ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Multiset.mem_Ioi
@[simp] lemma mem_Ioi : x ∈ Ioi a ↔ a < x 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α}
Goal: x ∈ Multiset.Ioi a ↔ a < x
-- Multiset.mem_Iic
@[simp] lemma mem_Iic : x ∈ Iic b ↔ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {b x : α}
Goal: x ∈ Multiset.Iic b ↔ x ≤ b
-- Multiset.mem_Iio
@[simp] lemma mem_Iio : x ∈ Iio b ↔ x < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {b x : α}
Goal: x ∈ Multiset.Iio b ↔ x < b
-- Multiset.nodup_Icc
theorem nodup_Icc : (Icc a b).Nodup 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Multiset.Icc a b).Nodup
-- Multiset.nodup_Ico
theorem nodup_Ico : (Ico a b).Nodup 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Multiset.Ico a b).Nodup
-- Multiset.nodup_Ioc
theorem nodup_Ioc : (Ioc a b).Nodup 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Multiset.Ioc a b).Nodup
-- Multiset.nodup_Ioo
theorem nodup_Ioo : (Ioo a b).Nodup 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: (Multiset.Ioo a b).Nodup
-- Multiset.Icc_eq_zero_iff
@[simp]
theorem Icc_eq_zero_iff : Icc a b = 0 ↔ ¬a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Multiset.Icc a b = 0 ↔ ¬a ≤ b
-- Multiset.Ico_eq_zero_iff
@[simp]
theorem Ico_eq_zero_iff : Ico a b = 0 ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Multiset.Ico a b = 0 ↔ ¬a < b
-- Multiset.Ioc_eq_zero_iff
@[simp]
theorem Ioc_eq_zero_iff : Ioc a b = 0 ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: Multiset.Ioc a b = 0 ↔ ¬a < b
-- Multiset.Ioo_eq_zero_iff
@[simp]
theorem Ioo_eq_zero_iff [DenselyOrdered α] : Ioo a b = 0 ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DenselyOrdered α]
Goal: Multiset.Ioo a b = 0 ↔ ¬a < b
-- Multiset.Ioo_eq_zero
@[simp]
theorem Ioo_eq_zero (h : ¬a < b) : Ioo a b = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : ¬a < b)
Goal: Multiset.Ioo a b = 0
-- Multiset.Icc_eq_zero_of_lt
@[simp]
theorem Icc_eq_zero_of_lt (h : b < a) : Icc a b = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b < a)
Goal: Multiset.Icc a b = 0
-- Multiset.Ico_eq_zero_of_le
@[simp]
theorem Ico_eq_zero_of_le (h : b ≤ a) : Ico a b = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Multiset.Ico a b = 0
-- Multiset.Ioc_eq_zero_of_le
@[simp]
theorem Ioc_eq_zero_of_le (h : b ≤ a) : Ioc a b = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Multiset.Ioc a b = 0
-- Multiset.Ioo_eq_zero_of_le
@[simp]
theorem Ioo_eq_zero_of_le (h : b ≤ a) : Ioo a b = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: Multiset.Ioo a b = 0
-- Multiset.Ico_self
theorem Ico_self : Ico a a = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Multiset.Ico a a = 0
-- Multiset.Ioc_self
theorem Ioc_self : Ioc a a = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Multiset.Ioc a a = 0
-- Multiset.Ioo_self
theorem Ioo_self : Ioo a a = 0 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a : α)
Goal: Multiset.Ioo a a = 0
-- Multiset.left_mem_Icc
theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∈ Multiset.Icc a b ↔ a ≤ b
-- Multiset.left_mem_Ico
theorem left_mem_Ico : a ∈ Ico a b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∈ Multiset.Ico a b ↔ a < b
-- Multiset.right_mem_Icc
theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∈ Multiset.Icc a b ↔ a ≤ b
-- Multiset.right_mem_Ioc
theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∈ Multiset.Ioc a b ↔ a < b
-- Multiset.left_not_mem_Ioc
theorem left_not_mem_Ioc : a ∉ Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∉ Multiset.Ioc a b
-- Multiset.left_not_mem_Ioo
theorem left_not_mem_Ioo : a ∉ Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∉ Multiset.Ioo a b
