ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Codisjoint.himp_inf_cancel_left
theorem Codisjoint.himp_inf_cancel_left (h : Codisjoint a b) : b ⇨ a ⊓ b = a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α} (h : Codisjoint a b)
Goal: b ⇨ a ⊓ b = a
-- Codisjoint.himp_le_of_right_le
theorem Codisjoint.himp_le_of_right_le (hac : Codisjoint a c) (hba : b ≤ a) : c ⇨ b ≤ a 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b c : α} (hac : Codisjoint a c) (hba : b ≤ a)
Goal: c ⇨ b ≤ a
-- le_himp_himp
theorem le_himp_himp : a ≤ (a ⇨ b) ⇨ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α}
Goal: a ≤ (a ⇨ b) ⇨ b
-- himp_eq_himp_iff
@[simp] lemma himp_eq_himp_iff : b ⇨ a = a ⇨ b ↔ a = b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α}
Goal: b ⇨ a = a ⇨ b ↔ a = b
-- himp_ne_himp_iff
lemma himp_ne_himp_iff : b ⇨ a ≠ a ⇨ b ↔ a ≠ b 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b : α}
Goal: b ⇨ a ≠ a ⇨ b ↔ a ≠ b
-- himp_triangle
theorem himp_triangle (a b c : α) : (a ⇨ b) ⊓ (b ⇨ c) ≤ a ⇨ c 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] (a b c : α)
Goal: (a ⇨ b) ⊓ (b ⇨ c) ≤ a ⇨ c
-- himp_inf_himp_cancel
theorem himp_inf_himp_cancel (hba : b ≤ a) (hcb : c ≤ b) : (a ⇨ b) ⊓ (b ⇨ c) = a ⇨ c 
Arguments: {α : Type u_2} [GeneralizedHeytingAlgebra α] {a b c : α} (hba : b ≤ a) (hcb : c ≤ b)
Goal: (a ⇨ b) ⊓ (b ⇨ c) = a ⇨ c
-- sdiff_le_iff
@[simp]
theorem sdiff_le_iff : a \ b ≤ c ↔ a ≤ b ⊔ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ b ≤ c ↔ a ≤ b ⊔ c
-- sdiff_le_iff'
theorem sdiff_le_iff' : a \ b ≤ c ↔ a ≤ c ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ b ≤ c ↔ a ≤ c ⊔ b
-- sdiff_le_comm
theorem sdiff_le_comm : a \ b ≤ c ↔ a \ c ≤ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ b ≤ c ↔ a \ c ≤ b
-- sdiff_le
theorem sdiff_le : a \ b ≤ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b ≤ a
-- Disjoint.disjoint_sdiff_left
theorem Disjoint.disjoint_sdiff_left (h : Disjoint a b) : Disjoint (a \ c) b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : Disjoint a b)
Goal: Disjoint (a \ c) b
-- Disjoint.disjoint_sdiff_right
theorem Disjoint.disjoint_sdiff_right (h : Disjoint a b) : Disjoint a (b \ c) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : Disjoint a b)
Goal: Disjoint a (b \ c)
-- sdiff_le_iff_left
theorem sdiff_le_iff_left : a \ b ≤ b ↔ a ≤ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b ≤ b ↔ a ≤ b
-- sdiff_self
@[simp]
theorem sdiff_self : a \ a = ⊥ 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α}
Goal: a \ a = ⊥
-- le_sup_sdiff
theorem le_sup_sdiff : a ≤ b ⊔ a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ≤ b ⊔ a \ b
-- le_sdiff_sup
theorem le_sdiff_sup : a ≤ a \ b ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ≤ a \ b ⊔ b
-- sup_sdiff_left
theorem sup_sdiff_left : a ⊔ a \ b = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ⊔ a \ b = a
-- sup_sdiff_right
theorem sup_sdiff_right : a \ b ⊔ a = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b ⊔ a = a
-- inf_sdiff_left
theorem inf_sdiff_left : a \ b ⊓ a = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b ⊓ a = a \ b
-- inf_sdiff_right
theorem inf_sdiff_right : a ⊓ a \ b = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a ⊓ a \ b = a \ b
