Is there a proof of existence of time? It seems to me that there is no such thing as time. There  is only movement in the universe and we compare our own movement to a different object to have a sense of time. It can be a clock or a atomic vibration.
Does this view of time work within the current framework of phsics?
Do physicists have an explanation/proof about time's existence?
 A: Your question seems to be related to the "problem of time", which apparently plagues canonical quantum gravity (as, say, opposed to superstring theory). The Problem of time Wikipedia entry doesn't exist anymore (although there is still lots to Google for). Perhaps for good reason. I really cannot comment on such theories, but I'll just add this reference (a rather random pick from what Google offers): Claus Kiefer, "Concept of time in canonical quantum gravity and string theory". And I found this:

The Inexistence of Time
[T]here are those like the English physicist Julian Barbour (b. 1937) who think that this theory [I believe this refers to canonical quantum gravity] helps spell the end of time. He believes that the formalism is telling us something deep — namely, that time doesn't exist. This is of course reminiscent of McTaggart's and Gödel's views on time. And as with McTaggart and Gödel, we must ask . . .

Is the conclusion that time doesn't exist?
Or is it only that time as we might want to understand it doesn't exist?

Perhaps Barbour, a relationalist and conventionalist about time, can be read as saying not that time doesn't exist, but that time is a skimpier entity than you thought (for instance, not tensed, not Newtonian, not linear and not even fundamental).

Introducing Time: A Graphic Guide by Craig Callender and Ralph Edney (p. 168)
Also see Time: Time as "unreal" and Eternalism (philosophy of time): Relation to physics on Wikipedia.
NB: Time is also discussed extensively in the entries Time and The Experience and Perception of Time in the SEP. 
NB 2: The question refers to "proof" and "existence". As I understand it, science isn't about proof, and existence is primarily a philosophical concept: Ontology. If Philosophy SE would be as well populated as Physics SE, I would strongly support the migration of this question.
A: It's easy to get mixed up between time and the flow of time, and I think you've done this in your question.
Take time first: since 1905 we describe any event as a spacetime point and label it with four co-ordinates ($t$, $x$, $y$, $z$). Saying that time doesn't exist means we can ignore the time co-ordinate and label everything by just it's spatial co-ordinates ($x$, $y$, $z$), which is contradiction with observations. The time co-ordinate obviously exists and be used to distinguish events that happen at the same place but at different times.
Now consider the flow of time: actually this is a tough concept, and relativity makes it tougher. We all think we know what we mean by the flow of time because we experience time passing. To take your example of movement, we describe this as the change of position with time, $d\vec{r}/dt$, where we regard time as somehow fundamental. I'm guessing that this is what you're questioning i.e. whether the flow of time is somehow fundamental.
I don't think there is a good answer to this. To talk about the flow of time you'd have to ask what it was flowing relative to. In relativity we can define the flow of co-ordinate time relative to proper time, $dt/d\tau$, and indeed you find that this is variable depending on the observer and in some circumstances (e.g. at black hole event horizons) co-ordinate time can stop altogether. But then you'd have to ask whether proper time was flowing. You could argue that proper time is just a way of parameterising a trajectory and doesn't flow in the way we think time flows.
At this point I'm kind of stuck for anything further to say. If I interpret your question correctly then you do have a point that just because we observe change of position with time (i.e. movement) this doesn't necessarily mean time is flowing in the way we think. However I'm not sure this conclusion is terribly useful, and possibly it's just semantics.
