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import NNG.Levels.AdvAddition.Level_4
Game "NNG"
World "AdvAddition"
Level 5
Title "add_right_cancel"
open MyNat
Introduction
"
The theorem `add_right_cancel` is the theorem that you can cancel on the right
when you're doing addition -- if `a + t = b + t` then `a = b`.
"
Statement MyNat.add_right_cancel
"On the set of natural numbers, addition has the right cancellation property.
In other words, if there are natural numbers $a, b$ and $c$ such that
$$ a + t = b + t, $$
then we have $a = b$."
(a t b : ℕ) : a + t = b + t → a = b := by
Hint (hidden := true) "You should start with `intro`."
intro h
Hint "I'd recommend now induction on `t`. Don't forget that `rw [add_zero] at h` can be used
to do rewriting of hypotheses rather than the goal."
induction t with d hd
rw [add_zero] at h
rw [add_zero] at h
exact h
apply hd
rw [add_succ] at h
rw [add_succ] at h
Hint (hidden := true) "At this point `succ_inj` might be useful."
exact succ_inj h
-- TODO: Display at this level after induction is confusing: old assumption floating in context