feat: port Order.LatticeIntervals (#1070)

Lots of renames, but easy.

I've added names for all instances

Co-authored-by: Chris Hughes <33847686+ChrisHughes24@users.noreply.github.com>

Author: mcdoll

Mathlib.lean

@@ -335,6 +335,7 @@ import Mathlib.Order.Hom.Set
 import Mathlib.Order.InitialSeg
 import Mathlib.Order.Iterate
 import Mathlib.Order.Lattice
+import Mathlib.Order.LatticeIntervals
 import Mathlib.Order.Max
 import Mathlib.Order.MinMax
 import Mathlib.Order.Monotone.Basic

Mathlib/Order/LatticeIntervals.lean

@@ -0,0 +1,183 @@
+/-
+Copyright (c) 2020 Aaron Anderson. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Aaron Anderson
+! This file was ported from Lean 3 source module order.lattice_intervals
+! leanprover-community/mathlib commit d012cd09a9b256d870751284dd6a29882b0be105
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Order.Bounds.Basic
+
+/-!
+# Intervals in Lattices
+
+In this file, we provide instances of lattice structures on intervals within lattices.
+Some of them depend on the order of the endpoints of the interval, and thus are not made
+global instances. These are probably not all of the lattice instances that could be placed on these
+intervals, but more can be added easily along the same lines when needed.
+
+## Main definitions
+
+In the following, `*` can represent either `c`, `o`, or `i`.
+  * `Set.Ic*.order_bot`
+  * `Set.Ii*.semillatice_inf`
+  * `Set.I*c.order_top`
+  * `Set.I*c.semillatice_inf`
+  * `Set.I**.lattice`
+  * `Set.Iic.bounded_order`, within an `order_bot`
+  * `Set.Ici.bounded_order`, within an `order_top`
+-/
+
+
+variable {α : Type _}
+
+namespace Set
+
+namespace Ico
+
+instance semilatticeInf [SemilatticeInf α] {a b : α} : SemilatticeInf (Ico a b) :=
+  Subtype.semilatticeInf fun _ _ hx hy => ⟨le_inf hx.1 hy.1, lt_of_le_of_lt inf_le_left hx.2⟩
+
+/-- `Ico a b` has a bottom element whenever `a < b`. -/
+@[reducible]
+protected def orderBot [PartialOrder α] {a b : α} (h : a < b) : OrderBot (Ico a b) :=
+  (isLeast_Ico h).orderBot
+#align set.Ico.order_bot Set.Ico.orderBot
+
+end Ico
+
+namespace Iio
+
+instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Iio a) :=
+  Subtype.semilatticeInf fun _ _ hx _ => lt_of_le_of_lt inf_le_left hx
+
+end Iio
+
+namespace Ioc
+
+instance semilatticeSup [SemilatticeSup α] {a b : α} : SemilatticeSup (Ioc a b) :=
+  Subtype.semilatticeSup fun _ _ hx hy => ⟨lt_of_lt_of_le hx.1 le_sup_left, sup_le hx.2 hy.2⟩
+
+/-- `Ioc a b` has a top element whenever `a < b`. -/
+@[reducible]
+protected def orderTop [PartialOrder α] {a b : α} (h : a < b) : OrderTop (Ioc a b) :=
+  (isGreatest_Ioc h).orderTop
+#align set.Ioc.order_top Set.Ioc.orderTop
+
+end Ioc
+
+namespace Ioi
+
+instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ioi a) :=
+  Subtype.semilatticeSup fun _ _ hx _ => lt_of_lt_of_le hx le_sup_left
+
+end Ioi
+
+namespace Iic
+
+instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Iic a) :=
+  Subtype.semilatticeInf fun _ _ hx _ => le_trans inf_le_left hx
+
+instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Iic a) :=
+  Subtype.semilatticeSup fun _ _ hx hy => sup_le hx hy
+
