ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Order.gc_pred_succ
lemma gc_pred_succ : GaloisConnection (pred : α → α) succ 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [PredOrder α]
Goal: GaloisConnection Order.pred Order.succ
-- Order.succ_pred_of_not_isMin
@[simp]
theorem succ_pred_of_not_isMin (h : ¬IsMin a) : succ (pred a) = a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] {a : α} (h : ¬IsMin a)
Goal: Order.succ (Order.pred a) = a
-- Order.pred_succ_of_not_isMax
@[simp]
theorem pred_succ_of_not_isMax (h : ¬IsMax a) : pred (succ a) = a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] {a : α} (h : ¬IsMax a)
Goal: Order.pred (Order.succ a) = a
-- Order.succ_pred
theorem succ_pred [NoMinOrder α] (a : α) : succ (pred a) = a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Order.succ (Order.pred a) = a
-- Order.pred_succ
theorem pred_succ [NoMaxOrder α] (a : α) : pred (succ a) = a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] [NoMaxOrder α] (a : α)
Goal: Order.pred (Order.succ a) = a
-- Order.pred_succ_iterate_of_not_isMax
theorem pred_succ_iterate_of_not_isMax (i : α) (n : ℕ) (hin : ¬IsMax (succ^[n - 1] i)) :
    pred^[n] (succ^[n] i) = i 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] (i : α) (n : ℕ) (hin : ¬IsMax (Order.succ^[n - 1] i))
Goal: Order.pred^[n] (Order.succ^[n] i) = i
-- Order.succ_pred_iterate_of_not_isMin
theorem succ_pred_iterate_of_not_isMin (i : α) (n : ℕ) (hin : ¬IsMin (pred^[n - 1] i)) :
    succ^[n] (pred^[n] i) = i 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [PredOrder α] (i : α) (n : ℕ) (hin : ¬IsMin (Order.pred^[n - 1] i))
Goal: Order.succ^[n] (Order.pred^[n] i) = i
-- WithTop.succ_coe_top
@[simp]
theorem succ_coe_top : succ ↑(⊤ : α) = (⊤ : WithTop α) 
Arguments: {α : Type u_1} [DecidableEq α] [PartialOrder α] [OrderTop α] [SuccOrder α]
Goal: Order.succ ↑⊤ = ⊤
-- WithTop.succ_coe_of_ne_top
theorem succ_coe_of_ne_top {a : α} (h : a ≠ ⊤) : succ (↑a : WithTop α) = ↑(succ a) 
Arguments: {α : Type u_1} [DecidableEq α] [PartialOrder α] [OrderTop α] [SuccOrder α] {a : α} (h : a ≠ ⊤)
Goal: Order.succ ↑a = ↑(Order.succ a)
-- WithTop.pred_top
@[simp]
theorem pred_top : pred (⊤ : WithTop α) = ↑(⊤ : α) 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] [PredOrder α]
Goal: Order.pred ⊤ = ↑⊤
-- WithTop.pred_coe
@[simp]
theorem pred_coe (a : α) : pred (↑a : WithTop α) = ↑(pred a) 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] [PredOrder α] (a : α)
Goal: Order.pred ↑a = ↑(Order.pred a)
-- WithTop.pred_untop
@[simp]
theorem pred_untop :
    ∀ (a : WithTop α) (ha : a ≠ ⊤),
      pred (a.untop ha) = (pred a).untop (by induction a <;> simp)
  
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] [PredOrder α] (a : WithTop α) (ha : a ≠ ⊤)
Goal: Order.pred (a.untop ha) = (Order.pred a).untop ⋯
-- WithTop.succ_coe
@[simp]
theorem succ_coe (a : α) : succ (↑a : WithTop α) = ↑(succ a) 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] [SuccOrder α] (a : α)
Goal: Order.succ ↑a = ↑(Order.succ a)
-- WithBot.succ_bot
@[simp]
theorem succ_bot : succ (⊥ : WithBot α) = ↑(⊥ : α) 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [SuccOrder α]
Goal: Order.succ ⊥ = ↑⊥
-- WithBot.succ_coe
@[simp]
theorem succ_coe (a : α) : succ (↑a : WithBot α) = ↑(succ a) 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [SuccOrder α] (a : α)
Goal: Order.succ ↑a = ↑(Order.succ a)
