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

• OrderDual α : A type synonym reversing the meaning of all inequalities, with notation αᵒᵈ.
• AsLinearOrder α: A type synonym to promote PartialOrder α to LinearOrder α using IsTotal α (≤).

Transferring orders

• Order.Preimage, Preorder.lift: Transfers a (pre)order on β to an order on α using a function f : α → β.
• PartialOrder.lift, LinearOrder.lift: Transfers a partial (resp., linear) order on β to a partial (resp., linear) order on α using an injective function f.

Extra class

• Sup: type class for the ⊔ notation
• Inf: type class for the ⊓ notation
• HasCompl: type class for the ᶜ notation
• DenselyOrdered: An order with no gap, i.e. for any two elements a < b there exists c such that a < c < b.

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

• expand module docs
• automatic construction of dual definitions / theorems

Tags

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

theorem le_trans' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a ≤ b → a ≤ c

theorem lt_trans' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b < c → a < b → a < c

theorem lt_of_le_of_lt' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a < b → a < c

theorem lt_of_lt_of_le' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b < c → a ≤ b → a < c

theorem ge_antisymm {α : Type u} [PartialOrder α] {a : α} {b : α} : a ≤ b → b ≤ a → b = a

theorem lt_of_le_of_ne' {α : Type u} [PartialOrder α] {a : α} {b : α} : a ≤ b → b ≠ a → a < b

theorem Ne.lt_of_le {α : Type u} [PartialOrder α] {a : α} {b : α} : a ≠ b → a ≤ b → a < b

theorem Ne.lt_of_le' {α : Type u} [PartialOrder α] {a : α} {b : α} : b ≠ a → a ≤ b → a < 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 ≤ b → b ≤ c → a ≤ c

Alias of le_trans.

---

The relation ≤ on a preorder is transitive.

theorem LE.le.trans' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a ≤ b → a ≤ c

Alias of le_trans'.

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

Alias of lt_of_le_of_lt.

theorem LE.le.trans_lt' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a < b → a < c

Alias of lt_of_le_of_lt'.

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

Alias of le_antisymm.

theorem LE.le.antisymm' {α : Type u} [PartialOrder α] {a : α} {b : α} : a ≤ b → b ≤ a → b = a

Alias of ge_antisymm.

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

Alias of lt_of_le_of_ne.

theorem LE.le.lt_of_ne' {α : Type u} [PartialOrder α] {a : α} {b : α} : a ≤ b → b ≠ a → a < 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 ≤ a → a < b

Alias of lt_of_le_not_le.

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

Alias of le_of_lt.

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

Alias of lt_trans.

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

Alias of lt_trans'.

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

Alias of lt_of_lt_of_le.

theorem LT.lt.trans_le' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b < c → a ≤ b → a < 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 = b → a ≤ b

Alias of le_of_eq.

theorem le_rfl {α : Type u} [Preorder α] {a : α} : a ≤ a

A version of le_refl where the argument is implicit

@[simp]

theorem lt_self_iff_false {α : Type u} [Preorder α] (x : α) : x < x ↔ False

theorem le_of_le_of_eq' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a = b → a ≤ c

theorem le_of_eq_of_le' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b = c → a ≤ b → a ≤ c

theorem lt_of_lt_of_eq' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b < c → a = b → a < c

theorem lt_of_eq_of_lt' {α : Type u} [Preorder α] {a : α} {b : α} {c : α} : b = c → a < b → a < c

theorem LE.le.trans_eq {α : Type u_1} {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} [Preorder α] {a : α} {b : α} {c : α} : b ≤ c → a = b → a ≤ c

Alias of le_of_le_of_eq'.

theorem LT.lt.trans_eq {α : Type u_1} {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} [Preorder α] {a : α} {b : α} {c : α} : b < c → a = b → a < c

Alias of lt_of_lt_of_eq'.

theorem Eq.trans_le {α : Type u_1} {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} [Preorder α] {a : α} {b : α} {c : α} : b = c → a ≤ b → a ≤ c

Alias of le_of_eq_of_le'.

theorem Eq.trans_lt {α : Type u_1} {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} [Preorder α] {a : α} {b : α} {c : α} : b = c → a < b → a < c

Alias of lt_of_eq_of_lt'.

