mathlib documentation

data.finset.locally_finite

Intervals as finsets #

This file provides basic results about all the finset.Ixx, which are defined in order.locally_finite.

TODO #

This file was originally only about finset.Ico a b where a b : ℕ. No care has yet been taken to generalize these lemmas properly and many lemmas about Icc, Ioc, Ioo are missing. In general, what's to do is taking the lemmas in data.x.intervals and abstract away the concrete structure.

@[simp]
theorem finset.nonempty_Icc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.nonempty_Ico {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.nonempty_Ioc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.nonempty_Ioo {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} [densely_ordered α] :
@[simp]
theorem finset.Icc_eq_empty_iff {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.Ico_eq_empty_iff {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.Ioc_eq_empty_iff {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.Ioo_eq_empty_iff {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} [densely_ordered α] :
theorem finset.Icc_eq_empty {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
¬a bfinset.Icc a b =

Alias of Icc_eq_empty_iff.

theorem finset.Ico_eq_empty {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
¬a < bfinset.Ico a b =

Alias of Ico_eq_empty_iff.

theorem finset.Ioc_eq_empty {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
¬a < bfinset.Ioc a b =

Alias of Ioc_eq_empty_iff.

@[simp]
theorem finset.Ioo_eq_empty {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} (h : ¬a < b) :
@[simp]
theorem finset.Icc_eq_empty_of_lt {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} (h : b < a) :
@[simp]
theorem finset.Ico_eq_empty_of_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} (h : b a) :
@[simp]
theorem finset.Ioc_eq_empty_of_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} (h : b a) :
@[simp]
theorem finset.Ioo_eq_empty_of_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} (h : b a) :
@[simp]
theorem finset.Ico_self {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) :
@[simp]
theorem finset.Ioc_self {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) :
@[simp]
theorem finset.Ioo_self {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) :
theorem finset.left_mem_Icc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
a finset.Icc a b a b
theorem finset.left_mem_Ico {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
a finset.Ico a b a < b
theorem finset.right_mem_Icc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
b finset.Icc a b a b
theorem finset.right_mem_Ioc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
b finset.Ioc a b a < b
@[simp]
theorem finset.left_not_mem_Ioc {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.left_not_mem_Ioo {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.right_not_mem_Ico {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
@[simp]
theorem finset.right_not_mem_Ioo {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} :
theorem finset.Ico_filter_lt_of_le_left {α : Type u_1} [preorder α] [locally_finite_order α] {a b c : α} [decidable_pred (λ (_x : α), _x < c)] (hca : c a) :
finset.filter (λ (x : α), x < c) (finset.Ico a b) =
theorem finset.Ico_filter_lt_of_right_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b c : α} [decidable_pred (λ (_x : α), _x < c)] (hbc : b c) :
finset.filter (λ (x : α), x < c) (finset.Ico a b) = finset.Ico a b
theorem finset.Ico_filter_lt_of_le_right {α : Type u_1} [preorder α] [locally_finite_order α] {a b c : α} [decidable_pred (λ (_x : α), _x < c)] (hcb : c b) :
finset.filter (λ (x : α), x < c) (finset.Ico a b) = finset.Ico a c
theorem finset.Ico_filter_le_of_le_left {α : Type u_1} [preorder α] [locally_finite_order α] {a b c : α} [decidable_pred (has_le.le c)] (hca : c a) :
finset.filter (λ (x : α), c x) (finset.Ico a b) = finset.Ico a b
theorem finset.Ico_filter_le_of_right_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} [decidable_pred (has_le.le b)] :
finset.filter (λ (x : α), b x) (finset.Ico a b) =
theorem finset.Ico_filter_le_of_left_le {α : Type u_1} [preorder α] [locally_finite_order α] {a b c : α} [decidable_pred (has_le.le c)] (hac : a c) :
finset.filter (λ (x : α), c x) (finset.Ico a b) = finset.Ico c b
def set.fintype_of_mem_bounds {α : Type u_1} [preorder α] [locally_finite_order α] {a b : α} {s : set α} [decidable_pred (λ (_x : α), _x s)] (ha : a lower_bounds s) (hb : b upper_bounds s) :

