ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Filter.vadd_filter.instNeBot
@[to_additive vadd_filter.instNeBot]
lemma smul_filter.instNeBot [NeBot f] : NeBot (a • f) 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {f : Filter β} {a : α} [f.NeBot]
Goal: (a +ᵥ f).NeBot
-- Filter.smul_filter.instNeBot
@[to_additive vadd_filter.instNeBot]
lemma smul_filter.instNeBot [NeBot f] : NeBot (a • f) 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {f : Filter β} {a : α} [f.NeBot]
Goal: (a • f).NeBot
-- Filter.smul_filter_le_smul_filter
@[to_additive]
theorem smul_filter_le_smul_filter (hf : f₁ ≤ f₂) : a • f₁ ≤ a • f₂ 
Arguments: {α : Type u_2} {β : Type u_3} [SMul α β] {f₁ f₂ : Filter β} {a : α} (hf : f₁ ≤ f₂)
Goal: a • f₁ ≤ a • f₂
-- Filter.vadd_filter_le_vadd_filter
@[to_additive]
theorem smul_filter_le_smul_filter (hf : f₁ ≤ f₂) : a • f₁ ≤ a • f₂ 
Arguments: {α : Type u_2} {β : Type u_3} [VAdd α β] {f₁ f₂ : Filter β} {a : α} (hf : f₁ ≤ f₂)
Goal: a +ᵥ f₁ ≤ a +ᵥ f₂
-- Filter.NeBot.smul_zero_nonneg
theorem NeBot.smul_zero_nonneg (hf : f.NeBot) : 0 ≤ f • (0 : Filter β) 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {f : Filter α} (hf : f.NeBot)
Goal: 0 ≤ f • 0
-- Filter.NeBot.zero_smul_nonneg
theorem NeBot.zero_smul_nonneg (hg : g.NeBot) : 0 ≤ (0 : Filter α) • g 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {g : Filter β} (hg : g.NeBot)
Goal: 0 ≤ 0 • g
-- Filter.zero_smul_filter_nonpos
theorem zero_smul_filter_nonpos : (0 : α) • g ≤ 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {g : Filter β}
Goal: 0 • g ≤ 0
-- Filter.zero_smul_filter
theorem zero_smul_filter (hg : g.NeBot) : (0 : α) • g = 0 
Arguments: {α : Type u_2} {β : Type u_3} [Zero α] [Zero β] [SMulWithZero α β] {g : Filter β} (hg : g.NeBot)
Goal: 0 • g = 0
-- NoBotOrder.exists_not_ge
class NoBotOrder (α : Type*) [LE α] : Prop where
    exists_not_ge (a : α) : ∃ b, ¬a ≤ b
Arguments: {α : Type u_3} [LE α] [self : NoBotOrder α] (a : α)
Goal: ∃ b, ¬a ≤ b
-- NoTopOrder.exists_not_le
class NoTopOrder (α : Type*) [LE α] : Prop where
    exists_not_le (a : α) : ∃ b, ¬b ≤ a
Arguments: {α : Type u_3} [LE α] [self : NoTopOrder α] (a : α)
Goal: ∃ b, ¬b ≤ a
-- NoMinOrder.exists_lt
class NoMinOrder (α : Type*) [LT α] : Prop where
    exists_lt (a : α) : ∃ b, b < a
Arguments: {α : Type u_3} [LT α] [self : NoMinOrder α] (a : α)
Goal: ∃ b, b < a
-- NoMaxOrder.exists_gt
class NoMaxOrder (α : Type*) [LT α] : Prop where
    exists_gt (a : α) : ∃ b, a < b
Arguments: {α : Type u_3} [LT α] [self : NoMaxOrder α] (a : α)
Goal: ∃ b, a < b
-- NoBotOrder.to_noMinOrder
theorem NoBotOrder.to_noMinOrder (α : Type*) [LinearOrder α] [NoBotOrder α] : NoMinOrder α 
Arguments: (α : Type u_3) [LinearOrder α] [NoBotOrder α]
Goal: NoMinOrder α
-- NoTopOrder.to_noMaxOrder
theorem NoTopOrder.to_noMaxOrder (α : Type*) [LinearOrder α] [NoTopOrder α] : NoMaxOrder α 
Arguments: (α : Type u_3) [LinearOrder α] [NoTopOrder α]
Goal: NoMaxOrder α
-- noBotOrder_iff_noMinOrder
theorem noBotOrder_iff_noMinOrder (α : Type*) [LinearOrder α] : NoBotOrder α ↔ NoMinOrder α 
Arguments: (α : Type u_3) [LinearOrder α]
Goal: NoBotOrder α ↔ NoMinOrder α
-- noTopOrder_iff_noMaxOrder
