Intervals as multisets

This file defines intervals as multisets.

Main declarations

In a LocallyFiniteOrder,

• Multiset.Icc: Closed-closed interval as a multiset.
• Multiset.Ico: Closed-open interval as a multiset.
• Multiset.Ioc: Open-closed interval as a multiset.
• Multiset.Ioo: Open-open interval as a multiset.

In a LocallyFiniteOrderTop,

• Multiset.Ici: Closed-infinite interval as a multiset.
• Multiset.Ioi: Open-infinite interval as a multiset.

In a LocallyFiniteOrderBot,

• Multiset.Iic: Infinite-open interval as a multiset.
• Multiset.Iio: Infinite-closed interval as a multiset.

TODO

Do we really need this file at all? (March 2024)

def Multiset.Icc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α) : Multiset α

The multiset of elements x such that a ≤ x and x ≤ b. Basically Set.Icc a b as a multiset.

Equations

• Multiset.Icc a b = (Finset.Icc a b).val

def Multiset.Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α) : Multiset α

The multiset of elements x such that a ≤ x and x < b. Basically Set.Ico a b as a multiset.

Equations

• Multiset.Ico a b = (Finset.Ico a b).val

def Multiset.Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α) : Multiset α

The multiset of elements x such that a < x and x ≤ b. Basically Set.Ioc a b as a multiset.

Equations

• Multiset.Ioc a b = (Finset.Ioc a b).val

def Multiset.Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α) : Multiset α

The multiset of elements x such that a < x and x < b. Basically Set.Ioo a b as a multiset.

Equations

• Multiset.Ioo a b = (Finset.Ioo a b).val

@[simp]

theorem Multiset.mem_Icc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α} : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b

@[simp]

theorem Multiset.mem_Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α} : x ∈ Ico a b ↔ a ≤ x ∧ x < b

@[simp]

theorem Multiset.mem_Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α} : x ∈ Ioc a b ↔ a < x ∧ x ≤ b

@[simp]

theorem Multiset.mem_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α} : x ∈ Ioo a b ↔ a < x ∧ x < b

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

The multiset of elements x such that a ≤ x. Basically Set.Ici a as a multiset.

Equations

• Multiset.Ici a = (Finset.Ici a).val

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

The multiset of elements x such that a < x. Basically Set.Ioi a as a multiset.

Equations

• Multiset.Ioi a = (Finset.Ioi a).val

@[simp]

theorem Multiset.mem_Ici {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α} : x ∈ Ici a ↔ a ≤ x

@[simp]

theorem Multiset.mem_Ioi {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α} : x ∈ Ioi a ↔ a < x

def Multiset.Iic {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (b : α) : Multiset α

The multiset of elements x such that x ≤ b. Basically Set.Iic b as a multiset.

Equations

• Multiset.Iic b = (Finset.Iic b).val

def Multiset.Iio {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (b : α) : Multiset α

The multiset of elements x such that x < b. Basically Set.Iio b as a multiset.

Equations

• Multiset.Iio b = (Finset.Iio b).val

@[simp]

theorem Multiset.mem_Iic {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {b x : α} : x ∈ Iic b ↔ x ≤ b

@[simp]

theorem Multiset.mem_Iio {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {b x : α} : x ∈ Iio b ↔ x < b

theorem Multiset.nodup_Icc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : (Icc a b).Nodup

theorem Multiset.nodup_Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : (Ico a b).Nodup

theorem Multiset.nodup_Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : (Ioc a b).Nodup

theorem Multiset.nodup_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : (Ioo a b).Nodup

@[simp]

theorem Multiset.Icc_eq_zero_iff {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : Icc a b = 0 ↔ ¬a ≤ b

@[simp]

theorem Multiset.Ico_eq_zero_iff {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : Ico a b = 0 ↔ ¬a < b

@[simp]

theorem Multiset.Ioc_eq_zero_iff {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : Ioc a b = 0 ↔ ¬a < b

@[simp]

theorem Multiset.Ioo_eq_zero_iff {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DenselyOrdered α] : Ioo a b = 0 ↔ ¬a < b

theorem Multiset.Icc_eq_zero {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : ¬a ≤ b → Icc a b = 0

Alias of the reverse direction of Multiset.Icc_eq_zero_iff.

theorem Multiset.Ico_eq_zero {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : ¬a < b → Ico a b = 0

Alias of the reverse direction of Multiset.Ico_eq_zero_iff.

theorem Multiset.Ioc_eq_zero {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : ¬a < b → Ioc a b = 0

Alias of the reverse direction of Multiset.Ioc_eq_zero_iff.

