b54a9ec9b9
We swap the arguments for `Membership.mem` so that when proceeded by a `SetLike` coercion, as is often the case in Mathlib, the resulting expression is recognized as eta expanded and reduce for many computations. The most beneficial outcome is that the discrimination tree keys for instances and simp lemmas concerning subsets become more robust resulting in more efficient searches. Closes `RFC` #4932 --------- Co-authored-by: Kim Morrison <kim@tqft.net> Co-authored-by: Henrik Böving <hargonix@gmail.com>
298 lines
7.2 KiB
Text
298 lines
7.2 KiB
Text
section Mathlib.Init.Order.Defs
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universe u
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variable {α : Type u}
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class Preorder (α : Type u) extends LE α, LT α
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theorem le_trans [Preorder α] : ∀ {a b c : α}, a ≤ b → b ≤ c → a ≤ c := sorry
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class PartialOrder (α : Type u) extends Preorder α where
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le_antisymm : ∀ a b : α, a ≤ b → b ≤ a → a = b
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end Mathlib.Init.Order.Defs
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section Mathlib.Init.Set
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set_option autoImplicit true
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def Set (α : Type u) := α → Prop
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namespace Set
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protected def Mem (s : Set α) (a : α) : Prop :=
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s a
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instance : Membership α (Set α) :=
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⟨Set.Mem⟩
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def image (f : α → β) (s : Set α) : Set β := fun b => ∃ a, ∃ (_ : a ∈ s), f a = b
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end Set
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end Mathlib.Init.Set
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section Mathlib.Data.Subtype
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attribute [coe] Subtype.val
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end Mathlib.Data.Subtype
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section Mathlib.Order.Notation
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class Sup (α : Type _) where
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sup : α → α → α
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class Inf (α : Type _) where
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inf : α → α → α
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@[inherit_doc]
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infixl:68 " ⊔ " => Sup.sup
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@[inherit_doc]
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infixl:69 " ⊓ " => Inf.inf
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class Top (α : Type _) where
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top : α
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class Bot (α : Type _) where
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bot : α
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notation "⊤" => Top.top
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notation "⊥" => Bot.bot
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end Mathlib.Order.Notation
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section Mathlib.Data.Set.Defs
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universe u
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namespace Set
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variable {α : Type u}
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@[coe, reducible] def Elem (s : Set α) : Type u := {x // x ∈ s}
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instance : CoeSort (Set α) (Type u) := ⟨Elem⟩
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end Set
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end Mathlib.Data.Set.Defs
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section Mathlib.Order.Basic
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universe u
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variable {α : Type u}
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def Preorder.lift {α β} [Preorder β] (f : α → β) : Preorder α where
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le x y := f x ≤ f y
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lt x y := f x < f y
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def PartialOrder.lift {α β} [PartialOrder β] (f : α → β) : PartialOrder α :=
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{ Preorder.lift f with le_antisymm := sorry }
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namespace Subtype
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instance le [LE α] {p : α → Prop} : LE (Subtype p) :=
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⟨fun x y ↦ (x : α) ≤ y⟩
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instance lt [LT α] {p : α → Prop} : LT (Subtype p) :=
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⟨fun x y ↦ (x : α) < y⟩
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instance partialOrder [PartialOrder α] (p : α → Prop) : PartialOrder (Subtype p) :=
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PartialOrder.lift (fun (a : Subtype p) ↦ (a : α))
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end Subtype
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end Mathlib.Order.Basic
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section Mathlib.Order.Lattice
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universe u
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variable {α : Type u}
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class SemilatticeSup (α : Type u) extends Sup α, PartialOrder α where
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protected le_sup_left : ∀ a b : α, a ≤ a ⊔ b
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protected le_sup_right : ∀ a b : α, b ≤ a ⊔ b
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protected sup_le : ∀ a b c : α, a ≤ c → b ≤ c → a ⊔ b ≤ c
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section SemilatticeSup
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variable [SemilatticeSup α] {a b c : α}
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theorem sup_le : a ≤ c → b ≤ c → a ⊔ b ≤ c := sorry
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end SemilatticeSup
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class SemilatticeInf (α : Type u) extends Inf α, PartialOrder α where
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protected inf_le_left : ∀ a b : α, a ⊓ b ≤ a
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protected inf_le_right : ∀ a b : α, a ⊓ b ≤ b
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protected le_inf : ∀ a b c : α, a ≤ b → a ≤ c → a ≤ b ⊓ c
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section SemilatticeInf
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variable [SemilatticeInf α] {a b c d : α}
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theorem inf_le_left : a ⊓ b ≤ a := sorry
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theorem le_inf_iff : a ≤ b ⊓ c ↔ a ≤ b ∧ a ≤ c := sorry
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end SemilatticeInf
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class Lattice (α : Type u) extends SemilatticeSup α, SemilatticeInf α
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namespace Subtype
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protected def semilatticeSup [SemilatticeSup α] {P : α → Prop}
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(Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) :
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SemilatticeSup { x : α // P x } where
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sup x y := ⟨x.1 ⊔ y.1, sorry⟩
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le_sup_left _ _ := sorry
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le_sup_right _ _ := sorry
