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54 lines
1.3 KiB
Plaintext
54 lines
1.3 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 4
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Title "apply"
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open MyNat
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Introduction
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"
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Let's do the same level again:
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We are given a proof $p$ of $P$ and our goal is to find a proof 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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Our goal is to prove $U$. But $l:T\\implies U$ is
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an implication which we are assuming, so it would suffice to prove $T$.
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Tell Lean this by starting the proof below with
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`apply l,`
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and notice that our assumptions don't change but *the goal changes*
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from `⊢ U` to `⊢ T`.
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Keep `apply`ing implications until your goal is `P`, and try not
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to get lost! 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 a proof of $P$).
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"
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Statement
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"We can solve a maze."
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(P Q R S T U: Prop) (p : P) (h : P → Q) (i : Q → R)
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(j : Q → T) (k : S → T) (l : T → U) : U := by
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apply l
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apply j
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apply h
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exact p
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DisabledTactic «have»
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Conclusion
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"
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"
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