theorem noTopOrder_iff_noMaxOrder (α : Type*) [LinearOrder α] : NoTopOrder α ↔ NoMaxOrder α 
Arguments: (α : Type u_3) [LinearOrder α]
Goal: NoTopOrder α ↔ NoMaxOrder α
-- NoMinOrder.not_acc
theorem NoMinOrder.not_acc [LT α] [NoMinOrder α] (a : α) : ¬Acc (· < ·) a 
Arguments: {α : Type u_1} [LT α] [NoMinOrder α] (a : α)
Goal: ¬Acc (fun x x_1 => x < x_1) a
-- NoMaxOrder.not_acc
theorem NoMaxOrder.not_acc [LT α] [NoMaxOrder α] (a : α) : ¬Acc (· > ·) a 
Arguments: {α : Type u_1} [LT α] [NoMaxOrder α] (a : α)
Goal: ¬Acc (fun x x_1 => x > x_1) a
-- not_isBot
@[simp]
theorem not_isBot [NoBotOrder α] (a : α) : ¬IsBot a 
Arguments: {α : Type u_1} [LE α] [NoBotOrder α] (a : α)
Goal: ¬IsBot a
-- not_isTop
@[simp]
theorem not_isTop [NoTopOrder α] (a : α) : ¬IsTop a 
Arguments: {α : Type u_1} [LE α] [NoTopOrder α] (a : α)
Goal: ¬IsTop a
-- IsBot.isMin
protected theorem IsBot.isMin (h : IsBot a) : IsMin a 
Arguments: {α : Type u_1} [LE α] {a : α} (h : IsBot a)
Goal: IsMin a
-- IsTop.isMax
protected theorem IsTop.isMax (h : IsTop a) : IsMax a 
Arguments: {α : Type u_1} [LE α] {a : α} (h : IsTop a)
Goal: IsMax a
-- isBot_toDual_iff
@[simp]
theorem isBot_toDual_iff : IsBot (toDual a) ↔ IsTop a 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: IsBot (OrderDual.toDual a) ↔ IsTop a
-- isTop_toDual_iff
@[simp]
theorem isTop_toDual_iff : IsTop (toDual a) ↔ IsBot a 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: IsTop (OrderDual.toDual a) ↔ IsBot a
-- isMin_toDual_iff
@[simp]
theorem isMin_toDual_iff : IsMin (toDual a) ↔ IsMax a 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: IsMin (OrderDual.toDual a) ↔ IsMax a
-- isMax_toDual_iff
@[simp]
theorem isMax_toDual_iff : IsMax (toDual a) ↔ IsMin a 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: IsMax (OrderDual.toDual a) ↔ IsMin a
-- isBot_ofDual_iff
@[simp]
theorem isBot_ofDual_iff {a : αᵒᵈ} : IsBot (ofDual a) ↔ IsTop a 
Arguments: {α : Type u_1} [LE α] {a : αᵒᵈ}
Goal: IsBot (OrderDual.ofDual a) ↔ IsTop a
-- isTop_ofDual_iff
@[simp]
theorem isTop_ofDual_iff {a : αᵒᵈ} : IsTop (ofDual a) ↔ IsBot a 
Arguments: {α : Type u_1} [LE α] {a : αᵒᵈ}
Goal: IsTop (OrderDual.ofDual a) ↔ IsBot a
-- isMin_ofDual_iff
@[simp]
theorem isMin_ofDual_iff {a : αᵒᵈ} : IsMin (ofDual a) ↔ IsMax a 
Arguments: {α : Type u_1} [LE α] {a : αᵒᵈ}
Goal: IsMin (OrderDual.ofDual a) ↔ IsMax a
-- isMax_ofDual_iff
@[simp]
theorem isMax_ofDual_iff {a : αᵒᵈ} : IsMax (ofDual a) ↔ IsMin a 
Arguments: {α : Type u_1} [LE α] {a : αᵒᵈ}
Goal: IsMax (OrderDual.ofDual a) ↔ IsMin a
-- IsBot.mono
theorem IsBot.mono (ha : IsBot a) (h : b ≤ a) : IsBot b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (ha : IsBot a) (h : b ≤ a)
Goal: IsBot b
-- IsTop.mono
theorem IsTop.mono (ha : IsTop a) (h : a ≤ b) : IsTop b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (ha : IsTop a) (h : a ≤ b)
Goal: IsTop b
-- IsMin.mono
theorem IsMin.mono (ha : IsMin a) (h : b ≤ a) : IsMin b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (ha : IsMin a) (h : b ≤ a)
Goal: IsMin b
-- IsMax.mono
theorem IsMax.mono (ha : IsMax a) (h : a ≤ b) : IsMax b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (ha : IsMax a) (h : a ≤ b)
