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 ⌀ |
|---|---|---|---|---|---|---|
Set.OrdConnected.apply_wcovBy_apply_iff (f : α ↪o β) (h : (range f).OrdConnected) :
f a ⩿ f b ↔ a ⩿ b :=
⟨fun h2 => h2.of_image f, fun hab => hab.image f h⟩
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | Set.OrdConnected.apply_wcovBy_apply_iff | null |
apply_wcovBy_apply_iff {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
e a ⩿ e b ↔ a ⩿ b :=
(ordConnected_range (e : α ≃o β)).apply_wcovBy_apply_iff ((e : α ≃o β) : α ↪o β)
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | apply_wcovBy_apply_iff | null |
toDual_wcovBy_toDual_iff : toDual b ⩿ toDual a ↔ a ⩿ b :=
and_congr_right' <| forall_congr' fun _ => forall_swap
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | toDual_wcovBy_toDual_iff | null |
ofDual_wcovBy_ofDual_iff {a b : αᵒᵈ} : ofDual a ⩿ ofDual b ↔ b ⩿ a :=
and_congr_right' <| forall_congr' fun _ => forall_swap
alias ⟨_, WCovBy.toDual⟩ := toDual_wcovBy_toDual_iff
alias ⟨_, WCovBy.ofDual⟩ := ofDual_wcovBy_ofDual_iff | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | ofDual_wcovBy_ofDual_iff | null |
OrderEmbedding.wcovBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
(f : α ↪o β) {x y : α} (h : f x ⩿ f y) : x ⩿ y := by
use f.le_iff_le.1 h.1
intro a
rw [← f.lt_iff_lt, ← f.lt_iff_lt]
apply h.2 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | OrderEmbedding.wcovBy_of_apply | null |
OrderIso.map_wcovBy {α β : Type*} [Preorder α] [Preorder β]
(f : α ≃o β) {x y : α} : f x ⩿ f y ↔ x ⩿ y := by
use f.toOrderEmbedding.wcovBy_of_apply
conv_lhs => rw [← f.symm_apply_apply x, ← f.symm_apply_apply y]
exact f.symm.toOrderEmbedding.wcovBy_of_apply | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | OrderIso.map_wcovBy | null |
WCovBy.eq_or_eq (h : a ⩿ b) (h2 : a ≤ c) (h3 : c ≤ b) : c = a ∨ c = b := by
rcases h2.eq_or_lt with (h2 | h2); · exact Or.inl h2.symm
rcases h3.eq_or_lt with (h3 | h3); · exact Or.inr h3
exact (h.2 h2 h3).elim | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.eq_or_eq | null |
wcovBy_iff_le_and_eq_or_eq : a ⩿ b ↔ a ≤ b ∧ ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b :=
⟨fun h => ⟨h.le, fun _ => h.eq_or_eq⟩, And.rec wcovBy_of_eq_or_eq⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_le_and_eq_or_eq | An `iff` version of `WCovBy.eq_or_eq` and `wcovBy_of_eq_or_eq`. |
WCovBy.le_and_le_iff (h : a ⩿ b) : a ≤ c ∧ c ≤ b ↔ c = a ∨ c = b := by
refine ⟨fun h2 => h.eq_or_eq h2.1 h2.2, ?_⟩; rintro (rfl | rfl)
exacts [⟨le_rfl, h.le⟩, ⟨h.le, le_rfl⟩] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.le_and_le_iff | null |
WCovBy.Icc_eq (h : a ⩿ b) : Icc a b = {a, b} := by
ext c
exact h.le_and_le_iff | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.Icc_eq | null |
WCovBy.Ico_subset (h : a ⩿ b) : Ico a b ⊆ {a} := by
rw [← Icc_diff_right, h.Icc_eq, diff_singleton_subset_iff, pair_comm] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.Ico_subset | null |
WCovBy.Ioc_subset (h : a ⩿ b) : Ioc a b ⊆ {b} := by