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

theorem Eq.not_gt {α : Type u} [Preorder α] {x : α} {y : α} (h : x = y) : ¬y < x

@[simp]

theorem le_of_subsingleton {α : Type u} [Preorder α] {a : α} {b : α} [Subsingleton α] : a ≤ b

theorem not_lt_of_subsingleton {α : Type u} [Preorder α] {a : α} {b : α} [Subsingleton α] : ¬a < b

theorem LE.le.ge {α : Type u} [LE α] {x : α} {y : α} (h : x ≤ y) : y ≥ x

theorem LE.le.lt_iff_ne {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : a < b ↔ a ≠ b

theorem LE.le.gt_iff_ne {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : a < b ↔ b ≠ a

theorem LE.le.not_lt_iff_eq {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : ¬a < b ↔ a = b

theorem LE.le.not_gt_iff_eq {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : ¬a < b ↔ b = a

theorem LE.le.le_iff_eq {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : b ≤ a ↔ b = a

theorem LE.le.ge_iff_eq {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : b ≤ a ↔ a = b

theorem LE.le.lt_or_le {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a ≤ b) (c : α) : a < c ∨ c ≤ b

theorem LE.le.le_or_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c < b

theorem LE.le.le_or_le {α : Type u} [LinearOrder α] {a : α} {b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c ≤ b

theorem LT.lt.gt {α : Type u} [LT α] {x : α} {y : α} (h : x < y) : y > x

theorem LT.lt.false {α : Type u} [Preorder α] {x : α} : x < x → False

theorem LT.lt.ne' {α : Type u} [Preorder α] {x : α} {y : α} (h : x < y) : y ≠ x

theorem LT.lt.lt_or_lt {α : Type u} [LinearOrder α] {x : α} {y : α} (h : x < y) (z : α) : x < z ∨ z < y

theorem GE.ge.le {α : Type u} [LE α] {x : α} {y : α} (h : x ≥ y) : y ≤ x

theorem GT.gt.lt {α : Type u} [LT α] {x : α} {y : α} (h : x > y) : y < x

theorem ge_of_eq {α : Type u} [Preorder α] {a : α} {b : α} (h : a = b) : a ≥ b

theorem not_le_of_lt {α : Type u} [Preorder α] {a : α} {b : α} (h : a < b) : ¬b ≤ a

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

theorem LE.le.not_lt {α : Type u} [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} [Preorder α] {a : α} {b : α} (h : ¬a ≤ b) : a ≠ b

theorem Decidable.le_iff_eq_or_lt {α : Type u} [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} [PartialOrder α] {a : α} {b : α} : a ≤ b ↔ a = b ∨ a < b

theorem lt_iff_le_and_ne {α : Type u} [PartialOrder α] {a : α} {b : α} : a < b ↔ a ≤ b ∧ a ≠ b

theorem eq_iff_not_lt_of_le {α : Type u_3} [PartialOrder α] {x : α} {y : α} : x ≤ y → y = x ↔ ¬x < y

theorem Decidable.eq_iff_le_not_lt {α : Type u} [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} [PartialOrder α] {a : α} {b : α} : a = b ↔ a ≤ b ∧ ¬a < b

theorem eq_or_lt_of_le {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : a = b ∨ a < b

theorem eq_or_gt_of_le {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≤ b) : b = a ∨ a < b

theorem gt_or_eq_of_le {α : Type u} [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} [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} [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} [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} [PartialOrder α] {a : α} {b : α} (hab : a ≤ b) (hba : ¬a < b) : a = b

theorem eq_of_ge_of_not_gt {α : Type u} [PartialOrder α] {a : α} {b : α} (hab : a ≤ b) (hba : ¬a < b) : b = a

theorem LE.le.eq_of_not_lt {α : Type u} [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} [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} [PartialOrder α] {a : α} {b : α} (h : a ≠ b) : a ≤ b ↔ a < b

theorem Ne.not_le_or_not_le {α : Type u} [PartialOrder α] {a : α} {b : α} (h : a ≠ b) : ¬a ≤ b ∨ ¬b ≤ a

theorem Decidable.ne_iff_lt_iff_le {α : Type u} [PartialOrder α] [DecidableEq α] {a : α} {b : α} : (a ≠ b ↔ a < b) ↔ a ≤ b

@[simp]

theorem ne_iff_lt_iff_le {α : Type u} [PartialOrder α] {a : α} {b : α} : (a ≠ b ↔ a < b) ↔ a ≤ b

theorem min_def' {α : Type u} [LinearOrder α] (a : α) (b : α) : min a b = if b ≤ a then b else a

theorem max_def' {α : Type u} [LinearOrder α] (a : α) (b : α) : max a b = if b ≤ a then a else b

theorem lt_of_not_le {α : Type u} [LinearOrder α] {a : α} {b : α} (h : ¬b ≤ a) : a < b

theorem lt_iff_not_le {α : Type u} [LinearOrder α] {x : α} {y : α} : x < y ↔ ¬y ≤ x

theorem Ne.lt_or_lt {α : Type u} [LinearOrder α] {x : α} {y : α} (h : x ≠ y) : x < y ∨ y < x