Goal: IsMax b
-- IsMin.not_lt
theorem IsMin.not_lt (h : IsMin a) : ¬b < a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : IsMin a)
Goal: ¬b < a
-- IsMax.not_lt
theorem IsMax.not_lt (h : IsMax a) : ¬a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : IsMax a)
Goal: ¬a < b
-- not_isMin_of_lt
@[simp]
theorem not_isMin_of_lt (h : b < a) : ¬IsMin a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : b < a)
Goal: ¬IsMin a
-- not_isMax_of_lt
@[simp]
theorem not_isMax_of_lt (h : a < b) : ¬IsMax a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a < b)
Goal: ¬IsMax a
-- isMin_iff_forall_not_lt
theorem isMin_iff_forall_not_lt : IsMin a ↔ ∀ b, ¬b < a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: IsMin a ↔ ∀ (b : α), ¬b < a
-- isMax_iff_forall_not_lt
theorem isMax_iff_forall_not_lt : IsMax a ↔ ∀ b, ¬a < b 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: IsMax a ↔ ∀ (b : α), ¬a < b
-- not_isMin_iff
@[simp]
theorem not_isMin_iff : ¬IsMin a ↔ ∃ b, b < a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: ¬IsMin a ↔ ∃ b, b < a
-- not_isMax_iff
@[simp]
theorem not_isMax_iff : ¬IsMax a ↔ ∃ b, a < b 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: ¬IsMax a ↔ ∃ b, a < b
-- not_isMin
@[simp]
theorem not_isMin [NoMinOrder α] (a : α) : ¬IsMin a 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α)
Goal: ¬IsMin a
-- not_isMax
@[simp]
theorem not_isMax [NoMaxOrder α] (a : α) : ¬IsMax a 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α)
Goal: ¬IsMax a
-- Subsingleton.isBot
protected theorem isBot (a : α) : IsBot a 
Arguments: {α : Type u_1} [Preorder α] [Subsingleton α] (a : α)
Goal: IsBot a
-- Subsingleton.isTop
protected theorem isTop (a : α) : IsTop a 
Arguments: {α : Type u_1} [Preorder α] [Subsingleton α] (a : α)
Goal: IsTop a
-- Subsingleton.isMin
protected theorem isMin (a : α) : IsMin a 
Arguments: {α : Type u_1} [Preorder α] [Subsingleton α] (a : α)
Goal: IsMin a
-- Subsingleton.isMax
protected theorem isMax (a : α) : IsMax a 
Arguments: {α : Type u_1} [Preorder α] [Subsingleton α] (a : α)
Goal: IsMax a
-- IsMin.eq_of_le
protected theorem IsMin.eq_of_le (ha : IsMin a) (h : b ≤ a) : b = a 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (ha : IsMin a) (h : b ≤ a)
Goal: b = a
-- IsMin.eq_of_ge
protected theorem IsMin.eq_of_ge (ha : IsMin a) (h : b ≤ a) : a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (ha : IsMin a) (h : b ≤ a)
Goal: a = b
-- IsMax.eq_of_le
protected theorem IsMax.eq_of_le (ha : IsMax a) (h : a ≤ b) : a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (ha : IsMax a) (h : a ≤ b)
Goal: a = b
-- IsMax.eq_of_ge
protected theorem IsMax.eq_of_ge (ha : IsMax a) (h : a ≤ b) : b = a 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (ha : IsMax a) (h : a ≤ b)
Goal: b = a
-- IsBot.prod_mk
theorem IsBot.prod_mk (ha : IsBot a) (hb : IsBot b) : IsBot (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : β} (ha : IsBot a) (hb : IsBot b)
Goal: IsBot (a, b)
-- IsTop.prod_mk
theorem IsTop.prod_mk (ha : IsTop a) (hb : IsTop b) : IsTop (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : β} (ha : IsTop a) (hb : IsTop b)
Goal: IsTop (a, b)
