Documentation

Mathlib.Order.Basic

Basic definitions about and < #

This file proves basic results about orders, provides extensive dot notation, defines useful order classes and allows to transfer order instances.

Type synonyms #

Transferring orders #

Extra class #

Notes #

and < are highly favored over and > in mathlib. The reason is that we can formulate all lemmas using /<, and rw has trouble unifying and . Hence choosing one direction spares us useless duplication. This is enforced by a linter. See Note [nolint_ge] for more infos.

Dot notation is particularly useful on (LE.le) and < (LT.lt). To that end, we provide many aliases to dot notation-less lemmas. For example, le_trans is aliased with LE.le.trans and can be used to construct hab.trans hbc : a ≤ c when hab : a ≤ b, hbc : b ≤ c, lt_of_le_of_lt is aliased as LE.le.trans_lt and can be used to construct hab.trans hbc : a < c when hab : a ≤ b, hbc : b < c.

TODO #

Tags #

preorder, order, partial order, poset, linear order, chain

theorem le_trans' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca ba c
theorem lt_trans' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca < ba < c
theorem lt_of_le_of_lt' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca < ba < c
theorem lt_of_lt_of_le' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca ba < c
theorem ge_antisymm {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a bb ab = a
theorem lt_of_le_of_ne' {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a bb aa < b
theorem Ne.lt_of_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a ba ba < b
theorem Ne.lt_of_le' {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
b aa ba < b
theorem LE.ext {α : Type u} (x : LE α) (y : LE α) (le : LE.le = LE.le) :
x = y
theorem LE.ext_iff {α : Type u} (x : LE α) (y : LE α) :
x = y LE.le = LE.le
theorem LE.le.trans {α : Type u} [Preorder α] {a : α} {b : α} {c : α} :
a bb ca c

Alias of le_trans.


The relation on a preorder is transitive.

theorem LE.le.trans' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca ba c

Alias of le_trans'.

theorem LE.le.trans_lt {α : Type u} [Preorder α] {a : α} {b : α} {c : α} :
a bb < ca < c

Alias of lt_of_le_of_lt.

theorem LE.le.trans_lt' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca < ba < c

Alias of lt_of_le_of_lt'.

theorem LE.le.antisymm {α : Type u} [PartialOrder α] {a : α} {b : α} :
a bb aa = b

Alias of le_antisymm.

theorem LE.le.antisymm' {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a bb ab = a

Alias of ge_antisymm.

theorem LE.le.lt_of_ne {α : Type u} [PartialOrder α] {a : α} {b : α} :
a ba ba < b

Alias of lt_of_le_of_ne.

theorem LE.le.lt_of_ne' {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a bb aa < b

Alias of lt_of_le_of_ne'.

theorem LE.le.lt_of_not_le {α : Type u} [Preorder α] {a : α} {b : α} :
a b¬b aa < b

Alias of lt_of_le_not_le.

theorem LE.le.lt_or_eq {α : Type u} [PartialOrder α] {a : α} {b : α} :
a ba < b a = b

Alias of lt_or_eq_of_le.

theorem LE.le.lt_or_eq_dec {α : Type u} [PartialOrder α] [DecidableRel fun (x x_1 : α) => x x_1] {a : α} {b : α} (hab : a b) :
a < b a = b

Alias of Decidable.lt_or_eq_of_le.

theorem LT.lt.le {α : Type u} [Preorder α] {a : α} {b : α} :
a < ba b

Alias of le_of_lt.

theorem LT.lt.trans {α : Type u} [Preorder α] {a : α} {b : α} {c : α} :
a < bb < ca < c

Alias of lt_trans.

theorem LT.lt.trans' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca < ba < c

Alias of lt_trans'.

theorem LT.lt.trans_le {α : Type u} [Preorder α] {a : α} {b : α} {c : α} :
a < bb ca < c

Alias of lt_of_lt_of_le.

theorem LT.lt.trans_le' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca ba < c

Alias of lt_of_lt_of_le'.

theorem LT.lt.ne {α : Type u} [Preorder α] {a : α} {b : α} (h : a < b) :
a b

Alias of ne_of_lt.

theorem LT.lt.asymm {α : Type u} [Preorder α] {a : α} {b : α} (h : a < b) :
¬b < a

Alias of lt_asymm.

theorem LT.lt.not_lt {α : Type u} [Preorder α] {a : α} {b : α} (h : a < b) :
¬b < a

Alias of lt_asymm.

theorem Eq.le {α : Type u} [Preorder α] {a : α} {b : α} :
a = ba b

Alias of le_of_eq.

@[simp]
theorem lt_self_iff_false {α : Type u_2} [Preorder α] (x : α) :
x < x False
theorem le_of_le_of_eq' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca = ba c
theorem le_of_eq_of_le' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b = ca ba c
theorem lt_of_lt_of_eq' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca = ba < c
theorem lt_of_eq_of_lt' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b = ca < ba < c
theorem LE.le.trans_eq {α : Type u} {a : α} {b : α} {c : α} [LE α] (h₁ : a b) (h₂ : b = c) :
a c

Alias of le_of_le_of_eq.

theorem LE.le.trans_eq' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b ca = ba c

Alias of le_of_le_of_eq'.

theorem LT.lt.trans_eq {α : Type u} {a : α} {b : α} {c : α} [LT α] (h₁ : a < b) (h₂ : b = c) :
a < c

Alias of lt_of_lt_of_eq.

theorem LT.lt.trans_eq' {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b < ca = ba < c

Alias of lt_of_lt_of_eq'.

theorem Eq.trans_le {α : Type u} {a : α} {b : α} {c : α} [LE α] (h₁ : a = b) (h₂ : b c) :
a c

Alias of le_of_eq_of_le.

theorem Eq.trans_ge {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b = ca ba c

