ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Directed.inf_iSup_eq
protected theorem Directed.inf_iSup_eq (h : Directed (· ≤ ·) f) :
    (a ⊓ ⨆ i, f i) = ⨆ i, a ⊓ f i 
Arguments: {ι : Sort u_1} {α : Type u_2} {f : ι → α} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} (h : Directed (fun x x_1 => x ≤ x_1) f)
Goal: a ⊓ ⨆ i, f i = ⨆ i, a ⊓ f i
-- Directed.iSup_inf_eq
protected theorem Directed.iSup_inf_eq (h : Directed (· ≤ ·) f) :
    (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a 
Arguments: {ι : Sort u_1} {α : Type u_2} {f : ι → α} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} (h : Directed (fun x x_1 => x ≤ x_1) f)
Goal: (⨆ i, f i) ⊓ a = ⨆ i, f i ⊓ a
-- DirectedOn.disjoint_sSup_right
protected theorem DirectedOn.disjoint_sSup_right (h : DirectedOn (· ≤ ·) s) :
    Disjoint a (sSup s) ↔ ∀ ⦃b⦄, b ∈ s → Disjoint a b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α} (h : DirectedOn (fun x x_1 => x ≤ x_1) s)
Goal: Disjoint a (sSup s) ↔ ∀ ⦃b : α⦄, b ∈ s → Disjoint a b
-- DirectedOn.disjoint_sSup_left
protected theorem DirectedOn.disjoint_sSup_left (h : DirectedOn (· ≤ ·) s) :
    Disjoint (sSup s) a ↔ ∀ ⦃b⦄, b ∈ s → Disjoint b a 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α} (h : DirectedOn (fun x x_1 => x ≤ x_1) s)
Goal: Disjoint (sSup s) a ↔ ∀ ⦃b : α⦄, b ∈ s → Disjoint b a
-- Directed.disjoint_iSup_right
protected theorem Directed.disjoint_iSup_right (h : Directed (· ≤ ·) f) :
    Disjoint a (⨆ i, f i) ↔ ∀ i, Disjoint a (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {f : ι → α} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} (h : Directed (fun x x_1 => x ≤ x_1) f)
Goal: Disjoint a (⨆ i, f i) ↔ ∀ (i : ι), Disjoint a (f i)
-- Directed.disjoint_iSup_left
protected theorem Directed.disjoint_iSup_left (h : Directed (· ≤ ·) f) :
    Disjoint (⨆ i, f i) a ↔ ∀ i, Disjoint (f i) a 
Arguments: {ι : Sort u_1} {α : Type u_2} {f : ι → α} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} (h : Directed (fun x x_1 => x ≤ x_1) f)
Goal: Disjoint (⨆ i, f i) a ↔ ∀ (i : ι), Disjoint (f i) a
-- inf_sSup_eq_iSup_inf_sup_finset
theorem inf_sSup_eq_iSup_inf_sup_finset :
    a ⊓ sSup s = ⨆ (t : Finset α) (_ : ↑t ⊆ s), a ⊓ t.sup id 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α}
Goal: a ⊓ sSup s = ⨆ t, ⨆ (_ : ↑t ⊆ s), a ⊓ t.sup id
-- CompleteLattice.setIndependent_iff_finite
theorem CompleteLattice.setIndependent_iff_finite {s : Set α} :
    CompleteLattice.SetIndependent s ↔
      ∀ t : Finset α, ↑t ⊆ s → CompleteLattice.SetIndependent (↑t : Set α) 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {s : Set α}
Goal: CompleteLattice.SetIndependent s ↔ ∀ (t : Finset α), ↑t ⊆ s → CompleteLattice.SetIndependent ↑t
-- CompleteLattice.independent_iff_supIndep_of_injOn
lemma CompleteLattice.independent_iff_supIndep_of_injOn {ι : Type*} {f : ι → α}
    (hf : InjOn f {i | f i ≠ ⊥}) :
    CompleteLattice.Independent f ↔ ∀ (s : Finset ι), s.SupIndep f 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {ι : Type u_3} {f : ι → α} (hf : Set.InjOn f {i | f i ≠ ⊥})
Goal: CompleteLattice.Independent f ↔ ∀ (s : Finset ι), s.SupIndep f
-- CompleteLattice.setIndependent_iUnion_of_directed
theorem CompleteLattice.setIndependent_iUnion_of_directed {η : Type*} {s : η → Set α}
