ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Antitone.neg
@[to_additive]
theorem Antitone.inv (hf : Antitone f) : Monotone fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} (hf : Antitone f)
Goal: Monotone fun x => -f x
-- MonotoneOn.neg
@[to_additive]
theorem MonotoneOn.inv (hf : MonotoneOn f s) : AntitoneOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : MonotoneOn f s)
Goal: AntitoneOn (fun x => -f x) s
-- MonotoneOn.inv
@[to_additive]
theorem MonotoneOn.inv (hf : MonotoneOn f s) : AntitoneOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : MonotoneOn f s)
Goal: AntitoneOn (fun x => (f x)⁻¹) s
-- AntitoneOn.inv
@[to_additive]
theorem AntitoneOn.inv (hf : AntitoneOn f s) : MonotoneOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : AntitoneOn f s)
Goal: MonotoneOn (fun x => (f x)⁻¹) s
-- AntitoneOn.neg
@[to_additive]
theorem AntitoneOn.inv (hf : AntitoneOn f s) : MonotoneOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : AntitoneOn f s)
Goal: MonotoneOn (fun x => -f x) s
-- StrictMono.inv
@[to_additive]
theorem StrictMono.inv (hf : StrictMono f) : StrictAnti fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} (hf : StrictMono f)
Goal: StrictAnti fun x => (f x)⁻¹
-- StrictMono.neg
@[to_additive]
theorem StrictMono.inv (hf : StrictMono f) : StrictAnti fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} (hf : StrictMono f)
Goal: StrictAnti fun x => -f x
-- StrictAnti.neg
@[to_additive]
theorem StrictAnti.inv (hf : StrictAnti f) : StrictMono fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} (hf : StrictAnti f)
Goal: StrictMono fun x => -f x
-- StrictAnti.inv
@[to_additive]
theorem StrictAnti.inv (hf : StrictAnti f) : StrictMono fun x => (f x)⁻¹ 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} (hf : StrictAnti f)
Goal: StrictMono fun x => (f x)⁻¹
-- StrictMonoOn.neg
@[to_additive]
theorem StrictMonoOn.inv (hf : StrictMonoOn f s) : StrictAntiOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : StrictMonoOn f s)
Goal: StrictAntiOn (fun x => -f x) s
-- StrictMonoOn.inv
@[to_additive]
theorem StrictMonoOn.inv (hf : StrictMonoOn f s) : StrictAntiOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : StrictMonoOn f s)
Goal: StrictAntiOn (fun x => (f x)⁻¹) s
-- StrictAntiOn.inv
@[to_additive]
theorem StrictAntiOn.inv (hf : StrictAntiOn f s) : StrictMonoOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [Group α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : StrictAntiOn f s)
Goal: StrictMonoOn (fun x => (f x)⁻¹) s
-- StrictAntiOn.neg
@[to_additive]
theorem StrictAntiOn.inv (hf : StrictAntiOn f s) : StrictMonoOn (fun x => (f x)⁻¹) s 
Arguments: {α : Type u} {β : Type u_1} [AddGroup α] [Preorder 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] [Preorder β] {f : β → α} {s : Set β} (hf : StrictAntiOn f s)
Goal: StrictMonoOn (fun x => -f x) s
-- Nonneg.bot_eq
theorem bot_eq [Preorder α] {a : α} : (⊥ : { x : α // a ≤ x }) = ⟨a, le_rfl⟩ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: ⊥ = ⟨a, ⋯⟩
-- Nonneg.coe_zero
@[simp, norm_cast]