-- WithBot.succ_unbot
@[simp]
theorem succ_unbot :
    ∀ (a : WithBot α) (ha : a ≠ ⊥),
      succ (a.unbot ha) = (succ a).unbot (by induction a <;> simp)
  
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [SuccOrder α] (a : WithBot α) (ha : a ≠ ⊥)
Goal: Order.succ (a.unbot ha) = (Order.succ a).unbot ⋯
-- WithBot.pred_coe_bot
@[simp]
theorem pred_coe_bot : pred ↑(⊥ : α) = (⊥ : WithBot α) 
Arguments: {α : Type u_1} [DecidableEq α] [PartialOrder α] [OrderBot α] [PredOrder α]
Goal: Order.pred ↑⊥ = ⊥
-- WithBot.pred_coe_of_ne_bot
theorem pred_coe_of_ne_bot {a : α} (h : a ≠ ⊥) : pred (↑a : WithBot α) = ↑(pred a) 
Arguments: {α : Type u_1} [DecidableEq α] [PartialOrder α] [OrderBot α] [PredOrder α] {a : α} (h : a ≠ ⊥)
Goal: Order.pred ↑a = ↑(Order.pred a)
-- WithBot.pred_coe
@[simp]
theorem pred_coe (a : α) : pred (↑a : WithBot α) = ↑(pred a) 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] [PredOrder α] (a : α)
Goal: Order.pred ↑a = ↑(Order.pred a)
-- IsSuccArchimedean.exists_succ_iterate_of_le
class IsSuccArchimedean (α : Type*) [Preorder α] [SuccOrder α] : Prop where
    exists_succ_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, succ^[n] a = b
Arguments: {α : Type u_3} [Preorder α] [SuccOrder α] [self : IsSuccArchimedean α] {a b : α} (h : a ≤ b)
Goal: ∃ n, Order.succ^[n] a = b
-- IsPredArchimedean.exists_pred_iterate_of_le
class IsPredArchimedean (α : Type*) [Preorder α] [PredOrder α] : Prop where
    exists_pred_iterate_of_le {a b : α} (h : a ≤ b) : ∃ n, pred^[n] b = a
Arguments: {α : Type u_3} [Preorder α] [PredOrder α] [self : IsPredArchimedean α] {a b : α} (h : a ≤ b)
Goal: ∃ n, Order.pred^[n] b = a
-- LE.le.exists_succ_iterate
theorem LE.le.exists_succ_iterate (h : a ≤ b) : ∃ n, succ^[n] a = b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {a b : α} (h : a ≤ b)
Goal: ∃ n, Order.succ^[n] a = b
-- exists_succ_iterate_iff_le
theorem exists_succ_iterate_iff_le : (∃ n, succ^[n] a = b) ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {a b : α}
Goal: (∃ n, Order.succ^[n] a = b) ↔ a ≤ b
-- Succ.rec
@[elab_as_elim]
theorem Succ.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, m ≤ n → P n → P (succ n)) ⦃n : α⦄
    (hmn : m ≤ n) : P n 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ (n : α), m ≤ n → P n → P (Order.succ n)) ⦃n : α⦄ (hmn : m ≤ n)
Goal: P n
-- Succ.rec_iff
theorem Succ.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) {a b : α} (h : a ≤ b) :
    p a ↔ p b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [IsSuccArchimedean α] {p : α → Prop} (hsucc : ∀ (a : α), p a ↔ p (Order.succ a)) {a b : α} (h : a ≤ b)
Goal: p a ↔ p b
-- LE.le.exists_pred_iterate
theorem LE.le.exists_pred_iterate (h : a ≤ b) : ∃ n, pred^[n] b = a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {a b : α} (h : a ≤ b)
Goal: ∃ n, Order.pred^[n] b = a
-- exists_pred_iterate_iff_le
theorem exists_pred_iterate_iff_le : (∃ n, pred^[n] b = a) ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {a b : α}
Goal: (∃ n, Order.pred^[n] b = a) ↔ a ≤ b
-- Pred.rec
@[elab_as_elim]
theorem Pred.rec {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n, n ≤ m → P n → P (pred n)) ⦃n : α⦄
    (hmn : n ≤ m) : P n 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {P : α → Prop} {m : α} (h0 : P m) (h1 : ∀ n ≤ m, P n → P (Order.pred n)) ⦃n : α⦄ (hmn : n ≤ m)
Goal: P n
-- Pred.rec_iff
