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49 lines
1.1 KiB
Plaintext
49 lines
1.1 KiB
Plaintext
import NNG.Metadata
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import NNG.MyNat.Addition
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import Std.Tactic.RCases
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Game "NNG"
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World "AdvProposition"
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Level 2
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Title "the `cases` tactic"
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open MyNat
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Introduction
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"
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If `P ∧ Q` is in the goal, then we can make progress with `split`.
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But what if `P ∧ Q` is a hypothesis? In this case, the `cases` tactic will enable
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us to extract proofs of `P` and `Q` from this hypothesis.
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The lemma below asks us to prove `P ∧ Q → Q ∧ P`, that is,
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symmetry of the \"and\" relation. The obvious first move is
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`intro h,`
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because the goal is an implication and this tactic is guaranteed
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to make progress. Now `h : P ∧ Q` is a hypothesis, and
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`cases h with p q,`
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will change `h`, the proof of `P ∧ Q`, into two proofs `p : P`
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and `q : Q`. From there, `split` and `exact` will get you home.
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"
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set_option tactic.hygienic false
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Statement --and_symm
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"If $P$ and $Q$ are true/false statements, then $P\\land Q\\implies Q\\land P$. "
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(P Q : Prop) : P ∧ Q → Q ∧ P := by
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intro h
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rcases h with ⟨p, q⟩
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constructor
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exact q
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exact p
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NewTactic rcases
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Conclusion
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"
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"
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