ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- disjoint_symmDiff_inf
theorem disjoint_symmDiff_inf : Disjoint (a ∆ b) (a ⊓ b) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Disjoint (a ∆ b) (a ⊓ b)
-- inf_symmDiff_distrib_left
theorem inf_symmDiff_distrib_left : a ⊓ b ∆ c = (a ⊓ b) ∆ (a ⊓ c) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ⊓ b ∆ c = (a ⊓ b) ∆ (a ⊓ c)
-- inf_symmDiff_distrib_right
theorem inf_symmDiff_distrib_right : a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ b ⊓ c = (a ⊓ c) ∆ (b ⊓ c)
-- sdiff_symmDiff
theorem sdiff_symmDiff : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ a ⊓ c \ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ a ⊓ c \ b
-- sdiff_symmDiff'
theorem sdiff_symmDiff' : c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ (a ⊔ b) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: c \ a ∆ b = c ⊓ a ⊓ b ⊔ c \ (a ⊔ b)
-- symmDiff_sdiff_left
@[simp]
theorem symmDiff_sdiff_left : a ∆ b \ a = b \ a 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ∆ b \ a = b \ a
-- symmDiff_sdiff_right
@[simp]
theorem symmDiff_sdiff_right : a ∆ b \ b = a \ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ∆ b \ b = a \ b
-- sdiff_symmDiff_left
@[simp]
theorem sdiff_symmDiff_left : a \ a ∆ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a \ a ∆ b = a ⊓ b
-- sdiff_symmDiff_right
@[simp]
theorem sdiff_symmDiff_right : b \ a ∆ b = a ⊓ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: b \ a ∆ b = a ⊓ b
-- symmDiff_eq_sup
theorem symmDiff_eq_sup : a ∆ b = a ⊔ b ↔ Disjoint a b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ∆ b = a ⊔ b ↔ Disjoint a b
-- le_symmDiff_iff_left
@[simp]
theorem le_symmDiff_iff_left : a ≤ a ∆ b ↔ Disjoint a b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ≤ a ∆ b ↔ Disjoint a b
-- le_symmDiff_iff_right
@[simp]
theorem le_symmDiff_iff_right : b ≤ a ∆ b ↔ Disjoint a b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: b ≤ a ∆ b ↔ Disjoint a b
-- symmDiff_symmDiff_left
theorem symmDiff_symmDiff_left :
    a ∆ b ∆ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ b ∆ c = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c
-- symmDiff_symmDiff_right
theorem symmDiff_symmDiff_right :
    a ∆ (b ∆ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ (b ∆ c) = a \ (b ⊔ c) ⊔ b \ (a ⊔ c) ⊔ c \ (a ⊔ b) ⊔ a ⊓ b ⊓ c
-- symmDiff_assoc
theorem symmDiff_assoc : a ∆ b ∆ c = a ∆ (b ∆ c) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ b ∆ c = a ∆ (b ∆ c)
-- symmDiff_left_comm
theorem symmDiff_left_comm : a ∆ (b ∆ c) = b ∆ (a ∆ c) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ (b ∆ c) = b ∆ (a ∆ c)
-- symmDiff_right_comm
theorem symmDiff_right_comm : a ∆ b ∆ c = a ∆ c ∆ b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c : α)
Goal: a ∆ b ∆ c = a ∆ c ∆ b
-- symmDiff_symmDiff_symmDiff_comm
theorem symmDiff_symmDiff_symmDiff_comm : a ∆ b ∆ (c ∆ d) = a ∆ c ∆ (b ∆ d) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b c d : α)
Goal: a ∆ b ∆ (c ∆ d) = a ∆ c ∆ (b ∆ d)
-- symmDiff_symmDiff_cancel_left
@[simp]
theorem symmDiff_symmDiff_cancel_left : a ∆ (a ∆ b) = b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ∆ (a ∆ b) = b
-- symmDiff_symmDiff_cancel_right
@[simp]
