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Sound and Complete Axiomatisations of CallByValue Control Operators
 Math. Structures Comput. Sci
, 1994
"... ion. Let \Gamma; x : oe ` M : ø . [[x : oe:M ]] = : (oe * ø ) * 0: (x : oe:M ) = by CApp : (x:C ø (k : ø * 0:k M )) = : (x:C ø ([[M ]])) = j oe*ø (x:C([[M ]])) which is the required expression since x:C oe ([[M ]]) is the abstraction of [[M ]] wrt. x in V. Application. Let \Gamma ` M : oe * ø ..."
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ion. Let \Gamma; x : oe ` M : ø . [[x : oe:M ]] = : (oe * ø ) * 0: (x : oe:M ) = by CApp : (x:C ø (k : ø * 0:k M )) = : (x:C ø ([[M ]])) = j oe*ø (x:C([[M ]])) which is the required expression since x:C oe ([[M ]]) is the abstraction of [[M ]] wrt. x in V. Application. Let \Gamma ` M : oe * ø and \Gamma ` N : oe. [[M N ]] = by definition of [[:]] : ø * 0: (M N ) = by App : ((m:m N ) M ) = by Conv :(m: (m N )) M = by BetaV :(k:k M ) (m: (m N )) = by Ass :(k:k M ) (m:(n:( (m n))) N ) = by BetaV :(k:k M ) (m:(l:l N ) (n:( (m n)))) = by definition of app. app([[M ]]; [[N ]]) Complete axiomatisations of control operators 17 A operator. Let \Gamma ` M : 0. [[A oe (M )]] = by definition : oe * 0: (A oe (M )) = by AAbs :A 0 (M ) = by A 0 Id :M = by Ident :(x : 0:x) M = by BetaV :(k:k M )(x : 0:x) = A oe ([[M ]]) C operator. If \Gamma ` M : (oe * 0) * 0 then [[C oe (M )]] = : oe * 0: C(M ) = by CNat :C 0 (k : 0 * 0:M (x : oe:k ( x))) = by 0Endo and Ident :C 0...