@[simp]

theorem lt_or_lt_iff_ne {α : Type u} [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} [LinearOrder α] {a : α} {b : α} : ¬a < b ↔ a = b ∨ b < a

theorem exists_ge_of_linear {α : Type u} [LinearOrder α] (a : α) (b : α) : ∃ (c : α), a ≤ c ∧ b ≤ c

theorem lt_imp_lt_of_le_imp_le {α : Type u} {β : Type u_3} [LinearOrder α] [Preorder β] {a : α} {b : α} {c : β} {d : β} (H : a ≤ b → c ≤ d) (h : d < c) : b < a

theorem le_imp_le_iff_lt_imp_lt {α : Type u} {β : Type u_3} [LinearOrder α] [LinearOrder β] {a : α} {b : α} {c : β} {d : β} : a ≤ b → c ≤ d ↔ d < c → b < a

theorem lt_iff_lt_of_le_iff_le' {α : Type u} {β : Type u_3} [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} {β : Type u_3} [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} {β : Type u_3} [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} [PartialOrder α] {a : α} {b : α} (H : ∀ (c : α), c ≤ a ↔ c ≤ b) : a = b

theorem le_of_forall_le {α : Type u} [Preorder α] {a : α} {b : α} (H : ∀ (c : α), c ≤ a → c ≤ b) : a ≤ b

theorem le_of_forall_le' {α : Type u} [Preorder α] {a : α} {b : α} (H : ∀ (c : α), a ≤ c → b ≤ c) : b ≤ a

theorem le_of_forall_lt {α : Type u} [LinearOrder α] {a : α} {b : α} (H : ∀ (c : α), c < a → c < b) : a ≤ b

theorem forall_lt_iff_le {α : Type u} [LinearOrder α] {a : α} {b : α} : (∀ ⦃c : α⦄, c < a → c < b) ↔ a ≤ b

theorem le_of_forall_lt' {α : Type u} [LinearOrder α] {a : α} {b : α} (H : ∀ (c : α), a < c → b < c) : b ≤ a

theorem forall_lt_iff_le' {α : Type u} [LinearOrder α] {a : α} {b : α} : (∀ ⦃c : α⦄, a < c → b < c) ↔ b ≤ a

theorem eq_of_forall_ge_iff {α : Type u} [PartialOrder α] {a : α} {b : α} (H : ∀ (c : α), a ≤ c ↔ b ≤ c) : a = b

theorem eq_of_forall_lt_iff {α : Type u} [LinearOrder α] {a : α} {b : α} (h : ∀ (c : α), c < a ↔ c < b) : a = b

theorem eq_of_forall_gt_iff {α : Type u} [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} {β : Type v} [PartialOrder β] (r : α → α → Prop) [IsSymm α r] {f : α → β} (h : ∀ (a b : α), r a b → f a ≤ f b) {a : α} {b : α} : r a b → f 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} {a : α} {b : α} {c : α} {d : α} [Preorder α] (hca : c ≤ a) (hbd : b ≤ d) : a ≤ b → c ≤ d

monotonicity of ≤ with respect to →

theorem commutative_of_le {α : Type u} {β : Type v} [PartialOrder α] {f : β → β → α} (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} [PartialOrder α] {f : α → α → α} (comm : Commutative f) (assoc : ∀ (a b c : α), f (f a b) c ≤ f a (f b c)) : Associative f

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.toLE_injective {α : Type u_3} : Function.Injective (@Preorder.toLE α)

theorem PartialOrder.toPreorder_injective {α : Type u_3} : Function.Injective (@PartialOrder.toPreorder α)

theorem LinearOrder.toPartialOrder_injective {α : Type u_3} : Function.Injective (@LinearOrder.toPartialOrder α)

theorem Preorder.ext {α : Type u_3} {A : Preorder α} {B : Preorder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) : A = B

theorem PartialOrder.ext {α : Type u_3} {A : PartialOrder α} {B : PartialOrder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) : A = B

theorem LinearOrder.ext {α : Type u_3} {A : LinearOrder α} {B : LinearOrder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y) : A = B

def Order.Preimage {α : Sort u_3} {β : Sort u_4} (f : α → β) (s : β → β → Prop) (x : α) (y : α) : Prop

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

• (f ⁻¹'o s) x y = s (f x) (f y)

def «term_⁻¹'o_» : Lean.TrailingParserDescr

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

• «term_⁻¹'o_» = Lean.ParserDescr.trailingNode `term_⁻¹'o_ 80 80 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⁻¹'o ") (Lean.ParserDescr.cat `term 81))

instance Order.Preimage.decidable {α : Sort u_3} {β : Sort u_4} (f : α → β) (s : β → β → Prop) [H : DecidableRel s] : DecidableRel (f ⁻¹'o s)

The preimage of a decidable order is decidable.

