ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- WCovBy.inf_eq
theorem WCovBy.inf_eq (hca : c ⩿ a) (hcb : c ⩿ b) (hab : a ≠ b) : a ⊓ b = c 
Arguments: {α : Type u_1} [SemilatticeInf α] {a b c : α} (hca : c ⩿ a) (hcb : c ⩿ b) (hab : a ≠ b)
Goal: a ⊓ b = c
-- CovBy.lt
theorem CovBy.lt (h : a ⋖ b) : a < b 
Arguments: {α : Type u_1} [LT α] {a b : α} (h : a ⋖ b)
Goal: a < b
-- not_covBy_iff
theorem not_covBy_iff (h : a < b) : ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b 
Arguments: {α : Type u_1} [LT α] {a b : α} (h : a < b)
Goal: ¬a ⋖ b ↔ ∃ c, a < c ∧ c < b
-- not_covBy
theorem not_covBy [DenselyOrdered α] : ¬a ⋖ b 
Arguments: {α : Type u_1} [LT α] {a b : α} [DenselyOrdered α]
Goal: ¬a ⋖ b
-- denselyOrdered_iff_forall_not_covBy
theorem denselyOrdered_iff_forall_not_covBy : DenselyOrdered α ↔ ∀ a b : α, ¬a ⋖ b 
Arguments: {α : Type u_1} [LT α]
Goal: DenselyOrdered α ↔ ∀ (a b : α), ¬a ⋖ b
-- toDual_covBy_toDual_iff
@[simp]
theorem toDual_covBy_toDual_iff : toDual b ⋖ toDual a ↔ a ⋖ b 
Arguments: {α : Type u_1} [LT α] {a b : α}
Goal: OrderDual.toDual b ⋖ OrderDual.toDual a ↔ a ⋖ b
-- ofDual_covBy_ofDual_iff
@[simp]
theorem ofDual_covBy_ofDual_iff {a b : αᵒᵈ} : ofDual a ⋖ ofDual b ↔ b ⋖ a 
Arguments: {α : Type u_1} [LT α] {a b : αᵒᵈ}
Goal: OrderDual.ofDual a ⋖ OrderDual.ofDual b ↔ b ⋖ a
-- CovBy.le
theorem CovBy.le (h : a ⋖ b) : a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b)
Goal: a ≤ b
-- CovBy.ne
protected theorem CovBy.ne (h : a ⋖ b) : a ≠ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b)
Goal: a ≠ b
-- CovBy.ne'
theorem CovBy.ne' (h : a ⋖ b) : b ≠ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b)
Goal: b ≠ a
-- CovBy.wcovBy
protected theorem CovBy.wcovBy (h : a ⋖ b) : a ⩿ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b)
Goal: a ⩿ b
-- WCovBy.covBy_of_not_le
theorem WCovBy.covBy_of_not_le (h : a ⩿ b) (h2 : ¬b ≤ a) : a ⋖ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⩿ b) (h2 : ¬b ≤ a)
Goal: a ⋖ b
-- WCovBy.covBy_of_lt
theorem WCovBy.covBy_of_lt (h : a ⩿ b) (h2 : a < b) : a ⋖ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⩿ b) (h2 : a < b)
Goal: a ⋖ b
-- CovBy.of_le_of_lt
lemma CovBy.of_le_of_lt (hac : a ⋖ c) (hab : a ≤ b) (hbc : b < c) : b ⋖ c 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hac : a ⋖ c) (hab : a ≤ b) (hbc : b < c)
Goal: b ⋖ c
-- CovBy.of_lt_of_le
lemma CovBy.of_lt_of_le (hac : a ⋖ c) (hab : a < b) (hbc : b ≤ c) : a ⋖ b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hac : a ⋖ c) (hab : a < b) (hbc : b ≤ c)
Goal: a ⋖ b
-- not_covBy_of_lt_of_lt
theorem not_covBy_of_lt_of_lt (h₁ : a < b) (h₂ : b < c) : ¬a ⋖ c 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (h₁ : a < b) (h₂ : b < c)
Goal: ¬a ⋖ c
-- covBy_iff_wcovBy_and_lt
theorem covBy_iff_wcovBy_and_lt : a ⋖ b ↔ a ⩿ b ∧ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ⋖ b ↔ a ⩿ b ∧ a < b