-- IsMin.prod_mk
theorem IsMin.prod_mk (ha : IsMin a) (hb : IsMin b) : IsMin (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : β} (ha : IsMin a) (hb : IsMin b)
Goal: IsMin (a, b)
-- IsMax.prod_mk
theorem IsMax.prod_mk (ha : IsMax a) (hb : IsMax b) : IsMax (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {a : α} {b : β} (ha : IsMax a) (hb : IsMax b)
Goal: IsMax (a, b)
-- IsBot.fst
theorem IsBot.fst (hx : IsBot x) : IsBot x.1 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsBot x)
Goal: IsBot x.1
-- IsBot.snd
theorem IsBot.snd (hx : IsBot x) : IsBot x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsBot x)
Goal: IsBot x.2
-- IsTop.fst
theorem IsTop.fst (hx : IsTop x) : IsTop x.1 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsTop x)
Goal: IsTop x.1
-- IsTop.snd
theorem IsTop.snd (hx : IsTop x) : IsTop x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsTop x)
Goal: IsTop x.2
-- IsMin.fst
theorem IsMin.fst (hx : IsMin x) : IsMin x.1 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsMin x)
Goal: IsMin x.1
-- IsMin.snd
theorem IsMin.snd (hx : IsMin x) : IsMin x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsMin x)
Goal: IsMin x.2
-- IsMax.fst
theorem IsMax.fst (hx : IsMax x) : IsMax x.1 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsMax x)
Goal: IsMax x.1
-- IsMax.snd
theorem IsMax.snd (hx : IsMax x) : IsMax x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β} (hx : IsMax x)
Goal: IsMax x.2
-- Prod.isBot_iff
theorem Prod.isBot_iff : IsBot x ↔ IsBot x.1 ∧ IsBot x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β}
Goal: IsBot x ↔ IsBot x.1 ∧ IsBot x.2
-- Prod.isTop_iff
theorem Prod.isTop_iff : IsTop x ↔ IsTop x.1 ∧ IsTop x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β}
Goal: IsTop x ↔ IsTop x.1 ∧ IsTop x.2
-- Prod.isMin_iff
theorem Prod.isMin_iff : IsMin x ↔ IsMin x.1 ∧ IsMin x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β}
Goal: IsMin x ↔ IsMin x.1 ∧ IsMin x.2
-- Prod.isMax_iff
theorem Prod.isMax_iff : IsMax x ↔ IsMax x.1 ∧ IsMax x.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {x : α × β}
Goal: IsMax x ↔ IsMax x.1 ∧ IsMax x.2
-- RelIso.coe_one
@[simp]
theorem coe_one : ((1 : r ≃r r) : α → α) = id 
Arguments: {α : Type u_1} {r : α → α → Prop}
Goal: ⇑1 = id
-- RelIso.coe_mul
@[simp]
theorem coe_mul (e₁ e₂ : r ≃r r) : ((e₁ * e₂) : α → α) = e₁ ∘ e₂ 
Arguments: {α : Type u_1} {r : α → α → Prop} (e₁ e₂ : r ≃r r)
Goal: ⇑(e₁ * e₂) = ⇑e₁ ∘ ⇑e₂
-- RelIso.mul_apply
theorem mul_apply (e₁ e₂ : r ≃r r) (x : α) : (e₁ * e₂) x = e₁ (e₂ x) 
Arguments: {α : Type u_1} {r : α → α → Prop} (e₁ e₂ : r ≃r r) (x : α)
Goal: (e₁ * e₂) x = e₁ (e₂ x)
-- RelIso.inv_apply_self
@[simp]
theorem inv_apply_self (e : r ≃r r) (x) : e⁻¹ (e x) = x 
Arguments: {α : Type u_1} {r : α → α → Prop} (e : r ≃r r) (x : α)
Goal: e⁻¹ (e x) = x
-- RelIso.apply_inv_self
@[simp]
theorem apply_inv_self (e : r ≃r r) (x) : e (e⁻¹ x) = x 
Arguments: {α : Type u_1} {r : α → α → Prop} (e : r ≃r r) (x : α)
Goal: e (e⁻¹ x) = x
-- RelHomClass.map_inf
theorem map_inf [SemilatticeInf α] [LinearOrder β] [FunLike F β α]
    [RelHomClass F (· < ·) (· < ·)] (a : F) (m n : β) :
