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import NNG.Levels.Multiplication.Level_4
Game "NNG"
World "Multiplication"
Level 5
Title "mul_assoc"
open MyNat
Introduction
"
We now have enough to prove that multiplication is associative.
## Random tactic hint
You can do `repeat rw [mul_succ]` to repeat a tactic as often as possible.
"
Statement MyNat.mul_assoc
"Multiplication is associative.
In other words, for all natural numbers $a$, $b$ and $c$, we have
$(ab)c = a(bc)$."
(a b c : ℕ) : (a * b) * c = a * (b * c) := by
induction c with d hd
· repeat rw [mul_zero]
rfl
· rw [mul_succ]
rw [mul_succ]
rw [hd]
rw [mul_add]
rfl
NewTactic «repeat»
LemmaTab "Mul"
-- TODO: old version introduced `rwa` here, but it would need to be modified