Equations

• Order.Preimage.decidable f s x✝ x = H (f x✝) (f x)

Order dual

def OrderDual (α : Type u_3) : Type u_3

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

Equations

• αᵒᵈ = α

def «term_ᵒᵈ» : Lean.TrailingParserDescr

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

Equations

• «term_ᵒᵈ» = Lean.ParserDescr.trailingNode `term_ᵒᵈ 1024 0 (Lean.ParserDescr.symbol "ᵒᵈ")

instance OrderDual.instNonemptyOrderDual (α : Type u_3) [h : Nonempty α] : Nonempty αᵒᵈ

Equations

• (_ : Nonempty αᵒᵈ) = h

instance OrderDual.instSubsingletonOrderDual (α : Type u_3) [h : Subsingleton α] : Subsingleton αᵒᵈ

Equations

• (_ : Subsingleton αᵒᵈ) = h

instance OrderDual.instLEOrderDual (α : Type u_3) [LE α] : LE αᵒᵈ

Equations

• OrderDual.instLEOrderDual α = { le := fun (x y : α) => y ≤ x }

instance OrderDual.instLTOrderDual (α : Type u_3) [LT α] : LT αᵒᵈ

Equations

• OrderDual.instLTOrderDual α = { lt := fun (x y : α) => y < x }

instance OrderDual.instPreorder (α : Type u_3) [Preorder α] : Preorder αᵒᵈ

Equations

• OrderDual.instPreorder α = Preorder.mk (_ : ∀ (x : αᵒᵈ), x ≤ x) (_ : ∀ (x x_1 x_2 : αᵒᵈ), x ≤ x_1 → x_1 ≤ x_2 → x_2 ≤ x)

instance OrderDual.instPartialOrder (α : Type u_3) [PartialOrder α] : PartialOrder αᵒᵈ

Equations

• OrderDual.instPartialOrder α = let src := inferInstanceAs (Preorder αᵒᵈ); PartialOrder.mk (_ : ∀ (a b : αᵒᵈ), a ≤ b → b ≤ a → a = b)

instance OrderDual.instLinearOrder (α : Type u_3) [LinearOrder α] : LinearOrder αᵒᵈ

Equations

• OrderDual.instLinearOrder α = let src := inferInstanceAs (PartialOrder αᵒᵈ); LinearOrder.mk (_ : ∀ (a b : α), b ≤ a ∨ a ≤ b) inferInstance decidableEqOfDecidableLE inferInstance

instance OrderDual.instForAllInhabitedOrderDual {α : Type u} [Inhabited α] : Inhabited αᵒᵈ

Equations

• OrderDual.instForAllInhabitedOrderDual = x

theorem OrderDual.Preorder.dual_dual (α : Type u_3) [H : Preorder α] : OrderDual.instPreorder αᵒᵈ = H

theorem OrderDual.instPartialOrder.dual_dual (α : Type u_3) [H : PartialOrder α] : OrderDual.instPartialOrder αᵒᵈ = H

theorem OrderDual.instLinearOrder.dual_dual (α : Type u_3) [H : LinearOrder α] : OrderDual.instLinearOrder αᵒᵈ = H

HasCompl

class HasCompl (α : Type u_3) : Type u_3

Set / lattice complement

• compl : α → α

  Set / lattice complement

def «term_ᶜ» : Lean.TrailingParserDescr

Set / lattice complement

Equations

• «term_ᶜ» = Lean.ParserDescr.trailingNode `term_ᶜ 1024 1024 (Lean.ParserDescr.symbol "ᶜ")

instance Prop.hasCompl : HasCompl Prop

Equations

• Prop.hasCompl = { compl := Not }

instance Pi.hasCompl {ι : Type u} {α : ι → Type v} [(i : ι) → HasCompl (α i)] : HasCompl ((i : ι) → α i)

Equations

• Pi.hasCompl = { compl := fun (x : (i : ι) → α i) (i : ι) => (x i)ᶜ }

theorem Pi.compl_def {ι : Type u} {α : ι → Type v} [(i : ι) → HasCompl (α i)] (x : (i : ι) → α i) : xᶜ = fun (i : ι) => (x i)ᶜ

@[simp]

theorem Pi.compl_apply {ι : Type u} {α : ι → Type v} [(i : ι) → HasCompl (α i)] (x : (i : ι) → α i) (i : ι) : xᶜ i = (x i)ᶜ

instance IsIrrefl.compl {α : Type u} (r : α → α → Prop) [IsIrrefl α r] : IsRefl α rᶜ

Equations

• (_ : IsRefl α rᶜ) = (_ : IsRefl α rᶜ)

instance IsRefl.compl {α : Type u} (r : α → α → Prop) [IsRefl α r] : IsIrrefl α rᶜ

Equations

• (_ : IsIrrefl α rᶜ) = (_ : IsIrrefl α rᶜ)

Order instances on the function space

instance Pi.hasLe {ι : Type u} {α : ι → Type v} [(i : ι) → LE (α i)] : LE ((i : ι) → α i)

