ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Order.isMax_iterate_succ_of_eq_of_ne
theorem isMax_iterate_succ_of_eq_of_ne {n m : ℕ} (h_eq : succ^[n] a = succ^[m] a)
    (h_ne : n ≠ m) : IsMax (succ^[n] a) 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α} {n m : ℕ} (h_eq : Order.succ^[n] a = Order.succ^[m] a) (h_ne : n ≠ m)
Goal: IsMax (Order.succ^[n] a)
-- Order.Iio_succ_of_not_isMax
theorem Iio_succ_of_not_isMax (ha : ¬IsMax a) : Iio (succ a) = Iic a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α} (ha : ¬IsMax a)
Goal: Set.Iio (Order.succ a) = Set.Iic a
-- Order.Ici_succ_of_not_isMax
theorem Ici_succ_of_not_isMax (ha : ¬IsMax a) : Ici (succ a) = Ioi a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α} (ha : ¬IsMax a)
Goal: Set.Ici (Order.succ a) = Set.Ioi a
-- Order.Ico_succ_right_of_not_isMax
theorem Ico_succ_right_of_not_isMax (hb : ¬IsMax b) : Ico a (succ b) = Icc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (hb : ¬IsMax b)
Goal: Set.Ico a (Order.succ b) = Set.Icc a b
-- Order.Ioo_succ_right_of_not_isMax
theorem Ioo_succ_right_of_not_isMax (hb : ¬IsMax b) : Ioo a (succ b) = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (hb : ¬IsMax b)
Goal: Set.Ioo a (Order.succ b) = Set.Ioc a b
-- Order.Icc_succ_left_of_not_isMax
theorem Icc_succ_left_of_not_isMax (ha : ¬IsMax a) : Icc (succ a) b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a)
Goal: Set.Icc (Order.succ a) b = Set.Ioc a b
-- Order.Ico_succ_left_of_not_isMax
theorem Ico_succ_left_of_not_isMax (ha : ¬IsMax a) : Ico (succ a) b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a)
Goal: Set.Ico (Order.succ a) b = Set.Ioo a b
-- Order.lt_succ
theorem lt_succ (a : α) : a < succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a : α)
Goal: a < Order.succ a
-- Order.lt_succ_iff
@[simp]
theorem lt_succ_iff : a < succ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: a < Order.succ b ↔ a ≤ b
-- Order.succ_le_iff
@[simp]
theorem succ_le_iff : succ a ≤ b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a ≤ b ↔ a < b
-- Order.succ_le_succ_iff
theorem succ_le_succ_iff : succ a ≤ succ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a ≤ Order.succ b ↔ a ≤ b
-- Order.succ_lt_succ_iff
theorem succ_lt_succ_iff : succ a < succ b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a < Order.succ b ↔ a < b
-- Order.succ_strictMono
theorem succ_strictMono : StrictMono (succ : α → α) 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α]
Goal: StrictMono Order.succ
-- Order.covBy_succ
theorem covBy_succ (a : α) : a ⋖ succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a : α)
Goal: a ⋖ Order.succ a
-- Order.Iio_succ
@[simp]
theorem Iio_succ (a : α) : Iio (succ a) = Iic a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a : α)
Goal: Set.Iio (Order.succ a) = Set.Iic a
-- Order.Ici_succ
@[simp]
theorem Ici_succ (a : α) : Ici (succ a) = Ioi a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a : α)
Goal: Set.Ici (Order.succ a) = Set.Ioi a
-- Order.Ico_succ_right
@[simp]
theorem Ico_succ_right (a b : α) : Ico a (succ b) = Icc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a b : α)
Goal: Set.Ico a (Order.succ b) = Set.Icc a b
-- Order.Ioo_succ_right