theorem Pred.rec_iff {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) {a b : α} (h : a ≤ b) :
    p a ↔ p b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [IsPredArchimedean α] {p : α → Prop} (hsucc : ∀ (a : α), p a ↔ p (Order.pred a)) {a b : α} (h : a ≤ b)
Goal: p a ↔ p b
-- succ_max
lemma succ_max (a b : α) : succ (max a b) = max (succ a) (succ b) 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] (a b : α)
Goal: Order.succ (max a b) = max (Order.succ a) (Order.succ b)
-- succ_min
lemma succ_min (a b : α) : succ (min a b) = min (succ a) (succ b) 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] (a b : α)
Goal: Order.succ (min a b) = min (Order.succ a) (Order.succ b)
-- exists_succ_iterate_or
theorem exists_succ_iterate_or : (∃ n, succ^[n] a = b) ∨ ∃ n, succ^[n] b = a 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] {a b : α}
Goal: (∃ n, Order.succ^[n] a = b) ∨ ∃ n, Order.succ^[n] b = a
-- Succ.rec_linear
theorem Succ.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (succ a)) (a b : α) : p a ↔ p b 
Arguments: {α : Type u_1} [LinearOrder α] [SuccOrder α] [IsSuccArchimedean α] {p : α → Prop} (hsucc : ∀ (a : α), p a ↔ p (Order.succ a)) (a b : α)
Goal: p a ↔ p b
-- pred_max
lemma pred_max (a b : α) : pred (max a b) = max (pred a) (pred b) 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] (a b : α)
Goal: Order.pred (max a b) = max (Order.pred a) (Order.pred b)
-- pred_min
lemma pred_min (a b : α) : pred (min a b) = min (pred a) (pred b) 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] (a b : α)
Goal: Order.pred (min a b) = min (Order.pred a) (Order.pred b)
-- exists_pred_iterate_or
theorem exists_pred_iterate_or : (∃ n, pred^[n] b = a) ∨ ∃ n, pred^[n] a = b 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] {a b : α}
Goal: (∃ n, Order.pred^[n] b = a) ∨ ∃ n, Order.pred^[n] a = b
-- Pred.rec_linear
theorem Pred.rec_linear {p : α → Prop} (hsucc : ∀ a, p a ↔ p (pred a)) (a b : α) : p a ↔ p b 
Arguments: {α : Type u_1} [LinearOrder α] [PredOrder α] [IsPredArchimedean α] {p : α → Prop} (hsucc : ∀ (a : α), p a ↔ p (Order.pred a)) (a b : α)
Goal: p a ↔ p b
-- StrictMono.not_bddAbove_range
lemma StrictMono.not_bddAbove_range [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β]
    (hf : StrictMono f) : ¬ BddAbove (Set.range f) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : α → β} [NoMaxOrder α] [SuccOrder β] [IsSuccArchimedean β] (hf : StrictMono f)
Goal: ¬BddAbove (Set.range f)
-- StrictMono.not_bddBelow_range
lemma StrictMono.not_bddBelow_range [NoMinOrder α] [PredOrder β] [IsPredArchimedean β]
    (hf : StrictMono f) : ¬ BddBelow (Set.range f) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : α → β} [NoMinOrder α] [PredOrder β] [IsPredArchimedean β] (hf : StrictMono f)
Goal: ¬BddBelow (Set.range f)
-- StrictAnti.not_bddAbove_range
lemma StrictAnti.not_bddAbove_range [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β]
    (hf : StrictAnti f) : ¬ BddAbove (Set.range f) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : α → β} [NoMinOrder α] [SuccOrder β] [IsSuccArchimedean β] (hf : StrictAnti f)
Goal: ¬BddAbove (Set.range f)
-- StrictAnti.not_bddBelow_range
lemma StrictAnti.not_bddBelow_range [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β]
    (hf : StrictAnti f) : ¬ BddBelow (Set.range f) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Nonempty α] [Preorder β] {f : α → β} [NoMaxOrder α] [PredOrder β] [IsPredArchimedean β] (hf : StrictAnti f)
