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Formally proving facts in the refinement algebra

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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  1. Formally proving facts in the refinement algebra VladShcherbina IlyaMaryassov Alexander Kogtenkov Alexander Myltsev PavelShapkin Sergey Paramonov Mentor: Sir Tony Hoare

  2. 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:)

  3. Theory • Concise • one binary relation • few operations • few axioms • Formal reasoning is unaccustomed • Intuition could be deceptive

  4. 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)

  5. Thanks • to Thomas Thümand Oliver Schwarz for introduction to Coq • to John Wickersonfor introduction to Isabelle

  6. 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.

  7. 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’

  8. Coq demo time (***********) (* v *) (* <O___,, *) (* \VV/ *) (* // *) (* *) (***********)

  9. Statistics • ~30 theorems • ~500 lines of Coq definitions and proofs • 5-60 minutes per proof (given the proof plan) • 2 inaccuracies found

  10. (************) (* ???? *) (* ?? ?? *) (* ?? *) (* ?? *) (* *) (* ?? *) (************) (***********) (* v *) (* <O___,, *) (* \VV/ *) (* // *) (* *) (***********) https://github.com/ Vlad-Shcherbina/ TheoryOfRefinement

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