-- covBy_iff_wcovBy_and_not_le
theorem covBy_iff_wcovBy_and_not_le : a ⋖ b ↔ a ⩿ b ∧ ¬b ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ⋖ b ↔ a ⩿ b ∧ ¬b ≤ a
-- wcovBy_iff_covBy_or_le_and_le
theorem wcovBy_iff_covBy_or_le_and_le : a ⩿ b ↔ a ⋖ b ∨ a ≤ b ∧ b ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ⩿ b ↔ a ⋖ b ∨ a ≤ b ∧ b ≤ a
-- AntisymmRel.trans_covBy
theorem AntisymmRel.trans_covBy (hab : AntisymmRel (· ≤ ·) a b) (hbc : b ⋖ c) : a ⋖ c 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hab : AntisymmRel (fun x x_1 => x ≤ x_1) a b) (hbc : b ⋖ c)
Goal: a ⋖ c
-- covBy_congr_left
theorem covBy_congr_left (hab : AntisymmRel (· ≤ ·) a b) : a ⋖ c ↔ b ⋖ c 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hab : AntisymmRel (fun x x_1 => x ≤ x_1) a b)
Goal: a ⋖ c ↔ b ⋖ c
-- CovBy.trans_antisymmRel
theorem CovBy.trans_antisymmRel (hab : a ⋖ b) (hbc : AntisymmRel (· ≤ ·) b c) : a ⋖ c 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hab : a ⋖ b) (hbc : AntisymmRel (fun x x_1 => x ≤ x_1) b c)
Goal: a ⋖ c
-- covBy_congr_right
theorem covBy_congr_right (hab : AntisymmRel (· ≤ ·) a b) : c ⋖ a ↔ c ⋖ b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hab : AntisymmRel (fun x x_1 => x ≤ x_1) a b)
Goal: c ⋖ a ↔ c ⋖ b
-- CovBy.Ioo_eq
theorem CovBy.Ioo_eq (h : a ⋖ b) : Ioo a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ⋖ b)
Goal: Set.Ioo a b = ∅
-- covBy_iff_Ioo_eq
theorem covBy_iff_Ioo_eq : a ⋖ b ↔ a < b ∧ Ioo a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ⋖ b ↔ a < b ∧ Set.Ioo a b = ∅
-- CovBy.of_image
theorem CovBy.of_image (f : α ↪o β) (h : f a ⋖ f b) : a ⋖ b 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : α} (f : α ↪o β) (h : f a ⋖ f b)
Goal: a ⋖ b
-- CovBy.image
theorem CovBy.image (f : α ↪o β) (hab : a ⋖ b) (h : (range f).OrdConnected) : f a ⋖ f b 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : α} (f : α ↪o β) (hab : a ⋖ b) (h : (Set.range ⇑f).OrdConnected)
Goal: f a ⋖ f b
-- Set.OrdConnected.apply_covBy_apply_iff
theorem Set.OrdConnected.apply_covBy_apply_iff (f : α ↪o β) (h : (range f).OrdConnected) :
    f a ⋖ f b ↔ a ⋖ b 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : α} (f : α ↪o β) (h : (Set.range ⇑f).OrdConnected)
Goal: f a ⋖ f b ↔ a ⋖ b
-- apply_covBy_apply_iff
@[simp]
theorem apply_covBy_apply_iff {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
    e a ⋖ e b ↔ a ⋖ b 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a b : α} {E : Type u_3} [EquivLike E α β] [OrderIsoClass E α β] (e : E)
Goal: e a ⋖ e b ↔ a ⋖ b
-- covBy_of_eq_or_eq
theorem covBy_of_eq_or_eq (hab : a < b) (h : ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b) : a ⋖ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (hab : a < b) (h : ∀ (c : α), a ≤ c → c ≤ b → c = a ∨ c = b)
Goal: a ⋖ b
-- WCovBy.covBy_of_ne
