Unbundled relation classes

In this file we prove some properties of Is* classes defined in Init.Algebra.Classes. The main difference between these classes and the usual order classes (Preorder etc) is that usual classes extend LE and/or LT while these classes take a relation as an explicit argument.

theorem of_eq {α : Type u} {r : α → α → Prop} [IsRefl α r] {a : α} {b : α} : a = b → r a b

theorem comm {α : Type u} {r : α → α → Prop} [IsSymm α r] {a : α} {b : α} : r a b ↔ r b a

theorem antisymm' {α : Type u} {r : α → α → Prop} [IsAntisymm α r] {a : α} {b : α} : r a b → r b a → b = a

theorem antisymm_iff {α : Type u} {r : α → α → Prop} [IsRefl α r] [IsAntisymm α r] {a : α} {b : α} : r a b ∧ r b a ↔ a = b

@[elab_without_expected_type]

theorem antisymm_of {α : Type u} (r : α → α → Prop) [IsAntisymm α r] {a : α} {b : α} : r a b → r b a → a = b

A version of antisymm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

@[elab_without_expected_type]

theorem antisymm_of' {α : Type u} (r : α → α → Prop) [IsAntisymm α r] {a : α} {b : α} : r a b → r b a → b = a

A version of antisymm' with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem comm_of {α : Type u} (r : α → α → Prop) [IsSymm α r] {a : α} {b : α} : r a b ↔ r b a

A version of comm with r explicit.

This lemma matches the lemmas from lean core in Init.Algebra.Classes, but is missing there.

theorem IsRefl.swap {α : Type u} (r : α → α → Prop) [IsRefl α r] : IsRefl α (Function.swap r)

theorem IsIrrefl.swap {α : Type u} (r : α → α → Prop) [IsIrrefl α r] : IsIrrefl α (Function.swap r)

theorem IsTrans.swap {α : Type u} (r : α → α → Prop) [IsTrans α r] : IsTrans α (Function.swap r)

theorem IsAntisymm.swap {α : Type u} (r : α → α → Prop) [IsAntisymm α r] : IsAntisymm α (Function.swap r)

theorem IsAsymm.swap {α : Type u} (r : α → α → Prop) [IsAsymm α r] : IsAsymm α (Function.swap r)

theorem IsTotal.swap {α : Type u} (r : α → α → Prop) [IsTotal α r] : IsTotal α (Function.swap r)

theorem IsTrichotomous.swap {α : Type u} (r : α → α → Prop) [IsTrichotomous α r] : IsTrichotomous α (Function.swap r)

theorem IsPreorder.swap {α : Type u} (r : α → α → Prop) [IsPreorder α r] : IsPreorder α (Function.swap r)

theorem IsStrictOrder.swap {α : Type u} (r : α → α → Prop) [IsStrictOrder α r] : IsStrictOrder α (Function.swap r)

theorem IsPartialOrder.swap {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] : IsPartialOrder α (Function.swap r)

theorem IsTotalPreorder.swap {α : Type u} (r : α → α → Prop) [IsTotalPreorder α r] : IsTotalPreorder α (Function.swap r)

theorem IsLinearOrder.swap {α : Type u} (r : α → α → Prop) [IsLinearOrder α r] : IsLinearOrder α (Function.swap r)

theorem IsAsymm.isAntisymm {α : Type u} (r : α → α → Prop) [IsAsymm α r] : IsAntisymm α r

theorem IsAsymm.isIrrefl {α : Type u} {r : α → α → Prop} [IsAsymm α r] : IsIrrefl α r

theorem IsTotal.isTrichotomous {α : Type u} (r : α → α → Prop) [IsTotal α r] : IsTrichotomous α r

instance IsTotal.to_isRefl {α : Type u} (r : α → α → Prop) [IsTotal α r] : IsRefl α r

Equations

• (_ : IsRefl α r) = (_ : IsRefl α r)

theorem ne_of_irrefl {α : Type u} {r : α → α → Prop} [IsIrrefl α r] {x : α} {y : α} : r x y → x ≠ y

theorem ne_of_irrefl' {α : Type u} {r : α → α → Prop} [IsIrrefl α r] {x : α} {y : α} : r x y → y ≠ x

theorem not_rel_of_subsingleton {α : Type u} (r : α → α → Prop) [IsIrrefl α r] [Subsingleton α] (x : α) (y : α) : ¬r x y

theorem rel_of_subsingleton {α : Type u} (r : α → α → Prop) [IsRefl α r] [Subsingleton α] (x : α) (y : α) : r x y

@[simp]

theorem empty_relation_apply {α : Type u} (a : α) (b : α) : EmptyRelation a b ↔ False

theorem eq_empty_relation {α : Type u} (r : α → α → Prop) [IsIrrefl α r] [Subsingleton α] : r = EmptyRelation

instance instIsIrreflEmptyRelation {α : Type u} : IsIrrefl α EmptyRelation

Equations

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

theorem trans_trichotomous_left {α : Type u} {r : α → α → Prop} [IsTrans α r] [IsTrichotomous α r] {a : α} {b : α} {c : α} : ¬r b a → r b c → r a c

theorem trans_trichotomous_right {α : Type u} {r : α → α → Prop} [IsTrans α r] [IsTrichotomous α r] {a : α} {b : α} {c : α} : r a b → ¬r c b → r a c

theorem transitive_of_trans {α : Type u} (r : α → α → Prop) [IsTrans α r] : Transitive r

theorem extensional_of_trichotomous_of_irrefl {α : Type u} (r : α → α → Prop) [IsTrichotomous α r] [IsIrrefl α r] {a : α} {b : α} (H : ∀ (x : α), r x a ↔ r x b) : a = b

In a trichotomous irreflexive order, every element is determined by the set of predecessors.