@[simp]
theorem Ioo_succ_right (a b : α) : Ioo a (succ b) = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a b : α)
Goal: Set.Ioo a (Order.succ b) = Set.Ioc a b
-- Order.Icc_succ_left
@[simp]
theorem Icc_succ_left (a b : α) : Icc (succ a) b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a b : α)
Goal: Set.Icc (Order.succ a) b = Set.Ioc a b
-- Order.Ico_succ_left
@[simp]
theorem Ico_succ_left (a b : α) : Ico (succ a) b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [NoMaxOrder α] (a b : α)
Goal: Set.Ico (Order.succ a) b = Set.Ioo a b
-- Order.succ_eq_iff_isMax
@[simp]
theorem succ_eq_iff_isMax : succ a = a ↔ IsMax a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α}
Goal: Order.succ a = a ↔ IsMax a
-- Order.succ_eq_succ_iff_of_not_isMax
theorem succ_eq_succ_iff_of_not_isMax (ha : ¬IsMax a) (hb : ¬IsMax b) :
    succ a = succ b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a) (hb : ¬IsMax b)
Goal: Order.succ a = Order.succ b ↔ a = b
-- Order.le_le_succ_iff
theorem le_le_succ_iff : a ≤ b ∧ b ≤ succ a ↔ b = a ∨ b = succ a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α}
Goal: a ≤ b ∧ b ≤ Order.succ a ↔ b = a ∨ b = Order.succ a
-- Order.succ_eq_of_covBy
lemma succ_eq_of_covBy (h : a ⋖ b) : succ a = b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (h : a ⋖ b)
Goal: Order.succ a = b
-- Order.le_succ_iff_eq_or_le
theorem le_succ_iff_eq_or_le : a ≤ succ b ↔ a = succ b ∨ a ≤ b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α}
Goal: a ≤ Order.succ b ↔ a = Order.succ b ∨ a ≤ b
-- Order.lt_succ_iff_eq_or_lt_of_not_isMax
theorem lt_succ_iff_eq_or_lt_of_not_isMax (hb : ¬IsMax b) : a < succ b ↔ a = b ∨ a < b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (hb : ¬IsMax b)
Goal: a < Order.succ b ↔ a = b ∨ a < b
-- Order.Iic_succ
theorem Iic_succ (a : α) : Iic (succ a) = insert (succ a) (Iic a) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] (a : α)
Goal: Set.Iic (Order.succ a) = insert (Order.succ a) (Set.Iic a)
-- Order.Icc_succ_right
theorem Icc_succ_right (h : a ≤ succ b) : Icc a (succ b) = insert (succ b) (Icc a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (h : a ≤ Order.succ b)
Goal: Set.Icc a (Order.succ b) = insert (Order.succ b) (Set.Icc a b)
-- Order.Ioc_succ_right
theorem Ioc_succ_right (h : a < succ b) : Ioc a (succ b) = insert (succ b) (Ioc a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (h : a < Order.succ b)
Goal: Set.Ioc a (Order.succ b) = insert (Order.succ b) (Set.Ioc a b)
-- Order.Iio_succ_eq_insert_of_not_isMax
theorem Iio_succ_eq_insert_of_not_isMax (h : ¬IsMax a) : Iio (succ a) = insert a (Iio a) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α} (h : ¬IsMax a)
Goal: Set.Iio (Order.succ a) = insert a (Set.Iio a)
-- Order.Ico_succ_right_eq_insert_of_not_isMax
theorem Ico_succ_right_eq_insert_of_not_isMax (h₁ : a ≤ b) (h₂ : ¬IsMax b) :
    Ico a (succ b) = insert b (Ico a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (h₁ : a ≤ b) (h₂ : ¬IsMax b)
Goal: Set.Ico a (Order.succ b) = insert b (Set.Ico a b)
-- Order.Ioo_succ_right_eq_insert_of_not_isMax
theorem Ioo_succ_right_eq_insert_of_not_isMax (h₁ : a < b) (h₂ : ¬IsMax b) :
