fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
unbounded_lt_of_unbounded_le [Preorder α] (h : Unbounded (· ≤ ·) s) : Unbounded (· < ·) s :=
h.rel_mono fun _ _ => le_of_lt | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_of_unbounded_le | null |
bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
Bounded (· ≤ ·) s ↔ Bounded (· < ·) s := by
refine ⟨fun h => ?_, bounded_le_of_bounded_lt⟩
obtain ⟨a, ha⟩ := h
obtain ⟨b, hb⟩ := exists_gt a
exact ⟨b, fun c hc => lt_of_le_of_lt (ha c hc) hb⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_iff_bounded_lt | null |
unbounded_lt_iff_unbounded_le [Preorder α] [NoMaxOrder α] :
Unbounded (· < ·) s ↔ Unbounded (· ≤ ·) s := by
simp_rw [← not_bounded_iff, bounded_le_iff_bounded_lt]
/-! #### Greater and greater or equal -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_iff_unbounded_le | null |
bounded_ge_of_bounded_gt [Preorder α] (h : Bounded (· > ·) s) : Bounded (· ≥ ·) s :=
let ⟨a, ha⟩ := h
⟨a, fun b hb => le_of_lt (ha b hb)⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_of_bounded_gt | null |
unbounded_gt_of_unbounded_ge [Preorder α] (h : Unbounded (· ≥ ·) s) : Unbounded (· > ·) s :=
fun a =>
let ⟨b, hb, hba⟩ := h a
⟨b, hb, fun hba' => hba (le_of_lt hba')⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_gt_of_unbounded_ge | null |
bounded_ge_iff_bounded_gt [Preorder α] [NoMinOrder α] :
Bounded (· ≥ ·) s ↔ Bounded (· > ·) s :=
@bounded_le_iff_bounded_lt αᵒᵈ _ _ _ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_iff_bounded_gt | null |
unbounded_gt_iff_unbounded_ge [Preorder α] [NoMinOrder α] :
Unbounded (· > ·) s ↔ Unbounded (· ≥ ·) s :=
@unbounded_lt_iff_unbounded_le αᵒᵈ _ _ _
/-! ### The universal set -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_gt_iff_unbounded_ge | null |
unbounded_le_univ [LE α] [NoTopOrder α] : Unbounded (· ≤ ·) (@Set.univ α) := fun a =>
let ⟨b, hb⟩ := exists_not_le a
⟨b, ⟨⟩, hb⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_univ | null |
unbounded_lt_univ [Preorder α] [NoTopOrder α] : Unbounded (· < ·) (@Set.univ α) :=
unbounded_lt_of_unbounded_le unbounded_le_univ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_univ | null |
unbounded_ge_univ [LE α] [NoBotOrder α] : Unbounded (· ≥ ·) (@Set.univ α) := fun a =>
let ⟨b, hb⟩ := exists_not_ge a
⟨b, ⟨⟩, hb⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_ge_univ | null |
unbounded_gt_univ [Preorder α] [NoBotOrder α] : Unbounded (· > ·) (@Set.univ α) :=
unbounded_gt_of_unbounded_ge unbounded_ge_univ
/-! ### Bounded and unbounded intervals -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_gt_univ | null |
bounded_self (a : α) : Bounded r { b | r b a } :=
⟨a, fun _ => id⟩
/-! #### Half-open bounded intervals -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_self | null |
bounded_lt_Iio [Preorder α] (a : α) : Bounded (· < ·) (Iio a) :=
bounded_self a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Iio | null |
bounded_le_Iio [Preorder α] (a : α) : Bounded (· ≤ ·) (Iio a) :=
bounded_le_of_bounded_lt (bounded_lt_Iio a) | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Iio | null |
bounded_le_Iic [Preorder α] (a : α) : Bounded (· ≤ ·) (Iic a) :=
bounded_self a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Iic | null |
bounded_lt_Iic [Preorder α] [NoMaxOrder α] (a : α) : Bounded (· < ·) (Iic a) := by
simp only [← bounded_le_iff_bounded_lt, bounded_le_Iic] | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Iic | null |
bounded_gt_Ioi [Preorder α] (a : α) : Bounded (· > ·) (Ioi a) :=
bounded_self a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Ioi | null |
bounded_ge_Ioi [Preorder α] (a : α) : Bounded (· ≥ ·) (Ioi a) :=
bounded_ge_of_bounded_gt (bounded_gt_Ioi a) | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Ioi | null |
bounded_ge_Ici [Preorder α] (a : α) : Bounded (· ≥ ·) (Ici a) :=
bounded_self a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Ici | null |
