import NNG.Levels.Multiplication.Level_3

Game "NNG"
World "Multiplication"
Level 4
Title "mul_add"

open MyNat

Introduction
"
Where are we going? Well we want to prove `mul_comm`
and `mul_assoc`, i.e. that `a * b = b * a` and
`(a * b) * c = a * (b * c)`. But we *also* want to
establish the way multiplication interacts with addition,
i.e. we want to prove that we can \"expand out the brackets\"
and show `a * (b + c) = (a * b) + (a * c)`.
The technical term for this is \"left distributivity of
multiplication over addition\" (there is also right distributivity,
which we'll get to later).

Note the name of this proof -- `mul_add`. And note the left
hand side -- `a * (b + c)`, a multiplication and then an addition.
I think `mul_add` is much easier to remember than \"`left_distrib`\",
an alternative name for the proof of this lemma.
"

Statement MyNat.mul_add
"Multiplication is distributive over addition.
In other words, for all natural numbers $a$, $b$ and $t$, we have
$t(a + b) = ta + tb$."
    (t a b : ℕ) : t * (a + b) = t * a + t * b := by
  induction b with d hd
  · rw [add_zero, mul_zero, add_zero]
    rfl
  · rw [add_succ]
    rw [mul_succ]
    rw [hd]
    rw [mul_succ]
    rw [add_assoc]
    rfl

LemmaTab "Mul"