protected theorem coe_zero [Zero α] [Preorder α] : ((0 : { x : α // 0 ≤ x }) : α) = 0 
Arguments: {α : Type u_1} [Zero α] [Preorder α]
Goal: ↑0 = 0
-- Nonneg.mk_eq_zero
@[simp]
theorem mk_eq_zero [Zero α] [Preorder α] {x : α} (hx : 0 ≤ x) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) = 0 ↔ x = 0 
Arguments: {α : Type u_1} [Zero α] [Preorder α] {x : α} (hx : 0 ≤ x)
Goal: ⟨x, hx⟩ = 0 ↔ x = 0
-- Nonneg.mk_add_mk
@[simp]
theorem mk_add_mk [AddZeroClass α] [Preorder α] [CovariantClass α α (· + ·) (· ≤ ·)] {x y : α}
    (hx : 0 ≤ x) (hy : 0 ≤ y) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) + ⟨y, hy⟩ = ⟨x + y, add_nonneg hx hy⟩ 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y)
Goal: ⟨x, hx⟩ + ⟨y, hy⟩ = ⟨x + y, ⋯⟩
-- Nonneg.coe_add
@[simp, norm_cast]
protected theorem coe_add [AddZeroClass α] [Preorder α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (a b : { x : α // 0 ≤ x }) : ((a + b : { x : α // 0 ≤ x }) : α) = a + b 
Arguments: {α : Type u_1} [AddZeroClass α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : { x // 0 ≤ x })
Goal: ↑(a + b) = ↑a + ↑b
-- Nonneg.nsmul_mk
@[simp]
theorem nsmul_mk [AddMonoid α] [Preorder α] [CovariantClass α α (· + ·) (· ≤ ·)] (n : ℕ) {x : α}
    (hx : 0 ≤ x) : (n • (⟨x, hx⟩ : { x : α // 0 ≤ x })) = ⟨n • x, nsmul_nonneg hx n⟩ 
Arguments: {α : Type u_1} [AddMonoid α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (n : ℕ) {x : α} (hx : 0 ≤ x)
Goal: n • ⟨x, hx⟩ = ⟨n • x, ⋯⟩
-- Nonneg.coe_nsmul
@[simp, norm_cast]
protected theorem coe_nsmul [AddMonoid α] [Preorder α] [CovariantClass α α (· + ·) (· ≤ ·)]
    (n : ℕ) (a : { x : α // 0 ≤ x }) : ((n • a : { x : α // 0 ≤ x }) : α) = n • (a : α) 
Arguments: {α : Type u_1} [AddMonoid α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (n : ℕ) (a : { x // 0 ≤ x })
Goal: ↑(n • a) = n • ↑a
-- Nonneg.coe_one
@[simp, norm_cast]
protected theorem coe_one : ((1 : { x : α // 0 ≤ x }) : α) = 1 
Arguments: {α : Type u_1} [Zero α] [One α] [LE α] [ZeroLEOneClass α]
Goal: ↑1 = 1
-- Nonneg.mk_eq_one
@[simp]
theorem mk_eq_one {x : α} (hx : 0 ≤ x) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) = 1 ↔ x = 1 
Arguments: {α : Type u_1} [Zero α] [One α] [LE α] [ZeroLEOneClass α] {x : α} (hx : 0 ≤ x)
Goal: ⟨x, hx⟩ = 1 ↔ x = 1
-- Nonneg.coe_mul
@[simp, norm_cast]
protected theorem coe_mul (a b : { x : α // 0 ≤ x }) :
    ((a * b : { x : α // 0 ≤ x }) : α) = a * b 
Arguments: {α : Type u_1} [MulZeroClass α] [Preorder α] [PosMulMono α] (a b : { x // 0 ≤ x })
Goal: ↑(a * b) = ↑a * ↑b
-- Nonneg.mk_mul_mk
@[simp]
theorem mk_mul_mk {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) * ⟨y, hy⟩ = ⟨x * y, mul_nonneg hx hy⟩ 
Arguments: {α : Type u_1} [MulZeroClass α] [Preorder α] [PosMulMono α] {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y)
Goal: ⟨x, hx⟩ * ⟨y, hy⟩ = ⟨x * y, ⋯⟩
-- Nonneg.nsmul_coe
@[norm_cast]
theorem nsmul_coe (n : ℕ) (r : { x : α // 0 ≤ x }) :
    ↑(n • r) = n • (r : α) 
Arguments: {α : Type u_1} [AddMonoid α] [Preorder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (n : ℕ) (r : { x // 0 ≤ x })
