ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- LinearOrder.min_def
class LinearOrder (α : Type u) extends PartialOrder α, Min α, Max α, Ord α :=
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
  min := fun a b => if a ≤ b then a else b
  max := fun a b => if a ≤ b then b else a
    min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl
    max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl
  compare a b := compareOfLessAndEq a b
    compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by
    compareOfLessAndEq_rfl
Arguments: {α : Type u} [self : LinearOrder α] (a b : α)
Goal: min a b = if a ≤ b then a else b
-- LinearOrder.le_total
class LinearOrder (α : Type u) extends PartialOrder α, Min α, Max α, Ord α :=
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
  min := fun a b => if a ≤ b then a else b
  max := fun a b => if a ≤ b then b else a
    min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl
    max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl
  compare a b := compareOfLessAndEq a b
    compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by
    compareOfLessAndEq_rfl
Arguments: {α : Type u} [self : LinearOrder α] (a b : α)
Goal: a ≤ b ∨ b ≤ a
-- LinearOrder.max_def
class LinearOrder (α : Type u) extends PartialOrder α, Min α, Max α, Ord α :=
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
  min := fun a b => if a ≤ b then a else b
  max := fun a b => if a ≤ b then b else a
    min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl
    max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl
  compare a b := compareOfLessAndEq a b
    compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by
    compareOfLessAndEq_rfl
Arguments: {α : Type u} [self : LinearOrder α] (a b : α)
Goal: max a b = if a ≤ b then b else a
-- LinearOrder.compare_eq_compareOfLessAndEq
class LinearOrder (α : Type u) extends PartialOrder α, Min α, Max α, Ord α :=
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
  min := fun a b => if a ≤ b then a else b
  max := fun a b => if a ≤ b then b else a
    min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl
    max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl
  compare a b := compareOfLessAndEq a b
    compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b := by
    compareOfLessAndEq_rfl
Arguments: {α : Type u} [self : LinearOrder α] (a b : α)
Goal: compare a b = compareOfLessAndEq a b
-- le_total
theorem le_total : ∀ a b : α, a ≤ b ∨ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a ≤ b ∨ b ≤ a
-- le_of_not_ge
theorem le_of_not_ge {a b : α} : ¬a ≥ b → a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: ¬a ≥ b → a ≤ b
-- le_of_not_le
theorem le_of_not_le {a b : α} : ¬a ≤ b → b ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: ¬a ≤ b → b ≤ a
-- not_lt_of_gt
theorem not_lt_of_gt {a b : α} (h : a > b) : ¬a < b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a > b)
Goal: ¬a < b
-- lt_trichotomy
theorem lt_trichotomy (a b : α) : a < b ∨ a = b ∨ b < a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a < b ∨ a = b ∨ b < a
-- le_of_not_lt
theorem le_of_not_lt {a b : α} (h : ¬b < a) : a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : ¬b < a)
Goal: a ≤ b
-- le_of_not_gt
theorem le_of_not_gt {a b : α} : ¬a > b → a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: ¬a > b → a ≤ b
-- lt_of_not_ge
theorem lt_of_not_ge {a b : α} (h : ¬a ≥ b) : a < b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : ¬a ≥ b)
Goal: a < b
-- lt_or_le
theorem lt_or_le (a b : α) : a < b ∨ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a < b ∨ b ≤ a
-- le_or_lt
theorem le_or_lt (a b : α) : a ≤ b ∨ b < a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a ≤ b ∨ b < a