    a (m ⊓ n) = a m ⊓ a n 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_5} [SemilatticeInf α] [LinearOrder β] [FunLike F β α] [RelHomClass F (fun x x_1 => x < x_1) fun x x_1 => x < x_1] (a : F) (m n : β)
Goal: a (m ⊓ n) = a m ⊓ a n
-- RelHomClass.map_sup
theorem map_sup [SemilatticeSup α] [LinearOrder β] [FunLike F β α]
    [RelHomClass F (· > ·) (· > ·)] (a : F) (m n : β) :
    a (m ⊔ n) = a m ⊔ a n 
Arguments: {α : Type u_1} {β : Type u_2} {F : Type u_5} [SemilatticeSup α] [LinearOrder β] [FunLike F β α] [RelHomClass F (fun x x_1 => x > x_1) fun x x_1 => x > x_1] (a : F) (m n : β)
Goal: a (m ⊔ n) = a m ⊔ a n
-- RelIso.range_eq
@[simp]
theorem range_eq (e : r ≃r s) : Set.range e = Set.univ 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s)
Goal: Set.range ⇑e = Set.univ
-- subrel_val
@[simp]
theorem subrel_val (r : α → α → Prop) (p : Set α) {a b} : Subrel r p a b ↔ r a.1 b.1 
Arguments: {α : Type u_1} (r : α → α → Prop) (p : Set α) {a b : ↑p}
Goal: Subrel r p a b ↔ r ↑a ↑b
-- Subrel.relEmbedding_apply
@[simp]
theorem relEmbedding_apply (r : α → α → Prop) (p a) : Subrel.relEmbedding r p a = a.1 
Arguments: {α : Type u_1} (r : α → α → Prop) (p : Set α) (a : ↑p)
Goal: (Subrel.relEmbedding r p) a = ↑a
-- RelEmbedding.codRestrict_apply
@[simp]
theorem RelEmbedding.codRestrict_apply (p) (f : r ↪r s) (H a) :
    RelEmbedding.codRestrict p f H a = ⟨f a, H a⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : Set β) (f : r ↪r s) (H : ∀ (a : α), f a ∈ p) (a : α)
Goal: (RelEmbedding.codRestrict p f H) a = ⟨f a, ⋯⟩
-- RelIso.image_eq_preimage_symm
theorem RelIso.image_eq_preimage_symm (e : r ≃r s) (t : Set α) : e '' t = e.symm ⁻¹' t 
Arguments: {α : Type u_5} {β : Type u_6} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (t : Set α)
Goal: ⇑e '' t = ⇑e.symm ⁻¹' t
-- RelIso.preimage_eq_image_symm
theorem RelIso.preimage_eq_image_symm (e : r ≃r s) (t : Set β) : e ⁻¹' t = e.symm '' t 
Arguments: {α : Type u_5} {β : Type u_6} {r : α → α → Prop} {s : β → β → Prop} (e : r ≃r s) (t : Set β)
Goal: ⇑e ⁻¹' t = ⇑e.symm '' t
-- extend_partialOrder
theorem extend_partialOrder {α : Type u} (r : α → α → Prop) [IsPartialOrder α r] :
    ∃ s : α → α → Prop, IsLinearOrder α s ∧ r ≤ s 
Arguments: {α : Type u} (r : α → α → Prop) [IsPartialOrder α r]
Goal: ∃ s, IsLinearOrder α s ∧ r ≤ s
-- HeytingHom.map_bot'
structure HeytingHom (α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] extends
  LatticeHom α β where
    protected map_bot' : toFun ⊥ = ⊥
    protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [HeytingAlgebra α] [HeytingAlgebra β] (self : HeytingHom α β)
Goal: self.toFun ⊥ = ⊥
-- HeytingHom.map_himp'
structure HeytingHom (α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] extends
  LatticeHom α β where
    protected map_bot' : toFun ⊥ = ⊥
    protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [HeytingAlgebra α] [HeytingAlgebra β] (self : HeytingHom α β) (a b : α)
Goal: self.toFun (a ⇨ b) = self.toFun a ⇨ self.toFun b
-- CoheytingHom.map_top'
structure CoheytingHom (α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] extends
  LatticeHom α β where
    protected map_top' : toFun ⊤ = ⊤
