ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.arcsin_of_one_le
theorem arcsin_of_one_le {x : ℝ} (hx : 1 ≤ x) : arcsin x = π / 2 
Arguments: {x : ℝ} (hx : 1 ≤ x)
Goal: Real.arcsin x = π / 2
-- Real.arcsin_neg_one
theorem arcsin_neg_one : arcsin (-1) = -(π / 2) 
Goal: Real.arcsin (-1) = -(π / 2)
-- Real.arcsin_of_le_neg_one
theorem arcsin_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arcsin x = -(π / 2) 
Arguments: {x : ℝ} (hx : x ≤ -1)
Goal: Real.arcsin x = -(π / 2)
-- Real.arcsin_neg
@[simp]
theorem arcsin_neg (x : ℝ) : arcsin (-x) = -arcsin x 
Arguments: (x : ℝ)
Goal: Real.arcsin (-x) = -Real.arcsin x
-- Real.arcsin_le_iff_le_sin
theorem arcsin_le_iff_le_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) :
    arcsin x ≤ y ↔ x ≤ sin y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Icc (-1) 1) (hy : y ∈ Set.Icc (-(π / 2)) (π / 2))
Goal: Real.arcsin x ≤ y ↔ x ≤ Real.sin y
-- Real.arcsin_le_iff_le_sin'
theorem arcsin_le_iff_le_sin' {x y : ℝ} (hy : y ∈ Ico (-(π / 2)) (π / 2)) :
    arcsin x ≤ y ↔ x ≤ sin y 
Arguments: {x y : ℝ} (hy : y ∈ Set.Ico (-(π / 2)) (π / 2))
Goal: Real.arcsin x ≤ y ↔ x ≤ Real.sin y
-- Real.le_arcsin_iff_sin_le
theorem le_arcsin_iff_sin_le {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) :
    x ≤ arcsin y ↔ sin x ≤ y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Icc (-(π / 2)) (π / 2)) (hy : y ∈ Set.Icc (-1) 1)
Goal: x ≤ Real.arcsin y ↔ Real.sin x ≤ y
-- Real.le_arcsin_iff_sin_le'
theorem le_arcsin_iff_sin_le' {x y : ℝ} (hx : x ∈ Ioc (-(π / 2)) (π / 2)) :
    x ≤ arcsin y ↔ sin x ≤ y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Ioc (-(π / 2)) (π / 2))
Goal: x ≤ Real.arcsin y ↔ Real.sin x ≤ y
-- Real.arcsin_lt_iff_lt_sin
theorem arcsin_lt_iff_lt_sin {x y : ℝ} (hx : x ∈ Icc (-1 : ℝ) 1) (hy : y ∈ Icc (-(π / 2)) (π / 2)) :
    arcsin x < y ↔ x < sin y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Icc (-1) 1) (hy : y ∈ Set.Icc (-(π / 2)) (π / 2))
Goal: Real.arcsin x < y ↔ x < Real.sin y
-- Real.arcsin_lt_iff_lt_sin'
theorem arcsin_lt_iff_lt_sin' {x y : ℝ} (hy : y ∈ Ioc (-(π / 2)) (π / 2)) :
    arcsin x < y ↔ x < sin y 
Arguments: {x y : ℝ} (hy : y ∈ Set.Ioc (-(π / 2)) (π / 2))
Goal: Real.arcsin x < y ↔ x < Real.sin y
-- Real.lt_arcsin_iff_sin_lt
theorem lt_arcsin_iff_sin_lt {x y : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) (hy : y ∈ Icc (-1 : ℝ) 1) :
    x < arcsin y ↔ sin x < y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Icc (-(π / 2)) (π / 2)) (hy : y ∈ Set.Icc (-1) 1)
Goal: x < Real.arcsin y ↔ Real.sin x < y
-- Real.lt_arcsin_iff_sin_lt'
theorem lt_arcsin_iff_sin_lt' {x y : ℝ} (hx : x ∈ Ico (-(π / 2)) (π / 2)) :
    x < arcsin y ↔ sin x < y 
Arguments: {x y : ℝ} (hx : x ∈ Set.Ico (-(π / 2)) (π / 2))
Goal: x < Real.arcsin y ↔ Real.sin x < y
-- Real.arcsin_eq_iff_eq_sin
theorem arcsin_eq_iff_eq_sin {x y : ℝ} (hy : y ∈ Ioo (-(π / 2)) (π / 2)) :
    arcsin x = y ↔ x = sin y 
Arguments: {x y : ℝ} (hy : y ∈ Set.Ioo (-(π / 2)) (π / 2))
Goal: Real.arcsin x = y ↔ x = Real.sin y