-- lt_or_ge
theorem lt_or_ge : ∀ a b : α, a < b ∨ a ≥ b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a < b ∨ a ≥ b
-- le_or_gt
theorem le_or_gt : ∀ a b : α, a ≤ b ∨ a > b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a ≤ b ∨ a > b
-- lt_or_gt_of_ne
theorem lt_or_gt_of_ne {a b : α} (h : a ≠ b) : a < b ∨ a > b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a ≠ b)
Goal: a < b ∨ a > b
-- ne_iff_lt_or_gt
theorem ne_iff_lt_or_gt {a b : α} : a ≠ b ↔ a < b ∨ a > b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: a ≠ b ↔ a < b ∨ a > b
-- lt_iff_not_ge
theorem lt_iff_not_ge (x y : α) : x < y ↔ ¬x ≥ y 
Arguments: {α : Type u} [LinearOrder α] (x y : α)
Goal: x < y ↔ ¬x ≥ y
-- not_lt
@[simp]
theorem not_lt {a b : α} : ¬a < b ↔ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: ¬a < b ↔ b ≤ a
-- not_le
@[simp]
theorem not_le {a b : α} : ¬a ≤ b ↔ b < a 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: ¬a ≤ b ↔ b < a
-- eq_or_lt_of_not_lt
theorem eq_or_lt_of_not_lt {a b : α} (h : ¬a < b) : a = b ∨ b < a 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : ¬a < b)
Goal: a = b ∨ b < a
-- le_imp_le_of_lt_imp_lt
theorem le_imp_le_of_lt_imp_lt {β} [Preorder α] [LinearOrder β] {a b : α} {c d : β}
    (H : d < c → b < a) (h : a ≤ b) : c ≤ d 
Arguments: {α : Type u} {β : Type u_1} [Preorder α] [LinearOrder β] {a b : α} {c d : β} (H : d < c → b < a) (h : a ≤ b)
Goal: c ≤ d
-- compare_lt_iff_lt
theorem compare_lt_iff_lt {a b : α} : (compare a b = .lt) ↔ a < b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: compare a b = Ordering.lt ↔ a < b
-- compare_gt_iff_gt
theorem compare_gt_iff_gt {a b : α} : (compare a b = .gt) ↔ a > b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: compare a b = Ordering.gt ↔ a > b
-- compare_eq_iff_eq
theorem compare_eq_iff_eq {a b : α} : (compare a b = .eq) ↔ a = b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: compare a b = Ordering.eq ↔ a = b
-- compare_le_iff_le
theorem compare_le_iff_le {a b : α} : (compare a b ≠ .gt) ↔ a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: compare a b ≠ Ordering.gt ↔ a ≤ b
-- compare_ge_iff_ge
theorem compare_ge_iff_ge {a b : α} : (compare a b ≠ .lt) ↔ a ≥ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: compare a b ≠ Ordering.lt ↔ a ≥ b
-- compare_iff
theorem compare_iff (a b : α) {o : Ordering} : compare a b = o ↔ o.toRel a b 
Arguments: {α : Type u} [LinearOrder α] (a b : α) {o : Ordering}
Goal: compare a b = o ↔ o.toRel a b
-- maximals_subset
theorem maximals_subset : maximals r s ⊆ s 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α)
Goal: maximals r s ⊆ s
-- minimals_subset
theorem minimals_subset : minimals r s ⊆ s 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α)
Goal: minimals r s ⊆ s
-- maximals_empty
@[simp]
theorem maximals_empty : maximals r ∅ = ∅ 
Arguments: {α : Type u_1} (r : α → α → Prop)
Goal: maximals r ∅ = ∅
-- minimals_empty
@[simp]
theorem minimals_empty : minimals r ∅ = ∅ 
Arguments: {α : Type u_1} (r : α → α → Prop)
Goal: minimals r ∅ = ∅
-- maximals_singleton
@[simp]
theorem maximals_singleton : maximals r {a} = {a} 
Arguments: {α : Type u_1} (r : α → α → Prop) (a : α)
Goal: maximals r {a} = {a}
-- minimals_singleton
@[simp]
theorem minimals_singleton : minimals r {a} = {a} 
Arguments: {α : Type u_1} (r : α → α → Prop) (a : α)
Goal: minimals r {a} = {a}
-- maximals_swap
theorem maximals_swap : maximals (swap r) s = minimals r s 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α)
Goal: maximals (Function.swap r) s = minimals r s
-- minimals_swap
theorem minimals_swap : minimals (swap r) s = maximals r s 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α)
Goal: minimals (Function.swap r) s = maximals r s
