import NNG.Levels.Multiplication
import NNG.Levels.AdvAddition

Game "NNG"
World "AdvMultiplication"
Level 1
Title "mul_pos"

open MyNat

Introduction
"
## Tricks

1) if your goal is `X ≠ Y` then `intro h` will give you `h : X = Y` and
a goal of `False`. This is because `X ≠ Y` *means* `(X = Y) → False`.
Conversely if your goal is `False` and you have `h : X ≠ Y` as a hypothesis
then `apply h` will turn the goal into `X = Y`.

2) if `hab : succ (3 * x + 2 * y + 1) = 0` is a hypothesis and your goal is `False`,
then `exact succ_ne_zero _ hab` will solve the goal, because Lean will figure
out that `_` is supposed to be `3 * x + 2 * y + 1`.

"
-- TODO: cases
-- Recall that if `b : ℕ` is a hypothesis and you do `cases b with n`,
-- your one goal will split into two goals,
-- namely the cases `b = 0` and `b = succ n`. So `cases` here is like
-- a weaker version of induction (you don't get the inductive hypothesis).


Statement
"The product of two non-zero natural numbers is non-zero."
    (a b : ℕ) : a ≠ 0 → b ≠ 0 → a * b ≠ 0 := by
  intro ha hb
  intro hab
  induction b with b
  apply hb
  rfl
  rw [mul_succ] at hab
  apply ha
  induction a with a
  rfl
  rw [add_succ] at hab
  exfalso
  exact succ_ne_zero _ hab

Conclusion
"

"