-- sup_sdiff_self
@[simp]
theorem sup_sdiff_self (a b : α) : a ⊔ b \ a = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a ⊔ b \ a = a ⊔ b
-- sdiff_sup_self
@[simp]
theorem sdiff_sup_self (a b : α) : b \ a ⊔ a = b ⊔ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: b \ a ⊔ a = b ⊔ a
-- sup_sdiff_eq_sup
theorem sup_sdiff_eq_sup (h : c ≤ a) : a ⊔ b \ c = a ⊔ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : c ≤ a)
Goal: a ⊔ b \ c = a ⊔ b
-- sup_sdiff_cancel'
theorem sup_sdiff_cancel' (hab : a ≤ b) (hbc : b ≤ c) : b ⊔ c \ a = c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
Goal: b ⊔ c \ a = c
-- sup_sdiff_cancel_right
theorem sup_sdiff_cancel_right (h : a ≤ b) : a ⊔ b \ a = b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: a ⊔ b \ a = b
-- sdiff_sup_cancel
theorem sdiff_sup_cancel (h : b ≤ a) : a \ b ⊔ b = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : b ≤ a)
Goal: a \ b ⊔ b = a
-- sup_le_of_le_sdiff_left
theorem sup_le_of_le_sdiff_left (h : b ≤ c \ a) (hac : a ≤ c) : a ⊔ b ≤ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : b ≤ c \ a) (hac : a ≤ c)
Goal: a ⊔ b ≤ c
-- sup_le_of_le_sdiff_right
theorem sup_le_of_le_sdiff_right (h : a ≤ c \ b) (hbc : b ≤ c) : a ⊔ b ≤ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : a ≤ c \ b) (hbc : b ≤ c)
Goal: a ⊔ b ≤ c
-- sdiff_eq_bot_iff
@[simp]
theorem sdiff_eq_bot_iff : a \ b = ⊥ ↔ a ≤ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b = ⊥ ↔ a ≤ b
-- sdiff_bot
@[simp]
theorem sdiff_bot : a \ ⊥ = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α}
Goal: a \ ⊥ = a
-- bot_sdiff
@[simp]
theorem bot_sdiff : ⊥ \ a = ⊥ 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a : α}
Goal: ⊥ \ a = ⊥
-- sdiff_sdiff_sdiff_le_sdiff
theorem sdiff_sdiff_sdiff_le_sdiff : (a \ b) \ (a \ c) ≤ c \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: (a \ b) \ (a \ c) ≤ c \ b
-- le_sup_sdiff_sup_sdiff
@[simp]
theorem le_sup_sdiff_sup_sdiff : a ≤ b ⊔ (a \ c ⊔ c \ b) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a ≤ b ⊔ (a \ c ⊔ c \ b)
-- sdiff_sdiff
theorem sdiff_sdiff (a b c : α) : (a \ b) \ c = a \ (b ⊔ c) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: (a \ b) \ c = a \ (b ⊔ c)
-- sdiff_sdiff_left
theorem sdiff_sdiff_left : (a \ b) \ c = a \ (b ⊔ c) 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: (a \ b) \ c = a \ (b ⊔ c)
-- sdiff_right_comm
theorem sdiff_right_comm (a b c : α) : (a \ b) \ c = (a \ c) \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: (a \ b) \ c = (a \ c) \ b
-- sdiff_sdiff_comm
theorem sdiff_sdiff_comm : (a \ b) \ c = (a \ c) \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: (a \ b) \ c = (a \ c) \ b
-- sdiff_idem
@[simp]
theorem sdiff_idem : (a \ b) \ b = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: (a \ b) \ b = a \ b
-- sdiff_sdiff_self
@[simp]
theorem sdiff_sdiff_self : (a \ b) \ a = ⊥ 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: (a \ b) \ a = ⊥
-- sup_sdiff_distrib
theorem sup_sdiff_distrib (a b c : α) : (a ⊔ b) \ c = a \ c ⊔ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: (a ⊔ b) \ c = a \ c ⊔ b \ c
-- sdiff_inf_distrib
theorem sdiff_inf_distrib (a b c : α) : a \ (b ⊓ c) = a \ b ⊔ a \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: a \ (b ⊓ c) = a \ b ⊔ a \ c
