ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsAntichain.to_dual
theorem to_dual [LE α] (hs : IsAntichain (· ≤ ·) s) : @IsAntichain αᵒᵈ (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LE α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsAntichain.to_dual_iff
theorem to_dual_iff [LE α] : IsAntichain (· ≤ ·) s ↔ @IsAntichain αᵒᵈ (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [LE α]
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ↔ IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsAntichain.image_compl
theorem image_compl [BooleanAlgebra α] (hs : IsAntichain (· ≤ ·) s) :
    IsAntichain (· ≤ ·) (compl '' s) 
Arguments: {α : Type u_1} {s : Set α} [BooleanAlgebra α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (compl '' s)
-- IsAntichain.preimage_compl
theorem preimage_compl [BooleanAlgebra α] (hs : IsAntichain (· ≤ ·) s) :
    IsAntichain (· ≤ ·) (compl ⁻¹' s) 
Arguments: {α : Type u_1} {s : Set α} [BooleanAlgebra α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) (compl ⁻¹' s)
-- isAntichain_singleton
theorem isAntichain_singleton (a : α) (r : α → α → Prop) : IsAntichain r {a} 
Arguments: {α : Type u_1} (a : α) (r : α → α → Prop)
Goal: IsAntichain r {a}
-- Set.Subsingleton.isAntichain
theorem Set.Subsingleton.isAntichain (hs : s.Subsingleton) (r : α → α → Prop) : IsAntichain r s 
Arguments: {α : Type u_1} {s : Set α} (hs : s.Subsingleton) (r : α → α → Prop)
Goal: IsAntichain r s
-- IsAntichain.not_lt
theorem IsAntichain.not_lt (hs : IsAntichain (· ≤ ·) s) (ha : a ∈ s) (hb : b ∈ s) : ¬a < b 
Arguments: {α : Type u_1} {s : Set α} {a b : α} [Preorder α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s) (ha : a ∈ s) (hb : b ∈ s)
Goal: ¬a < b
-- isAntichain_and_least_iff
theorem isAntichain_and_least_iff : IsAntichain (· ≤ ·) s ∧ IsLeast s a ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α]
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ∧ IsLeast s a ↔ s = {a}
-- isAntichain_and_greatest_iff
theorem isAntichain_and_greatest_iff : IsAntichain (· ≤ ·) s ∧ IsGreatest s a ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α]
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ∧ IsGreatest s a ↔ s = {a}
-- IsAntichain.least_iff
theorem IsAntichain.least_iff (hs : IsAntichain (· ≤ ·) s) : IsLeast s a ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsLeast s a ↔ s = {a}
-- IsAntichain.greatest_iff
theorem IsAntichain.greatest_iff (hs : IsAntichain (· ≤ ·) s) : IsGreatest s a ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsGreatest s a ↔ s = {a}
-- IsLeast.antichain_iff
theorem IsLeast.antichain_iff (hs : IsLeast s a) : IsAntichain (· ≤ ·) s ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (hs : IsLeast s a)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ↔ s = {a}
-- IsGreatest.antichain_iff
theorem IsGreatest.antichain_iff (hs : IsGreatest s a) : IsAntichain (· ≤ ·) s ↔ s = {a} 
