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feat(Order/PartialSups): add exists_partialSups_eq #34029

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feat(Order/PartialSups): add exists_partialSups_eq #34029

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20 changes: 20 additions & 0 deletions Mathlib/Order/PartialSups.lean
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Expand Up @@ -9,6 +9,7 @@ public import Mathlib.Data.Set.Finite.Lattice
public import Mathlib.Order.ConditionallyCompleteLattice.Indexed
public import Mathlib.Order.Interval.Finset.Nat
public import Mathlib.Order.SuccPred.Basic
import Mathlib.Data.Finset.Max

/-!
# The monotone sequence of partial supremums of a sequence
Expand Down Expand Up @@ -286,3 +287,22 @@ lemma partialSups_eq_biUnion_range (s : ℕ → Set α) (n : ℕ) :
simp [partialSups_eq_biSup, Nat.lt_succ_iff]

end Set

section LinearOrder
/-!
### Functions taking values on some `LinearOrder`.
-/

variable [Preorder ι] [LocallyFiniteOrderBot ι] [LinearOrder α]

theorem exists_partialSups_eq (f : ι → α) (i : ι) :
∃ j ≤ i, partialSups f i = f j := by
obtain ⟨j, hj⟩ : ∃ j ∈ Finset.Iic i, ∀ k ∈ Finset.Iic i, f k ≤ f j :=
Finset.exists_max_image _ _ ⟨i, Finset.mem_Iic.mpr le_rfl⟩
simp_all only [Finset.mem_Iic, partialSups, OrderHom.coe_mk]
use j, hj.1
apply le_antisymm
· exact Finset.sup'_le _ _ fun k hk => hj.2 k (Finset.mem_Iic.1 hk)
· exact Finset.le_sup' _ (Finset.mem_Iic.2 hj.1 )

end LinearOrder
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