    (hs : Directed (· ⊆ ·) s) (h : ∀ i, CompleteLattice.SetIndependent (s i)) :
    CompleteLattice.SetIndependent (⋃ i, s i) 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {η : Type u_3} {s : η → Set α} (hs : Directed (fun x x_1 => x ⊆ x_1) s) (h : ∀ (i : η), CompleteLattice.SetIndependent (s i))
Goal: CompleteLattice.SetIndependent (⋃ i, s i)
-- CompleteLattice.independent_sUnion_of_directed
theorem CompleteLattice.independent_sUnion_of_directed {s : Set (Set α)} (hs : DirectedOn (· ⊆ ·) s)
    (h : ∀ a ∈ s, CompleteLattice.SetIndependent a) : CompleteLattice.SetIndependent (⋃₀ s) 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {s : Set (Set α)} (hs : DirectedOn (fun x x_1 => x ⊆ x_1) s) (h : ∀ a ∈ s, CompleteLattice.SetIndependent a)
Goal: CompleteLattice.SetIndependent (⋃₀ s)
-- CompleteLattice.isCompactlyGenerated_of_wellFounded
theorem isCompactlyGenerated_of_wellFounded (h : WellFounded ((· > ·) : α → α → Prop)) :
    IsCompactlyGenerated α 
Arguments: {α : Type u_2} [CompleteLattice α] (h : WellFounded fun x x_1 => x > x_1)
Goal: IsCompactlyGenerated α
-- CompleteLattice.Iic_coatomic_of_compact_element
theorem Iic_coatomic_of_compact_element {k : α} (h : IsCompactElement k) :
    IsCoatomic (Set.Iic k) 
Arguments: {α : Type u_2} [CompleteLattice α] {k : α} (h : CompleteLattice.IsCompactElement k)
Goal: IsCoatomic ↑(Set.Iic k)
-- CompleteLattice.coatomic_of_top_compact
theorem coatomic_of_top_compact (h : IsCompactElement (⊤ : α)) : IsCoatomic α 
Arguments: {α : Type u_2} [CompleteLattice α] (h : CompleteLattice.IsCompactElement ⊤)
Goal: IsCoatomic α
-- exists_setIndependent_isCompl_sSup_atoms
theorem exists_setIndependent_isCompl_sSup_atoms (h : sSup { a : α | IsAtom a } = ⊤) (b : α) :
    ∃ s : Set α, CompleteLattice.SetIndependent s ∧
    IsCompl b (sSup s) ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a 
Arguments: {α : Type u_2} [CompleteLattice α] [IsModularLattice α] [IsCompactlyGenerated α] (h : sSup {a | IsAtom a} = ⊤) (b : α)
Goal: ∃ s, CompleteLattice.SetIndependent s ∧ IsCompl b (sSup s) ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a
-- exists_setIndependent_of_sSup_atoms_eq_top
theorem exists_setIndependent_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
    ∃ s : Set α, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a⦄, a ∈ s → IsAtom a 
Arguments: {α : Type u_2} [CompleteLattice α] [IsModularLattice α] [IsCompactlyGenerated α] (h : sSup {a | IsAtom a} = ⊤)
Goal: ∃ s, CompleteLattice.SetIndependent s ∧ sSup s = ⊤ ∧ ∀ ⦃a : α⦄, a ∈ s → IsAtom a
-- complementedLattice_of_sSup_atoms_eq_top
theorem complementedLattice_of_sSup_atoms_eq_top (h : sSup { a : α | IsAtom a } = ⊤) :
    ComplementedLattice α 
Arguments: {α : Type u_2} [CompleteLattice α] [IsModularLattice α] [IsCompactlyGenerated α] (h : sSup {a | IsAtom a} = ⊤)
Goal: ComplementedLattice α
-- complementedLattice_of_isAtomistic
theorem complementedLattice_of_isAtomistic [IsAtomistic α] : ComplementedLattice α 
Arguments: {α : Type u_2} [CompleteLattice α] [IsModularLattice α] [IsCompactlyGenerated α] [IsAtomistic α]
Goal: ComplementedLattice α
-- complementedLattice_iff_isAtomistic
theorem complementedLattice_iff_isAtomistic : ComplementedLattice α ↔ IsAtomistic α 