-- sup_sdiff
theorem sup_sdiff : (a ⊔ b) \ c = a \ c ⊔ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: (a ⊔ b) \ c = a \ c ⊔ b \ c
-- sup_sdiff_right_self
@[simp]
theorem sup_sdiff_right_self : (a ⊔ b) \ b = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: (a ⊔ b) \ b = a \ b
-- sup_sdiff_left_self
@[simp]
theorem sup_sdiff_left_self : (a ⊔ b) \ a = b \ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: (a ⊔ b) \ a = b \ a
-- sdiff_le_sdiff_right
@[gcongr]
theorem sdiff_le_sdiff_right (h : a ≤ b) : a \ c ≤ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : a ≤ b)
Goal: a \ c ≤ b \ c
-- sdiff_le_sdiff_left
@[gcongr]
theorem sdiff_le_sdiff_left (h : a ≤ b) : c \ b ≤ c \ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : a ≤ b)
Goal: c \ b ≤ c \ a
-- sdiff_le_sdiff
@[gcongr]
theorem sdiff_le_sdiff (hab : a ≤ b) (hcd : c ≤ d) : a \ d ≤ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c d : α} (hab : a ≤ b) (hcd : c ≤ d)
Goal: a \ d ≤ b \ c
-- sdiff_inf
theorem sdiff_inf : a \ (b ⊓ c) = a \ b ⊔ a \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ (b ⊓ c) = a \ b ⊔ a \ c
-- sdiff_inf_self_left
@[simp]
theorem sdiff_inf_self_left (a b : α) : a \ (a ⊓ b) = a \ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: a \ (a ⊓ b) = a \ b
-- sdiff_inf_self_right
@[simp]
theorem sdiff_inf_self_right (a b : α) : b \ (a ⊓ b) = b \ a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b : α)
Goal: b \ (a ⊓ b) = b \ a
-- Disjoint.sdiff_eq_left
theorem Disjoint.sdiff_eq_left (h : Disjoint a b) : a \ b = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : Disjoint a b)
Goal: a \ b = a
-- Disjoint.sdiff_eq_right
theorem Disjoint.sdiff_eq_right (h : Disjoint a b) : b \ a = b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : Disjoint a b)
Goal: b \ a = b
-- Disjoint.sup_sdiff_cancel_left
theorem Disjoint.sup_sdiff_cancel_left (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
-- Disjoint.sup_sdiff_cancel_right
theorem Disjoint.sup_sdiff_cancel_right (h : Disjoint a b) : (a ⊔ b) \ b = a 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α} (h : Disjoint a b)
Goal: (a ⊔ b) \ b = a
-- Disjoint.le_sdiff_of_le_left
theorem Disjoint.le_sdiff_of_le_left (hac : Disjoint a c) (hab : a ≤ b) : a ≤ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (hac : Disjoint a c) (hab : a ≤ b)
Goal: a ≤ b \ c
-- sdiff_sdiff_le
theorem sdiff_sdiff_le : a \ (a \ b) ≤ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ (a \ b) ≤ b
-- sdiff_eq_sdiff_iff
@[simp] lemma sdiff_eq_sdiff_iff : a \ b = b \ a ↔ a = b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b = b \ a ↔ a = b
-- sdiff_ne_sdiff_iff
lemma sdiff_ne_sdiff_iff : a \ b ≠ b \ a ↔ a ≠ b 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b : α}
Goal: a \ b ≠ b \ a ↔ a ≠ b
-- sdiff_triangle
theorem sdiff_triangle (a b c : α) : a \ c ≤ a \ b ⊔ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] (a b c : α)
Goal: a \ c ≤ a \ b ⊔ b \ c
-- sdiff_sup_sdiff_cancel
theorem sdiff_sup_sdiff_cancel (hba : b ≤ a) (hcb : c ≤ b) : a \ b ⊔ b \ c = a \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (hba : b ≤ a) (hcb : c ≤ b)