-- Real.arcsin_nonneg
@[simp]
theorem arcsin_nonneg {x : ℝ} : 0 ≤ arcsin x ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: 0 ≤ Real.arcsin x ↔ 0 ≤ x
-- Real.arcsin_nonpos
@[simp]
theorem arcsin_nonpos {x : ℝ} : arcsin x ≤ 0 ↔ x ≤ 0 
Arguments: {x : ℝ}
Goal: Real.arcsin x ≤ 0 ↔ x ≤ 0
-- Real.arcsin_eq_zero_iff
@[simp]
theorem arcsin_eq_zero_iff {x : ℝ} : arcsin x = 0 ↔ x = 0 
Arguments: {x : ℝ}
Goal: Real.arcsin x = 0 ↔ x = 0
-- Real.zero_eq_arcsin_iff
@[simp]
theorem zero_eq_arcsin_iff {x} : 0 = arcsin x ↔ x = 0 
Arguments: {x : ℝ}
Goal: 0 = Real.arcsin x ↔ x = 0
-- Real.arcsin_pos
@[simp]
theorem arcsin_pos {x : ℝ} : 0 < arcsin x ↔ 0 < x 
Arguments: {x : ℝ}
Goal: 0 < Real.arcsin x ↔ 0 < x
-- Real.arcsin_lt_zero
@[simp]
theorem arcsin_lt_zero {x : ℝ} : arcsin x < 0 ↔ x < 0 
Arguments: {x : ℝ}
Goal: Real.arcsin x < 0 ↔ x < 0
-- Real.arcsin_lt_pi_div_two
@[simp]
theorem arcsin_lt_pi_div_two {x : ℝ} : arcsin x < π / 2 ↔ x < 1 
Arguments: {x : ℝ}
Goal: Real.arcsin x < π / 2 ↔ x < 1
-- Real.neg_pi_div_two_lt_arcsin
@[simp]
theorem neg_pi_div_two_lt_arcsin {x : ℝ} : -(π / 2) < arcsin x ↔ -1 < x 
Arguments: {x : ℝ}
Goal: -(π / 2) < Real.arcsin x ↔ -1 < x
-- Real.arcsin_eq_pi_div_two
@[simp]
theorem arcsin_eq_pi_div_two {x : ℝ} : arcsin x = π / 2 ↔ 1 ≤ x 
Arguments: {x : ℝ}
Goal: Real.arcsin x = π / 2 ↔ 1 ≤ x
-- Real.pi_div_two_eq_arcsin
@[simp]
theorem pi_div_two_eq_arcsin {x} : π / 2 = arcsin x ↔ 1 ≤ x 
Arguments: {x : ℝ}
Goal: π / 2 = Real.arcsin x ↔ 1 ≤ x
-- Real.pi_div_two_le_arcsin
@[simp]
theorem pi_div_two_le_arcsin {x} : π / 2 ≤ arcsin x ↔ 1 ≤ x 
Arguments: {x : ℝ}
Goal: π / 2 ≤ Real.arcsin x ↔ 1 ≤ x
-- Real.arcsin_eq_neg_pi_div_two
@[simp]
theorem arcsin_eq_neg_pi_div_two {x : ℝ} : arcsin x = -(π / 2) ↔ x ≤ -1 
Arguments: {x : ℝ}
Goal: Real.arcsin x = -(π / 2) ↔ x ≤ -1
-- Real.neg_pi_div_two_eq_arcsin
@[simp]
theorem neg_pi_div_two_eq_arcsin {x} : -(π / 2) = arcsin x ↔ x ≤ -1 
Arguments: {x : ℝ}
Goal: -(π / 2) = Real.arcsin x ↔ x ≤ -1
-- Real.arcsin_le_neg_pi_div_two
@[simp]
theorem arcsin_le_neg_pi_div_two {x} : arcsin x ≤ -(π / 2) ↔ x ≤ -1 
Arguments: {x : ℝ}
Goal: Real.arcsin x ≤ -(π / 2) ↔ x ≤ -1
-- Real.pi_div_four_le_arcsin
@[simp]
theorem pi_div_four_le_arcsin {x} : π / 4 ≤ arcsin x ↔ √2 / 2 ≤ x 
Arguments: {x : ℝ}
Goal: π / 4 ≤ Real.arcsin x ↔ √2 / 2 ≤ x
-- Real.mapsTo_sin_Ioo
theorem mapsTo_sin_Ioo : MapsTo sin (Ioo (-(π / 2)) (π / 2)) (Ioo (-1) 1) 
Goal: Set.MapsTo Real.sin (Set.Ioo (-(π / 2)) (π / 2)) (Set.Ioo (-1) 1)
-- Real.cos_arcsin_nonneg
theorem cos_arcsin_nonneg (x : ℝ) : 0 ≤ cos (arcsin x) 
Arguments: (x : ℝ)
Goal: 0 ≤ Real.cos (Real.arcsin x)
-- Real.cos_arcsin
theorem cos_arcsin (x : ℝ) : cos (arcsin x) = √(1 - x ^ 2) 
Arguments: (x : ℝ)
Goal: Real.cos (Real.arcsin x) = √(1 - x ^ 2)
-- Real.tan_arcsin
theorem tan_arcsin (x : ℝ) : tan (arcsin x) = x / √(1 - x ^ 2) 
Arguments: (x : ℝ)
Goal: Real.tan (Real.arcsin x) = x / √(1 - x ^ 2)
-- Real.arccos_eq_pi_div_two_sub_arcsin