@[simp]

theorem Multiset.Ioo_eq_zero {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : ¬a < b) : Ioo a b = 0

@[simp]

theorem Multiset.Icc_eq_zero_of_lt {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b < a) : Icc a b = 0

@[simp]

theorem Multiset.Ico_eq_zero_of_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a) : Ico a b = 0

@[simp]

theorem Multiset.Ioc_eq_zero_of_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a) : Ioc a b = 0

@[simp]

theorem Multiset.Ioo_eq_zero_of_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a) : Ioo a b = 0

theorem Multiset.Ico_self {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α) : Ico a a = 0

theorem Multiset.Ioc_self {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α) : Ioc a a = 0

theorem Multiset.Ioo_self {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α) : Ioo a a = 0

theorem Multiset.left_mem_Icc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∈ Icc a b ↔ a ≤ b

theorem Multiset.left_mem_Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∈ Ico a b ↔ a < b

theorem Multiset.right_mem_Icc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∈ Icc a b ↔ a ≤ b

theorem Multiset.right_mem_Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∈ Ioc a b ↔ a < b

theorem Multiset.left_notMem_Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∉ Ioc a b

@[deprecated Multiset.left_notMem_Ioc (since := "2025-05-23")]

theorem Multiset.left_not_mem_Ioc {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∉ Ioc a b

Alias of Multiset.left_notMem_Ioc.

theorem Multiset.left_notMem_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∉ Ioo a b

@[deprecated Multiset.left_notMem_Ioo (since := "2025-05-23")]

theorem Multiset.left_not_mem_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : a ∉ Ioo a b

Alias of Multiset.left_notMem_Ioo.

theorem Multiset.right_notMem_Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∉ Ico a b

@[deprecated Multiset.right_notMem_Ico (since := "2025-05-23")]

theorem Multiset.right_not_mem_Ico {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∉ Ico a b

Alias of Multiset.right_notMem_Ico.

theorem Multiset.right_notMem_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∉ Ioo a b

@[deprecated Multiset.right_notMem_Ioo (since := "2025-05-23")]

theorem Multiset.right_not_mem_Ioo {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} : b ∉ Ioo a b

Alias of Multiset.right_notMem_Ioo.

theorem Multiset.Ico_filter_lt_of_le_left {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun (x : α) => x < c] (hca : c ≤ a) : filter (fun (x : α) => x < c) (Ico a b) = ∅

theorem Multiset.Ico_filter_lt_of_right_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun (x : α) => x < c] (hbc : b ≤ c) : filter (fun (x : α) => x < c) (Ico a b) = Ico a b

theorem Multiset.Ico_filter_lt_of_le_right {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun (x : α) => x < c] (hcb : c ≤ b) : filter (fun (x : α) => x < c) (Ico a b) = Ico a c

theorem Multiset.Ico_filter_le_of_le_left {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun (x : α) => c ≤ x] (hca : c ≤ a) : filter (fun (x : α) => c ≤ x) (Ico a b) = Ico a b

theorem Multiset.Ico_filter_le_of_right_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun (x : α) => b ≤ x] : filter (fun (x : α) => b ≤ x) (Ico a b) = ∅

theorem Multiset.Ico_filter_le_of_left_le {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun (x : α) => c ≤ x] (hac : a ≤ c) : filter (fun (x : α) => c ≤ x) (Ico a b) = Ico c b

@[simp]

theorem Multiset.Icc_self {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a : α) : Icc a a = {a}

theorem Multiset.Ico_cons_right {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a ≤ b) : b ::ₘ Ico a b = Icc a b

theorem Multiset.Ioo_cons_left {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a < b) : a ::ₘ Ioo a b = Ico a b

theorem Multiset.Ico_disjoint_Ico {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b c d : α} (h : b ≤ c) : Disjoint (Ico a b) (Ico c d)

@[simp]

theorem Multiset.Ico_inter_Ico_of_le {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] {a b c d : α} (h : b ≤ c) : Ico a b ∩ Ico c d = 0

theorem Multiset.Ico_filter_le_left {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun (x : α) => x ≤ a] (hab : a < b) : filter (fun (x : α) => x ≤ a) (Ico a b) = {a}

theorem Multiset.card_Ico_eq_card_Icc_sub_one {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α) : (Ico a b).card = (Icc a b).card - 1

theorem Multiset.card_Ioc_eq_card_Icc_sub_one {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α) : (Ioc a b).card = (Icc a b).card - 1

theorem Multiset.card_Ioo_eq_card_Ico_sub_one {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α) : (Ioo a b).card = (Ico a b).card - 1

theorem Multiset.card_Ioo_eq_card_Icc_sub_two {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α) : (Ioo a b).card = (Icc a b).card - 2

theorem Multiset.Ico_subset_Ico_iff {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) : Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂

theorem Multiset.Ico_add_Ico_eq_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) : Ico a b + Ico b c = Ico a c

theorem Multiset.Ico_inter_Ico {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α} : Ico a b ∩ Ico c d = Ico (max a c) (min b d)

@[simp]

theorem Multiset.Ico_filter_lt {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α) : filter (fun (x : α) => x < c) (Ico a b) = Ico a (min b c)

@[simp]

theorem Multiset.Ico_filter_le {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α) : filter (fun (x : α) => c ≤ x) (Ico a b) = Ico (max a c) b

@[simp]

theorem Multiset.Ico_sub_Ico_left {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α) : Ico a b - Ico a c = Ico (max a c) b

@[simp]

theorem Multiset.Ico_sub_Ico_right {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α) : Ico a b - Ico c b = Ico a (min b c)