mathlib documentation

order.cover

The covering relation #

This file defines the covering relation in an order. b is said to cover a if a < b and there is no element in between. We say that b weakly covers a if a ≤ b and there is no element between a and b. In a partial order this is equivalent to a ⋖ b ∨ a = b, in a preorder this is equivalent to a ⋖ b ∨ (a ≤ b ∧ b ≤ a)

Notation #

def wcovby {α : Type u_1} [preorder α] (a b : α) :
Prop

wcovby a b means that a = b or b covers a. This means that a ≤ b and there is no element in between.

Equations
Instances for wcovby
theorem wcovby.le {α : Type u_1} [preorder α] {a b : α} (h : a ⩿ b) :
a b
theorem wcovby.refl {α : Type u_1} [preorder α] (a : α) :
a ⩿ a
theorem wcovby.rfl {α : Type u_1} [preorder α] {a : α} :
a ⩿ a
@[protected]
theorem eq.wcovby {α : Type u_1} [preorder α] {a b : α} (h : a = b) :
a ⩿ b
theorem wcovby_of_le_of_le {α : Type u_1} [preorder α] {a b : α} (h1 : a b) (h2 : b a) :
a ⩿ b
theorem has_le.le.wcovby_of_le {α : Type u_1} [preorder α] {a b : α} (h1 : a b) (h2 : b a) :
a ⩿ b

Alias of wcovby_of_le_of_le`.

theorem wcovby.wcovby_iff_le {α : Type u_1} [preorder α] {a b : α} (hab : a ⩿ b) :
b ⩿ a b a
theorem wcovby_of_eq_or_eq {α : Type u_1} [preorder α] {a b : α} (hab : a b) (h : ∀ (c : α), a cc bc = a c = b) :
a ⩿ b
theorem not_wcovby_iff {α : Type u_1} [preorder α] {a b : α} (h : a b) :
¬a ⩿ b ∃ (c : α), a < c c < b

If a ≤ b, then b does not cover a iff there's an element in between.

@[protected, instance]
def wcovby.is_refl {α : Type u_1} [preorder α] :
theorem wcovby.Ioo_eq {α : Type u_1} [preorder α] {a b : α} (h : a ⩿ b) :
theorem wcovby.of_image {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (h : f a ⩿ f b) :
a ⩿ b
theorem wcovby.image {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (hab : a ⩿ b) (h : (set.range f).ord_connected) :
f a ⩿ f b
theorem set.ord_connected.apply_wcovby_apply_iff {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (h : (set.range f).ord_connected) :
f a ⩿ f b a ⩿ b
@[simp]
theorem apply_wcovby_apply_iff {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} {E : Type u_3} [order_iso_class E α β] (e : E) :
e a ⩿ e b a ⩿ b
@[simp]
theorem to_dual_wcovby_to_dual_iff {α : Type u_1} [preorder α] {a b : α} :
@[simp]
theorem wcovby.to_dual {α : Type u_1} [preorder α] {a b : α} :

Alias of the reverse direction of to_dual_wcovby_to_dual_iff`.

theorem wcovby.of_dual {α : Type u_1} [preorder α] {a b : αᵒᵈ} :

Alias of the reverse direction of of_dual_wcovby_of_dual_iff`.

theorem wcovby.eq_or_eq {α : Type u_1} [partial_order α] {a b c : α} (h : a ⩿ b) (h2 : a c) (h3 : c b) :
c = a c = b
theorem wcovby.le_and_le_iff {α : Type u_1} [partial_order α] {a b c : α} (h : a ⩿ b) :
a c c b c = a c = b
theorem wcovby.Icc_eq {α : Type u_1} [partial_order α] {a b : α} (h : a ⩿ b) :
set.Icc a b = {a, b}
theorem wcovby.Ico_subset {α : Type u_1} [partial_order α] {a b : α} (h : a ⩿ b) :
set.Ico a b {a}
theorem wcovby.Ioc_subset {α : Type u_1} [partial_order α] {a b : α} (h : a ⩿ b) :
set.Ioc a b {b}
def covby {α : Type u_1} [has_lt α] (a b : α) :
Prop

covby a b means that b covers a: a < b and there is no element in between.

Equations
Instances for covby
theorem covby.lt {α : Type u_1} [has_lt α] {a b : α} (h : a b) :
a < b
theorem not_covby_iff {α : Type u_1} [has_lt α] {a b : α} (h : a < b) :
¬a b ∃ (c : α), a < c c < b

If a < b, then b does not cover a iff there's an element in between.

theorem exists_lt_lt_of_not_covby {α : Type u_1} [has_lt α] {a b : α} (h : a < b) :
¬a b(∃ (c : α), a < c c < b)

Alias of the forward direction of not_covby_iff`.

theorem has_lt.lt.exists_lt_lt {α : Type u_1} [has_lt α] {a b : α} (h : a < b) :
¬a b(∃ (c : α), a < c c < b)

