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import NNG.Metadata
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import NNG.MyNat.Addition
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import NNG.Levels.Addition.Level_3
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Game "NNG"
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World "Addition"
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Level 4
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Title "`add_comm` (boss level)"
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open MyNat
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namespace AdditionWorld
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theorem succ_add (a b : ℕ) : succ a + b = succ (a + b) := by
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induction b with d hd
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· rw [add_zero]
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rw [add_zero]
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rfl
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· rw [add_succ]
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rw [hd]
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rw [add_succ]
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rfl
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Introduction
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"
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[boss battle music]
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Look in your inventory to see the proofs you have available.
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These should be enough.
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"
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Statement add_comm
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"On the set of natural numbers, addition is commutative.
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In other words, for all natural numbers $a$ and $b$, we have
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$a + b = b + a$."
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(a b : ℕ) : a + b = b + a := by
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Branch
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induction a with d hd
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· rw [zero_add]
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rw [add_zero]
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rfl
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· rw [succ_add]
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rw [hd]
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rw [add_succ]
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rfl
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induction b with d hd
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· rw [zero_add]
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rw [add_zero]
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rfl
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· rw [add_succ]
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rw [hd]
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rw [succ_add]
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rfl
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NewLemma MyNat.succ_add
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Conclusion
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"
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If you got this far -- nice! You're nearly ready to make a choice:
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Multiplication World or Function World. But there are just a couple
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more useful lemmas in Addition World which you should prove first.
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Press on to level 5.
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"
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