Alias of le_of_eq_of_le'.

theorem Eq.trans_lt {α : Type u} {a : α} {b : α} {c : α} [LT α] (h₁ : a = b) (h₂ : b < c) :
a < c

Alias of lt_of_eq_of_lt.

theorem Eq.trans_gt {α : Type u_2} [Preorder α] {a : α} {b : α} {c : α} :
b = ca < ba < c

Alias of lt_of_eq_of_lt'.

theorem Eq.ge {α : Type u_2} [Preorder α] {x : α} {y : α} (h : x = y) :
y x

If x = y then y ≤ x. Note: this lemma uses y ≤ x instead of x ≥ y, because le is used almost exclusively in mathlib.

theorem Eq.not_lt {α : Type u_2} [Preorder α] {x : α} {y : α} (h : x = y) :
¬x < y
theorem Eq.not_gt {α : Type u_2} [Preorder α] {x : α} {y : α} (h : x = y) :
¬y < x
@[simp]
theorem le_of_subsingleton {α : Type u_2} [Preorder α] {a : α} {b : α} [Subsingleton α] :
a b
theorem not_lt_of_subsingleton {α : Type u_2} [Preorder α] {a : α} {b : α} [Subsingleton α] :
¬a < b
theorem LE.le.ge {α : Type u_2} [LE α] {x : α} {y : α} (h : x y) :
y x
theorem LE.le.lt_iff_ne {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a < b a b
theorem LE.le.gt_iff_ne {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a < b b a
theorem LE.le.not_lt_iff_eq {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
¬a < b a = b
theorem LE.le.not_gt_iff_eq {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
¬a < b b = a
theorem LE.le.le_iff_eq {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
b a b = a
theorem LE.le.ge_iff_eq {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
b a a = b
theorem LE.le.lt_or_le {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : a b) (c : α) :
a < c c b
theorem LE.le.le_or_lt {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : a b) (c : α) :
a c c < b
theorem LE.le.le_or_le {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : a b) (c : α) :
a c c b
theorem LT.lt.gt {α : Type u_2} [LT α] {x : α} {y : α} (h : x < y) :
y > x
theorem LT.lt.false {α : Type u_2} [Preorder α] {x : α} :
x < xFalse
theorem LT.lt.ne' {α : Type u_2} [Preorder α] {x : α} {y : α} (h : x < y) :
y x
theorem LT.lt.lt_or_lt {α : Type u_2} [LinearOrder α] {x : α} {y : α} (h : x < y) (z : α) :
x < z z < y
theorem GE.ge.le {α : Type u_2} [LE α] {x : α} {y : α} (h : x y) :
y x
theorem GT.gt.lt {α : Type u_2} [LT α] {x : α} {y : α} (h : x > y) :
y < x
theorem ge_of_eq {α : Type u_2} [Preorder α] {a : α} {b : α} (h : a = b) :
a b
theorem not_le_of_lt {α : Type u_2} [Preorder α] {a : α} {b : α} (h : a < b) :
¬b a
theorem LT.lt.not_le {α : Type u_2} [Preorder α] {a : α} {b : α} (h : a < b) :
¬b a

Alias of not_le_of_lt.

theorem not_lt_of_le {α : Type u_2} [Preorder α] {a : α} {b : α} (h : a b) :
¬b < a
theorem LE.le.not_lt {α : Type u_2} [Preorder α] {a : α} {b : α} (h : a b) :
¬b < a

Alias of not_lt_of_le.

theorem ne_of_not_le {α : Type u_2} [Preorder α] {a : α} {b : α} (h : ¬a b) :
a b
theorem Decidable.le_iff_eq_or_lt {α : Type u_2} [PartialOrder α] [DecidableRel fun (x x_1 : α) => x x_1] {a : α} {b : α} :
a b a = b a < b
theorem le_iff_eq_or_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a b a = b a < b
theorem lt_iff_le_and_ne {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a < b a b a b
theorem eq_iff_not_lt_of_le {α : Type u_2} [PartialOrder α] {x : α} {y : α} :
x yy = x ¬x < y
theorem Decidable.eq_iff_le_not_lt {α : Type u_2} [PartialOrder α] [DecidableRel fun (x x_1 : α) => x x_1] {a : α} {b : α} :
a = b a b ¬a < b
theorem eq_iff_le_not_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
a = b a b ¬a < b
theorem eq_or_lt_of_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a = b a < b
theorem eq_or_gt_of_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
b = a a < b
theorem gt_or_eq_of_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a < b b = a
theorem LE.le.eq_or_lt_dec {α : Type u} [PartialOrder α] [DecidableRel fun (x x_1 : α) => x x_1] {a : α} {b : α} (hab : a b) :
a = b a < b

Alias of Decidable.eq_or_lt_of_le.

theorem LE.le.eq_or_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a = b a < b

Alias of eq_or_lt_of_le.

theorem LE.le.eq_or_gt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
b = a a < b

Alias of eq_or_gt_of_le.

theorem LE.le.gt_or_eq {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a < b b = a

Alias of gt_or_eq_of_le.

theorem eq_of_le_of_not_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (hab : a b) (hba : ¬a < b) :
a = b
theorem eq_of_ge_of_not_gt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (hab : a b) (hba : ¬a < b) :
b = a
theorem LE.le.eq_of_not_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (hab : a b) (hba : ¬a < b) :
a = b

Alias of eq_of_le_of_not_lt.

theorem LE.le.eq_of_not_gt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (hab : a b) (hba : ¬a < b) :
b = a

Alias of eq_of_ge_of_not_gt.