Equations

• Pi.hasLe = { le := fun (x y : (i : ι) → α i) => ∀ (i : ι), x i ≤ y i }

theorem Pi.le_def {ι : Type u} {α : ι → Type v} [(i : ι) → LE (α i)] {x : (i : ι) → α i} {y : (i : ι) → α i} : x ≤ y ↔ ∀ (i : ι), x i ≤ y i

instance Pi.preorder {ι : Type u} {α : ι → Type v} [(i : ι) → Preorder (α i)] : Preorder ((i : ι) → α i)

Equations

• Pi.preorder = let src := inferInstanceAs (LE ((i : ι) → α i)); Preorder.mk (_ : ∀ (a : (i : ι) → α i) (i : ι), a i ≤ a i) (_ : ∀ (a b c : (i : ι) → α i), a ≤ b → b ≤ c → ∀ (i : ι), a i ≤ c i)

theorem Pi.lt_def {ι : Type u} {α : ι → Type v} [(i : ι) → Preorder (α i)] {x : (i : ι) → α i} {y : (i : ι) → α i} : x < y ↔ x ≤ y ∧ ∃ (i : ι), x i < y i

instance Pi.partialOrder {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → PartialOrder (π i)] : PartialOrder ((i : ι) → π i)

Equations

• Pi.partialOrder = let src := Pi.preorder; PartialOrder.mk (_ : ∀ (x x_1 : (i : ι) → π i), x ≤ x_1 → x_1 ≤ x → x = x_1)

def StrongLT {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → LT (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) : Prop

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

Equations

• StrongLT a b = ∀ (i : ι), a i < b i

theorem le_of_strongLT {ι : Type u_1} {π : ι → Type u_2} [(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 : ι → Type u_2} [(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 : ι → Type u_2} [(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) : StrongLT a c

theorem strongLT_of_le_of_strongLT {ι : Type u_1} {π : ι → Type u_2} [(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) : StrongLT a c

theorem StrongLT.le {ι : Type u_1} {π : ι → Type u_2} [(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 : ι → Type u_2} [(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 : ι → Type u_2} [(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) : StrongLT a c

Alias of strongLT_of_strongLT_of_le.

theorem LE.le.trans_strongLT {ι : Type u_1} {π : ι → Type u_2} [(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) : StrongLT a c

Alias of strongLT_of_le_of_strongLT.

theorem le_update_iff {ι : Type u_1} {π : ι → Type u_2} [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 ≠ i → x j ≤ y j

theorem update_le_iff {ι : Type u_1} {π : ι → Type u_2} [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 ≠ i → x j ≤ y j

theorem update_le_update_iff {ι : Type u_1} {π : ι → Type u_2} [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 ≠ i → x j ≤ y j

@[simp]

theorem update_le_update_iff' {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} {b : π i} : Function.update x i a ≤ Function.update x i b ↔ a ≤ b

@[simp]

theorem update_lt_update_iff {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι] [(i : ι) → Preorder (π i)] {x : (i : ι) → π i} {i : ι} {a : π i} {b : π i} : Function.update x i a < Function.update x i b ↔ a < b

@[simp]

theorem le_update_self_iff {ι : Type u_1} {π : ι → Type u_2} [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 : ι → Type u_2} [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 : ι → Type u_2} [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 : ι → Type u_2} [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} {α : ι → Type v} [(i : ι) → SDiff (α i)] : SDiff ((i : ι) → α i)

Equations

• Pi.sdiff = { sdiff := fun (x y : (i : ι) → α i) (i : ι) => x i \ y i }

theorem Pi.sdiff_def {ι : Type u} {α : ι → Type v} [(i : ι) → SDiff (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) : x \ y = fun (i : ι) => x i \ y i

@[simp]

theorem Pi.sdiff_apply {ι : Type u} {α : ι → Type v} [(i : ι) → SDiff (α i)] (x : (i : ι) → α i) (y : (i : ι) → α i) (i : ι) : (x \ y) i = x i \ y i

@[simp]

theorem Function.const_le_const {α : Type u} {β : Type v} [Preorder α] [Nonempty β] {a : α} {b : α} : Function.const β a ≤ Function.const β b ↔ a ≤ b