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sup_le _ _ _ h1 h2 := sorry
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protected def semilatticeInf [SemilatticeInf α] {P : α → Prop}
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(Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) :
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SemilatticeInf { x : α // P x } where
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inf x y := ⟨x.1 ⊓ y.1, sorry⟩
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inf_le_left _ _ := sorry
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inf_le_right _ _ := sorry
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le_inf _ _ _ h1 h2 := sorry
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end Subtype
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end Mathlib.Order.Lattice
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section Mathlib.Order.BoundedOrder
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universe u
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class OrderTop (α : Type u) [LE α] extends Top α where
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le_top : ∀ a : α, a ≤ ⊤
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class OrderBot (α : Type u) [LE α] extends Bot α where
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bot_le : ∀ a : α, ⊥ ≤ a
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end Mathlib.Order.BoundedOrder
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section Mathlib.Data.Set.Intervals.Basic
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variable {α : Type _} [Preorder α]
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def Set.Iic (b : α) : Set α := fun x => x ≤ b
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end Mathlib.Data.Set.Intervals.Basic
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section Mathlib.Order.SetNotation
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universe u
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variable {α : Type u}
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class SupSet (α : Type _) where
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sSup : Set α → α
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class InfSet (α : Type _) where
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sInf : Set α → α
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export SupSet (sSup)
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export InfSet (sInf)
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end Mathlib.Order.SetNotation
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section Mathlib.Order.CompleteLattice
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variable {α : Type _}
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class CompleteSemilatticeSup (α : Type _) extends PartialOrder α, SupSet α where
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sSup_le : ∀ s a, (∀ b ∈ s, b ≤ a) → sSup s ≤ a
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section
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variable [CompleteSemilatticeSup α] {s t : Set α} {a b : α}
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theorem sSup_le : (∀ b ∈ s, b ≤ a) → sSup s ≤ a := sorry
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end
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class CompleteSemilatticeInf (α : Type _) extends PartialOrder α, InfSet α where
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le_sInf : ∀ s a, (∀ b ∈ s, a ≤ b) → a ≤ sInf s
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section
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variable [CompleteSemilatticeInf α] {s t : Set α} {a b : α}
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theorem le_sInf : (∀ b ∈ s, a ≤ b) → a ≤ sInf s := sorry
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end
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class CompleteLattice (α : Type _) extends Lattice α, CompleteSemilatticeSup α,
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CompleteSemilatticeInf α, Top α, Bot α where
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protected le_top : ∀ x : α, x ≤ ⊤
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protected bot_le : ∀ x : α, ⊥ ≤ x
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instance (priority := 100) CompleteLattice.toOrderTop [h : CompleteLattice α] : OrderTop α :=
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{ h with }
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instance (priority := 100) CompleteLattice.toOrderBot [h : CompleteLattice α] : OrderBot α :=
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{ h with }
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end Mathlib.Order.CompleteLattice
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section Mathlib.Order.LatticeIntervals
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variable {α : Type _}
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namespace Set
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namespace Iic
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instance semilatticeInf [SemilatticeInf α] {a : α} : SemilatticeInf (Iic a) :=
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Subtype.semilatticeInf fun _ _ hx _ => le_trans inf_le_left hx
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instance semilatticeSup [SemilatticeSup α] {a : α} : SemilatticeSup (Iic a) :=
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Subtype.semilatticeSup fun _ _ hx hy => sup_le hx hy
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instance [Lattice α] {a : α} : Lattice (Iic a) :=
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{ Iic.semilatticeInf, Iic.semilatticeSup with }
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instance orderTop [Preorder α] {a : α} : OrderTop (Iic a) := sorry
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instance orderBot [Preorder α] [OrderBot α] {a : α} : OrderBot (Iic a) := sorry
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end Iic
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end Set
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end Mathlib.Order.LatticeIntervals
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section Mathlib.Order.CompleteLatticeIntervals
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variable {α : Type _}
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namespace Set.Iic
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variable [CompleteLattice α] {a : α}
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def instCompleteLattice : CompleteLattice (Iic a) where
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sSup S := ⟨sSup (Set.image (fun x : Iic a => (x : α)) S), sorry⟩
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sInf S := ⟨a ⊓ sInf (Set.image (fun x : Iic a => (x : α)) S), sorry⟩
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sSup_le S b hb := sSup_le <| fun c' ⟨c, hc, hc'⟩ ↦ hc' ▸ hb c hc
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le_sInf S b hb := le_inf_iff.mpr
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⟨b.property, le_sInf fun d' ⟨d, hd, hd'⟩ ↦ hd' ▸ hb d hd⟩
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le_top := sorry
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bot_le := sorry
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example : CompleteLattice (Iic a) where
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sSup S := ⟨sSup (Set.image (fun x : Iic a => (x : α)) S), sorry⟩
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sInf S := ⟨a ⊓ sInf (Set.image (fun x : Iic a => (x : α)) S), sorry⟩
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sSup_le S b hb := sSup_le (α := α) <| fun c' ⟨c, hc, hc'⟩ ↦ hc' ▸ hb c hc
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le_sInf S b hb := (le_inf_iff (α := α)).mpr
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⟨b.property, le_sInf fun d' ⟨d, hd, hd'⟩ ↦ hd' ▸ hb d hd⟩
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le_top := sorry
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bot_le := sorry
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end Set.Iic
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end Mathlib.Order.CompleteLatticeIntervals
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