import NNG.Metadata
import NNG.MyNat.Addition

Game "NNG"
World "Proposition"
Level 2
Title "intro"

open MyNat

Introduction
"
Let's prove an implication. Let $P$ be a true/false statement,
and let's prove that $P\\implies P$. You can see that the goal of this level is
`P → P`. Constructing a term
of type `P → P` (which is what solving this goal *means*) in this
case amounts to proving that $P\\implies P$, and computer scientists
think of this as coming up with a function which sends proofs of $P$
to proofs of $P$.

To define an implication $P\\implies Q$ we need to choose an arbitrary
proof $p : P$ of $P$ and then, perhaps using $p$, construct a proof
of $Q$.  The Lean way to say \"let's assume $P$ is true\" is `intro p`,
i.e., \"let's assume we have a proof of $P$\".

## Note for worriers.

Those of you who know
something about the subtle differences between truth and provability
discovered by Goedel -- these are not relevant here. Imagine we are
working in a fixed model of mathematics, and when I say \"proof\"
I actually mean \"truth in the model\", or \"proof in the metatheory\".

## Rule of thumb: 

If your goal is to prove `P → Q` (i.e. that $P\\implies Q$)
then `intro p`, meaning \"assume $p$ is a proof of $P$\", will make progress.

"

Statement
"If $P$ is a proposition then $P\\implies P$."
    {P : Prop} : P → P := by
  Hint "
  To solve this goal, you have to come up with a function from
  `P` (thought of as the set of proofs of $P$!) to itself. Start with

  ```
  intro p
  ```
  "
  intro p
  Hint "
  Our job now is to construct a proof of $P$. But ${p}$ is a proof of $P$.
  So

  `exact {p}`

  will close the goal. Note that `exact P` will not work -- don't
  confuse a true/false statement (which could be false!) with a proof.
  We will stick with the convention of capital letters for propositions
  and small letters for proofs.
  "
  exact p