theorem symmDiff_symmDiff_cancel_right : b ∆ a ∆ a = b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: b ∆ a ∆ a = b
-- symmDiff_symmDiff_self'
@[simp]
theorem symmDiff_symmDiff_self' : a ∆ b ∆ a = b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: a ∆ b ∆ a = b
-- symmDiff_left_involutive
theorem symmDiff_left_involutive (a : α) : Involutive (· ∆ a) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Involutive fun x => x ∆ a
-- symmDiff_right_involutive
theorem symmDiff_right_involutive (a : α) : Involutive (a ∆ ·) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Involutive fun x => a ∆ x
-- symmDiff_left_injective
theorem symmDiff_left_injective (a : α) : Injective (· ∆ a) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Injective fun x => x ∆ a
-- symmDiff_right_injective
theorem symmDiff_right_injective (a : α) : Injective (a ∆ ·) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Injective fun x => a ∆ x
-- symmDiff_left_surjective
theorem symmDiff_left_surjective (a : α) : Surjective (· ∆ a) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Surjective fun x => x ∆ a
-- symmDiff_right_surjective
theorem symmDiff_right_surjective (a : α) : Surjective (a ∆ ·) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] (a : α)
Goal: Function.Surjective fun x => a ∆ x
-- symmDiff_left_inj
@[simp]
theorem symmDiff_left_inj : a ∆ b = c ∆ b ↔ a = c 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α}
Goal: a ∆ b = c ∆ b ↔ a = c
-- symmDiff_right_inj
@[simp]
theorem symmDiff_right_inj : a ∆ b = a ∆ c ↔ b = c 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α}
Goal: a ∆ b = a ∆ c ↔ b = c
-- symmDiff_eq_left
@[simp]
theorem symmDiff_eq_left : a ∆ b = a ↔ b = ⊥ 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b : α}
Goal: a ∆ b = a ↔ b = ⊥
-- symmDiff_eq_right
@[simp]
theorem symmDiff_eq_right : a ∆ b = b ↔ a = ⊥ 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b : α}
Goal: a ∆ b = b ↔ a = ⊥
-- Disjoint.symmDiff_left
protected theorem Disjoint.symmDiff_left (ha : Disjoint a c) (hb : Disjoint b c) :
    Disjoint (a ∆ b) c 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} (ha : Disjoint a c) (hb : Disjoint b c)
Goal: Disjoint (a ∆ b) c
-- Disjoint.symmDiff_right
protected theorem Disjoint.symmDiff_right (ha : Disjoint a b) (hb : Disjoint a c) :
    Disjoint a (b ∆ c) 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} (ha : Disjoint a b) (hb : Disjoint a c)
Goal: Disjoint a (b ∆ c)
-- symmDiff_eq_iff_sdiff_eq
theorem symmDiff_eq_iff_sdiff_eq (ha : a ≤ c) : a ∆ b = c ↔ c \ a = b 
Arguments: {α : Type u_2} [GeneralizedBooleanAlgebra α] {a b c : α} (ha : a ≤ c)
Goal: a ∆ b = c ↔ c \ a = b
-- inf_himp_bihimp
@[simp]
theorem inf_himp_bihimp : a ⇔ b ⇨ a ⊓ b = a ⊔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ⇨ a ⊓ b = a ⊔ b
-- codisjoint_bihimp_sup
theorem codisjoint_bihimp_sup : Codisjoint (a ⇔ b) (a ⊔ b) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: Codisjoint (a ⇔ b) (a ⊔ b)
-- himp_bihimp_left
@[simp]
theorem himp_bihimp_left : a ⇨ a ⇔ b = a ⇨ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇨ a ⇔ b = a ⇨ b
-- himp_bihimp_right
@[simp]
theorem himp_bihimp_right : b ⇨ a ⇔ b = b ⇨ a 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: b ⇨ a ⇔ b = b ⇨ a
-- bihimp_himp_left