Arguments: {α : Type u_2} [CompleteLattice α] [IsModularLattice α] [IsCompactlyGenerated α]
Goal: ComplementedLattice α ↔ IsAtomistic α
-- Set.mem_ordConnectedComponent
theorem mem_ordConnectedComponent : y ∈ ordConnectedComponent s x ↔ [[x, y]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y : α}
Goal: y ∈ s.ordConnectedComponent x ↔ [[x, y]] ⊆ s
-- Set.dual_ordConnectedComponent
theorem dual_ordConnectedComponent :
    ordConnectedComponent (ofDual ⁻¹' s) (toDual x) = ofDual ⁻¹' ordConnectedComponent s x 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α}
Goal: (⇑OrderDual.ofDual ⁻¹' s).ordConnectedComponent (OrderDual.toDual x) = ⇑OrderDual.ofDual ⁻¹' s.ordConnectedComponent x
-- Set.ordConnectedComponent_subset
theorem ordConnectedComponent_subset : ordConnectedComponent s x ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α}
Goal: s.ordConnectedComponent x ⊆ s
-- Set.subset_ordConnectedComponent
theorem subset_ordConnectedComponent {t} [h : OrdConnected s] (hs : x ∈ s) (ht : s ⊆ t) :
    s ⊆ ordConnectedComponent t x 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {t : Set α} [h : s.OrdConnected] (hs : x ∈ s) (ht : s ⊆ t)
Goal: s ⊆ t.ordConnectedComponent x
-- Set.self_mem_ordConnectedComponent
@[simp]
theorem self_mem_ordConnectedComponent : x ∈ ordConnectedComponent s x ↔ x ∈ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α}
Goal: x ∈ s.ordConnectedComponent x ↔ x ∈ s
-- Set.nonempty_ordConnectedComponent
@[simp]
theorem nonempty_ordConnectedComponent : (ordConnectedComponent s x).Nonempty ↔ x ∈ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α}
Goal: (s.ordConnectedComponent x).Nonempty ↔ x ∈ s
-- Set.ordConnectedComponent_eq_empty
@[simp]
theorem ordConnectedComponent_eq_empty : ordConnectedComponent s x = ∅ ↔ x ∉ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α}
Goal: s.ordConnectedComponent x = ∅ ↔ x ∉ s
-- Set.ordConnectedComponent_empty
@[simp]
theorem ordConnectedComponent_empty : ordConnectedComponent ∅ x = ∅ 
Arguments: {α : Type u_1} [LinearOrder α] {x : α}
Goal: ∅.ordConnectedComponent x = ∅
-- Set.ordConnectedComponent_univ
@[simp]
theorem ordConnectedComponent_univ : ordConnectedComponent univ x = univ 
Arguments: {α : Type u_1} [LinearOrder α] {x : α}
Goal: Set.univ.ordConnectedComponent x = Set.univ
-- Set.ordConnectedComponent_inter
theorem ordConnectedComponent_inter (s t : Set α) (x : α) :
    ordConnectedComponent (s ∩ t) x = ordConnectedComponent s x ∩ ordConnectedComponent t x 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α) (x : α)
Goal: (s ∩ t).ordConnectedComponent x = s.ordConnectedComponent x ∩ t.ordConnectedComponent x
-- Set.mem_ordConnectedComponent_comm
theorem mem_ordConnectedComponent_comm :
    y ∈ ordConnectedComponent s x ↔ x ∈ ordConnectedComponent s y 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y : α}
Goal: y ∈ s.ordConnectedComponent x ↔ x ∈ s.ordConnectedComponent y
-- Set.mem_ordConnectedComponent_trans
theorem mem_ordConnectedComponent_trans (hxy : y ∈ ordConnectedComponent s x)
    (hyz : z ∈ ordConnectedComponent s y) : z ∈ ordConnectedComponent s x 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y z : α} (hxy : y ∈ s.ordConnectedComponent x) (hyz : z ∈ s.ordConnectedComponent y)
Goal: z ∈ s.ordConnectedComponent x
-- Set.ordConnectedComponent_eq