theorem Ne.le_iff_lt {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
a b a < b
theorem Ne.not_le_or_not_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} (h : a b) :
¬a b ¬b a
theorem Decidable.ne_iff_lt_iff_le {α : Type u_2} [PartialOrder α] [DecidableEq α] {a : α} {b : α} :
(a b a < b) a b
@[simp]
theorem ne_iff_lt_iff_le {α : Type u_2} [PartialOrder α] {a : α} {b : α} :
(a b a < b) a b
theorem min_def' {α : Type u_2} [LinearOrder α] (a : α) (b : α) :
min a b = if b a then b else a
theorem max_def' {α : Type u_2} [LinearOrder α] (a : α) (b : α) :
max a b = if b a then a else b
theorem lt_of_not_le {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : ¬b a) :
a < b
theorem lt_iff_not_le {α : Type u_2} [LinearOrder α] {x : α} {y : α} :
x < y ¬y x
theorem Ne.lt_or_lt {α : Type u_2} [LinearOrder α] {x : α} {y : α} (h : x y) :
x < y y < x
@[simp]
theorem lt_or_lt_iff_ne {α : Type u_2} [LinearOrder α] {x : α} {y : α} :
x < y y < x x y

A version of ne_iff_lt_or_gt with LHS and RHS reversed.

theorem not_lt_iff_eq_or_lt {α : Type u_2} [LinearOrder α] {a : α} {b : α} :
¬a < b a = b b < a
theorem exists_ge_of_linear {α : Type u_2} [LinearOrder α] (a : α) (b : α) :
∃ (c : α), a c b c
theorem exists_forall_ge_and {α : Type u_2} [LinearOrder α] {p : αProp} {q : αProp} :
(∃ (i : α), ∀ (j : α), j ip j)(∃ (i : α), ∀ (j : α), j iq j)∃ (i : α), ∀ (j : α), j ip j q j
theorem lt_imp_lt_of_le_imp_le {α : Type u_2} {β : Type u_5} [LinearOrder α] [Preorder β] {a : α} {b : α} {c : β} {d : β} (H : a bc d) (h : d < c) :
b < a
theorem le_imp_le_iff_lt_imp_lt {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a : α} {b : α} {c : β} {d : β} :
a bc d d < cb < a
theorem lt_iff_lt_of_le_iff_le' {α : Type u_2} {β : Type u_5} [Preorder α] [Preorder β] {a : α} {b : α} {c : β} {d : β} (H : a b c d) (H' : b a d c) :
b < a d < c
theorem lt_iff_lt_of_le_iff_le {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a : α} {b : α} {c : β} {d : β} (H : a b c d) :
b < a d < c
theorem le_iff_le_iff_lt_iff_lt {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a : α} {b : α} {c : β} {d : β} :
(a b c d) (b < a d < c)
theorem eq_of_forall_le_iff {α : Type u_2} [PartialOrder α] {a : α} {b : α} (H : ∀ (c : α), c a c b) :
a = b
theorem le_of_forall_le {α : Type u_2} [Preorder α] {a : α} {b : α} (H : ∀ (c : α), c ac b) :
a b
theorem le_of_forall_le' {α : Type u_2} [Preorder α] {a : α} {b : α} (H : ∀ (c : α), a cb c) :
b a
theorem le_of_forall_lt {α : Type u_2} [LinearOrder α] {a : α} {b : α} (H : ∀ (c : α), c < ac < b) :
a b
theorem forall_lt_iff_le {α : Type u_2} [LinearOrder α] {a : α} {b : α} :
(∀ ⦃c : α⦄, c < ac < b) a b
theorem le_of_forall_lt' {α : Type u_2} [LinearOrder α] {a : α} {b : α} (H : ∀ (c : α), a < cb < c) :
b a
theorem forall_lt_iff_le' {α : Type u_2} [LinearOrder α] {a : α} {b : α} :
(∀ ⦃c : α⦄, a < cb < c) b a
theorem eq_of_forall_ge_iff {α : Type u_2} [PartialOrder α] {a : α} {b : α} (H : ∀ (c : α), a c b c) :
a = b
theorem eq_of_forall_lt_iff {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : ∀ (c : α), c < a c < b) :
a = b
theorem eq_of_forall_gt_iff {α : Type u_2} [LinearOrder α] {a : α} {b : α} (h : ∀ (c : α), a < c b < c) :
a = b
theorem rel_imp_eq_of_rel_imp_le {α : Type u_2} {β : Type u_3} [PartialOrder β] (r : ααProp) [IsSymm α r] {f : αβ} (h : ∀ (a b : α), r a bf a f b) {a : α} {b : α} :
r a bf a = f b

A symmetric relation implies two values are equal, when it implies they're less-equal.

theorem le_implies_le_of_le_of_le {α : Type u_2} {a : α} {b : α} {c : α} {d : α} [Preorder α] (hca : c a) (hbd : b d) :
a bc d

monotonicity of with respect to

theorem commutative_of_le {α : Type u_2} {β : Type u_3} [PartialOrder α] {f : ββα} (comm : ∀ (a b : β), f a b f b a) (a : β) (b : β) :
f a b = f b a

To prove commutativity of a binary operation , we only to check a ○ b ≤ b ○ a for all a, b.

theorem associative_of_commutative_of_le {α : Type u_2} [PartialOrder α] {f : ααα} (comm : Commutative f) (assoc : ∀ (a b c : α), f (f a b) c f a (f b c)) :

To prove associativity of a commutative binary operation , we only to check (a ○ b) ○ c ≤ a ○ (b ○ c) for all a, b, c.

theorem Preorder.ext {α : Type u_2} {A : Preorder α} {B : Preorder α} (H : ∀ (x y : α), x y x y) :
A = B
theorem PartialOrder.ext {α : Type u_2} {A : PartialOrder α} {B : PartialOrder α} (H : ∀ (x y : α), x y x y) :
A = B
theorem LinearOrder.ext {α : Type u_2} {A : LinearOrder α} {B : LinearOrder α} (H : ∀ (x y : α), x y x y) :
A = B
def Order.Preimage {α : Type u_2} {β : Type u_3} (f : αβ) (s : ββProp) (x : α) (y : α) :

Given a relation R on β and a function f : α → β, the preimage relation on α is defined by x ≤ y ↔ f x ≤ f y. It is the unique relation on α making f a RelEmbedding (assuming f is injective).