theorem arccos_eq_pi_div_two_sub_arcsin (x : ℝ) : arccos x = π / 2 - arcsin x 
Arguments: (x : ℝ)
Goal: Real.arccos x = π / 2 - Real.arcsin x
-- Real.arcsin_eq_pi_div_two_sub_arccos
theorem arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x 
Arguments: (x : ℝ)
Goal: Real.arcsin x = π / 2 - Real.arccos x
-- Real.arccos_le_pi
theorem arccos_le_pi (x : ℝ) : arccos x ≤ π 
Arguments: (x : ℝ)
Goal: Real.arccos x ≤ π
-- Real.arccos_nonneg
theorem arccos_nonneg (x : ℝ) : 0 ≤ arccos x 
Arguments: (x : ℝ)
Goal: 0 ≤ Real.arccos x
-- Real.arccos_pos
@[simp]
theorem arccos_pos {x : ℝ} : 0 < arccos x ↔ x < 1 
Arguments: {x : ℝ}
Goal: 0 < Real.arccos x ↔ x < 1
-- Real.cos_arccos
theorem cos_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arccos x) = x 
Arguments: {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1)
Goal: Real.cos (Real.arccos x) = x
-- Real.arccos_cos
theorem arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x 
Arguments: {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π)
Goal: Real.arccos (Real.cos x) = x
-- Real.arccos_eq_of_eq_cos
lemma arccos_eq_of_eq_cos (hy₀ : 0 ≤ y) (hy₁ : y ≤ π) (hxy : x = cos y) : arccos x = y 
Arguments: {x y : ℝ} (hy₀ : 0 ≤ y) (hy₁ : y ≤ π) (hxy : x = Real.cos y)
Goal: Real.arccos x = y
-- Real.strictAntiOn_arccos
theorem strictAntiOn_arccos : StrictAntiOn arccos (Icc (-1) 1) 
Goal: StrictAntiOn Real.arccos (Set.Icc (-1) 1)
-- Real.arccos_injOn
theorem arccos_injOn : InjOn arccos (Icc (-1) 1) 
Goal: Set.InjOn Real.arccos (Set.Icc (-1) 1)
-- Real.arccos_inj
theorem arccos_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) :
    arccos x = arccos y ↔ x = y 
Arguments: {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1)
Goal: Real.arccos x = Real.arccos y ↔ x = y
-- Real.arccos_zero
@[simp]
theorem arccos_zero : arccos 0 = π / 2 
Goal: Real.arccos 0 = π / 2
-- Real.arccos_one
@[simp]
theorem arccos_one : arccos 1 = 0 
Goal: Real.arccos 1 = 0
-- Real.arccos_neg_one
@[simp]
theorem arccos_neg_one : arccos (-1) = π 
Goal: Real.arccos (-1) = π
-- Real.arccos_eq_zero
@[simp]
theorem arccos_eq_zero {x} : arccos x = 0 ↔ 1 ≤ x 
Arguments: {x : ℝ}
Goal: Real.arccos x = 0 ↔ 1 ≤ x
-- Real.arccos_eq_pi_div_two
@[simp]
theorem arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 
Arguments: {x : ℝ}
Goal: Real.arccos x = π / 2 ↔ x = 0
-- Real.arccos_eq_pi
@[simp]
theorem arccos_eq_pi {x} : arccos x = π ↔ x ≤ -1 
Arguments: {x : ℝ}
Goal: Real.arccos x = π ↔ x ≤ -1
-- Real.arccos_neg
theorem arccos_neg (x : ℝ) : arccos (-x) = π - arccos x 
Arguments: (x : ℝ)
Goal: Real.arccos (-x) = π - Real.arccos x
-- Real.arccos_of_one_le
theorem arccos_of_one_le {x : ℝ} (hx : 1 ≤ x) : arccos x = 0 
Arguments: {x : ℝ} (hx : 1 ≤ x)
Goal: Real.arccos x = 0
-- Real.arccos_of_le_neg_one
theorem arccos_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arccos x = π 
Arguments: {x : ℝ} (hx : x ≤ -1)
Goal: Real.arccos x = π
-- Real.sin_arccos
theorem sin_arccos (x : ℝ) : sin (arccos x) = √(1 - x ^ 2) 
Arguments: (x : ℝ)
Goal: Real.sin (Real.arccos x) = √(1 - x ^ 2)