rw [← Icc_diff_left, h.Icc_eq, diff_singleton_subset_iff] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.Ioc_subset | null |
WCovBy.sup_eq (hac : a ⩿ c) (hbc : b ⩿ c) (hab : a ≠ b) : a ⊔ b = c :=
(sup_le hac.le hbc.le).eq_of_not_lt fun h =>
hab.lt_sup_or_lt_sup.elim (fun h' => hac.2 h' h) fun h' => hbc.2 h' h | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.sup_eq | null |
WCovBy.inf_eq (hca : c ⩿ a) (hcb : c ⩿ b) (hab : a ≠ b) : a ⊓ b = c :=
(le_inf hca.le hcb.le).eq_of_not_lt' fun h => hab.inf_lt_or_inf_lt.elim (hca.2 h) (hcb.2 h) | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.inf_eq | null |
CovBy.lt (h : a ⋖ b) : a < b :=
h.1 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.lt | null |
not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b := by
simp_rw [CovBy, h, true_and, not_forall, exists_prop, not_not]
alias ⟨exists_lt_lt_of_not_covBy, _⟩ := not_covBy_iff
alias LT.lt.exists_lt_lt := exists_lt_lt_of_not_covBy | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | not_covBy_iff | If `a < b`, then `b` does not cover `a` iff there's an element in between. |
not_covBy [DenselyOrdered α] : ¬a ⋖ b := fun h =>
let ⟨_, hc⟩ := exists_between h.1
h.2 hc.1 hc.2 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | not_covBy | In a dense order, nothing covers anything. |
denselyOrdered_iff_forall_not_covBy : DenselyOrdered α ↔ ∀ a b : α, ¬a ⋖ b :=
⟨fun h _ _ => @not_covBy _ _ _ _ h, fun h =>
⟨fun _ _ hab => exists_lt_lt_of_not_covBy hab <| h _ _⟩⟩
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | denselyOrdered_iff_forall_not_covBy | null |
toDual_covBy_toDual_iff : toDual b ⋖ toDual a ↔ a ⋖ b :=
and_congr_right' <| forall_congr' fun _ => forall_swap
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | toDual_covBy_toDual_iff | null |
ofDual_covBy_ofDual_iff {a b : αᵒᵈ} : ofDual a ⋖ ofDual b ↔ b ⋖ a :=
and_congr_right' <| forall_congr' fun _ => forall_swap
alias ⟨_, CovBy.toDual⟩ := toDual_covBy_toDual_iff
alias ⟨_, CovBy.ofDual⟩ := ofDual_covBy_ofDual_iff | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | ofDual_covBy_ofDual_iff | null |
CovBy.le (h : a ⋖ b) : a ≤ b :=
h.1.le | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.le | null |
protected CovBy.ne (h : a ⋖ b) : a ≠ b :=
h.lt.ne | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.ne | null |
CovBy.ne' (h : a ⋖ b) : b ≠ a :=
h.lt.ne' | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.ne' | null |
protected CovBy.wcovBy (h : a ⋖ b) : a ⩿ b :=
⟨h.le, h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.wcovBy | null |
WCovBy.covBy_of_not_le (h : a ⩿ b) (h2 : ¬b ≤ a) : a ⋖ b :=
⟨h.le.lt_of_not_ge h2, h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.covBy_of_not_le | null |
WCovBy.covBy_of_lt (h : a ⩿ b) (h2 : a < b) : a ⋖ b :=
⟨h2, h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.covBy_of_lt | null |
CovBy.of_le_of_lt (hac : a ⋖ c) (hab : a ≤ b) (hbc : b < c) : b ⋖ c :=
⟨hbc, fun _x hbx hxc ↦ hac.2 (hab.trans_lt hbx) hxc⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.of_le_of_lt | null |
