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76 lines
2.7 KiB
Plaintext
76 lines
2.7 KiB
Plaintext
import NNG.Metadata
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import NNG.MyNat.Addition
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Game "NNG"
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World "Function"
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Level 8
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Title "(P → Q) → ((Q → empty) → (P → empty))"
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open MyNat
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Introduction
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"
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Level 8 is the same as level 7, except we have replaced the
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set $F$ with the empty set $\\emptyset$. The same proof will work (after all, our
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previous proof worked for all sets, and the empty set is a set).
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But note that if you start with `intro f, intro h, intro p,`
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(which can incidentally be shortened to `intros f h p`),
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then the local context looks like this:
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```
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P Q : Type,
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f : P → Q,
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h : Q → empty,
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p : P
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⊢ empty
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```
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and your job is to construct an element of the empty set!
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This on the face of it seems hard, but what is going on is that
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our hypotheses (we have an element of $P$, and functions $P\\to Q$
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and $Q\\to\\emptyset$) are themselves contradictory, so
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I guess we are doing some kind of proof by contradiction at this point? However,
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if your next line is `apply h` then all of a sudden the goal
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seems like it might be possible again. If this is confusing, note
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that the proof of the previous world worked for all sets $F$, so in particular
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it worked for the empty set, you just probably weren't really thinking about
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this case explicitly beforehand. [Technical note to constructivists: I know
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that we are not doing a proof by contradiction. But how else do you explain
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to a classical mathematician that their goal is to prove something false
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and this is OK because their hypotheses don't add up?]
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"
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Statement
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"Whatever the sets $P$ and $Q$ are, we
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make an element of $\\operatorname{Hom}(\\operatorname{Hom}(P,Q),
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\\operatorname{Hom}(\\operatorname{Hom}(Q,\\emptyset),\\operatorname{Hom}(P,\\emptyset)))$."
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(P Q : Type) : (P → Q) → ((Q → Empty) → (P → Empty)) := by
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intros f h p
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Hint "
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Note that now your job is to construct an element of the empty set!
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This on the face of it seems hard, but what is going on is that
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our hypotheses (we have an element of $P$, and functions $P\\to Q$
|
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and $Q\\to\\emptyset$) are themselves contradictory, so
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I guess we are doing some kind of proof by contradiction at this point? However,
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if your next line is
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`apply {h}`
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then all of a sudden the goal
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seems like it might be possible again. If this is confusing, note
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that the proof of the previous world worked for all sets $F$, so in particular
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it worked for the empty set, you just probably weren't really thinking about
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this case explicitly beforehand. [Technical note to constructivists: I know
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that we are not doing a proof by contradiction. But how else do you explain
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to a classical mathematician that their goal is to prove something false
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and this is OK because their hypotheses don't add up?]
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"
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apply h
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apply f
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exact p
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Conclusion
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"
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"
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