-- Real.arccos_le_pi_div_two
@[simp]
theorem arccos_le_pi_div_two {x} : arccos x ≤ π / 2 ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: Real.arccos x ≤ π / 2 ↔ 0 ≤ x
-- Real.arccos_lt_pi_div_two
@[simp]
theorem arccos_lt_pi_div_two {x : ℝ} : arccos x < π / 2 ↔ 0 < x 
Arguments: {x : ℝ}
Goal: Real.arccos x < π / 2 ↔ 0 < x
-- Real.arccos_le_pi_div_four
@[simp]
theorem arccos_le_pi_div_four {x} : arccos x ≤ π / 4 ↔ √2 / 2 ≤ x 
Arguments: {x : ℝ}
Goal: Real.arccos x ≤ π / 4 ↔ √2 / 2 ≤ x
-- Real.continuous_arccos
@[continuity]
theorem continuous_arccos : Continuous arccos 
Goal: Continuous Real.arccos
-- Real.tan_arccos
theorem tan_arccos (x : ℝ) : tan (arccos x) = √(1 - x ^ 2) / x 
Arguments: (x : ℝ)
Goal: Real.tan (Real.arccos x) = √(1 - x ^ 2) / x
-- Real.arccos_eq_arcsin
theorem arccos_eq_arcsin {x : ℝ} (h : 0 ≤ x) : arccos x = arcsin (√(1 - x ^ 2)) 
Arguments: {x : ℝ} (h : 0 ≤ x)
Goal: Real.arccos x = Real.arcsin √(1 - x ^ 2)
-- Real.arcsin_eq_arccos
theorem arcsin_eq_arccos {x : ℝ} (h : 0 ≤ x) : arcsin x = arccos (√(1 - x ^ 2)) 
Arguments: {x : ℝ} (h : 0 ≤ x)
Goal: Real.arcsin x = Real.arccos √(1 - x ^ 2)
-- Polynomial.tendsto_div_exp_atTop
theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) 
Arguments: (p : Polynomial ℝ)
Goal: Filter.Tendsto (fun x => Polynomial.eval x p / rexp x) Filter.atTop (𝓝 0)
-- EReal.log_exp
@[simp] lemma EReal.log_exp (x : EReal) : log (exp x) = x 
Arguments: (x : EReal)
Goal: x.exp.log = x
-- ENNReal.exp_log
@[simp] lemma ENNReal.exp_log (x : ℝ≥0∞) : exp (log x) = x 
Arguments: (x : ℝ≥0∞)
Goal: x.log.exp = x
-- EReal.exp_nmul
lemma exp_nmul (x : EReal) (n : ℕ) : exp (n * x) = (exp x) ^ n 
Arguments: (x : EReal) (n : ℕ)
Goal: (↑n * x).exp = x.exp ^ n
-- EReal.exp_mul
lemma exp_mul (x : EReal) (y : ℝ) : exp (x * y) = (exp x) ^ y 
Arguments: (x : EReal) (y : ℝ)
Goal: (x * ↑y).exp = x.exp ^ y
-- ENNReal.logOrderIso_apply
@[simp] lemma logOrderIso_apply (x : ℝ≥0∞) : logOrderIso x = log x 
Arguments: (x : ℝ≥0∞)
Goal: ENNReal.logOrderIso x = x.log
-- EReal.expOrderIso_apply
@[simp] lemma _root_.EReal.expOrderIso_apply (x : EReal) : expOrderIso x = exp x 
Arguments: (x : EReal)
Goal: EReal.expOrderIso x = x.exp
-- ENNReal.logOrderIso_symm
@[simp] lemma logOrderIso_symm : logOrderIso.symm = expOrderIso 
Goal: ENNReal.logOrderIso.symm = EReal.expOrderIso
-- EReal.expOrderIso_symm
@[simp] lemma _root_.EReal.expOrderIso_symm : expOrderIso.symm = logOrderIso 
Goal: EReal.expOrderIso.symm = ENNReal.logOrderIso
-- ENNReal.logHomeomorph_apply
@[simp] lemma logHomeomorph_apply (x : ℝ≥0∞) : logHomeomorph x = log x 
Arguments: (x : ℝ≥0∞)
Goal: ENNReal.logHomeomorph x = x.log
-- EReal.expHomeomorph_apply
@[simp] lemma _root_.EReal.expHomeomorph_apply (x : EReal) : expHomeomorph x = exp x 
Arguments: (x : EReal)
Goal: EReal.expHomeomorph x = x.exp
-- ENNReal.logHomeomorph_symm
@[simp] lemma logHomeomorph_symm : logHomeomorph.symm = expHomeomorph 
Goal: ENNReal.logHomeomorph.symm = EReal.expHomeomorph