    Ioo a (succ b) = insert b (Ioo a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} (h₁ : a < b) (h₂ : ¬IsMax b)
Goal: Set.Ioo a (Order.succ b) = insert b (Set.Ioo a b)
-- Order.succ_eq_succ_iff
@[simp]
theorem succ_eq_succ_iff : succ a = succ b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a = Order.succ b ↔ a = b
-- Order.succ_injective
theorem succ_injective : Injective (succ : α → α) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [NoMaxOrder α]
Goal: Function.Injective Order.succ
-- Order.succ_ne_succ_iff
theorem succ_ne_succ_iff : succ a ≠ succ b ↔ a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a ≠ Order.succ b ↔ a ≠ b
-- Order.lt_succ_iff_eq_or_lt
theorem lt_succ_iff_eq_or_lt : a < succ b ↔ a = b ∨ a < b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: a < Order.succ b ↔ a = b ∨ a < b
-- Order.succ_eq_iff_covBy
theorem succ_eq_iff_covBy : succ a = b ↔ a ⋖ b 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α]
Goal: Order.succ a = b ↔ a ⋖ b
-- Order.Iio_succ_eq_insert
theorem Iio_succ_eq_insert (a : α) : Iio (succ a) = insert a (Iio a) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [NoMaxOrder α] (a : α)
Goal: Set.Iio (Order.succ a) = insert a (Set.Iio a)
-- Order.Ico_succ_right_eq_insert
theorem Ico_succ_right_eq_insert (h : a ≤ b) : Ico a (succ b) = insert b (Ico a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α] (h : a ≤ b)
Goal: Set.Ico a (Order.succ b) = insert b (Set.Ico a b)
-- Order.Ioo_succ_right_eq_insert
theorem Ioo_succ_right_eq_insert (h : a < b) : Ioo a (succ b) = insert b (Ioo a b) 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a b : α} [NoMaxOrder α] (h : a < b)
Goal: Set.Ioo a (Order.succ b) = insert b (Set.Ioo a b)
-- Order.succ_top
@[simp]
theorem succ_top : succ (⊤ : α) = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [OrderTop α]
Goal: Order.succ ⊤ = ⊤
-- Order.succ_le_iff_eq_top
theorem succ_le_iff_eq_top : succ a ≤ a ↔ a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α} [OrderTop α]
Goal: Order.succ a ≤ a ↔ a = ⊤
-- Order.lt_succ_iff_ne_top
theorem lt_succ_iff_ne_top : a < succ a ↔ a ≠ ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α} [OrderTop α]
Goal: a < Order.succ a ↔ a ≠ ⊤
-- Order.lt_succ_bot_iff
theorem lt_succ_bot_iff [NoMaxOrder α] : a < succ ⊥ ↔ a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α} [OrderBot α] [NoMaxOrder α]
Goal: a < Order.succ ⊥ ↔ a = ⊥
-- Order.le_succ_bot_iff
theorem le_succ_bot_iff : a ≤ succ ⊥ ↔ a = ⊥ ∨ a = succ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] {a : α} [OrderBot α]
Goal: a ≤ Order.succ ⊥ ↔ a = ⊥ ∨ a = Order.succ ⊥
-- Order.bot_lt_succ
theorem bot_lt_succ (a : α) : ⊥ < succ a 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [OrderBot α] [Nontrivial α] (a : α)
Goal: ⊥ < Order.succ a
-- Order.succ_ne_bot
theorem succ_ne_bot (a : α) : succ a ≠ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [SuccOrder α] [OrderBot α] [Nontrivial α] (a : α)
Goal: Order.succ a ≠ ⊥
-- Order.succ_eq_iInf
theorem succ_eq_iInf (a : α) : succ a = ⨅ (b) (_ : a < b), b 
Arguments: {α : Type u_1} [CompleteLattice α] [SuccOrder α] (a : α)
Goal: Order.succ a = ⨅ b, ⨅ (_ : a < b), b
-- Order.pred_le