A set with upper and lower bounds in a locally finite order is a fintype

Equations
theorem bdd_below.finite_of_bdd_above {α : Type u_1} [preorder α] [locally_finite_order α] {s : set α} (h₀ : bdd_below s) (h₁ : bdd_above s) :
theorem finset.filter_lt_lt_eq_Ioo {α : Type u_1} [preorder α] [locally_finite_order α] (a b : α) [fintype α] [decidable_pred (λ (j : α), a < j j < b)] :
finset.filter (λ (j : α), a < j j < b) finset.univ = finset.Ioo a b
theorem finset.filter_lt_le_eq_Ioc {α : Type u_1} [preorder α] [locally_finite_order α] (a b : α) [fintype α] [decidable_pred (λ (j : α), a < j j b)] :
finset.filter (λ (j : α), a < j j b) finset.univ = finset.Ioc a b
theorem finset.filter_le_lt_eq_Ico {α : Type u_1} [preorder α] [locally_finite_order α] (a b : α) [fintype α] [decidable_pred (λ (j : α), a j j < b)] :
finset.filter (λ (j : α), a j j < b) finset.univ = finset.Ico a b
theorem finset.filter_le_le_eq_Icc {α : Type u_1} [preorder α] [locally_finite_order α] (a b : α) [fintype α] [decidable_pred (λ (j : α), a j j b)] :
finset.filter (λ (j : α), a j j b) finset.univ = finset.Icc a b
theorem finset.filter_lt_eq_Ioi {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) [fintype α] [order_top α] [decidable_pred (has_lt.lt a)] :
finset.filter (λ (j : α), a < j) finset.univ = finset.Ioi a
theorem finset.filter_le_eq_Ici {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) [fintype α] [order_top α] [decidable_pred (has_le.le a)] :
theorem finset.filter_gt_eq_Iio {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) [fintype α] [order_bot α] [decidable_pred (λ (_x : α), _x < a)] :
finset.filter (λ (j : α), j < a) finset.univ = finset.Iio a
theorem finset.filter_ge_eq_Iic {α : Type u_1} [preorder α] [locally_finite_order α] (a : α) [fintype α] [order_bot α] [decidable_pred (λ (_x : α), _x a)] :
@[simp]
theorem finset.Icc_self {α : Type u_1} [partial_order α] [locally_finite_order α] (a : α) :
finset.Icc a a = {a}
@[simp]
theorem finset.Icc_eq_singleton_iff {α : Type u_1} [partial_order α] [locally_finite_order α] {a b c : α} :
finset.Icc a b = {c} a = c b = c
theorem finset.Icc_erase_left {α : Type u_1} [partial_order α] [locally_finite_order α] {a b : α} [decidable_eq α] :
theorem finset.Icc_erase_right {α : Type u_1} [partial_order α] [locally_finite_order α] {a b : α} [decidable_eq α] :
theorem finset.Ico_insert_right {α : Type u_1} [partial_order α] [locally_finite_order α] {a b : α} [decidable_eq α] (h : a b) :
theorem finset.Ioo_insert_left {α : Type u_1} [partial_order α] [locally_finite_order α] {a b : α} [decidable_eq α] (h : a < b) :
@[simp]
theorem finset.Ico_inter_Ico_consecutive {α : Type u_1} [partial_order α] [locally_finite_order α] [decidable_eq α] (a b c : α) :
theorem finset.Ico_disjoint_Ico_consecutive {α : Type u_1} [partial_order α] [locally_finite_order α] [decidable_eq α] (a b c : α) :
theorem finset.Ico_filter_le_left {α : Type u_1} [partial_order α] [locally_finite_order α] {a b : α} [decidable_pred (λ (_x : α), _x a)] (hab : a < b) :