-- EReal.expHomeomorph_symm
@[simp] lemma _root_.EReal.expHomeomorph_symm : expHomeomorph.symm = logHomeomorph 
Goal: EReal.expHomeomorph.symm = ENNReal.logHomeomorph
-- ENNReal.continuous_log
@[continuity, fun_prop]
lemma continuous_log : Continuous log 
Goal: Continuous ENNReal.log
-- ENNReal.continuous_exp
@[continuity, fun_prop]
lemma continuous_exp : Continuous exp 
Goal: Continuous EReal.exp
-- ENNReal.measurable_log
@[measurability, fun_prop]
lemma measurable_log : Measurable log 
Goal: Measurable ENNReal.log
-- EReal.measurable_exp
@[measurability, fun_prop]
lemma _root_.EReal.measurable_exp : Measurable exp 
Goal: Measurable EReal.exp
-- Measurable.ennreal_log
@[measurability, fun_prop]
lemma _root_.Measurable.ennreal_log {α : Type*} {_ : MeasurableSpace α}
    {f : α → ℝ≥0∞} (hf : Measurable f) :
    Measurable fun x ↦ log (f x) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {f : α → ℝ≥0∞}, Measurable f → Measurable fun x => (f x).log
-- Measurable.ereal_exp
@[measurability, fun_prop]
lemma _root_.Measurable.ereal_exp {α : Type*} {_ : MeasurableSpace α}
    {f : α → EReal} (hf : Measurable f) :
    Measurable fun x ↦ exp (f x) 
Arguments: {α : Type u_1}
Goal: ∀ {x : MeasurableSpace α} {f : α → EReal}, Measurable f → Measurable fun x => (f x).exp
-- Real.deriv_arcsin_aux
theorem deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ arcsin x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1)
Goal: HasStrictDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x ∧ ContDiffAt ℝ ⊤ Real.arcsin x
-- Real.hasStrictDerivAt_arcsin
theorem hasStrictDerivAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arcsin (1 / √(1 - x ^ 2)) x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1)
Goal: HasStrictDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x
-- Real.hasDerivAt_arcsin
theorem hasDerivAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasDerivAt arcsin (1 / √(1 - x ^ 2)) x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1)
Goal: HasDerivAt Real.arcsin (1 / √(1 - x ^ 2)) x
-- Real.contDiffAt_arcsin
theorem contDiffAt_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} : ContDiffAt ℝ n arcsin x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞}
Goal: ContDiffAt ℝ n Real.arcsin x
-- Real.hasDerivWithinAt_arcsin_Ici
theorem hasDerivWithinAt_arcsin_Ici {x : ℝ} (h : x ≠ -1) :
    HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Ici x) x 
Arguments: {x : ℝ} (h : x ≠ -1)
Goal: HasDerivWithinAt Real.arcsin (1 / √(1 - x ^ 2)) (Set.Ici x) x
-- Real.hasDerivWithinAt_arcsin_Iic
theorem hasDerivWithinAt_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
    HasDerivWithinAt arcsin (1 / √(1 - x ^ 2)) (Iic x) x 
Arguments: {x : ℝ} (h : x ≠ 1)
Goal: HasDerivWithinAt Real.arcsin (1 / √(1 - x ^ 2)) (Set.Iic x) x
-- Real.differentiableWithinAt_arcsin_Ici
theorem differentiableWithinAt_arcsin_Ici {x : ℝ} :
    DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1 
Arguments: {x : ℝ}
Goal: DifferentiableWithinAt ℝ Real.arcsin (Set.Ici x) x ↔ x ≠ -1