@[reducible]

def partialOrderOfSO {α : Type u} (r : α → α → Prop) [IsStrictOrder α r] : PartialOrder α

Construct a partial order from an isStrictOrder relation.

See note [reducible non-instances].

Equations

• partialOrderOfSO r = PartialOrder.mk (_ : ∀ (x y : α), x ≤ y → y ≤ x → x = y)

@[reducible]

def linearOrderOfSTO {α : Type u} (r : α → α → Prop) [IsStrictTotalOrder α r] [(x y : α) → Decidable ¬r x y] : LinearOrder α

Construct a linear order from an IsStrictTotalOrder relation.

See note [reducible non-instances].

Equations

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

theorem IsStrictTotalOrder.swap {α : Type u} (r : α → α → Prop) [IsStrictTotalOrder α r] : IsStrictTotalOrder α (Function.swap r)

Order connection

class IsOrderConnected (α : Type u) (lt : α → α → Prop) : Prop

A connected order is one satisfying the condition a < c → a < b ∨ b < c. This is recognizable as an intuitionistic substitute for a ≤ b ∨ b ≤ a on the constructive reals, and is also known as negative transitivity, since the contrapositive asserts transitivity of the relation ¬ a < b.

• conn : ∀ (a b c : α), lt a c → lt a b ∨ lt b c

  A connected order is one satisfying the condition a < c → a < b ∨ b < c.

theorem IsOrderConnected.neg_trans {α : Type u} {r : α → α → Prop} [IsOrderConnected α r] {a : α} {b : α} {c : α} (h₁ : ¬r a b) (h₂ : ¬r b c) : ¬r a c

theorem isStrictWeakOrder_of_isOrderConnected {α : Type u} {r : α → α → Prop} [IsAsymm α r] [IsOrderConnected α r] : IsStrictWeakOrder α r

instance isStrictOrderConnected_of_isStrictTotalOrder {α : Type u} {r : α → α → Prop} [IsStrictTotalOrder α r] : IsOrderConnected α r

Equations

• (_ : IsOrderConnected α r) = (_ : IsOrderConnected α r)

instance isStrictTotalOrder_of_isStrictTotalOrder {α : Type u} {r : α → α → Prop} [IsStrictTotalOrder α r] : IsStrictWeakOrder α r

Equations

• (_ : IsStrictWeakOrder α r) = (_ : IsStrictWeakOrder α r)

Well-order

theorem IsWellFounded_iff (α : Type u) (r : α → α → Prop) : IsWellFounded α r ↔ WellFounded r

class IsWellFounded (α : Type u) (r : α → α → Prop) : Prop

A well-founded relation. Not to be confused with IsWellOrder.

• wf : WellFounded r

  The relation is WellFounded, as a proposition.

instance WellFoundedRelation.isWellFounded {α : Type u} [h : WellFoundedRelation α] : IsWellFounded α WellFoundedRelation.rel

Equations

• (_ : IsWellFounded α WellFoundedRelation.rel) = (_ : IsWellFounded α WellFoundedRelation.rel)

theorem WellFoundedRelation.asymmetric {α : Sort u_1} [WellFoundedRelation α] {a : α} {b : α} : WellFoundedRelation.rel a b → ¬WellFoundedRelation.rel b a

theorem WellFounded.prod_lex {α : Type u} {β : Type v} {ra : α → α → Prop} {rb : β → β → Prop} (ha : WellFounded ra) (hb : WellFounded rb) : WellFounded (Prod.Lex ra rb)

theorem IsWellFounded.induction {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {C : α → Prop} (a : α) : (∀ (x : α), (∀ (y : α), r y x → C y) → C x) → C a

Induction on a well-founded relation.

theorem IsWellFounded.apply {α : Type u} (r : α → α → Prop) [IsWellFounded α r] (a : α) : Acc r a

All values are accessible under the well-founded relation.

def IsWellFounded.fix {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {C : α → Sort u_1} : ((x : α) → ((y : α) → r y x → C y) → C x) → (x : α) → C x

Creates data, given a way to generate a value from all that compare as less under a well-founded relation. See also IsWellFounded.fix_eq.

Equations

• IsWellFounded.fix r = WellFounded.fix (_ : WellFounded fun (y x : α) => r y x)

theorem IsWellFounded.fix_eq {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {C : α → Sort u_1} (F : (x : α) → ((y : α) → r y x → C y) → C x) (x : α) : IsWellFounded.fix r F x = F x fun (y : α) (x : r y x) => IsWellFounded.fix r F y

The value from IsWellFounded.fix is built from the previous ones as specified.

def IsWellFounded.toWellFoundedRelation {α : Type u} (r : α → α → Prop) [IsWellFounded α r] : WellFoundedRelation α

Derive a WellFoundedRelation instance from an isWellFounded instance.