bounded_gt_Ici [Preorder α] [NoMinOrder α] (a : α) : Bounded (· > ·) (Ici a) := by
simp only [← bounded_ge_iff_bounded_gt, bounded_ge_Ici]
/-! #### Other bounded intervals -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Ici | null |
bounded_lt_Ioo [Preorder α] (a b : α) : Bounded (· < ·) (Ioo a b) :=
(bounded_lt_Iio b).mono Set.Ioo_subset_Iio_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Ioo | null |
bounded_lt_Ico [Preorder α] (a b : α) : Bounded (· < ·) (Ico a b) :=
(bounded_lt_Iio b).mono Set.Ico_subset_Iio_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Ico | null |
bounded_lt_Ioc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Ioc a b) :=
(bounded_lt_Iic b).mono Set.Ioc_subset_Iic_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Ioc | null |
bounded_lt_Icc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Icc a b) :=
(bounded_lt_Iic b).mono Set.Icc_subset_Iic_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_Icc | null |
bounded_le_Ioo [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioo a b) :=
(bounded_le_Iio b).mono Set.Ioo_subset_Iio_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Ioo | null |
bounded_le_Ico [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ico a b) :=
(bounded_le_Iio b).mono Set.Ico_subset_Iio_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Ico | null |
bounded_le_Ioc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioc a b) :=
(bounded_le_Iic b).mono Set.Ioc_subset_Iic_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Ioc | null |
bounded_le_Icc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Icc a b) :=
(bounded_le_Iic b).mono Set.Icc_subset_Iic_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_Icc | null |
bounded_gt_Ioo [Preorder α] (a b : α) : Bounded (· > ·) (Ioo a b) :=
(bounded_gt_Ioi a).mono Set.Ioo_subset_Ioi_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Ioo | null |
bounded_gt_Ioc [Preorder α] (a b : α) : Bounded (· > ·) (Ioc a b) :=
(bounded_gt_Ioi a).mono Set.Ioc_subset_Ioi_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Ioc | null |
bounded_gt_Ico [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Ico a b) :=
(bounded_gt_Ici a).mono Set.Ico_subset_Ici_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Ico | null |
bounded_gt_Icc [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Icc a b) :=
(bounded_gt_Ici a).mono Set.Icc_subset_Ici_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_Icc | null |
bounded_ge_Ioo [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioo a b) :=
(bounded_ge_Ioi a).mono Set.Ioo_subset_Ioi_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Ioo | null |
bounded_ge_Ioc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ioc a b) :=
(bounded_ge_Ioi a).mono Set.Ioc_subset_Ioi_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Ioc | null |
bounded_ge_Ico [Preorder α] (a b : α) : Bounded (· ≥ ·) (Ico a b) :=
(bounded_ge_Ici a).mono Set.Ico_subset_Ici_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Ico | null |
bounded_ge_Icc [Preorder α] (a b : α) : Bounded (· ≥ ·) (Icc a b) :=
(bounded_ge_Ici a).mono Set.Icc_subset_Ici_self
/-! #### Unbounded intervals -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_Icc | null |
unbounded_le_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
Unbounded (· ≤ ·) (Ioi a) := fun b =>
let ⟨c, hc⟩ := exists_gt (a ⊔ b)
⟨c, le_sup_left.trans_lt hc, (le_sup_right.trans_lt hc).not_ge⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_Ioi | null |
unbounded_le_Ici [SemilatticeSup α] [NoMaxOrder α] (a : α) :
Unbounded (· ≤ ·) (Ici a) :=
(unbounded_le_Ioi a).mono Set.Ioi_subset_Ici_self | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_Ici | null |
unbounded_lt_Ioi [SemilatticeSup α] [NoMaxOrder α] (a : α) :
Unbounded (· < ·) (Ioi a) :=
unbounded_lt_of_unbounded_le (unbounded_le_Ioi a) | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_Ioi | null |
