feat(data/set/finite): add lemmas relating definitions of infinite sets #18617
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Includes the following lemmas
* `infinite.exists_nat_lt`,
* `arb_large_infinite`
* `infinite_iff_arb_large`
Which state the following
```lean
lemma infinite.exists_nat_lt {s : set ℕ} (hs : s.infinite) (n : ℕ) : ∃ m ∈ s, n < m
lemma arb_large_infinite {S : set ℕ} : S.infinite → ∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ m > n)
lemma infinite_iff_arb_large {S : set ℕ } : S.infinite ↔ (∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ m > n))
```
dignissimus
changed the title
eric-wieser
changed the title
eric-wieser
reviewed
Mar 18, 2023
Comment on lines
+997
to
+1002
| lemma arb_large_infinite {S : set ℕ} : S.infinite → ∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ m > n) := | ||
| begin | ||
| intros h n, | ||
| apply bex_def.mp, | ||
| apply set.infinite.exists_nat_lt h, | ||
| end |
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This lemma is pretty much identical to infinite.exists_nat_lt, so I don't think it's worth having.
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After removing arb_large_infinite, could I then replace the definition of infinite_iff_arb_large with the following?
lemma infinite_iff_arb_large {S : set ℕ } : S.infinite ↔ ∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ n < m) :=
begin
split,
{ intro hS,
have exists_nat_lt := hS.exists_nat_lt,
finish, },
{ exact infinite_exists_nat_lt, }
end
eric-wieser
reviewed
Mar 18, 2023
| apply set.infinite.exists_nat_lt h, | ||
| end | ||
|
|
||
| lemma infinite_exists_nat_lt {S : set ℕ} : (∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ m > n)) → S.infinite := |
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I think this would be a better name and statement
Suggested change
| lemma infinite_exists_nat_lt {S : set ℕ} : (∀ (n : ℕ ), ∃ (m : ℕ), (m ∈ S ∧ m > n)) → S.infinite := | |
| lemma infinite_of_forall_exists_nat_lt {S : set ℕ} (h : ∀ n, ∃ m ∈ s, n < m) : S.infinite := |
|
Oh, I'm afraid our CI doesn't run on forks. You'll need to ask on Zulip for permission to push directly to a branch in our repository, and then reopen this PR. |
|
Sure, will do. Thank you! |
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Add lemmas relating definitions of infinite sets
Includes the following lemmas
infinite.exists_nat_lt,arb_large_infiniteinfinite_iff_arb_largeWhich state the following