Equations
Instances For

    Given a relation R on β and a function f : α → β, the preimage relation on α is defined by x ≤ y ↔ f x ≤ f y. It is the unique relation on α making f a RelEmbedding (assuming f is injective).

    Equations
    Instances For
      instance Order.Preimage.decidable {α : Type u_2} {β : Type u_3} (f : αβ) (s : ββProp) [H : DecidableRel s] :

      The preimage of a decidable order is decidable.

      Equations
      @[simp]
      theorem ltByCases_lt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : x < y) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} :
      ltByCases x y h₁ h₂ h₃ = h₁ h
      @[simp]
      theorem ltByCases_gt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : y < x) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} :
      ltByCases x y h₁ h₂ h₃ = h₃ h
      @[simp]
      theorem ltByCases_eq {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : x = y) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} :
      ltByCases x y h₁ h₂ h₃ = h₂ h
      theorem ltByCases_not_lt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : ¬x < y) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} (p : optParam (¬y < xx = y) ) :
      ltByCases x y h₁ h₂ h₃ = if h' : y < x then h₃ h' else h₂
      theorem ltByCases_not_gt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : ¬y < x) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} (p : optParam (¬x < yx = y) ) :
      ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₂
      theorem ltByCases_ne {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} (h : x y) {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} (p : optParam (¬x < yy < x) ) :
      ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₃
      theorem ltByCases_comm {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} (p : optParam (y = xx = y) ) :
      ltByCases x y h₁ h₂ h₃ = ltByCases y x h₃ (h₂ p) h₁
      theorem eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt {α : Type u_2} [LinearOrder α] {x : α} {y : α} {x' : α} {y' : α} (ltc : x < y x' < y') (gtc : y < x y' < x') :
      x = y x' = y'
      theorem ltByCases_rec {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} (p : P) (hlt : ∀ (h : x < y), h₁ h = p) (heq : ∀ (h : x = y), h₂ h = p) (hgt : ∀ (h : y < x), h₃ h = p) :
      ltByCases x y h₁ h₂ h₃ = p
      theorem ltByCases_eq_iff {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} {p : P} :
      ltByCases x y h₁ h₂ h₃ = p (∃ (h : x < y), h₁ h = p) (∃ (h : x = y), h₂ h = p) ∃ (h : y < x), h₃ h = p
      theorem ltByCases_congr {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {x' : α} {y' : α} {h₁ : x < yP} {h₂ : x = yP} {h₃ : y < xP} {h₁' : x' < y'P} {h₂' : x' = y'P} {h₃' : y' < x'P} (ltc : x < y x' < y') (gtc : y < x y' < x') (hh'₁ : ∀ (h : x' < y'), h₁ = h₁' h) (hh'₂ : ∀ (h : x' = y'), h₂ = h₂' h) (hh'₃ : ∀ (h : y' < x'), h₃ = h₃' h) :
      ltByCases x y h₁ h₂ h₃ = ltByCases x' y' h₁' h₂' h₃'
      @[reducible, inline]
      abbrev ltTrichotomy {α : Type u_2} [LinearOrder α] {P : Sort u_5} (x : α) (y : α) (p : P) (q : P) (r : P) :
      P

      Perform a case-split on the ordering of x and y in a decidable linear order, non-dependently.

      Equations
      Instances For
        @[simp]
        theorem ltTrichotomy_lt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : x < y) :
        ltTrichotomy x y p q r = p
        @[simp]
        theorem ltTrichotomy_gt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : y < x) :
        ltTrichotomy x y p q r = r
        @[simp]
        theorem ltTrichotomy_eq {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : x = y) :
        ltTrichotomy x y p q r = q
        theorem ltTrichotomy_not_lt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : ¬x < y) :
        ltTrichotomy x y p q r = if y < x then r else q
        theorem ltTrichotomy_not_gt {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : ¬y < x) :
        ltTrichotomy x y p q r = if x < y then p else q
        theorem ltTrichotomy_ne {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} (h : x y) :
        ltTrichotomy x y p q r = if x < y then p else r
        theorem ltTrichotomy_comm {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} :
        ltTrichotomy x y p q r = ltTrichotomy y x r q p
        theorem ltTrichotomy_self {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} :
        ltTrichotomy x y p p p = p
        theorem ltTrichotomy_eq_iff {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} {s : P} :
        ltTrichotomy x y p q r = s x < y p = s x = y q = s y < x r = s
        theorem ltTrichotomy_congr {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x : α} {y : α} {p : P} {q : P} {r : P} {x' : α} {y' : α} {p' : P} {q' : P} {r' : P} (ltc : x < y x' < y') (gtc : y < x y' < x') (hh'₁ : x' < y'p = p') (hh'₂ : x' = y'q = q') (hh'₃ : y' < x'r = r') :
        ltTrichotomy x y p q r = ltTrichotomy x' y' p' q' r'

        Order dual #

        def OrderDual (α : Type u_5) :
        Type u_5

        Type synonym to equip a type with the dual order: means and < means >. αᵒᵈ is notation for OrderDual α.

        Equations
        Instances For

          Type synonym to equip a type with the dual order: means and < means >. αᵒᵈ is notation for OrderDual α.