-- Real.differentiableWithinAt_arcsin_Iic
theorem differentiableWithinAt_arcsin_Iic {x : ℝ} :
    DifferentiableWithinAt ℝ arcsin (Iic x) x ↔ x ≠ 1 
Arguments: {x : ℝ}
Goal: DifferentiableWithinAt ℝ Real.arcsin (Set.Iic x) x ↔ x ≠ 1
-- Real.differentiableAt_arcsin
theorem differentiableAt_arcsin {x : ℝ} : DifferentiableAt ℝ arcsin x ↔ x ≠ -1 ∧ x ≠ 1 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.arcsin x ↔ x ≠ -1 ∧ x ≠ 1
-- Real.deriv_arcsin
@[simp]
theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) 
Goal: deriv Real.arcsin = fun x => 1 / √(1 - x ^ 2)
-- Real.differentiableOn_arcsin
theorem differentiableOn_arcsin : DifferentiableOn ℝ arcsin {-1, 1}ᶜ 
Goal: DifferentiableOn ℝ Real.arcsin {-1, 1}ᶜ
-- Real.contDiffOn_arcsin
theorem contDiffOn_arcsin {n : ℕ∞} : ContDiffOn ℝ n arcsin {-1, 1}ᶜ 
Arguments: {n : ℕ∞}
Goal: ContDiffOn ℝ n Real.arcsin {-1, 1}ᶜ
-- Real.contDiffAt_arcsin_iff
theorem contDiffAt_arcsin_iff {x : ℝ} {n : ℕ∞} : ContDiffAt ℝ n arcsin x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1 
Arguments: {x : ℝ} {n : ℕ∞}
Goal: ContDiffAt ℝ n Real.arcsin x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1
-- Real.hasStrictDerivAt_arccos
theorem hasStrictDerivAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasStrictDerivAt arccos (-(1 / √(1 - x ^ 2))) x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1)
Goal: HasStrictDerivAt Real.arccos (-(1 / √(1 - x ^ 2))) x
-- Real.hasDerivAt_arccos
theorem hasDerivAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
    HasDerivAt arccos (-(1 / √(1 - x ^ 2))) x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1)
Goal: HasDerivAt Real.arccos (-(1 / √(1 - x ^ 2))) x
-- Real.contDiffAt_arccos
theorem contDiffAt_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} : ContDiffAt ℝ n arccos x 
Arguments: {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞}
Goal: ContDiffAt ℝ n Real.arccos x
-- Real.hasDerivWithinAt_arccos_Ici
theorem hasDerivWithinAt_arccos_Ici {x : ℝ} (h : x ≠ -1) :
    HasDerivWithinAt arccos (-(1 / √(1 - x ^ 2))) (Ici x) x 
Arguments: {x : ℝ} (h : x ≠ -1)
Goal: HasDerivWithinAt Real.arccos (-(1 / √(1 - x ^ 2))) (Set.Ici x) x
-- Real.hasDerivWithinAt_arccos_Iic
theorem hasDerivWithinAt_arccos_Iic {x : ℝ} (h : x ≠ 1) :
    HasDerivWithinAt arccos (-(1 / √(1 - x ^ 2))) (Iic x) x 
Arguments: {x : ℝ} (h : x ≠ 1)
Goal: HasDerivWithinAt Real.arccos (-(1 / √(1 - x ^ 2))) (Set.Iic x) x
-- Real.differentiableWithinAt_arccos_Ici
theorem differentiableWithinAt_arccos_Ici {x : ℝ} :
    DifferentiableWithinAt ℝ arccos (Ici x) x ↔ x ≠ -1 
Arguments: {x : ℝ}
Goal: DifferentiableWithinAt ℝ Real.arccos (Set.Ici x) x ↔ x ≠ -1
-- Real.differentiableWithinAt_arccos_Iic
theorem differentiableWithinAt_arccos_Iic {x : ℝ} :
    DifferentiableWithinAt ℝ arccos (Iic x) x ↔ x ≠ 1 
Arguments: {x : ℝ}
Goal: DifferentiableWithinAt ℝ Real.arccos (Set.Iic x) x ↔ x ≠ 1
-- Real.differentiableAt_arccos
theorem differentiableAt_arccos {x : ℝ} : DifferentiableAt ℝ arccos x ↔ x ≠ -1 ∧ x ≠ 1 