unbounded_lt_Ici [SemilatticeSup α] (a : α) : Unbounded (· < ·) (Ici a) := fun b =>
⟨a ⊔ b, le_sup_left, le_sup_right.not_gt⟩
/-! ### Bounded initial segments -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_Ici | null |
bounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
Bounded r (s ∩ { b | ¬r b a }) ↔ Bounded r s := by
refine ⟨?_, Bounded.mono inter_subset_left⟩
rintro ⟨b, hb⟩
obtain ⟨m, hm⟩ := H a b
exact ⟨m, fun c hc => hm c (or_iff_not_imp_left.2 fun hca => hb c ⟨hc, hca⟩)⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_inter_not | null |
unbounded_inter_not (H : ∀ a b, ∃ m, ∀ c, r c a ∨ r c b → r c m) (a : α) :
Unbounded r (s ∩ { b | ¬r b a }) ↔ Unbounded r s := by
simp_rw [← not_bounded_iff, bounded_inter_not H]
/-! #### Less or equal -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_inter_not | null |
bounded_le_inter_not_le [SemilatticeSup α] (a : α) :
Bounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Bounded (· ≤ ·) s :=
bounded_inter_not (fun x y => ⟨x ⊔ y, fun _ h => h.elim le_sup_of_le_left le_sup_of_le_right⟩) a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_inter_not_le | null |
unbounded_le_inter_not_le [SemilatticeSup α] (a : α) :
Unbounded (· ≤ ·) (s ∩ { b | ¬b ≤ a }) ↔ Unbounded (· ≤ ·) s := by
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_le_inter_not_le a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_inter_not_le | null |
bounded_le_inter_lt [LinearOrder α] (a : α) :
Bounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Bounded (· ≤ ·) s := by
simp_rw [← not_le, bounded_le_inter_not_le] | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_inter_lt | null |
unbounded_le_inter_lt [LinearOrder α] (a : α) :
Unbounded (· ≤ ·) (s ∩ { b | a < b }) ↔ Unbounded (· ≤ ·) s := by
convert @unbounded_le_inter_not_le _ s _ a
exact lt_iff_not_ge | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_inter_lt | null |
bounded_le_inter_le [LinearOrder α] (a : α) :
Bounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· ≤ ·) s := by
refine ⟨?_, Bounded.mono Set.inter_subset_left⟩
rw [← @bounded_le_inter_lt _ s _ a]
exact Bounded.mono fun x ⟨hx, hx'⟩ => ⟨hx, le_of_lt hx'⟩ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_le_inter_le | null |
unbounded_le_inter_le [LinearOrder α] (a : α) :
Unbounded (· ≤ ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· ≤ ·) s := by
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_le_inter_le a
/-! #### Less than -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_le_inter_le | null |
bounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
Bounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Bounded (· < ·) s :=
bounded_inter_not (fun x y => ⟨x ⊔ y, fun _ h => h.elim lt_sup_of_lt_left lt_sup_of_lt_right⟩) a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_inter_not_lt | null |
unbounded_lt_inter_not_lt [SemilatticeSup α] (a : α) :
Unbounded (· < ·) (s ∩ { b | ¬b < a }) ↔ Unbounded (· < ·) s := by
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_lt_inter_not_lt a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_inter_not_lt | null |
bounded_lt_inter_le [LinearOrder α] (a : α) :
Bounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Bounded (· < ·) s := by
convert @bounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_inter_le | null |
unbounded_lt_inter_le [LinearOrder α] (a : α) :
Unbounded (· < ·) (s ∩ { b | a ≤ b }) ↔ Unbounded (· < ·) s := by
convert @unbounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_inter_le | null |
bounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
Bounded (· < ·) (s ∩ { b | a < b }) ↔ Bounded (· < ·) s := by
rw [← bounded_le_iff_bounded_lt, ← bounded_le_iff_bounded_lt]
exact bounded_le_inter_lt a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_lt_inter_lt | null |
unbounded_lt_inter_lt [LinearOrder α] [NoMaxOrder α] (a : α) :
