenTheory, Holide and Dedukti

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Translation

translate ( Γ ` ϕ = ψsym

Γ ` ψ = ϕ) = Sym|A||t1||t2|

translate( Γ ` s = t ∆ ` t = u transΓ ∪∆ ` s = u

) =Trans|A||x ||y ||z ||D1||D2|, where D1 is the proof of s = tand D1 is the proof of t = u

translate( Γ ` ϕ ∆ ` ψproveHyp

Γ ∪ (∆ \ {ϕ}) ` ψ) =

ProveHyp|x ||y ||D1|(λx : ||ψ||.|D2|), where D1 is the proofof ϕ and D2 is the proof of ψ.

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Translation (Continued)

Sym, Trans and ProveHyp have types as follows:

Sym : Πα : type.Πx , y : termα. proof (eq bool x y)→proof (eq bool y x)

Trans : Πα : type.Πx , y , z : termα proof (eq α x y)→proof (eq α y z)→ proof (eq α x z)

ProveHyp : Πx , y : term bool .proof x → (proof x →proof y)→ proof y

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Benchmarks: Translation

Figure: The Size of Article Files and Translation Time

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Benchmarks: Checking

Figure: Th Size of Article Files and Proof Checking Time

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Evaluation

Fully verified all the libraries in OpenTheory.Little difference between version 5 and version 6.The size of proof articles got reduced by around 7%. Theproof checking time reduced by around 5%.These benchmarks were generated on a 64-bit Intel Corei5-4590 CPU @3.30GHz ×4 PC with 3.8GB RAM.Proof checking results are included in ProofCloud

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ProofCloud

1 the first proof retrievel engine.2 1700+ pages of proofs and proof packages with analysis.3 the first complete set of proof checking results about

OpenTheory.4 where are the proofs?5 which proofs are constructive?6 what axioms are used in the proof?7 how long does it take to verify these proofs?

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Version 0

A representation of the proof analysis and proof checking re-sults of HOLALA for my internship.

Figure: Index Page of ProofCloud (version 0)

It was based on Google Custom Search.

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Version 1

A proof search engine which represents the analysis and proofchecking results of the basic package of OpenTheory.


It’s version 2 now!!!

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Version 2

More packages; a better interface; complete verification results;faster search engine


It’s using Swiftype now!

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Version 2: Verification

Figure: A verification page of ProofCloud

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Version 2: A Proof Page

Figure: A proof page of ProofCloud

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Version 2: A Proof Page

Figure: A proof page of ProofCloud

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Demo

So far there are 6 packages: base, stream, probability,natural-bits, natural-divides and natural-prime.Try it out yourself:http://airobert.github.io/proofcloud

search "ADD" for example.

http://airobert.github.io/proofcloud

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Statistical Results

1209 proofs in the standard library.541 constructive proofs v.s. 668 classical proofs44.75% of them constructive proofs.

In contrast, the natural-divides package has only 10 construc-tive proofs out of 136 proofs, making only 7.35% of them con-structive.

Do any of these these 668 proofs have a constructive form?

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Some Results that used to be there: StructuralAnalysis

the combination of the subst and eqmp rule takes over 45% ofall the inferences rules.

Inference Rules Countsubst 93667eqmp 92617appthm 53155proveHyp 47728betaConv 21485absThm 15096. . . . . .

assume 16986Overall 413207

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Structural Results

The frequency of subst and eqmp combined is over 45%

Figure: Frequency of Main Inference Rules of OpenTheory Articles

And more for each individual theorem or combine with MachineLearning?

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ProofCloud 3?

Side-by-side proof comparison on the flyMore OpenTheory packages and even Coq, Mizar . . .libraries?Better GUIBetter searching accuracyMore text description of theorems (at least a proper name)A proof analysis engine combined with machine learning asa feature suggestion engine after comparing thousands ofproofs?Funding is the key!

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Conclusion and Acknowledgement

An overview of recent updates of OpenTheory, Holide,Dedukti, ProofCloudProof checking benchmarksThe first proof search engineA cool user interface of OpenTheoryThanks to Dr. Gilles Dowek, Dr. Ali Assaf, Dr. Joe Hurd,Mr. Frédéric Gilbert and Mr. Nigel Sham and Mr. DianlinShen.

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Thank you!

I am looking for a master thesis supervisor and opportunitiesfor doctoral studies related but not limited to those provers!Thanks for your attension!

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