ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- leOnePart_eq_one
@[to_additive (attr := simp)]
lemma leOnePart_eq_one : a⁻ᵐ = 1 ↔ 1 ≤ a 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ᵐ = 1 ↔ 1 ≤ a
-- negPart_pos
@[to_additive (attr := simp) negPart_pos] lemma one_lt_ltOnePart (ha : a < 1) : 1 < a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (ha : a < 0)
Goal: 0 < a⁻
-- one_lt_ltOnePart
@[to_additive (attr := simp) negPart_pos] lemma one_lt_ltOnePart (ha : a < 1) : 1 < a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (ha : a < 1)
Goal: 1 < a⁻ᵐ
-- posPart_sub_negPart
@[to_additive (attr := simp)] lemma oneLePart_div_leOnePart (a : α) : a⁺ᵐ / a⁻ᵐ = a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ - a⁻ = a
-- oneLePart_div_leOnePart
@[to_additive (attr := simp)] lemma oneLePart_div_leOnePart (a : α) : a⁺ᵐ / a⁻ᵐ = a 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ᵐ / a⁻ᵐ = a
-- leOnePart_div_oneLePart
@[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ᵐ / a⁺ᵐ = a⁻¹
-- negPart_sub_posPart
@[to_additive (attr := simp)] lemma leOnePart_div_oneLePart (a : α) : a⁻ᵐ / a⁺ᵐ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ - a⁺ = -a
-- negPart_anti
@[to_additive] lemma leOnePart_anti : Antitone (leOnePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: Antitone negPart
-- leOnePart_anti
@[to_additive] lemma leOnePart_anti : Antitone (leOnePart : α → α) 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: Antitone leOnePart
-- negPart_eq_neg_inf_zero
@[to_additive]
lemma leOnePart_eq_inv_inf_one (a : α) : a⁻ᵐ = (a ⊓ 1)⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ = -(a ⊓ 0)
-- leOnePart_eq_inv_inf_one
@[to_additive]
lemma leOnePart_eq_inv_inf_one (a : α) : a⁻ᵐ = (a ⊓ 1)⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ᵐ = (a ⊓ 1)⁻¹
-- posPart_add_negPart
@[to_additive] lemma oneLePart_mul_leOnePart (a : α) : a⁺ᵐ * a⁻ᵐ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ + a⁻ = |a|
-- oneLePart_mul_leOnePart
@[to_additive] lemma oneLePart_mul_leOnePart (a : α) : a⁺ᵐ * a⁻ᵐ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ᵐ * a⁻ᵐ = mabs a
-- leOnePart_mul_oneLePart
@[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ᵐ * a⁺ᵐ = mabs a
-- negPart_add_posPart
@[to_additive] lemma leOnePart_mul_oneLePart (a : α) : a⁻ᵐ * a⁺ᵐ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁻ + a⁺ = |a|
-- oneLePart_inf_leOnePart_eq_one
@[to_additive] lemma oneLePart_inf_leOnePart_eq_one (a : α) : a⁺ᵐ ⊓ a⁻ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ᵐ ⊓ a⁻ᵐ = 1
-- posPart_inf_negPart_eq_zero
@[to_additive] lemma oneLePart_inf_leOnePart_eq_one (a : α) : a⁺ᵐ ⊓ a⁻ᵐ = 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a⁺ ⊓ a⁻ = 0
-- sup_eq_mul_oneLePart_div
@[to_additive] lemma sup_eq_mul_oneLePart_div (a b : α) : a ⊔ b = b * (a / b)⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊔ b = b * (a / b)⁺ᵐ
-- sup_eq_add_posPart_sub
@[to_additive] lemma sup_eq_mul_oneLePart_div (a b : α) : a ⊔ b = b * (a / b)⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊔ b = b + (a - b)⁺
