ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Real.real_sqrt_lt_nat_sqrt_succ
theorem real_sqrt_lt_nat_sqrt_succ {a : ℕ} : √(a : ℝ) < Nat.sqrt a + 1 
Arguments: {a : ℕ}
Goal: √↑a < ↑a.sqrt + 1
-- Real.real_sqrt_le_nat_sqrt_succ
theorem real_sqrt_le_nat_sqrt_succ {a : ℕ} : √(a : ℝ) ≤ Nat.sqrt a + 1 
Arguments: {a : ℕ}
Goal: √↑a ≤ ↑a.sqrt + 1
-- Real.floor_real_sqrt_eq_nat_sqrt
@[simp]
theorem floor_real_sqrt_eq_nat_sqrt {a : ℕ} : ⌊√(a : ℝ)⌋ = Nat.sqrt a 
Arguments: {a : ℕ}
Goal: ⌊√↑a⌋ = ↑a.sqrt
-- Real.nat_floor_real_sqrt_eq_nat_sqrt
@[simp]
theorem nat_floor_real_sqrt_eq_nat_sqrt {a : ℕ} : ⌊√(a : ℝ)⌋₊ = Nat.sqrt a 
Arguments: {a : ℕ}
Goal: ⌊√↑a⌋₊ = a.sqrt
-- Real.sqrt_one_add_le
theorem sqrt_one_add_le (h : -1 ≤ x) : √(1 + x) ≤ 1 + x / 2 
Arguments: {x : ℝ} (h : -1 ≤ x)
Goal: √(1 + x) ≤ 1 + x / 2
-- Filter.Tendsto.sqrt
theorem Filter.Tendsto.sqrt {f : α → ℝ} {l : Filter α} {x : ℝ} (h : Tendsto f l (𝓝 x)) :
    Tendsto (fun x => √(f x)) l (𝓝 (√x)) 
Arguments: {α : Type u_1} {f : α → ℝ} {l : Filter α} {x : ℝ} (h : Filter.Tendsto f l (𝓝 x))
Goal: Filter.Tendsto (fun x => √(f x)) l (𝓝 √x)
-- ContinuousWithinAt.sqrt
nonrec theorem ContinuousWithinAt.sqrt (h : ContinuousWithinAt f s x) :
    ContinuousWithinAt (fun x => √(f x)) s x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} {x : α} (h : ContinuousWithinAt f s x)
Goal: ContinuousWithinAt (fun x => √(f x)) s x
-- ContinuousAt.sqrt
@[fun_prop]
nonrec theorem ContinuousAt.sqrt (h : ContinuousAt f x) : ContinuousAt (fun x => √(f x)) x 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {x : α} (h : ContinuousAt f x)
Goal: ContinuousAt (fun x => √(f x)) x
-- ContinuousOn.sqrt
@[fun_prop]
theorem ContinuousOn.sqrt (h : ContinuousOn f s) : ContinuousOn (fun x => √(f x)) s 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} (h : ContinuousOn f s)
Goal: ContinuousOn (fun x => √(f x)) s
-- Continuous.sqrt
@[continuity, fun_prop]
theorem Continuous.sqrt (h : Continuous f) : Continuous fun x => √(f x) 
Arguments: {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} (h : Continuous f)
Goal: Continuous fun x => √(f x)
-- NNReal.sum_mul_le_sqrt_mul_sqrt
lemma sum_mul_le_sqrt_mul_sqrt (s : Finset ι) (f g : ι → ℝ≥0) :
    ∑ i ∈ s, f i * g i ≤ sqrt (∑ i ∈ s, f i ^ 2) * sqrt (∑ i ∈ s, g i ^ 2) 