Unbounded (· < ·) (s ∩ { b | a < b }) ↔ Unbounded (· < ·) s := by
rw [← not_bounded_iff, ← not_bounded_iff, not_iff_not]
exact bounded_lt_inter_lt a
/-! #### Greater or equal -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_lt_inter_lt | null |
bounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
Bounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Bounded (· ≥ ·) s :=
@bounded_le_inter_not_le αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_inter_not_ge | null |
unbounded_ge_inter_not_ge [SemilatticeInf α] (a : α) :
Unbounded (· ≥ ·) (s ∩ { b | ¬a ≤ b }) ↔ Unbounded (· ≥ ·) s :=
@unbounded_le_inter_not_le αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_ge_inter_not_ge | null |
bounded_ge_inter_gt [LinearOrder α] (a : α) :
Bounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Bounded (· ≥ ·) s :=
@bounded_le_inter_lt αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_inter_gt | null |
unbounded_ge_inter_gt [LinearOrder α] (a : α) :
Unbounded (· ≥ ·) (s ∩ { b | b < a }) ↔ Unbounded (· ≥ ·) s :=
@unbounded_le_inter_lt αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_ge_inter_gt | null |
bounded_ge_inter_ge [LinearOrder α] (a : α) :
Bounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· ≥ ·) s :=
@bounded_le_inter_le αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_ge_inter_ge | null |
unbounded_ge_iff_unbounded_inter_ge [LinearOrder α] (a : α) :
Unbounded (· ≥ ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· ≥ ·) s :=
@unbounded_le_inter_le αᵒᵈ s _ a
/-! #### Greater than -/ | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_ge_iff_unbounded_inter_ge | null |
bounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
Bounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Bounded (· > ·) s :=
@bounded_lt_inter_not_lt αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_inter_not_gt | null |
unbounded_gt_inter_not_gt [SemilatticeInf α] (a : α) :
Unbounded (· > ·) (s ∩ { b | ¬a < b }) ↔ Unbounded (· > ·) s :=
@unbounded_lt_inter_not_lt αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_gt_inter_not_gt | null |
bounded_gt_inter_ge [LinearOrder α] (a : α) :
Bounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Bounded (· > ·) s :=
@bounded_lt_inter_le αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_inter_ge | null |
unbounded_inter_ge [LinearOrder α] (a : α) :
Unbounded (· > ·) (s ∩ { b | b ≤ a }) ↔ Unbounded (· > ·) s :=
@unbounded_lt_inter_le αᵒᵈ s _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_inter_ge | null |
bounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
Bounded (· > ·) (s ∩ { b | b < a }) ↔ Bounded (· > ·) s :=
@bounded_lt_inter_lt αᵒᵈ s _ _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | bounded_gt_inter_gt | null |
unbounded_gt_inter_gt [LinearOrder α] [NoMinOrder α] (a : α) :
Unbounded (· > ·) (s ∩ { b | b < a }) ↔ Unbounded (· > ·) s :=
@unbounded_lt_inter_lt αᵒᵈ s _ _ a | theorem | Order | [
"Mathlib.Order.RelClasses",
"Mathlib.Order.Interval.Set.Basic"
] | Mathlib/Order/Bounded.lean | unbounded_gt_inter_gt | null |
Btw (α : Type*) where
/-- Betweenness for circular orders. `btw a b c` states that `b` is between `a` and `c` (in that
order). -/
btw : α → α → α → Prop
export Btw (btw) | class | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | Btw | Syntax typeclass for a betweenness relation. |
SBtw (α : Type*) where
/-- Strict betweenness for circular orders. `sbtw a b c` states that `b` is strictly between `a`
and `c` (in that order). -/
sbtw : α → α → α → Prop
export SBtw (sbtw) | class | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | SBtw | Syntax typeclass for a strict betweenness relation. |
CircularPreorder (α : Type*) extends Btw α, SBtw α where
/-- `a` is between `a` and `a`. -/
btw_refl (a : α) : btw a a a
/-- If `b` is between `a` and `c`, then `c` is between `b` and `a`.