A: I think Bertrand Russell's attitude to this is worth explaining.  His main motivation for doing Logic was to ground the Physics (of his day...he kind of gave up after General Relativity changed everything...)
Objects exist.  That star over there.  That neutron over here.  Physical concepts are linguistic abstractions we use to describe the objects and to assert that some objects stand in this or that relation to some other (your name here) object.  But the abstractions do not exist.  Time is not an object, therefore time does not exist either in the sense in which real physical objects or the sense in which real psychological states (e.g., perceptions you have) of you exist.  
But.  Time is objective in that any system of linguistic abstractions that would truthfully describe the physicsl universe has to have some concept in it doing all the same things that "time" does in an English-language book of Physics.  And no other things that it does not.  You can't get along without it, and there is no close-but-not-exactly-the-same substitute, either.
Now I will explain Einstein's point of view.
Time does not exist, because time and space do not exist, they are just the forms in which we perceive the material world.  If all matter were gone, space-time would be gone, too.
These are not necessarily my own opinions.  If I ever grow up, I will figure out if they are right or not.  But, in my opinion, one of the problems is that "existence" cannot be defined explicitly.  In Physics, we name objects that exist, and of course can do experiments to decide if a star exists or not, but we assert that it exists simply by naming it.  (This was Wittgenstein's attitude, too.)  Now I will close by pointing out you cannot do an experiment to decide whether "time" exists.  Our experience making physics theories has convinced us that it is objective in the sense I explained above: no theory that describes every physical truth can do without it or use a non-equivalent substitute.  But we cannot prove this indispensability by experiment.
A: Fundamental quantities in physics are defined using operational definitions, which means definitions that spell out the operations needed in order to measure them. Time is what a clock measures. The definition would be objectionable if it depended on the type of clock, e.g., if relativistic time dilation were different for different types of clocks. But in fact, clock-comparison experiments give null results (see http://relativity.livingreviews.org/Articles/lrr-2005-5/ , section 5.2).
A: My favorite POV on this I derived from reading Hawking's bestseller.
Consider yourself in two possible states $A$ and $B$ of the universe.
The more information about state $A$ does your mind(brain, ...) in state $B$ possess(have records of, ...) the more you feel in state $B$ that state $A$ is past(remembered, ...)
Thus time is about having information. You percieve something as past if you remember it, and "remember" can be only formally described as "have information about".
Hawking uses this to explain the second law of thermodynamics: the more exhaustive description of $A$ does $B$ contain, the more its entropy must exceed that of $A$. Hence the states you feel like past tend to have less entropy than your current state.
This is nice also because it can explain things like deja vu or nirvana :)
A: An analogue argument may be: is there proof of the existence of three dimensional space
I argue that if dx/dy dx/dz dy/dz dy/dx dz/dx dz/dy  were all identically equal to zero, there would be no way of defining three dimensional space. It is the changes in the map of the globe that allow us to speak of a map and a differentiation of one space point (x,y,z) from another (x',y',z').
I will extend the argument to time, that it is changes in time that allow to distinguish its existence. dx/dt and/or dy/dt and/or dz/dt define time's existence. If all these differentials are identically zero through all space points no time can be defined, (a timeless view)  in the same manner that no space variables have a meaning if all space differentials are zero.
A: 
There is no such thing as time

