A Theory of Generalised Substitutions

A Practical Single Refinement Method for B

“…In particular, the bounded angelic and demonic choice operators are denoted respectively by " " and " ". Like ordinary generalised substitutions [1,8], extended substitutions can naturally express heterogeneous computations (those whose starting and final state spaces are distinct). This merely requires that their passive (read frame) variables are all associated with the starting state space, while their active (write frame) variables are all associated with the final state space 1 .…”

“…Notice that the variable u in cod(S ) is just a placeholder for sets of possible final states of the monotonic computation characterised by S , in the same way that the primed frame variables in the before-after predicate prd(T ) of an ordinary generalised substitution T in [8] are collectively just a placeholder for individual possible final states of the conjunctive computation characterised by T .…”

“…An ordinary generalised substitution is characterised by its frame, its termination predicate trm and its before-after predicate prd [8], whereas in contrast an extended substitution is characterised by its frame and its cod alone without 2 It is called a power co-predicate to distinguish it from its dual the power predicate pod(S ) = df ¬ [S ] s / ∈ u also defined in [10], which with the frame s provides an alternative full characterisation of S .…”

Formalising Generalised Substitutions

“…In [8] Dunne considered the generalised substitutions used in the B method. He developed the notion of the frame of a substitution, the variables "involved" in it, and defined generalised substitution operations in terms of frames and weakest preconditions.…”

“…These results are proved using the Isabelle/HOL theorem prover, see [5]. We have also performed, in Isabelle, proofs of other results of Dunne in [8], of which details are in the Appendix.…”