Equations

• IsWellFounded.toWellFoundedRelation r = { rel := r, wf := (_ : WellFounded r) }

theorem WellFounded.asymmetric {α : Sort u_1} {r : α → α → Prop} (h : WellFounded r) (a : α) (b : α) : r a b → ¬r b a

instance instIsAsymm {α : Type u} (r : α → α → Prop) [IsWellFounded α r] : IsAsymm α r

Equations

• (_ : IsAsymm α r) = (_ : IsAsymm α r)

instance instIsIrrefl {α : Type u} (r : α → α → Prop) [IsWellFounded α r] : IsIrrefl α r

Equations

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

instance instIsWellFoundedTransGen {α : Type u} (r : α → α → Prop) [i : IsWellFounded α r] : IsWellFounded α (Relation.TransGen r)

Equations

• (_ : IsWellFounded α (Relation.TransGen r)) = (_ : IsWellFounded α (Relation.TransGen r))

@[reducible]

def WellFoundedLT (α : Type u_1) [LT α] : Prop

A class for a well founded relation <.

Equations

• WellFoundedLT α = IsWellFounded α fun (x x_1 : α) => x < x_1

@[reducible]

def WellFoundedGT (α : Type u_1) [LT α] : Prop

A class for a well founded relation >.

Equations

• WellFoundedGT α = IsWellFounded α fun (x x_1 : α) => x > x_1

theorem wellFounded_lt {α : Type u} [LT α] [WellFoundedLT α] : WellFounded fun (x x_1 : α) => x < x_1

theorem wellFounded_gt {α : Type u} [LT α] [WellFoundedGT α] : WellFounded fun (x x_1 : α) => x > x_1

instance instWellFoundedGTOrderDualInstLTOrderDual (α : Type u_1) [LT α] [h : WellFoundedLT α] : WellFoundedGT αᵒᵈ

Equations

• (_ : WellFoundedGT αᵒᵈ) = h

instance instWellFoundedLTOrderDualInstLTOrderDual (α : Type u_1) [LT α] [h : WellFoundedGT α] : WellFoundedLT αᵒᵈ

Equations

• (_ : WellFoundedLT αᵒᵈ) = h

theorem wellFoundedGT_dual_iff (α : Type u_1) [LT α] : WellFoundedGT αᵒᵈ ↔ WellFoundedLT α

theorem wellFoundedLT_dual_iff (α : Type u_1) [LT α] : WellFoundedLT αᵒᵈ ↔ WellFoundedGT α

class IsWellOrder (α : Type u) (r : α → α → Prop) extends IsTrichotomous , IsTrans , IsWellFounded : Prop

A well order is a well-founded linear order.

instance instIsStrictTotalOrder {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] : IsStrictTotalOrder α r

Equations

• (_ : IsStrictTotalOrder α r) = (_ : IsStrictTotalOrder α r)

instance instIsTrichotomous {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] : IsTrichotomous α r

Equations

• (_ : IsTrichotomous α r) = (_ : IsTrichotomous α r)

instance instIsTrans_1 {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] : IsTrans α r

Equations

• (_ : IsTrans α r) = (_ : IsTrans α r)

instance instIsIrrefl_1 {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] : IsIrrefl α r

Equations

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

instance instIsAsymm_1 {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] : IsAsymm α r

Equations

• (_ : IsAsymm α r) = (_ : IsAsymm α r)

theorem WellFoundedLT.induction {α : Type u} [LT α] [WellFoundedLT α] {C : α → Prop} (a : α) : (∀ (x : α), (∀ (y : α), y < x → C y) → C x) → C a

Inducts on a well-founded < relation.

theorem WellFoundedLT.apply {α : Type u} [LT α] [WellFoundedLT α] (a : α) : Acc (fun (x x_1 : α) => x < x_1) a

All values are accessible under the well-founded <.

def WellFoundedLT.fix {α : Type u} [LT α] [WellFoundedLT α] {C : α → Sort u_1} : ((x : α) → ((y : α) → y < x → C y) → C x) → (x : α) → C x

Creates data, given a way to generate a value from all that compare as lesser. See also WellFoundedLT.fix_eq.

Equations

• WellFoundedLT.fix = IsWellFounded.fix fun (x x_1 : α) => x < x_1

theorem WellFoundedLT.fix_eq {α : Type u} [LT α] [WellFoundedLT α] {C : α → Sort u_1} (F : (x : α) → ((y : α) → y < x → C y) → C x) (x : α) : WellFoundedLT.fix F x = F x fun (y : α) (x : y < x) => WellFoundedLT.fix F y

The value from WellFoundedLT.fix is built from the previous ones as specified.

def WellFoundedLT.toWellFoundedRelation {α : Type u} [LT α] [WellFoundedLT α] : WellFoundedRelation α

Derive a WellFoundedRelation instance from a WellFoundedLT instance.

Equations

• WellFoundedLT.toWellFoundedRelation = IsWellFounded.toWellFoundedRelation fun (x x_1 : α) => x < x_1

theorem WellFoundedGT.induction {α : Type u} [LT α] [WellFoundedGT α] {C : α → Prop} (a : α) : (∀ (x : α), (∀ (y : α), x < y → C y) → C x) → C a

Inducts on a well-founded > relation.

theorem WellFoundedGT.apply {α : Type u} [LT α] [WellFoundedGT α] (a : α) : Acc (fun (x x_1 : α) => x > x_1) a

All values are accessible under the well-founded >.

def WellFoundedGT.fix {α : Type u} [LT α] [WellFoundedGT α] {C : α → Sort u_1} : ((x : α) → ((y : α) → x < y → C y) → C x) → (x : α) → C x

Creates data, given a way to generate a value from all that compare as greater. See also WellFoundedGT.fix_eq.