@[simp]

theorem Function.const_lt_const {α : Type u} {β : Type v} [Preorder α] [Nonempty β] {a : α} {b : α} : Function.const β a < Function.const β b ↔ a < b

min/max recursors

theorem min_rec {α : Type u} [LinearOrder α] {p : α → Prop} {x : α} {y : α} (hx : x ≤ y → p x) (hy : y ≤ x → p y) : p (min x y)

theorem max_rec {α : Type u} [LinearOrder α] {p : α → Prop} {x : α} {y : α} (hx : y ≤ x → p x) (hy : x ≤ y → p y) : p (max x y)

theorem min_rec' {α : Type u} [LinearOrder α] {x : α} {y : α} (p : α → Prop) (hx : p x) (hy : p y) : p (min x y)

theorem max_rec' {α : Type u} [LinearOrder α] {x : α} {y : α} (p : α → Prop) (hx : p x) (hy : p y) : p (max x y)

theorem min_def_lt {α : Type u} [LinearOrder α] (x : α) (y : α) : min x y = if x < y then x else y

theorem max_def_lt {α : Type u} [LinearOrder α] (x : α) (y : α) : max x y = if x < y then y else x

Sup and Inf

theorem Sup.ext {α : Type u} (x : Sup α) (y : Sup α) (sup : Sup.sup = Sup.sup) : x = y

theorem Sup.ext_iff {α : Type u} (x : Sup α) (y : Sup α) : x = y ↔ Sup.sup = Sup.sup

class Sup (α : Type u) : Type u

Typeclass for the ⊔ (\lub) notation

• sup : α → α → α

  Least upper bound (\lub notation)

theorem Inf.ext {α : Type u} (x : Inf α) (y : Inf α) (inf : Inf.inf = Inf.inf) : x = y

theorem Inf.ext_iff {α : Type u} (x : Inf α) (y : Inf α) : x = y ↔ Inf.inf = Inf.inf

class Inf (α : Type u) : Type u

Typeclass for the ⊓ (\glb) notation

• inf : α → α → α

  Greatest lower bound (\glb notation)

def «term_⊔_» : Lean.TrailingParserDescr

Least upper bound (\lub notation)

Equations

• «term_⊔_» = Lean.ParserDescr.trailingNode `term_⊔_ 68 68 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊔ ") (Lean.ParserDescr.cat `term 69))

def «term_⊓_» : Lean.TrailingParserDescr

Greatest lower bound (\glb notation)

Equations

• «term_⊓_» = Lean.ParserDescr.trailingNode `term_⊓_ 69 69 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊓ ") (Lean.ParserDescr.cat `term 70))

Lifts of order instances

@[reducible]

def Preorder.lift {α : Type u_3} {β : Type u_4} [Preorder β] (f : α → β) : Preorder α

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

Equations

• Preorder.lift f = Preorder.mk (_ : ∀ (x : α), f x ≤ f x) (_ : ∀ (x x_1 x_2 : α), f x ≤ f x_1 → f x_1 ≤ f x_2 → f x ≤ f x_2)

@[reducible]

def PartialOrder.lift {α : Type u_3} {β : Type u_4} [PartialOrder β] (f : α → β) (inj : Function.Injective f) : PartialOrder α

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

Equations

• PartialOrder.lift f inj = let src := Preorder.lift f; PartialOrder.mk (_ : ∀ (x x_1 : α), x ≤ x_1 → x_1 ≤ x → x = x_1)

theorem compare_of_injective_eq_compareOfLessAndEq {α : Type u} {β : Type v} (a : α) (b : α) [LinearOrder β] [DecidableEq α] (f : α → β) (inj : Function.Injective f) [Decidable (a < b)] : compare (f a) (f b) = compareOfLessAndEq a b

@[reducible]

def LinearOrder.lift {α : Type u_3} {β : Type u_4} [LinearOrder β] [Sup α] [Inf α] (f : α → β) (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)) : LinearOrder α

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.

@[reducible]

def LinearOrder.lift' {α : Type u_3} {β : Type u_4} [LinearOrder β] (f : α → β) (inj : Function.Injective f) : LinearOrder α

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

• One or more equations did not get rendered due to their size.

@[reducible]

def LinearOrder.liftWithOrd {α : Type u_3} {β : Type u_4} [LinearOrder β] [Sup α] [Inf α] [Ord α] (f : α → β) (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)) : LinearOrder α

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.

@[reducible]

def LinearOrder.liftWithOrd' {α : Type u_3} {β : Type u_4} [LinearOrder β] [Ord α] (f : α → β) (inj : Function.Injective f) (compare_f : ∀ (a b : α), compare a b = compare (f a) (f b)) : LinearOrder α

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

• One or more equations did not get rendered due to their size.