Alias of exists_lt_lt_of_not_covby`.

theorem not_covby {α : Type u_1} [has_lt α] {a b : α} [densely_ordered α] :
¬a b

In a dense order, nothing covers anything.

theorem densely_ordered_iff_forall_not_covby {α : Type u_1} [has_lt α] :
densely_ordered α ∀ (a b : α), ¬a b
@[simp]
theorem to_dual_covby_to_dual_iff {α : Type u_1} [has_lt α] {a b : α} :
@[simp]
theorem of_dual_covby_of_dual_iff {α : Type u_1} [has_lt α] {a b : αᵒᵈ} :
theorem covby.to_dual {α : Type u_1} [has_lt α] {a b : α} :

Alias of the reverse direction of to_dual_covby_to_dual_iff`.

theorem covby.of_dual {α : Type u_1} [has_lt α] {a b : αᵒᵈ} :

Alias of the reverse direction of of_dual_covby_of_dual_iff`.

theorem covby.le {α : Type u_1} [preorder α] {a b : α} (h : a b) :
a b
@[protected]
theorem covby.ne {α : Type u_1} [preorder α] {a b : α} (h : a b) :
a b
theorem covby.ne' {α : Type u_1} [preorder α] {a b : α} (h : a b) :
b a
@[protected]
theorem covby.wcovby {α : Type u_1} [preorder α] {a b : α} (h : a b) :
a ⩿ b
theorem wcovby.covby_of_not_le {α : Type u_1} [preorder α] {a b : α} (h : a ⩿ b) (h2 : ¬b a) :
a b
theorem wcovby.covby_of_lt {α : Type u_1} [preorder α] {a b : α} (h : a ⩿ b) (h2 : a < b) :
a b
theorem covby_iff_wcovby_and_lt {α : Type u_1} [preorder α] {a b : α} :
a b a ⩿ b a < b
theorem covby_iff_wcovby_and_not_le {α : Type u_1} [preorder α] {a b : α} :
a b a ⩿ b ¬b a
theorem wcovby_iff_covby_or_le_and_le {α : Type u_1} [preorder α] {a b : α} :
a ⩿ b a b a b b a
@[protected, instance]
def covby.is_irrefl {α : Type u_1} [preorder α] :
theorem covby.Ioo_eq {α : Type u_1} [preorder α] {a b : α} (h : a b) :
theorem covby.of_image {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (h : f a f b) :
a b
theorem covby.image {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (hab : a b) (h : (set.range f).ord_connected) :
f a f b
theorem set.ord_connected.apply_covby_apply_iff {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} (f : α ↪o β) (h : (set.range f).ord_connected) :
f a f b a b
@[simp]
theorem apply_covby_apply_iff {α : Type u_1} {β : Type u_2} [preorder α] [preorder β] {a b : α} {E : Type u_3} [order_iso_class E α β] (e : E) :
e a e b a b
theorem wcovby.covby_of_ne {α : Type u_1} [partial_order α] {a b : α} (h : a ⩿ b) (h2 : a b) :
a b
theorem covby_iff_wcovby_and_ne {α : Type u_1} [partial_order α] {a b : α} :
a b a ⩿ b a b
theorem wcovby_iff_covby_or_eq {α : Type u_1} [partial_order α] {a b : α} :
a ⩿ b a b a = b
theorem covby.Ico_eq {α : Type u_1} [partial_order α] {a b : α} (h : a b) :
set.Ico a b = {a}
theorem covby.Ioc_eq {α : Type u_1} [partial_order α] {a b : α} (h : a b) :
set.Ioc a b = {b}
theorem covby.Icc_eq {α : Type u_1} [partial_order α] {a b : α} (h : a b) :
set.Icc a b = {a, b}
theorem covby.Ioi_eq {α : Type u_1} [linear_order α] {a b : α} (h : a b) :
theorem covby.Iio_eq {α : Type u_1} [linear_order α] {a b : α} (h : a b) :
theorem wcovby.le_of_lt {α : Type u_1} [linear_order α] {a b c : α} (hab : a ⩿ b) (hcb : c < b) :
c a
theorem wcovby.ge_of_gt {α : Type u_1} [linear_order α] {a b c : α} (hab : a ⩿ b) (hac : a < c) :
b c
theorem covby.le_of_lt {α : Type u_1} [linear_order α] {a b c : α} (hab : a b) :
c < bc a
theorem covby.ge_of_gt {α : Type u_1} [linear_order α] {a b c : α} (hab : a b) :
a < cb c
theorem covby.unique_left {α : Type u_1} [linear_order α] {a b c : α} (ha : a c) (hb : b c) :
a = b
theorem covby.unique_right {α : Type u_1} [linear_order α] {a b c : α} (hb : a b) (hc : a c) :
b = c
theorem set.wcovby_insert {α : Type u_1} (x : α) (s : set α) :
theorem set.covby_insert {α : Type u_1} {x : α} {s : set α} (hx : x s) :