-- mem_minimals_iff_forall_lt_not_mem'
theorem mem_minimals_iff_forall_lt_not_mem'
    (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
    x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt y x → y ∉ s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {x : α} (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt]
Goal: x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, rlt y x → y ∉ s
-- mem_maximals_iff_forall_lt_not_mem'
theorem mem_maximals_iff_forall_lt_not_mem'
    (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt] :
    x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, rlt x y → y ∉ s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {x : α} (rlt : α → α → Prop) [IsNonstrictStrictOrder α r rlt]
Goal: x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, rlt x y → y ∉ s
-- eq_of_mem_maximals
theorem eq_of_mem_maximals (ha : a ∈ maximals r s) (hb : b ∈ s) (h : r a b) : a = b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a b : α} [IsAntisymm α r] (ha : a ∈ maximals r s) (hb : b ∈ s) (h : r a b)
Goal: a = b
-- eq_of_mem_minimals
theorem eq_of_mem_minimals (ha : a ∈ minimals r s) (hb : b ∈ s) (h : r b a) : a = b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a b : α} [IsAntisymm α r] (ha : a ∈ minimals r s) (hb : b ∈ s) (h : r b a)
Goal: a = b
-- mem_maximals_iff
theorem mem_maximals_iff : x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, y ∈ s → r x y → x = y 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {x : α} [IsAntisymm α r]
Goal: x ∈ maximals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r x y → x = y
-- mem_maximals_setOf_iff
theorem mem_maximals_setOf_iff {P : α → Prop} :
    x ∈ maximals r (setOf P) ↔ P x ∧ ∀ ⦃y⦄, P y → r x y → x = y 
Arguments: {α : Type u_1} {r : α → α → Prop} {x : α} [IsAntisymm α r] {P : α → Prop}
Goal: x ∈ maximals r (setOf P) ↔ P x ∧ ∀ ⦃y : α⦄, P y → r x y → x = y
-- mem_minimals_iff
theorem mem_minimals_iff : x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y⦄, y ∈ s → r y x → x = y 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {x : α} [IsAntisymm α r]
Goal: x ∈ minimals r s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y ∈ s → r y x → x = y
-- mem_minimals_setOf_iff
theorem mem_minimals_setOf_iff {P : α → Prop} :
    x ∈ minimals r (setOf P) ↔ P x ∧ ∀ ⦃y⦄, P y → r y x → x = y 
Arguments: {α : Type u_1} {r : α → α → Prop} {x : α} [IsAntisymm α r] {P : α → Prop}
Goal: x ∈ minimals r (setOf P) ↔ P x ∧ ∀ ⦃y : α⦄, P y → r y x → x = y
-- minimals_eq_minimals_of_subset_of_forall
theorem minimals_eq_minimals_of_subset_of_forall [IsTrans α r] (hts : t ⊆ s)
    (h : ∀ x ∈ s, ∃ y ∈ t, r y x) : minimals r s = minimals r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} [IsAntisymm α r] [IsTrans α r] (hts : t ⊆ s) (h : ∀ x ∈ s, ∃ y ∈ t, r y x)
Goal: minimals r s = minimals r t
-- maximals_eq_maximals_of_subset_of_forall
theorem maximals_eq_maximals_of_subset_of_forall [IsTrans α r] (hts : t ⊆ s)
    (h : ∀ x ∈ s, ∃ y ∈ t, r x y) : maximals r s = maximals r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} [IsAntisymm α r] [IsTrans α r] (hts : t ⊆ s) (h : ∀ x ∈ s, ∃ y ∈ t, r x y)
Goal: maximals r s = maximals r t
-- maximals_antichain
theorem maximals_antichain : IsAntichain r (maximals r s) 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α) [IsAntisymm α r]
Goal: IsAntichain r (maximals r s)
-- minimals_antichain
theorem minimals_antichain : IsAntichain r (minimals r s) 
Arguments: {α : Type u_1} (r : α → α → Prop) (s : Set α) [IsAntisymm α r]
Goal: IsAntichain r (minimals r s)
-- mem_minimals_iff_forall_ssubset_not_mem
theorem mem_minimals_iff_forall_ssubset_not_mem {x : Set α} (s : Set (Set α)) :
