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import NNG.Levels.AdvAddition.Level_2
Game "NNG"
World "AdvAddition"
Level 3
Title "succ_eq_succ_of_eq"
open MyNat
Introduction
"
We are going to prove something completely obvious: if $a=b$ then
$\\operatorname{succ}(a)=\\operatorname{succ}(b)$. This is *not* `succ_inj`!
"
Statement MyNat.succ_eq_succ_of_eq
"For all naturals $a$ and $b$, $a=b\\implies \\operatorname{succ}(a)=\\operatorname{succ}(b)$."
{a b : ℕ} : a = b → succ a = succ b := by
Hint "This is trivial -- we can just rewrite our proof of `a=b`.
But how do we get to that proof? Use the `intro` tactic."
intro h
Hint "Now we can indeed just `rw` `a` to `b`."
rw [h]
Hint (hidden := true) "Recall that `rfl` closes these goals."