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[Merged by Bors] - feat(data/nat/pairing): add some nat.pair lemmas #8740

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[Merged by Bors] - feat(data/nat/pairing): add some nat.pair lemmas #8740

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cb132c2
initial commit
JasonKYi Aug 18, 2021
4b80f6d
generalize and change rhs
JasonKYi Aug 18, 2021
0ab470d
change to supr_unpair and infi unpair
JasonKYi Aug 18, 2021
16d2b83
golf
JasonKYi Aug 18, 2021
cdf9327
add set version
JasonKYi Aug 18, 2021
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20 changes: 20 additions & 0 deletions src/data/nat/pairing.lean
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Expand Up @@ -125,10 +125,30 @@ end
end nat
open nat

section complete_lattice

lemma supr_unpair {α} [complete_lattice α] (f : ℕ → ℕ → α) :
(⨆ n : ℕ, f n.unpair.1 n.unpair.2) = ⨆ i j : ℕ, f i j :=
by rw [← (supr_prod : (⨆ i : ℕ × ℕ, f i.1 i.2) = _), ← nat.surjective_unpair.supr_comp]

lemma infi_unpair {α} [complete_lattice α] (f : ℕ → ℕ → α) :
(⨅ n : ℕ, f n.unpair.1 n.unpair.2) = ⨅ i j : ℕ, f i j :=
supr_unpair (show ℕ → ℕ → order_dual α, from f)
Comment on lines +130 to +136
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In your original application did you want the Union version? If so, you should probably add that one below these (proved in terms of them).

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Can I not directly apply the supr version in my use case?

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Not with a rw. It will work with an apply


end complete_lattice

namespace set

lemma Union_unpair_prod {α β} {s : ℕ → set α} {t : ℕ → set β} :
(⋃ n : ℕ, (s n.unpair.fst).prod (t n.unpair.snd)) = (⋃ n, s n).prod (⋃ n, t n) :=
by { rw [← Union_prod], convert surjective_unpair.Union_comp _, refl }

lemma Union_unpair {α} (f : ℕ → ℕ → set α) :
(⋃ n : ℕ, f n.unpair.1 n.unpair.2) = ⋃ i j : ℕ, f i j :=
supr_unpair f

lemma Inter_unpair {α} (f : ℕ → ℕ → set α) :
(⋂ n : ℕ, f n.unpair.1 n.unpair.2) = ⋂ i j : ℕ, f i j :=
infi_unpair f

end set