Goal: a \ b ⊔ b \ c = a \ c
-- sdiff_le_sdiff_of_sup_le_sup_left
theorem sdiff_le_sdiff_of_sup_le_sup_left (h : c ⊔ a ≤ c ⊔ b) : a \ c ≤ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : c ⊔ a ≤ c ⊔ b)
Goal: a \ c ≤ b \ c
-- sdiff_le_sdiff_of_sup_le_sup_right
theorem sdiff_le_sdiff_of_sup_le_sup_right (h : a ⊔ c ≤ b ⊔ c) : a \ c ≤ b \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α} (h : a ⊔ c ≤ b ⊔ c)
Goal: a \ c ≤ b \ c
-- inf_sdiff_sup_left
@[simp]
theorem inf_sdiff_sup_left : a \ c ⊓ (a ⊔ b) = a \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ c ⊓ (a ⊔ b) = a \ c
-- inf_sdiff_sup_right
@[simp]
theorem inf_sdiff_sup_right : a \ c ⊓ (b ⊔ a) = a \ c 
Arguments: {α : Type u_2} [GeneralizedCoheytingAlgebra α] {a b c : α}
Goal: a \ c ⊓ (b ⊔ a) = a \ c
-- himp_bot
@[simp]
theorem himp_bot (a : α) : a ⇨ ⊥ = aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: a ⇨ ⊥ = aᶜ
-- bot_himp
@[simp]
theorem bot_himp (a : α) : ⊥ ⇨ a = ⊤ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: ⊥ ⇨ a = ⊤
-- compl_sup_distrib
theorem compl_sup_distrib (a b : α) : (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a b : α)
Goal: (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ
-- compl_sup
@[simp]
theorem compl_sup : (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: (a ⊔ b)ᶜ = aᶜ ⊓ bᶜ
-- compl_le_himp
theorem compl_le_himp : aᶜ ≤ a ⇨ b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: aᶜ ≤ a ⇨ b
-- compl_sup_le_himp
theorem compl_sup_le_himp : aᶜ ⊔ b ≤ a ⇨ b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: aᶜ ⊔ b ≤ a ⇨ b
-- sup_compl_le_himp
theorem sup_compl_le_himp : b ⊔ aᶜ ≤ a ⇨ b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: b ⊔ aᶜ ≤ a ⇨ b
-- himp_compl
@[simp]
theorem himp_compl (a : α) : a ⇨ aᶜ = aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: a ⇨ aᶜ = aᶜ
-- himp_compl_comm
theorem himp_compl_comm (a b : α) : a ⇨ bᶜ = b ⇨ aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a b : α)
Goal: a ⇨ bᶜ = b ⇨ aᶜ
-- le_compl_iff_disjoint_right
theorem le_compl_iff_disjoint_right : a ≤ bᶜ ↔ Disjoint a b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: a ≤ bᶜ ↔ Disjoint a b
-- le_compl_iff_disjoint_left
theorem le_compl_iff_disjoint_left : a ≤ bᶜ ↔ Disjoint b a 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: a ≤ bᶜ ↔ Disjoint b a
-- le_compl_comm
theorem le_compl_comm : a ≤ bᶜ ↔ b ≤ aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: a ≤ bᶜ ↔ b ≤ aᶜ
-- disjoint_compl_left
theorem disjoint_compl_left : Disjoint aᶜ a 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: Disjoint aᶜ a
-- disjoint_compl_right
theorem disjoint_compl_right : Disjoint a aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: Disjoint a aᶜ
-- LE.le.disjoint_compl_left
theorem LE.le.disjoint_compl_left (h : b ≤ a) : Disjoint aᶜ b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : b ≤ a)
Goal: Disjoint aᶜ b
-- LE.le.disjoint_compl_right
theorem LE.le.disjoint_compl_right (h : a ≤ b) : Disjoint a bᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: Disjoint a bᶜ
-- IsCompl.compl_eq
theorem IsCompl.compl_eq (h : IsCompl a b) : aᶜ = b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: aᶜ = b
-- IsCompl.eq_compl