theorem ordConnectedComponent_eq (h : [[x, y]] ⊆ s) :
    ordConnectedComponent s x = ordConnectedComponent s y 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y : α} (h : [[x, y]] ⊆ s)
Goal: s.ordConnectedComponent x = s.ordConnectedComponent y
-- Set.ordConnectedProj_mem_ordConnectedComponent
theorem ordConnectedProj_mem_ordConnectedComponent (s : Set α) (x : s) :
    ordConnectedProj s x ∈ ordConnectedComponent s x 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α) (x : ↑s)
Goal: s.ordConnectedProj x ∈ s.ordConnectedComponent ↑x
-- Set.mem_ordConnectedComponent_ordConnectedProj
theorem mem_ordConnectedComponent_ordConnectedProj (s : Set α) (x : s) :
    ↑x ∈ ordConnectedComponent s (ordConnectedProj s x) 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α) (x : ↑s)
Goal: ↑x ∈ s.ordConnectedComponent (s.ordConnectedProj x)
-- Set.ordConnectedComponent_ordConnectedProj
@[simp]
theorem ordConnectedComponent_ordConnectedProj (s : Set α) (x : s) :
    ordConnectedComponent s (ordConnectedProj s x) = ordConnectedComponent s x 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α) (x : ↑s)
Goal: s.ordConnectedComponent (s.ordConnectedProj x) = s.ordConnectedComponent ↑x
-- Set.ordConnectedProj_eq
@[simp]
theorem ordConnectedProj_eq {x y : s} :
    ordConnectedProj s x = ordConnectedProj s y ↔ [[(x : α), y]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y : ↑s}
Goal: s.ordConnectedProj x = s.ordConnectedProj y ↔ [[↑x, ↑y]] ⊆ s
-- Set.dual_ordConnectedSection
theorem dual_ordConnectedSection (s : Set α) :
    ordConnectedSection (ofDual ⁻¹' s) = ofDual ⁻¹' ordConnectedSection s 
Arguments: {α : Type u_1} [LinearOrder α] (s : Set α)
Goal: (⇑OrderDual.ofDual ⁻¹' s).ordConnectedSection = ⇑OrderDual.ofDual ⁻¹' s.ordConnectedSection
-- Set.ordConnectedSection_subset
theorem ordConnectedSection_subset : ordConnectedSection s ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α}
Goal: s.ordConnectedSection ⊆ s
-- Set.eq_of_mem_ordConnectedSection_of_uIcc_subset
theorem eq_of_mem_ordConnectedSection_of_uIcc_subset (hx : x ∈ ordConnectedSection s)
    (hy : y ∈ ordConnectedSection s) (h : [[x, y]] ⊆ s) : x = y 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x y : α} (hx : x ∈ s.ordConnectedSection) (hy : y ∈ s.ordConnectedSection) (h : [[x, y]] ⊆ s)
Goal: x = y
-- Set.ordSeparatingSet_comm
theorem ordSeparatingSet_comm (s t : Set α) : ordSeparatingSet s t = ordSeparatingSet t s 
Arguments: {α : Type u_1} [LinearOrder α] (s t : Set α)
Goal: s.ordSeparatingSet t = t.ordSeparatingSet s
-- Set.disjoint_left_ordSeparatingSet
theorem disjoint_left_ordSeparatingSet : Disjoint s (ordSeparatingSet s t) 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α}
Goal: Disjoint s (s.ordSeparatingSet t)
-- Set.disjoint_right_ordSeparatingSet
theorem disjoint_right_ordSeparatingSet : Disjoint t (ordSeparatingSet s t) 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α}
Goal: Disjoint t (s.ordSeparatingSet t)
-- Set.dual_ordSeparatingSet
theorem dual_ordSeparatingSet :
    ordSeparatingSet (ofDual ⁻¹' s) (ofDual ⁻¹' t) = ofDual ⁻¹' ordSeparatingSet s t 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α}
Goal: (⇑OrderDual.ofDual ⁻¹' s).ordSeparatingSet (⇑OrderDual.ofDual ⁻¹' t) = ⇑OrderDual.ofDual ⁻¹' s.ordSeparatingSet t
