import NNG.Metadata
import NNG.MyNat.Addition

Game "NNG"
World "Proposition"
Level 4
Title "apply"

open MyNat

Introduction
"
Let's do the same level again:

![diagram](https://wwwf.imperial.ac.uk/~buzzard/xena/natural_number_game_images/implies_diag.jpg)

We are given a proof $p$ of $P$ and our goal is to find a proof of $U$, or
in other words to find a path through the maze that links $P$ to $U$.
In level 3 we solved this by using `have`s to move forward, from $P$
to $Q$ to $T$ to $U$. Using the `apply` tactic we can instead construct
the path backwards, moving from $U$ to $T$ to $Q$ to $P$.

Our goal is to prove $U$. But $l:T\\implies U$ is
an implication which we are assuming, so it would suffice to prove $T$.
Tell Lean this by starting the proof below with

`apply l,`

and notice that our assumptions don't change but *the goal changes*
from `⊢ U` to `⊢ T`. 

Keep `apply`ing implications until your goal is `P`, and try not
to get lost! Now solve this goal
with `exact p`. Note: you will need to learn the difference between
`exact p` (which works) and `exact P` (which doesn't, because $P$ is
not a proof of $P$).
"

Statement
"We can solve a maze."
    (P Q R S T U: Prop) (p : P) (h : P → Q) (i : Q → R)
    (j : Q → T) (k : S → T) (l : T → U) : U := by
  apply l
  apply j
  apply h
  exact p

DisabledTactic «have» 

Conclusion
"

"