49 lines
1.2 KiB
Lean4
49 lines
1.2 KiB
Lean4
import Game.Metadata
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import Game.MyNat.Addition
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World "AdvProposition"
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Level 2
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Title "the `rcases` 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 `constructor`.
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But what if `P ∧ Q` is a hypothesis? In this case, the `rcases` tactic will enable
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us to extract proofs of `P` and `Q` from this hypothesis.
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"
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/-- If $P$ and $Q$ are true/false statements, then $P\\land Q\\implies Q\\land P$. -/
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Statement -- and_symm
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(P Q : Prop) : P ∧ Q → Q ∧ P := by
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Hint "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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```
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intro h
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```
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because the goal is an implication and this tactic is guaranteed
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to make progress."
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intro h
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Hint "Now `{h} : P ∧ Q` is a hypothesis, and
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```
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rcases {h} with ⟨p, q⟩
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```
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will change `{h}`, the proof of `P ∧ Q`, into two proofs `p : P`
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and `q : Q`.
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You can write `⟨p, q⟩` with `\\<>` or `\\<` and `\\>`. Note that `rcases h` by itself will just
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automatically name the new assumptions."
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rcases h with ⟨p, q⟩
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Hint "Now a combination of `constructor` and `exact` will get you home."
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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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