    protected map_sdiff' : ∀ a b, toFun (a \ b) = toFun a \ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [CoheytingAlgebra α] [CoheytingAlgebra β] (self : CoheytingHom α β)
Goal: self.toFun ⊤ = ⊤
-- CoheytingHom.map_sdiff'
structure CoheytingHom (α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] extends
  LatticeHom α β where
    protected map_top' : toFun ⊤ = ⊤
    protected map_sdiff' : ∀ a b, toFun (a \ b) = toFun a \ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [CoheytingAlgebra α] [CoheytingAlgebra β] (self : CoheytingHom α β) (a b : α)
Goal: self.toFun (a \ b) = self.toFun a \ self.toFun b
-- BiheytingHom.map_himp'
structure BiheytingHom (α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] extends
  LatticeHom α β where
    protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b
    protected map_sdiff' : ∀ a b, toFun (a \ b) = toFun a \ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [BiheytingAlgebra α] [BiheytingAlgebra β] (self : BiheytingHom α β) (a b : α)
Goal: self.toFun (a ⇨ b) = self.toFun a ⇨ self.toFun b
-- BiheytingHom.map_sdiff'
structure BiheytingHom (α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] extends
  LatticeHom α β where
    protected map_himp' : ∀ a b, toFun (a ⇨ b) = toFun a ⇨ toFun b
    protected map_sdiff' : ∀ a b, toFun (a \ b) = toFun a \ toFun b
Arguments: {α : Type u_6} {β : Type u_7} [BiheytingAlgebra α] [BiheytingAlgebra β] (self : BiheytingHom α β) (a b : α)
Goal: self.toFun (a \ b) = self.toFun a \ self.toFun b
-- HeytingHomClass.map_bot
class HeytingHomClass (F α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_bot (f : F) : f ⊥ = ⊥
    map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β] [self : HeytingHomClass F α β] (f : F)
Goal: f ⊥ = ⊥
-- HeytingHomClass.map_himp
class HeytingHomClass (F α β : Type*) [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_bot (f : F) : f ⊥ = ⊥
    map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [HeytingAlgebra α] [HeytingAlgebra β] [FunLike F α β] [self : HeytingHomClass F α β] (f : F) (a b : α)
Goal: f (a ⇨ b) = f a ⇨ f b
-- CoheytingHomClass.map_top
class CoheytingHomClass (F α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_top (f : F) : f ⊤ = ⊤
    map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β] [self : CoheytingHomClass F α β] (f : F)
Goal: f ⊤ = ⊤
-- CoheytingHomClass.map_sdiff
class CoheytingHomClass (F α β : Type*) [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_top (f : F) : f ⊤ = ⊤
    map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [CoheytingAlgebra α] [CoheytingAlgebra β] [FunLike F α β] [self : CoheytingHomClass F α β] (f : F) (a b : α)
Goal: f (a \ b) = f a \ f b
-- BiheytingHomClass.map_sdiff
class BiheytingHomClass (F α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b
    map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β] [self : BiheytingHomClass F α β] (f : F) (a b : α)
Goal: f (a \ b) = f a \ f b
-- BiheytingHomClass.map_himp
class BiheytingHomClass (F α β : Type*) [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β]
  extends LatticeHomClass F α β : Prop where
    map_himp (f : F) : ∀ a b, f (a ⇨ b) = f a ⇨ f b