Arguments: {α : Type u_1} {s : Set α} {a : α} [Preorder α] (hs : IsGreatest s a)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ↔ s = {a}
-- IsAntichain.bot_mem_iff
theorem IsAntichain.bot_mem_iff [OrderBot α] (hs : IsAntichain (· ≤ ·) s) : ⊥ ∈ s ↔ s = {⊥} 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [OrderBot α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: ⊥ ∈ s ↔ s = {⊥}
-- IsAntichain.top_mem_iff
theorem IsAntichain.top_mem_iff [OrderTop α] (hs : IsAntichain (· ≤ ·) s) : ⊤ ∈ s ↔ s = {⊤} 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [OrderTop α] (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: ⊤ ∈ s ↔ s = {⊤}
-- isAntichain_iff_forall_not_lt
theorem isAntichain_iff_forall_not_lt :
    IsAntichain (· ≤ ·) s ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ¬a < b 
Arguments: {α : Type u_1} {s : Set α} [PartialOrder α]
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s ↔ ∀ ⦃a : α⦄, a ∈ s → ∀ ⦃b : α⦄, b ∈ s → ¬a < b
-- IsStrongAntichain.subset
protected theorem subset (hs : IsStrongAntichain r s) (h : t ⊆ s) : IsStrongAntichain r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (hs : IsStrongAntichain r s) (h : t ⊆ s)
Goal: IsStrongAntichain r t
-- IsStrongAntichain.mono
theorem mono (hs : IsStrongAntichain r₁ s) (h : r₂ ≤ r₁) : IsStrongAntichain r₂ s 
Arguments: {α : Type u_1} {r₁ r₂ : α → α → Prop} {s : Set α} (hs : IsStrongAntichain r₁ s) (h : r₂ ≤ r₁)
Goal: IsStrongAntichain r₂ s
-- IsStrongAntichain.eq
theorem eq (hs : IsStrongAntichain r s) {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hac : r a c)
    (hbc : r b c) : a = b 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} (hs : IsStrongAntichain r s) {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hac : r a c) (hbc : r b c)
Goal: a = b
-- IsStrongAntichain.isAntichain
protected theorem isAntichain [IsRefl α r] (h : IsStrongAntichain r s) : IsAntichain r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsRefl α r] (h : IsStrongAntichain r s)
Goal: IsAntichain r s
-- IsStrongAntichain.subsingleton
protected theorem subsingleton [IsDirected α r] (h : IsStrongAntichain r s) : s.Subsingleton 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsDirected α r] (h : IsStrongAntichain r s)
Goal: s.Subsingleton
-- IsStrongAntichain.flip
protected theorem flip [IsSymm α r] (hs : IsStrongAntichain r s) : IsStrongAntichain (flip r) s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsSymm α r] (hs : IsStrongAntichain r s)
Goal: IsStrongAntichain (flip r) s
-- IsStrongAntichain.swap
theorem swap [IsSymm α r] (hs : IsStrongAntichain r s) : IsStrongAntichain (swap r) s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} [IsSymm α r] (hs : IsStrongAntichain r s)
Goal: IsStrongAntichain (Function.swap r) s
-- IsStrongAntichain.image
theorem image (hs : IsStrongAntichain r s) {f : α → β} (hf : Surjective f)