theorem pred_le : ∀ a : α, pred a ≤ a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] (a : α)
Goal: Order.pred a ≤ a
-- Order.min_of_le_pred
theorem min_of_le_pred {a : α} : a ≤ pred a → IsMin a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α}
Goal: a ≤ Order.pred a → IsMin a
-- Order.le_pred_of_lt
theorem le_pred_of_lt {a b : α} : a < b → a ≤ pred b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α}
Goal: a < b → a ≤ Order.pred b
-- Order.le_of_pred_lt
theorem le_of_pred_lt {a b : α} : pred a < b → a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α}
Goal: Order.pred a < b → a ≤ b
-- Order.le_pred_iff_isMin
@[simp]
theorem le_pred_iff_isMin : a ≤ pred a ↔ IsMin a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α}
Goal: a ≤ Order.pred a ↔ IsMin a
-- Order.pred_lt_iff_not_isMin
@[simp]
theorem pred_lt_iff_not_isMin : pred a < a ↔ ¬IsMin a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α}
Goal: Order.pred a < a ↔ ¬IsMin a
-- Order.pred_wcovBy
theorem pred_wcovBy (a : α) : pred a ⩿ a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] (a : α)
Goal: Order.pred a ⩿ a
-- Order.pred_covBy_of_not_isMin
theorem pred_covBy_of_not_isMin (h : ¬IsMin a) : pred a ⋖ a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α} (h : ¬IsMin a)
Goal: Order.pred a ⋖ a
-- Order.pred_lt_iff_of_not_isMin
theorem pred_lt_iff_of_not_isMin (ha : ¬IsMin a) : pred a < b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a)
Goal: Order.pred a < b ↔ a ≤ b
-- Order.le_pred_iff_of_not_isMin
theorem le_pred_iff_of_not_isMin (ha : ¬IsMin a) : b ≤ pred a ↔ b < a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a)
Goal: b ≤ Order.pred a ↔ b < a
-- Order.pred_lt_pred_of_not_isMin
lemma pred_lt_pred_of_not_isMin (h : a < b) (ha : ¬ IsMin a) : pred a < pred b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (h : a < b) (ha : ¬IsMin a)
Goal: Order.pred a < Order.pred b
-- Order.pred_lt_pred_iff_of_not_isMin
theorem pred_lt_pred_iff_of_not_isMin (ha : ¬IsMin a) (hb : ¬IsMin b) :
    pred a < pred b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a) (hb : ¬IsMin b)
Goal: Order.pred a < Order.pred b ↔ a < b
-- Order.pred_le_pred_iff_of_not_isMin
theorem pred_le_pred_iff_of_not_isMin (ha : ¬IsMin a) (hb : ¬IsMin b) :
    pred a ≤ pred b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a) (hb : ¬IsMin b)
Goal: Order.pred a ≤ Order.pred b ↔ a ≤ b
-- Order.pred_le_pred
@[simp, mono]
theorem pred_le_pred {a b : α} (h : a ≤ b) : pred a ≤ pred b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (h : a ≤ b)
Goal: Order.pred a ≤ Order.pred b
-- Order.pred_mono
theorem pred_mono : Monotone (pred : α → α) 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α]
Goal: Monotone Order.pred
-- Order.pred_le_of_wcovBy
lemma pred_le_of_wcovBy (h : a ⩿ b) : pred b ≤ a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (h : a ⩿ b)
Goal: Order.pred b ≤ a
-- Order.pred_iterate_le
theorem pred_iterate_le (k : ℕ) (x : α) : pred^[k] x ≤ x 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] (k : ℕ) (x : α)
Goal: Order.pred^[k] x ≤ x
-- Order.isMin_iterate_pred_of_eq_of_lt
theorem isMin_iterate_pred_of_eq_of_lt {n m : ℕ} (h_eq : pred^[n] a = pred^[m] a)