    x ∈ minimals (· ⊆ ·) s ↔ x ∈ s ∧ ∀ ⦃y⦄, y ⊂ x → y ∉ s 
Arguments: {α : Type u_1} {x : Set α} (s : Set (Set α))
Goal: x ∈ minimals (fun x x_1 => x ⊆ x_1) s ↔ x ∈ s ∧ ∀ ⦃y : Set α⦄, y ⊂ x → y ∉ s
-- mem_minimals_iff_forall_lt_not_mem
theorem mem_minimals_iff_forall_lt_not_mem [Preorder α] {x : α} {s : Set α} :
    x ∈ minimals (· ≤ ·) s ↔ x ∈ s ∧ ∀ ⦃y⦄, y < x → y ∉ s 
Arguments: {α : Type u_1} [Preorder α] {x : α} {s : Set α}
Goal: x ∈ minimals (fun x x_1 => x ≤ x_1) s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, y < x → y ∉ s
-- mem_maximals_iff_forall_ssubset_not_mem
theorem mem_maximals_iff_forall_ssubset_not_mem {x : Set α} {s : Set (Set α)} :
    x ∈ maximals (· ⊆ ·) s ↔ x ∈ s ∧ ∀ ⦃y⦄, x ⊂ y → y ∉ s 
Arguments: {α : Type u_1} {x : Set α} {s : Set (Set α)}
Goal: x ∈ maximals (fun x x_1 => x ⊆ x_1) s ↔ x ∈ s ∧ ∀ ⦃y : Set α⦄, x ⊂ y → y ∉ s
-- mem_maximals_iff_forall_lt_not_mem
theorem mem_maximals_iff_forall_lt_not_mem [Preorder α] {x : α} {s : Set α} :
    x ∈ maximals (· ≤ ·) s ↔ x ∈ s ∧ ∀ ⦃y⦄, x < y → y ∉ s 
Arguments: {α : Type u_1} [Preorder α] {x : α} {s : Set α}
Goal: x ∈ maximals (fun x x_1 => x ≤ x_1) s ↔ x ∈ s ∧ ∀ ⦃y : α⦄, x < y → y ∉ s
-- Set.mem_maximals_iff_forall_insert
theorem Set.mem_maximals_iff_forall_insert {P : Set α → Prop} (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
    s ∈ maximals (· ⊆ ·) {t | P t} ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s) 
Arguments: {α : Type u_1} {s : Set α} {P : Set α → Prop} (hP : ∀ ⦃s t : Set α⦄, P t → s ⊆ t → P s)
Goal: s ∈ maximals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∉ s, ¬P (insert x s)
-- Set.mem_minimals_iff_forall_diff_singleton
theorem Set.mem_minimals_iff_forall_diff_singleton {P : Set α → Prop}
    (hP : ∀ ⦃s t⦄, P s → s ⊆ t → P t) :
    s ∈ minimals (· ⊆ ·) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬ P (s \ {x}) 
Arguments: {α : Type u_1} {s : Set α} {P : Set α → Prop} (hP : ∀ ⦃s t : Set α⦄, P s → s ⊆ t → P t)
Goal: s ∈ minimals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬P (s \ {x})
-- maximals_of_symm
theorem maximals_of_symm [IsSymm α r] : maximals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsSymm α r]
Goal: maximals r s = s
-- minimals_of_symm
theorem minimals_of_symm [IsSymm α r] : minimals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsSymm α r]
Goal: minimals r s = s
-- maximals_eq_minimals
theorem maximals_eq_minimals [IsSymm α r] : maximals r s = minimals r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsSymm α r]
Goal: maximals r s = minimals r s
-- Set.Subsingleton.maximals_eq
theorem Set.Subsingleton.maximals_eq (h : s.Subsingleton) : maximals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (h : s.Subsingleton)
Goal: maximals r s = s
-- Set.Subsingleton.minimals_eq
theorem Set.Subsingleton.minimals_eq (h : s.Subsingleton) : minimals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (h : s.Subsingleton)
Goal: minimals r s = s
-- maximals_mono
theorem maximals_mono [IsAntisymm α r₂] (h : ∀ a b, r₁ a b → r₂ a b) :
    maximals r₂ s ⊆ maximals r₁ s 
Arguments: {α : Type u_1} {r₁ r₂ : α → α → Prop} {s : Set α} [IsAntisymm α r₂] (h : ∀ (a b : α), r₁ a b → r₂ a b)
Goal: maximals r₂ s ⊆ maximals r₁ s
-- minimals_mono
theorem minimals_mono [IsAntisymm α r₂] (h : ∀ a b, r₁ a b → r₂ a b) :
    minimals r₂ s ⊆ minimals r₁ s 
Arguments: {α : Type u_1} {r₁ r₂ : α → α → Prop} {s : Set α} [IsAntisymm α r₂] (h : ∀ (a b : α), r₁ a b → r₂ a b)
Goal: minimals r₂ s ⊆ minimals r₁ s
-- maximals_union
theorem maximals_union : maximals r (s ∪ t) ⊆ maximals r s ∪ maximals r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: maximals r (s ∪ t) ⊆ maximals r s ∪ maximals r t
-- minimals_union