@[simp]
theorem bihimp_himp_left : a ⇔ b ⇨ a = a ⊔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ⇨ a = a ⊔ b
-- bihimp_himp_right
@[simp]
theorem bihimp_himp_right : a ⇔ b ⇨ b = a ⊔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ⇨ b = a ⊔ b
-- bihimp_eq_inf
@[simp]
theorem bihimp_eq_inf : a ⇔ b = a ⊓ b ↔ Codisjoint a b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b = a ⊓ b ↔ Codisjoint a b
-- bihimp_le_iff_left
@[simp]
theorem bihimp_le_iff_left : a ⇔ b ≤ a ↔ Codisjoint a b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ≤ a ↔ Codisjoint a b
-- bihimp_le_iff_right
@[simp]
theorem bihimp_le_iff_right : a ⇔ b ≤ b ↔ Codisjoint a b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ≤ b ↔ Codisjoint a b
-- bihimp_assoc
theorem bihimp_assoc : a ⇔ b ⇔ c = a ⇔ (b ⇔ c) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b c : α)
Goal: a ⇔ b ⇔ c = a ⇔ (b ⇔ c)
-- bihimp_left_comm
theorem bihimp_left_comm : a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b c : α)
Goal: a ⇔ (b ⇔ c) = b ⇔ (a ⇔ c)
-- bihimp_right_comm
theorem bihimp_right_comm : a ⇔ b ⇔ c = a ⇔ c ⇔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b c : α)
Goal: a ⇔ b ⇔ c = a ⇔ c ⇔ b
-- bihimp_bihimp_bihimp_comm
theorem bihimp_bihimp_bihimp_comm : a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b c d : α)
Goal: a ⇔ b ⇔ (c ⇔ d) = a ⇔ c ⇔ (b ⇔ d)
-- bihimp_bihimp_cancel_left
@[simp]
theorem bihimp_bihimp_cancel_left : a ⇔ (a ⇔ b) = b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ (a ⇔ b) = b
-- bihimp_bihimp_cancel_right
@[simp]
theorem bihimp_bihimp_cancel_right : b ⇔ a ⇔ a = b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: b ⇔ a ⇔ a = b
-- bihimp_bihimp_self
@[simp]
theorem bihimp_bihimp_self : a ⇔ b ⇔ a = b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b ⇔ a = b
-- bihimp_left_involutive
theorem bihimp_left_involutive (a : α) : Involutive (· ⇔ a) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Involutive fun x => x ⇔ a
-- bihimp_right_involutive
theorem bihimp_right_involutive (a : α) : Involutive (a ⇔ ·) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Involutive fun x => a ⇔ x
-- bihimp_left_injective
theorem bihimp_left_injective (a : α) : Injective (· ⇔ a) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Injective fun x => x ⇔ a
-- bihimp_right_injective
theorem bihimp_right_injective (a : α) : Injective (a ⇔ ·) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Injective fun x => a ⇔ x
-- bihimp_left_surjective
theorem bihimp_left_surjective (a : α) : Surjective (· ⇔ a) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Surjective fun x => x ⇔ a
-- bihimp_right_surjective
theorem bihimp_right_surjective (a : α) : Surjective (a ⇔ ·) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: Function.Surjective fun x => a ⇔ x
-- bihimp_left_inj
@[simp]
theorem bihimp_left_inj : a ⇔ b = c ⇔ b ↔ a = c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α}
Goal: a ⇔ b = c ⇔ b ↔ a = c
-- bihimp_right_inj
@[simp]
theorem bihimp_right_inj : a ⇔ b = a ⇔ c ↔ b = c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α}
Goal: a ⇔ b = a ⇔ c ↔ b = c
-- bihimp_eq_left
@[simp]
theorem bihimp_eq_left : a ⇔ b = a ↔ b = ⊤ 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b : α}