    (h : ∀ a b, r' (f a) (f b) → r a b) : IsStrongAntichain r' (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {r' : β → β → Prop} {s : Set α} (hs : IsStrongAntichain r s) {f : α → β} (hf : Function.Surjective f) (h : ∀ (a b : α), r' (f a) (f b) → r a b)
Goal: IsStrongAntichain r' (f '' s)
-- IsStrongAntichain.preimage
theorem preimage (hs : IsStrongAntichain r s) {f : β → α} (hf : Injective f)
    (h : ∀ a b, r' a b → r (f a) (f b)) : IsStrongAntichain 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 : IsStrongAntichain r s) {f : β → α} (hf : Function.Injective f) (h : ∀ (a b : β), r' a b → r (f a) (f b))
Goal: IsStrongAntichain r' (f ⁻¹' s)
-- isStrongAntichain_insert
theorem _root_.isStrongAntichain_insert :
    IsStrongAntichain r (insert a s) ↔
      IsStrongAntichain r s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ∀ c, ¬r a c ∨ ¬r b c 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α}
Goal: IsStrongAntichain r (insert a s) ↔ IsStrongAntichain r s ∧ ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ∀ (c : α), ¬r a c ∨ ¬r b c
-- IsStrongAntichain.insert
protected theorem insert (hs : IsStrongAntichain r s)
    (h : ∀ ⦃b⦄, b ∈ s → a ≠ b → ∀ c, ¬r a c ∨ ¬r b c) : IsStrongAntichain r (insert a s) 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {a : α} (hs : IsStrongAntichain r s) (h : ∀ ⦃b : α⦄, b ∈ s → a ≠ b → ∀ (c : α), ¬r a c ∨ ¬r b c)
Goal: IsStrongAntichain r (insert a s)
-- Set.Subsingleton.isStrongAntichain
theorem Set.Subsingleton.isStrongAntichain (hs : s.Subsingleton) (r : α → α → Prop) :
    IsStrongAntichain r s 
Arguments: {α : Type u_1} {s : Set α} (hs : s.Subsingleton) (r : α → α → Prop)
Goal: IsStrongAntichain r s
-- IsAntichain.of_strictMonoOn_antitoneOn
lemma IsAntichain.of_strictMonoOn_antitoneOn (hf : StrictMonoOn f s) (hf' : AntitoneOn f s) :
    IsAntichain (· ≤ ·) s 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {f : α → β} {s : Set α} (hf : StrictMonoOn f s) (hf' : AntitoneOn f s)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsAntichain.of_monotoneOn_strictAntiOn
lemma IsAntichain.of_monotoneOn_strictAntiOn (hf : MonotoneOn f s) (hf' : StrictAntiOn f s) :
    IsAntichain (· ≤ ·) s 
Arguments: {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] {f : α → β} {s : Set α} (hf : MonotoneOn f s) (hf' : StrictAntiOn f s)
Goal: IsAntichain (fun x x_1 => x ≤ x_1) s
-- IsWeakAntichain.subset
protected theorem subset (hs : IsWeakAntichain s) : t ⊆ s → IsWeakAntichain t 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s t : Set ((i : ι) → α i)} (hs : IsWeakAntichain s)
Goal: t ⊆ s → IsWeakAntichain t
-- IsWeakAntichain.eq
protected theorem eq (hs : IsWeakAntichain s) : a ∈ s → b ∈ s → a ≺ b → a = b 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ((i : ι) → α i)} {a b : (i : ι) → α i} (hs : IsWeakAntichain s)
Goal: a ∈ s → b ∈ s → StrongLT a b → a = b
-- IsWeakAntichain.insert
protected theorem insert (hs : IsWeakAntichain s) :
    (∀ ⦃b⦄, b ∈ s → a ≠ b → ¬b ≺ a) →