Goal: ↑(n • r) = n • ↑r
-- Nonneg.coe_natCast
@[simp, norm_cast]
protected theorem coe_natCast (n : ℕ) : ((↑n : { x : α // 0 ≤ x }) : α) = n 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass α] (n : ℕ)
Goal: ↑↑n = ↑n
-- Nonneg.mk_natCast
@[simp]
theorem mk_natCast (n : ℕ) : (⟨n, n.cast_nonneg'⟩ : { x : α // 0 ≤ x }) = n 
Arguments: {α : Type u_1} [AddMonoidWithOne α] [PartialOrder α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [ZeroLEOneClass α] (n : ℕ)
Goal: ⟨↑n, ⋯⟩ = ↑n
-- Nonneg.pow_nonneg
@[simp]
theorem pow_nonneg {a : α} (H : 0 ≤ a) : ∀ n : ℕ, 0 ≤ a ^ n
  
Arguments: {α : Type u_1} [MonoidWithZero α] [Preorder α] [ZeroLEOneClass α] [PosMulMono α] {a : α} (H : 0 ≤ a) (n : ℕ)
Goal: 0 ≤ a ^ n
-- Nonneg.coe_pow
@[simp, norm_cast]
protected theorem coe_pow (a : { x : α // 0 ≤ x }) (n : ℕ) :
    (↑(a ^ n) : α) = (a : α) ^ n 
Arguments: {α : Type u_1} [MonoidWithZero α] [Preorder α] [ZeroLEOneClass α] [PosMulMono α] (a : { x // 0 ≤ x }) (n : ℕ)
Goal: ↑(a ^ n) = ↑a ^ n
-- Nonneg.mk_pow
@[simp]
theorem mk_pow {x : α} (hx : 0 ≤ x) (n : ℕ) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) ^ n = ⟨x ^ n, pow_nonneg hx n⟩ 
Arguments: {α : Type u_1} [MonoidWithZero α] [Preorder α] [ZeroLEOneClass α] [PosMulMono α] {x : α} (hx : 0 ≤ x) (n : ℕ)
Goal: ⟨x, hx⟩ ^ n = ⟨x ^ n, ⋯⟩
-- Nonneg.coe_toNonneg
@[simp]
theorem coe_toNonneg {a : α} : (toNonneg a : α) = max a 0 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] {a : α}
Goal: ↑(Nonneg.toNonneg a) = max a 0
-- Nonneg.toNonneg_of_nonneg
@[simp]
theorem toNonneg_of_nonneg {a : α} (h : 0 ≤ a) : toNonneg a = ⟨a, h⟩ 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] {a : α} (h : 0 ≤ a)
Goal: Nonneg.toNonneg a = ⟨a, h⟩
-- Nonneg.toNonneg_coe
@[simp]
theorem toNonneg_coe {a : { x : α // 0 ≤ x }} : toNonneg (a : α) = a 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] {a : { x // 0 ≤ x }}
Goal: Nonneg.toNonneg ↑a = a
-- Nonneg.toNonneg_le
@[simp]
theorem toNonneg_le {a : α} {b : { x : α // 0 ≤ x }} : toNonneg a ≤ b ↔ a ≤ b 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] {a : α} {b : { x // 0 ≤ x }}
Goal: Nonneg.toNonneg a ≤ b ↔ a ≤ ↑b
-- Nonneg.toNonneg_lt
@[simp]
theorem toNonneg_lt {a : { x : α // 0 ≤ x }} {b : α} : a < toNonneg b ↔ ↑a < b 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] {a : { x // 0 ≤ x }} {b : α}
Goal: a < Nonneg.toNonneg b ↔ ↑a < b
-- Nonneg.mk_sub_mk
@[simp]
theorem mk_sub_mk [Sub α] {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y) :
    (⟨x, hx⟩ : { x : α // 0 ≤ x }) - ⟨y, hy⟩ = toNonneg (x - y) 
Arguments: {α : Type u_1} [Zero α] [LinearOrder α] [Sub α] {x y : α} (hx : 0 ≤ x) (hy : 0 ≤ y)
Goal: ⟨x, hx⟩ - ⟨y, hy⟩ = Nonneg.toNonneg (x - y)
-- one_add_mul_le_pow'
lemma one_add_mul_le_pow' (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a)) (H : 0 ≤ 2 + a) :
    ∀ n : ℕ, 1 + n * a ≤ (1 + a) ^ n
  
Arguments: {R : Type u_1} [OrderedSemiring R] {a : R} (Hsq : 0 ≤ a * a) (Hsq' : 0 ≤ (1 + a) * (1 + a)) (H : 0 ≤ 2 + a) (n : ℕ)
Goal: 1 + ↑n * a ≤ (1 + a) ^ n
