  In this paper we present an assume-guarantee specification theory (aka
interface theory from [14]) for modular synthesis and verification of real-time
systems with critical timing constraints. It is a further step of our earlier
work [10] which achieved an elegant algebraic specification theory for
real-time systems endowed with the capability to freeze time. In this paper we
relinquish such (unrealisable) capability and target more realistic systems
without the ability to stop time.
  Our theory, in a combined process-algebraic and reactive-synthesis style,
provides the operations of parallel composition for system integration, logical
conjunction/disjunction for viewpoint fusion and independent development, and
quotient for incremental synthesis.
  We show that a substitutive refinement preorder, which is a coarsening of the
pre-congruence in [10], constitutes the weakest pre-congruence preserving
freedom of incompatibility errors. The coarsening requires a shift in the focus
of our theory to a more game-theoretical treatment, where the coarsening
constitutes a reactive synthesis game named normalisation and is efficiently
implementable by a novel local bot-backpropagation algorithm.
  Previously, timed concurrent games have been studied in [1,14,13], where one
of the key concern is the removal of time-blocking strategies by applying blame
assignment [13]. Our timed games also have the issue of time-blocking
strategies, which may arise through the composition of specifications. However,
due to our distinctively different formulation of timed games, we have
discovered another elegant solution to the problem without blame assignment.
Our solution utilises a second reactive synthesis game called realisation,
which is dual to normalisation and implementable by the dual local
top-backpropagation algorithm.
