The refinement calculus provides a methodology for transforming an abstract specification into a concrete implementation, by following a succession of refinement rules. These rules have been mechanized in theorem-provers, thus providing a formal and rigorous way to prove that a given program refines another one. In a previous work, we have extended this mechanization for object-oriented programs, where the memory is represented as a graph, and we have integrated our approach within the rCOS tool, a model-driven software development tool providing a refinement language. Hence, for any refinement step, the tool automatically generates the corresponding proof obligations and the user can manually discharge them, using a provided library of refinement lemmas. In this work, we propose an approach to automate the search of possible refinement rules from a program to another, using the rewriting tool Maude. Each refinement rule in Maude is associated with the corresponding lemma in Isabelle, thus allowing the tool to automatically generate the Isabelle proof when a refinement rule can be automatically found. The user can add a new refinement rule by providing the corresponding Maude rule and Isabelle lemma.
We report a detailed ab initio investigation on hydrogen bonding, geometry, electronic structure, and lattice dynamics of ice under a large high pressure range, including the ice X phase (55-380GPa), the previous theoretically proposed higher-pressure phase ice XIIIM (Refs. 1-2) (380GPa), ice XV (a new structure we derived from ice XIIIM) (300-380GPa), as well as the ambient pressure low-temperature phase ice XI. Different from many other materials, the band gap of ice X is found to be increasing linearly with pressure from 55GPa up to 290GPa, the electronic density of states (DOS) shows that the valence bands have a tendency of red shift (move to lower energies) referring to the Fermi energy while the conduction bands have a blue shift (move to higher energies). This behavior is interpreted as the high pressure induced change of s-p charge transfers between hydrogen and oxygen. It is found that ice X exists in the pressure range from 75GPa to about 290GPa. Beyond 300GPa, a new hydrogen-bonding structure with 50% hydrogen atoms in symmetric positions in O-H-O bonds and the other half being asymmetric, ice XV, is identified. The physical mechanism for this broken symmetry in hydrogen bonding is revealed.