Equations

• WellFoundedGT.fix = IsWellFounded.fix fun (x x_1 : α) => x > x_1

theorem WellFoundedGT.fix_eq {α : Type u} [LT α] [WellFoundedGT α] {C : α → Sort u_1} (F : (x : α) → ((y : α) → x < y → C y) → C x) (x : α) : WellFoundedGT.fix F x = F x fun (y : α) (x : x < y) => WellFoundedGT.fix F y

The value from WellFoundedGT.fix is built from the successive ones as specified.

def WellFoundedGT.toWellFoundedRelation {α : Type u} [LT α] [WellFoundedGT α] : WellFoundedRelation α

Derive a WellFoundedRelation instance from a WellFoundedGT instance.

Equations

• WellFoundedGT.toWellFoundedRelation = IsWellFounded.toWellFoundedRelation fun (x x_1 : α) => x > x_1

noncomputable def IsWellOrder.linearOrder {α : Type u} (r : α → α → Prop) [IsWellOrder α r] : LinearOrder α

Construct a decidable linear order from a well-founded linear order.

Equations

• IsWellOrder.linearOrder r = linearOrderOfSTO r

def IsWellOrder.toHasWellFounded {α : Type u} [LT α] [hwo : IsWellOrder α fun (x x_1 : α) => x < x_1] : WellFoundedRelation α

Derive a WellFoundedRelation instance from an IsWellOrder instance.

Equations

• IsWellOrder.toHasWellFounded = { rel := fun (x x_1 : α) => x < x_1, wf := (_ : WellFounded fun (x x_1 : α) => x < x_1) }

theorem Subsingleton.isWellOrder {α : Type u} [Subsingleton α] (r : α → α → Prop) [hr : IsIrrefl α r] : IsWellOrder α r

instance instIsWellOrderEmptyRelation {α : Type u} [Subsingleton α] : IsWellOrder α EmptyRelation

Equations

• (_ : IsWellOrder α EmptyRelation) = (_ : IsWellOrder α EmptyRelation)

instance instIsWellOrder {α : Type u} [IsEmpty α] (r : α → α → Prop) : IsWellOrder α r

Equations

• (_ : IsWellOrder α r) = (_ : IsWellOrder α r)

instance instIsWellFoundedProdLex {α : Type u} {β : Type v} {r : α → α → Prop} {s : β → β → Prop} [IsWellFounded α r] [IsWellFounded β s] : IsWellFounded (α × β) (Prod.Lex r s)

Equations

• (_ : IsWellFounded (α × β) (Prod.Lex r s)) = (_ : IsWellFounded (α × β) (Prod.Lex r s))

instance instIsWellOrderProdLex {α : Type u} {β : Type v} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r] [IsWellOrder β s] : IsWellOrder (α × β) (Prod.Lex r s)

Equations

• (_ : IsWellOrder (α × β) (Prod.Lex r s)) = (_ : IsWellOrder (α × β) (Prod.Lex r s))

instance instIsWellFoundedInvImage {α : Type u} {β : Type v} (r : α → α → Prop) [IsWellFounded α r] (f : β → α) : IsWellFounded β (InvImage r f)

Equations

• (_ : IsWellFounded β (InvImage r f)) = (_ : IsWellFounded β (InvImage r f))

instance instIsWellFoundedInvImageNatLt {α : Type u} (f : α → ℕ) : IsWellFounded α (InvImage (fun (x x_1 : ℕ) => x < x_1) f)

Equations

• (_ : IsWellFounded α (InvImage (fun (x x_1 : ℕ) => x < x_1) f)) = (_ : IsWellFounded α (InvImage (fun (x x_1 : ℕ) => x < x_1) f))

theorem Subrelation.isWellFounded {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {s : α → α → Prop} (h : Subrelation s r) : IsWellFounded α s

instance Prod.wellFoundedLT {α : Type u} {β : Type v} [PartialOrder α] [WellFoundedLT α] [Preorder β] [WellFoundedLT β] : WellFoundedLT (α × β)

Equations

• (_ : WellFoundedLT (α × β)) = (_ : IsWellFounded (α × β) fun (x x_1 : α × β) => x < x_1)

instance Prod.wellFoundedGT {α : Type u} {β : Type v} [PartialOrder α] [WellFoundedGT α] [Preorder β] [WellFoundedGT β] : WellFoundedGT (α × β)

Equations

• (_ : WellFoundedGT (α × β)) = (_ : WellFoundedLT (αᵒᵈ × βᵒᵈ))

def Set.Unbounded {α : Type u} (r : α → α → Prop) (s : Set α) : Prop

An unbounded or cofinal set.

Equations

• Set.Unbounded r s = ∀ (a : α), ∃ (b : α), b ∈ s ∧ ¬r b a

def Set.Bounded {α : Type u} (r : α → α → Prop) (s : Set α) : Prop

A bounded or final set. Not to be confused with Bornology.IsBounded.