-- Set.disjoint_ordT5Nhd
theorem disjoint_ordT5Nhd : Disjoint (ordT5Nhd s t) (ordT5Nhd t s) 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α}
Goal: Disjoint (s.ordT5Nhd t) (t.ordT5Nhd s)
-- symmDiff_def
theorem symmDiff_def [Sup α] [SDiff α] (a b : α) : a ∆ b = a \ b ⊔ b \ a 
Arguments: {α : Type u_2} [Sup α] [SDiff α] (a b : α)
Goal: a ∆ b = a \ b ⊔ b \ a
-- bihimp_def
theorem bihimp_def [Inf α] [HImp α] (a b : α) : a ⇔ b = (b ⇨ a) ⊓ (a ⇨ b) 
Arguments: {α : Type u_2} [Inf α] [HImp α] (a b : α)
Goal: a ⇔ b = (b ⇨ a) ⊓ (a ⇨ b)
-- symmDiff_eq_Xor'
theorem symmDiff_eq_Xor' (p q : Prop) : p ∆ q = Xor' p q 
Arguments: (p q : Prop)
Goal: p ∆ q = Xor' p q
-- bihimp_iff_iff
@[simp]
theorem bihimp_iff_iff {p q : Prop} : p ⇔ q ↔ (p ↔ q) 
Arguments: {p q : Prop}
Goal: p ⇔ q ↔ (p ↔ q)
-- Bool.symmDiff_eq_xor
@[simp]
theorem Bool.symmDiff_eq_xor : ∀ p q : Bool, p ∆ q = xor p q 
Arguments: (p q : Bool)
Goal: p ∆ q = xor p q
-- toDual_symmDiff
@[simp]
theorem toDual_symmDiff : toDual (a ∆ b) = toDual a ⇔ toDual b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: OrderDual.toDual (a ∆ b) = OrderDual.toDual a ⇔ OrderDual.toDual b
-- ofDual_bihimp
@[simp]
theorem ofDual_bihimp (a b : αᵒᵈ) : ofDual (a ⇔ b) = ofDual a ∆ ofDual b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a ⇔ b) = OrderDual.ofDual a ∆ OrderDual.ofDual b
-- symmDiff_comm
theorem symmDiff_comm : a ∆ b = b ∆ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ b = b ∆ a
-- symmDiff_self
@[simp]
theorem symmDiff_self : a ∆ a = ⊥ 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α)
Goal: a ∆ a = ⊥
-- symmDiff_bot
@[simp]
theorem symmDiff_bot : a ∆ ⊥ = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α)
Goal: a ∆ ⊥ = a
-- bot_symmDiff
@[simp]
theorem bot_symmDiff : ⊥ ∆ a = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a : α)
Goal: ⊥ ∆ a = a
-- symmDiff_eq_bot
@[simp]
theorem symmDiff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ∆ b = ⊥ ↔ a = b
-- symmDiff_of_le
theorem symmDiff_of_le {a b : α} (h : a ≤ b) : a ∆ b = b \ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: a ∆ b = b \ a
-- symmDiff_of_ge
theorem symmDiff_of_ge {a b : α} (h : b ≤ a) : a ∆ b = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : b ≤ a)
Goal: a ∆ b = a \ b
-- symmDiff_le
theorem symmDiff_le {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c) : a ∆ b ≤ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (ha : a ≤ b ⊔ c) (hb : b ≤ a ⊔ c)
Goal: a ∆ b ≤ c
-- symmDiff_le_iff
theorem symmDiff_le_iff {a b c : α} : a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a ∆ b ≤ c ↔ a ≤ b ⊔ c ∧ b ≤ a ⊔ c
-- symmDiff_le_sup
@[simp]
theorem symmDiff_le_sup {a b : α} : a ∆ b ≤ a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ∆ b ≤ a ⊔ b
-- symmDiff_eq_sup_sdiff_inf
theorem symmDiff_eq_sup_sdiff_inf : a ∆ b = (a ⊔ b) \ (a ⊓ b) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ b = (a ⊔ b) \ (a ⊓ b)
-- Disjoint.symmDiff_eq_sup
theorem Disjoint.symmDiff_eq_sup {a b : α} (h : Disjoint a b) : a ∆ b = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : Disjoint a b)
Goal: a ∆ b = a ⊔ b
-- symmDiff_sdiff
theorem symmDiff_sdiff : a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: a ∆ b \ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c)