-- inf_eq_sub_posPart_sub
@[to_additive] lemma inf_eq_div_oneLePart_div (a b : α) : a ⊓ b = a / (a / b)⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊓ b = a - (a - b)⁺
-- inf_eq_div_oneLePart_div
@[to_additive] lemma inf_eq_div_oneLePart_div (a b : α) : a ⊓ b = a / (a / b)⁺ᵐ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊓ b = a / (a / b)⁺ᵐ
-- le_iff_oneLePart_leOnePart
@[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ
-- le_iff_posPart_negPart
@[to_additive] lemma le_iff_oneLePart_leOnePart (a b : α) : a ≤ b ↔ a⁺ᵐ ≤ b⁺ᵐ ∧ b⁻ᵐ ≤ a⁻ᵐ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ≤ b ↔ a⁺ ≤ b⁺ ∧ b⁻ ≤ a⁻
-- mabs_mul_eq_oneLePart_sq
@[to_additive abs_add_eq_two_nsmul_posPart]
lemma mabs_mul_eq_oneLePart_sq (a : α) : |a|ₘ * a = a⁺ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: mabs a * a = a⁺ᵐ ^ 2
-- abs_add_eq_two_nsmul_posPart
@[to_additive abs_add_eq_two_nsmul_posPart]
lemma mabs_mul_eq_oneLePart_sq (a : α) : |a|ₘ * a = a⁺ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: |a| + a = 2 • a⁺
-- add_abs_eq_two_nsmul_posPart
@[to_additive add_abs_eq_two_nsmul_posPart]
lemma mul_mabs_eq_oneLePart_sq (a : α) : a * |a|ₘ = a⁺ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a + |a| = 2 • a⁺
-- mul_mabs_eq_oneLePart_sq
@[to_additive add_abs_eq_two_nsmul_posPart]
lemma mul_mabs_eq_oneLePart_sq (a : α) : a * |a|ₘ = a⁺ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a * mabs a = a⁺ᵐ ^ 2
-- mabs_div_eq_leOnePart_sq
@[to_additive abs_sub_eq_two_nsmul_negPart]
lemma mabs_div_eq_leOnePart_sq (a : α) : |a|ₘ / a = a⁻ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: mabs a / a = a⁻ᵐ ^ 2
-- abs_sub_eq_two_nsmul_negPart
@[to_additive abs_sub_eq_two_nsmul_negPart]
lemma mabs_div_eq_leOnePart_sq (a : α) : |a|ₘ / a = a⁻ᵐ ^ 2 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: |a| - a = 2 • a⁻
-- div_mabs_eq_inv_leOnePart_sq
@[to_additive sub_abs_eq_neg_two_nsmul_negPart]
lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a / mabs a = (a⁻ᵐ ^ 2)⁻¹
-- sub_abs_eq_neg_two_nsmul_negPart
@[to_additive sub_abs_eq_neg_two_nsmul_negPart]
lemma div_mabs_eq_inv_leOnePart_sq (a : α) : a / |a|ₘ = (a⁻ᵐ ^ 2)⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: a - |a| = -(2 • a⁻)
-- posPart_eq_ite
@[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 
Arguments: {α : Type u_1} [LinearOrder α] [AddGroup α] {a : α}
Goal: a⁺ = if 0 ≤ a then a else 0
-- oneLePart_eq_ite
@[to_additive] lemma oneLePart_eq_ite : a⁺ᵐ = if 1 ≤ a then a else 1 
Arguments: {α : Type u_1} [LinearOrder α] [Group α] {a : α}
Goal: a⁺ᵐ = if 1 ≤ a then a else 1
-- one_lt_oneLePart_iff
@[to_additive (attr := simp) posPart_pos_iff] lemma one_lt_oneLePart_iff : 1 < a⁺ᵐ ↔ 1 < a 
Arguments: {α : Type u_1} [LinearOrder α] [Group α] {a : α}
Goal: 1 < a⁺ᵐ ↔ 1 < a
-- posPart_pos_iff
@[to_additive (attr := simp) posPart_pos_iff] lemma one_lt_oneLePart_iff : 1 < a⁺ᵐ ↔ 1 < a 
Arguments: {α : Type u_1} [LinearOrder α] [AddGroup α] {a : α}