Goal: ¬BddBelow (Set.range f)
-- Succ.rec_bot
theorem Succ.rec_bot (p : α → Prop) (hbot : p ⊥) (hsucc : ∀ a, p a → p (succ a)) (a : α) : p a 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α] (p : α → Prop) (hbot : p ⊥) (hsucc : ∀ (a : α), p a → p (Order.succ a)) (a : α)
Goal: p a
-- Pred.rec_top
theorem Pred.rec_top (p : α → Prop) (htop : p ⊤) (hpred : ∀ a, p a → p (pred a)) (a : α) : p a 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] [PredOrder α] [IsPredArchimedean α] (p : α → Prop) (htop : p ⊤) (hpred : ∀ (a : α), p a → p (Order.pred a)) (a : α)
Goal: p a
-- SuccOrder.forall_ne_bot_iff
lemma SuccOrder.forall_ne_bot_iff
    [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α]
    (P : α → Prop) :
    (∀ i, i ≠ ⊥ → P i) ↔ (∀ i, P (SuccOrder.succ i)) 
Arguments: {α : Type u_1} [Nontrivial α] [PartialOrder α] [OrderBot α] [SuccOrder α] [IsSuccArchimedean α] (P : α → Prop)
Goal: (∀ (i : α), i ≠ ⊥ → P i) ↔ ∀ (i : α), P (SuccOrder.succ i)
-- Set.OrdConnected.out'
class OrdConnected (s : Set α) : Prop where
    out' ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s) : Icc x y ⊆ s
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [self : s.OrdConnected] ⦃x : α⦄ (hx : x ∈ s) ⦃y : α⦄ (hy : y ∈ s)
Goal: Set.Icc x y ⊆ s
-- Set.OrdConnected.out
theorem OrdConnected.out (h : OrdConnected s) : ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), Icc x y ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (h : s.OrdConnected) ⦃x : α⦄
Goal: x ∈ s → ∀ ⦃y : α⦄, y ∈ s → Set.Icc x y ⊆ s
-- Set.ordConnected_def
theorem ordConnected_def : OrdConnected s ↔ ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), Icc x y ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: s.OrdConnected ↔ ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → Set.Icc x y ⊆ s
-- Set.ordConnected_iff
theorem ordConnected_iff : OrdConnected s ↔ ∀ x ∈ s, ∀ y ∈ s, x ≤ y → Icc x y ⊆ s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: s.OrdConnected ↔ ∀ x ∈ s, ∀ y ∈ s, x ≤ y → Set.Icc x y ⊆ s
-- Set.ordConnected_of_Ioo
theorem ordConnected_of_Ioo {α : Type*} [PartialOrder α] {s : Set α}
    (hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Ioo x y ⊆ s) : OrdConnected s 
Arguments: {α : Type u_3} [PartialOrder α] {s : Set α} (hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Set.Ioo x y ⊆ s)
Goal: s.OrdConnected
-- Set.OrdConnected.preimage_mono
theorem OrdConnected.preimage_mono {f : β → α} (hs : OrdConnected s) (hf : Monotone f) :
    OrdConnected (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} {f : β → α} (hs : s.OrdConnected) (hf : Monotone f)
Goal: (f ⁻¹' s).OrdConnected
-- Set.OrdConnected.preimage_anti
theorem OrdConnected.preimage_anti {f : β → α} (hs : OrdConnected s) (hf : Antitone f) :
    OrdConnected (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : Set α} {f : β → α} (hs : s.OrdConnected) (hf : Antitone f)
Goal: (f ⁻¹' s).OrdConnected
-- Set.Icc_subset
protected theorem Icc_subset (s : Set α) [hs : OrdConnected s] {x y} (hx : x ∈ s) (hy : y ∈ s) :
    Icc x y ⊆ s 
Arguments: {α : Type u_1} [Preorder α] (s : Set α) [hs : s.OrdConnected] {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: Set.Icc x y ⊆ s
-- OrderEmbedding.image_Icc
theorem image_Icc (e : α ↪o β) (he : OrdConnected (range e)) (x y : α) :
