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import NNG.Levels.AdvAddition.Level_8
Game "NNG"
World "AdvAddition"
Level 9
Title "succ_ne_zero"
open MyNat
Introduction
"
Levels 9 to 13 introduce the last axiom of Peano, namely
that $0\\not=\\operatorname{succ}(a)$. The proof of this is called `zero_ne_succ a`.
```
zero_ne_succ (a : ¬) : 0 ≠ succ a
```
$X\\ne Y$ is *defined to mean* $X = Y\\implies{\\tt False}$, similar to how `¬A` was defined.
"
-- TODO: I dont think there is a `symmetry` tactic anymore.
-- The `symmetry` tactic will turn any goal of the form `R x y` into `R y x`,
-- if `R` is a symmetric binary relation (for example `=` or `≠`).
-- In particular, you can prove `succ_ne_zero` below by first using
-- `symmetry` and then `exact zero_ne_succ a`.
Statement MyNat.succ_ne_zero
"Zero is not the successor of any natural number."
(a : ℕ) : succ a ≠ 0 := by
Hint "You have several options how to start. One would be to recall that `≠` is defined as
`(· = ·) → False` and start with `intro`. Or do `rw [Ne, Not]` to explicitely remove the
`≠`. Or you could use the lemma `Ne.symm (a b : ℕ) : a ≠ b → b ≠ a` which I just added to your