CovBy.of_lt_of_le (hac : a ⋖ c) (hab : a < b) (hbc : b ≤ c) : a ⋖ b :=
⟨hab, fun _x hax hxb ↦ hac.2 hax <| hxb.trans_le hbc⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.of_lt_of_le | null |
not_covBy_of_lt_of_lt (h₁ : a < b) (h₂ : b < c) : ¬a ⋖ c :=
(not_covBy_iff (h₁.trans h₂)).2 ⟨b, h₁, h₂⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | not_covBy_of_lt_of_lt | null |
covBy_iff_wcovBy_and_lt : a ⋖ b ↔ a ⩿ b ∧ a < b :=
⟨fun h => ⟨h.wcovBy, h.lt⟩, fun h => h.1.covBy_of_lt h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_wcovBy_and_lt | null |
covBy_iff_wcovBy_and_not_le : a ⋖ b ↔ a ⩿ b ∧ ¬b ≤ a :=
⟨fun h => ⟨h.wcovBy, h.lt.not_ge⟩, fun h => h.1.covBy_of_not_le h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_wcovBy_and_not_le | null |
wcovBy_iff_covBy_or_le_and_le : a ⩿ b ↔ a ⋖ b ∨ a ≤ b ∧ b ≤ a :=
⟨fun h => or_iff_not_imp_right.mpr fun h' => h.covBy_of_not_le fun hba => h' ⟨h.le, hba⟩,
fun h' => h'.elim (fun h => h.wcovBy) fun h => h.1.wcovBy_of_le h.2⟩
alias ⟨WCovBy.covBy_or_le_and_le, _⟩ := wcovBy_iff_covBy_or_le_and_le | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_covBy_or_le_and_le | null |
AntisymmRel.trans_covBy (hab : AntisymmRel (· ≤ ·) a b) (hbc : b ⋖ c) : a ⋖ c :=
⟨hab.1.trans_lt hbc.lt, fun _ had hdc => hbc.2 (hab.2.trans_lt had) hdc⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | AntisymmRel.trans_covBy | null |
covBy_congr_left (hab : AntisymmRel (· ≤ ·) a b) : a ⋖ c ↔ b ⋖ c :=
⟨hab.symm.trans_covBy, hab.trans_covBy⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_congr_left | null |
CovBy.trans_antisymmRel (hab : a ⋖ b) (hbc : AntisymmRel (· ≤ ·) b c) : a ⋖ c :=
⟨hab.lt.trans_le hbc.1, fun _ had hdb => hab.2 had <| hdb.trans_le hbc.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.trans_antisymmRel | null |
covBy_congr_right (hab : AntisymmRel (· ≤ ·) a b) : c ⋖ a ↔ c ⋖ b :=
⟨fun h => h.trans_antisymmRel hab, fun h => h.trans_antisymmRel hab.symm⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_congr_right | null |
CovBy.isIrrefl : IsIrrefl α (· ⋖ ·) :=
⟨fun _ ha => ha.ne rfl⟩ | instance | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.isIrrefl | null |
CovBy.Ioo_eq (h : a ⋖ b) : Ioo a b = ∅ :=
h.wcovBy.Ioo_eq | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Ioo_eq | null |
covBy_iff_Ioo_eq : a ⋖ b ↔ a < b ∧ Ioo a b = ∅ :=
and_congr_right' <| by simp [eq_empty_iff_forall_notMem] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_Ioo_eq | null |
CovBy.of_image (f : α ↪o β) (h : f a ⋖ f b) : a ⋖ b :=
⟨f.lt_iff_lt.mp h.lt, fun _ hac hcb => h.2 (f.lt_iff_lt.mpr hac) (f.lt_iff_lt.mpr hcb)⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.of_image | null |
CovBy.image (f : α ↪o β) (hab : a ⋖ b) (h : (range f).OrdConnected) : f a ⋖ f b :=
(hab.wcovBy.image f h).covBy_of_lt <| f.strictMono hab.lt | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.image | null |
Set.OrdConnected.apply_covBy_apply_iff (f : α ↪o β) (h : (range f).OrdConnected) :
f a ⋖ f b ↔ a ⋖ b :=
⟨CovBy.of_image f, fun hab => hab.image f h⟩