-- Multiset.right_not_mem_Ico
theorem right_not_mem_Ico : b ∉ Ico a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∉ Multiset.Ico a b
-- Multiset.right_not_mem_Ioo
theorem right_not_mem_Ioo : b ∉ Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∉ Multiset.Ioo a b
-- Multiset.Ico_filter_lt_of_le_left
theorem Ico_filter_lt_of_le_left [DecidablePred (· < c)] (hca : c ≤ a) :
    ((Ico a b).filter fun x => x < c) = ∅ 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hca : c ≤ a)
Goal: Multiset.filter (fun x => x < c) (Multiset.Ico a b) = ∅
-- Multiset.Ico_filter_lt_of_right_le
theorem Ico_filter_lt_of_right_le [DecidablePred (· < c)] (hbc : b ≤ c) :
    ((Ico a b).filter fun x => x < c) = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hbc : b ≤ c)
Goal: Multiset.filter (fun x => x < c) (Multiset.Ico a b) = Multiset.Ico a b
-- Multiset.Ico_filter_lt_of_le_right
theorem Ico_filter_lt_of_le_right [DecidablePred (· < c)] (hcb : c ≤ b) :
    ((Ico a b).filter fun x => x < c) = Ico a c 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => x < c] (hcb : c ≤ b)
Goal: Multiset.filter (fun x => x < c) (Multiset.Ico a b) = Multiset.Ico a c
-- Multiset.Ico_filter_le_of_le_left
theorem Ico_filter_le_of_le_left [DecidablePred (c ≤ ·)] (hca : c ≤ a) :
    ((Ico a b).filter fun x => c ≤ x) = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => c ≤ x] (hca : c ≤ a)
Goal: Multiset.filter (fun x => c ≤ x) (Multiset.Ico a b) = Multiset.Ico a b
-- Multiset.Ico_filter_le_of_right_le
theorem Ico_filter_le_of_right_le [DecidablePred (b ≤ ·)] :
    ((Ico a b).filter fun x => b ≤ x) = ∅ 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun x => b ≤ x]
Goal: Multiset.filter (fun x => b ≤ x) (Multiset.Ico a b) = ∅
-- Multiset.Ico_filter_le_of_left_le
theorem Ico_filter_le_of_left_le [DecidablePred (c ≤ ·)] (hac : a ≤ c) :
    ((Ico a b).filter fun x => c ≤ x) = Ico c b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b c : α} [DecidablePred fun x => c ≤ x] (hac : a ≤ c)
Goal: Multiset.filter (fun x => c ≤ x) (Multiset.Ico a b) = Multiset.Ico c b
-- Multiset.Icc_self
@[simp]
theorem Icc_self (a : α) : Icc a a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a : α)
Goal: Multiset.Icc a a = {a}
-- Multiset.Ico_cons_right
theorem Ico_cons_right (h : a ≤ b) : b ::ₘ Ico a b = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a ≤ b)
Goal: b ::ₘ Multiset.Ico a b = Multiset.Icc a b
-- Multiset.Ioo_cons_left
theorem Ioo_cons_left (h : a < b) : a ::ₘ Ioo a b = Ico a b 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a < b)
Goal: a ::ₘ Multiset.Ioo a b = Multiset.Ico a b
-- Multiset.Ico_disjoint_Ico
theorem Ico_disjoint_Ico {a b c d : α} (h : b ≤ c) : (Ico a b).Disjoint (Ico c d) 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b c d : α} (h : b ≤ c)
Goal: (Multiset.Ico a b).Disjoint (Multiset.Ico c d)
-- Multiset.Ico_inter_Ico_of_le
@[simp]
theorem Ico_inter_Ico_of_le [DecidableEq α] {a b c d : α} (h : b ≤ c) : Ico a b ∩ Ico c d = 0 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] {a b c d : α} (h : b ≤ c)
Goal: Multiset.Ico a b ∩ Multiset.Ico c d = 0
-- Multiset.Ico_filter_le_left
theorem Ico_filter_le_left {a b : α} [DecidablePred (· ≤ a)] (hab : a < b) :
    ((Ico a b).filter fun x => x ≤ a) = {a} 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun x => x ≤ a] (hab : a < b)
Goal: Multiset.filter (fun x => x ≤ a) (Multiset.Ico a b) = {a}
-- Multiset.card_Ico_eq_card_Icc_sub_one
theorem card_Ico_eq_card_Icc_sub_one (a b : α) : card (Ico a b) = card (Icc a b) - 1 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: Multiset.card (Multiset.Ico a b) = Multiset.card (Multiset.Icc a b) - 1
-- Multiset.card_Ioc_eq_card_Icc_sub_one