theorem WCovBy.covBy_of_ne (h : a ⩿ b) (h2 : a ≠ b) : a ⋖ b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⩿ b) (h2 : a ≠ b)
Goal: a ⋖ b
-- covBy_iff_wcovBy_and_ne
theorem covBy_iff_wcovBy_and_ne : a ⋖ b ↔ a ⩿ b ∧ a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: a ⋖ b ↔ a ⩿ b ∧ a ≠ b
-- wcovBy_iff_covBy_or_eq
theorem wcovBy_iff_covBy_or_eq : a ⩿ b ↔ a ⋖ b ∨ a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: a ⩿ b ↔ a ⋖ b ∨ a = b
-- wcovBy_iff_eq_or_covBy
theorem wcovBy_iff_eq_or_covBy : a ⩿ b ↔ a = b ∨ a ⋖ b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: a ⩿ b ↔ a = b ∨ a ⋖ b
-- CovBy.eq_or_eq
theorem CovBy.eq_or_eq (h : a ⋖ b) (h2 : a ≤ c) (h3 : c ≤ b) : c = a ∨ c = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b c : α} (h : a ⋖ b) (h2 : a ≤ c) (h3 : c ≤ b)
Goal: c = a ∨ c = b
-- covBy_iff_lt_and_eq_or_eq
theorem covBy_iff_lt_and_eq_or_eq : a ⋖ b ↔ a < b ∧ ∀ c, a ≤ c → c ≤ b → c = a ∨ c = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: a ⋖ b ↔ a < b ∧ ∀ (c : α), a ≤ c → c ≤ b → c = a ∨ c = b
-- CovBy.Ico_eq
theorem CovBy.Ico_eq (h : a ⋖ b) : Ico a b = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⋖ b)
Goal: Set.Ico a b = {a}
-- CovBy.Ioc_eq
theorem CovBy.Ioc_eq (h : a ⋖ b) : Ioc a b = {b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⋖ b)
Goal: Set.Ioc a b = {b}
-- CovBy.Icc_eq
theorem CovBy.Icc_eq (h : a ⋖ b) : Icc a b = {a, b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ⋖ b)
Goal: Set.Icc a b = {a, b}
-- CovBy.Ioi_eq
theorem CovBy.Ioi_eq (h : a ⋖ b) : Ioi a = Ici b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ⋖ b)
Goal: Set.Ioi a = Set.Ici b
-- CovBy.Iio_eq
theorem CovBy.Iio_eq (h : a ⋖ b) : Iio b = Iic a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ⋖ b)
Goal: Set.Iio b = Set.Iic a
-- WCovBy.le_of_lt
theorem WCovBy.le_of_lt (hab : a ⩿ b) (hcb : c < b) : c ≤ a 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⩿ b) (hcb : c < b)
Goal: c ≤ a
-- WCovBy.ge_of_gt
theorem WCovBy.ge_of_gt (hab : a ⩿ b) (hac : a < c) : b ≤ c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⩿ b) (hac : a < c)
Goal: b ≤ c
-- CovBy.le_of_lt
theorem CovBy.le_of_lt (hab : a ⋖ b) : c < b → c ≤ a 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⋖ b)
Goal: c < b → c ≤ a
-- CovBy.ge_of_gt
theorem CovBy.ge_of_gt (hab : a ⋖ b) : a < c → b ≤ c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (hab : a ⋖ b)
Goal: a < c → b ≤ c
-- CovBy.unique_left
theorem CovBy.unique_left (ha : a ⋖ c) (hb : b ⋖ c) : a = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (ha : a ⋖ c) (hb : b ⋖ c)
Goal: a = b
-- CovBy.unique_right
theorem CovBy.unique_right (hb : a ⋖ b) (hc : a ⋖ c) : b = c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (hb : a ⋖ b) (hc : a ⋖ c)
Goal: b = c
-- CovBy.eq_of_between
theorem CovBy.eq_of_between {x : α} (hab : a ⋖ b) (hbc : b ⋖ c) (hax : a < x) (hxc : x < c) :
