Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Branched coveringIn mathematics, a branched covering is a map that is almost a covering map, except on a small set. In topology, a map is a branched covering if it is a covering map everywhere except for a nowhere dense set known as the branch set. Examples include the map from a wedge of circles to a single circle, where the map is a homeomorphism on each circle. In algebraic geometry, the term branched covering is used to describe morphisms from an algebraic variety to another one , the two dimensions being the same, and the typical fibre of being of dimension 0.