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | Set.OrdConnected.apply_covBy_apply_iff | null |
apply_covBy_apply_iff {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
e a ⋖ e b ↔ a ⋖ b :=
(ordConnected_range (e : α ≃o β)).apply_covBy_apply_iff ((e : α ≃o β) : α ↪o β) | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | apply_covBy_apply_iff | null |
covBy_of_eq_or_eq (hab : a < b) (h : ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b) : a ⋖ b :=
⟨hab, fun c ha hb => (h c ha.le hb.le).elim ha.ne' hb.ne⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_of_eq_or_eq | null |
OrderEmbedding.covBy_of_apply {α β : Type*} [Preorder α] [Preorder β]
(f : α ↪o β) {x y : α} (h : f x ⋖ f y) : x ⋖ y := by
use f.lt_iff_lt.1 h.1
intro a
rw [← f.lt_iff_lt, ← f.lt_iff_lt]
apply h.2 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | OrderEmbedding.covBy_of_apply | null |
OrderIso.map_covBy {α β : Type*} [Preorder α] [Preorder β]
(f : α ≃o β) {x y : α} : f x ⋖ f y ↔ x ⋖ y := by
use f.toOrderEmbedding.covBy_of_apply
conv_lhs => rw [← f.symm_apply_apply x, ← f.symm_apply_apply y]
exact f.symm.toOrderEmbedding.covBy_of_apply | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | OrderIso.map_covBy | null |
WCovBy.covBy_of_ne (h : a ⩿ b) (h2 : a ≠ b) : a ⋖ b :=
⟨h.le.lt_of_ne h2, h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.covBy_of_ne | null |
covBy_iff_wcovBy_and_ne : a ⋖ b ↔ a ⩿ b ∧ a ≠ b :=
⟨fun h => ⟨h.wcovBy, h.ne⟩, fun h => h.1.covBy_of_ne h.2⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_wcovBy_and_ne | null |
wcovBy_iff_covBy_or_eq : a ⩿ b ↔ a ⋖ b ∨ a = b := by
rw [le_antisymm_iff, wcovBy_iff_covBy_or_le_and_le] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_covBy_or_eq | null |
wcovBy_iff_eq_or_covBy : a ⩿ b ↔ a = b ∨ a ⋖ b :=
wcovBy_iff_covBy_or_eq.trans or_comm
alias ⟨WCovBy.covBy_or_eq, _⟩ := wcovBy_iff_covBy_or_eq
alias ⟨WCovBy.eq_or_covBy, _⟩ := wcovBy_iff_eq_or_covBy | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_eq_or_covBy | null |
CovBy.eq_or_eq (h : a ⋖ b) (h2 : a ≤ c) (h3 : c ≤ b) : c = a ∨ c = b :=
h.wcovBy.eq_or_eq h2 h3 | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.eq_or_eq | null |
covBy_iff_lt_and_eq_or_eq : a ⋖ b ↔ a < b ∧ ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b :=
⟨fun h => ⟨h.lt, fun _ => h.eq_or_eq⟩, And.rec covBy_of_eq_or_eq⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_lt_and_eq_or_eq | An `iff` version of `CovBy.eq_or_eq` and `covBy_of_eq_or_eq`. |
CovBy.Ico_eq (h : a ⋖ b) : Ico a b = {a} := by
rw [← Ioo_union_left h.lt, h.Ioo_eq, empty_union] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Ico_eq | null |
CovBy.Ioc_eq (h : a ⋖ b) : Ioc a b = {b} := by
rw [← Ioo_union_right h.lt, h.Ioo_eq, empty_union] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Ioc_eq | null |
CovBy.Icc_eq (h : a ⋖ b) : Icc a b = {a, b} :=
h.wcovBy.Icc_eq | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Icc_eq | null |
CovBy.Ioi_eq (h : a ⋖ b) : Ioi a = Ici b := by
rw [← Ioo_union_Ici_eq_Ioi h.lt, h.Ioo_eq, empty_union] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Ioi_eq | null |
