# What a Proof Is

The word "proof" makes a lot of people freeze - it sounds like a wall of Greek symbols only
mathematicians are allowed behind. It isn't. A proof is something you already do informally every time
you convince someone of a conclusion they can't wriggle out of: you start from things they already
accept, take steps they can't object to, and arrive somewhere they now have to agree with. That's the
whole idea. A proof is an argument with the gaps removed.

This guide demystifies it. You'll see what a proof actually is (and how it differs from "evidence"),
meet the handful of standard techniques that cover almost everything - direct proof, contradiction,
contrapositive, cases, the single counterexample - and then unlock the one that feels like magic until
it doesn't: **induction**, which lets a few lines prove something true for infinitely many cases at
once. If you've written a recursive function, you already think the way induction works.

## How to read this
- **Curious what proof even means?** [Phase 1](01-what-a-proof-actually-is.md) is the reframe.
- **Want the toolkit?** Read in order - Phase 2 is the techniques, Phase 3 is induction.

## The phases
1. **[What a Proof Actually Is](01-what-a-proof-actually-is.md)** - a gap-free chain from accepted
   truths to a conclusion, and why that's stronger than evidence.
2. **[The Main Proof Techniques](02-the-main-proof-techniques.md)** - direct, contradiction,
   contrapositive, cases, and disproof by counterexample.
3. **[Proof by Induction](03-proof-by-induction.md)** - the domino principle, and why it's the same
   shape as recursion.

> This builds on validity from [What Logic Actually Is](/guides/what-logic-actually-is) and the
> contrapositive from [Propositional Logic](/guides/propositional-logic). Last in the Logic foundations:
> spotting fallacies.