      (∀ ⦃b⦄, b ∈ s → a ≠ b → ¬a ≺ b) → IsWeakAntichain (insert a s) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ((i : ι) → α i)} {a : (i : ι) → α i} (hs : IsWeakAntichain s)
Goal: (∀ ⦃b : (i : ι) → α i⦄, b ∈ s → a ≠ b → ¬StrongLT b a) →
  (∀ ⦃b : (i : ι) → α i⦄, b ∈ s → a ≠ b → ¬StrongLT a b) → IsWeakAntichain (insert a s)
-- isWeakAntichain_insert
theorem _root_.isWeakAntichain_insert :
    IsWeakAntichain (insert a s) ↔ IsWeakAntichain s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬a ≺ b ∧ ¬b ≺ a 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ((i : ι) → α i)} {a : (i : ι) → α i}
Goal: IsWeakAntichain (insert a s) ↔ IsWeakAntichain s ∧ ∀ ⦃b : (i : ι) → α i⦄, b ∈ s → a ≠ b → ¬StrongLT a b ∧ ¬StrongLT b a
-- IsAntichain.isWeakAntichain
protected theorem IsAntichain.isWeakAntichain (hs : IsAntichain (· ≤ ·) s) : IsWeakAntichain s 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ((i : ι) → α i)} (hs : IsAntichain (fun x x_1 => x ≤ x_1) s)
Goal: IsWeakAntichain s
-- Set.Subsingleton.isWeakAntichain
theorem Set.Subsingleton.isWeakAntichain (hs : s.Subsingleton) : IsWeakAntichain s 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ((i : ι) → α i)} (hs : s.Subsingleton)
Goal: IsWeakAntichain s
-- GradeOrder.covBy_grade
class GradeOrder (𝕆 α : Type*) [Preorder 𝕆] [Preorder α] where
    protected grade : α → 𝕆
    grade_strictMono : StrictMono grade
    covBy_grade ⦃a b : α⦄ : a ⋖ b → grade a ⋖ grade b
Arguments: {𝕆 : Type u_5} {α : Type u_6} [Preorder 𝕆] [Preorder α] [self : GradeOrder 𝕆 α] ⦃a b : α⦄
Goal: a ⋖ b → GradeOrder.grade a ⋖ GradeOrder.grade b
-- GradeOrder.grade_strictMono
class GradeOrder (𝕆 α : Type*) [Preorder 𝕆] [Preorder α] where
    protected grade : α → 𝕆
    grade_strictMono : StrictMono grade
    covBy_grade ⦃a b : α⦄ : a ⋖ b → grade a ⋖ grade b
Arguments: {𝕆 : Type u_5} {α : Type u_6} [Preorder 𝕆] [Preorder α] [self : GradeOrder 𝕆 α]
Goal: StrictMono GradeOrder.grade
-- GradeMinOrder.isMin_grade
class GradeMinOrder (𝕆 α : Type*) [Preorder 𝕆] [Preorder α] extends GradeOrder 𝕆 α where
    isMin_grade ⦃a : α⦄ : IsMin a → IsMin (grade a)
Arguments: {𝕆 : Type u_5} {α : Type u_6} [Preorder 𝕆] [Preorder α] [self : GradeMinOrder 𝕆 α] ⦃a : α⦄
Goal: IsMin a → IsMin (GradeOrder.grade a)
-- GradeMaxOrder.isMax_grade
class GradeMaxOrder (𝕆 α : Type*) [Preorder 𝕆] [Preorder α] extends GradeOrder 𝕆 α where
    isMax_grade ⦃a : α⦄ : IsMax a → IsMax (grade a)
Arguments: {𝕆 : Type u_5} {α : Type u_6} [Preorder 𝕆] [Preorder α] [self : GradeMaxOrder 𝕆 α] ⦃a : α⦄
Goal: IsMax a → IsMax (GradeOrder.grade a)
-- CovBy.grade
protected theorem CovBy.grade (h : a ⋖ b) : grade 𝕆 a ⋖ grade 𝕆 b 
Arguments: (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] {a b : α} (h : a ⋖ b)
Goal: grade 𝕆 a ⋖ grade 𝕆 b
-- grade_strictMono
theorem grade_strictMono : StrictMono (grade 𝕆 : α → 𝕆) 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α]