Goal: 0 < a⁺ ↔ 0 < a
-- posPart_eq_of_posPart_pos
@[to_additive posPart_eq_of_posPart_pos]
lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a 
Arguments: {α : Type u_1} [LinearOrder α] [AddGroup α] {a : α} (ha : 0 < a⁺)
Goal: a⁺ = a
-- oneLePart_of_one_lt_oneLePart
@[to_additive posPart_eq_of_posPart_pos]
lemma oneLePart_of_one_lt_oneLePart (ha : 1 < a⁺ᵐ) : a⁺ᵐ = a 
Arguments: {α : Type u_1} [LinearOrder α] [Group α] {a : α} (ha : 1 < a⁺ᵐ )
Goal: a⁺ᵐ = a
-- leOnePart_eq_ite
@[to_additive] lemma leOnePart_eq_ite : a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1 
Arguments: {α : Type u_1} [LinearOrder α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1
-- negPart_eq_ite
@[to_additive] lemma leOnePart_eq_ite : a⁻ᵐ = if a ≤ 1 then a⁻¹ else 1 
Arguments: {α : Type u_1} [LinearOrder α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: a⁻ = if a ≤ 0 then -a else 0
-- negPart_pos_iff
@[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 
Arguments: {α : Type u_1} [LinearOrder α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: 0 < a⁻ ↔ a < 0
-- one_lt_ltOnePart_iff
@[to_additive (attr := simp) negPart_pos_iff] lemma one_lt_ltOnePart_iff : 1 < a⁻ᵐ ↔ a < 1 
Arguments: {α : Type u_1} [LinearOrder α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: 1 < a⁻ᵐ ↔ a < 1
-- Pi.posPart_apply
@[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ i = (f i)⁺ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i) (i : ι)
Goal: f⁺ i = (f i)⁺
-- Pi.oneLePart_apply
@[to_additive (attr := simp)] lemma oneLePart_apply (f : ∀ i, α i) (i : ι) : f⁺ i = (f i)⁺ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i) (i : ι)
Goal: f⁺ i = (f i)⁺
-- Pi.leOnePart_apply
@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ i = (f i)⁻ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i) (i : ι)
Goal: f⁻ i = (f i)⁻
-- Pi.negPart_apply
@[to_additive (attr := simp)] lemma leOnePart_apply (f : ∀ i, α i) (i : ι) : f⁻ i = (f i)⁻ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i) (i : ι)
Goal: f⁻ i = (f i)⁻
-- Pi.posPart_def
@[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ = fun i ↦ (f i)⁺ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i)
Goal: f⁺ = fun i => (f i)⁺
-- Pi.oneLePart_def
@[to_additive] lemma oneLePart_def (f : ∀ i, α i) : f⁺ = fun i ↦ (f i)⁺ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i)
Goal: f⁺ = fun i => (f i)⁺
-- Pi.leOnePart_def
@[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ = fun i ↦ (f i)⁻ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i)
Goal: f⁻ = fun i => (f i)⁻
-- Pi.negPart_def
@[to_additive] lemma leOnePart_def (f : ∀ i, α i) : f⁻ = fun i ↦ (f i)⁻ 
Arguments: {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → Lattice (α i)] [(i : ι) → AddCommGroup (α i)] (f : (i : ι) → α i)
Goal: f⁻ = fun i => (f i)⁻
-- Rat.divInt_nonneg_iff_of_pos_right
@[simp] lemma divInt_nonneg_iff_of_pos_right {a b : ℤ} (hb : 0 < b) : 0 ≤ a /. b ↔ 0 ≤ a 
Arguments: {a b : ℤ} (hb : 0 < b)
Goal: 0 ≤ Rat.divInt a b ↔ 0 ≤ a
