Principal upper/lower sets #

The results in this file all assume that the underlying type is equipped with at least a preorder.

Main declarations #

UpperSet.Ici: Principal upper set. Set.Ici as an upper set.
UpperSet.Ioi: Strict principal upper set. Set.Ioi as an upper set.
LowerSet.Iic: Principal lower set. Set.Iic as a lower set.
LowerSet.Iio: Strict principal lower set. Set.Iio as a lower set.

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def UpperSet.Ici {α : Type u_1} [Preorder α] (a : α) :

UpperSet α

Principal upper set. Set.Ici as an upper set. The smallest upper set containing a given element.

Equations

UpperSet.Ici a = { carrier := Set.Ici a, upper' := ⋯ }

Instances For

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def UpperSet.Ioi {α : Type u_1} [Preorder α] (a : α) :

UpperSet α

Strict principal upper set. Set.Ioi as an upper set.

Equations

UpperSet.Ioi a = { carrier := Set.Ioi a, upper' := ⋯ }

Instances For

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@[simp]

theorem UpperSet.coe_Ici {α : Type u_1} [Preorder α] (a : α) :

↑(Ici a) = Set.Ici a

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@[simp]

theorem UpperSet.coe_Ioi {α : Type u_1} [Preorder α] (a : α) :

↑(Ioi a) = Set.Ioi a

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@[simp]

theorem UpperSet.mem_Ici_iff {α : Type u_1} [Preorder α] {a b : α} :

b ∈ Ici a ↔ a ≤ b

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@[simp]

theorem UpperSet.mem_Ioi_iff {α : Type u_1} [Preorder α] {a b : α} :

b ∈ Ioi a ↔ a < b

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@[simp]

theorem UpperSet.map_Ici {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) :

(map f) (Ici a) = Ici (f a)

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@[simp]

theorem UpperSet.map_Ioi {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) :

(map f) (Ioi a) = Ioi (f a)

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theorem UpperSet.Ici_le_Ioi {α : Type u_1} [Preorder α] (a : α) :

Ici a ≤ Ioi a

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@[simp]

theorem UpperSet.Ici_bot {α : Type u_1} [Preorder α] [OrderBot α] :

Ici ⊥ = ⊥

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@[simp]

theorem UpperSet.Ioi_top {α : Type u_1} [Preorder α] [OrderTop α] :

Ioi ⊤ = ⊤

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@[simp]

theorem UpperSet.Ici_ne_top {α : Type u_1} [Preorder α] {a : α} :

Ici a ≠ ⊤

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@[simp]

theorem UpperSet.Ici_lt_top {α : Type u_1} [Preorder α] {a : α} :

Ici a < ⊤

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@[simp]

theorem UpperSet.le_Ici {α : Type u_1} [Preorder α] {s : UpperSet α} {a : α} :

s ≤ Ici a ↔ a ∈ s

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theorem UpperSet.Ici_injective {α : Type u_1} [PartialOrder α] :

Function.Injective Ici

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@[simp]

theorem UpperSet.Ici_inj {α : Type u_1} [PartialOrder α] {a b : α} :

Ici a = Ici b ↔ a = b

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theorem UpperSet.Ici_ne_Ici {α : Type u_1} [PartialOrder α] {a b : α} :

Ici a ≠ Ici b ↔ a ≠ b

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@[simp]

theorem UpperSet.Ici_sup {α : Type u_1} [SemilatticeSup α] (a b : α) :

Ici (a ⊔ b) = Ici a ⊔ Ici b

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@[simp]

theorem UpperSet.Ici_sSup {α : Type u_1} [CompleteLattice α] (S : Set α) :

Ici (sSup S) = ⨆ a ∈ S, Ici a

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@[simp]

theorem UpperSet.Ici_iSup {α : Type u_1} {ι : Sort u_3} [CompleteLattice α] (f : ι → α) :

Ici (⨆ (i : ι), f i) = ⨆ (i : ι), Ici (f i)

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theorem UpperSet.Ici_iSup₂ {α : Type u_1} {ι : Sort u_3} {κ : ι → Sort u_4} [CompleteLattice α] (f : (i : ι) → κ i → α) :

Ici (⨆ (i : ι), ⨆ (j : κ i), f i j) = ⨆ (i : ι), ⨆ (j : κ i), Ici (f i j)

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def LowerSet.Iic {α : Type u_1} [Preorder α] (a : α) :

LowerSet α

Principal lower set. Set.Iic as a lower set. The smallest lower set containing a given element.