Arguments: {x : ℝ}
Goal: DifferentiableAt ℝ Real.arccos x ↔ x ≠ -1 ∧ x ≠ 1
-- Real.deriv_arccos
@[simp]
theorem deriv_arccos : deriv arccos = fun x => -(1 / √(1 - x ^ 2)) 
Goal: deriv Real.arccos = fun x => -(1 / √(1 - x ^ 2))
-- Real.differentiableOn_arccos
theorem differentiableOn_arccos : DifferentiableOn ℝ arccos {-1, 1}ᶜ 
Goal: DifferentiableOn ℝ Real.arccos {-1, 1}ᶜ
-- Real.contDiffOn_arccos
theorem contDiffOn_arccos {n : ℕ∞} : ContDiffOn ℝ n arccos {-1, 1}ᶜ 
Arguments: {n : ℕ∞}
Goal: ContDiffOn ℝ n Real.arccos {-1, 1}ᶜ
-- Real.contDiffAt_arccos_iff
theorem contDiffAt_arccos_iff {x : ℝ} {n : ℕ∞} :
    ContDiffAt ℝ n arccos x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1 
Arguments: {x : ℝ} {n : ℕ∞}
Goal: ContDiffAt ℝ n Real.arccos x ↔ n = 0 ∨ x ≠ -1 ∧ x ≠ 1
-- gronwallBound_K0
theorem gronwallBound_K0 (δ ε : ℝ) : gronwallBound δ 0 ε = fun x => δ + ε * x 
Arguments: (δ ε : ℝ)
Goal: gronwallBound δ 0 ε = fun x => δ + ε * x
-- gronwallBound_of_K_ne_0
theorem gronwallBound_of_K_ne_0 {δ K ε : ℝ} (hK : K ≠ 0) :
    gronwallBound δ K ε = fun x => δ * exp (K * x) + ε / K * (exp (K * x) - 1) 
Arguments: {δ K ε : ℝ} (hK : K ≠ 0)
Goal: gronwallBound δ K ε = fun x => δ * rexp (K * x) + ε / K * (rexp (K * x) - 1)
-- hasDerivAt_gronwallBound
theorem hasDerivAt_gronwallBound (δ K ε x : ℝ) :
    HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x 
Arguments: (δ K ε x : ℝ)
Goal: HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x
-- hasDerivAt_gronwallBound_shift
theorem hasDerivAt_gronwallBound_shift (δ K ε x a : ℝ) :
    HasDerivAt (fun y => gronwallBound δ K ε (y - a)) (K * gronwallBound δ K ε (x - a) + ε) x 
Arguments: (δ K ε x a : ℝ)
Goal: HasDerivAt (fun y => gronwallBound δ K ε (y - a)) (K * gronwallBound δ K ε (x - a) + ε) x
-- gronwallBound_x0
theorem gronwallBound_x0 (δ K ε : ℝ) : gronwallBound δ K ε 0 = δ 
Arguments: (δ K ε : ℝ)
Goal: gronwallBound δ K ε 0 = δ
-- gronwallBound_ε0
theorem gronwallBound_ε0 (δ K x : ℝ) : gronwallBound δ K 0 x = δ * exp (K * x) 
Arguments: (δ K x : ℝ)
Goal: gronwallBound δ K 0 x = δ * rexp (K * x)
-- gronwallBound_ε0_δ0
theorem gronwallBound_ε0_δ0 (K x : ℝ) : gronwallBound 0 K 0 x = 0 
Arguments: (K x : ℝ)
Goal: gronwallBound 0 K 0 x = 0
-- gronwallBound_continuous_ε
theorem gronwallBound_continuous_ε (δ K x : ℝ) : Continuous fun ε => gronwallBound δ K ε x 
Arguments: (δ K x : ℝ)
Goal: Continuous fun ε => gronwallBound δ K ε x
-- le_gronwallBound_of_liminf_deriv_right_le
theorem le_gronwallBound_of_liminf_deriv_right_le {f f' : ℝ → ℝ} {δ K ε : ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r)
    (ha : f a ≤ δ) (bound : ∀ x ∈ Ico a b, f' x ≤ K * f x + ε) :
    ∀ x ∈ Icc a b, f x ≤ gronwallBound δ K ε (x - a) 
Arguments: {f f' : ℝ → ℝ} {δ K ε a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, (z - x)⁻¹ * (f z - f x) < r) (ha : f a ≤ δ) (bound : ∀ x ∈ Set.Ico a b, f' x ≤ K * f x + ε) (x : ℝ)
Goal: x ∈ Set.Icc a b → f x ≤ gronwallBound δ K ε (x - a)
