Theorems · Definition · order theory
Function.argminOn
{α : Type u_1} → {β : Type u_2} → (α → β) → [inst : LT β] → [WellFoundedLT β] → (s : Set α) → s.Nonempty → αGiven a function f : α → β where β carries a well-founded <, and a non-empty subset s
of α, this is an element of s whose image under f is minimal in the sense of
Function.not_lt_argminOn.
See also Set.Finite.exists_minimalFor and related lemmas for the case when α or s is finite.
TODO Consider removing this definition in favour of exists_minimalFor_of_wellFoundedLT.
- Defined in
- Mathlib.Order.WellFounded
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- LTWellFoundedLT
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Set.Nonemptystatement and proof · cited by 2,627
- WellFoundedLTstatement and proof · cited by 491
- WellFounded.minproof · cited by 33
Cited by9
Results whose statement or proof uses this declaration.
- Function.argminOn_memstatement · cited by 7
- Caratheodory.minCardFinsetOfMemConvexHullproof · cited by 6
- Function.argminOn_lestatement · cited by 5
- exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_powproof · cited by 2
- Function.not_lt_argminOnstatement · cited by 2
- sInf_eq_argmin_onstatement · cited by 1
- Function.isMinimalFor_argminOnstatement and proof · cited by 0
- Function.argminOn.congr_simpstatement and proof · cited by 0