-- symmDiff_sdiff_inf
@[simp]
theorem symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ b \ (a ⊓ b) = a ∆ b
-- symmDiff_sdiff_eq_sup
@[simp]
theorem symmDiff_sdiff_eq_sup : a ∆ (b \ a) = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ (b \ a) = a ⊔ b
-- sdiff_symmDiff_eq_sup
@[simp]
theorem sdiff_symmDiff_eq_sup : (a \ b) ∆ b = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: (a \ b) ∆ b = a ⊔ b
-- symmDiff_sup_inf
@[simp]
theorem symmDiff_sup_inf : a ∆ b ⊔ a ⊓ b = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ b ⊔ a ⊓ b = a ⊔ b
-- inf_sup_symmDiff
@[simp]
theorem inf_sup_symmDiff : a ⊓ b ⊔ a ∆ b = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ⊓ b ⊔ a ∆ b = a ⊔ b
-- symmDiff_symmDiff_inf
@[simp]
theorem symmDiff_symmDiff_inf : a ∆ b ∆ (a ⊓ b) = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ∆ b ∆ (a ⊓ b) = a ⊔ b
-- inf_symmDiff_symmDiff
@[simp]
theorem inf_symmDiff_symmDiff : (a ⊓ b) ∆ (a ∆ b) = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: (a ⊓ b) ∆ (a ∆ b) = a ⊔ b
-- symmDiff_triangle
theorem symmDiff_triangle : a ∆ c ≤ a ∆ b ⊔ b ∆ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: a ∆ c ≤ a ∆ b ⊔ b ∆ c
-- le_symmDiff_sup_right
theorem le_symmDiff_sup_right (a b : α) : a ≤ (a ∆ b) ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ≤ a ∆ b ⊔ b
-- le_symmDiff_sup_left
theorem le_symmDiff_sup_left (a b : α) : b ≤ (a ∆ b) ⊔ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: b ≤ a ∆ b ⊔ a
-- toDual_bihimp
@[simp]
theorem toDual_bihimp : toDual (a ⇔ b) = toDual a ∆ toDual b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: OrderDual.toDual (a ⇔ b) = OrderDual.toDual a ∆ OrderDual.toDual b
-- ofDual_symmDiff
@[simp]
theorem ofDual_symmDiff (a b : αᵒᵈ) : ofDual (a ∆ b) = ofDual a ⇔ ofDual b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a ∆ b) = OrderDual.ofDual a ⇔ OrderDual.ofDual b
-- bihimp_comm
theorem bihimp_comm : a ⇔ b = b ⇔ a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⇔ b = b ⇔ a
-- bihimp_self
@[simp]
theorem bihimp_self : a ⇔ a = ⊤ 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α)
Goal: a ⇔ a = ⊤
-- bihimp_top
@[simp]
theorem bihimp_top : a ⇔ ⊤ = a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α)
Goal: a ⇔ ⊤ = a
-- top_bihimp
@[simp]
theorem top_bihimp : ⊤ ⇔ a = a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a : α)
Goal: ⊤ ⇔ a = a
-- bihimp_eq_top
@[simp]
theorem bihimp_eq_top {a b : α} : a ⇔ b = ⊤ ↔ a = b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α}
Goal: a ⇔ b = ⊤ ↔ a = b
-- bihimp_of_le
theorem bihimp_of_le {a b : α} (h : a ≤ b) : a ⇔ b = b ⇨ a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: a ⇔ b = b ⇨ a
-- bihimp_of_ge
theorem bihimp_of_ge {a b : α} (h : b ≤ a) : a ⇔ b = a ⇨ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : b ≤ a)
Goal: a ⇔ b = a ⇨ b
-- le_bihimp
theorem le_bihimp {a b c : α} (hb : a ⊓ b ≤ c) (hc : a ⊓ c ≤ b) : a ≤ b ⇔ c 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b c : α} (hb : a ⊓ b ≤ c) (hc : a ⊓ c ≤ b)
Goal: a ≤ b ⇔ c
-- le_bihimp_iff
theorem le_bihimp_iff {a b c : α} : a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b c : α}
Goal: a ≤ b ⇔ c ↔ a ⊓ b ≤ c ∧ a ⊓ c ≤ b
-- inf_le_bihimp
@[simp]
theorem inf_le_bihimp {a b : α} : a ⊓ b ≤ a ⇔ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α}
