feat(data/finset): new union, set difference, singleton lemmas #1702
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b-mehta
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b-mehta
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ChrisHughes24
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Nov 17, 2019
src/data/finset.lean
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| @@ -546,6 +555,16 @@ set.ext $ λ _, mem_sdiff | |||
| @[simp] lemma to_set_sdiff (s t : finset α) : (s \ t).to_set = s.to_set \ t.to_set := | |||
| by apply finset.coe_sdiff | |||
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| theorem union_sdiff_self_eq_union (s t : finset α) : s ∪ (t \ s) = s ∪ t := | |||
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This could be @[simp] with a left and right version
src/data/finset.lean
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| lemma union_sdiff_symm (s t : finset α) : s ∪ (t \ s) = t ∪ (s \ t) := | ||
| by rw [union_sdiff_self_eq_union, union_sdiff_self_eq_union, union_comm] | ||
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| lemma sdiff_empty_iff_subset (s t : finset α) : s \ t = ∅ ↔ s ⊆ t := |
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Suggested change
| lemma sdiff_empty_iff_subset (s t : finset α) : s \ t = ∅ ↔ s ⊆ t := | |
| lemma sdiff_eq_empty_iff_subset {s t : finset α} : s \ t = ∅ ↔ s ⊆ t := |
src/data/finset.lean
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| @@ -171,6 +171,15 @@ theorem singleton_inj {a b : α} : ι a = ι b ↔ a = b := | |||
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| @[simp] lemma coe_singleton (a : α) : ↑(ι a) = ({a} : set α) := rfl | |||
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| lemma singleton_iff_unique_mem (s : finset α) : (∃ a, s = finset.singleton a) ↔ ∃! a, a ∈ s := | |||
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This might be better stated in a slightly stronger way without the exists.
lemma eq_singleton_iff {s : finset α} {a : α} :
s = finset.singleton a ↔ a ∈ s ∧ ∀ x ∈ s, x = aThat way we know both the as are the same, and the stated theorem should be a trivial consequence of this, by simp only [eq_singleton_iff_unique_mem, exists_unique] does the trick.
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Thanks for the suggestions, I've implemented them all. |
ChrisHughes24
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Nov 17, 2019
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anrddh
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May 15, 2020
…rover-community#1702) * Singleton iff unique element lemma * Set difference lemmas * Changes from review
anrddh
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to anrddh/mathlib
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May 16, 2020
…rover-community#1702) * Singleton iff unique element lemma * Set difference lemmas * Changes from review
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