theorem card_Ioc_eq_card_Icc_sub_one (a b : α) : card (Ioc a b) = card (Icc a b) - 1 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: Multiset.card (Multiset.Ioc a b) = Multiset.card (Multiset.Icc a b) - 1
-- Multiset.card_Ioo_eq_card_Ico_sub_one
theorem card_Ioo_eq_card_Ico_sub_one (a b : α) : card (Ioo a b) = card (Ico a b) - 1 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: Multiset.card (Multiset.Ioo a b) = Multiset.card (Multiset.Ico a b) - 1
-- Multiset.card_Ioo_eq_card_Icc_sub_two
theorem card_Ioo_eq_card_Icc_sub_two (a b : α) : card (Ioo a b) = card (Icc a b) - 2 
Arguments: {α : Type u_1} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: Multiset.card (Multiset.Ioo a b) = Multiset.card (Multiset.Icc a b) - 2
-- Multiset.Ico_subset_Ico_iff
theorem Ico_subset_Ico_iff {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) :
    Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁)
Goal: Multiset.Ico a₁ b₁ ⊆ Multiset.Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Multiset.Ico_add_Ico_eq_Ico
theorem Ico_add_Ico_eq_Ico {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
    Ico a b + Ico b c = Ico a c 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
Goal: Multiset.Ico a b + Multiset.Ico b c = Multiset.Ico a c
-- Multiset.Ico_inter_Ico
theorem Ico_inter_Ico : Ico a b ∩ Ico c d = Ico (max a c) (min b d) 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α}
Goal: Multiset.Ico a b ∩ Multiset.Ico c d = Multiset.Ico (max a c) (min b d)
-- Multiset.Ico_filter_lt
@[simp]
theorem Ico_filter_lt (a b c : α) : ((Ico a b).filter fun x => x < c) = Ico a (min b c) 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.filter (fun x => x < c) (Multiset.Ico a b) = Multiset.Ico a (min b c)
-- Multiset.Ico_filter_le
@[simp]
theorem Ico_filter_le (a b c : α) : ((Ico a b).filter fun x => c ≤ x) = Ico (max a c) b 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.filter (fun x => c ≤ x) (Multiset.Ico a b) = Multiset.Ico (max a c) b
-- Multiset.Ico_sub_Ico_left
@[simp]
theorem Ico_sub_Ico_left (a b c : α) : Ico a b - Ico a c = Ico (max a c) b 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.Ico a b - Multiset.Ico a c = Multiset.Ico (max a c) b
-- Multiset.Ico_sub_Ico_right
@[simp]
theorem Ico_sub_Ico_right (a b c : α) : Ico a b - Ico c b = Ico a (min b c) 
Arguments: {α : Type u_1} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Multiset.Ico a b - Multiset.Ico c b = Multiset.Ico a (min b c)
-- _private.Mathlib.Order.Interval.Finset.Box.0.Finset.Icc_neg_mono
private lemma Icc_neg_mono : Monotone fun n : ℕ ↦ Icc (-n : α) n 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α]
Goal: Monotone fun n => Finset.Icc (-↑n) ↑n
-- Finset.box_zero
@[simp] lemma box_zero : (box 0 : Finset α) = {0} 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α]
Goal: Finset.box 0 = {0}
-- Finset.box_succ_eq_sdiff
lemma box_succ_eq_sdiff (n : ℕ) :
    box (n + 1) = Icc (-n.succ : α) n.succ \ Icc (-n) n 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] (n : ℕ)
Goal: Finset.box (n + 1) = Finset.Icc (-↑n.succ) ↑n.succ \ Finset.Icc (-↑n) ↑n
-- Finset.disjoint_box_succ_prod
lemma disjoint_box_succ_prod (n : ℕ) : Disjoint (box (n + 1)) (Icc (-n : α) n) 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] (n : ℕ)
Goal: Disjoint (Finset.box (n + 1)) (Finset.Icc (-↑n) ↑n)
-- Finset.box_succ_union_prod
@[simp] lemma box_succ_union_prod (n : ℕ) :
    box (n + 1) ∪ Icc (-n : α) n = Icc (-n.succ : α) n.succ 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] (n : ℕ)
Goal: Finset.box (n + 1) ∪ Finset.Icc (-↑n) ↑n = Finset.Icc (-↑n.succ) ↑n.succ
-- Finset.box_succ_disjUnion
lemma box_succ_disjUnion (n : ℕ) :