-- one_add_mul_le_pow
lemma one_add_mul_le_pow (H : -2 ≤ a) (n : ℕ) : 1 + n * a ≤ (1 + a) ^ n 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a : R} (H : -2 ≤ a) (n : ℕ)
Goal: 1 + ↑n * a ≤ (1 + a) ^ n
-- one_add_mul_sub_le_pow
lemma one_add_mul_sub_le_pow (H : -1 ≤ a) (n : ℕ) : 1 + n * (a - 1) ≤ a ^ n 
Arguments: {R : Type u_1} [LinearOrderedRing R] {a : R} (H : -1 ≤ a) (n : ℕ)
Goal: 1 + ↑n * (a - 1) ≤ a ^ n
-- sub_toIcoDiv_zsmul_mem_Ico
theorem sub_toIcoDiv_zsmul_mem_Ico (a b : α) : b - toIcoDiv hp a b • p ∈ Set.Ico a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIcoDiv hp a b • p ∈ Set.Ico a (a + p)
-- toIcoDiv_eq_of_sub_zsmul_mem_Ico
theorem toIcoDiv_eq_of_sub_zsmul_mem_Ico (h : b - n • p ∈ Set.Ico a (a + p)) :
    toIcoDiv hp a b = n 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α} {n : ℤ} (h : b - n • p ∈ Set.Ico a (a + p))
Goal: toIcoDiv hp a b = n
-- sub_toIocDiv_zsmul_mem_Ioc
theorem sub_toIocDiv_zsmul_mem_Ioc (a b : α) : b - toIocDiv hp a b • p ∈ Set.Ioc a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIocDiv hp a b • p ∈ Set.Ioc a (a + p)
-- toIocDiv_eq_of_sub_zsmul_mem_Ioc
theorem toIocDiv_eq_of_sub_zsmul_mem_Ioc (h : b - n • p ∈ Set.Ioc a (a + p)) :
    toIocDiv hp a b = n 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α} {n : ℤ} (h : b - n • p ∈ Set.Ioc a (a + p))
Goal: toIocDiv hp a b = n
-- toIcoMod_mem_Ico
theorem toIcoMod_mem_Ico (a b : α) : toIcoMod hp a b ∈ Set.Ico a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b ∈ Set.Ico a (a + p)
-- toIcoMod_mem_Ico'
theorem toIcoMod_mem_Ico' (b : α) : toIcoMod hp 0 b ∈ Set.Ico 0 p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (b : α)
Goal: toIcoMod hp 0 b ∈ Set.Ico 0 p
-- toIocMod_mem_Ioc
theorem toIocMod_mem_Ioc (a b : α) : toIocMod hp a b ∈ Set.Ioc a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b ∈ Set.Ioc a (a + p)
-- left_le_toIcoMod
theorem left_le_toIcoMod (a b : α) : a ≤ toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: a ≤ toIcoMod hp a b
-- left_lt_toIocMod
theorem left_lt_toIocMod (a b : α) : a < toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: a < toIocMod hp a b
-- toIcoMod_lt_right
theorem toIcoMod_lt_right (a b : α) : toIcoMod hp a b < a + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b < a + p
-- toIocMod_le_right
theorem toIocMod_le_right (a b : α) : toIocMod hp a b ≤ a + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b ≤ a + p
-- self_sub_toIcoDiv_zsmul
@[simp]
theorem self_sub_toIcoDiv_zsmul (a b : α) : b - toIcoDiv hp a b • p = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIcoDiv hp a b • p = toIcoMod hp a b
-- self_sub_toIocDiv_zsmul
@[simp]
theorem self_sub_toIocDiv_zsmul (a b : α) : b - toIocDiv hp a b • p = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIocDiv hp a b • p = toIocMod hp a b
-- toIcoDiv_zsmul_sub_self
@[simp]
theorem toIcoDiv_zsmul_sub_self (a b : α) : toIcoDiv hp a b • p - b = -toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a b • p - b = -toIcoMod hp a b
-- toIocDiv_zsmul_sub_self
@[simp]