    (h_lt : n < m) : IsMin (pred^[n] a) 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α} {n m : ℕ} (h_eq : Order.pred^[n] a = Order.pred^[m] a) (h_lt : n < m)
Goal: IsMin (Order.pred^[n] a)
-- Order.isMin_iterate_pred_of_eq_of_ne
theorem isMin_iterate_pred_of_eq_of_ne {n m : ℕ} (h_eq : pred^[n] a = pred^[m] a)
    (h_ne : n ≠ m) : IsMin (pred^[n] a) 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α} {n m : ℕ} (h_eq : Order.pred^[n] a = Order.pred^[m] a) (h_ne : n ≠ m)
Goal: IsMin (Order.pred^[n] a)
-- Order.Ioi_pred_of_not_isMin
theorem Ioi_pred_of_not_isMin (ha : ¬IsMin a) : Ioi (pred a) = Ici a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α} (ha : ¬IsMin a)
Goal: Set.Ioi (Order.pred a) = Set.Ici a
-- Order.Iic_pred_of_not_isMin
theorem Iic_pred_of_not_isMin (ha : ¬IsMin a) : Iic (pred a) = Iio a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a : α} (ha : ¬IsMin a)
Goal: Set.Iic (Order.pred a) = Set.Iio a
-- Order.Ioc_pred_left_of_not_isMin
theorem Ioc_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioc (pred a) b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a)
Goal: Set.Ioc (Order.pred a) b = Set.Icc a b
-- Order.Ioo_pred_left_of_not_isMin
theorem Ioo_pred_left_of_not_isMin (ha : ¬IsMin a) : Ioo (pred a) b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin a)
Goal: Set.Ioo (Order.pred a) b = Set.Ico a b
-- Order.Icc_pred_right_of_not_isMin
theorem Icc_pred_right_of_not_isMin (ha : ¬IsMin b) : Icc a (pred b) = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin b)
Goal: Set.Icc a (Order.pred b) = Set.Ico a b
-- Order.Ioc_pred_right_of_not_isMin
theorem Ioc_pred_right_of_not_isMin (ha : ¬IsMin b) : Ioc a (pred b) = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} (ha : ¬IsMin b)
Goal: Set.Ioc a (Order.pred b) = Set.Ioo a b
-- Order.pred_lt
theorem pred_lt (a : α) : pred a < a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Order.pred a < a
-- Order.pred_lt_iff
@[simp]
theorem pred_lt_iff : pred a < b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a < b ↔ a ≤ b
-- Order.le_pred_iff
@[simp]
theorem le_pred_iff : a ≤ pred b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: a ≤ Order.pred b ↔ a < b
-- Order.pred_le_pred_iff
theorem pred_le_pred_iff : pred a ≤ pred b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a ≤ Order.pred b ↔ a ≤ b
-- Order.pred_lt_pred_iff
theorem pred_lt_pred_iff : pred a < pred b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a < Order.pred b ↔ a < b
-- Order.pred_strictMono
theorem pred_strictMono : StrictMono (pred : α → α) 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α]
Goal: StrictMono Order.pred
-- Order.pred_covBy
theorem pred_covBy (a : α) : pred a ⋖ a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Order.pred a ⋖ a
-- Order.Ioi_pred
@[simp]
theorem Ioi_pred (a : α) : Ioi (pred a) = Ici a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Set.Ioi (Order.pred a) = Set.Ici a
-- Order.Iic_pred
@[simp]
theorem Iic_pred (a : α) : Iic (pred a) = Iio a 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Set.Iic (Order.pred a) = Set.Iio a
-- Order.Ioc_pred_left
@[simp]
theorem Ioc_pred_left (a b : α) : Ioc (pred a) b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a b : α)
