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73 lines
2.4 KiB
Plaintext
73 lines
2.4 KiB
Plaintext
import NNG.Metadata
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import NNG.Levels.Addition.Level_6
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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 1
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Title "the `exact` tactic"
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open MyNat
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Introduction
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"
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## A new kind of goal.
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All through addition world, our goals have been theorems,
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and it was our job to find the proofs.
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**The levels in function world aren't theorems**. This is the only world where
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the levels aren't theorems in fact. In function world the object of a level
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is to create an element of the set in the goal. The goal will look like `Goal: X`
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with $X$ a set and you get rid of the goal by constructing an element of $X$.
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I don't know if you noticed this, but you finished
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essentially every goal of Addition World (and Multiplication World and Power World,
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if you played them) with `rfl`.
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This tactic is no use to us here.
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We are going to have to learn a new way of solving goals – the `exact` tactic.
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## The `exact` tactic
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If you can explicitly see how to make an element of your goal set,
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i.e. you have a formula for it, then you can just write `exact <formula>`
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and this will close the goal.
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"
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Statement
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"If $P$ is true, and $P\\implies Q$ is also true, then $Q$ is true."
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(P Q : Prop) (p : P) (h : P → Q) : Q := by
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Hint
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"In this situation, we have sets $P$ and $Q$ (but Lean calls them types),
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and an element $p$ of $P$ (written `p : P`
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but meaning $p\\in P$). We also have a function $h$ from $P$ to $Q$,
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and our goal is to construct an
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element of the set $Q$. It's clear what to do *mathematically* to solve
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this goal -- we can
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make an element of $Q$ by applying the function $h$ to
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the element $p$. But how to do it in Lean? There are at least two ways
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to explain this idea to Lean,
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and here we will learn about one of them, namely the method which
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uses the `exact` tactic.
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Concretely, `h p` is an element of type `Q`, so you can use `exact h p` to use it.
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Note that while in mathematics you might write $h(p)$, in Lean you always avoid brackets
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for function application: `h p`. Brackets are only used for grouping elements, for
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example for repeated funciton application, you could write `g (h p)`.
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"
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Hint (hidden := true) "
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**Important note**: Note that `exact h P,` won't work (with a capital $P$);
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this is a common error I see from beginners.
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$P$ is not an element of $P$, it's $p$ that is an element of $P$.
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So try `exact h p`.
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"
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exact h p
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NewTactic exact
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Conclusion
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"
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"
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