End ForwardTrace.
Module Partial := ForwardTrace (ProgramState.Partial).
Module Total := ForwardTrace (ProgramState.Total).

Lemma Total2Partial_forward_trace_with_test
{state: Type}
{state_R: Relation state}
(P: state -> Prop):
Included _
(Total.forward_trace_with_test P)
(Partial.forward_trace_with_test P).