Goal: Set.Ioc (Order.pred a) b = Set.Icc a b
-- Order.Ioo_pred_left
@[simp]
theorem Ioo_pred_left (a b : α) : Ioo (pred a) b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a b : α)
Goal: Set.Ioo (Order.pred a) b = Set.Ico a b
-- Order.Icc_pred_right
@[simp]
theorem Icc_pred_right (a b : α) : Icc a (pred b) = Ico a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a b : α)
Goal: Set.Icc a (Order.pred b) = Set.Ico a b
-- Order.Ioc_pred_right
@[simp]
theorem Ioc_pred_right (a b : α) : Ioc a (pred b) = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [PredOrder α] [NoMinOrder α] (a b : α)
Goal: Set.Ioc a (Order.pred b) = Set.Ioo a b
-- Order.pred_eq_iff_isMin
@[simp]
theorem pred_eq_iff_isMin : pred a = a ↔ IsMin a 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α}
Goal: Order.pred a = a ↔ IsMin a
-- Order.pred_eq_pred_iff_of_not_isMin
theorem pred_eq_pred_iff_of_not_isMin (ha : ¬IsMin a) (hb : ¬IsMin b) :
    pred a = pred b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} (ha : ¬IsMin a) (hb : ¬IsMin b)
Goal: Order.pred a = Order.pred b ↔ a = b
-- Order.pred_le_le_iff
theorem pred_le_le_iff {a b : α} : pred a ≤ b ∧ b ≤ a ↔ b = a ∨ b = pred a 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α}
Goal: Order.pred a ≤ b ∧ b ≤ a ↔ b = a ∨ b = Order.pred a
-- Order.pred_eq_of_covBy
lemma pred_eq_of_covBy (h : a ⋖ b) : pred b = a 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} (h : a ⋖ b)
Goal: Order.pred b = a
-- Order.pred_le_iff_eq_or_le
theorem pred_le_iff_eq_or_le : pred a ≤ b ↔ b = pred a ∨ a ≤ b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α}
Goal: Order.pred a ≤ b ↔ b = Order.pred a ∨ a ≤ b
-- Order.pred_lt_iff_eq_or_lt_of_not_isMin
theorem pred_lt_iff_eq_or_lt_of_not_isMin (ha : ¬IsMin a) : pred a < b ↔ a = b ∨ a < b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} (ha : ¬IsMin a)
Goal: Order.pred a < b ↔ a = b ∨ a < b
-- Order.Ici_pred
theorem Ici_pred (a : α) : Ici (pred a) = insert (pred a) (Ici a) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] (a : α)
Goal: Set.Ici (Order.pred a) = insert (Order.pred a) (Set.Ici a)
-- Order.Ioi_pred_eq_insert_of_not_isMin
theorem Ioi_pred_eq_insert_of_not_isMin (ha : ¬IsMin a) : Ioi (pred a) = insert a (Ioi a) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α} (ha : ¬IsMin a)
Goal: Set.Ioi (Order.pred a) = insert a (Set.Ioi a)
-- Order.Icc_pred_left
theorem Icc_pred_left (h : pred a ≤ b) : Icc (pred a) b = insert (pred a) (Icc a b) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} (h : Order.pred a ≤ b)
Goal: Set.Icc (Order.pred a) b = insert (Order.pred a) (Set.Icc a b)
-- Order.Ico_pred_left
theorem Ico_pred_left (h : pred a < b) : Ico (pred a) b = insert (pred a) (Ico a b) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} (h : Order.pred a < b)
Goal: Set.Ico (Order.pred a) b = insert (Order.pred a) (Set.Ico a b)
-- Order.pred_eq_pred_iff
@[simp]
theorem pred_eq_pred_iff : pred a = pred b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a = Order.pred b ↔ a = b
-- Order.pred_injective
theorem pred_injective : Injective (pred : α → α) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [NoMinOrder α]
Goal: Function.Injective Order.pred
