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import NNG.Levels.Addition.Level_1
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import NNG.Levels.Addition.Level_2
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import NNG.Levels.Addition.Level_3
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import NNG.Levels.Addition.Level_4
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import NNG.Levels.Addition.Level_5
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import NNG.Levels.Addition.Level_6
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Game "NNG"
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World "Addition"
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Title "Addition World"
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Introduction
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"
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Welcome to Addition World. If you've done all four levels in tutorial world
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and know about `rewrite` and `rfl`, then you're in the right place. Here's
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a reminder of the things you're now equipped with which we'll need in this world.
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## Data:
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* a type called `ℕ` or `MyNat`.
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* a term `0 : ℕ`, interpreted as the number zero.
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* a function `succ : ℕ → ℕ`, with `succ n` interpreted as \"the number after `n`\".
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* Usual numerical notation `0,1,2` etc. (although `2` onwards will be of no use to us until much later ;-) ).
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* Addition (with notation `a + b`).
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## Theorems:
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* `add_zero (a : ℕ) : a + 0 = a`. Use with `rewrite [add_zero]`.
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* `add_succ (a b : ℕ) : a + succ(b) = succ(a + b)`. Use with `rewrite [add_succ]`.
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* The principle of mathematical induction. Use with `induction` (which we learn about in this chapter).
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## Tactics:
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* `rfl` : proves goals of the form `X = X`.
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* `rewrite [h]` : if `h` is a proof of `A = B`, changes all `A`'s in the goal to `B`'s.
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* `induction n with d hd` : we're going to learn this right now.
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You will also find all this information in your Inventory to read the documentation.
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"
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