Goal: StrictMono (grade 𝕆)
-- covBy_iff_lt_covBy_grade
theorem covBy_iff_lt_covBy_grade : a ⋖ b ↔ a < b ∧ grade 𝕆 a ⋖ grade 𝕆 b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] {a b : α}
Goal: a ⋖ b ↔ a < b ∧ grade 𝕆 a ⋖ grade 𝕆 b
-- IsMin.grade
protected theorem IsMin.grade (h : IsMin a) : IsMin (grade 𝕆 a) 
Arguments: (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMinOrder 𝕆 α] {a : α} (h : IsMin a)
Goal: IsMin (grade 𝕆 a)
-- isMin_grade_iff
@[simp]
theorem isMin_grade_iff : IsMin (grade 𝕆 a) ↔ IsMin a 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMinOrder 𝕆 α] {a : α}
Goal: IsMin (grade 𝕆 a) ↔ IsMin a
-- IsMax.grade
protected theorem IsMax.grade (h : IsMax a) : IsMax (grade 𝕆 a) 
Arguments: (𝕆 : Type u_1) {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMaxOrder 𝕆 α] {a : α} (h : IsMax a)
Goal: IsMax (grade 𝕆 a)
-- isMax_grade_iff
@[simp]
theorem isMax_grade_iff : IsMax (grade 𝕆 a) ↔ IsMax a 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeMaxOrder 𝕆 α] {a : α}
Goal: IsMax (grade 𝕆 a) ↔ IsMax a
-- grade_mono
theorem grade_mono [PartialOrder α] [GradeOrder 𝕆 α] : Monotone (grade 𝕆 : α → 𝕆) 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [PartialOrder α] [GradeOrder 𝕆 α]
Goal: Monotone (grade 𝕆)
-- grade_injective
theorem grade_injective : Function.Injective (grade 𝕆 : α → 𝕆) 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α]
Goal: Function.Injective (grade 𝕆)
-- grade_le_grade_iff
@[simp]
theorem grade_le_grade_iff : grade 𝕆 a ≤ grade 𝕆 b ↔ a ≤ b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α}
Goal: grade 𝕆 a ≤ grade 𝕆 b ↔ a ≤ b
-- grade_lt_grade_iff
@[simp]
theorem grade_lt_grade_iff : grade 𝕆 a < grade 𝕆 b ↔ a < b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α}
Goal: grade 𝕆 a < grade 𝕆 b ↔ a < b
-- grade_eq_grade_iff
@[simp]
theorem grade_eq_grade_iff : grade 𝕆 a = grade 𝕆 b ↔ a = b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α}
Goal: grade 𝕆 a = grade 𝕆 b ↔ a = b
-- grade_ne_grade_iff
theorem grade_ne_grade_iff : grade 𝕆 a ≠ grade 𝕆 b ↔ a ≠ b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α}
Goal: grade 𝕆 a ≠ grade 𝕆 b ↔ a ≠ b
-- grade_covBy_grade_iff
theorem grade_covBy_grade_iff : grade 𝕆 a ⋖ grade 𝕆 b ↔ a ⋖ b 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [LinearOrder α] [GradeOrder 𝕆 α] {a b : α}
Goal: grade 𝕆 a ⋖ grade 𝕆 b ↔ a ⋖ b
-- grade_bot
@[simp]
theorem grade_bot [OrderBot 𝕆] [OrderBot α] [GradeMinOrder 𝕆 α] : grade 𝕆 (⊥ : α) = ⊥ 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [PartialOrder 𝕆] [Preorder α] [OrderBot 𝕆] [OrderBot α] [GradeMinOrder 𝕆 α]
Goal: grade 𝕆 ⊥ = ⊥
-- grade_top
@[simp]
theorem grade_top [OrderTop 𝕆] [OrderTop α] [GradeMaxOrder 𝕆 α] : grade 𝕆 (⊤ : α) = ⊤ 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [PartialOrder 𝕆] [Preorder α] [OrderTop 𝕆] [OrderTop α] [GradeMaxOrder 𝕆 α]
Goal: grade 𝕆 ⊤ = ⊤
-- grade_self
@[simp]
theorem grade_self (a : α) : grade α a = a 
