lean4game/server/nng/NNG/Levels/Addition/Level_5.lean

import NNG.Metadata
import NNG.MyNat.Addition
import NNG.Levels.Addition.Level_4

Game "NNG"
World "Addition"
Level 5
Title "succ_eq_add_one"

open MyNat

namespace AdditionWorld

theorem add_comm (a b : ℕ) : a + b = b + a := by
  induction b with d hd
  · rw [zero_add]
    rw [add_zero]
    rfl
  · rw [add_succ]
    rw [hd]
    rw [succ_add]
    rfl

theorem one_eq_succ_zero : (1 : ℕ) = succ 0 := by simp only

NewLemma MyNat.add_comm MyNat.one_eq_succ_zero

Introduction
"
I've just added `one_eq_succ_zero` (a proof of $1 = \\operatorname{succ}(0)$)
to your list of theorems; this is true
by definition of $1$, but we didn't need it until now.

Levels 5 and 6 are the two last levels in Addition World.
Level 5 involves the number $1$. When you see a $1$ in your goal,
you can write `rw [one_eq_succ_zero]` to get back
to something which only mentions `0`. This is a good move because $0$ is easier for us to
manipulate than $1$ right now, because we have
some theorems about $0$ (`zero_add`, `add_zero`), but, other than `1 = succ 0`,
no theorems at all which mention $1$. Let's prove one now.
"

Statement --MyNat.succ_eq_add_one
"For any natural number $n$, we have
$ \\operatorname{succ}(n) = n+1$ ."
    (n : ℕ) : succ n = n + 1 := by
  rw [one_eq_succ_zero]
  rw [add_succ]
  rw [add_zero]
  rfl

Conclusion
"
Well done! On to the last level!
"