    e '' Icc x y = Icc (e x) (e y) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range ⇑e).OrdConnected) (x y : α)
Goal: ⇑e '' Set.Icc x y = Set.Icc (e x) (e y)
-- OrderEmbedding.image_Ico
theorem image_Ico (e : α ↪o β) (he : OrdConnected (range e)) (x y : α) :
    e '' Ico x y = Ico (e x) (e y) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range ⇑e).OrdConnected) (x y : α)
Goal: ⇑e '' Set.Ico x y = Set.Ico (e x) (e y)
-- OrderEmbedding.image_Ioc
theorem image_Ioc (e : α ↪o β) (he : OrdConnected (range e)) (x y : α) :
    e '' Ioc x y = Ioc (e x) (e y) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range ⇑e).OrdConnected) (x y : α)
Goal: ⇑e '' Set.Ioc x y = Set.Ioc (e x) (e y)
-- OrderEmbedding.image_Ioo
theorem image_Ioo (e : α ↪o β) (he : OrdConnected (range e)) (x y : α) :
    e '' Ioo x y = Ioo (e x) (e y) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (he : (Set.range ⇑e).OrdConnected) (x y : α)
Goal: ⇑e '' Set.Ioo x y = Set.Ioo (e x) (e y)
-- Set.image_subtype_val_Icc
@[simp]
lemma image_subtype_val_Icc {s : Set α} [OrdConnected s] (x y : s) :
    Subtype.val '' Icc x y = Icc x.1 y 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x y : ↑s)
Goal: Subtype.val '' Set.Icc x y = Set.Icc ↑x ↑y
-- Set.image_subtype_val_Ico
@[simp]
lemma image_subtype_val_Ico {s : Set α} [OrdConnected s] (x y : s) :
    Subtype.val '' Ico x y = Ico x.1 y 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x y : ↑s)
Goal: Subtype.val '' Set.Ico x y = Set.Ico ↑x ↑y
-- Set.image_subtype_val_Ioc
@[simp]
lemma image_subtype_val_Ioc {s : Set α} [OrdConnected s] (x y : s) :
    Subtype.val '' Ioc x y = Ioc x.1 y 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x y : ↑s)
Goal: Subtype.val '' Set.Ioc x y = Set.Ioc ↑x ↑y
-- Set.image_subtype_val_Ioo
@[simp]
lemma image_subtype_val_Ioo {s : Set α} [OrdConnected s] (x y : s) :
    Subtype.val '' Ioo x y = Ioo x.1 y 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected] (x y : ↑s)
Goal: Subtype.val '' Set.Ioo x y = Set.Ioo ↑x ↑y
-- Set.OrdConnected.inter
theorem OrdConnected.inter {s t : Set α} (hs : OrdConnected s) (ht : OrdConnected t) :
    OrdConnected (s ∩ t) 
Arguments: {α : Type u_1} [Preorder α] {s t : Set α} (hs : s.OrdConnected) (ht : t.OrdConnected)
Goal: (s ∩ t).OrdConnected
-- Set.OrdConnected.dual
theorem OrdConnected.dual {s : Set α} (hs : OrdConnected s) :
    OrdConnected (OrderDual.ofDual ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} (hs : s.OrdConnected)
Goal: (⇑OrderDual.ofDual ⁻¹' s).OrdConnected
-- Set.ordConnected_dual
theorem ordConnected_dual {s : Set α} : OrdConnected (OrderDual.ofDual ⁻¹' s) ↔ OrdConnected s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: (⇑OrderDual.ofDual ⁻¹' s).OrdConnected ↔ s.OrdConnected
-- Set.ordConnected_sInter
theorem ordConnected_sInter {S : Set (Set α)} (hS : ∀ s ∈ S, OrdConnected s) :
    OrdConnected (⋂₀ S) 
Arguments: {α : Type u_1} [Preorder α] {S : Set (Set α)} (hS : ∀ s ∈ S, s.OrdConnected)
Goal: (⋂₀ S).OrdConnected
-- Set.ordConnected_iInter
theorem ordConnected_iInter {ι : Sort*} {s : ι → Set α} (hs : ∀ i, OrdConnected (s i)) :
    OrdConnected (⋂ i, s i) 
Arguments: {α : Type u_1} [Preorder α] {ι : Sort u_3} {s : ι → Set α} (hs : ∀ (i : ι), (s i).OrdConnected)
Goal: (⋂ i, s i).OrdConnected
-- Set.ordConnected_biInter
theorem ordConnected_biInter {ι : Sort*} {p : ι → Prop} {s : ∀ i, p i → Set α}
