57 lines
1.2 KiB
Lean4
57 lines
1.2 KiB
Lean4
import Game.Metadata
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Game "NNG"
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World "Function"
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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 a function, 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 sets, and our goal is to construct a function
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which takes an element of $P$ and outputs a function from $Q$ to $P$.
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We don't know anything at all about the sets $P$ and $Q$, so initially
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this seems like a bit of a tall order. But let's give it a go.
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"
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Statement
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"We define an element of $\\operatorname{Hom}(P,\\operatorname{Hom}(Q,P))$."
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(P Q : Type) : P → (Q → P) := by
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Hint "Our goal is `P → X` for some set $X=\\operatorname\{Hom}(Q,P)$, and if our
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goal is to construct a function then we almost always want to use the
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`intro` tactic from level 2, Lean's version of \"let $p\\in P$ be arbitrary.\"
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So let's start with
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```
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intro p
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```"
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intro p
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Hint "
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We now have an arbitrary element $p\\in P$ and we are supposed to be constructing
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a function $Q\to P$. Well, how about the constant function, which
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sends everything to $p$?
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This will work. So let $q\\in Q$ be arbitrary:
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```
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intro q
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```"
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intro q
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Hint "and then let's output `p`.
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```
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exact p
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```"
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exact p
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Conclusion
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"
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"
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