CovBy.Iio_eq (h : a ⋖ b) : Iio b = Iic a := by
rw [← Iic_union_Ioo_eq_Iio h.lt, h.Ioo_eq, union_empty] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.Iio_eq | null |
WCovBy.le_of_lt (hab : a ⩿ b) (hcb : c < b) : c ≤ a :=
not_lt.1 fun hac => hab.2 hac hcb | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.le_of_lt | null |
WCovBy.ge_of_gt (hab : a ⩿ b) (hac : a < c) : b ≤ c :=
not_lt.1 <| hab.2 hac | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | WCovBy.ge_of_gt | null |
CovBy.le_of_lt (hab : a ⋖ b) : c < b → c ≤ a :=
hab.wcovBy.le_of_lt | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.le_of_lt | null |
CovBy.ge_of_gt (hab : a ⋖ b) : a < c → b ≤ c :=
hab.wcovBy.ge_of_gt | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.ge_of_gt | null |
CovBy.unique_left (ha : a ⋖ c) (hb : b ⋖ c) : a = b :=
(hb.le_of_lt ha.lt).antisymm <| ha.le_of_lt hb.lt | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.unique_left | null |
CovBy.unique_right (hb : a ⋖ b) (hc : a ⋖ c) : b = c :=
(hb.ge_of_gt hc.lt).antisymm <| hc.ge_of_gt hb.lt | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.unique_right | null |
CovBy.eq_of_between {x : α} (hab : a ⋖ b) (hbc : b ⋖ c) (hax : a < x) (hxc : x < c) :
x = b :=
le_antisymm (le_of_not_gt fun h => hbc.2 h hxc) (le_of_not_gt <| hab.2 hax) | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | CovBy.eq_of_between | If `a`, `b`, `c` are consecutive and `a < x < c` then `x = b`. |
covBy_iff_lt_iff_le_left {x y : α} : x ⋖ y ↔ ∀ {z}, z < y ↔ z ≤ x where
mp := fun hx _z ↦ ⟨hx.le_of_lt, fun hz ↦ hz.trans_lt hx.lt⟩
mpr := fun H ↦ ⟨H.2 le_rfl, fun _z hx hz ↦ (H.1 hz).not_gt hx⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_lt_iff_le_left | null |
covBy_iff_le_iff_lt_left {x y : α} : x ⋖ y ↔ ∀ {z}, z ≤ x ↔ z < y := by
simp_rw [covBy_iff_lt_iff_le_left, iff_comm] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_le_iff_lt_left | null |
covBy_iff_lt_iff_le_right {x y : α} : x ⋖ y ↔ ∀ {z}, x < z ↔ y ≤ z := by
trans ∀ {z}, ¬ z ≤ x ↔ ¬ z < y
· simp_rw [covBy_iff_le_iff_lt_left, not_iff_not]
· simp | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_lt_iff_le_right | null |
covBy_iff_le_iff_lt_right {x y : α} : x ⋖ y ↔ ∀ {z}, y ≤ z ↔ x < z := by
simp_rw [covBy_iff_lt_iff_le_right, iff_comm]
alias ⟨CovBy.lt_iff_le_left, _⟩ := covBy_iff_lt_iff_le_left
alias ⟨CovBy.le_iff_lt_left, _⟩ := covBy_iff_le_iff_lt_left
alias ⟨CovBy.lt_iff_le_right, _⟩ := covBy_iff_lt_iff_le_right
alias ⟨CovBy.le_i... | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_le_iff_lt_right | null |
LT.lt.exists_disjoint_Iio_Ioi (h : a < b) :
∃ a' > a, ∃ b' < b, ∀ x < a', ∀ y > b', x < y := by
by_cases h' : a ⋖ b
· exact ⟨b, h, a, h, fun x hx y hy => hx.trans_le <| h'.ge_of_gt hy⟩
· rcases h.exists_lt_lt h' with ⟨c, ha, hb⟩
exact ⟨c, ha, c, hb, fun _ h₁ _ => lt_trans h₁⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | LT.lt.exists_disjoint_Iio_Ioi | If `a < b` then there exist `a' > a` and `b' < b` such that `Set.Iio a'` is strictly to the left