Equations

• Set.Bounded r s = ∃ (a : α), ∀ (b : α), b ∈ s → r b a

@[simp]

theorem Set.not_bounded_iff {α : Type u} {r : α → α → Prop} (s : Set α) : ¬Set.Bounded r s ↔ Set.Unbounded r s

@[simp]

theorem Set.not_unbounded_iff {α : Type u} {r : α → α → Prop} (s : Set α) : ¬Set.Unbounded r s ↔ Set.Bounded r s

theorem Set.unbounded_of_isEmpty {α : Type u} [IsEmpty α] {r : α → α → Prop} (s : Set α) : Set.Unbounded r s

instance Prod.isRefl_preimage_fst {α : Type u} {r : α → α → Prop} [IsRefl α r] : IsRefl (α × α) (Prod.fst ⁻¹'o r)

Equations

• (_ : IsRefl (α × α) (Prod.fst ⁻¹'o r)) = (_ : IsRefl (α × α) (Prod.fst ⁻¹'o r))

instance Prod.isRefl_preimage_snd {α : Type u} {r : α → α → Prop} [IsRefl α r] : IsRefl (α × α) (Prod.snd ⁻¹'o r)

Equations

• (_ : IsRefl (α × α) (Prod.snd ⁻¹'o r)) = (_ : IsRefl (α × α) (Prod.snd ⁻¹'o r))

instance Prod.isTrans_preimage_fst {α : Type u} {r : α → α → Prop} [IsTrans α r] : IsTrans (α × α) (Prod.fst ⁻¹'o r)

Equations

• (_ : IsTrans (α × α) (Prod.fst ⁻¹'o r)) = (_ : IsTrans (α × α) (Prod.fst ⁻¹'o r))

instance Prod.isTrans_preimage_snd {α : Type u} {r : α → α → Prop} [IsTrans α r] : IsTrans (α × α) (Prod.snd ⁻¹'o r)

Equations

• (_ : IsTrans (α × α) (Prod.snd ⁻¹'o r)) = (_ : IsTrans (α × α) (Prod.snd ⁻¹'o r))

Strict-non strict relations

class IsNonstrictStrictOrder (α : Type u_1) (r : semiOutParam (α → α → Prop)) (s : α → α → Prop) : Prop

An unbundled relation class stating that r is the nonstrict relation corresponding to the strict relation s. Compare Preorder.lt_iff_le_not_le. This is mostly meant to provide dot notation on (⊆) and (⊂).

• right_iff_left_not_left : ∀ (a b : α), s✝ a b ↔ r a b ∧ ¬r b a

  The relation r is the nonstrict relation corresponding to the strict relation s.

theorem right_iff_left_not_left {α : Type u} {r : α → α → Prop} {s : α → α → Prop} [IsNonstrictStrictOrder α r s] {a : α} {b : α} : s a b ↔ r a b ∧ ¬r b a

theorem right_iff_left_not_left_of {α : Type u} (r : α → α → Prop) (s : α → α → Prop) [IsNonstrictStrictOrder α r s] {a : α} {b : α} : s a b ↔ r a b ∧ ¬r b a

A version of right_iff_left_not_left with explicit r and s.

instance instIsIrrefl_2 {α : Type u} {r : α → α → Prop} {s : α → α → Prop} [IsNonstrictStrictOrder α r s] : IsIrrefl α s

Equations

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

⊆ and ⊂

theorem subset_of_eq_of_subset {α : Type u} [HasSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b ⊆ c) : a ⊆ c

theorem subset_of_subset_of_eq {α : Type u} [HasSubset α] {a : α} {b : α} {c : α} (hab : a ⊆ b) (hbc : b = c) : a ⊆ c

theorem subset_refl {α : Type u} [HasSubset α] [IsRefl α fun (x x_1 : α) => x ⊆ x_1] (a : α) : a ⊆ a

theorem subset_rfl {α : Type u} [HasSubset α] {a : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ a

theorem subset_of_eq {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : a = b → a ⊆ b

theorem superset_of_eq {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : a = b → b ⊆ a

theorem ne_of_not_subset {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : ¬a ⊆ b → a ≠ b

theorem ne_of_not_superset {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : ¬a ⊆ b → b ≠ a

theorem subset_trans {α : Type u} [HasSubset α] [IsTrans α fun (x x_1 : α) => x ⊆ x_1] {a : α} {b : α} {c : α} : a ⊆ b → b ⊆ c → a ⊆ c

theorem subset_antisymm {α : Type u} [HasSubset α] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ b → b ⊆ a → a = b

theorem superset_antisymm {α : Type u} [HasSubset α] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ b → b ⊆ a → b = a

theorem Eq.trans_subset {α : Type u} [HasSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b ⊆ c) : a ⊆ c

Alias of subset_of_eq_of_subset.

theorem HasSubset.subset.trans_eq {α : Type u} [HasSubset α] {a : α} {b : α} {c : α} (hab : a ⊆ b) (hbc : b = c) : a ⊆ c

Alias of subset_of_subset_of_eq.

theorem Eq.subset' {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : a = b → a ⊆ b

Alias of subset_of_eq.

theorem Eq.superset {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] : a = b → b ⊆ a

Alias of superset_of_eq.

theorem HasSubset.Subset.trans {α : Type u} [HasSubset α] [IsTrans α fun (x x_1 : α) => x ⊆ x_1] {a : α} {b : α} {c : α} : a ⊆ b → b ⊆ c → a ⊆ c

Alias of subset_trans.

theorem HasSubset.Subset.antisymm {α : Type u} [HasSubset α] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ b → b ⊆ a → a = b

Alias of subset_antisymm.

theorem HasSubset.Subset.antisymm' {α : Type u} [HasSubset α] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ b → b ⊆ a → b = a

Alias of superset_antisymm.