          Equations
          Instances For
            instance OrderDual.instNonempty (α : Type u_5) [h : Nonempty α] :
            Equations
            • = h
            Equations
            • = h
            instance OrderDual.instLE (α : Type u_5) [LE α] :
            Equations
            instance OrderDual.instLT (α : Type u_5) [LT α] :
            Equations
            Equations
            Equations
            • OrderDual.instInhabited = x

            HasCompl #

            Equations
            instance Pi.hasCompl {ι : Type u_1} {π : ιType u_4} [(i : ι) → HasCompl (π i)] :
            HasCompl ((i : ι) → π i)
            Equations
            • Pi.hasCompl = { compl := fun (x : (i : ι) → π i) (i : ι) => (x i) }
            theorem Pi.compl_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → HasCompl (π i)] (x : (i : ι) → π i) :
            x = fun (i : ι) => (x i)
            @[simp]
            theorem Pi.compl_apply {ι : Type u_1} {π : ιType u_4} [(i : ι) → HasCompl (π i)] (x : (i : ι) → π i) (i : ι) :
            x i = (x i)
            instance IsIrrefl.compl {α : Type u_2} (r : ααProp) [IsIrrefl α r] :
            Equations
            • =
            instance IsRefl.compl {α : Type u_2} (r : ααProp) [IsRefl α r] :
            Equations
            • =

            Order instances on the function space #

            instance Pi.hasLe {ι : Type u_1} {π : ιType u_4} [(i : ι) → LE (π i)] :
            LE ((i : ι) → π i)
            Equations
            • Pi.hasLe = { le := fun (x y : (i : ι) → π i) => ∀ (i : ι), x i y i }
            theorem Pi.le_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → LE (π i)] {x : (i : ι) → π i} {y : (i : ι) → π i} :
            x y ∀ (i : ι), x i y i
            instance Pi.preorder {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] :
            Preorder ((i : ι) → π i)
            Equations
            theorem Pi.lt_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {y : (i : ι) → π i} :
            x < y x y ∃ (i : ι), x i < y i
            instance Pi.partialOrder {ι : Type u_1} {π : ιType u_4} [(i : ι) → PartialOrder (π i)] :
            PartialOrder ((i : ι) → π i)
            Equations
            @[simp]
            theorem Sum.elim_le_elim_iff {β : Type u_3} {α₁ : Type u_5} {α₂ : Type u_6} [LE β] {u₁ : α₁β} {v₁ : α₁β} {u₂ : α₂β} {v₂ : α₂β} :
            Sum.elim u₁ u₂ Sum.elim v₁ v₂ u₁ v₁ u₂ v₂
            theorem Sum.const_le_elim_iff {β : Type u_3} {α₁ : Type u_5} {α₂ : Type u_6} [LE β] {b : β} {v₁ : α₁β} {v₂ : α₂β} :
            Function.const (α₁ α₂) b Sum.elim v₁ v₂ Function.const α₁ b v₁ Function.const α₂ b v₂
            theorem Sum.elim_le_const_iff {β : Type u_3} {α₁ : Type u_5} {α₂ : Type u_6} [LE β] {b : β} {u₁ : α₁β} {u₂ : α₂β} :
            Sum.elim u₁ u₂ Function.const (α₁ α₂) b u₁ Function.const α₁ b u₂ Function.const α₂ b
            def StrongLT {ι : Type u_1} {π : ιType u_4} [(i : ι) → LT (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :

            A function a is strongly less than a function b if a i < b i for all i.

            Equations
            Instances For
              theorem le_of_strongLT {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} (h : StrongLT a b) :
              a b
              theorem lt_of_strongLT {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} [Nonempty ι] (h : StrongLT a b) :
              a < b
              theorem strongLT_of_strongLT_of_le {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} {c : (i : ι) → π i} (hab : StrongLT a b) (hbc : b c) :
              theorem strongLT_of_le_of_strongLT {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} {c : (i : ι) → π i} (hab : a b) (hbc : StrongLT b c) :
              theorem StrongLT.le {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} (h : StrongLT a b) :
              a b

              Alias of le_of_strongLT.

              theorem StrongLT.lt {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} [Nonempty ι] (h : StrongLT a b) :
              a < b

              Alias of lt_of_strongLT.

              theorem StrongLT.trans_le {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} {c : (i : ι) → π i} (hab : StrongLT a b) (hbc : b c) :

              Alias of strongLT_of_strongLT_of_le.

              theorem LE.le.trans_strongLT {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] {a : (i : ι) → π i} {b : (i : ι) → π i} {c : (i : ι) → π i} (hab : a b) (hbc : StrongLT b c) :

              Alias of strongLT_of_le_of_strongLT.

              theorem le_update_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {y : (i : ι) → π i} {i : ι} {a : π i} :
              x Function.update y i a x i a ∀ (j : ι), j ix j y j
              theorem update_le_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {y : (i : ι) → π i} {i : ι} {a : π i} :
              Function.update x i a y a y i ∀ (j : ι), j ix j y j
              theorem update_le_update_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {y : (i : ι) → π i} {i : ι} {a : π i} {b : π i} :
              Function.update x i a Function.update y i b a b ∀ (j : ι), j ix j y j
              @[simp]
              theorem update_le_update_iff' {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} {b : π i} :
              @[simp]
              theorem update_lt_update_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} {b : π i} :
              @[simp]
              theorem le_update_self_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} :
              x Function.update x i a x i a
              @[simp]
              theorem update_le_self_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} :
              Function.update x i a x a x i
              @[simp]
              theorem lt_update_self_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} :
              x < Function.update x i a x i < a
              @[simp]
              theorem update_lt_self_iff {ι : Type u_1} {π : ιType u_4} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} :
              Function.update x i a < x a < x i
              instance Pi.sdiff {ι : Type u_1} {π : ιType u_4} [(i : ι) → SDiff (π i)] :
              SDiff ((i : ι) → π i)
              Equations
              • Pi.sdiff = { sdiff := fun (x y : (i : ι) → π i) (i : ι) => x i \ y i }
              theorem Pi.sdiff_def {ι : Type u_1} {π : ιType u_4} [(i : ι) → SDiff (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) :
              x \ y = fun (i : ι) => x i \ y i
              @[simp]
              theorem Pi.sdiff_apply {ι : Type u_1} {π : ιType u_4} [(i : ι) → SDiff (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) (i : ι) :
              (x \ y) i = x i \ y i
              @[simp]
              theorem Function.const_le_const {α : Type u_2} {β : Type u_3} [Preorder α] [Nonempty β] {a : α} {b : α} :
              @[simp]
              theorem Function.const_lt_const {α : Type u_2} {β : Type u_3} [Preorder α] [Nonempty β] {a : α} {b : α} :