    (box (n + 1)).disjUnion (Icc (-n : α) n) (disjoint_box_succ_prod _) =
      Icc (-n.succ : α) n.succ 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] (n : ℕ)
Goal: (Finset.box (n + 1)).disjUnion (Finset.Icc (-↑n) ↑n) ⋯ = Finset.Icc (-↑n.succ) ↑n.succ
-- Finset.zero_mem_box
@[simp] lemma zero_mem_box : (0 : α) ∈ box n ↔ n = 0 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] {n : ℕ}
Goal: 0 ∈ Finset.box n ↔ n = 0
-- Finset.eq_zero_iff_eq_zero_of_mem_box
lemma eq_zero_iff_eq_zero_of_mem_box  {x : α} (hx : x ∈ box n) : x = 0 ↔ n = 0 
Arguments: {α : Type u_1} [OrderedRing α] [LocallyFiniteOrder α] [DecidableEq α] {n : ℕ} {x : α} (hx : x ∈ Finset.box n)
Goal: x = 0 ↔ n = 0
-- Prod.card_box_succ
@[simp] lemma card_box_succ (n : ℕ) :
    (box (n + 1) : Finset (α × β)).card =
      (Icc (-n.succ : α) n.succ).card * (Icc (-n.succ : β) n.succ).card -
        (Icc (-n : α) n).card * (Icc (-n : β) n).card 
Arguments: {α : Type u_1} {β : Type u_2} [OrderedRing α] [OrderedRing β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableEq α] [DecidableEq β] [DecidableRel fun x x_1 => x ≤ x_1] (n : ℕ)
Goal: (Finset.box (n + 1)).card =
  (Finset.Icc (-↑n.succ) ↑n.succ).card * (Finset.Icc (-↑n.succ) ↑n.succ).card -
    (Finset.Icc (-↑n) ↑n).card * (Finset.Icc (-↑n) ↑n).card
-- Int.card_box
lemma card_box : ∀ {n}, n ≠ 0 → (box n : Finset (ℤ × ℤ)).card = 8 * n
  
Arguments: {n : ℕ}
Goal: n ≠ 0 → (Finset.box n).card = 8 * n
-- Int.mem_box
@[simp] lemma mem_box : ∀ {n}, x ∈ box n ↔ max x.1.natAbs x.2.natAbs = n
  
Arguments: {x : ℤ × ℤ} {n : ℕ}
Goal: x ∈ Finset.box n ↔ max x.1.natAbs x.2.natAbs = n
-- Int.existsUnique_mem_box
lemma existsUnique_mem_box (x : ℤ × ℤ) : ∃! n : ℕ, x ∈ box n 
Arguments: (x : ℤ × ℤ)
Goal: ∃! n, x ∈ Finset.box n
-- strictMono_of_odd_strictMonoOn_nonneg
theorem strictMono_of_odd_strictMonoOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : StrictMonoOn f (Ici 0)) : StrictMono f 
Arguments: {G : Type u_1} {H : Type u_2} [LinearOrderedAddCommGroup G] [OrderedAddCommGroup H] {f : G → H} (h₁ : ∀ (x : G), f (-x) = -f x) (h₂ : StrictMonoOn f (Set.Ici 0))
Goal: StrictMono f
-- strictAnti_of_odd_strictAntiOn_nonneg
theorem strictAnti_of_odd_strictAntiOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : StrictAntiOn f (Ici 0)) : StrictAnti f 
Arguments: {G : Type u_1} {H : Type u_2} [LinearOrderedAddCommGroup G] [OrderedAddCommGroup H] {f : G → H} (h₁ : ∀ (x : G), f (-x) = -f x) (h₂ : StrictAntiOn f (Set.Ici 0))
Goal: StrictAnti f
-- monotone_of_odd_of_monotoneOn_nonneg
theorem monotone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : MonotoneOn f (Ici 0)) : Monotone f 
Arguments: {G : Type u_1} {H : Type u_2} [LinearOrderedAddCommGroup G] [OrderedAddCommGroup H] {f : G → H} (h₁ : ∀ (x : G), f (-x) = -f x) (h₂ : MonotoneOn f (Set.Ici 0))
Goal: Monotone f
-- antitone_of_odd_of_monotoneOn_nonneg
theorem antitone_of_odd_of_monotoneOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
    (h₂ : AntitoneOn f (Ici 0)) : Antitone f 
Arguments: {G : Type u_1} {H : Type u_2} [LinearOrderedAddCommGroup G] [OrderedAddCommGroup H] {f : G → H} (h₁ : ∀ (x : G), f (-x) = -f x) (h₂ : AntitoneOn f (Set.Ici 0))
Goal: Antitone f
-- Symmetric.compl
protected theorem Symmetric.compl (h : Symmetric r) : Symmetric rᶜ 
Arguments: {α : Type u_1} {r : α → α → Prop} (h : Symmetric r)
Goal: Symmetric rᶜ
-- IsAntichain.subset
protected theorem subset (hs : IsAntichain r s) (h : t ⊆ s) : IsAntichain r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (hs : IsAntichain r s) (h : t ⊆ s)
Goal: IsAntichain r t
-- IsAntichain.mono
theorem mono (hs : IsAntichain r₁ s) (h : r₂ ≤ r₁) : IsAntichain r₂ s 