theorem subset_antisymm_iff {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a = b ↔ a ⊆ b ∧ b ⊆ a

theorem superset_antisymm_iff {α : Type u} [HasSubset α] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a = b ↔ b ⊆ a ∧ a ⊆ b

theorem ssubset_of_eq_of_ssubset {α : Type u} [HasSSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b ⊂ c) : a ⊂ c

theorem ssubset_of_ssubset_of_eq {α : Type u} [HasSSubset α] {a : α} {b : α} {c : α} (hab : a ⊂ b) (hbc : b = c) : a ⊂ c

theorem ssubset_irrefl {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] (a : α) : ¬a ⊂ a

theorem ssubset_irrfl {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} : ¬a ⊂ a

theorem ne_of_ssubset {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → a ≠ b

theorem ne_of_ssuperset {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → b ≠ a

theorem ssubset_trans {α : Type u} [HasSSubset α] [IsTrans α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} : a ⊂ b → b ⊂ c → a ⊂ c

theorem ssubset_asymm {α : Type u} [HasSSubset α] [IsAsymm α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → ¬b ⊂ a

theorem Eq.trans_ssubset {α : Type u} [HasSSubset α] {a : α} {b : α} {c : α} (hab : a = b) (hbc : b ⊂ c) : a ⊂ c

Alias of ssubset_of_eq_of_ssubset.

theorem HasSSubset.SSubset.trans_eq {α : Type u} [HasSSubset α] {a : α} {b : α} {c : α} (hab : a ⊂ b) (hbc : b = c) : a ⊂ c

Alias of ssubset_of_ssubset_of_eq.

theorem HasSSubset.SSubset.false {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} : ¬a ⊂ a

Alias of ssubset_irrfl.

theorem HasSSubset.SSubset.ne {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → a ≠ b

Alias of ne_of_ssubset.

theorem HasSSubset.SSubset.ne' {α : Type u} [HasSSubset α] [IsIrrefl α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → b ≠ a

Alias of ne_of_ssuperset.

theorem HasSSubset.SSubset.trans {α : Type u} [HasSSubset α] [IsTrans α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} : a ⊂ b → b ⊂ c → a ⊂ c

Alias of ssubset_trans.

theorem HasSSubset.SSubset.asymm {α : Type u} [HasSSubset α] [IsAsymm α fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b → ¬b ⊂ a

Alias of ssubset_asymm.

theorem ssubset_iff_subset_not_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} : a ⊂ b ↔ a ⊆ b ∧ ¬b ⊆ a

theorem subset_of_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊂ b) : a ⊆ b

theorem not_subset_of_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊂ b) : ¬b ⊆ a

theorem not_ssubset_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊆ b) : ¬b ⊂ a

theorem ssubset_of_subset_not_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h₁ : a ⊆ b) (h₂ : ¬b ⊆ a) : a ⊂ b

theorem HasSSubset.SSubset.subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊂ b) : a ⊆ b

Alias of subset_of_ssubset.

theorem HasSSubset.SSubset.not_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊂ b) : ¬b ⊆ a

Alias of not_subset_of_ssubset.

theorem HasSubset.Subset.not_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h : a ⊆ b) : ¬b ⊂ a

Alias of not_ssubset_of_subset.

theorem HasSubset.Subset.ssubset_of_not_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} (h₁ : a ⊆ b) (h₂ : ¬b ⊆ a) : a ⊂ b

Alias of ssubset_of_subset_not_subset.

theorem ssubset_of_subset_of_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} [IsTrans α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : b ⊂ c) : a ⊂ c

theorem ssubset_of_ssubset_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} [IsTrans α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊂ b) (h₂ : b ⊆ c) : a ⊂ c

theorem ssubset_of_subset_of_ne {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : a ≠ b) : a ⊂ b

theorem ssubset_of_ne_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ≠ b) (h₂ : a ⊆ b) : a ⊂ b

theorem eq_or_ssubset_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h : a ⊆ b) : a = b ∨ a ⊂ b

theorem ssubset_or_eq_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h : a ⊆ b) : a ⊂ b ∨ a = b

theorem HasSubset.Subset.trans_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} [IsTrans α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : b ⊂ c) : a ⊂ c

Alias of ssubset_of_subset_of_ssubset.

theorem HasSSubset.SSubset.trans_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} {c : α} [IsTrans α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊂ b) (h₂ : b ⊆ c) : a ⊂ c

Alias of ssubset_of_ssubset_of_subset.

theorem HasSubset.Subset.ssubset_of_ne {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : a ≠ b) : a ⊂ b

Alias of ssubset_of_subset_of_ne.

theorem Ne.ssubset_of_subset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h₁ : a ≠ b) (h₂ : a ⊆ b) : a ⊂ b

Alias of ssubset_of_ne_of_subset.

theorem HasSubset.Subset.eq_or_ssubset {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h : a ⊆ b) : a = b ∨ a ⊂ b

Alias of eq_or_ssubset_of_subset.

theorem HasSubset.Subset.ssubset_or_eq {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] (h : a ⊆ b) : a ⊂ b ∨ a = b

Alias of ssubset_or_eq_of_subset.

theorem ssubset_iff_subset_ne {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊂ b ↔ a ⊆ b ∧ a ≠ b

theorem subset_iff_ssubset_or_eq {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun (x x_1 : α) => x ⊆ x_1) fun (x x_1 : α) => x ⊂ x_1] {a : α} {b : α} [IsRefl α fun (x x_1 : α) => x ⊆ x_1] [IsAntisymm α fun (x x_1 : α) => x ⊆ x_1] : a ⊆ b ↔ a ⊂ b ∨ a = b

