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import TestGame.Metadata
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Level 4
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Title "Addition"
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Introduction
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"
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Peano defined addition `a + b` by induction on `b`, or,
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more precisely, by *recursion* on `b`. He first explained how to add 0 to a number:
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this is the base case.
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* `add_zero (a : ℕ) : a + 0 = a`
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We will call this theorem `add_zero`. It has just appeared in your inventory!
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Mathematicians sometimes call it \"Lemma 2.1\" or \"Hypothesis P6\" or something. But
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computer scientists call it `add_zero` because it tells you
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what the answer to \"$x$ add zero\" is. It's a *much* better name than \"Lemma 2.1\".
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Even better, we can use the rewrite tactic with `add_zero`.
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If you ever see `x + 0` in your goal, `rewrite [add_zero]` will simplify it to `x`.
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This is because `add_zero` is a proof that `x + 0 = x` (more precisely,
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`add_zero x` is a proof that `x + 0 = x` but Lean can figure out the `x` from the context).
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Now here's the inductive step. If you know how to add `d` to `a`, then
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Peano tells you how to add `succ(d)` to `a`. It looks like this:
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* `add_succ (a d : ℕ) : a + succ(d) = succ (a + d)`
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What's going on here is that we assume `a + d` is already
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defined, and we define `a + succ(d)` to be the number after it.
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This is also in your inventory now -- `add_succ` tells you
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how to add a successor to something. If you ever see `... + succ ...`
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in your goal, you should be able to use `rewrite [add_succ]` to make
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progress. Here is a simple example where we shall see both. Let's prove
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that $x$ add the number after $0$ is the number after $x$.
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Observe that the goal mentions `... + succ ...`. So type
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`rewrite [add_succ]`
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and hit enter; see the goal change.
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"
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Statement (a : ℕ ) : a + succ 0 = succ a := by
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rewrite [add_succ]
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rewrite [add_zero]
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rfl
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Message (a : ℕ) : succ (a + 0) = succ a => "
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Do you see that the goal now mentions ` ... + 0 ...`? So type
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`rewrite [add_zero]`
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and try to finish the level alone from there.
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"
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Conclusion "Congratulations for completing your fourth level! This is the end of the tutorial part
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of the game. Serious things start in the next level."
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Tactics rfl rewrite
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Lemmas add_succ add_zero |