of `Set.Ioi b'`. |
@[simp] wcovBy_iff : ∀ {a b : Bool}, a ⩿ b ↔ a ≤ b := by unfold WCovBy; decide
@[simp] theorem covBy_iff : ∀ {a b : Bool}, a ⋖ b ↔ a < b := by unfold CovBy; decide | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff | null |
instDecidableRelWCovBy : DecidableRel (· ⩿ · : Bool → Bool → Prop) := fun _ _ ↦
decidable_of_iff _ wcovBy_iff.symm | instance | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | instDecidableRelWCovBy | null |
instDecidableRelCovBy : DecidableRel (· ⋖ · : Bool → Bool → Prop) := fun _ _ ↦
decidable_of_iff _ covBy_iff.symm | instance | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | instDecidableRelCovBy | null |
@[simp] wcovBy_insert (x : α) (s : Set α) : s ⩿ insert x s := by
refine wcovBy_of_eq_or_eq (subset_insert x s) fun t hst h2t => ?_
by_cases h : x ∈ t
· exact Or.inr (subset_antisymm h2t <| insert_subset_iff.mpr ⟨h, hst⟩)
· refine Or.inl (subset_antisymm ?_ hst)
rwa [← diff_singleton_eq_self h, diff_singleto... | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_insert | null |
_root_.CovBy.exists_set_insert (h : s ⋖ t) : ∃ a ∉ s, insert a s = t :=
let ⟨a, ha, hst⟩ := ssubset_iff_insert.1 h.lt
⟨a, ha, (hst.eq_of_not_ssuperset <| h.2 <| ssubset_insert ha).symm⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | _root_.CovBy.exists_set_insert | null |
_root_.CovBy.exists_set_sdiff_singleton (h : s ⋖ t) : ∃ a ∈ t, t \ {a} = s :=
let ⟨a, ha, hst⟩ := ssubset_iff_sdiff_singleton.1 h.lt
⟨a, ha, (hst.eq_of_not_ssubset fun h' ↦ h.2 h' <|
sdiff_lt (singleton_subset_iff.2 ha) <| singleton_ne_empty _).symm⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | _root_.CovBy.exists_set_sdiff_singleton | null |
covBy_iff_exists_insert : s ⋖ t ↔ ∃ a ∉ s, insert a s = t :=
⟨CovBy.exists_set_insert, by rintro ⟨a, ha, rfl⟩; exact covBy_insert ha⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_exists_insert | null |
covBy_iff_exists_sdiff_singleton : s ⋖ t ↔ ∃ a ∈ t, t \ {a} = s :=
⟨CovBy.exists_set_sdiff_singleton, by rintro ⟨a, ha, rfl⟩; exact sdiff_singleton_covBy ha⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_exists_sdiff_singleton | null |
wcovBy_eq_reflGen_covBy [PartialOrder α] : (· ⩿ · : α → α → Prop) = ReflGen (· ⋖ ·) := by
ext x y; simp_rw [wcovBy_iff_eq_or_covBy, @eq_comm _ x, reflGen_iff] | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_eq_reflGen_covBy | null |
transGen_wcovBy_eq_reflTransGen_covBy [PartialOrder α] :
TransGen (· ⩿ · : α → α → Prop) = ReflTransGen (· ⋖ ·) := by
rw [wcovBy_eq_reflGen_covBy, transGen_reflGen] | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | transGen_wcovBy_eq_reflTransGen_covBy | null |
reflTransGen_wcovBy_eq_reflTransGen_covBy [PartialOrder α] :
ReflTransGen (· ⩿ · : α → α → Prop) = ReflTransGen (· ⋖ ·) := by
rw [wcovBy_eq_reflGen_covBy, reflTransGen_reflGen] | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | reflTransGen_wcovBy_eq_reflTransGen_covBy | null |
@[simp]
swap_wcovBy_swap : x.swap ⩿ y.swap ↔ x ⩿ y :=
apply_wcovBy_apply_iff (OrderIso.prodComm : α × β ≃o β × α)