theorem toIocDiv_zsmul_sub_self (a b : α) : toIocDiv hp a b • p - b = -toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a b • p - b = -toIocMod hp a b
-- toIcoMod_sub_self
@[simp]
theorem toIcoMod_sub_self (a b : α) : toIcoMod hp a b - b = -toIcoDiv hp a b • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b - b = -toIcoDiv hp a b • p
-- toIocMod_sub_self
@[simp]
theorem toIocMod_sub_self (a b : α) : toIocMod hp a b - b = -toIocDiv hp a b • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b - b = -toIocDiv hp a b • p
-- self_sub_toIcoMod
@[simp]
theorem self_sub_toIcoMod (a b : α) : b - toIcoMod hp a b = toIcoDiv hp a b • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIcoMod hp a b = toIcoDiv hp a b • p
-- self_sub_toIocMod
@[simp]
theorem self_sub_toIocMod (a b : α) : b - toIocMod hp a b = toIocDiv hp a b • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: b - toIocMod hp a b = toIocDiv hp a b • p
-- toIcoMod_add_toIcoDiv_zsmul
@[simp]
theorem toIcoMod_add_toIcoDiv_zsmul (a b : α) : toIcoMod hp a b + toIcoDiv hp a b • p = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b + toIcoDiv hp a b • p = b
-- toIocMod_add_toIocDiv_zsmul
@[simp]
theorem toIocMod_add_toIocDiv_zsmul (a b : α) : toIocMod hp a b + toIocDiv hp a b • p = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b + toIocDiv hp a b • p = b
-- toIcoDiv_zsmul_sub_toIcoMod
@[simp]
theorem toIcoDiv_zsmul_sub_toIcoMod (a b : α) : toIcoDiv hp a b • p + toIcoMod hp a b = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a b • p + toIcoMod hp a b = b
-- toIocDiv_zsmul_sub_toIocMod
@[simp]
theorem toIocDiv_zsmul_sub_toIocMod (a b : α) : toIocDiv hp a b • p + toIocMod hp a b = b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a b • p + toIocMod hp a b = b
-- toIcoMod_eq_iff
theorem toIcoMod_eq_iff : toIcoMod hp a b = c ↔ c ∈ Set.Ico a (a + p) ∧ ∃ z : ℤ, b = c + z • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α}
Goal: toIcoMod hp a b = c ↔ c ∈ Set.Ico a (a + p) ∧ ∃ z, b = c + z • p
-- toIocMod_eq_iff
theorem toIocMod_eq_iff : toIocMod hp a b = c ↔ c ∈ Set.Ioc a (a + p) ∧ ∃ z : ℤ, b = c + z • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α}
Goal: toIocMod hp a b = c ↔ c ∈ Set.Ioc a (a + p) ∧ ∃ z, b = c + z • p
-- toIcoDiv_apply_left
@[simp]
theorem toIcoDiv_apply_left (a : α) : toIcoDiv hp a a = 0 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIcoDiv hp a a = 0
-- toIocDiv_apply_left
@[simp]
theorem toIocDiv_apply_left (a : α) : toIocDiv hp a a = -1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIocDiv hp a a = -1
-- toIcoMod_apply_left
@[simp]
theorem toIcoMod_apply_left (a : α) : toIcoMod hp a a = a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIcoMod hp a a = a
-- toIocMod_apply_left
@[simp]
theorem toIocMod_apply_left (a : α) : toIocMod hp a a = a + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIocMod hp a a = a + p
-- toIcoDiv_apply_right
theorem toIcoDiv_apply_right (a : α) : toIcoDiv hp a (a + p) = 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIcoDiv hp a (a + p) = 1