Goal: a ⊓ b ≤ a ⇔ b
-- bihimp_eq_inf_himp_inf
theorem bihimp_eq_inf_himp_inf : a ⇔ b = a ⊔ b ⇨ a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⇔ b = a ⊔ b ⇨ a ⊓ b
-- Codisjoint.bihimp_eq_inf
theorem Codisjoint.bihimp_eq_inf {a b : α} (h : Codisjoint a b) : a ⇔ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : Codisjoint a b)
Goal: a ⇔ b = a ⊓ b
-- himp_bihimp
theorem himp_bihimp : a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c) 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b c : α)
Goal: a ⇨ b ⇔ c = (a ⊓ c ⇨ b) ⊓ (a ⊓ b ⇨ c)
-- sup_himp_bihimp
@[simp]
theorem sup_himp_bihimp : a ⊔ b ⇨ a ⇔ b = a ⇔ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⊔ b ⇨ a ⇔ b = a ⇔ b
-- bihimp_himp_eq_inf
@[simp]
theorem bihimp_himp_eq_inf : a ⇔ (a ⇨ b) = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⇔ (a ⇨ b) = a ⊓ b
-- himp_bihimp_eq_inf
@[simp]
theorem himp_bihimp_eq_inf : (b ⇨ a) ⇔ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: (b ⇨ a) ⇔ b = a ⊓ b
-- bihimp_inf_sup
@[simp]
theorem bihimp_inf_sup : a ⇔ b ⊓ (a ⊔ b) = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⇔ b ⊓ (a ⊔ b) = a ⊓ b
-- sup_inf_bihimp
@[simp]
theorem sup_inf_bihimp : (a ⊔ b) ⊓ a ⇔ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: (a ⊔ b) ⊓ a ⇔ b = a ⊓ b
-- bihimp_bihimp_sup
@[simp]
theorem bihimp_bihimp_sup : a ⇔ b ⇔ (a ⊔ b) = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: a ⇔ b ⇔ (a ⊔ b) = a ⊓ b
-- sup_bihimp_bihimp
@[simp]
theorem sup_bihimp_bihimp : (a ⊔ b) ⇔ (a ⇔ b) = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b : α)
Goal: (a ⊔ b) ⇔ (a ⇔ b) = a ⊓ b
-- bihimp_triangle
theorem bihimp_triangle : a ⇔ b ⊓ b ⇔ c ≤ a ⇔ c 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b c : α)
Goal: a ⇔ b ⊓ b ⇔ c ≤ a ⇔ c
-- symmDiff_top'
@[simp]
theorem symmDiff_top' : a ∆ ⊤ = ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: a ∆ ⊤ = ￢a
-- top_symmDiff'
@[simp]
theorem top_symmDiff' : ⊤ ∆ a = ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: ⊤ ∆ a = ￢a
-- hnot_symmDiff_self
@[simp]
theorem hnot_symmDiff_self : (￢a) ∆ a = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: (￢a) ∆ a = ⊤
-- symmDiff_hnot_self
@[simp]
theorem symmDiff_hnot_self : a ∆ (￢a) = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: a ∆ (￢a) = ⊤
-- IsCompl.symmDiff_eq_top
theorem IsCompl.symmDiff_eq_top {a b : α} (h : IsCompl a b) : a ∆ b = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: a ∆ b = ⊤
-- bihimp_bot
@[simp]
theorem bihimp_bot : a ⇔ ⊥ = aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: a ⇔ ⊥ = aᶜ
-- bot_bihimp
@[simp]
theorem bot_bihimp : ⊥ ⇔ a = aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: ⊥ ⇔ a = aᶜ
-- compl_bihimp_self
@[simp]
theorem compl_bihimp_self : aᶜ ⇔ a = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: aᶜ ⇔ a = ⊥
-- bihimp_hnot_self
@[simp]
theorem bihimp_hnot_self : a ⇔ aᶜ = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: a ⇔ aᶜ = ⊥
-- IsCompl.bihimp_eq_bot
theorem IsCompl.bihimp_eq_bot {a b : α} (h : IsCompl a b) : a ⇔ b = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: a ⇔ b = ⊥
-- sup_sdiff_symmDiff
@[simp]
theorem sup_sdiff_symmDiff : (a ⊔ b) \ a ∆ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: (a ⊔ b) \ a ∆ b = a ⊓ b