-- Rat.divInt_nonneg
@[simp] lemma divInt_nonneg {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b) : 0 ≤ a /. b 
Arguments: {a b : ℤ} (ha : 0 ≤ a) (hb : 0 ≤ b)
Goal: 0 ≤ Rat.divInt a b
-- Rat.mkRat_nonneg
@[simp] lemma mkRat_nonneg {a : ℤ} (ha : 0 ≤ a) (b : ℕ) : 0 ≤ mkRat a b 
Arguments: {a : ℤ} (ha : 0 ≤ a) (b : ℕ)
Goal: 0 ≤ mkRat a b
-- Rat.ofScientific_nonneg
theorem ofScientific_nonneg (m : ℕ) (s : Bool) (e : ℕ) :
    0 ≤ Rat.ofScientific m s e 
Arguments: (m : ℕ) (s : Bool) (e : ℕ)
Goal: 0 ≤ Rat.ofScientific m s e
-- NNRat.cast_ofScientific
@[simp, norm_cast]
theorem _root_.NNRat.cast_ofScientific {K} [NNRatCast K] (m : ℕ) (s : Bool) (e : ℕ) :
    (OfScientific.ofScientific m s e : ℚ≥0) = (OfScientific.ofScientific m s e : K) 
Arguments: {K : Type u_1} [NNRatCast K] (m : ℕ) (s : Bool) (e : ℕ)
Goal: ↑(OfScientific.ofScientific m s e) = OfScientific.ofScientific m s e
-- Rat.add_nonneg
protected lemma add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b 
Arguments: {a b : ℚ}
Goal: 0 ≤ a → 0 ≤ b → 0 ≤ a + b
-- Rat.mul_nonneg
protected lemma mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b 
Arguments: {a b : ℚ}
Goal: 0 ≤ a → 0 ≤ b → 0 ≤ a * b
-- Rat.le_iff_sub_nonneg
protected theorem le_iff_sub_nonneg (a b : ℚ) : a ≤ b ↔ 0 ≤ b - a 
Arguments: (a b : ℚ)
Goal: a ≤ b ↔ 0 ≤ b - a
-- Rat.divInt_le_divInt
protected lemma divInt_le_divInt {a b c d : ℤ} (b0 : 0 < b) (d0 : 0 < d) :
    a /. b ≤ c /. d ↔ a * d ≤ c * b 
Arguments: {a b c d : ℤ} (b0 : 0 < b) (d0 : 0 < d)
Goal: Rat.divInt a b ≤ Rat.divInt c d ↔ a * d ≤ c * b
-- Rat.le_total
protected lemma le_total : a ≤ b ∨ b ≤ a 
Arguments: {a b : ℚ}
Goal: a ≤ b ∨ b ≤ a
-- Rat.not_le
protected theorem not_le {a b : ℚ} : ¬a ≤ b ↔ b < a 
Arguments: {a b : ℚ}
Goal: ¬a ≤ b ↔ b < a
-- Rat.le_def
protected lemma le_def : p ≤ q ↔ p.num * q.den ≤ q.num * p.den 
Arguments: {p q : ℚ}
Goal: p ≤ q ↔ p.num * ↑q.den ≤ q.num * ↑p.den
-- Rat.lt_def
protected lemma lt_def : p < q ↔ p.num * q.den < q.num * p.den 
Arguments: {p q : ℚ}
Goal: p < q ↔ p.num * ↑q.den < q.num * ↑p.den
-- Rat.add_le_add_left
protected theorem add_le_add_left {a b c : ℚ} : c + a ≤ c + b ↔ a ≤ b 
Arguments: {a b c : ℚ}
Goal: c + a ≤ c + b ↔ a ≤ b
-- Rat.num_nonpos
@[simp] lemma num_nonpos {a : ℚ} : a.num ≤ 0 ↔ a ≤ 0 
Arguments: {a : ℚ}
Goal: a.num ≤ 0 ↔ a ≤ 0
-- Rat.num_pos
@[simp] lemma num_pos {a : ℚ} : 0 < a.num ↔ 0 < a 
Arguments: {a : ℚ}
Goal: 0 < a.num ↔ 0 < a
-- Rat.num_neg
@[simp] lemma num_neg {a : ℚ} : a.num < 0 ↔ a < 0 
Arguments: {a : ℚ}
Goal: a.num < 0 ↔ a < 0
-- Rat.div_lt_div_iff_mul_lt_mul
theorem div_lt_div_iff_mul_lt_mul {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d) :
    (a : ℚ) / b < c / d ↔ a * d < c * b 
Arguments: {a b c d : ℤ} (b_pos : 0 < b) (d_pos : 0 < d)
Goal: ↑a / ↑b < ↑c / ↑d ↔ a * d < c * b
-- Rat.lt_one_iff_num_lt_denom
theorem lt_one_iff_num_lt_denom {q : ℚ} : q < 1 ↔ q.num < q.den 
Arguments: {q : ℚ}
Goal: q < 1 ↔ q.num < ↑q.den
-- Rat.abs_def
theorem abs_def (q : ℚ) : |q| = q.num.natAbs /. q.den 
Arguments: (q : ℚ)