    (hs : ∀ i hi, OrdConnected (s i hi)) : OrdConnected (⋂ (i) (hi), s i hi) 
Arguments: {α : Type u_1} [Preorder α] {ι : Sort u_3} {p : ι → Prop} {s : (i : ι) → p i → Set α} (hs : ∀ (i : ι) (hi : p i), (s i hi).OrdConnected)
Goal: (⋂ i, ⋂ (hi : p i), s i hi).OrdConnected
-- Set.ordConnected_pi
theorem ordConnected_pi {ι : Type*} {α : ι → Type*} [∀ i, Preorder (α i)] {s : Set ι}
    {t : ∀ i, Set (α i)} (h : ∀ i ∈ s, OrdConnected (t i)) : OrdConnected (s.pi t) 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Preorder (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} (h : ∀ i ∈ s, (t i).OrdConnected)
Goal: (s.pi t).OrdConnected
-- Set.ordConnected_Ici
@[instance]
theorem ordConnected_Ici {a : α} : OrdConnected (Ici a) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Ici a).OrdConnected
-- Set.ordConnected_Iic
@[instance]
theorem ordConnected_Iic {a : α} : OrdConnected (Iic a) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Iic a).OrdConnected
-- Set.ordConnected_Ioi
@[instance]
theorem ordConnected_Ioi {a : α} : OrdConnected (Ioi a) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Ioi a).OrdConnected
-- Set.ordConnected_Iio
@[instance]
theorem ordConnected_Iio {a : α} : OrdConnected (Iio a) 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Iio a).OrdConnected
-- Set.ordConnected_Icc
@[instance]
theorem ordConnected_Icc {a b : α} : OrdConnected (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Icc a b).OrdConnected
-- Set.ordConnected_Ico
@[instance]
theorem ordConnected_Ico {a b : α} : OrdConnected (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Ico a b).OrdConnected
-- Set.ordConnected_Ioc
@[instance]
theorem ordConnected_Ioc {a b : α} : OrdConnected (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Ioc a b).OrdConnected
-- Set.ordConnected_Ioo
@[instance]
theorem ordConnected_Ioo {a b : α} : OrdConnected (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Ioo a b).OrdConnected
-- Set.ordConnected_singleton
@[instance]
theorem ordConnected_singleton {α : Type*} [PartialOrder α] {a : α} :
    OrdConnected ({a} : Set α) 
Arguments: {α : Type u_3} [PartialOrder α] {a : α}
Goal: {a}.OrdConnected
-- Set.ordConnected_empty
@[instance]
theorem ordConnected_empty : OrdConnected (∅ : Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: ∅.OrdConnected
-- Set.ordConnected_univ
@[instance]
theorem ordConnected_univ : OrdConnected (univ : Set α) 
Arguments: {α : Type u_1} [Preorder α]
Goal: Set.univ.OrdConnected
-- Set.ordConnected_preimage
@[instance]
theorem ordConnected_preimage {F : Type*} [FunLike F α β] [OrderHomClass F α β] (f : F)
    {s : Set β} [hs : OrdConnected s] : OrdConnected (f ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {F : Type u_3} [FunLike F α β] [OrderHomClass F α β] (f : F) {s : Set β} [hs : s.OrdConnected]
Goal: (⇑f ⁻¹' s).OrdConnected
-- Set.ordConnected_image
@[instance]
theorem ordConnected_image {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) {s : Set α}
    [hs : OrdConnected s] : OrdConnected (e '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {E : Type u_3} [EquivLike E α β] [OrderIsoClass E α β] (e : E) {s : Set α} [hs : s.OrdConnected]
Goal: (⇑e '' s).OrdConnected
-- Set.ordConnected_range
@[instance]
theorem ordConnected_range {E : Type*} [EquivLike E α β] [OrderIsoClass E α β] (e : E) :