theorem IsCompl.eq_compl (h : IsCompl a b) : a = bᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: a = bᶜ
-- compl_unique
theorem compl_unique (h₀ : a ⊓ b = ⊥) (h₁ : a ⊔ b = ⊤) : aᶜ = b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h₀ : a ⊓ b = ⊥) (h₁ : a ⊔ b = ⊤)
Goal: aᶜ = b
-- inf_compl_self
@[simp]
theorem inf_compl_self (a : α) : a ⊓ aᶜ = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: a ⊓ aᶜ = ⊥
-- compl_inf_self
@[simp]
theorem compl_inf_self (a : α) : aᶜ ⊓ a = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: aᶜ ⊓ a = ⊥
-- inf_compl_eq_bot
theorem inf_compl_eq_bot : a ⊓ aᶜ = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: a ⊓ aᶜ = ⊥
-- compl_inf_eq_bot
theorem compl_inf_eq_bot : aᶜ ⊓ a = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: aᶜ ⊓ a = ⊥
-- compl_top
@[simp]
theorem compl_top : (⊤ : α)ᶜ = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α]
Goal: ⊤ᶜ = ⊥
-- compl_bot
@[simp]
theorem compl_bot : (⊥ : α)ᶜ = ⊤ 
Arguments: {α : Type u_2} [HeytingAlgebra α]
Goal: ⊥ᶜ = ⊤
-- le_compl_self
@[simp] theorem le_compl_self : a ≤ aᶜ ↔ a = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: a ≤ aᶜ ↔ a = ⊥
-- ne_compl_self
@[simp] theorem ne_compl_self [Nontrivial α] : a ≠ aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α} [Nontrivial α]
Goal: a ≠ aᶜ
-- compl_ne_self
@[simp] theorem compl_ne_self [Nontrivial α] : aᶜ ≠ a 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α} [Nontrivial α]
Goal: aᶜ ≠ a
-- lt_compl_self
@[simp] theorem lt_compl_self [Nontrivial α] : a < aᶜ ↔ a = ⊥ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α} [Nontrivial α]
Goal: a < aᶜ ↔ a = ⊥
-- le_compl_compl
theorem le_compl_compl : a ≤ aᶜᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a : α}
Goal: a ≤ aᶜᶜ
-- compl_anti
theorem compl_anti : Antitone (compl : α → α) 
Arguments: {α : Type u_2} [HeytingAlgebra α]
Goal: Antitone compl
-- compl_le_compl
@[gcongr]
theorem compl_le_compl (h : a ≤ b) : bᶜ ≤ aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: bᶜ ≤ aᶜ
-- compl_compl_compl
@[simp]
theorem compl_compl_compl (a : α) : aᶜᶜᶜ = aᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: aᶜᶜᶜ = aᶜ
-- disjoint_compl_compl_left_iff
@[simp]
theorem disjoint_compl_compl_left_iff : Disjoint aᶜᶜ b ↔ Disjoint a b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: Disjoint aᶜᶜ b ↔ Disjoint a b
-- disjoint_compl_compl_right_iff
@[simp]
theorem disjoint_compl_compl_right_iff : Disjoint a bᶜᶜ ↔ Disjoint a b 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: Disjoint a bᶜᶜ ↔ Disjoint a b
-- compl_sup_compl_le
theorem compl_sup_compl_le : aᶜ ⊔ bᶜ ≤ (a ⊓ b)ᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] {a b : α}
Goal: aᶜ ⊔ bᶜ ≤ (a ⊓ b)ᶜ
-- compl_compl_inf_distrib
theorem compl_compl_inf_distrib (a b : α) : (a ⊓ b)ᶜᶜ = aᶜᶜ ⊓ bᶜᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a b : α)
Goal: (a ⊓ b)ᶜᶜ = aᶜᶜ ⊓ bᶜᶜ
-- compl_compl_himp_distrib
theorem compl_compl_himp_distrib (a b : α) : (a ⇨ b)ᶜᶜ = aᶜᶜ ⇨ bᶜᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a b : α)
Goal: (a ⇨ b)ᶜᶜ = aᶜᶜ ⇨ bᶜᶜ
-- ofDual_hnot
@[simp]
theorem ofDual_hnot (a : αᵒᵈ) : ofDual (￢a) = (ofDual a)ᶜ 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : αᵒᵈ)
