A proof is a logical argument demonstrating that a specific statement, proposition, or mathematical formula is true. It consists of a set of assumptions, or premises, which are combined according to logical rules, to establish a valid conclusion. This validation can be achieved by direct proof that verifies the conclusion is true, or by indirect proof that establishes that it cannot be false.

The term proof is derived from the Latin probare, meaning to test. The Greek philosopher and mathematician Thales is said to have introduced the first proofs into mathematics about 600 b.c. A more complete mathematical system of testing, or proving, the truth of statements was set forth by the Greek mathematician Euclid in his geometry text, Elements, published around 300 b.c. As proposed by Euclid, a proof is a valid argument from true premises to arrive at a conclusion. It consists of a set of...