Goal: a ⇔ b = a ↔ b = ⊤
-- bihimp_eq_right
@[simp]
theorem bihimp_eq_right : a ⇔ b = b ↔ a = ⊤ 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b : α}
Goal: a ⇔ b = b ↔ a = ⊤
-- Codisjoint.bihimp_left
protected theorem Codisjoint.bihimp_left (ha : Codisjoint a c) (hb : Codisjoint b c) :
    Codisjoint (a ⇔ b) c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (ha : Codisjoint a c) (hb : Codisjoint b c)
Goal: Codisjoint (a ⇔ b) c
-- Codisjoint.bihimp_right
protected theorem Codisjoint.bihimp_right (ha : Codisjoint a b) (hb : Codisjoint a c) :
    Codisjoint a (b ⇔ c) 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (ha : Codisjoint a b) (hb : Codisjoint a c)
Goal: Codisjoint a (b ⇔ c)
-- symmDiff_eq
theorem symmDiff_eq : a ∆ b = a ⊓ bᶜ ⊔ b ⊓ aᶜ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ∆ b = a ⊓ bᶜ ⊔ b ⊓ aᶜ
-- bihimp_eq
theorem bihimp_eq : a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b = (a ⊔ bᶜ) ⊓ (b ⊔ aᶜ)
-- symmDiff_eq'
theorem symmDiff_eq' : a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ) 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ∆ b = (a ⊔ b) ⊓ (aᶜ ⊔ bᶜ)
-- bihimp_eq'
theorem bihimp_eq' : a ⇔ b = a ⊓ b ⊔ aᶜ ⊓ bᶜ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b = a ⊓ b ⊔ aᶜ ⊓ bᶜ
-- symmDiff_top
theorem symmDiff_top : a ∆ ⊤ = aᶜ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: a ∆ ⊤ = aᶜ
-- top_symmDiff
theorem top_symmDiff : ⊤ ∆ a = aᶜ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: ⊤ ∆ a = aᶜ
-- compl_symmDiff
@[simp]
theorem compl_symmDiff : (a ∆ b)ᶜ = a ⇔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: (a ∆ b)ᶜ = a ⇔ b
-- compl_bihimp
@[simp]
theorem compl_bihimp : (a ⇔ b)ᶜ = a ∆ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: (a ⇔ b)ᶜ = a ∆ b
-- compl_symmDiff_compl
@[simp]
theorem compl_symmDiff_compl : aᶜ ∆ bᶜ = a ∆ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: aᶜ ∆ bᶜ = a ∆ b
-- compl_bihimp_compl
@[simp]
theorem compl_bihimp_compl : aᶜ ⇔ bᶜ = a ⇔ b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: aᶜ ⇔ bᶜ = a ⇔ b
-- symmDiff_eq_top
@[simp]
theorem symmDiff_eq_top : a ∆ b = ⊤ ↔ IsCompl a b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ∆ b = ⊤ ↔ IsCompl a b
-- bihimp_eq_bot
@[simp]
theorem bihimp_eq_bot : a ⇔ b = ⊥ ↔ IsCompl a b 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b : α)
Goal: a ⇔ b = ⊥ ↔ IsCompl a b
-- compl_symmDiff_self
@[simp]
theorem compl_symmDiff_self : aᶜ ∆ a = ⊤ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: aᶜ ∆ a = ⊤
-- symmDiff_compl_self
@[simp]
theorem symmDiff_compl_self : a ∆ aᶜ = ⊤ 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a : α)
Goal: a ∆ aᶜ = ⊤
-- symmDiff_symmDiff_right'
theorem symmDiff_symmDiff_right' :
    a ∆ (b ∆ c) = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c 
Arguments: {α : Type u_2} [BooleanAlgebra α] (a b c : α)
Goal: a ∆ (b ∆ c) = a ⊓ b ⊓ c ⊔ a ⊓ bᶜ ⊓ cᶜ ⊔ aᶜ ⊓ b ⊓ cᶜ ⊔ aᶜ ⊓ bᶜ ⊓ c
-- Disjoint.le_symmDiff_sup_symmDiff_left
theorem Disjoint.le_symmDiff_sup_symmDiff_left (h : Disjoint a b) : c ≤ a ∆ c ⊔ b ∆ c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Disjoint a b)
Goal: c ≤ a ∆ c ⊔ b ∆ c
-- Disjoint.le_symmDiff_sup_symmDiff_right
theorem Disjoint.le_symmDiff_sup_symmDiff_right (h : Disjoint b c) : a ≤ a ∆ b ⊔ a ∆ c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Disjoint b c)