Goal: OrderDual.ofDual (￢a) = (OrderDual.ofDual a)ᶜ
-- toDual_compl
@[simp]
theorem toDual_compl (a : α) : toDual aᶜ = ￢toDual a 
Arguments: {α : Type u_2} [HeytingAlgebra α] (a : α)
Goal: OrderDual.toDual aᶜ = ￢OrderDual.toDual a
-- top_sdiff'
@[simp]
theorem top_sdiff' (a : α) : ⊤ \ a = ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: ⊤ \ a = ￢a
-- sdiff_top
@[simp]
theorem sdiff_top (a : α) : a \ ⊤ = ⊥ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: a \ ⊤ = ⊥
-- hnot_inf_distrib
theorem hnot_inf_distrib (a b : α) : ￢(a ⊓ b) = ￢a ⊔ ￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : α)
Goal: ￢(a ⊓ b) = ￢a ⊔ ￢b
-- sdiff_le_hnot
theorem sdiff_le_hnot : a \ b ≤ ￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: a \ b ≤ ￢b
-- sdiff_le_inf_hnot
theorem sdiff_le_inf_hnot : a \ b ≤ a ⊓ ￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: a \ b ≤ a ⊓ ￢b
-- hnot_sdiff
@[simp]
theorem hnot_sdiff (a : α) : ￢a \ a = ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: ￢a \ a = ￢a
-- hnot_sdiff_comm
theorem hnot_sdiff_comm (a b : α) : ￢a \ b = ￢b \ a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : α)
Goal: ￢a \ b = ￢b \ a
-- hnot_le_iff_codisjoint_right
theorem hnot_le_iff_codisjoint_right : ￢a ≤ b ↔ Codisjoint a b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: ￢a ≤ b ↔ Codisjoint a b
-- hnot_le_iff_codisjoint_left
theorem hnot_le_iff_codisjoint_left : ￢a ≤ b ↔ Codisjoint b a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: ￢a ≤ b ↔ Codisjoint b a
-- hnot_le_comm
theorem hnot_le_comm : ￢a ≤ b ↔ ￢b ≤ a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: ￢a ≤ b ↔ ￢b ≤ a
-- codisjoint_hnot_right
theorem codisjoint_hnot_right : Codisjoint a (￢a) 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a : α}
Goal: Codisjoint a (￢a)
-- codisjoint_hnot_left
theorem codisjoint_hnot_left : Codisjoint (￢a) a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a : α}
Goal: Codisjoint (￢a) a
-- LE.le.codisjoint_hnot_left
theorem LE.le.codisjoint_hnot_left (h : a ≤ b) : Codisjoint (￢a) b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: Codisjoint (￢a) b
-- LE.le.codisjoint_hnot_right
theorem LE.le.codisjoint_hnot_right (h : b ≤ a) : Codisjoint a (￢b) 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : b ≤ a)
Goal: Codisjoint a (￢b)
-- IsCompl.hnot_eq
theorem IsCompl.hnot_eq (h : IsCompl a b) : ￢a = b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: ￢a = b
-- IsCompl.eq_hnot
theorem IsCompl.eq_hnot (h : IsCompl a b) : a = ￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : IsCompl a b)
Goal: a = ￢b
-- sup_hnot_self
@[simp]
theorem sup_hnot_self (a : α) : a ⊔ ￢a = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: a ⊔ ￢a = ⊤
-- hnot_sup_self
@[simp]
theorem hnot_sup_self (a : α) : ￢a ⊔ a = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: ￢a ⊔ a = ⊤
-- hnot_bot
@[simp]
theorem hnot_bot : ￢(⊥ : α) = ⊤ 
Arguments: {α : Type u_2} [CoheytingAlgebra α]
Goal: ￢⊥ = ⊤
-- hnot_top
@[simp]
theorem hnot_top : ￢(⊤ : α) = ⊥ 
Arguments: {α : Type u_2} [CoheytingAlgebra α]
Goal: ￢⊤ = ⊥
-- hnot_hnot_le
theorem hnot_hnot_le : ￢￢a ≤ a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a : α}
Goal: ￢￢a ≤ a
-- hnot_anti