Conversion of bundled order typeclasses to unbundled relation typeclasses

instance instIsReflLeToLE {α : Type u} [Preorder α] : IsRefl α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsRefl α fun (x x_1 : α) => x ≤ x_1) = (_ : IsRefl α fun (x x_1 : α) => x ≤ x_1)

instance instIsReflGeToLE {α : Type u} [Preorder α] : IsRefl α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsRefl α fun (x x_1 : α) => x ≥ x_1) = (_ : IsRefl α (Function.swap fun (x x_1 : α) => x ≤ x_1))

instance instIsTransLeToLE {α : Type u} [Preorder α] : IsTrans α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsTrans α fun (x x_1 : α) => x ≤ x_1) = (_ : IsTrans α fun (x x_1 : α) => x ≤ x_1)

instance instIsTransGeToLE {α : Type u} [Preorder α] : IsTrans α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsTrans α fun (x x_1 : α) => x ≥ x_1) = (_ : IsTrans α (Function.swap fun (x x_1 : α) => x ≤ x_1))

instance instIsPreorderLeToLE {α : Type u} [Preorder α] : IsPreorder α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsPreorder α fun (x x_1 : α) => x ≤ x_1) = (_ : IsPreorder α fun (x x_1 : α) => x ≤ x_1)

instance instIsPreorderGeToLE {α : Type u} [Preorder α] : IsPreorder α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsPreorder α fun (x x_1 : α) => x ≥ x_1) = (_ : IsPreorder α fun (x x_1 : α) => x ≥ x_1)

instance instIsIrreflLtToLT {α : Type u} [Preorder α] : IsIrrefl α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsIrrefl α fun (x x_1 : α) => x < x_1) = (_ : IsIrrefl α fun (x x_1 : α) => x < x_1)

instance instIsIrreflGtToLT {α : Type u} [Preorder α] : IsIrrefl α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsIrrefl α fun (x x_1 : α) => x > x_1) = (_ : IsIrrefl α (Function.swap fun (x x_1 : α) => x < x_1))

instance instIsTransLtToLT {α : Type u} [Preorder α] : IsTrans α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsTrans α fun (x x_1 : α) => x < x_1) = (_ : IsTrans α fun (x x_1 : α) => x < x_1)

instance instIsTransGtToLT {α : Type u} [Preorder α] : IsTrans α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsTrans α fun (x x_1 : α) => x > x_1) = (_ : IsTrans α (Function.swap fun (x x_1 : α) => x < x_1))

instance instIsAsymmLtToLT {α : Type u} [Preorder α] : IsAsymm α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsAsymm α fun (x x_1 : α) => x < x_1) = (_ : IsAsymm α fun (x x_1 : α) => x < x_1)

instance instIsAsymmGtToLT {α : Type u} [Preorder α] : IsAsymm α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsAsymm α fun (x x_1 : α) => x > x_1) = (_ : IsAsymm α (Function.swap fun (x x_1 : α) => x < x_1))

instance instIsAntisymmLtToLT {α : Type u} [Preorder α] : IsAntisymm α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsAntisymm α fun (x x_1 : α) => x < x_1) = (_ : IsAntisymm α fun (x x_1 : α) => x < x_1)

instance instIsAntisymmGtToLT {α : Type u} [Preorder α] : IsAntisymm α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsAntisymm α fun (x x_1 : α) => x > x_1) = (_ : IsAntisymm α fun (x x_1 : α) => x > x_1)

instance instIsStrictOrderLtToLT {α : Type u} [Preorder α] : IsStrictOrder α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsStrictOrder α fun (x x_1 : α) => x < x_1) = (_ : IsStrictOrder α fun (x x_1 : α) => x < x_1)

instance instIsStrictOrderGtToLT {α : Type u} [Preorder α] : IsStrictOrder α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsStrictOrder α fun (x x_1 : α) => x > x_1) = (_ : IsStrictOrder α fun (x x_1 : α) => x > x_1)

instance instIsNonstrictStrictOrderLeToLELtToLT {α : Type u} [Preorder α] : IsNonstrictStrictOrder α (fun (x x_1 : α) => x ≤ x_1) fun (x x_1 : α) => x < x_1

Equations

• (_ : IsNonstrictStrictOrder α (fun (x x_1 : α) => x ≤ x_1) fun (x x_1 : α) => x < x_1) = (_ : IsNonstrictStrictOrder α (fun (x x_1 : α) => x ≤ x_1) fun (x x_1 : α) => x < x_1)

instance instIsAntisymmLeToLEToPreorder {α : Type u} [PartialOrder α] : IsAntisymm α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsAntisymm α fun (x x_1 : α) => x ≤ x_1) = (_ : IsAntisymm α fun (x x_1 : α) => x ≤ x_1)

instance instIsAntisymmGeToLEToPreorder {α : Type u} [PartialOrder α] : IsAntisymm α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsAntisymm α fun (x x_1 : α) => x ≥ x_1) = (_ : IsAntisymm α (Function.swap fun (x x_1 : α) => x ≤ x_1))

instance instIsPartialOrderLeToLEToPreorder {α : Type u} [PartialOrder α] : IsPartialOrder α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsPartialOrder α fun (x x_1 : α) => x ≤ x_1) = (_ : IsPartialOrder α fun (x x_1 : α) => x ≤ x_1)

instance instIsPartialOrderGeToLEToPreorder {α : Type u} [PartialOrder α] : IsPartialOrder α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsPartialOrder α fun (x x_1 : α) => x ≥ x_1) = (_ : IsPartialOrder α fun (x x_1 : α) => x ≥ x_1)

instance LE.isTotal {α : Type u} [LinearOrder α] : IsTotal α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsTotal α fun (x x_1 : α) => x ≤ x_1) = (_ : IsTotal α fun (x x_1 : α) => x ≤ x_1)