This is motivated by imagining three points on a circle. -/
btw_cyclic_left {a b c : α} : btw a b c → btw b c a
sbtw :=... | class | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | CircularPreorder | A circular preorder is the analogue of a preorder where you can loop around. `≤` and `<` are
replaced by ternary relations `btw` and `sbtw`. `btw` is reflexive and cyclic. `sbtw` is transitive. |
CircularPartialOrder (α : Type*) extends CircularPreorder α where
/-- If `b` is between `a` and `c` and also between `c` and `a`, then at least one pair of points
among `a`, `b`, `c` are identical. -/
btw_antisymm {a b c : α} : btw a b c → btw c b a → a = b ∨ b = c ∨ c = a
export CircularPartialOrder (btw_antisym... | class | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | CircularPartialOrder | A circular partial order is the analogue of a partial order where you can loop around. `≤` and
`<` are replaced by ternary relations `btw` and `sbtw`. `btw` is reflexive, cyclic and
antisymmetric. `sbtw` is transitive. |
CircularOrder (α : Type*) extends CircularPartialOrder α where
/-- For any triple of points, the second is between the other two one way or another. -/
btw_total : ∀ a b c : α, btw a b c ∨ btw c b a
export CircularOrder (btw_total)
/-! ### Circular preorders -/ | class | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | CircularOrder | A circular order is the analogue of a linear order where you can loop around. `≤` and `<` are
replaced by ternary relations `btw` and `sbtw`. `btw` is reflexive, cyclic, antisymmetric and total.
`sbtw` is transitive. |
btw_rfl {a : α} : btw a a a :=
btw_refl _ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_rfl | null |
Btw.btw.cyclic_left {a b c : α} (h : btw a b c) : btw b c a :=
btw_cyclic_left h | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | Btw.btw.cyclic_left | null |
btw_cyclic_right {a b c : α} (h : btw a b c) : btw c a b :=
h.cyclic_left.cyclic_left
alias Btw.btw.cyclic_right := btw_cyclic_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_cyclic_right | null |
btw_cyclic {a b c : α} : btw a b c ↔ btw c a b :=
⟨btw_cyclic_right, btw_cyclic_left⟩ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_cyclic | The order of the `↔` has been chosen so that `rw [btw_cyclic]` cycles to the right while
`rw [← btw_cyclic]` cycles to the left (thus following the prepended arrow). |
sbtw_iff_btw_not_btw {a b c : α} : sbtw a b c ↔ btw a b c ∧ ¬btw c b a :=
CircularPreorder.sbtw_iff_btw_not_btw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_iff_btw_not_btw | null |
btw_of_sbtw {a b c : α} (h : sbtw a b c) : btw a b c :=
(sbtw_iff_btw_not_btw.1 h).1
alias SBtw.sbtw.btw := btw_of_sbtw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_of_sbtw | null |
not_btw_of_sbtw {a b c : α} (h : sbtw a b c) : ¬btw c b a :=
(sbtw_iff_btw_not_btw.1 h).2
alias SBtw.sbtw.not_btw := not_btw_of_sbtw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | not_btw_of_sbtw | null |
not_sbtw_of_btw {a b c : α} (h : btw a b c) : ¬sbtw c b a := fun h' => h'.not_btw h
alias Btw.btw.not_sbtw := not_sbtw_of_btw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | not_sbtw_of_btw | null |
sbtw_of_btw_not_btw {a b c : α} (habc : btw a b c) (hcba : ¬btw c b a) : sbtw a b c :=
sbtw_iff_btw_not_btw.2 ⟨habc, hcba⟩
alias Btw.btw.sbtw_of_not_btw := sbtw_of_btw_not_btw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_of_btw_not_btw | null |
sbtw_cyclic_left {a b c : α} (h : sbtw a b c) : sbtw b c a :=
h.btw.cyclic_left.sbtw_of_not_btw fun h' => h.not_btw h'.cyclic_left