Arguments: {α : Type u_3} [Preorder α] (a : α)
Goal: grade α a = a
-- grade_toDual
@[simp]
theorem grade_toDual [GradeOrder 𝕆 α] (a : α) : grade 𝕆ᵒᵈ (toDual a) = toDual (grade 𝕆 a) 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] (a : α)
Goal: grade 𝕆ᵒᵈ (OrderDual.toDual a) = OrderDual.toDual (grade 𝕆 a)
-- grade_ofDual
@[simp]
theorem grade_ofDual [GradeOrder 𝕆 α] (a : αᵒᵈ) : grade 𝕆 (ofDual a) = ofDual (grade 𝕆ᵒᵈ a) 
Arguments: {𝕆 : Type u_1} {α : Type u_3} [Preorder 𝕆] [Preorder α] [GradeOrder 𝕆 α] (a : αᵒᵈ)
Goal: grade 𝕆 (OrderDual.ofDual a) = OrderDual.ofDual (grade 𝕆ᵒᵈ a)
-- GradeOrder.wellFoundedLT
theorem GradeOrder.wellFoundedLT (𝕆 : Type*) [Preorder 𝕆] [GradeOrder 𝕆 α]
    [WellFoundedLT 𝕆] : WellFoundedLT α 
Arguments: {α : Type u_3} [Preorder α] (𝕆 : Type u_5) [Preorder 𝕆] [GradeOrder 𝕆 α] [WellFoundedLT 𝕆]
Goal: WellFoundedLT α
-- GradeOrder.wellFoundedGT
theorem GradeOrder.wellFoundedGT (𝕆 : Type*) [Preorder 𝕆] [GradeOrder 𝕆 α]
    [WellFoundedGT 𝕆] : WellFoundedGT α 
Arguments: {α : Type u_3} [Preorder α] (𝕆 : Type u_5) [Preorder 𝕆] [GradeOrder 𝕆 α] [WellFoundedGT 𝕆]
Goal: WellFoundedGT α
-- OrderTop.le_top
class OrderTop (α : Type u) [LE α] extends Top α where
    le_top : ∀ a : α, a ≤ ⊤
Arguments: {α : Type u} [LE α] [self : OrderTop α] (a : α)
Goal: a ≤ ⊤
-- le_top
@[simp]
theorem le_top : a ≤ ⊤ 
Arguments: {α : Type u} [LE α] [OrderTop α] {a : α}
Goal: a ≤ ⊤
-- isTop_top
@[simp]
theorem isTop_top : IsTop (⊤ : α) 
Arguments: {α : Type u} [LE α] [OrderTop α]
Goal: IsTop ⊤
-- isMax_top
@[simp]
theorem isMax_top : IsMax (⊤ : α) 
Arguments: {α : Type u} [Preorder α] [OrderTop α]
Goal: IsMax ⊤
-- not_top_lt
@[simp]
theorem not_top_lt : ¬⊤ < a 
Arguments: {α : Type u} [Preorder α] [OrderTop α] {a : α}
Goal: ¬⊤ < a
-- ne_top_of_lt
theorem ne_top_of_lt (h : a < b) : a ≠ ⊤ 
Arguments: {α : Type u} [Preorder α] [OrderTop α] {a b : α} (h : a < b)
Goal: a ≠ ⊤
-- isMax_iff_eq_top
@[simp]
theorem isMax_iff_eq_top : IsMax a ↔ a = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: IsMax a ↔ a = ⊤
-- isTop_iff_eq_top
@[simp]
theorem isTop_iff_eq_top : IsTop a ↔ a = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: IsTop a ↔ a = ⊤
-- not_isMax_iff_ne_top
theorem not_isMax_iff_ne_top : ¬IsMax a ↔ a ≠ ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: ¬IsMax a ↔ a ≠ ⊤
-- not_isTop_iff_ne_top
theorem not_isTop_iff_ne_top : ¬IsTop a ↔ a ≠ ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: ¬IsTop a ↔ a ≠ ⊤
-- top_le_iff
@[simp]
theorem top_le_iff : ⊤ ≤ a ↔ a = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: ⊤ ≤ a ↔ a = ⊤
-- top_unique
theorem top_unique (h : ⊤ ≤ a) : a = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : ⊤ ≤ a)
Goal: a = ⊤
-- eq_top_iff
theorem eq_top_iff : a = ⊤ ↔ ⊤ ≤ a 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: a = ⊤ ↔ ⊤ ≤ a
-- eq_top_mono