instance instIsTotalGeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTotal α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsTotal α fun (x x_1 : α) => x ≥ x_1) = (_ : IsTotal α (Function.swap fun (x x_1 : α) => x ≤ x_1))

instance instIsTotalPreorderLeToLEToPreorderToPartialOrder_1 {α : Type u} [LinearOrder α] : IsTotalPreorder α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsTotalPreorder α fun (x x_1 : α) => x ≤ x_1) = (_ : IsTotalPreorder α fun (x x_1 : α) => x ≤ x_1)

instance instIsTotalPreorderGeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTotalPreorder α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsTotalPreorder α fun (x x_1 : α) => x ≥ x_1) = (_ : IsTotalPreorder α fun (x x_1 : α) => x ≥ x_1)

instance instIsLinearOrderLeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsLinearOrder α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsLinearOrder α fun (x x_1 : α) => x ≤ x_1) = (_ : IsLinearOrder α fun (x x_1 : α) => x ≤ x_1)

instance instIsLinearOrderGeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsLinearOrder α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsLinearOrder α fun (x x_1 : α) => x ≥ x_1) = (_ : IsLinearOrder α fun (x x_1 : α) => x ≥ x_1)

instance instIsTrichotomousLtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTrichotomous α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsTrichotomous α fun (x x_1 : α) => x < x_1) = (_ : IsTrichotomous α fun (x x_1 : α) => x < x_1)

instance instIsTrichotomousGtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTrichotomous α fun (x x_1 : α) => x > x_1

Equations

• (_ : IsTrichotomous α fun (x x_1 : α) => x > x_1) = (_ : IsTrichotomous α (Function.swap fun (x x_1 : α) => x < x_1))

instance instIsTrichotomousLeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTrichotomous α fun (x x_1 : α) => x ≤ x_1

Equations

• (_ : IsTrichotomous α fun (x x_1 : α) => x ≤ x_1) = (_ : IsTrichotomous α fun (x x_1 : α) => x ≤ x_1)

instance instIsTrichotomousGeToLEToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsTrichotomous α fun (x x_1 : α) => x ≥ x_1

Equations

• (_ : IsTrichotomous α fun (x x_1 : α) => x ≥ x_1) = (_ : IsTrichotomous α fun (x x_1 : α) => x ≥ x_1)

instance instIsStrictTotalOrderLtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsStrictTotalOrder α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsStrictTotalOrder α fun (x x_1 : α) => x < x_1) = (_ : IsStrictTotalOrder α fun (x x_1 : α) => x < x_1)

instance instIsOrderConnectedLtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsOrderConnected α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsOrderConnected α fun (x x_1 : α) => x < x_1) = (_ : IsOrderConnected α fun (x x_1 : α) => x < x_1)

instance instIsIncompTransLtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsIncompTrans α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsIncompTrans α fun (x x_1 : α) => x < x_1) = (_ : IsIncompTrans α fun (x x_1 : α) => x < x_1)

instance instIsStrictWeakOrderLtToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] : IsStrictWeakOrder α fun (x x_1 : α) => x < x_1

Equations

• (_ : IsStrictWeakOrder α fun (x x_1 : α) => x < x_1) = (_ : IsStrictWeakOrder α fun (x x_1 : α) => x < x_1)

theorem transitive_le {α : Type u} [Preorder α] : Transitive LE.le

theorem transitive_lt {α : Type u} [Preorder α] : Transitive LT.lt

theorem transitive_ge {α : Type u} [Preorder α] : Transitive GE.ge

theorem transitive_gt {α : Type u} [Preorder α] : Transitive GT.gt

instance OrderDual.isTotal_le {α : Type u} [LE α] [h : IsTotal α fun (x x_1 : α) => x ≤ x_1] : IsTotal αᵒᵈ fun (x x_1 : αᵒᵈ) => x ≤ x_1

Equations

• (_ : IsTotal αᵒᵈ fun (x x_1 : αᵒᵈ) => x ≤ x_1) = (_ : IsTotal α (Function.swap fun (x x_1 : αᵒᵈ) => x_1 ≤ x))

instance instWellFoundedLTNatInstLTNat : WellFoundedLT ℕ

Equations

• instWellFoundedLTNatInstLTNat = (_ : IsWellFounded ℕ fun (x x_1 : ℕ) => x < x_1)

instance Nat.lt.isWellOrder : IsWellOrder ℕ fun (x x_1 : ℕ) => x < x_1

Equations

• Nat.lt.isWellOrder = (_ : IsWellOrder ℕ fun (x x_1 : ℕ) => x < x_1)

instance instIsWellOrderOrderDualGtInstLTOrderDualToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] [h : IsWellOrder α fun (x x_1 : α) => x < x_1] : IsWellOrder αᵒᵈ fun (x x_1 : αᵒᵈ) => x > x_1

Equations

• (_ : IsWellOrder αᵒᵈ fun (x x_1 : αᵒᵈ) => x > x_1) = h

instance instIsWellOrderOrderDualLtInstLTOrderDualToLTToPreorderToPartialOrder {α : Type u} [LinearOrder α] [h : IsWellOrder α fun (x x_1 : α) => x > x_1] : IsWellOrder αᵒᵈ fun (x x_1 : αᵒᵈ) => x < x_1

Equations

• (_ : IsWellOrder αᵒᵈ fun (x x_1 : αᵒᵈ) => x < x_1) = h