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import NNG.Metadata
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import NNG.MyNat.Multiplication
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Game "NNG"
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World "Function"
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Level 2
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Title "the intro tactic"
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open MyNat
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Introduction
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"
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Let's make a function. Let's define the function on the natural
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numbers which sends a natural number $n$ to $3n+2$. You
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see that the goal is `ℕ → ℕ`. A mathematician
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might denote this set with some exotic name such as
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$\\operatorname{Hom}(\\mathbb{N},\\mathbb{N})$,
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but computer scientists use notation `X → Y` to denote the set of
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functions from `X` to `Y` and this name definitely has its merits.
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In type theory, `X → Y` is a type (the type of all functions from $X$ to $Y$),
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and `f : X → Y` means that `f` is a term
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of this type, i.e., $f$ is a function from $X$ to $Y$.
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To define a function $X\\to Y$ we need to choose an arbitrary
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element $x\\in X$ and then, perhaps using $x$, make an element of $Y$.
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The Lean tactic for \"let $x\\in X$ be arbitrary\" is `intro x`.
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"
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Statement
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"We define a function from ℕ to ℕ."
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: ℕ → ℕ := by
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Hint "To solve this goal,
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you have to come up with a function from `ℕ`
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to `ℕ`. Start with
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`intro n`"
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intro n
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Hint "Our job now is to construct a natural number, which is
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allowed to depend on ${n}$. We can do this using `exact` and
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writing a formula for the function we want to define. For example
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we imported addition and multiplication at the top of this file,
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so
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`exact 3 * {n} + 2`
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will close the goal, ultimately defining the function $f({n})=3{n}+2$."
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exact 3 * n + 2
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NewTactic intro
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Conclusion
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"
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## Rule of thumb:
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If your goal is `P → Q` then `intro p` will make progress.
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"
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