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69 lines
1.6 KiB
Plaintext
69 lines
1.6 KiB
Plaintext
import NNG.Metadata
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import NNG.MyNat.Theorems.Addition
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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 4
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Title "the `apply` tactic"
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open MyNat
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Introduction
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"Let's do the same level again:
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$$
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\\begin{CD}
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P @>{h}>> Q @>{i}>> R \\\\
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@. @V{j}VV \\\\
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S @>{k}>> T @>{l}>> U \\\\
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\\end{CD}
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$$
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We are given $p \\in P$ and our goal is to find an element of $U$, or
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in other words to find a path through the maze that links $P$ to $U$.
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In level 3 we solved this by using `have`s to move forward, from $P$
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to $Q$ to $T$ to $U$. Using the `apply` tactic we can instead construct
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the path backwards, moving from $U$ to $T$ to $Q$ to $P$.
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"
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Statement
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"Given an element of $P$ we can define an element of $U$."
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(P Q R S T U: Type)
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(p : P)
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(h : P → Q)
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(i : Q → R)
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(j : Q → T)
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(k : S → T)
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(l : T → U) : U :=
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by
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Hint "Our goal is to construct an element of the set $U$. But $l:T\\to U$ is
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a function, so it would suffice to construct an element of $T$. Tell
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Lean this by starting the proof below with
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`apply l`"
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apply l
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Hint "Notice that our assumptions don't change but *the goal changes*
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from `Goal: U` to `Goal: T`.
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Keep `apply`ing functions until your goal is `P`, and try not
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to get lost!"
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Branch
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apply k
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Hint "Looks like you got lost. \"Undo\" the last step."
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apply j
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apply h
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Hint " Now solve this goal
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with `exact p`. Note: you will need to learn the difference between
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`exact p` (which works) and `exact P` (which doesn't, because $P$ is
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not an element of $P$)."
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exact p
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NewTactic apply
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DisabledTactic «have»
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Conclusion
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"
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"
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