Subtype of an order

instance Subtype.le {α : Type u} [LE α] {p : α → Prop} : LE (Subtype p)

Equations

• Subtype.le = { le := fun (x y : Subtype p) => ↑x ≤ ↑y }

instance Subtype.lt {α : Type u} [LT α] {p : α → Prop} : LT (Subtype p)

Equations

• Subtype.lt = { lt := fun (x y : Subtype p) => ↑x < ↑y }

@[simp]

theorem Subtype.mk_le_mk {α : Type u} [LE α] {p : α → Prop} {x : α} {y : α} {hx : p x} {hy : p y} : { val := x, property := hx } ≤ { val := y, property := hy } ↔ x ≤ y

@[simp]

theorem Subtype.mk_lt_mk {α : Type u} [LT α] {p : α → Prop} {x : α} {y : α} {hx : p x} {hy : p y} : { val := x, property := hx } < { val := y, property := hy } ↔ x < y

@[simp]

theorem Subtype.coe_le_coe {α : Type u} [LE α] {p : α → Prop} {x : Subtype p} {y : Subtype p} : ↑x ≤ ↑y ↔ x ≤ y

@[simp]

theorem Subtype.coe_lt_coe {α : Type u} [LT α] {p : α → Prop} {x : Subtype p} {y : Subtype p} : ↑x < ↑y ↔ x < y

instance Subtype.preorder {α : Type u} [Preorder α] (p : α → Prop) : Preorder (Subtype p)

Equations

• Subtype.preorder p = Preorder.lift fun (a : Subtype p) => ↑a

instance Subtype.partialOrder {α : Type u} [PartialOrder α] (p : α → Prop) : PartialOrder (Subtype p)

Equations

• Subtype.partialOrder p = PartialOrder.lift (fun (a : Subtype p) => ↑a) (_ : Function.Injective fun (a : Subtype p) => ↑a)

instance Subtype.decidableLE {α : Type u} [Preorder α] [h : DecidableRel fun (x x_1 : α) => x ≤ x_1] {p : α → Prop} : DecidableRel fun (x x_1 : Subtype p) => x ≤ x_1

Equations

• Subtype.decidableLE a b = h ↑a ↑b

instance Subtype.decidableLT {α : Type u} [Preorder α] [h : DecidableRel fun (x x_1 : α) => x < x_1] {p : α → Prop} : DecidableRel fun (x x_1 : Subtype p) => x < x_1

Equations

• Subtype.decidableLT a b = h ↑a ↑b

instance Subtype.linearOrder {α : Type u} [LinearOrder α] (p : α → Prop) : LinearOrder (Subtype p)

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

• One or more equations did not get rendered due to their size.

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.instLEProd (α : Type u) (β : Type v) [LE α] [LE β] : LE (α × β)

Equations

• Prod.instLEProd α β = { le := fun (p q : α × β) => p.1 ≤ q.1 ∧ p.2 ≤ q.2 }

instance Prod.instDecidableLE (α : Type u) (β : Type v) [LE α] [LE β] (x : α × β) (y : α × β) [Decidable (x.1 ≤ y.1)] [Decidable (x.2 ≤ y.2)] : Decidable (x ≤ y)

Equations

• Prod.instDecidableLE α β x y = And.decidable

theorem Prod.le_def {α : Type u} {β : Type v} [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} {β : Type v} [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} {β : Type v} [LE α] [LE β] {x : α × β} {y : α × β} : Prod.swap x ≤ Prod.swap y ↔ x ≤ y

instance Prod.instPreorderProd (α : Type u) (β : Type v) [Preorder α] [Preorder β] : Preorder (α × β)

Equations

• Prod.instPreorderProd α β = let src := inferInstanceAs (LE (α × β)); Preorder.mk (_ : ∀ (x : α × β), x ≤ x) (_ : ∀ (x x_1 x_2 : α × β), x ≤ x_1 → x_1 ≤ x_2 → x ≤ x_2)

@[simp]

theorem Prod.swap_lt_swap {α : Type u} {β : Type v} [Preorder α] [Preorder β] {x : α × β} {y : α × β} : Prod.swap x < Prod.swap y ↔ x < y

theorem Prod.mk_le_mk_iff_left {α : Type u} {β : Type v} [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} {β : Type v} [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} {β : Type v} [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} {β : Type v} [Preorder α] [Preorder β] {a : α} {b₁ : β} {b₂ : β} : (a, b₁) < (a, b₂) ↔ b₁ < b₂

theorem Prod.lt_iff {α : Type u} {β : Type v} [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} {β : Type v} [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) (β : Type v) [PartialOrder α] [PartialOrder β] : PartialOrder (α × 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

• Prod.instPartialOrder α β = let src := inferInstanceAs (Preorder (α × β)); PartialOrder.mk (_ : ∀ (x x_1 : α × β), x ≤ x_1 → x_1 ≤ x → x = x_1)

Additional order classes

class DenselyOrdered (α : Type u) [LT α] : Prop

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.

theorem exists_between {α : Type u} [LT α] [DenselyOrdered α] {a₁ : α} {a₂ : α} : a₁ < a₂ → ∃ (a : α), a₁ < a ∧ a < a₂

instance OrderDual.denselyOrdered (α : Type u) [LT α] [h : DenselyOrdered α] : DenselyOrdered αᵒᵈ