Yes and no, "time" is just a consequence of one fundamental fact: 
There is cause and effect
because one comes before other always, we have a sequence of events and thus we can "number" those events, what gives some measurable physical quantity, which we call time, this also can be seen strictly within special theory of relativity' framework.
Of course, there is still lot of questions and mystery about time that this definition can't fully explain, anyway this explanation gives quite nice reason for time's one direction flow.
A: Following the wikipedia link to "Time:Time as Unreal" from Glenn the Udderboat's answer I read :-

Time as "unreal"
In 5th century BC Greece, Antiphon the Sophist, in a fragment preserved from his chief work On Truth, held that: "Time is not a reality (hypostasis), but a concept (noêma) or a measure (metron)."

If I can paraphrase your hypothesis: "It seems to me that there is no such thing as time"  as: "time is an abstract concept but not a material thing" then it would appear to be consistent with Antiphon's declaration. 
Let me assume that they are consistent.
Then, regarding your first question: "Does this view of time work within the current framework of phsics?" 
Antiphon's declaration is certainly consistent with the Galilean/Newtonian framework of classical physics. Here time in the sense of "relative time interval" is a quality used to describe intervals between events which occur in some changing system.  Relative times (epochs) of events can be quantified by reference to some standard measure of chronological time (tree-rings, volcanic eruptions, moon phases, Earth rotations, etc.. Relative time intervals (durations) can be measured using standard measures of time interval (clocks). Time is a fundamental dimension (along with Mass and Length) in Newtonian physics but note that Newton's Laws are all expressed in terms of ratios rather than absolute measures.  Where Time is used in Newtonian physics it is a generally used to quantify a speed, velocity or other rate of change.  Or it may be used as a multiplier to express the quantity of something which has accumulated/increased/decreased over a period of time e.g. $distance = velocity * time$.
In the Galilean/Newtonian framework it is always permissible to say that two events are absolutely simultaneous or absolutely sequential (one occurring before the other). This is not the case in Einsteinian physics (see other answers to this Question).
Regarding your second question "Do physicists have an explanation/proof about time's existence?" 
In the Galilean/Newtonian framework time is not a material thing but we can make measurements of changing things and pretend that these measurements are indicators of the current property of an imaginary, changing thing which we might call "chronological time".  So it might be considered that there is nothing further that we need to explain or prove about the imaginary thing called time.  
Regarding the so-called "Arrow of Time" which relates to the observation that certain complex phenomena (e.g. explosions) are observed to proceed in one particular "direction" and which lead some to suggest that there is a thing called time which can only evolve in one "direction" ...this can be dismissed as a spurious conclusion based on inadequate understanding of the evolution of the special complex dissipative systems which give rise to such "uni-directional" phenomena  (see any textbook on Complex Systems or Chaos for elaboration).
It is interesting to observe that many physical systems in our experience do seem to act like clocks which are keeping regular time with each other.  For example the periods of axial revolutions and orbits of bodies in the solar system seem very consistent over the span of human observations.  However this "co-regularity" is something of an illusion because variations (cyclic, chaotic and progressive) do occur but are so small that they are not obvious without careful, detailed measurements.
Regarding Einsteinian and other frameworks, whether explanations of time are required and what those explanations might be... I defer to others more knowledgeable about those frameworks.
On a general note Physics and other sciences can be seen as basically methods of describing our environment which help us to predict and control our environment.  In such a pragmatic scheme it may be useful to invent abstract concepts like Time and Space because they help us to describe our observations, make models and extract rules which we can then use for prediction and influence/control. Such concepts (Time, Space) do not have to exist as material things in order to be useful.  We can do without the ideas of physical Time and Space by using qualities of "when-ness" and "where-ness" which are clearly-definable, relative, measurable properties within any material system.
Accordingly it can be argued that attempts to investigate the physical nature of Time and Space are misguided and that Physical Reality can be described essentially as "(material) stuff moving about". One might be tempted to add something poetic like "...in the ever-changing Present" but that would refer us back to the triad of past-present-future which is merely a (useful) conceptual invention.
A: This paper is on linear existence. We could not conceive or understand events without linear timeframe. We quantify events as the past, present and the future. For example, we bought a car last week, and today I plan to drive it. And hopefully, tomorrow I plan to let my friend borrow it. Here, we are measuring time by the past, present and the future.  Before I started writing this topic I was going to read as much article on the topic of “time”’ but I decided not to do that because I didn’t what to be swayed by others ideology on the topic. For example, some would say "Time is just an illusion,” or “does time really exists.” We exist in a form of physical restriction and considering the possibility of other existence outside of our own is a difficult concept for anyone. Time is essential for all physically restricted being like ourselves and could not be denied of its existence. So, we view time as current or present and we could not explain what happen to the past. The present is the only thing tangible we have but event of the past does not disappear. For example, I parked my car on the curve five minute ago, and if I go outside right now I will find the car still park on the curve where I last parked it. So, if the past could not be quantified by time in linear fashion then the car should not be where I left it. The past event would be altered and what ever activities I did does not reflect on the present. All human effort would be governed by chaos. We as a physical being along with everything in our planet, earth and everything in this planet are all governed by time and we could not deny time existence since every action we have revolves around it. We expect to explain our present action based on our past activities. A police would detain an individual based on what they did in the past and we hold position of everything based on what took place in the past. For example, my computer is not a figment of my imagination it exists. Here I am typing on it. It was manufactured and build by someone last month, which assures me of the past existence and time being measured.