              min/max recursors #

              theorem min_rec {α : Type u_2} [LinearOrder α] {p : αProp} {x : α} {y : α} (hx : x yp x) (hy : y xp y) :
              p (min x y)
              theorem max_rec {α : Type u_2} [LinearOrder α] {p : αProp} {x : α} {y : α} (hx : y xp x) (hy : x yp y) :
              p (max x y)
              theorem min_rec' {α : Type u_2} [LinearOrder α] {x : α} {y : α} (p : αProp) (hx : p x) (hy : p y) :
              p (min x y)
              theorem max_rec' {α : Type u_2} [LinearOrder α] {x : α} {y : α} (p : αProp) (hx : p x) (hy : p y) :
              p (max x y)
              theorem min_def_lt {α : Type u_2} [LinearOrder α] (x : α) (y : α) :
              min x y = if x < y then x else y
              theorem max_def_lt {α : Type u_2} [LinearOrder α] (x : α) (y : α) :
              max x y = if x < y then y else x

              Lifts of order instances #

              @[reducible, inline]
              abbrev Preorder.lift {α : Type u_2} {β : Type u_3} [Preorder β] (f : αβ) :

              Transfer a Preorder on β to a Preorder on α using a function f : α → β. See note [reducible non-instances].

              Equations
              Instances For
                @[reducible, inline]
                abbrev PartialOrder.lift {α : Type u_2} {β : Type u_3} [PartialOrder β] (f : αβ) (inj : Function.Injective f) :

                Transfer a PartialOrder on β to a PartialOrder on α using an injective function f : α → β. See note [reducible non-instances].

                Equations
                Instances For
                  theorem compare_of_injective_eq_compareOfLessAndEq {α : Type u_2} {β : Type u_3} (a : α) (b : α) [LinearOrder β] [DecidableEq α] (f : αβ) (inj : Function.Injective f) [Decidable (a < b)] :
                  compare (f a) (f b) = compareOfLessAndEq a b
                  @[reducible, inline]
                  abbrev LinearOrder.lift {α : Type u_2} {β : Type u_3} [LinearOrder β] [Sup α] [Inf α] (f : αβ) (inj : Function.Injective f) (hsup : ∀ (x y : α), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : α), f (x y) = min (f x) (f y)) :

                  Transfer a LinearOrder on β to a LinearOrder on α using an injective function f : α → β. This version takes [Sup α] and [Inf α] as arguments, then uses them for max and min fields. See LinearOrder.lift' for a version that autogenerates min and max fields, and LinearOrder.liftWithOrd for one that does not auto-generate compare fields. See note [reducible non-instances].

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For
                    @[reducible, inline]
                    abbrev LinearOrder.lift' {α : Type u_2} {β : Type u_3} [LinearOrder β] (f : αβ) (inj : Function.Injective f) :

                    Transfer a LinearOrder on β to a LinearOrder on α using an injective function f : α → β. This version autogenerates min and max fields. See LinearOrder.lift for a version that takes [Sup α] and [Inf α], then uses them as max and min. See LinearOrder.liftWithOrd' for a version which does not auto-generate compare fields. See note [reducible non-instances].

                    Equations
                    Instances For
                      @[reducible, inline]
                      abbrev LinearOrder.liftWithOrd {α : Type u_2} {β : Type u_3} [LinearOrder β] [Sup α] [Inf α] [Ord α] (f : αβ) (inj : Function.Injective f) (hsup : ∀ (x y : α), f (x y) = max (f x) (f y)) (hinf : ∀ (x y : α), f (x y) = min (f x) (f y)) (compare_f : ∀ (a b : α), compare a b = compare (f a) (f b)) :

                      Transfer a LinearOrder on β to a LinearOrder on α using an injective function f : α → β. This version takes [Sup α] and [Inf α] as arguments, then uses them for max and min fields. It also takes [Ord α] as an argument and uses them for compare fields. See LinearOrder.lift for a version that autogenerates compare fields, and LinearOrder.liftWithOrd' for one that auto-generates min and max fields. fields. See note [reducible non-instances].

                      Equations
                      • One or more equations did not get rendered due to their size.
                      Instances For
                        @[reducible, inline]
                        abbrev LinearOrder.liftWithOrd' {α : Type u_2} {β : Type u_3} [LinearOrder β] [Ord α] (f : αβ) (inj : Function.Injective f) (compare_f : ∀ (a b : α), compare a b = compare (f a) (f b)) :

                        Transfer a LinearOrder on β to a LinearOrder on α using an injective function f : α → β. This version auto-generates min and max fields. It also takes [Ord α] as an argument and uses them for compare fields. See LinearOrder.lift for a version that autogenerates compare fields, and LinearOrder.liftWithOrd for one that doesn't auto-generate min and max fields. fields. See note [reducible non-instances].