@[simp] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | swap_wcovBy_swap | null |
swap_covBy_swap : x.swap ⋖ y.swap ↔ x ⋖ y :=
apply_covBy_apply_iff (OrderIso.prodComm : α × β ≃o β × α) | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | swap_covBy_swap | null |
fst_eq_or_snd_eq_of_wcovBy : x ⩿ y → x.1 = y.1 ∨ x.2 = y.2 := by
refine fun h => of_not_not fun hab => ?_
push_neg at hab
exact
h.2 (mk_lt_mk.2 <| Or.inl ⟨hab.1.lt_of_le h.1.1, le_rfl⟩)
(mk_lt_mk.2 <| Or.inr ⟨le_rfl, hab.2.lt_of_le h.1.2⟩) | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | fst_eq_or_snd_eq_of_wcovBy | null |
_root_.WCovBy.fst (h : x ⩿ y) : x.1 ⩿ y.1 :=
⟨h.1.1, fun _ h₁ h₂ => h.2 (mk_lt_mk_iff_left.2 h₁) ⟨⟨h₂.le, h.1.2⟩, fun hc => h₂.not_ge hc.1⟩⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | _root_.WCovBy.fst | null |
_root_.WCovBy.snd (h : x ⩿ y) : x.2 ⩿ y.2 :=
⟨h.1.2, fun _ h₁ h₂ => h.2 (mk_lt_mk_iff_right.2 h₁) ⟨⟨h.1.1, h₂.le⟩, fun hc => h₂.not_ge hc.2⟩⟩ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | _root_.WCovBy.snd | null |
mk_wcovBy_mk_iff_left : (a₁, b) ⩿ (a₂, b) ↔ a₁ ⩿ a₂ := by
refine ⟨WCovBy.fst, (And.imp mk_le_mk_iff_left.2) fun h c h₁ h₂ => ?_⟩
have : c.2 = b := h₂.le.2.antisymm h₁.le.2
rw [← @Prod.mk.eta _ _ c, this, mk_lt_mk_iff_left] at h₁ h₂
exact h h₁ h₂ | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_wcovBy_mk_iff_left | null |
mk_wcovBy_mk_iff_right : (a, b₁) ⩿ (a, b₂) ↔ b₁ ⩿ b₂ :=
swap_wcovBy_swap.trans mk_wcovBy_mk_iff_left | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_wcovBy_mk_iff_right | null |
mk_covBy_mk_iff_left : (a₁, b) ⋖ (a₂, b) ↔ a₁ ⋖ a₂ := by
simp_rw [covBy_iff_wcovBy_and_lt, mk_wcovBy_mk_iff_left, mk_lt_mk_iff_left] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_covBy_mk_iff_left | null |
mk_covBy_mk_iff_right : (a, b₁) ⋖ (a, b₂) ↔ b₁ ⋖ b₂ := by
simp_rw [covBy_iff_wcovBy_and_lt, mk_wcovBy_mk_iff_right, mk_lt_mk_iff_right] | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_covBy_mk_iff_right | null |
mk_wcovBy_mk_iff : (a₁, b₁) ⩿ (a₂, b₂) ↔ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂ := by
refine ⟨fun h => ?_, ?_⟩
· obtain rfl | rfl : a₁ = a₂ ∨ b₁ = b₂ := fst_eq_or_snd_eq_of_wcovBy h
· exact Or.inr ⟨mk_wcovBy_mk_iff_right.1 h, rfl⟩
· exact Or.inl ⟨mk_wcovBy_mk_iff_left.1 h, rfl⟩
· rintro (⟨h, rfl⟩ | ⟨h, rfl... | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_wcovBy_mk_iff | null |
mk_covBy_mk_iff : (a₁, b₁) ⋖ (a₂, b₂) ↔ a₁ ⋖ a₂ ∧ b₁ = b₂ ∨ b₁ ⋖ b₂ ∧ a₁ = a₂ := by
refine ⟨fun h => ?_, ?_⟩
· obtain rfl | rfl : a₁ = a₂ ∨ b₁ = b₂ := fst_eq_or_snd_eq_of_wcovBy h.wcovBy
· exact Or.inr ⟨mk_covBy_mk_iff_right.1 h, rfl⟩
· exact Or.inl ⟨mk_covBy_mk_iff_left.1 h, rfl⟩
· rintro (⟨h, rfl⟩ | ⟨h,... | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | mk_covBy_mk_iff | null |
wcovBy_iff : x ⩿ y ↔ x.1 ⩿ y.1 ∧ x.2 = y.2 ∨ x.2 ⩿ y.2 ∧ x.1 = y.1 := by
cases x
cases y
exact mk_wcovBy_mk_iff | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff | null |
covBy_iff : x ⋖ y ↔ x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1 := by
cases x
cases y
