Invariant vs Assertion vs lemma

Question

I am reading Distributed Algorithms by Nancy Lynch. I have come across lemmas, assertions and invariants, but I do not understand the difference between them. I think lemma means an intermediate result. Can someone please dexplain the definitions for each?

Answer 1

All terms describe formal claims that are true. The main difference is in how they are used.

A lemma is mathematical claim that is (generally) true. In textbooks, Lemma's are usually either proven, left as an exercise, or assumed to be prior knowledge. A lemma is distinguished from a theorem in that the lemma is usually a less important or intermediate result.

An assertion is a claim, again supposed to be true. An assertion can be made in a proof, but also by a computer program or algorithm. In the latter case, the assertion is often tested, i.e. a computer program checks if the claim is true.

An invariant is a property of some (mathematical) process that remains true over time. For example, a loop invariant is a property that remains true during the execution of a loop.