Arguments: {α : Type u_1} {r₁ r₂ : α → α → Prop} {s : Set α} (hs : IsAntichain r₁ s) (h : r₂ ≤ r₁)
Goal: IsAntichain r₂ s
-- IsAntichain.mono_on
theorem mono_on (hs : IsAntichain r₁ s) (h : s.Pairwise fun ⦃a b⦄ => r₂ a b → r₁ a b) :
    IsAntichain r₂ s 
Arguments: {α : Type u_1} {r₁ r₂ : α → α → Prop} {s : Set α} (hs : IsAntichain r₁ s) (h : s.Pairwise fun ⦃a b⦄ => r₂ a b → r₁ a b)
Goal: IsAntichain r₂ s
-- IsAntichain.eq
protected theorem eq (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r a b) :
    a = b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r a b)
Goal: a = b
-- IsAntichain.eq'
protected theorem eq' (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r b a) :
    a = b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r b a)
Goal: a = b
-- IsAntichain.isAntisymm
protected theorem isAntisymm (h : IsAntichain r univ) : IsAntisymm α r 
Arguments: {α : Type u_1} {r : α → α → Prop} (h : IsAntichain r Set.univ)
Goal: IsAntisymm α r
-- IsAntichain.subsingleton
protected theorem subsingleton [IsTrichotomous α r] (h : IsAntichain r s) : s.Subsingleton 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsTrichotomous α r] (h : IsAntichain r s)
Goal: s.Subsingleton
-- IsAntichain.flip
protected theorem flip (hs : IsAntichain r s) : IsAntichain (flip r) s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (hs : IsAntichain r s)
Goal: IsAntichain (flip r) s
-- IsAntichain.swap
theorem swap (hs : IsAntichain r s) : IsAntichain (swap r) s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (hs : IsAntichain r s)
Goal: IsAntichain (Function.swap r) s
-- IsAntichain.image
theorem image (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b⦄, r' (f a) (f b) → r a b) :
    IsAntichain r' (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b : α⦄, r' (f a) (f b) → r a b)
Goal: IsAntichain r' (f '' s)
-- IsAntichain.preimage
theorem preimage (hs : IsAntichain r s) {f : β → α} (hf : Injective f)
    (h : ∀ ⦃a b⦄, r' a b → r (f a) (f b)) : IsAntichain r' (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} (hs : IsAntichain r s) {f : β → α} (hf : Function.Injective f) (h : ∀ ⦃a b : β⦄, r' a b → r (f a) (f b))
Goal: IsAntichain r' (f ⁻¹' s)
-- isAntichain_insert
theorem _root_.isAntichain_insert :
    IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b ∧ ¬r b a 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α}
Goal: IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ¬r a b ∧ ¬r b a
-- IsAntichain.insert
protected theorem insert (hs : IsAntichain r s) (hl : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r b a)
    (hr : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b) : IsAntichain r (insert a s) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α} (hs : IsAntichain r s) (hl : ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ¬r b a) (hr : ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ¬r a b)
Goal: IsAntichain r (insert a s)
-- isAntichain_insert_of_symmetric
theorem _root_.isAntichain_insert_of_symmetric (hr : Symmetric r) :
    IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α} (hr : Symmetric r)
Goal: IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ¬r a b
-- IsAntichain.insert_of_symmetric
theorem insert_of_symmetric (hs : IsAntichain r s) (hr : Symmetric r)
    (h : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b) : IsAntichain r (insert a s) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α} (hs : IsAntichain r s) (hr : Symmetric r) (h : ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ¬r a b)
Goal: IsAntichain r (insert a s)
-- IsAntichain.image_relEmbedding
theorem image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} (hs : IsAntichain r s) (φ : r ↪r r')
