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License: Other
Kleene Algebra (Kozen axioms)
License: Other
Roughly these rules which depend on a proof ax≡xb
bisimulation : ∀ {a b x : A} → a ∙ x ≡ x ∙ b
→ a * ∙ x ≡ x ∙ b *
bisimulation {a} {b} {x} ax≡xb = {! !}
And then something like this (but I should double check):
-- sliding rule
sliding : ∀ {a b : A} → (a ∙ b) * ≡ a ∙ (b ∙ a) *
sliding {a} {b} = {! !}
-- denesting rule
denesting : ∀ {x y z : A} → (x + y) * ≡ x * ∙ (y ∙ x *) *
denesting = {! !}
-- https://pdfs.semanticscholar.org/6f33/978012f794cc5fff60506db730e6a15185e7.pdf -- Introduction to Kleene Algebra Lecture 7, Equational Theory of Kleene Algebra
-- https://pdfs.semanticscholar.org/58e7/4d6c5c462887f7e1ebb9b092554397b89b3b.pdf -- Kleene Algebra Completeness
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l01-intro.pdf -- Course Roadmap
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l02-axioms.pdf -- Introduction to Kleene Algebra Lecture 2, Axioms of Kleene Algebra
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l03-alt.pdf -- Alternative Axiomatizations
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l04-eqthy.pdf -- Characterizing the Equational Theory
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l06-adj.pdf -- Relations Among Algebras
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l07-complete.pdf -- Equational Theory of Kleene Algebra
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l08-complete.pdf -- Finite Automata
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l09-complete.pdf -- Completeness
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l10-complexity.pdf -- Complexity of the Equational Theory
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l19-kat.pdf -- Kleene Algebra with Tests
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l20-katmodels.pdf -- Models of KAT
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l21-katcomplete.pdf -- Completeness of KAT
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l22-katHoare.pdf -- KAT and Hoare Logic
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l23-katcomplete.pdf -- Completeness of KAT for the Hoare Theory of Relational Models
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l24-complexity.pdf -- Complexity of Kleene Algebra with Tests
-- http://www.cs.cornell.edu/courses/cs786/2004sp/Lectures/l25-phlcomplexity.pdf -- Complexity of PHL
-- https://www.cs.cornell.edu/~kozen/Papers/ka.pdf -- A Completeness Theorem for Kleene Algebras and the Algebra of Regular Events
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