theorem eq_top_mono (h : a ≤ b) (h₂ : a = ⊤) : b = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a b : α} (h : a ≤ b) (h₂ : a = ⊤)
Goal: b = ⊤
-- lt_top_iff_ne_top
theorem lt_top_iff_ne_top : a < ⊤ ↔ a ≠ ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: a < ⊤ ↔ a ≠ ⊤
-- not_lt_top_iff
@[simp]
theorem not_lt_top_iff : ¬a < ⊤ ↔ a = ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α}
Goal: ¬a < ⊤ ↔ a = ⊤
-- eq_top_or_lt_top
theorem eq_top_or_lt_top (a : α) : a = ⊤ ∨ a < ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] (a : α)
Goal: a = ⊤ ∨ a < ⊤
-- Ne.lt_top
theorem Ne.lt_top (h : a ≠ ⊤) : a < ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : a ≠ ⊤)
Goal: a < ⊤
-- Ne.lt_top'
theorem Ne.lt_top' (h : ⊤ ≠ a) : a < ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a : α} (h : ⊤ ≠ a)
Goal: a < ⊤
-- ne_top_of_le_ne_top
theorem ne_top_of_le_ne_top (hb : b ≠ ⊤) (hab : a ≤ b) : a ≠ ⊤ 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] {a b : α} (hb : b ≠ ⊤) (hab : a ≤ b)
Goal: a ≠ ⊤
-- StrictMono.apply_eq_top_iff
theorem StrictMono.apply_eq_top_iff (hf : StrictMono f) : f a = f ⊤ ↔ a = ⊤ 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [OrderTop α] [Preorder β] {f : α → β} {a : α} (hf : StrictMono f)
Goal: f a = f ⊤ ↔ a = ⊤
-- StrictAnti.apply_eq_top_iff
theorem StrictAnti.apply_eq_top_iff (hf : StrictAnti f) : f a = f ⊤ ↔ a = ⊤ 
Arguments: {α : Type u} {β : Type v} [PartialOrder α] [OrderTop α] [Preorder β] {f : α → β} {a : α} (hf : StrictAnti f)
Goal: f a = f ⊤ ↔ a = ⊤
-- not_isMin_top
theorem not_isMin_top : ¬IsMin (⊤ : α) 
Arguments: {α : Type u} [PartialOrder α] [OrderTop α] [Nontrivial α]
Goal: ¬IsMin ⊤
-- StrictMono.maximal_preimage_top
theorem StrictMono.maximal_preimage_top [LinearOrder α] [Preorder β] [OrderTop β] {f : α → β}
    (H : StrictMono f) {a} (h_top : f a = ⊤) (x : α) : x ≤ a 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [Preorder β] [OrderTop β] {f : α → β} (H : StrictMono f) {a : α} (h_top : f a = ⊤) (x : α)
Goal: x ≤ a
-- OrderTop.ext_top
theorem OrderTop.ext_top {α} {hA : PartialOrder α} (A : OrderTop α) {hB : PartialOrder α}
    (B : OrderTop α) (H : ∀ x y : α, (haveI := hA; x ≤ y) ↔ x ≤ y) :
    (@Top.top α (@OrderTop.toTop α hA.toLE A)) = (@Top.top α (@OrderTop.toTop α hB.toLE B)) 
Arguments: {α : Type u_3} {hA : PartialOrder α} (A : OrderTop α) {hB : PartialOrder α} (B : OrderTop α) (H : ∀ (x y : α), x ≤ y ↔ x ≤ y)
Goal: ⊤ = ⊤
-- OrderBot.bot_le
class OrderBot (α : Type u) [LE α] extends Bot α where
    bot_le : ∀ a : α, ⊥ ≤ a
Arguments: {α : Type u} [LE α] [self : OrderBot α] (a : α)
Goal: ⊥ ≤ a
-- bot_le
@[simp]
theorem bot_le : ⊥ ≤ a 
Arguments: {α : Type u} [LE α] [OrderBot α] {a : α}
Goal: ⊥ ≤ a
-- isBot_bot
@[simp]
theorem isBot_bot : IsBot (⊥ : α) 
Arguments: {α : Type u} [LE α] [OrderBot α]
Goal: IsBot ⊥
-- OrderDual.ofDual_bot
@[simp]
theorem ofDual_bot [Top α] : ofDual ⊥ = (⊤ : α) 
Arguments: (α : Type u) [Top α]
Goal: OrderDual.ofDual ⊥ = ⊤