    x = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c x : α} (hab : a ⋖ b) (hbc : b ⋖ c) (hax : a < x) (hxc : x < c)
Goal: x = b
-- LT.lt.exists_disjoint_Iio_Ioi
lemma LT.lt.exists_disjoint_Iio_Ioi (h : a < b) :
    ∃ a' > a, ∃ b' < b, ∀ x < a', ∀ y > b', x < y 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: ∃ a' > a, ∃ b' < b, ∀ x < a', ∀ y > b', x < y
-- Set.wcovBy_insert
@[simp] lemma wcovBy_insert (x : α) (s : Set α) : s ⩿ insert x s 
Arguments: {α : Type u_1} (x : α) (s : Set α)
Goal: s ⩿ insert x s
-- Set.sdiff_singleton_wcovBy
@[simp] lemma sdiff_singleton_wcovBy (s : Set α) (a : α) : s \ {a} ⩿ s 
Arguments: {α : Type u_1} (s : Set α) (a : α)
Goal: s \ {a} ⩿ s
-- Set.covBy_insert
@[simp] lemma covBy_insert (ha : a ∉ s) : s ⋖ insert a s 
Arguments: {α : Type u_1} {s : Set α} {a : α} (ha : a ∉ s)
Goal: s ⋖ insert a s
-- Set.sdiff_singleton_covBy
@[simp] lemma sdiff_singleton_covBy (ha : a ∈ s) : s \ {a} ⋖ s 
Arguments: {α : Type u_1} {s : Set α} {a : α} (ha : a ∈ s)
Goal: s \ {a} ⋖ s
-- CovBy.exists_set_insert
lemma _root_.CovBy.exists_set_insert (h : s ⋖ t) : ∃ a ∉ s, insert a s = t 
Arguments: {α : Type u_1} {s t : Set α} (h : s ⋖ t)
Goal: ∃ a ∉ s, insert a s = t
-- CovBy.exists_set_sdiff_singleton
lemma _root_.CovBy.exists_set_sdiff_singleton (h : s ⋖ t) : ∃ a ∈ t, t \ {a} =  s 
Arguments: {α : Type u_1} {s t : Set α} (h : s ⋖ t)
Goal: ∃ a ∈ t, t \ {a} = s
-- Set.covBy_iff_exists_insert
lemma covBy_iff_exists_insert : s ⋖ t ↔ ∃ a ∉ s, insert a s = t 
Arguments: {α : Type u_1} {s t : Set α}
Goal: s ⋖ t ↔ ∃ a ∉ s, insert a s = t
-- Set.covBy_iff_exists_sdiff_singleton
lemma covBy_iff_exists_sdiff_singleton : s ⋖ t ↔ ∃ a ∈ t, t \ {a} = s 
Arguments: {α : Type u_1} {s t : Set α}
Goal: s ⋖ t ↔ ∃ a ∈ t, t \ {a} = s
-- wcovBy_eq_reflGen_covBy
lemma wcovBy_eq_reflGen_covBy [PartialOrder α] : ((· : α) ⩿ ·) = ReflGen (· ⋖ ·) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: (fun x x_1 => x ⩿ x_1) = Relation.ReflGen fun x x_1 => x ⋖ x_1
-- transGen_wcovBy_eq_reflTransGen_covBy
lemma transGen_wcovBy_eq_reflTransGen_covBy [PartialOrder α] :
    TransGen ((· : α) ⩿ ·) = ReflTransGen (· ⋖ ·) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: (Relation.TransGen fun x x_1 => x ⩿ x_1) = Relation.ReflTransGen fun x x_1 => x ⋖ x_1
-- reflTransGen_wcovBy_eq_reflTransGen_covBy
lemma reflTransGen_wcovBy_eq_reflTransGen_covBy [PartialOrder α] :
    ReflTransGen ((· : α) ⩿ ·) = ReflTransGen (· ⋖ ·) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: (Relation.ReflTransGen fun x x_1 => x ⩿ x_1) = Relation.ReflTransGen fun x x_1 => x ⋖ x_1
-- Prod.swap_wcovBy_swap
@[simp]
theorem swap_wcovBy_swap : x.swap ⩿ y.swap ↔ x ⩿ y 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β}
Goal: x.swap ⩿ y.swap ↔ x ⩿ y
-- Prod.swap_covBy_swap
@[simp]
theorem swap_covBy_swap : x.swap ⋖ y.swap ↔ x ⋖ y 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β}