    map_sdiff (f : F) : ∀ a b, f (a \ b) = f a \ f b
Arguments: {F : Type u_6} {α : Type u_7} {β : Type u_8} [BiheytingAlgebra α] [BiheytingAlgebra β] [FunLike F α β] [self : BiheytingHomClass F α β] (f : F) (a b : α)
Goal: f (a ⇨ b) = f a ⇨ f b
-- BoundedLatticeHomClass.toBiheytingHomClass
lemma BoundedLatticeHomClass.toBiheytingHomClass [BooleanAlgebra α] [BooleanAlgebra β]
    [BoundedLatticeHomClass F α β] : BiheytingHomClass F α β 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [BooleanAlgebra α] [BooleanAlgebra β] [BoundedLatticeHomClass F α β]
Goal: BiheytingHomClass F α β
-- map_compl
@[simp]
theorem map_compl (a : α) : f aᶜ = (f a)ᶜ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F) (a : α)
Goal: f aᶜ = (f a)ᶜ
-- map_bihimp
@[simp]
theorem map_bihimp (a b : α) : f (a ⇔ b) = f a ⇔ f b 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [HeytingAlgebra α] [HeytingAlgebra β] [HeytingHomClass F α β] (f : F) (a b : α)
Goal: f (bihimp a b) = bihimp (f a) (f b)
-- map_hnot
@[simp]
theorem map_hnot (a : α) : f (￢a) = ￢f a 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F) (a : α)
Goal: f (￢a) = ￢f a
-- map_symmDiff
@[simp]
theorem map_symmDiff (a b : α) : f (a ∆ b) = f a ∆ f b 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [FunLike F α β] [CoheytingAlgebra α] [CoheytingAlgebra β] [CoheytingHomClass F α β] (f : F) (a b : α)
Goal: f (symmDiff a b) = symmDiff (f a) (f b)
-- HeytingHom.toFun_eq_coe
theorem toFun_eq_coe {f : HeytingHom α β} : f.toFun = ⇑f 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] {f : HeytingHom α β}
Goal: f.toFun = ⇑f
-- HeytingHom.toFun_eq_coe_aux
@[simp]
theorem toFun_eq_coe_aux {f : HeytingHom α β} : (↑f.toLatticeHom) = ⇑f 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] {f : HeytingHom α β}
Goal: ⇑f.toLatticeHom = ⇑f
-- HeytingHom.ext
@[ext]
theorem ext {f g : HeytingHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] {f g : HeytingHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- HeytingHom.coe_copy
@[simp]
theorem coe_copy (f : HeytingHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- HeytingHom.copy_eq
theorem copy_eq (f : HeytingHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [HeytingAlgebra α] [HeytingAlgebra β] (f : HeytingHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- HeytingHom.coe_id
@[simp]
theorem coe_id : ⇑(HeytingHom.id α) = id 
Arguments: (α : Type u_2) [HeytingAlgebra α]
Goal: ⇑(HeytingHom.id α) = id
-- HeytingHom.id_apply
@[simp]
theorem id_apply (a : α) : HeytingHom.id α a = a 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: (HeytingHom.id α) a = a
-- HeytingHom.coe_comp
@[simp]
theorem coe_comp (f : HeytingHom β γ) (g : HeytingHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] (f : HeytingHom β γ) (g : HeytingHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- HeytingHom.comp_apply
@[simp]
theorem comp_apply (f : HeytingHom β γ) (g : HeytingHom α β) (a : α) : f.comp g a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [HeytingAlgebra α] [HeytingAlgebra β] [HeytingAlgebra γ] (f : HeytingHom β γ) (g : HeytingHom α β) (a : α)
Goal: (f.comp g) a = f (g a)