Arguments: {ι : Type u_2} (s : Finset ι) (f g : ι → ℝ≥0)
Goal: ∑ i ∈ s, f i * g i ≤ NNReal.sqrt (∑ i ∈ s, f i ^ 2) * NNReal.sqrt (∑ i ∈ s, g i ^ 2)
-- NNReal.sum_sqrt_mul_sqrt_le
lemma sum_sqrt_mul_sqrt_le (s : Finset ι) (f g : ι → ℝ≥0) :
    ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i) 
Arguments: {ι : Type u_2} (s : Finset ι) (f g : ι → ℝ≥0)
Goal: ∑ i ∈ s, NNReal.sqrt (f i) * NNReal.sqrt (g i) ≤ NNReal.sqrt (∑ i ∈ s, f i) * NNReal.sqrt (∑ i ∈ s, g i)
-- Real.sum_mul_le_sqrt_mul_sqrt
lemma sum_mul_le_sqrt_mul_sqrt (s : Finset ι) (f g : ι → ℝ) :
    ∑ i ∈ s, f i * g i ≤ √(∑ i ∈ s, f i ^ 2) * √(∑ i ∈ s, g i ^ 2) 
Arguments: {ι : Type u_2} (s : Finset ι) (f g : ι → ℝ)
Goal: ∑ i ∈ s, f i * g i ≤ √(∑ i ∈ s, f i ^ 2) * √(∑ i ∈ s, g i ^ 2)
-- Real.sum_sqrt_mul_sqrt_le
lemma sum_sqrt_mul_sqrt_le (s : Finset ι) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) :
    ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i) 
Arguments: {ι : Type u_2} {f g : ι → ℝ} (s : Finset ι) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i)
Goal: ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)
-- Hyperreal.coe_eq_coe
@[simp, norm_cast]
theorem coe_eq_coe {x y : ℝ} : (x : ℝ*) = y ↔ x = y 
Arguments: {x y : ℝ}
Goal: ↑x = ↑y ↔ x = y
-- Hyperreal.coe_ne_coe
theorem coe_ne_coe {x y : ℝ} : (x : ℝ*) ≠ y ↔ x ≠ y 
Arguments: {x y : ℝ}
Goal: ↑x ≠ ↑y ↔ x ≠ y
-- Hyperreal.coe_eq_zero
@[simp, norm_cast]
theorem coe_eq_zero {x : ℝ} : (x : ℝ*) = 0 ↔ x = 0 
Arguments: {x : ℝ}
Goal: ↑x = 0 ↔ x = 0
-- Hyperreal.coe_eq_one
@[simp, norm_cast]
theorem coe_eq_one {x : ℝ} : (x : ℝ*) = 1 ↔ x = 1 
Arguments: {x : ℝ}
Goal: ↑x = 1 ↔ x = 1
-- Hyperreal.coe_ne_zero
@[norm_cast]
theorem coe_ne_zero {x : ℝ} : (x : ℝ*) ≠ 0 ↔ x ≠ 0 
Arguments: {x : ℝ}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- Hyperreal.coe_ne_one
@[norm_cast]
theorem coe_ne_one {x : ℝ} : (x : ℝ*) ≠ 1 ↔ x ≠ 1 
Arguments: {x : ℝ}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- Hyperreal.coe_one
@[simp, norm_cast]
theorem coe_one : ↑(1 : ℝ) = (1 : ℝ*) 
Goal: ↑1 = 1
-- Hyperreal.coe_zero
@[simp, norm_cast]
theorem coe_zero : ↑(0 : ℝ) = (0 : ℝ*) 
Goal: ↑0 = 0
-- Hyperreal.coe_inv
@[simp, norm_cast]
theorem coe_inv (x : ℝ) : ↑x⁻¹ = (x⁻¹ : ℝ*) 
Arguments: (x : ℝ)
Goal: ↑x⁻¹ = (↑x)⁻¹
