I've not read all the comments, but i have a question/suggestion.
May it be that we are missign the coordinization of TIME and FUEL?
Let's make more simple:
10:00 AM : LONDON
11:00 AM : PARIS
12:AM : ROME
In this case TIME is not a "continous variable", it's discrete.
So, to be able to move to algebra to make calculus we need a function to move TIME frome discrete to continous, as we did with the POSTISIONS (cities).
EDIT: to clarify. TIME in the timesheet is only a set of values. We can call them TIME_A, TIME_B, TIME_C. We need to apply a coordinization to TIME and FUEL as well. I think that the misleading is that that function is identity.
So to answer Erik question:
F :: (A -> B) -> (FA -> FB)
i think A -> B is the lookup table itself.
Then FA -> FB is the function that allows to compute position from time in the algebric way.
In this way F, the coordinization, should be well defined.
P.S.: sorry if i mispelled somthing, but is the first time ever that i talk math in english