【Prove】 ,  【try】 ,  【test】 ,  【demonstrate】  are comparable when they mean to establish a given or an implied contention or reach a convincing conclusion by such appropriate means as evidence, argument, or experiment.

The same distinctions in implications and connotations are evident in their corresponding nouns proof, trial,  【test】 , demonstration when they denote the process or the means by which a contention is established or a convincing conclusion is reached.

【Prove】  and proof (see also INDICATE REASON 1 ) are the most widely useful of these terms, employable not only in reference to contentions and conclusions, but also in reference to persons or things whose quality (as of strength, genuineness, or fitness) is in question.

When used in reference to contentions or to conclusions reached by study, they imply that evidence sufficient in amount and sufficiently reliable in its character has been adduced to bring conviction of the truth of the contentions or conclusions and to make other contentions or other conclusions untenable.

But  【prove】  and proof when used in reference to persons or things about which there is doubt in some particular imply the settlement of this doubt or the establishing of certainty of his or its quality by subjecting the thing to an experiment or by giving the person a chance to manifest his quality in experience, or by such means as assaying, verifying, or checking.

【Try】  and trial (see also ATTEMPT TRIAL 2 ) carry implications from their earliest senses of to separate, or the separation of, the good from the bad in a person or thing, and therefore stress not the conclusion reached but the process by which the guilt or innocence of a person is definitely proved, or a thing’s genuineness or falsity, its worth or worthlessness, or its degree of strength or validity is definitely established.

【Test】 , both as a verb and as a noun, implies a putting to decisive proof by means of experiment, use, experience, or comparison with a high standard, or through subjection to a thorough examination or trial for the sake of such proof or a determination of the facts.

【Demonstrate】  and demonstration (see also SHOW 1 ) imply the conclusive proof of a contention or the reaching of a conclusion about which there can be no doubt. In such use,  【prove】  and  【demonstrate】  and their corresponding nouns are not distinguishable except that in  【demonstrate】 the emphasis is upon the resulting certainty or formality of method.