theorem minimals_union : minimals r (s ∪ t) ⊆ minimals r s ∪ minimals r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: minimals r (s ∪ t) ⊆ minimals r s ∪ minimals r t
-- maximals_inter_subset
theorem maximals_inter_subset : maximals r s ∩ t ⊆ maximals r (s ∩ t) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: maximals r s ∩ t ⊆ maximals r (s ∩ t)
-- minimals_inter_subset
theorem minimals_inter_subset : minimals r s ∩ t ⊆ minimals r (s ∩ t) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: minimals r s ∩ t ⊆ minimals r (s ∩ t)
-- inter_maximals_subset
theorem inter_maximals_subset : s ∩ maximals r t ⊆ maximals r (s ∩ t) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: s ∩ maximals r t ⊆ maximals r (s ∩ t)
-- inter_minimals_subset
theorem inter_minimals_subset : s ∩ minimals r t ⊆ minimals r (s ∩ t) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α}
Goal: s ∩ minimals r t ⊆ minimals r (s ∩ t)
-- IsAntichain.maximals_eq
theorem IsAntichain.maximals_eq (h : IsAntichain r s) : maximals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (h : IsAntichain r s)
Goal: maximals r s = s
-- IsAntichain.minimals_eq
theorem IsAntichain.minimals_eq (h : IsAntichain r s) : minimals r s = s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (h : IsAntichain r s)
Goal: minimals r s = s
-- maximals_idem
@[simp]
theorem maximals_idem : maximals r (maximals r s) = maximals r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α}
Goal: maximals r (maximals r s) = maximals r s
-- minimals_idem
@[simp]
theorem minimals_idem : minimals r (minimals r s) = minimals r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α}
Goal: minimals r (minimals r s) = minimals r s
-- IsAntichain.max_maximals
theorem IsAntichain.max_maximals (ht : IsAntichain r t) (h : maximals r s ⊆ t)
    (hs : ∀ ⦃a⦄, a ∈ t → ∃ b ∈ maximals r s, r b a) : maximals r s = t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (ht : IsAntichain r t) (h : maximals r s ⊆ t) (hs : ∀ ⦃a : α⦄, a ∈ t → ∃ b ∈ maximals r s, r b a)
Goal: maximals r s = t
-- IsAntichain.max_minimals
theorem IsAntichain.max_minimals (ht : IsAntichain r t) (h : minimals r s ⊆ t)
    (hs : ∀ ⦃a⦄, a ∈ t → ∃ b ∈ minimals r s, r a b) : minimals r s = t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (ht : IsAntichain r t) (h : minimals r s ⊆ t) (hs : ∀ ⦃a : α⦄, a ∈ t → ∃ b ∈ minimals r s, r a b)
Goal: minimals r s = t
-- IsLeast.mem_minimals
theorem IsLeast.mem_minimals [Preorder α] (h : IsLeast s a) : a ∈ minimals (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (h : IsLeast s a)
Goal: a ∈ minimals (fun x x_1 => x ≤ x_1) s
-- IsGreatest.mem_maximals
theorem IsGreatest.mem_maximals [Preorder α] (h : IsGreatest s a) : a ∈ maximals (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (h : IsGreatest s a)
Goal: a ∈ maximals (fun x x_1 => x ≤ x_1) s
-- IsLeast.minimals_eq
theorem IsLeast.minimals_eq [PartialOrder α] (h : IsLeast s a) : minimals (· ≤ ·) s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [PartialOrder α] (h : IsLeast s a)
Goal: minimals (fun x x_1 => x ≤ x_1) s = {a}
-- IsGreatest.maximals_eq
theorem IsGreatest.maximals_eq [PartialOrder α] (h : IsGreatest s a) : maximals (· ≤ ·) s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [PartialOrder α] (h : IsGreatest s a)
Goal: maximals (fun x x_1 => x ≤ x_1) s = {a}
-- IsAntichain.minimals_upperClosure
theorem IsAntichain.minimals_upperClosure [PartialOrder α] (hs : IsAntichain (· ≤ ·) s) :
    minimals (· ≤ ·) (upperClosure s : Set α) = s 
Arguments: {α : Type u_1} {s : Set α} [PartialOrder α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: minimals (fun x x_1 => x ≤ x_1) ↑(upperClosure s) = s
-- IsAntichain.maximals_lowerClosure
theorem IsAntichain.maximals_lowerClosure [PartialOrder α] (hs : IsAntichain (· ≤ ·) s) :
