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46 lines
1.2 KiB
Plaintext
46 lines
1.2 KiB
Plaintext
import NNG.Metadata
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import NNG.MyNat.Addition
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Game "NNG"
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World "AdvProposition"
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Level 4
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Title ""
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open MyNat
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Introduction
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"
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The mathematical statement $P\\iff Q$ is equivalent to $(P\\implies Q)\\land(Q\\implies P)$. The `rcases`
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and `constructor` tactics work on hypotheses and goals (respectively) of the form `P ↔ Q`. If you need
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to write an `↔` arrow you can do so by typing `\\iff`, but you shouldn't need to.
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"
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Statement --iff_trans
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"If $P$, $Q$ and $R$ are true/false statements, then
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$P\\iff Q$ and $Q\\iff R$ together imply $P\\iff R$."
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(P Q R : Prop) : (P ↔ Q) → (Q ↔ R) → (P ↔ R) := by
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Hint "Similar to \"and\", you can use `intro` and `rcases` to add the `P ↔ Q` to your
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assumptions and split it into its constituent parts."
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Branch
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intro hpq
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intro hqr
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Hint "Now you want to use `rcases {hpq} with ⟨pq, qp⟩`."
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rcases hpq with ⟨hpq, hqp⟩
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rcases hqr with ⟨hqr, hrq⟩
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intro ⟨pq, qp⟩
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intro ⟨qr, rq⟩
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Hint "If you want to prove an iff-statement, you can use `constructor` to split it
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into its two implications."
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constructor
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· intro p
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apply qr
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apply pq
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exact p
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· intro r
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apply qp
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apply rq
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exact r
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NewDefinition Iff
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