import NNG.Metadata
import NNG.MyNat.Addition
import Std.Tactic.RCases

Game "NNG"
World "AdvProposition"
Level 5
Title "`iff_trans` easter eggs."

open MyNat

Introduction
"
Let's try `iff_trans` again. Try proving it in other ways.

### A trick.

Instead of using `cases` on `h : P ↔ Q` you can just access the proofs of `P → Q` and `Q → P`
directly with `h.1` and `h.2`. So you can solve this level with

```
intros hpq hqr, 
split,
intro p,
apply hqr.1,
...
```

### Another trick

Instead of using `cases` on `h : P ↔ Q`, you can just `rw h`, and this will change all `P`s to `Q`s
in the goal. You can use this to create a much shorter proof. Note that
this is an argument for *not* running the `cases` tactic on an iff statement;
you cannot rewrite one-way implications, but you can rewrite two-way implications.

### Another trick

`cc` works on this sort of goal too.
"

Statement --iff_trans
"If $P$, $Q$ and $R$ are true/false statements, then `P ↔ Q` and `Q ↔ R` together imply `P ↔ R`.
"
    (P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) := by
  intro hpq hqr
  constructor
  intro p
  apply hqr.1
  apply hpq.1
  assumption
  intro r
  apply hpq.2
  apply hqr.2
  assumption

Conclusion
"

"