alias SBtw.sbtw.cyclic_left := sbtw_cyclic_left | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_cyclic_left | null |
sbtw_cyclic_right {a b c : α} (h : sbtw a b c) : sbtw c a b :=
h.cyclic_left.cyclic_left
alias SBtw.sbtw.cyclic_right := sbtw_cyclic_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_cyclic_right | null |
sbtw_cyclic {a b c : α} : sbtw a b c ↔ sbtw c a b :=
⟨sbtw_cyclic_right, sbtw_cyclic_left⟩ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_cyclic | The order of the `↔` has been chosen so that `rw [sbtw_cyclic]` cycles to the right while
`rw [← sbtw_cyclic]` cycles to the left (thus following the prepended arrow). |
SBtw.sbtw.trans_left {a b c d : α} (h : sbtw a b c) : sbtw b d c → sbtw a d c :=
sbtw_trans_left h | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | SBtw.sbtw.trans_left | null |
sbtw_trans_right {a b c d : α} (hbc : sbtw a b c) (hcd : sbtw a c d) : sbtw a b d :=
(hbc.cyclic_left.trans_left hcd.cyclic_left).cyclic_right
alias SBtw.sbtw.trans_right := sbtw_trans_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_trans_right | null |
sbtw_asymm {a b c : α} (h : sbtw a b c) : ¬sbtw c b a :=
h.btw.not_sbtw
alias SBtw.sbtw.not_sbtw := sbtw_asymm | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_asymm | null |
sbtw_irrefl_left_right {a b : α} : ¬sbtw a b a := fun h => h.not_btw h.btw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_irrefl_left_right | null |
sbtw_irrefl_left {a b : α} : ¬sbtw a a b := fun h => sbtw_irrefl_left_right h.cyclic_left | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_irrefl_left | null |
sbtw_irrefl_right {a b : α} : ¬sbtw a b b := fun h => sbtw_irrefl_left_right h.cyclic_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_irrefl_right | null |
sbtw_irrefl (a : α) : ¬sbtw a a a :=
sbtw_irrefl_left_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_irrefl | null |
Btw.btw.antisymm {a b c : α} (h : btw a b c) : btw c b a → a = b ∨ b = c ∨ c = a :=
btw_antisymm h | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | Btw.btw.antisymm | null |
btw_refl_left_right (a b : α) : btw a b a :=
or_self_iff.1 (btw_total a b a) | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_refl_left_right | null |
btw_rfl_left_right {a b : α} : btw a b a :=
btw_refl_left_right _ _ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_rfl_left_right | null |
btw_refl_left (a b : α) : btw a a b :=
btw_rfl_left_right.cyclic_right | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_refl_left | null |
btw_rfl_left {a b : α} : btw a a b :=
btw_refl_left _ _ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_rfl_left | null |
btw_refl_right (a b : α) : btw a b b :=
btw_rfl_left_right.cyclic_left | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_refl_right | null |
btw_rfl_right {a b : α} : btw a b b :=
btw_refl_right _ _ | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_rfl_right | null |
sbtw_iff_not_btw {a b c : α} : sbtw a b c ↔ ¬btw c b a := by
rw [sbtw_iff_btw_not_btw]
exact and_iff_right_of_imp (btw_total _ _ _).resolve_left | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | sbtw_iff_not_btw | null |
btw_iff_not_sbtw {a b c : α} : btw a b c ↔ ¬sbtw c b a :=
iff_not_comm.1 sbtw_iff_not_btw | theorem | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | btw_iff_not_sbtw | null |
cIcc (a b : α) : Set α :=
{ x | btw a x b } | def | Order | [
"Mathlib.Order.Lattice",
"Mathlib.Tactic.Order"
] | Mathlib/Order/Circular.lean | cIcc | Closed-closed circular interval |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.