exact mk_covBy_mk_iff | theorem | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff | null |
_root_.WCovBy.eval (h : a ⩿ b) (i : ι) : a i ⩿ b i := by
classical
refine ⟨h.1 i, fun ci h₁ h₂ ↦ ?_⟩
have hcb : Function.update a i ci ≤ b := by simpa [update_le_iff, h₂.le] using fun j hj ↦ h.1 j
refine h.2 (by simpa) (lt_of_le_not_ge hcb ?_)
simp [le_update_iff, h₂.not_ge] | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | _root_.WCovBy.eval | null |
exists_forall_antisymmRel_of_covBy (h : a ⋖ b) :
∃ i, ∀ j ≠ i, AntisymmRel (· ≤ ·) (a j) (b j) := by
classical
simp only [CovBy, Pi.lt_def, not_and, and_imp, forall_exists_index, not_exists] at h
obtain ⟨⟨hab, ⟨i, hi⟩⟩, h⟩ := h
refine ⟨i, fun j hj ↦ ?_⟩
let c : (i : ι) → α i := Function.update a i (b i)
... | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | exists_forall_antisymmRel_of_covBy | null |
exists_forall_antisymmRel_of_wcovBy [Nonempty ι] (h : a ⩿ b) :
∃ i, ∀ j ≠ i, AntisymmRel (· ≤ ·) (a j) (b j) := by
rw [wcovBy_iff_covBy_or_le_and_le] at h
obtain h | h := h
· exact exists_forall_antisymmRel_of_covBy h
· inhabit ι
exact ⟨default, fun j hj ↦ ⟨h.left j, h.right j⟩⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | exists_forall_antisymmRel_of_wcovBy | null |
wcovBy_iff_antisymmRel [Nonempty ι] :
a ⩿ b ↔ ∃ i, a i ⩿ b i ∧ ∀ j ≠ i, AntisymmRel (· ≤ ·) (a j) (b j) := by
constructor
· intro h
obtain ⟨i, hi⟩ := exists_forall_antisymmRel_of_wcovBy h
exact ⟨i, h.eval _, hi⟩
rintro ⟨i, hab, h⟩
refine ⟨fun j ↦ (eq_or_ne j i).elim (· ▸ hab.1) (h j · |>.1), fun c h... | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | wcovBy_iff_antisymmRel | A characterisation of the `WCovBy` relation in products of preorders. See `Pi.wcovBy_iff` for the
(more common) version in products of partial orders. |
covBy_iff_antisymmRel :
a ⋖ b ↔ ∃ i, a i ⋖ b i ∧ ∀ j ≠ i, AntisymmRel (· ≤ ·) (a j) (b j) := by
constructor
· intro h
obtain ⟨j, hj⟩ := (Pi.lt_def.1 h.1).2
have : Nonempty ι := ⟨j⟩
obtain ⟨i, hi⟩ := exists_forall_antisymmRel_of_wcovBy h.wcovBy
obtain rfl : i = j := by_contra fun this ↦ (hi j (Ne... | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | covBy_iff_antisymmRel | A characterisation of the `CovBy` relation in products of preorders. See `Pi.covBy_iff` for the
(more common) version in products of partial orders. |
exists_forall_eq_of_covBy (h : a ⋖ b) : ∃ i, ∀ j ≠ i, a j = b j := by
obtain ⟨i, hi⟩ := exists_forall_antisymmRel_of_covBy h
exact ⟨i, fun j hj ↦ AntisymmRel.eq (hi _ hj)⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | exists_forall_eq_of_covBy | null |
exists_forall_eq_of_wcovBy [Nonempty ι] (h : a ⩿ b) : ∃ i, ∀ j ≠ i, a j = b j := by
obtain ⟨i, hi⟩ := exists_forall_antisymmRel_of_wcovBy h
exact ⟨i, fun j hj ↦ AntisymmRel.eq (hi _ hj)⟩ | lemma | Order | [
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Hom.WithTopBot",
"Mathlib.Order.Interval.Set.OrdConnected",
"Mathlib.Order.Interval.Set.WithBotTop"
] | Mathlib/Order/Cover.lean | exists_forall_eq_of_wcovBy | null |
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