Goal: IsAntichain r' (⇑φ '' s)
-- IsAntichain.preimage_relEmbedding
theorem preimage_relEmbedding {t : Set β} (ht : IsAntichain r' t) (φ : r ↪r r') :
    IsAntichain r (φ ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {t : Set β} (ht : IsAntichain r' t) (φ : r ↪r r')
Goal: IsAntichain r (⇑φ ⁻¹' t)
-- IsAntichain.image_relIso
theorem image_relIso (hs : IsAntichain r s) (φ : r ≃r r') : IsAntichain r' (φ '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} (hs : IsAntichain r s) (φ : r ≃r r')
Goal: IsAntichain r' (⇑φ '' s)
-- IsAntichain.preimage_relIso
theorem preimage_relIso {t : Set β} (hs : IsAntichain r' t) (φ : r ≃r r') :
    IsAntichain r (φ ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {t : Set β} (hs : IsAntichain r' t) (φ : r ≃r r')
Goal: IsAntichain r (⇑φ ⁻¹' t)
-- IsAntichain.image_relEmbedding_iff
theorem image_relEmbedding_iff {φ : r ↪r r'} : IsAntichain r' (φ '' s) ↔ IsAntichain r s 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} {φ : r ↪r r'}
Goal: IsAntichain r' (⇑φ '' s) ↔ IsAntichain r s
-- IsAntichain.image_relIso_iff
theorem image_relIso_iff {φ : r ≃r r'} : IsAntichain r' (φ '' s) ↔ IsAntichain r s 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} {φ : r ≃r r'}
Goal: IsAntichain r' (⇑φ '' s) ↔ IsAntichain r s
-- IsAntichain.image_embedding
theorem image_embedding [LE α] [LE β] (hs : IsAntichain (· ≤ ·) s) (φ : α ↪o β) :
    IsAntichain (· ≤ ·) (φ '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} [LE α] [LE β] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s) (φ : α ↪o β)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ '' s)
-- IsAntichain.preimage_embedding
theorem preimage_embedding [LE α] [LE β] {t : Set β} (ht : IsAntichain (· ≤ ·) t) (φ : α ↪o β) :
    IsAntichain (· ≤ ·) (φ ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} [LE α] [LE β] {t : Set β} (ht : IsAntichain (fun x x_1 => x ≤ x_1) t) (φ : α ↪o β)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ ⁻¹' t)
-- IsAntichain.image_embedding_iff
theorem image_embedding_iff [LE α] [LE β] {φ : α ↪o β} :
    IsAntichain (· ≤ ·) (φ '' s) ↔ IsAntichain (· ≤ ·) s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} [LE α] [LE β] {φ : α ↪o β}
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ '' s) ↔ IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsAntichain.image_iso
theorem image_iso [LE α] [LE β] (hs : IsAntichain (· ≤ ·) s) (φ : α ≃o β) :
    IsAntichain (· ≤ ·) (φ '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} [LE α] [LE β] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s) (φ : α ≃o β)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ '' s)
-- IsAntichain.image_iso_iff
theorem image_iso_iff [LE α] [LE β] {φ : α ≃o β} :
    IsAntichain (· ≤ ·) (φ '' s) ↔ IsAntichain (· ≤ ·) s 
Arguments: {α : Type u_1} {β : Type u_2} {s : Set α} [LE α] [LE β] {φ : α ≃o β}
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ '' s) ↔ IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsAntichain.preimage_iso
theorem preimage_iso [LE α] [LE β] {t : Set β} (ht : IsAntichain (· ≤ ·) t) (φ : α ≃o β) :
    IsAntichain (· ≤ ·) (φ ⁻¹' t) 
Arguments: {α : Type u_1} {β : Type u_2} [LE α] [LE β] {t : Set β} (ht : IsAntichain (fun x x_1 => x ≤ x_1) t) (φ : α ≃o β)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ ⁻¹' t)
-- IsAntichain.preimage_iso_iff
theorem preimage_iso_iff [LE α] [LE β] {t : Set β} {φ : α ≃o β} :
    IsAntichain (· ≤ ·) (φ ⁻¹' t) ↔ IsAntichain (· ≤ ·) t 
Arguments: {α : Type u_1} {β : Type u_2} [LE α] [LE β] {t : Set β} {φ : α ≃o β}
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (⇑φ ⁻¹' t) ↔ IsAntichain (fun x x_1 => x ≤ x_1) t