Defs.lean

/-
Copyright (c) 2026 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
module

public import Mathlib.Order.Bounds.Defs
public import Mathlib.Order.Directed
public import Mathlib.Order.SetNotation

/-! # Conditionally complete partial orders

This file defines conditionally compelte partial orders, which are partial orders where every
nonempty, directed set which is bounded above has a least upper bound. This class extends `SupSet`
and the requirement is that `sSup` must be the least upper bound.

In fact, this file defines *three* classes:

+ `ConditionallyCompletePartialOrderSup` for partial orders with suprema,
+ `ConditionallyCompletePartialOrderInf` for partial orders with infima, and
+ `ConditionallyCompletePartialOrder` for partial orders with both suprema and infima

One common use case for these classes is the order on a von Neumann algebra, or W⋆-algebra.
In addition, `ℂ` can be made to satisfy this class (one must provide a suitable `SupSet` instance),
with the order `w ≤ z ↔ w.re ≤ z.re ∧ w.im = z.im`, which is available in the `ComplexOrder`
namespace.

## TODO

+ Write more `csSup`/`ciSup` lemmas for this class with `DirectedOn`/`Directed` assumptions.
  Then prove the existing lemmas for conditionally complete lattices in terms of these.

-/

--TODO: We could mimic more `sSup`/`iSup` lemmas

@[expose] public section

variable {ι : Sort*} {α : Type*}

/-- Conditionally complete partial orders (with infima) are partial orders
where every nonempty, directed set which is bounded below has a greatest lower bound. -/
class ConditionallyCompletePartialOrderInf (α : Type*)
    extends PartialOrder α, InfSet α where
  /-- For each nonempty, directed set `s` which is bounded below, `sInf s` is
  the greatest lower bound of `s`. -/
  isGLB_csInf_of_directed :
    ∀ s, DirectedOn (· ≥ ·) s → s.Nonempty → BddBelow s → IsGLB s (sInf s)

/-- Conditionally complete partial orders (with suprema) are partial orders
where every nonempty, directed set which is bounded above has a least upper bound. -/
@[to_dual existing]
class ConditionallyCompletePartialOrderSup (α : Type*)
    extends PartialOrder α, SupSet α where
  /-- For each nonempty, directed set `s` which is bounded above, `sSup s` is
  the least upper bound of `s`. -/
  isLUB_csSup_of_directed :
    ∀ s, DirectedOn (· ≤ ·) s → s.Nonempty → BddAbove s → IsLUB s (sSup s)

/-- Conditionally complete partial orders (with suprema and infimae) are partial orders
where every nonempty, directed set which is bounded above (respectively, below) has a
least upper (respectively, greatest lower) bound. -/
class ConditionallyCompletePartialOrder (α : Type*)
    extends ConditionallyCompletePartialOrderSup α, ConditionallyCompletePartialOrderInf α where

variable [ConditionallyCompletePartialOrderSup α]
variable {f : ι → α} {s : Set α} {a : α}

@[to_dual]
protected lemma DirectedOn.isLUB_csSup (h_dir : DirectedOn (· ≤ ·) s)
    (h_non : s.Nonempty) (h_bdd : BddAbove s) : IsLUB s (sSup s) :=
  ConditionallyCompletePartialOrderSup.isLUB_csSup_of_directed s h_dir h_non h_bdd

@[to_dual csInf_le]
protected lemma DirectedOn.le_csSup (hs : DirectedOn (· ≤ ·) s)
    (h_bdd : BddAbove s) (ha : a ∈ s) : a ≤ sSup s :=
  (hs.isLUB_csSup ⟨a, ha⟩ h_bdd).1 ha

@[to_dual le_csInf]
protected lemma DirectedOn.csSup_le (hd : DirectedOn (· ≤ ·) s) (h_non : s.Nonempty)
    (ha : ∀ b ∈ s, b ≤ a) : sSup s ≤ a :=
  (hd.isLUB_csSup h_non ⟨a, ha⟩).2 ha

@[to_dual ciInf_le]
protected lemma Directed.le_ciSup (hf : Directed (· ≤ ·) f)
    (hf_bdd : BddAbove (Set.range f)) (i : ι) : f i ≤ ⨆ j, f j :=
  hf.directedOn_range.le_csSup  hf_bdd <| Set.mem_range_self _

@[to_dual le_ciInf]
protected lemma Directed.ciSup_le [Nonempty ι] (hf : Directed (· ≤ ·) f)
    (ha : ∀ i, f i ≤ a) : ⨆ i, f i ≤ a :=
  hf.directedOn_range.csSup_le (Set.range_nonempty _) <| Set.forall_mem_range.2 ha