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import NNG.Levels.Addition.Level_3
Game "NNG"
World "Addition"
Level 4
Title "`add_comm` (boss level)"
open MyNat
Introduction
"
[boss battle music]
Look in your inventory to see the proofs you have available.
These should be enough.
"
Statement MyNat.add_comm
"On the set of natural numbers, addition is commutative.
In other words, for all natural numbers $a$ and $b$, we have
$a + b = b + a$."
(a b : ℕ) : a + b = b + a := by
Hint (hidden := true) "You might want to start by induction."
Branch
induction a with d hd
· rw [zero_add]
rw [add_zero]
rfl
· rw [succ_add]
rw [hd]
rw [add_succ]
rfl
induction b with d hd
· rw [zero_add]
rw [add_zero]
rfl
· rw [add_succ]
rw [hd]
rw [succ_add]
rfl
LemmaTab "Add"
Conclusion
"
If you got this far -- nice! You're nearly ready to make a choice:
Multiplication World or Function World. But there are just a couple
more useful lemmas in Addition World which you should prove first.