Goal: |q| = Rat.divInt ↑q.num.natAbs ↑q.den
-- Int.cast_mono
lemma cast_mono : Monotone (Int.cast : ℤ → R) 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R]
Goal: Monotone Int.cast
-- Int.cast_nonneg
@[simp] lemma cast_nonneg : ∀ {n : ℤ}, (0 : R) ≤ n ↔ 0 ≤ n
  
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {n : ℤ}
Goal: 0 ≤ ↑n ↔ 0 ≤ n
-- Int.cast_le
@[simp, norm_cast] lemma cast_le : (m : R) ≤ n ↔ m ≤ n 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {m n : ℤ}
Goal: ↑m ≤ ↑n ↔ m ≤ n
-- Int.cast_strictMono
lemma cast_strictMono : StrictMono (fun x : ℤ => (x : R)) 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1]
Goal: StrictMono fun x => ↑x
-- Int.cast_lt
@[simp, norm_cast] lemma cast_lt : (m : R) < n ↔ m < n 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {m n : ℤ}
Goal: ↑m < ↑n ↔ m < n
-- Int.cast_nonpos
@[simp] lemma cast_nonpos : (n : R) ≤ 0 ↔ n ≤ 0 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {n : ℤ}
Goal: ↑n ≤ 0 ↔ n ≤ 0
-- Int.cast_pos
@[simp] lemma cast_pos : (0 : R) < n ↔ 0 < n 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {n : ℤ}
Goal: 0 < ↑n ↔ 0 < n
-- Int.cast_lt_zero
@[simp] lemma cast_lt_zero : (n : R) < 0 ↔ n < 0 
Arguments: {R : Type u_1} [AddCommGroupWithOne R] [PartialOrder R] [CovariantClass R R (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass R] [NeZero 1] {n : ℤ}
Goal: ↑n < 0 ↔ n < 0
-- Int.cast_min
@[simp, norm_cast]
lemma cast_min : ↑(min a b) = (min a b : R) 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a b : ℤ}
Goal: ↑(min a b) = min ↑a ↑b
-- Int.cast_max
@[simp, norm_cast]
lemma cast_max : (↑(max a b) : R) = max (a : R) (b : R) 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a b : ℤ}
Goal: ↑(max a b) = max ↑a ↑b
-- Int.cast_abs
@[simp, norm_cast]
lemma cast_abs : (↑|a| : R) = |(a : R)| 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a : ℤ}
Goal: ↑|a| = |↑a|
-- Int.cast_one_le_of_pos
lemma cast_one_le_of_pos (h : 0 < a) : (1 : R) ≤ a 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a : ℤ} (h : 0 < a)
Goal: 1 ≤ ↑a
-- Int.cast_le_neg_one_of_neg
lemma cast_le_neg_one_of_neg (h : a < 0) : (a : R) ≤ -1 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a : ℤ} (h : a < 0)
Goal: ↑a ≤ -1
-- Int.cast_le_neg_one_or_one_le_cast_of_ne_zero
lemma cast_le_neg_one_or_one_le_cast_of_ne_zero (hn : n ≠ 0) : (n : R) ≤ -1 ∨ 1 ≤ (n : R) 
Arguments: (R : Type u_1) [LinearOrderedRing R] {n : ℤ} (hn : n ≠ 0)
Goal: ↑n ≤ -1 ∨ 1 ≤ ↑n
-- Int.nneg_mul_add_sq_of_abs_le_one
lemma nneg_mul_add_sq_of_abs_le_one (n : ℤ) (hx : |x| ≤ 1) : (0 : R) ≤ n * x + n * n 
Arguments: {R : Type u_1} [LinearOrderedRing R] {x : R} (n : ℤ) (hx : |x| ≤ 1)
Goal: 0 ≤ ↑n * x + ↑n * ↑n
-- Int.cast_natAbs
lemma cast_natAbs : (n.natAbs : R) = |n| 
Arguments: {R : Type u_1} [LinearOrderedRing R] {n : ℤ}
Goal: ↑n.natAbs = ↑|n|
-- toDual_intCast
@[simp] lemma toDual_intCast [IntCast R] (n : ℤ) : toDual (n : R) = n 
Arguments: {R : Type u_1} [IntCast R] (n : ℤ)
Goal: OrderDual.toDual ↑n = ↑n
-- ofDual_intCast