-- OrderDual.ofDual_top
@[simp]
theorem ofDual_top [Bot α] : ofDual ⊤ = (⊥ : α) 
Arguments: (α : Type u) [Bot α]
Goal: OrderDual.ofDual ⊤ = ⊥
-- OrderDual.toDual_bot
@[simp]
theorem toDual_bot [Bot α] : toDual (⊥ : α) = ⊤ 
Arguments: (α : Type u) [Bot α]
Goal: OrderDual.toDual ⊥ = ⊤
-- OrderDual.toDual_top
@[simp]
theorem toDual_top [Top α] : toDual (⊤ : α) = ⊥ 
Arguments: (α : Type u) [Top α]
Goal: OrderDual.toDual ⊤ = ⊥
-- isMin_bot
@[simp]
theorem isMin_bot : IsMin (⊥ : α) 
Arguments: {α : Type u} [Preorder α] [OrderBot α]
Goal: IsMin ⊥
-- not_lt_bot
@[simp]
theorem not_lt_bot : ¬a < ⊥ 
Arguments: {α : Type u} [Preorder α] [OrderBot α] {a : α}
Goal: ¬a < ⊥
-- ne_bot_of_gt
theorem ne_bot_of_gt (h : a < b) : b ≠ ⊥ 
Arguments: {α : Type u} [Preorder α] [OrderBot α] {a b : α} (h : a < b)
Goal: b ≠ ⊥
-- isMin_iff_eq_bot
@[simp]
theorem isMin_iff_eq_bot : IsMin a ↔ a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: IsMin a ↔ a = ⊥
-- isBot_iff_eq_bot
@[simp]
theorem isBot_iff_eq_bot : IsBot a ↔ a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: IsBot a ↔ a = ⊥
-- not_isMin_iff_ne_bot
theorem not_isMin_iff_ne_bot : ¬IsMin a ↔ a ≠ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: ¬IsMin a ↔ a ≠ ⊥
-- not_isBot_iff_ne_bot
theorem not_isBot_iff_ne_bot : ¬IsBot a ↔ a ≠ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: ¬IsBot a ↔ a ≠ ⊥
-- le_bot_iff
@[simp]
theorem le_bot_iff : a ≤ ⊥ ↔ a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: a ≤ ⊥ ↔ a = ⊥
-- bot_unique
theorem bot_unique (h : a ≤ ⊥) : a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : a ≤ ⊥)
Goal: a = ⊥
-- eq_bot_iff
theorem eq_bot_iff : a = ⊥ ↔ a ≤ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: a = ⊥ ↔ a ≤ ⊥
-- eq_bot_mono
theorem eq_bot_mono (h : a ≤ b) (h₂ : b = ⊥) : a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a b : α} (h : a ≤ b) (h₂ : b = ⊥)
Goal: a = ⊥
-- bot_lt_iff_ne_bot
theorem bot_lt_iff_ne_bot : ⊥ < a ↔ a ≠ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: ⊥ < a ↔ a ≠ ⊥
-- not_bot_lt_iff
@[simp]
theorem not_bot_lt_iff : ¬⊥ < a ↔ a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α}
Goal: ¬⊥ < a ↔ a = ⊥
-- eq_bot_or_bot_lt
theorem eq_bot_or_bot_lt (a : α) : a = ⊥ ∨ ⊥ < a 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] (a : α)
Goal: a = ⊥ ∨ ⊥ < a
-- eq_bot_of_minimal
theorem eq_bot_of_minimal (h : ∀ b, ¬b < a) : a = ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : ∀ (b : α), ¬b < a)
Goal: a = ⊥
-- Ne.bot_lt
theorem Ne.bot_lt (h : a ≠ ⊥) : ⊥ < a 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : a ≠ ⊥)
Goal: ⊥ < a
-- Ne.bot_lt'
theorem Ne.bot_lt' (h : ⊥ ≠ a) : ⊥ < a 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a : α} (h : ⊥ ≠ a)
Goal: ⊥ < a
-- ne_bot_of_le_ne_bot
theorem ne_bot_of_le_ne_bot (hb : b ≠ ⊥) (hab : b ≤ a) : a ≠ ⊥ 
Arguments: {α : Type u} [PartialOrder α] [OrderBot α] {a b : α} (hb : b ≠ ⊥) (hab : b ≤ a)
Goal: a ≠ ⊥