Equations

LowerSet.Iic a = { carrier := Set.Iic a, lower' := ⋯ }

Instances For

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def LowerSet.Iio {α : Type u_1} [Preorder α] (a : α) :

LowerSet α

Strict principal lower set. Set.Iio as a lower set.

Equations

LowerSet.Iio a = { carrier := Set.Iio a, lower' := ⋯ }

Instances For

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@[simp]

theorem LowerSet.coe_Iic {α : Type u_1} [Preorder α] (a : α) :

↑(Iic a) = Set.Iic a

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@[simp]

theorem LowerSet.coe_Iio {α : Type u_1} [Preorder α] (a : α) :

↑(Iio a) = Set.Iio a

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@[simp]

theorem LowerSet.mem_Iic_iff {α : Type u_1} [Preorder α] {a b : α} :

b ∈ Iic a ↔ b ≤ a

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@[simp]

theorem LowerSet.mem_Iio_iff {α : Type u_1} [Preorder α] {a b : α} :

b ∈ Iio a ↔ b < a

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@[simp]

theorem LowerSet.map_Iic {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) :

(map f) (Iic a) = Iic (f a)

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@[simp]

theorem LowerSet.map_Iio {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α) :

(map f) (Iio a) = Iio (f a)

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theorem LowerSet.Ioi_le_Ici {α : Type u_1} [Preorder α] (a : α) :

Set.Ioi a ≤ Set.Ici a

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@[simp]

theorem LowerSet.Iic_top {α : Type u_1} [Preorder α] [OrderTop α] :

Iic ⊤ = ⊤

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@[simp]

theorem LowerSet.Iio_bot {α : Type u_1} [Preorder α] [OrderBot α] :

Iio ⊥ = ⊥

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@[simp]

theorem LowerSet.Iic_ne_bot {α : Type u_1} [Preorder α] {a : α} :

Iic a ≠ ⊥

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@[simp]

theorem LowerSet.bot_lt_Iic {α : Type u_1} [Preorder α] {a : α} :

⊥ < Iic a

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@[simp]

theorem LowerSet.Iic_le {α : Type u_1} [Preorder α] {s : LowerSet α} {a : α} :

Iic a ≤ s ↔ a ∈ s

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theorem LowerSet.Iic_injective {α : Type u_1} [PartialOrder α] :

Function.Injective Iic

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@[simp]

theorem LowerSet.Iic_inj {α : Type u_1} [PartialOrder α] {a b : α} :

Iic a = Iic b ↔ a = b

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theorem LowerSet.Iic_ne_Iic {α : Type u_1} [PartialOrder α] {a b : α} :

Iic a ≠ Iic b ↔ a ≠ b

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@[simp]

theorem LowerSet.Iic_inf {α : Type u_1} [SemilatticeInf α] (a b : α) :

Iic (a ⊓ b) = Iic a ⊓ Iic b

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@[simp]

theorem LowerSet.Iic_sInf {α : Type u_1} [CompleteLattice α] (S : Set α) :

Iic (sInf S) = ⨅ a ∈ S, Iic a

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@[simp]

theorem LowerSet.Iic_iInf {α : Type u_1} {ι : Sort u_3} [CompleteLattice α] (f : ι → α) :

Iic (⨅ (i : ι), f i) = ⨅ (i : ι), Iic (f i)

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theorem LowerSet.Iic_iInf₂ {α : Type u_1} {ι : Sort u_3} {κ : ι → Sort u_4} [CompleteLattice α] (f : (i : ι) → κ i → α) :

Iic (⨅ (i : ι), ⨅ (j : κ i), f i j) = ⨅ (i : ι), ⨅ (j : κ i), Iic (f i j)

Documentation

Mathlib.Order.UpperLower.Principal

Principal upper/lower sets #

Main declarations #