Goal: a ≤ a ∆ b ⊔ a ∆ c
-- Codisjoint.bihimp_inf_bihimp_le_left
theorem Codisjoint.bihimp_inf_bihimp_le_left (h : Codisjoint a b) : a ⇔ c ⊓ b ⇔ c ≤ c 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Codisjoint a b)
Goal: a ⇔ c ⊓ b ⇔ c ≤ c
-- Codisjoint.bihimp_inf_bihimp_le_right
theorem Codisjoint.bihimp_inf_bihimp_le_right (h : Codisjoint b c) : a ⇔ b ⊓ a ⇔ c ≤ a 
Arguments: {α : Type u_2} [BooleanAlgebra α] {a b c : α} (h : Codisjoint b c)
Goal: a ⇔ b ⊓ a ⇔ c ≤ a
-- symmDiff_fst
@[simp]
theorem symmDiff_fst [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β]
    (a b : α × β) : (a ∆ b).1 = a.1 ∆ b.1 
Arguments: {α : Type u_2} {β : Type u_3} [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β] (a b : α × β)
Goal: (a ∆ b).1 = a.1 ∆ b.1
-- symmDiff_snd
@[simp]
theorem symmDiff_snd [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β]
    (a b : α × β) : (a ∆ b).2 = a.2 ∆ b.2 
Arguments: {α : Type u_2} {β : Type u_3} [GeneralizedCoheytingAlgebra α] [GeneralizedCoheytingAlgebra β] (a b : α × β)
Goal: (a ∆ b).2 = a.2 ∆ b.2
-- bihimp_fst
@[simp]
theorem bihimp_fst [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) :
    (a ⇔ b).1 = a.1 ⇔ b.1 
Arguments: {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β)
Goal: (a ⇔ b).1 = a.1 ⇔ b.1
-- bihimp_snd
@[simp]
theorem bihimp_snd [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β) :
    (a ⇔ b).2 = a.2 ⇔ b.2 
Arguments: {α : Type u_2} {β : Type u_3} [GeneralizedHeytingAlgebra α] [GeneralizedHeytingAlgebra β] (a b : α × β)
Goal: (a ⇔ b).2 = a.2 ⇔ b.2
-- Pi.symmDiff_def
theorem symmDiff_def [∀ i, GeneralizedCoheytingAlgebra (π i)] (a b : ∀ i, π i) :
    a ∆ b = fun i => a i ∆ b i 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a b : (i : ι) → π i)
Goal: a ∆ b = fun i => a i ∆ b i
-- Pi.bihimp_def
theorem bihimp_def [∀ i, GeneralizedHeytingAlgebra (π i)] (a b : ∀ i, π i) :
    a ⇔ b = fun i => a i ⇔ b i 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a b : (i : ι) → π i)
Goal: a ⇔ b = fun i => a i ⇔ b i
-- Pi.symmDiff_apply
@[simp]
theorem symmDiff_apply [∀ i, GeneralizedCoheytingAlgebra (π i)] (a b : ∀ i, π i) (i : ι) :
    (a ∆ b) i = a i ∆ b i 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → GeneralizedCoheytingAlgebra (π i)] (a b : (i : ι) → π i) (i : ι)
Goal: (a ∆ b) i = a i ∆ b i
-- Pi.bihimp_apply
@[simp]
theorem bihimp_apply [∀ i, GeneralizedHeytingAlgebra (π i)] (a b : ∀ i, π i) (i : ι) :
    (a ⇔ b) i = a i ⇔ b i 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → GeneralizedHeytingAlgebra (π i)] (a b : (i : ι) → π i) (i : ι)
Goal: (a ⇔ b) i = a i ⇔ b i
-- SuccOrder.succ_le_of_lt
@[ext]
class SuccOrder (α : Type*) [Preorder α] where
    succ : α → α
    le_succ : ∀ a, a ≤ succ a
    max_of_succ_le {a} : succ a ≤ a → IsMax a
    succ_le_of_lt {a b} : a < b → succ a ≤ b
    le_of_lt_succ {a b} : a < succ b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : SuccOrder α] {a b : α}
Goal: a < b → SuccOrder.succ a ≤ b
-- SuccOrder.le_of_lt_succ
@[ext]
class SuccOrder (α : Type*) [Preorder α] where
    succ : α → α
    le_succ : ∀ a, a ≤ succ a
    max_of_succ_le {a} : succ a ≤ a → IsMax a
    succ_le_of_lt {a b} : a < b → succ a ≤ b