theorem hnot_anti : Antitone (hnot : α → α) 
Arguments: {α : Type u_2} [CoheytingAlgebra α]
Goal: Antitone hnot
-- hnot_le_hnot
theorem hnot_le_hnot (h : a ≤ b) : ￢b ≤ ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α} (h : a ≤ b)
Goal: ￢b ≤ ￢a
-- hnot_hnot_hnot
@[simp]
theorem hnot_hnot_hnot (a : α) : ￢￢￢a = ￢a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: ￢￢￢a = ￢a
-- codisjoint_hnot_hnot_left_iff
@[simp]
theorem codisjoint_hnot_hnot_left_iff : Codisjoint (￢￢a) b ↔ Codisjoint a b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: Codisjoint (￢￢a) b ↔ Codisjoint a b
-- codisjoint_hnot_hnot_right_iff
@[simp]
theorem codisjoint_hnot_hnot_right_iff : Codisjoint a (￢￢b) ↔ Codisjoint a b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: Codisjoint a (￢￢b) ↔ Codisjoint a b
-- le_hnot_inf_hnot
theorem le_hnot_inf_hnot : ￢(a ⊔ b) ≤ ￢a ⊓ ￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] {a b : α}
Goal: ￢(a ⊔ b) ≤ ￢a ⊓ ￢b
-- hnot_hnot_sup_distrib
theorem hnot_hnot_sup_distrib (a b : α) : ￢￢(a ⊔ b) = ￢￢a ⊔ ￢￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : α)
Goal: ￢￢(a ⊔ b) = ￢￢a ⊔ ￢￢b
-- hnot_hnot_sdiff_distrib
theorem hnot_hnot_sdiff_distrib (a b : α) : ￢￢(a \ b) = ￢￢a \ ￢￢b 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : α)
Goal: ￢￢(a \ b) = ￢￢a \ ￢￢b
-- ofDual_compl
@[simp]
theorem ofDual_compl (a : αᵒᵈ) : ofDual aᶜ = ￢ofDual a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : αᵒᵈ)
Goal: OrderDual.ofDual aᶜ = ￢OrderDual.ofDual a
-- ofDual_himp
@[simp]
theorem ofDual_himp (a b : αᵒᵈ) : ofDual (a ⇨ b) = ofDual b \ ofDual a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a ⇨ b) = OrderDual.ofDual b \ OrderDual.ofDual a
-- toDual_hnot
@[simp]
theorem toDual_hnot (a : α) : toDual (￢a) = (toDual a)ᶜ 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a : α)
Goal: OrderDual.toDual (￢a) = (OrderDual.toDual a)ᶜ
-- toDual_sdiff
@[simp]
theorem toDual_sdiff (a b : α) : toDual (a \ b) = toDual b ⇨ toDual a 
Arguments: {α : Type u_2} [CoheytingAlgebra α] (a b : α)
Goal: OrderDual.toDual (a \ b) = OrderDual.toDual b ⇨ OrderDual.toDual a
-- compl_le_hnot
theorem compl_le_hnot : aᶜ ≤ ￢a 
Arguments: {α : Type u_2} [BiheytingAlgebra α] {a : α}
Goal: aᶜ ≤ ￢a
-- himp_iff_imp
@[simp]
theorem himp_iff_imp (p q : Prop) : p ⇨ q ↔ p → q 
Arguments: (p q : Prop)
Goal: p ⇨ q ↔ p → q
-- compl_iff_not
@[simp]
theorem compl_iff_not (p : Prop) : pᶜ ↔ ¬p 
Arguments: (p : Prop)
Goal: pᶜ ↔ ¬p
-- PUnit.top_eq
@[simp]
theorem top_eq : (⊤ : PUnit) = unit 
Goal: ⊤ = PUnit.unit
-- PUnit.bot_eq
@[simp]
theorem bot_eq : (⊥ : PUnit) = unit 
Goal: ⊥ = PUnit.unit
-- PUnit.sup_eq
@[simp, nolint simpNF]
theorem sup_eq : a ⊔ b = unit 
Arguments: (a b : PUnit.{u + 1})
Goal: a ⊔ b = PUnit.unit
-- PUnit.inf_eq
@[simp, nolint simpNF]
theorem inf_eq : a ⊓ b = unit 
Arguments: (a b : PUnit.{u + 1})
Goal: a ⊓ b = PUnit.unit
-- PUnit.compl_eq
@[simp]
theorem compl_eq : aᶜ = unit 
Arguments: (a : PUnit.{u + 1})
Goal: aᶜ = PUnit.unit
-- PUnit.sdiff_eq
@[simp, nolint simpNF]
theorem sdiff_eq : a \ b = unit 
Arguments: (a b : PUnit.{u + 1})
Goal: a \ b = PUnit.unit
-- PUnit.hnot_eq
@[simp, nolint simpNF]
theorem hnot_eq : ￢a = unit 
Arguments: (a : PUnit.{u + 1})
Goal: ￢a = PUnit.unit