Goal: x.swap ⋖ y.swap ↔ x ⋖ y
-- Prod.fst_eq_or_snd_eq_of_wcovBy
theorem fst_eq_or_snd_eq_of_wcovBy : x ⩿ y → x.1 = y.1 ∨ x.2 = y.2 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β}
Goal: x ⩿ y → x.1 = y.1 ∨ x.2 = y.2
-- WCovBy.fst
theorem _root_.WCovBy.fst (h : x ⩿ y) : x.1 ⩿ y.1 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β} (h : x ⩿ y)
Goal: x.1 ⩿ y.1
-- WCovBy.snd
theorem _root_.WCovBy.snd (h : x ⩿ y) : x.2 ⩿ y.2 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β} (h : x ⩿ y)
Goal: x.2 ⩿ y.2
-- Prod.mk_wcovBy_mk_iff_left
theorem mk_wcovBy_mk_iff_left : (a₁, b) ⩿ (a₂, b) ↔ a₁ ⩿ a₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ a₂ : α} {b : β}
Goal: (a₁, b) ⩿ (a₂, b) ↔ a₁ ⩿ a₂
-- Prod.mk_wcovBy_mk_iff_right
theorem mk_wcovBy_mk_iff_right : (a, b₁) ⩿ (a, b₂) ↔ b₁ ⩿ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a : α} {b₁ b₂ : β}
Goal: (a, b₁) ⩿ (a, b₂) ↔ b₁ ⩿ b₂
-- Prod.mk_covBy_mk_iff_left
theorem mk_covBy_mk_iff_left : (a₁, b) ⋖ (a₂, b) ↔ a₁ ⋖ a₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ a₂ : α} {b : β}
Goal: (a₁, b) ⋖ (a₂, b) ↔ a₁ ⋖ a₂
-- Prod.mk_covBy_mk_iff_right
theorem mk_covBy_mk_iff_right : (a, b₁) ⋖ (a, b₂) ↔ b₁ ⋖ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a : α} {b₁ b₂ : β}
Goal: (a, b₁) ⋖ (a, b₂) ↔ b₁ ⋖ b₂
-- Prod.mk_wcovBy_mk_iff
theorem mk_wcovBy_mk_iff : (a₁, b₁) ⩿ (a₂, b₂) ↔ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ a₂ : α} {b₁ b₂ : β}
Goal: (a₁, b₁) ⩿ (a₂, b₂) ↔ a₁ ⩿ a₂ ∧ b₁ = b₂ ∨ b₁ ⩿ b₂ ∧ a₁ = a₂
-- Prod.mk_covBy_mk_iff
theorem mk_covBy_mk_iff : (a₁, b₁) ⋖ (a₂, b₂) ↔ a₁ ⋖ a₂ ∧ b₁ = b₂ ∨ b₁ ⋖ b₂ ∧ a₁ = a₂ 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {a₁ a₂ : α} {b₁ b₂ : β}
Goal: (a₁, b₁) ⋖ (a₂, b₂) ↔ a₁ ⋖ a₂ ∧ b₁ = b₂ ∨ b₁ ⋖ b₂ ∧ a₁ = a₂
-- Prod.wcovBy_iff
theorem wcovBy_iff : x ⩿ y ↔ x.1 ⩿ y.1 ∧ x.2 = y.2 ∨ x.2 ⩿ y.2 ∧ x.1 = y.1 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β}
Goal: x ⩿ y ↔ x.1 ⩿ y.1 ∧ x.2 = y.2 ∨ x.2 ⩿ y.2 ∧ x.1 = y.1
-- Prod.covBy_iff
theorem covBy_iff : x ⋖ y ↔ x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {x y : α × β}
Goal: x ⋖ y ↔ x.1 ⋖ y.1 ∧ x.2 = y.2 ∨ x.2 ⋖ y.2 ∧ x.1 = y.1
-- OrderIso.upperBounds_image
theorem upperBounds_image {s : Set α} : upperBounds (f '' s) = f '' upperBounds s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α}
Goal: upperBounds (⇑f '' s) = ⇑f '' upperBounds s
-- OrderIso.lowerBounds_image
theorem lowerBounds_image {s : Set α} : lowerBounds (f '' s) = f '' lowerBounds s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α}
Goal: lowerBounds (⇑f '' s) = ⇑f '' lowerBounds s
-- OrderIso.isLUB_image
@[simp]
theorem isLUB_image {s : Set α} {x : β} : IsLUB (f '' s) x ↔ IsLUB s (f.symm x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α} {x : β}
Goal: IsLUB (⇑f '' s) x ↔ IsLUB s (f.symm x)