    le_of_lt_succ {a b} : a < succ b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : SuccOrder α] {a b : α}
Goal: a < SuccOrder.succ b → a ≤ b
-- SuccOrder.le_succ
@[ext]
class SuccOrder (α : Type*) [Preorder α] where
    succ : α → α
    le_succ : ∀ a, a ≤ succ a
    max_of_succ_le {a} : succ a ≤ a → IsMax a
    succ_le_of_lt {a b} : a < b → succ a ≤ b
    le_of_lt_succ {a b} : a < succ b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : SuccOrder α] (a : α)
Goal: a ≤ SuccOrder.succ a
-- SuccOrder.max_of_succ_le
@[ext]
class SuccOrder (α : Type*) [Preorder α] where
    succ : α → α
    le_succ : ∀ a, a ≤ succ a
    max_of_succ_le {a} : succ a ≤ a → IsMax a
    succ_le_of_lt {a b} : a < b → succ a ≤ b
    le_of_lt_succ {a b} : a < succ b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : SuccOrder α] {a : α}
Goal: SuccOrder.succ a ≤ a → IsMax a
-- PredOrder.min_of_le_pred
@[ext]
class PredOrder (α : Type*) [Preorder α] where
    pred : α → α
    pred_le : ∀ a, pred a ≤ a
    min_of_le_pred {a} : a ≤ pred a → IsMin a
    le_pred_of_lt {a b} : a < b → a ≤ pred b
    le_of_pred_lt {a b} : pred a < b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : PredOrder α] {a : α}
Goal: a ≤ PredOrder.pred a → IsMin a
-- PredOrder.le_of_pred_lt
@[ext]
class PredOrder (α : Type*) [Preorder α] where
    pred : α → α
    pred_le : ∀ a, pred a ≤ a
    min_of_le_pred {a} : a ≤ pred a → IsMin a
    le_pred_of_lt {a b} : a < b → a ≤ pred b
    le_of_pred_lt {a b} : pred a < b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : PredOrder α] {a b : α}
Goal: PredOrder.pred a < b → a ≤ b
-- PredOrder.le_pred_of_lt
@[ext]
class PredOrder (α : Type*) [Preorder α] where
    pred : α → α
    pred_le : ∀ a, pred a ≤ a
    min_of_le_pred {a} : a ≤ pred a → IsMin a
    le_pred_of_lt {a b} : a < b → a ≤ pred b
    le_of_pred_lt {a b} : pred a < b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : PredOrder α] {a b : α}
Goal: a < b → a ≤ PredOrder.pred b
-- PredOrder.pred_le
@[ext]
class PredOrder (α : Type*) [Preorder α] where
    pred : α → α
    pred_le : ∀ a, pred a ≤ a
    min_of_le_pred {a} : a ≤ pred a → IsMin a
    le_pred_of_lt {a b} : a < b → a ≤ pred b
    le_of_pred_lt {a b} : pred a < b → a ≤ b
Arguments: {α : Type u_3} [Preorder α] [self : PredOrder α] (a : α)
Goal: PredOrder.pred a ≤ a
-- SuccOrder.ofCore_succ
@[simps]
def SuccOrder.ofCore (succ : α → α) (hn : ∀ {a}, ¬IsMax a → ∀ b, a < b ↔ succ a ≤ b)
    (hm : ∀ a, IsMax a → succ a = a) : SuccOrder α 
Arguments: {α : Type u_1} [LinearOrder α] (succ : α → α) (hn : ∀ {a : α}, ¬IsMax a → ∀ (b : α), a < b ↔ succ a ≤ b) (hm : ∀ (a : α), IsMax a → succ a = a)
Goal: ∀ (a : α), SuccOrder.succ a = succ a
-- PredOrder.ofCore_pred
@[simps]
def PredOrder.ofCore {α} [LinearOrder α] (pred : α → α)
    (hn : ∀ {a}, ¬IsMin a → ∀ b, b ≤ pred a ↔ b < a) (hm : ∀ a, IsMin a → pred a = a) :
    PredOrder α 
Arguments: {α : Type u_3} [LinearOrder α] (pred : α → α) (hn : ∀ {a : α}, ¬IsMin a → ∀ (b : α), b ≤ pred a ↔ b < a) (hm : ∀ (a : α), IsMin a → pred a = a)
Goal: ∀ (a : α), PredOrder.pred a = pred a
-- Order.le_succ
theorem le_succ : ∀ a : α, a ≤ succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] (a : α)
Goal: a ≤ Order.succ a
-- Order.max_of_succ_le
