import NNG.Metadata

Game "NNG"
World "Function"
Level 9
Title ""

open MyNat

Introduction
"
I asked around on Zulip and apparently there is not a tactic for this, perhaps because
this level is rather artificial. In world 6 we will see a variant of this example
which can be solved by a tactic. It would be an interesting project to make a tactic
which could solve this sort of level in Lean.

You can of course work both forwards and backwards, or you could crack and draw a picture.
"

Statement
"Given a bunch of functions, we can define another one."
    (A B C D E F G H I J K L : Type)
    (f1 : A → B) (f2 : B → E) (f3 : E → D) (f4 : D → A) (f5 : E → F)
    (f6 : F → C) (f7 : B → C) (f8 : F → G) (f9 : G → J) (f10 : I → J)
    (f11 : J → I) (f12 : I → H) (f13 : E → H) (f14 : H → K) (f15 : I → L) : A → L := by
  Hint "In any case, start with `intro a`!"
  intro a
  Hint "Now use a combination of `have` and `apply`."
  apply f15
  apply f11
  apply f9
  apply f8
  apply f5
  apply f2
  apply f1
  exact a


Conclusion
"
That's the end of Function World! Next it's Proposition world, and the tactics you've learnt in Function World are enough
to solve all nine levels! In fact, the levels in Proposition world might look strangely familiar$\\ldots$.
"