-- OrderIso.isLUB_image'
theorem isLUB_image' {s : Set α} {x : α} : IsLUB (f '' s) (f x) ↔ IsLUB s x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α} {x : α}
Goal: IsLUB (⇑f '' s) (f x) ↔ IsLUB s x
-- OrderIso.isGLB_image
@[simp]
theorem isGLB_image {s : Set α} {x : β} : IsGLB (f '' s) x ↔ IsGLB s (f.symm x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α} {x : β}
Goal: IsGLB (⇑f '' s) x ↔ IsGLB s (f.symm x)
-- OrderIso.isGLB_image'
theorem isGLB_image' {s : Set α} {x : α} : IsGLB (f '' s) (f x) ↔ IsGLB s x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set α} {x : α}
Goal: IsGLB (⇑f '' s) (f x) ↔ IsGLB s x
-- OrderIso.isLUB_preimage
@[simp]
theorem isLUB_preimage {s : Set β} {x : α} : IsLUB (f ⁻¹' s) x ↔ IsLUB s (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set β} {x : α}
Goal: IsLUB (⇑f ⁻¹' s) x ↔ IsLUB s (f x)
-- OrderIso.isLUB_preimage'
theorem isLUB_preimage' {s : Set β} {x : β} : IsLUB (f ⁻¹' s) (f.symm x) ↔ IsLUB s x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set β} {x : β}
Goal: IsLUB (⇑f ⁻¹' s) (f.symm x) ↔ IsLUB s x
-- OrderIso.isGLB_preimage
@[simp]
theorem isGLB_preimage {s : Set β} {x : α} : IsGLB (f ⁻¹' s) x ↔ IsGLB s (f x) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set β} {x : α}
Goal: IsGLB (⇑f ⁻¹' s) x ↔ IsGLB s (f x)
-- OrderIso.isGLB_preimage'
theorem isGLB_preimage' {s : Set β} {x : β} : IsGLB (f ⁻¹' s) (f.symm x) ↔ IsGLB s x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) {s : Set β} {x : β}
Goal: IsGLB (⇑f ⁻¹' s) (f.symm x) ↔ IsGLB s x
-- Order.exists_between_finsets
theorem exists_between_finsets {α : Type*} [LinearOrder α] [DenselyOrdered α] [NoMinOrder α]
    [NoMaxOrder α] [nonem : Nonempty α] (lo hi : Finset α) (lo_lt_hi : ∀ x ∈ lo, ∀ y ∈ hi, x < y) :
    ∃ m : α, (∀ x ∈ lo, x < m) ∧ ∀ y ∈ hi, m < y 
Arguments: {α : Type u_1} [LinearOrder α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] [nonem : Nonempty α] (lo hi : Finset α) (lo_lt_hi : ∀ x ∈ lo, ∀ y ∈ hi, x < y)
Goal: ∃ m, (∀ x ∈ lo, x < m) ∧ ∀ y ∈ hi, m < y
-- Order.PartialIso.exists_across
theorem exists_across [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
    (f : PartialIso α β) (a : α) :
    ∃ b : β, ∀ p ∈ f.val, cmp (Prod.fst p) a = cmp (Prod.snd p) b 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] (f : Order.PartialIso α β) (a : α)
Goal: ∃ b, ∀ p ∈ ↑f, cmp p.1 a = cmp p.2 b
-- Order.embedding_from_countable_to_dense
theorem embedding_from_countable_to_dense [Countable α] [DenselyOrdered β] [Nontrivial β] :
    Nonempty (α ↪o β) 
Arguments: (α : Type u_1) (β : Type u_2) [LinearOrder α] [LinearOrder β] [Countable α] [DenselyOrdered β] [Nontrivial β]
Goal: Nonempty (α ↪o β)
-- Order.iso_of_countable_dense
theorem iso_of_countable_dense [Countable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α]
