Extended natural numbers form a complete linear order

This instance is not in Data.ENat.Basic to avoid dependency on Finsets.

We also restate some lemmas about WithTop for ENat to have versions that use Nat.cast instead of WithTop.some.

noncomputable instance instCompleteLinearOrderENat : CompleteLinearOrder ℕ∞

Equations

• instCompleteLinearOrderENat = inferInstanceAs (CompleteLinearOrder (WithTop ℕ))

theorem ENat.iSup_coe_eq_top {ι : Sort u_1} {f : ι → ℕ} : ⨆ (i : ι), ↑(f i) = ⊤ ↔ ¬BddAbove (Set.range f)

theorem ENat.iSup_coe_ne_top {ι : Sort u_1} {f : ι → ℕ} : ⨆ (i : ι), ↑(f i) ≠ ⊤ ↔ BddAbove (Set.range f)

theorem ENat.iSup_coe_lt_top {ι : Sort u_1} {f : ι → ℕ} : ⨆ (i : ι), ↑(f i) < ⊤ ↔ BddAbove (Set.range f)

theorem ENat.iInf_coe_eq_top {ι : Sort u_1} {f : ι → ℕ} : ⨅ (i : ι), ↑(f i) = ⊤ ↔ IsEmpty ι

theorem ENat.iInf_coe_ne_top {ι : Sort u_1} {f : ι → ℕ} : ⨅ (i : ι), ↑(f i) ≠ ⊤ ↔ Nonempty ι

theorem ENat.iInf_coe_lt_top {ι : Sort u_1} {f : ι → ℕ} : ⨅ (i : ι), ↑(f i) < ⊤ ↔ Nonempty ι

theorem ENat.coe_sSup {s : Set ℕ} : BddAbove s → ↑(sSup s) = ⨆ a ∈ s, ↑a

theorem ENat.coe_sInf {s : Set ℕ} (hs : s.Nonempty) : ↑(sInf s) = ⨅ a ∈ s, ↑a

theorem ENat.coe_iSup {ι : Sort u_1} {f : ι → ℕ} : BddAbove (Set.range f) → ↑(⨆ (i : ι), f i) = ⨆ (i : ι), ↑(f i)

theorem ENat.coe_iInf {ι : Sort u_1} {f : ι → ℕ} [Nonempty ι] : ↑(⨅ (i : ι), f i) = ⨅ (i : ι), ↑(f i)

@[simp]

theorem ENat.iInf_eq_top_of_isEmpty {ι : Sort u_1} {f : ι → ℕ} [IsEmpty ι] : ⨅ (i : ι), ↑(f i) = ⊤

theorem ENat.iInf_toNat {ι : Sort u_1} {f : ι → ℕ} : (⨅ (i : ι), ↑(f i)).toNat = ⨅ (i : ι), f i

theorem ENat.iInf_eq_zero {ι : Sort u_1} {f : ι → ℕ} : ⨅ (i : ι), ↑(f i) = 0 ↔ ∃ (i : ι), f i = 0

theorem ENat.sSup_eq_zero {s : Set ℕ∞} : sSup s = 0 ↔ ∀ a ∈ s, a = 0

theorem ENat.sInf_eq_zero {s : Set ℕ∞} : sInf s = 0 ↔ 0 ∈ s

theorem ENat.sSup_eq_zero' {s : Set ℕ∞} : sSup s = 0 ↔ s = ∅ ∨ s = {0}