Its an interesting one. Historicallly, one of the reasons why the Greek work is regarded as fundamental is the amount of work that they did to prove results and hence derive new theorems, and while its thought that a fair bit of this work was already known by other, older cultures, theres little/no evidence that they stressed the element of proof. However, learning geometry (for instance) in a Euclidean/deductive style isnt necessarily something that would encourage kids to take A level (and probably cause gcse kids to hide under their desks). Similarly induction has been mentioned a couple of times and even thats in further rather than regular maths.
Regular A level seems to cover deduction/exhaustion/counter example and contradiction, as well as cropping up a few times in the regular topics (trig identiites, arithmetic and geometric series). Which sounds a reasonable amount, but most of the contradiction questions/proofs are selected from a small set, most kids seem to approach trig identities by throwing random things together and seeing what sticks and the proofs of arithmetic and geometric series can be a couple of lines each using the gauss and recursive definition "tricks". Using the fuller notation usually used can get in the way of understanding what the terms mean.
I guess one of the problems with including more is the difficulty of examination. Assessing basic proofs generally involves selecting simple examples from a fairly small set. If youre assessing the proof of series (for instance), Id guess youre looking at things like
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 + ....
Its not easy to spot the value infinite sum (numerator is fibonacci, denominator is 2^n so geometric) as the "common" ratio is about 0.8 so takes a while to "converge". However, it would probably be too much "problem solving" for most kids to do it, unless a lot of signposting is used in the question.