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Formally proving facts in the refinement algebra PowerPoint PPT Presentation


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Formally proving facts in the refinement algebra. Vlad Shcherbina Ilya Maryassov Alexander Kogtenkov Alexander Myltsev Pavel Shapkin Sergey Paramonov Mentor: Sir Tony Hoare. Project motivation. Educational (get some experience with interactive theorem provers )

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Formally proving facts in the refinement algebra

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Formally proving facts in the refinement algebra l.jpg

Formally proving facts in the refinement algebra

VladShcherbina

IlyaMaryassov

Alexander Kogtenkov

Alexander Myltsev

PavelShapkin

Sergey Paramonov

Mentor: Sir Tony Hoare


Project motivation l.jpg

Project motivation

  • Educational (get some experience with interactive theorem provers)

  • Relevant to the school

    • provers are used in verification

    • the theory itself can be used in principle to reason about programs and specifications

  • It’s always nice to be absolutely sure

(almost:)


Theory l.jpg

Theory

  • Concise

    • one binary relation

    • few operations

    • few axioms

  • Formal reasoning is unaccustomed

  • Intuition could be deceptive


Interactive theorem provers l.jpg

Interactive theorem provers

  • Most proof steps are automated, but sometimes user intervention is required

    • to introduce useful lemma

    • to apply some nontrivial substitution

      ...

  • LCF-style (proof is correct by construction)


Thanks l.jpg

Thanks

  • to Thomas Thümand Oliver Schwarz for introduction to Coq

  • to John Wickersonfor introduction to Isabelle


Slide6 l.jpg

Coq

  • First order (for our purposes) intuitionistic logic

  • In the form of natural deduction

  • Proofs are constructed “backwards”

  • Proofs are spells, that are hard to comprehend without running Coq.


Example l.jpg

Example

  • Refinement relation ⊑ is partial

  • Binary operations ; and |

  • (definition) Milner transition: p -q-> r <=> (q; r) ⊑ p

  • Exchange law: (p | p’) ; (q| q’) ⊑ (p ; q) | (p’;q’)

    • Parallel rule for Milner transition:p -q-> r & p’ –q’-> r’ => => p|p’ –(q|q’)-> r|r’


Coq demo time l.jpg

Coq demo time

(***********)

(* v *)

(* <O___,, *)

(* \VV/ *)

(* // *)

(* *)

(***********)


Statistics l.jpg

Statistics

  • ~30 theorems

  • ~500 lines of Coq definitions and proofs

  • 5-60 minutes per proof (given the proof plan)

  • 2 inaccuracies found


Slide10 l.jpg

(************)

(* ???? *)

(* ?? ?? *)

(* ?? *)

(* ?? *)

(* *)

(* ?? *)

(************)

(***********)

(* v *)

(* <O___,, *)

(* \VV/ *)

(* // *)

(* *)

(***********)

https://github.com/

Vlad-Shcherbina/

TheoryOfRefinement


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