The phrase "actual proof" is correct and usable in written English.
It can be used when referring to evidence that is concrete and verifiable, as opposed to mere claims or assumptions.
Example: "The scientist presented actual proof of her hypothesis through rigorous experimentation and data analysis."
Alternatives: "concrete evidence" or "tangible proof."
No actual proof was needed to substantiate this.
It is not worth it for them if they have to provide actual proof".
There's no actual proof Mr. Giuliani is a downstate snob who hates driving three hours to have dinner at a place where the steak comes wrapped in cheese.
No actual proof or evidence – just quotes taken out of context and events (apparently Brown got angry once, shock horror!) exaggerated to an absurd degree.
Maureen Castaneda, a former Enron executive, last week went public with allegations of large-scale document shredding at Enron and provided actual proof.
"There is a big distinction between innuendo and rumour and actual proof," said Whittingdale told BBC Radio 4's Today programme on Monday.
While the Method shows that he arrived at the formulas for the surface area and volume of a sphere by "mechanical" reasoning involving infinitesimals, in his actual proofs of the results in Sphere and Cylinder he uses only the rigorous methods of successive finite approximation that had been invented by Eudoxus of Cnidus in the 4th century bce.
In the OASIS, users can go on quests that require work that are actual proof-of work.
In actual proof-theoretic practice, as well as explicitly in (Hilbert and Bernays, 1934; Bernays, 1930), these basic operations are generalized to operations defined by recursion, paradigmatically, primitive recursion, e.g., multiplication and exponentiation (see Parsons 1998 for a discussion of philosophical difficulties in relation to exponentiation).
However we end up characterizing gaps, it is undeniably the case that most actual proofs as presented by mathematicians have gaps.
(Actual proofs such as the ε-substitution had always been of such a kind: provide finitary procedures which eliminate transfinite elements from proofs of real statements, in particular, of 0 = 1).