    maximals (· ≤ ·) (lowerClosure s : Set α) = s 
Arguments: {α : Type u_1} {s : Set α} [PartialOrder α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: maximals (fun x x_1 => x ≤ x_1) ↑(lowerClosure s) = s
-- map_mem_minimals
theorem map_mem_minimals (ha : a ∈ minimals r x) : f a ∈ minimals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) {a : α} (ha : a ∈ minimals r x)
Goal: f a ∈ minimals s (f '' x)
-- map_mem_maximals
theorem map_mem_maximals (ha : a ∈ maximals r x) : f a ∈ maximals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) {a : α} (ha : a ∈ maximals r x)
Goal: f a ∈ maximals s (f '' x)
-- map_mem_minimals_iff
theorem map_mem_minimals_iff (ha : a ∈ x) : f a ∈ minimals s (f '' x) ↔ a ∈ minimals r x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) {a : α} (ha : a ∈ x)
Goal: f a ∈ minimals s (f '' x) ↔ a ∈ minimals r x
-- map_mem_maximals_iff
theorem map_mem_maximals_iff (ha : a ∈ x) : f a ∈ maximals s (f '' x) ↔ a ∈ maximals r x 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) {a : α} (ha : a ∈ x)
Goal: f a ∈ maximals s (f '' x) ↔ a ∈ maximals r x
-- image_minimals_of_rel_iff_rel
theorem image_minimals_of_rel_iff_rel : f '' minimals r x = minimals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a')))
Goal: f '' minimals r x = minimals s (f '' x)
-- image_maximals_of_rel_iff_rel
theorem image_maximals_of_rel_iff_rel : f '' maximals r x = maximals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a')))
Goal: f '' maximals r x = maximals s (f '' x)
-- RelEmbedding.image_minimals_eq
theorem RelEmbedding.image_minimals_eq (f : r ↪r s) (x : Set α) :
    f '' minimals r x = minimals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (x : Set α)
Goal: ⇑f '' minimals r x = minimals s (⇑f '' x)
-- RelEmbedding.image_maximals_eq
theorem RelEmbedding.image_maximals_eq (f : r ↪r s) (x : Set α) :
    f '' maximals r x = maximals s (f '' x) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (x : Set α)
Goal: ⇑f '' maximals r x = maximals s (⇑f '' x)
-- image_minimals_univ
theorem image_minimals_univ :
    Subtype.val '' minimals (· ≤ ·) (univ : Set s) = minimals (· ≤ ·) s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: Subtype.val '' minimals (fun x x_1 => x ≤ x_1) Set.univ = minimals (fun x x_1 => x ≤ x_1) s
-- image_maximals_univ
theorem image_maximals_univ :
    Subtype.val '' maximals (· ≤ ·) (univ : Set s) = maximals (· ≤ ·) s 
Arguments: {α : Type u_1} [LE α] {s : Set α}
Goal: Subtype.val '' maximals (fun x x_1 => x ≤ x_1) Set.univ = maximals (fun x x_1 => x ≤ x_1) s
-- OrderIso.map_mem_minimals
nonrec theorem OrderIso.map_mem_minimals (f : s ≃o t) {x : α}
    (hx : x ∈ minimals (· ≤ ·) s) : (f ⟨x, hx.1⟩).val ∈ minimals (· ≤ ·) t 
Arguments: {α : Type u_1} {β : Type u_2} [LE α] [LE β] {s : Set α} {t : Set β} (f : ↑s ≃o ↑t) {x : α} (hx : x ∈ minimals (fun x x_1 => x ≤ x_1) s)
Goal: ↑(f ⟨x, ⋯⟩) ∈ minimals (fun x x_1 => x ≤ x_1) t
-- OrderIso.map_mem_maximals
theorem OrderIso.map_mem_maximals (f : s ≃o t) {x : α}
    (hx : x ∈ maximals (· ≤ ·) s) : (f ⟨x, hx.1⟩).val ∈ maximals (· ≤ ·) t 
Arguments: {α : Type u_1} {β : Type u_2} [LE α] [LE β] {s : Set α} {t : Set β} (f : ↑s ≃o ↑t) {x : α} (hx : x ∈ maximals (fun x x_1 => x ≤ x_1) s)
Goal: ↑(f ⟨x, ⋯⟩) ∈ maximals (fun x x_1 => x ≤ x_1) t
-- inter_minimals_preimage_inter_eq_of_rel_iff_rel_on
theorem inter_minimals_preimage_inter_eq_of_rel_iff_rel_on
    {x : Set α} (hf : ∀ ⦃a a'⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (y : Set β) :
    x ∩ f ⁻¹' (minimals s ((f '' x) ∩ y)) = minimals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (y : Set β)
Goal: x ∩ f ⁻¹' minimals s (f '' x ∩ y) = minimals r (x ∩ f ⁻¹' y)