-- toIocDiv_apply_right
theorem toIocDiv_apply_right (a : α) : toIocDiv hp a (a + p) = 0 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIocDiv hp a (a + p) = 0
-- toIcoMod_apply_right
theorem toIcoMod_apply_right (a : α) : toIcoMod hp a (a + p) = a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIcoMod hp a (a + p) = a
-- toIocMod_apply_right
theorem toIocMod_apply_right (a : α) : toIocMod hp a (a + p) = a + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: toIocMod hp a (a + p) = a + p
-- toIcoDiv_add_zsmul
@[simp]
theorem toIcoDiv_add_zsmul (a b : α) (m : ℤ) : toIcoDiv hp a (b + m • p) = toIcoDiv hp a b + m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoDiv hp a (b + m • p) = toIcoDiv hp a b + m
-- toIcoDiv_add_zsmul'
@[simp]
theorem toIcoDiv_add_zsmul' (a b : α) (m : ℤ) :
    toIcoDiv hp (a + m • p) b = toIcoDiv hp a b - m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoDiv hp (a + m • p) b = toIcoDiv hp a b - m
-- toIocDiv_add_zsmul
@[simp]
theorem toIocDiv_add_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b + m • p) = toIocDiv hp a b + m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocDiv hp a (b + m • p) = toIocDiv hp a b + m
-- toIocDiv_add_zsmul'
@[simp]
theorem toIocDiv_add_zsmul' (a b : α) (m : ℤ) :
    toIocDiv hp (a + m • p) b = toIocDiv hp a b - m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocDiv hp (a + m • p) b = toIocDiv hp a b - m
-- toIcoDiv_zsmul_add
@[simp]
theorem toIcoDiv_zsmul_add (a b : α) (m : ℤ) : toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoDiv hp a (m • p + b) = m + toIcoDiv hp a b
-- toIocDiv_zsmul_add
@[simp]
theorem toIocDiv_zsmul_add (a b : α) (m : ℤ) : toIocDiv hp a (m • p + b) = m + toIocDiv hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocDiv hp a (m • p + b) = m + toIocDiv hp a b
-- toIcoDiv_sub_zsmul
@[simp]
theorem toIcoDiv_sub_zsmul (a b : α) (m : ℤ) : toIcoDiv hp a (b - m • p) = toIcoDiv hp a b - m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoDiv hp a (b - m • p) = toIcoDiv hp a b - m
-- toIcoDiv_sub_zsmul'
@[simp]
theorem toIcoDiv_sub_zsmul' (a b : α) (m : ℤ) :
    toIcoDiv hp (a - m • p) b = toIcoDiv hp a b + m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoDiv hp (a - m • p) b = toIcoDiv hp a b + m
-- toIocDiv_sub_zsmul
@[simp]
theorem toIocDiv_sub_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b - m • p) = toIocDiv hp a b - m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocDiv hp a (b - m • p) = toIocDiv hp a b - m
-- toIocDiv_sub_zsmul'
@[simp]
theorem toIocDiv_sub_zsmul' (a b : α) (m : ℤ) :
    toIocDiv hp (a - m • p) b = toIocDiv hp a b + m 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocDiv hp (a - m • p) b = toIocDiv hp a b + m
-- toIcoDiv_add_right
@[simp]
theorem toIcoDiv_add_right (a b : α) : toIcoDiv hp a (b + p) = toIcoDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a (b + p) = toIcoDiv hp a b + 1
-- toIcoDiv_add_right'
@[simp]
theorem toIcoDiv_add_right' (a b : α) : toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1