theorem max_of_succ_le {a : α} : succ a ≤ a → IsMax a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α}
Goal: Order.succ a ≤ a → IsMax a
-- Order.succ_le_of_lt
theorem succ_le_of_lt {a b : α} : a < b → succ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α}
Goal: a < b → Order.succ a ≤ b
-- Order.le_of_lt_succ
theorem le_of_lt_succ {a b : α} : a < succ b → a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α}
Goal: a < Order.succ b → a ≤ b
-- Order.succ_le_iff_isMax
@[simp]
theorem succ_le_iff_isMax : succ a ≤ a ↔ IsMax a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α}
Goal: Order.succ a ≤ a ↔ IsMax a
-- Order.lt_succ_iff_not_isMax
@[simp]
theorem lt_succ_iff_not_isMax : a < succ a ↔ ¬IsMax a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α}
Goal: a < Order.succ a ↔ ¬IsMax a
-- Order.wcovBy_succ
theorem wcovBy_succ (a : α) : a ⩿ succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] (a : α)
Goal: a ⩿ Order.succ a
-- Order.covBy_succ_of_not_isMax
theorem covBy_succ_of_not_isMax (h : ¬IsMax a) : a ⋖ succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α} (h : ¬IsMax a)
Goal: a ⋖ Order.succ a
-- Order.lt_succ_iff_of_not_isMax
theorem lt_succ_iff_of_not_isMax (ha : ¬IsMax a) : b < succ a ↔ b ≤ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a)
Goal: b < Order.succ a ↔ b ≤ a
-- Order.succ_le_iff_of_not_isMax
theorem succ_le_iff_of_not_isMax (ha : ¬IsMax a) : succ a ≤ b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a)
Goal: Order.succ a ≤ b ↔ a < b
-- Order.succ_lt_succ_of_not_isMax
lemma succ_lt_succ_of_not_isMax (h : a < b) (hb : ¬ IsMax b) : succ a < succ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (h : a < b) (hb : ¬IsMax b)
Goal: Order.succ a < Order.succ b
-- Order.succ_lt_succ_iff_of_not_isMax
theorem succ_lt_succ_iff_of_not_isMax (ha : ¬IsMax a) (hb : ¬IsMax b) :
    succ a < succ b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a) (hb : ¬IsMax b)
Goal: Order.succ a < Order.succ b ↔ a < b
-- Order.succ_le_succ_iff_of_not_isMax
theorem succ_le_succ_iff_of_not_isMax (ha : ¬IsMax a) (hb : ¬IsMax b) :
    succ a ≤ succ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (ha : ¬IsMax a) (hb : ¬IsMax b)
Goal: Order.succ a ≤ Order.succ b ↔ a ≤ b
-- Order.succ_le_succ
@[simp, mono]
theorem succ_le_succ (h : a ≤ b) : succ a ≤ succ b 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (h : a ≤ b)
Goal: Order.succ a ≤ Order.succ b
-- Order.succ_mono
theorem succ_mono : Monotone (succ : α → α) 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α]
Goal: Monotone Order.succ
-- Order.le_succ_of_wcovBy
lemma le_succ_of_wcovBy (h : a ⩿ b) : b ≤ succ a 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a b : α} (h : a ⩿ b)
Goal: b ≤ Order.succ a
-- Order.le_succ_iterate
theorem le_succ_iterate (k : ℕ) (x : α) : x ≤ succ^[k] x 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] (k : ℕ) (x : α)
Goal: x ≤ Order.succ^[k] x
-- Order.isMax_iterate_succ_of_eq_of_lt
theorem isMax_iterate_succ_of_eq_of_lt {n m : ℕ} (h_eq : succ^[n] a = succ^[m] a)
    (h_lt : n < m) : IsMax (succ^[n] a) 
Arguments: {α : Type u_1} [Preorder α] [SuccOrder α] {a : α} {n m : ℕ} (h_eq : Order.succ^[n] a = Order.succ^[m] a) (h_lt : n < m)
Goal: IsMax (Order.succ^[n] a)