                        Equations
                        Instances For

                          Subtype of an order #

                          instance Subtype.le {α : Type u_2} [LE α] {p : αProp} :
                          Equations
                          • Subtype.le = { le := fun (x y : Subtype p) => x y }
                          instance Subtype.lt {α : Type u_2} [LT α] {p : αProp} :
                          Equations
                          • Subtype.lt = { lt := fun (x y : Subtype p) => x < y }
                          @[simp]
                          theorem Subtype.mk_le_mk {α : Type u_2} [LE α] {p : αProp} {x : α} {y : α} {hx : p x} {hy : p y} :
                          x, hx y, hy x y
                          @[simp]
                          theorem Subtype.mk_lt_mk {α : Type u_2} [LT α] {p : αProp} {x : α} {y : α} {hx : p x} {hy : p y} :
                          x, hx < y, hy x < y
                          @[simp]
                          theorem Subtype.coe_le_coe {α : Type u_2} [LE α] {p : αProp} {x : Subtype p} {y : Subtype p} :
                          x y x y
                          @[simp]
                          theorem Subtype.coe_lt_coe {α : Type u_2} [LT α] {p : αProp} {x : Subtype p} {y : Subtype p} :
                          x < y x < y
                          instance Subtype.preorder {α : Type u_2} [Preorder α] (p : αProp) :
                          Equations
                          instance Subtype.partialOrder {α : Type u_2} [PartialOrder α] (p : αProp) :
                          Equations
                          instance Subtype.decidableLE {α : Type u_2} [Preorder α] [h : DecidableRel fun (x x_1 : α) => x x_1] {p : αProp} :
                          DecidableRel fun (x x_1 : Subtype p) => x x_1
                          Equations
                          • a.decidableLE b = h a b
                          instance Subtype.decidableLT {α : Type u_2} [Preorder α] [h : DecidableRel fun (x x_1 : α) => x < x_1] {p : αProp} :
                          DecidableRel fun (x x_1 : Subtype p) => x < x_1
                          Equations
                          • a.decidableLT b = h a b
                          instance Subtype.instLinearOrder {α : Type u_2} [LinearOrder α] (p : αProp) :

                          A subtype of a linear order is a linear order. We explicitly give the proofs of decidable equality and decidable order in order to ensure the decidability instances are all definitionally equal.

                          Equations

                          Pointwise order on α × β #

                          The lexicographic order is defined in Data.Prod.Lex, and the instances are available via the type synonym α ×ₗ β = α × β.

                          instance Prod.instLE_mathlib (α : Type u_5) (β : Type u_6) [LE α] [LE β] :
                          LE (α × β)
                          Equations
                          instance Prod.instDecidableLE (α : Type u_5) (β : Type u_6) [LE α] [LE β] (x : α × β) (y : α × β) [Decidable (x.1 y.1)] [Decidable (x.2 y.2)] :
                          Equations
                          theorem Prod.le_def {α : Type u_2} {β : Type u_3} [LE α] [LE β] {x : α × β} {y : α × β} :
                          x y x.1 y.1 x.2 y.2
                          @[simp]
                          theorem Prod.mk_le_mk {α : Type u_2} {β : Type u_3} [LE α] [LE β] {x₁ : α} {x₂ : α} {y₁ : β} {y₂ : β} :
                          (x₁, y₁) (x₂, y₂) x₁ x₂ y₁ y₂
                          @[simp]
                          theorem Prod.swap_le_swap {α : Type u_2} {β : Type u_3} [LE α] [LE β] {x : α × β} {y : α × β} :
                          x.swap y.swap x y
                          instance Prod.instPreorder (α : Type u_5) (β : Type u_6) [Preorder α] [Preorder β] :
                          Preorder (α × β)
                          Equations
                          @[simp]
                          theorem Prod.swap_lt_swap {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {x : α × β} {y : α × β} :
                          x.swap < y.swap x < y
                          theorem Prod.mk_le_mk_iff_left {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {a₁ : α} {a₂ : α} {b : β} :
                          (a₁, b) (a₂, b) a₁ a₂
                          theorem Prod.mk_le_mk_iff_right {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {a : α} {b₁ : β} {b₂ : β} :
                          (a, b₁) (a, b₂) b₁ b₂
                          theorem Prod.mk_lt_mk_iff_left {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {a₁ : α} {a₂ : α} {b : β} :
                          (a₁, b) < (a₂, b) a₁ < a₂
                          theorem Prod.mk_lt_mk_iff_right {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {a : α} {b₁ : β} {b₂ : β} :
                          (a, b₁) < (a, b₂) b₁ < b₂
                          theorem Prod.lt_iff {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {x : α × β} {y : α × β} :
                          x < y x.1 < y.1 x.2 y.2 x.1 y.1 x.2 < y.2
                          @[simp]
                          theorem Prod.mk_lt_mk {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] {a₁ : α} {a₂ : α} {b₁ : β} {b₂ : β} :
                          (a₁, b₁) < (a₂, b₂) a₁ < a₂ b₁ b₂ a₁ a₂ b₁ < b₂
                          instance Prod.instPartialOrder (α : Type u_5) (β : Type u_6) [PartialOrder α] [PartialOrder β] :

                          The pointwise partial order on a product. (The lexicographic ordering is defined in Order.Lexicographic, and the instances are available via the type synonym α ×ₗ β = α × β.)

                          Equations

                          Additional order classes #

                          class DenselyOrdered (α : Type u_5) [LT α] :

                          An order is dense if there is an element between any pair of distinct comparable elements.

                          • dense : ∀ (a₁ a₂ : α), a₁ < a₂∃ (a : α), a₁ < a a < a₂

                            An order is dense if there is an element between any pair of distinct elements.

                          Instances
                            theorem DenselyOrdered.dense {α : Type u_5} [LT α] [self : DenselyOrdered α] (a₁ : α) (a₂ : α) :
                            a₁ < a₂∃ (a : α), a₁ < a a < a₂

                            An order is dense if there is an element between any pair of distinct elements.

