# Assertion, proof, theorem and certificate

- An
**assertion**is any claimed statement. - A
**proof**is the logical argument that establishes an assertion's truth. - A
**theorem**is an assertion that has been proven to be true. - A
**certificate**is a piece of evidence for verifying an assertion.

## Assertion

An assertion is a declarative statement that is claimed to be true.

All prime numbers greater than 2 are odd.

## Proof

A proof is a logical argument that establishes the truth of an assertion.

A proof showing that the sum of two even numbers is always even.

## Certificate

A certificate is evidence or a witness that can be used to verify the truth of
an assertion. They provide a way to efficiently check the validity of a solution
without redoing the entire proof.

In the context of a graph coloring problem, a certificate might be a specific
coloring of the graph that demonstrates it can be colored with three colors.

## Theorem

A theorem is a statement that has been proven to be true based on axioms and
logical deductions. It is a specific type of assertion that is distinguished by
having a proof. Theorems are significant assertions that are established within
a formal system.

The Pythagorean Theorem, which states that in a right triangle, the square of
the length of the hypotenuse is equal to the sum of the squares of the lengths
of the other two sides.

This work by Pierre-Henry frÃ¶hring is licensed under CC BY-SA 4.0