+instance [Lattice α] {a : α} : Lattice (Iic a) :=
+  { Iic.semilatticeInf, Iic.semilatticeSup with }
+
+instance orderTop [Preorder α] {a : α} :
+    OrderTop (Iic a) where
+  top := ⟨a, le_refl a⟩
+  le_top x := x.prop
+
+@[simp]
+theorem coe_top [Preorder α] {a : α} : (⊤ : Iic a) = a :=
+  rfl
+#align set.Iic.coe_top Set.Iic.coe_top
+
+instance orderBot [Preorder α] [OrderBot α] {a : α} :
+    OrderBot (Iic a) where
+  bot := ⟨⊥, bot_le⟩
+  bot_le := fun ⟨_, _⟩ => Subtype.mk_le_mk.2 bot_le
+
+@[simp]
+theorem coe_bot [Preorder α] [OrderBot α] {a : α} : (⊥ : Iic a) = (⊥ : α) :=
+  rfl
+#align set.Iic.coe_bot Set.Iic.coe_bot
+
+instance [Preorder α] [OrderBot α] {a : α} : BoundedOrder (Iic a) :=
+  { Iic.orderTop, Iic.orderBot with }
+
+end Iic
+
+namespace Ici
+
+instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Ici a) :=
+  Subtype.semilatticeInf fun _ _ hx hy => le_inf hx hy
+
+instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Ici a) :=
+  Subtype.semilatticeSup fun _ _ hx _ => le_trans hx le_sup_left
+
+instance lattice [Lattice α] {a : α} : Lattice (Ici a) :=
+  { Ici.semilatticeInf, Ici.semilatticeSup with }
+
+instance distribLattice [DistribLattice α] {a : α} : DistribLattice (Ici a) :=
+  { Ici.lattice with le_sup_inf := fun _ _ _ => le_sup_inf }
+
+instance orderBot [Preorder α] {a : α} :
+    OrderBot (Ici a) where
+  bot := ⟨a, le_refl a⟩
+  bot_le x := x.prop
+
+@[simp]
+theorem coe_bot [Preorder α] {a : α} : ↑(⊥ : Ici a) = a :=
+  rfl
+#align set.Ici.coe_bot Set.Ici.coe_bot
+
+instance orderTop [Preorder α] [OrderTop α] {a : α} :
+    OrderTop (Ici a) where
+  top := ⟨⊤, le_top⟩
+  le_top := fun ⟨_, _⟩ => Subtype.mk_le_mk.2 le_top
+
+@[simp]
+theorem coe_top [Preorder α] [OrderTop α] {a : α} : ↑(⊤ : Ici a) = (⊤ : α) :=
+  rfl
+#align set.Ici.coe_top Set.Ici.coe_top
+
+instance boundedOrder [Preorder α] [OrderTop α] {a : α} : BoundedOrder (Ici a) :=
+  { Ici.orderTop, Ici.orderBot with }
+
+end Ici
+
+namespace Icc
+
+instance semilatticeInf [SemilatticeInf α] {a b : α} : SemilatticeInf (Icc a b) :=
+  Subtype.semilatticeInf fun _ _ hx hy => ⟨le_inf hx.1 hy.1, le_trans inf_le_left hx.2⟩
+
+instance semilatticeSup [SemilatticeSup α] {a b : α} : SemilatticeSup (Icc a b) :=
+  Subtype.semilatticeSup fun _ _ hx hy => ⟨le_trans hx.1 le_sup_left, sup_le hx.2 hy.2⟩
+
+instance lattice [Lattice α] {a b : α} : Lattice (Icc a b) :=
+  { Icc.semilatticeInf, Icc.semilatticeSup with }
+
+/-- `Icc a b` has a bottom element whenever `a ≤ b`. -/
+@[reducible]
+protected def orderBot [Preorder α] {a b : α} (h : a ≤ b) : OrderBot (Icc a b) :=
+  (isLeast_Icc h).orderBot
+#align set.Icc.order_bot Set.Icc.orderBot
+
+/-- `Icc a b` has a top element whenever `a ≤ b`. -/
+@[reducible]
+protected def orderTop [Preorder α] {a b : α} (h : a ≤ b) : OrderTop (Icc a b) :=
+  (isGreatest_Icc h).orderTop
+#align set.Icc.order_top Set.Icc.orderTop
+
+/-- `Icc a b` is a `bounded_order` whenever `a ≤ b`. -/
+@[reducible]
+protected def boundedOrder [Preorder α] {a b : α} (h : a ≤ b) : BoundedOrder (Icc a b) :=
+  { Icc.orderTop h, Icc.orderBot h with }
+#align set.Icc.bounded_order Set.Icc.boundedOrder
+
+end Icc
+
+end Set