Let ti and ti+1 be two consecutive times in the happening sequence for simplify(P)**. The**
set of active continuous effects over (ti, ti+1) is:
{Q |(t, a[d]) ∈ P,(ti, ti+1) ⊆ [t, t + d] and Q **is a continuous effect of** a}
and Cts is the set of **systems of continuous effects**:
Cts = {(C, ti, ti+1)| C is the set of active continuous effects over (ti, ti+1)}
The components of a continuous plan separate out the invariant conditions and continuous effects from the rest of the simple plan in order to allow correct application of the continuous updates and to allow confirmation that the invariants hold in the face of the continuous effects.

Definition 24 Trace Let I be a planning instance that includes continuous durative actions, P be a plan for I, (SP, Inv, Cts) **be the induced continuous plan for** P, {ti}i=0...k be the happening sequence for S and S0 be the initial state for I. The trace for P **is the**
sequence of states {Si}i=0...k+1 **defined as follows:**

- If there is no element (C, ti, ti+1) ∈ Cts then Si+1 **is the state resulting from applying**
the happening at ti in the simple plan SP **to the state** Si.

- If (C, ti, ti+1) ∈ Cts then let Ti be the the state formed by substituting f(ti+1 − ti) for the numeric part of state Si, where f **is the continuous update function defined by** C
for state Si. Then Si+1 is the state resulting from applying the happening at ti **in the**
simple plan SP to the state Ti. If f is undefined for any element in Cts **then so is**
the trace.
Definition 24 defines a trace in a similar fashion to the traces for simple plans and plans with durative actions. The key difference is the need to apply the continuous updates. These are handled by solving the systems of simultaneous differential equations across each interval in which they are active and then applying the result to update the numeric **values across**
that interval. Of course, this is easier to describe than it is to do, since **solving arbitrary**
simultaneous differential equations algorithmically is not generally possible. Under certain constraints this semantics can be implemented in order to confirm the validity of a plan automatically.

Definition 25 Invariant Safe Let I be a planning instance that includes continuous durative actions, P be a plan for I, (S, Inv, Cts) be the induced continuous plan for P and
{Si}i=0...k+1 be the trace for P. For each (C, ti, ti+1) ∈ Cts let fi **be the continuous update**
function defined by C for Si. P is invariant safe if, for each fi **that is defined and for each**
(Q, t, u) ∈ Inv such that [(ti, ti+1)] = I ⊆ (t, u), then ∀x ∈ I, Num(s, Q)(fi(x)) **where** s is the logical state in Si.

In this definition, the symbols [(..)] are used to mean that the interval I **can be closed or**
open at either end.