Theorems · Inductive type · order theory
LowerSet
(α : Type u_1) → [LE α] → Type u_1
A lower set in an order α is a set such that any element less than one of its
members is also a member. Also called down-set, downward-closed set.
- Defined in
- Mathlib.Order.Defs.Unbundled
- Cited by
- 230 results in Mathlib
- Foundations
- Depth 1 from the axioms, rests on 2 definitions · uses no axioms
- Assumes
- LE
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites0
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
Nothing in Mathlib beyond the foundations.
Cited by263
Results whose statement or proof uses this declaration.
- lowerClosurestatement · cited by 83
- LowerSet.Iicstatement · cited by 37
- LowerSet.extstatement and proof · cited by 29
- LowerSet.complstatement and proof · cited by 17
- UpperSet.complstatement · cited by 17
- LowerSet.lowerstatement and proof · cited by 13
- LowerSet.mapstatement and proof · cited by 12
- Order.Ideal.principalproof · cited by 12
- subset_lowerClosurestatement · cited by 12
- LowerSet.carrierstatement and proof · cited by 11
- LowerSet.erasestatement and proof · cited by 9
- Order.Ideal.toLowerSetstatement · cited by 9
Showing the 200 most cited of 263.