finset.filter (λ (x : α), x a) (finset.Ico a b) = {a}
theorem finset.card_Ico_eq_card_Icc_sub_one {α : Type u_1} [partial_order α] [locally_finite_order α] (a b : α) :
theorem finset.card_Ioc_eq_card_Icc_sub_one {α : Type u_1} [partial_order α] [locally_finite_order α] (a b : α) :
theorem finset.card_Ioo_eq_card_Ico_sub_one {α : Type u_1} [partial_order α] [locally_finite_order α] (a b : α) :
theorem finset.card_Ioo_eq_card_Icc_sub_two {α : Type u_1} [partial_order α] [locally_finite_order α] (a b : α) :
theorem finset.Ico_subset_Ico_iff {α : Type u_1} [linear_order α] [locally_finite_order α] {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) :
finset.Ico a₁ b₁ finset.Ico a₂ b₂ a₂ a₁ b₁ b₂
theorem finset.Ico_union_Ico_eq_Ico {α : Type u_1} [linear_order α] [locally_finite_order α] {a b c : α} (hab : a b) (hbc : b c) :
theorem finset.Ico_union_Ico' {α : Type u_1} [linear_order α] [locally_finite_order α] {a b c d : α} (hcb : c b) (had : a d) :
theorem finset.Ico_union_Ico {α : Type u_1} [linear_order α] [locally_finite_order α] {a b c d : α} (h₁ : min a b max c d) (h₂ : min c d max a b) :
theorem finset.Ico_inter_Ico {α : Type u_1} [linear_order α] [locally_finite_order α] {a b c d : α} :
@[simp]
theorem finset.Ico_filter_lt {α : Type u_1} [linear_order α] [locally_finite_order α] (a b c : α) :
finset.filter (λ (x : α), x < c) (finset.Ico a b) = finset.Ico a (min b c)
@[simp]
theorem finset.Ico_filter_le {α : Type u_1} [linear_order α] [locally_finite_order α] (a b c : α) :
finset.filter (λ (x : α), c x) (finset.Ico a b) = finset.Ico (max a c) b
@[simp]
theorem finset.Ico_diff_Ico_left {α : Type u_1} [linear_order α] [locally_finite_order α] (a b c : α) :
@[simp]
theorem finset.Ico_diff_Ico_right {α : Type u_1} [linear_order α] [locally_finite_order α] (a b c : α) :
theorem bdd_below.finite {α : Type u_1} [preorder α] [order_top α] [locally_finite_order α] {s : set α} (hs : bdd_below s) :
theorem bdd_above.finite {α : Type u_1} [preorder α] [order_bot α] [locally_finite_order α] {s : set α} (hs : bdd_above s) :
theorem finset.image_add_right_Icc {α : Type u_1} [ordered_cancel_add_comm_monoid α] [has_exists_add_of_le α] [decidable_eq α] [locally_finite_order α] (a b c : α) :
finset.image (λ (x : α), x + c) (finset.Icc a b) = finset.Icc (a + c) (b + c)
theorem finset.image_add_right_Ico {α : Type u_1} [ordered_cancel_add_comm_monoid α] [has_exists_add_of_le α] [decidable_eq α] [locally_finite_order α] (a b c : α) :
finset.image (λ (x : α), x + c) (finset.Ico a b) = finset.Ico (a + c) (b + c)
theorem finset.image_add_right_Ioc {α : Type u_1} [ordered_cancel_add_comm_monoid α] [has_exists_add_of_le α] [decidable_eq α] [locally_finite_order α] (a b c : α) :
finset.image (λ (x : α), x + c) (finset.Ioc a b) = finset.Ioc (a + c) (b + c)
theorem finset.image_add_right_Ioo {α : Type u_1} [ordered_cancel_add_comm_monoid α] [has_exists_add_of_le α] [decidable_eq α] [locally_finite_order α] (a b c : α) :
finset.image (λ (x : α), x + c) (finset.Ioo a b) = finset.Ioo (a + c) (b + c)