    [Nonempty α] [Countable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β] :
    Nonempty (α ≃o β) 
Arguments: (α : Type u_1) (β : Type u_2) [LinearOrder α] [LinearOrder β] [Countable α] [DenselyOrdered α] [NoMinOrder α] [NoMaxOrder α] [Nonempty α] [Countable β] [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
Goal: Nonempty (α ≃o β)
-- Fin.pi_lex_lt_cons_cons
lemma pi_lex_lt_cons_cons {x₀ y₀ : α 0} {x y : ∀ i : Fin n, α i.succ}
    (s : ∀ {i : Fin n.succ}, α i → α i → Prop) :
    Pi.Lex (· < ·) (@s) (Fin.cons x₀ x) (Fin.cons y₀ y) ↔
      s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (· < ·) (@fun i : Fin n ↦ @s i.succ) x y 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_1} {x₀ y₀ : α 0} {x y : (i : Fin n) → α i.succ} (s : {i : Fin n.succ} → α i → α i → Prop)
Goal: Pi.Lex (fun x x_1 => x < x_1) s (Fin.cons x₀ x) (Fin.cons y₀ y) ↔
  s x₀ y₀ ∨ x₀ = y₀ ∧ Pi.Lex (fun x x_1 => x < x_1) (fun i => s) x y
-- Fin.insertNth_mem_Icc
lemma insertNth_mem_Icc {i : Fin (n + 1)} {x : α i} {p : ∀ j, α (i.succAbove j)}
    {q₁ q₂ : ∀ j, α j} :
    i.insertNth x p ∈ Icc q₁ q₂ ↔
      x ∈ Icc (q₁ i) (q₂ i) ∧ p ∈ Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {i : Fin (n + 1)} {x : α i} {p : (j : Fin n) → α (i.succAbove j)} {q₁ q₂ : (j : Fin (n + 1)) → α j}
Goal: i.insertNth x p ∈ Set.Icc q₁ q₂ ↔
  x ∈ Set.Icc (q₁ i) (q₂ i) ∧ p ∈ Set.Icc (fun j => q₁ (i.succAbove j)) fun j => q₂ (i.succAbove j)
-- Fin.preimage_insertNth_Icc_of_mem
lemma preimage_insertNth_Icc_of_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}
    (hx : x ∈ Icc (q₁ i) (q₂ i)) :
    i.insertNth x ⁻¹' Icc q₁ q₂ = Icc (fun j ↦ q₁ (i.succAbove j)) fun j ↦ q₂ (i.succAbove j) 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {i : Fin (n + 1)} {x : α i} {q₁ q₂ : (j : Fin (n + 1)) → α j} (hx : x ∈ Set.Icc (q₁ i) (q₂ i))
Goal: i.insertNth x ⁻¹' Set.Icc q₁ q₂ = Set.Icc (fun j => q₁ (i.succAbove j)) fun j => q₂ (i.succAbove j)
-- Fin.preimage_insertNth_Icc_of_not_mem
lemma preimage_insertNth_Icc_of_not_mem {i : Fin (n + 1)} {x : α i} {q₁ q₂ : ∀ j, α j}
    (hx : x ∉ Icc (q₁ i) (q₂ i)) : i.insertNth x ⁻¹' Icc q₁ q₂ = ∅ 
Arguments: {n : ℕ} {α : Fin (n + 1) → Type u_1} [(i : Fin (n + 1)) → Preorder (α i)] {i : Fin (n + 1)} {x : α i} {q₁ q₂ : (j : Fin (n + 1)) → α j} (hx : x ∉ Set.Icc (q₁ i) (q₂ i))
Goal: i.insertNth x ⁻¹' Set.Icc q₁ q₂ = ∅
-- liftFun_vecCons
lemma liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :
    ((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f 
Arguments: {α : Type u_1} {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α}
Goal: ((fun x x_1 => x < x_1) ⇒ r) (Matrix.vecCons a f) (Matrix.vecCons a f) ↔ r a (f 0) ∧ ((fun x x_1 => x < x_1) ⇒ r) f f
-- strictMono_vecCons
@[simp] lemma strictMono_vecCons : StrictMono (vecCons a f) ↔ a < f 0 ∧ StrictMono f 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}
