import NNG.Metadata
import NNG.MyNat.Multiplication

Game "NNG"
World "Function"
Level 2
Title "the intro tactic"

open MyNat

Introduction
"
Let's make a function. Let's define the function on the natural
numbers which sends a natural number $n$ to $3n+2$. You
see that the goal is `ℕ → ℕ`. A mathematician
might denote this set with some exotic name such as
$\\operatorname{Hom}(\\mathbb{N},\\mathbb{N})$,
but computer scientists use notation `X → Y` to denote the set of
functions from `X` to `Y` and this name definitely has its merits.
In type theory, `X → Y` is a type (the type of all functions from $X$ to $Y$),
and `f : X → Y` means that `f` is a term
of this type, i.e., $f$ is a function from $X$ to $Y$.

To define a function $X\\to Y$ we need to choose an arbitrary
element $x\\in X$ and then, perhaps using $x$, make an element of $Y$.
The Lean tactic for \"let $x\\in X$ be arbitrary\" is `intro x`.
"

Statement
"We define a function from ℕ to ℕ."
    : ℕ → ℕ := by
  Hint "To solve this goal,
  you have to come up with a function from `ℕ`
  to `ℕ`. Start with

  `intro n`"
  intro n
  Hint "Our job now is to construct a natural number, which is
  allowed to depend on ${n}$. We can do this using `exact` and
  writing a formula for the function we want to define. For example
  we imported addition and multiplication at the top of this file,
  so

  `exact 3 * {n} + 2`

  will close the goal, ultimately defining the function $f({n})=3{n}+2$."
  exact 3 * n + 2

NewTactic intro

Conclusion
"
## Rule of thumb:

If your goal is `P → Q` then `intro p` will make progress.
"