    OrdConnected (range e) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {E : Type u_3} [EquivLike E α β] [OrderIsoClass E α β] (e : E)
Goal: (Set.range ⇑e).OrdConnected
-- Set.dual_ordConnected_iff
@[simp]
theorem dual_ordConnected_iff {s : Set α} : OrdConnected (ofDual ⁻¹' s) ↔ OrdConnected s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: (⇑OrderDual.ofDual ⁻¹' s).OrdConnected ↔ s.OrdConnected
-- Set.dual_ordConnected
@[instance]
theorem dual_ordConnected {s : Set α} [OrdConnected s] : OrdConnected (ofDual ⁻¹' s) 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} [s.OrdConnected]
Goal: (⇑OrderDual.ofDual ⁻¹' s).OrdConnected
-- IsAntichain.ordConnected
protected theorem _root_.IsAntichain.ordConnected (hs : IsAntichain (· ≤ ·) s) : s.OrdConnected 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α} (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: s.OrdConnected
-- Set.ordConnected_inter_Icc_of_subset
lemma ordConnected_inter_Icc_of_subset (h : Ioo x y ⊆ s) : OrdConnected (s ∩ Icc x y) 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α} {x y : α} (h : Set.Ioo x y ⊆ s)
Goal: (s ∩ Set.Icc x y).OrdConnected
-- Set.ordConnected_inter_Icc_iff
lemma ordConnected_inter_Icc_iff (hx : x ∈ s) (hy : y ∈ s) :
    OrdConnected (s ∩ Icc x y) ↔ Ioo x y ⊆ s 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: (s ∩ Set.Icc x y).OrdConnected ↔ Set.Ioo x y ⊆ s
-- Set.not_ordConnected_inter_Icc_iff
lemma not_ordConnected_inter_Icc_iff (hx : x ∈ s) (hy : y ∈ s) :
    ¬ OrdConnected (s ∩ Icc x y) ↔ ∃ z ∉ s, z ∈ Ioo x y 
Arguments: {α : Type u_1} [PartialOrder α] {s : Set α} {x y : α} (hx : x ∈ s) (hy : y ∈ s)
Goal: ¬(s ∩ Set.Icc x y).OrdConnected ↔ ∃ z ∉ s, z ∈ Set.Ioo x y
-- Set.ordConnected_uIcc
@[instance]
theorem ordConnected_uIcc {a b : α} : OrdConnected [[a, b]] 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: [[a, b]].OrdConnected
-- Set.ordConnected_uIoc
@[instance]
theorem ordConnected_uIoc {a b : α} : OrdConnected (Ι a b) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: (Ι a b).OrdConnected
-- Set.OrdConnected.uIcc_subset
theorem OrdConnected.uIcc_subset (hs : OrdConnected s) ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s) :
    [[x, y]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} (hs : s.OrdConnected) ⦃x : α⦄ (hx : x ∈ s) ⦃y : α⦄ (hy : y ∈ s)
Goal: [[x, y]] ⊆ s
-- Set.OrdConnected.uIoc_subset
theorem OrdConnected.uIoc_subset (hs : OrdConnected s) ⦃x⦄ (hx : x ∈ s) ⦃y⦄ (hy : y ∈ s) :
    Ι x y ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} (hs : s.OrdConnected) ⦃x : α⦄ (hx : x ∈ s) ⦃y : α⦄ (hy : y ∈ s)
Goal: Ι x y ⊆ s
-- Set.ordConnected_iff_uIcc_subset
theorem ordConnected_iff_uIcc_subset :
    OrdConnected s ↔ ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s), [[x, y]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α}
Goal: s.OrdConnected ↔ ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → [[x, y]] ⊆ s
-- Set.ordConnected_of_uIcc_subset_left
theorem ordConnected_of_uIcc_subset_left (h : ∀ y ∈ s, [[x, y]] ⊆ s) : OrdConnected s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (h : ∀ y ∈ s, [[x, y]] ⊆ s)
Goal: s.OrdConnected
-- Set.ordConnected_iff_uIcc_subset_left
theorem ordConnected_iff_uIcc_subset_left (hx : x ∈ s) :
    OrdConnected s ↔ ∀ ⦃y⦄, y ∈ s → [[x, y]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (hx : x ∈ s)
Goal: s.OrdConnected ↔ ∀ ⦃y : α⦄, y ∈ s → [[x, y]] ⊆ s