Equations

• (_ : DenselyOrdered αᵒᵈ) = (_ : DenselyOrdered αᵒᵈ)

@[simp]

theorem denselyOrdered_orderDual {α : Type u} [LT α] : DenselyOrdered αᵒᵈ ↔ DenselyOrdered α

instance instDenselyOrderedProdToLTInstPreorderProd {α : Type u} {β : Type v} [Preorder α] [Preorder β] [DenselyOrdered α] [DenselyOrdered β] : DenselyOrdered (α × β)

Equations

• (_ : DenselyOrdered (α × β)) = (_ : DenselyOrdered (α × β))

instance instDenselyOrderedForAllToLTPreorder {ι : Type u_1} {α : ι → Type u_3} [(i : ι) → Preorder (α i)] [∀ (i : ι), DenselyOrdered (α i)] : DenselyOrdered ((i : ι) → α i)

Equations

• (_ : DenselyOrdered ((i : ι) → α i)) = (_ : DenselyOrdered ((i : ι) → α i))

theorem le_of_forall_le_of_dense {α : Type u} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h : ∀ (a : α), a₂ < a → a₁ ≤ a) : a₁ ≤ a₂

theorem eq_of_le_of_forall_le_of_dense {α : Type u} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : ∀ (a : α), a₂ < a → a₁ ≤ a) : a₁ = a₂

theorem le_of_forall_ge_of_dense {α : Type u} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h : ∀ (a₃ : α), a₃ < a₁ → a₃ ≤ a₂) : a₁ ≤ a₂

theorem eq_of_le_of_forall_ge_of_dense {α : Type u} [LinearOrder α] [DenselyOrdered α] {a₁ : α} {a₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : ∀ (a₃ : α), a₃ < a₁ → a₃ ≤ a₂) : a₁ = a₂

theorem dense_or_discrete {α : Type u} [LinearOrder α] (a₁ : α) (a₂ : α) : (∃ (a : α), a₁ < a ∧ a < a₂) ∨ (∀ (a : α), a₁ < a → a₂ ≤ a) ∧ ∀ (a : α), a < a₂ → a ≤ a₁

theorem eq_or_eq_or_eq_of_forall_not_lt_lt {α : Type u} [LinearOrder α] (h : ∀ ⦃x y z : α⦄, x < y → y < z → False) (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.

instance PUnit.linearOrder : LinearOrder PUnit.{u_3 + 1}

Equations

• One or more equations did not get rendered due to their size.

theorem PUnit.max_eq (a : PUnit.{u + 1} ) (b : PUnit.{u + 1} ) : max a b = PUnit.unit

theorem PUnit.min_eq (a : PUnit.{u + 1} ) (b : PUnit.{u + 1} ) : min a b = PUnit.unit

theorem PUnit.le (a : PUnit.{u + 1} ) (b : PUnit.{u + 1} ) : a ≤ b

theorem PUnit.not_lt (a : PUnit.{u + 1} ) (b : PUnit.{u + 1} ) : ¬a < b

instance PUnit.instDenselyOrderedPUnitToLTToPreorderToPartialOrderLinearOrder : DenselyOrdered PUnit.{u_3 + 1}

Equations

• PUnit.instDenselyOrderedPUnitToLTToPreorderToPartialOrderLinearOrder = (_ : DenselyOrdered PUnit.{u_3 + 1} )

instance Prop.le : LE Prop

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

Equations

• Prop.le = { le := fun (x x_1 : Prop) => x → x_1 }

@[simp]

theorem le_Prop_eq : (fun (x x_1 : Prop) => x ≤ x_1) = fun (x x_1 : Prop) => x → x_1

theorem subrelation_iff_le {α : Type u} {r : α → α → Prop} {s : α → α → Prop} : Subrelation r s ↔ r ≤ s

instance Prop.partialOrder : PartialOrder Prop

Equations

• Prop.partialOrder = let src := Prop.le; PartialOrder.mk Prop.partialOrder.proof_2

Linear order from a total partial order

def AsLinearOrder (α : Type u) : Type u

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

Equations

• AsLinearOrder α = α

instance instInhabitedAsLinearOrder {α : Type u_3} [Inhabited α] : Inhabited (AsLinearOrder α)

Equations

• instInhabitedAsLinearOrder = { default := default }

noncomputable instance AsLinearOrder.linearOrder {α : Type u_3} [PartialOrder α] [IsTotal α fun (x x_1 : α) => x ≤ x_1] : LinearOrder (AsLinearOrder α)

Equations

• One or more equations did not get rendered due to their size.

theorem dite_nonneg {α : Type u} [Zero α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [One α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [Zero α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [One α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [Zero α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [One α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [Zero α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [One α] {p : Prop} [Decidable p] {a : p → α} {b : ¬p → α} [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} [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} [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} [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} [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} [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} [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} [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} [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