-- Order.pred_ne_pred_iff
theorem pred_ne_pred_iff : pred a ≠ pred b ↔ a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a ≠ Order.pred b ↔ a ≠ b
-- Order.pred_lt_iff_eq_or_lt
theorem pred_lt_iff_eq_or_lt : pred a < b ↔ a = b ∨ a < b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred a < b ↔ a = b ∨ a < b
-- Order.pred_eq_iff_covBy
theorem pred_eq_iff_covBy : pred b = a ↔ a ⋖ b 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α]
Goal: Order.pred b = a ↔ a ⋖ b
-- Order.Ioi_pred_eq_insert
theorem Ioi_pred_eq_insert (a : α) : Ioi (pred a) = insert a (Ioi a) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [NoMinOrder α] (a : α)
Goal: Set.Ioi (Order.pred a) = insert a (Set.Ioi a)
-- Order.Ico_pred_right_eq_insert
theorem Ico_pred_right_eq_insert (h : a ≤ b) : Ioc (pred a) b = insert a (Ioc a b) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α] (h : a ≤ b)
Goal: Set.Ioc (Order.pred a) b = insert a (Set.Ioc a b)
-- Order.Ioo_pred_right_eq_insert
theorem Ioo_pred_right_eq_insert (h : a < b) : Ioo (pred a) b = insert a (Ioo a b) 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a b : α} [NoMinOrder α] (h : a < b)
Goal: Set.Ioo (Order.pred a) b = insert a (Set.Ioo a b)
-- Order.pred_bot
@[simp]
theorem pred_bot : pred (⊥ : α) = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [OrderBot α]
Goal: Order.pred ⊥ = ⊥
-- Order.le_pred_iff_eq_bot
theorem le_pred_iff_eq_bot : a ≤ pred a ↔ a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α} [OrderBot α]
Goal: a ≤ Order.pred a ↔ a = ⊥
-- Order.pred_lt_iff_ne_bot
theorem pred_lt_iff_ne_bot : pred a < a ↔ a ≠ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α} [OrderBot α]
Goal: Order.pred a < a ↔ a ≠ ⊥
-- Order.pred_top_lt_iff
theorem pred_top_lt_iff [NoMinOrder α] : pred ⊤ < a ↔ a = ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α} [OrderTop α] [NoMinOrder α]
Goal: Order.pred ⊤ < a ↔ a = ⊤
-- Order.pred_top_le_iff
theorem pred_top_le_iff : pred ⊤ ≤ a ↔ a = ⊤ ∨ a = pred ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] {a : α} [OrderTop α]
Goal: Order.pred ⊤ ≤ a ↔ a = ⊤ ∨ a = Order.pred ⊤
-- Order.pred_lt_top
theorem pred_lt_top (a : α) : pred a < ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [OrderTop α] [Nontrivial α] (a : α)
Goal: Order.pred a < ⊤
-- Order.pred_ne_top
theorem pred_ne_top (a : α) : pred a ≠ ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [PredOrder α] [OrderTop α] [Nontrivial α] (a : α)
Goal: Order.pred a ≠ ⊤
-- Order.pred_eq_iSup
theorem pred_eq_iSup (a : α) : pred a = ⨆ (b) (_ : b < a), b 
Arguments: {α : Type u_1} [CompleteLattice α] [PredOrder α] (a : α)
Goal: Order.pred a = ⨆ b, ⨆ (_ : b < a), b
-- Order.le_succ_pred
lemma le_succ_pred (a : α) : a ≤ succ (pred a) 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [PredOrder α] (a : α)
Goal: a ≤ Order.succ (Order.pred a)
-- Order.pred_succ_le
lemma pred_succ_le (a : α) : pred (succ a) ≤ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [PredOrder α] (a : α)
Goal: Order.pred (Order.succ a) ≤ a
-- Order.pred_le_iff_le_succ
lemma pred_le_iff_le_succ : pred a ≤ b ↔ a ≤ succ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] [PredOrder α] {a b : α}
Goal: Order.pred a ≤ b ↔ a ≤ Order.succ b