                            theorem exists_between {α : Type u_2} [LT α] [DenselyOrdered α] {a₁ : α} {a₂ : α} :
                            a₁ < a₂∃ (a : α), a₁ < a a < a₂
                            Equations
                            • =
                            instance instDenselyOrderedProd {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] :
                            Equations
                            • =
                            instance instDenselyOrderedForall {ι : Type u_1} {π : ιType u_4} [(i : ι) → Preorder (π i)] [∀ (i : ι), DenselyOrdered (π i)] :
                            DenselyOrdered ((i : ι) → π i)
                            Equations
                            • =
                            theorem le_of_forall_le_of_dense {α : Type u_2} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h : ∀ (a : α), a₂ < aa₁ a) :
                            a₁ a₂
                            theorem eq_of_le_of_forall_le_of_dense {α : Type u_2} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h₁ : a₂ a₁) (h₂ : ∀ (a : α), a₂ < aa₁ a) :
                            a₁ = a₂
                            theorem le_of_forall_ge_of_dense {α : Type u_2} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h : ∀ (a₃ : α), a₃ < a₁a₃ a₂) :
                            a₁ a₂
                            theorem eq_of_le_of_forall_ge_of_dense {α : Type u_2} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h₁ : a₂ a₁) (h₂ : ∀ (a₃ : α), a₃ < a₁a₃ a₂) :
                            a₁ = a₂
                            theorem dense_or_discrete {α : Type u_2} [LinearOrder α] (a₁ : α) (a₂ : α) :
                            (∃ (a : α), a₁ < a a < a₂) (∀ (a : α), a₁ < aa₂ a) ∀ (a : α), a < a₂a a₁
                            theorem eq_or_eq_or_eq_of_forall_not_lt_lt {α : Type u_2} [LinearOrder α] (h : ∀ ⦃x y z : α⦄, x < yy < zFalse) (x : α) (y : α) (z : α) :
                            x = y y = z x = z

                            If a linear order has no elements x < y < z, then it has at most two elements.

                            Equations
                            • One or more equations did not get rendered due to their size.
                            instance Prop.le :

                            Propositions form a complete boolean algebra, where the relation is given by implication.

                            Equations
                            @[simp]
                            theorem le_Prop_eq :
                            (fun (x x_1 : Prop) => x x_1) = fun (x x_1 : Prop) => xx_1
                            theorem subrelation_iff_le {α : Type u_2} {r : ααProp} {s : ααProp} :

                            Linear order from a total partial order #

                            def AsLinearOrder (α : Type u_5) :
                            Type u_5

                            Type synonym to create an instance of LinearOrder from a PartialOrder and IsTotal α (≤)

                            Equations
                            Instances For
                              Equations
                              • instInhabitedAsLinearOrder = { default := default }
                              noncomputable instance AsLinearOrder.linearOrder {α : Type u_2} [PartialOrder α] [IsTotal α fun (x x_1 : α) => x x_1] :
                              Equations
                              • One or more equations did not get rendered due to their size.
                              theorem dite_nonneg {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LE α] (ha : ∀ (h : p), 0 a h) (hb : ∀ (h : ¬p), 0 b h) :
                              0 dite p a b
                              theorem one_le_dite {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LE α] (ha : ∀ (h : p), 1 a h) (hb : ∀ (h : ¬p), 1 b h) :
                              1 dite p a b
                              theorem dite_nonpos {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LE α] (ha : ∀ (h : p), a h 0) (hb : ∀ (h : ¬p), b h 0) :
                              dite p a b 0
                              theorem dite_le_one {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LE α] (ha : ∀ (h : p), a h 1) (hb : ∀ (h : ¬p), b h 1) :
                              dite p a b 1
                              theorem dite_pos {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LT α] (ha : ∀ (h : p), 0 < a h) (hb : ∀ (h : ¬p), 0 < b h) :
                              0 < dite p a b
                              theorem one_lt_dite {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LT α] (ha : ∀ (h : p), 1 < a h) (hb : ∀ (h : ¬p), 1 < b h) :
                              1 < dite p a b
                              theorem dite_neg {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LT α] (ha : ∀ (h : p), a h < 0) (hb : ∀ (h : ¬p), b h < 0) :
                              dite p a b < 0
                              theorem dite_lt_one {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : pα} {b : ¬pα} [LT α] (ha : ∀ (h : p), a h < 1) (hb : ∀ (h : ¬p), b h < 1) :
                              dite p a b < 1
                              theorem ite_nonneg {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : α} {b : α} [LE α] (ha : 0 a) (hb : 0 b) :
                              0 if p then a else b
                              theorem one_le_ite {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : α} {b : α} [LE α] (ha : 1 a) (hb : 1 b) :
                              1 if p then a else b
                              theorem ite_nonpos {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : α} {b : α} [LE α] (ha : a 0) (hb : b 0) :
                              (if p then a else b) 0
                              theorem ite_le_one {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : α} {b : α} [LE α] (ha : a 1) (hb : b 1) :
                              (if p then a else b) 1
                              theorem ite_pos {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : α} {b : α} [LT α] (ha : 0 < a) (hb : 0 < b) :
                              0 < if p then a else b
                              theorem one_lt_ite {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : α} {b : α} [LT α] (ha : 1 < a) (hb : 1 < b) :
                              1 < if p then a else b
                              theorem ite_neg {α : Type u_2} [Zero α] {p : Prop} [Decidable p] {a : α} {b : α} [LT α] (ha : a < 0) (hb : b < 0) :
                              (if p then a else b) < 0
                              theorem ite_lt_one {α : Type u_2} [One α] {p : Prop} [Decidable p] {a : α} {b : α} [LT α] (ha : a < 1) (hb : b < 1) :
                              (if p then a else b) < 1