-- norm_le_gronwallBound_of_norm_deriv_right_le
theorem norm_le_gronwallBound_of_norm_deriv_right_le {f f' : ℝ → E} {δ K ε : ℝ} {a b : ℝ}
    (hf : ContinuousOn f (Icc a b)) (hf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x)
    (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) :
    ∀ x ∈ Icc a b, ‖f x‖ ≤ gronwallBound δ K ε (x - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f f' : ℝ → E} {δ K ε a b : ℝ} (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ x ∈ Set.Ico a b, HasDerivWithinAt f (f' x) (Set.Ici x) x) (ha : ‖f a‖ ≤ δ) (bound : ∀ x ∈ Set.Ico a b, ‖f' x‖ ≤ K * ‖f x‖ + ε) (x : ℝ)
Goal: x ∈ Set.Icc a b → ‖f x‖ ≤ gronwallBound δ K ε (x - a)
-- dist_le_of_approx_trajectories_ODE_of_mem
theorem dist_le_of_approx_trajectories_ODE_of_mem
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (f' t) (Ici t) t)
    (f_bound : ∀ t ∈ Ico a b, dist (f' t) (v t (f t)) ≤ εf)
    (hfs : ∀ t ∈ Ico a b, f t ∈ s t)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (g' t) (Ici t) t)
    (g_bound : ∀ t ∈ Ico a b, dist (g' t) (v t (g t)) ≤ εg)
    (hgs : ∀ t ∈ Ico a b, g t ∈ s t)
    (ha : dist (f a) (g a) ≤ δ) :
    ∀ t ∈ Icc a b, dist (f t) (g t) ≤ gronwallBound δ K (εf + εg) (t - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {s : ℝ → Set E} {K : ℝ≥0} {f g f' g' : ℝ → E} {a b εf εg δ : ℝ} (hv : ∀ (t : ℝ), LipschitzOnWith K (v t) (s t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (f' t) (Set.Ici t) t) (f_bound : ∀ t ∈ Set.Ico a b, dist (f' t) (v t (f t)) ≤ εf) (hfs : ∀ t ∈ Set.Ico a b, f t ∈ s t) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (g' t) (Set.Ici t) t) (g_bound : ∀ t ∈ Set.Ico a b, dist (g' t) (v t (g t)) ≤ εg) (hgs : ∀ t ∈ Set.Ico a b, g t ∈ s t) (ha : dist (f a) (g a) ≤ δ) (t : ℝ)
Goal: t ∈ Set.Icc a b → dist (f t) (g t) ≤ gronwallBound δ (↑K) (εf + εg) (t - a)
-- dist_le_of_approx_trajectories_ODE
theorem dist_le_of_approx_trajectories_ODE
    (hv : ∀ t, LipschitzWith K (v t))
    (hf : ContinuousOn f (Icc a b))
    (hf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (f' t) (Ici t) t)
    (f_bound : ∀ t ∈ Ico a b, dist (f' t) (v t (f t)) ≤ εf)
    (hg : ContinuousOn g (Icc a b))
    (hg' : ∀ t ∈ Ico a b, HasDerivWithinAt g (g' t) (Ici t) t)
    (g_bound : ∀ t ∈ Ico a b, dist (g' t) (v t (g t)) ≤ εg)
    (ha : dist (f a) (g a) ≤ δ) :
    ∀ t ∈ Icc a b, dist (f t) (g t) ≤ gronwallBound δ K (εf + εg) (t - a) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {v : ℝ → E → E} {K : ℝ≥0} {f g f' g' : ℝ → E} {a b εf εg δ : ℝ} (hv : ∀ (t : ℝ), LipschitzWith K (v t)) (hf : ContinuousOn f (Set.Icc a b)) (hf' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt f (f' t) (Set.Ici t) t) (f_bound : ∀ t ∈ Set.Ico a b, dist (f' t) (v t (f t)) ≤ εf) (hg : ContinuousOn g (Set.Icc a b)) (hg' : ∀ t ∈ Set.Ico a b, HasDerivWithinAt g (g' t) (Set.Ici t) t) (g_bound : ∀ t ∈ Set.Ico a b, dist (g' t) (v t (g t)) ≤ εg) (ha : dist (f a) (g a) ≤ δ) (t : ℝ)
Goal: t ∈ Set.Icc a b → dist (f t) (g t) ≤ gronwallBound δ (↑K) (εf + εg) (t - a)