@[simp] lemma ofDual_intCast [IntCast R] (n : ℤ) : (ofDual n : R) = n 
Arguments: {R : Type u_1} [IntCast R] (n : ℤ)
Goal: OrderDual.ofDual ↑n = ↑n
-- toLex_intCast
@[simp] lemma toLex_intCast [IntCast R] (n : ℤ) : toLex (n : R) = n 
Arguments: {R : Type u_1} [IntCast R] (n : ℤ)
Goal: toLex ↑n = ↑n
-- ofLex_intCast
@[simp] lemma ofLex_intCast [IntCast R] (n : ℤ) : (ofLex n : R) = n 
Arguments: {R : Type u_1} [IntCast R] (n : ℤ)
Goal: ofLex ↑n = ↑n
-- Nat.isCompl_even_odd
lemma isCompl_even_odd : IsCompl { n : ℕ | Even n } { n | Odd n } 
Goal: IsCompl {n | Even n} {n | Odd n}
-- Left.neg_nonpos_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: -a ≤ 0 ↔ 0 ≤ a
-- Left.inv_le_one_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.inv_le_one_iff : a⁻¹ ≤ 1 ↔ 1 ≤ a 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: a⁻¹ ≤ 1 ↔ 1 ≤ a
-- Left.one_le_inv_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: 1 ≤ a⁻¹ ↔ a ≤ 1
-- Left.nonneg_neg_iff
@[to_additive (attr := simp) "Uses `left` co(ntra)variant."]
theorem Left.one_le_inv_iff : 1 ≤ a⁻¹ ↔ a ≤ 1 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a : α}
Goal: 0 ≤ -a ↔ a ≤ 0
-- le_inv_mul_iff_mul_le
@[to_additive (attr := simp)]
theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b ≤ a⁻¹ * c ↔ a * b ≤ c
-- le_neg_add_iff_add_le
@[to_additive (attr := simp)]
theorem le_inv_mul_iff_mul_le : b ≤ a⁻¹ * c ↔ a * b ≤ c 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b ≤ -a + c ↔ a + b ≤ c
-- neg_add_le_iff_le_add
@[to_additive (attr := simp)]
theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: -b + a ≤ c ↔ a ≤ b + c
-- inv_mul_le_iff_le_mul
@[to_additive (attr := simp)]
theorem inv_mul_le_iff_le_mul : b⁻¹ * a ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b c : α}
Goal: b⁻¹ * a ≤ c ↔ a ≤ b * c
-- inv_le_iff_one_le_mul'
@[to_additive neg_le_iff_add_nonneg']
theorem inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a⁻¹ ≤ b ↔ 1 ≤ a * b
-- neg_le_iff_add_nonneg'
@[to_additive neg_le_iff_add_nonneg']
theorem inv_le_iff_one_le_mul' : a⁻¹ ≤ b ↔ 1 ≤ a * b 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: -a ≤ b ↔ 0 ≤ a + b
-- le_inv_iff_mul_le_one_left
@[to_additive]
theorem le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a ≤ b⁻¹ ↔ b * a ≤ 1
-- le_neg_iff_add_nonpos_left
@[to_additive]
theorem le_inv_iff_mul_le_one_left : a ≤ b⁻¹ ↔ b * a ≤ 1 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a ≤ -b ↔ b + a ≤ 0
-- le_neg_add_iff_le
@[to_additive]
theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a 
Arguments: {α : Type u} [AddGroup α] [LE α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: 0 ≤ -b + a ↔ b ≤ a
-- le_inv_mul_iff_le
@[to_additive]
theorem le_inv_mul_iff_le : 1 ≤ b⁻¹ * a ↔ b ≤ a 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: 1 ≤ b⁻¹ * a ↔ b ≤ a
-- inv_mul_le_one_iff
@[to_additive]
theorem inv_mul_le_one_iff : a⁻¹ * b ≤ 1 ↔ b ≤ a 
Arguments: {α : Type u} [Group α] [LE α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] {a b : α}
Goal: a⁻¹ * b ≤ 1 ↔ b ≤ a