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81 lines
2.4 KiB
Plaintext
81 lines
2.4 KiB
Plaintext
import NNG.Metadata
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import NNG.MyNat.Addition
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Game "NNG"
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World "Proposition"
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Level 5
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Title "P → (Q → P)"
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open MyNat
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Introduction
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"
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In this level, our goal is to construct an implication, like in level 2.
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```
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⊢ P → (Q → P)
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```
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So $P$ and $Q$ are propositions, and our goal is to prove
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that $P\\implies(Q\\implies P)$.
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We don't know whether $P$, $Q$ are true or false, so initially
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this seems like a bit of a tall order. But let's give it a go. Delete
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the `sorry` and let's think about how to proceed.
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Our goal is `P → X` for some true/false statement $X$, and if our
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goal is to construct an implication then we almost always want to use the
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`intro` tactic from level 2, Lean's version of \"assume $P$\", or more precisely
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\"assume $p$ is a proof of $P$\". So let's start with
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`intro p,`
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and we then find ourselves in this state:
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```
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P Q : Prop,
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p : P
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⊢ Q → P
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```
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We now have a proof $p$ of $P$ and we are supposed to be constructing
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a proof of $Q\\implies P$. So let's assume that $Q$ is true and try
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and prove that $P$ is true. We assume $Q$ like this:
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`intro q,`
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and now we have to prove $P$, but have a proof handy:
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`exact p,`
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"
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Statement
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"For any propositions $P$ and $Q$, we always have
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$P\\implies(Q\\implies P)$."
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(P Q : Prop) : P → (Q → P) := by
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intro p
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intro q
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exact p
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Conclusion
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"
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A mathematician would treat the proposition $P\\implies(Q\\implies P)$
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as the same as the proposition $P\\land Q\\implies P$,
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because to give a proof of either of these is just to give a method which takes
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a proof of $P$ and a proof of $Q$, and returns a proof of $P$. Thinking of the
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goal as $P\\land Q\\implies P$ we see why it is provable.
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## Did you notice?
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I wrote `P → (Q → P)` but Lean just writes `P → Q → P`. This is because
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computer scientists adopt the convention that `→` is *right associative*,
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which is a fancy way of saying \"when we write `P → Q → R`, we mean `P → (Q → R)`.
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Mathematicians would never dream of writing something as ambiguous as
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$P\\implies Q\\implies R$ (they are not really interested in proving abstract
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propositions, they would rather work with concrete ones such as Fermat's Last Theorem),
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so they do not have a convention for where the brackets go. It's important to
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remember Lean's convention though, or else you will get confused. If your goal
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is `P → Q → R` then you need to know whether `intro h` will create `h : P` or `h : P → Q`.
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Make sure you understand which one.
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"
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