Documentation

Mathlib.Order.UpperLower.Principal

Principal upper/lower sets #

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

Main declarations #

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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.

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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.

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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.

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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.

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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)