Goal: StrictMono (Matrix.vecCons a f) ↔ a < f 0 ∧ StrictMono f
-- monotone_vecCons
@[simp]
lemma monotone_vecCons : Monotone (vecCons a f) ↔ a ≤ f 0 ∧ Monotone f 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}
Goal: Monotone (Matrix.vecCons a f) ↔ a ≤ f 0 ∧ Monotone f
-- monotone_vecEmpty
@[simp] lemma monotone_vecEmpty : Monotone ![a]
  
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Monotone ![a]
-- strictMono_vecEmpty
@[simp] lemma strictMono_vecEmpty : StrictMono ![a]
  
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: StrictMono ![a]
-- strictAnti_vecCons
@[simp] lemma strictAnti_vecCons : StrictAnti (vecCons a f) ↔ f 0 < a ∧ StrictAnti f 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}
Goal: StrictAnti (Matrix.vecCons a f) ↔ f 0 < a ∧ StrictAnti f
-- antitone_vecCons
@[simp] lemma antitone_vecCons : Antitone (vecCons a f) ↔ f 0 ≤ a ∧ Antitone f 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α}
Goal: Antitone (Matrix.vecCons a f) ↔ f 0 ≤ a ∧ Antitone f
-- antitone_vecEmpty
@[simp] lemma antitone_vecEmpty : Antitone (vecCons a vecEmpty)
  
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Antitone ![a]
-- strictAnti_vecEmpty
@[simp] lemma strictAnti_vecEmpty : StrictAnti (vecCons a vecEmpty)
  
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: StrictAnti ![a]
-- StrictMono.vecCons
lemma StrictMono.vecCons (hf : StrictMono f) (ha : a < f 0) : StrictMono (vecCons a f) 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} (hf : StrictMono f) (ha : a < f 0)
Goal: StrictMono (Matrix.vecCons a f)
-- StrictAnti.vecCons
lemma StrictAnti.vecCons (hf : StrictAnti f) (ha : f 0 < a) : StrictAnti (vecCons a f) 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} (hf : StrictAnti f) (ha : f 0 < a)
Goal: StrictAnti (Matrix.vecCons a f)
-- Monotone.vecCons
lemma Monotone.vecCons (hf : Monotone f) (ha : a ≤ f 0) : Monotone (vecCons a f) 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} (hf : Monotone f) (ha : a ≤ f 0)
Goal: Monotone (Matrix.vecCons a f)
-- Antitone.vecCons
lemma Antitone.vecCons (hf : Antitone f) (ha : f 0 ≤ a) : Antitone (vecCons a f) 
Arguments: {α : Type u_1} [Preorder α] {n : ℕ} {f : Fin (n + 1) → α} {a : α} (hf : Antitone f) (ha : f 0 ≤ a)
Goal: Antitone (Matrix.vecCons a f)
-- finSuccAboveOrderIso_apply
lemma finSuccAboveOrderIso_apply (p : Fin (n + 1)) (i : Fin n) :
    finSuccAboveOrderIso p i = ⟨p.succAbove i, p.succAbove_ne i⟩ 
Arguments: {n : ℕ} (p : Fin (n + 1)) (i : Fin n)
Goal: (finSuccAboveOrderIso p) i = ⟨p.succAbove i, ⋯⟩
-- finSuccAboveOrderIso_symm_apply_last
lemma finSuccAboveOrderIso_symm_apply_last (x : { x : Fin (n + 1) // x ≠ Fin.last n }) :
    (finSuccAboveOrderIso (Fin.last n)).symm x = Fin.castLT x.1 (Fin.val_lt_last x.2) 
Arguments: {n : ℕ} (x : { x // x ≠ Fin.last n })
Goal: (finSuccAboveOrderIso (Fin.last n)).symm x = (↑x).castLT ⋯