-- Hyperreal.coe_neg
@[simp, norm_cast]
theorem coe_neg (x : ℝ) : ↑(-x) = (-x : ℝ*) 
Arguments: (x : ℝ)
Goal: ↑(-x) = -↑x
-- Hyperreal.coe_add
@[simp, norm_cast]
theorem coe_add (x y : ℝ) : ↑(x + y) = (x + y : ℝ*) 
Arguments: (x y : ℝ)
Goal: ↑(x + y) = ↑x + ↑y
-- Hyperreal.coe_ofNat
@[simp, norm_cast]
theorem coe_ofNat (n : ℕ) [n.AtLeastTwo] :
    ((no_index (OfNat.ofNat n : ℝ)) : ℝ*) = OfNat.ofNat n 
Arguments: (n : ℕ) [n.AtLeastTwo]
Goal: ↑(OfNat.ofNat n) = OfNat.ofNat n
-- Hyperreal.coe_mul
@[simp, norm_cast]
theorem coe_mul (x y : ℝ) : ↑(x * y) = (x * y : ℝ*) 
Arguments: (x y : ℝ)
Goal: ↑(x * y) = ↑x * ↑y
-- Hyperreal.coe_div
@[simp, norm_cast]
theorem coe_div (x y : ℝ) : ↑(x / y) = (x / y : ℝ*) 
Arguments: (x y : ℝ)
Goal: ↑(x / y) = ↑x / ↑y
-- Hyperreal.coe_sub
@[simp, norm_cast]
theorem coe_sub (x y : ℝ) : ↑(x - y) = (x - y : ℝ*) 
Arguments: (x y : ℝ)
Goal: ↑(x - y) = ↑x - ↑y
-- Hyperreal.coe_le_coe
@[simp, norm_cast]
theorem coe_le_coe {x y : ℝ} : (x : ℝ*) ≤ y ↔ x ≤ y 
Arguments: {x y : ℝ}
Goal: ↑x ≤ ↑y ↔ x ≤ y
-- Hyperreal.coe_lt_coe
@[simp, norm_cast]
theorem coe_lt_coe {x y : ℝ} : (x : ℝ*) < y ↔ x < y 
Arguments: {x y : ℝ}
Goal: ↑x < ↑y ↔ x < y
-- Hyperreal.coe_nonneg
@[simp, norm_cast]
theorem coe_nonneg {x : ℝ} : 0 ≤ (x : ℝ*) ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: 0 ≤ ↑x ↔ 0 ≤ x
-- Hyperreal.coe_pos
@[simp, norm_cast]
theorem coe_pos {x : ℝ} : 0 < (x : ℝ*) ↔ 0 < x 
Arguments: {x : ℝ}
Goal: 0 < ↑x ↔ 0 < x
-- Hyperreal.coe_abs
@[simp, norm_cast]
theorem coe_abs (x : ℝ) : ((|x| : ℝ) : ℝ*) = |↑x| 
Arguments: (x : ℝ)
Goal: ↑|x| = |↑x|
-- Hyperreal.coe_max
@[simp, norm_cast]
theorem coe_max (x y : ℝ) : ((max x y : ℝ) : ℝ*) = max ↑x ↑y 
Arguments: (x y : ℝ)
Goal: ↑(max x y) = max ↑x ↑y
-- Hyperreal.coe_min
@[simp, norm_cast]
theorem coe_min (x y : ℝ) : ((min x y : ℝ) : ℝ*) = min ↑x ↑y 
Arguments: (x y : ℝ)
Goal: ↑(min x y) = min ↑x ↑y
-- Hyperreal.ofSeq_surjective
theorem ofSeq_surjective : Function.Surjective ofSeq 
Goal: Function.Surjective Hyperreal.ofSeq
-- Hyperreal.ofSeq_lt_ofSeq
theorem ofSeq_lt_ofSeq {f g : ℕ → ℝ} : ofSeq f < ofSeq g ↔ ∀ᶠ n in hyperfilter ℕ, f n < g n 
Arguments: {f g : ℕ → ℝ}
Goal: Hyperreal.ofSeq f < Hyperreal.ofSeq g ↔ ∀ᶠ (n : ℕ) in ↑(Filter.hyperfilter ℕ), f n < g n
-- Hyperreal.inv_omega
@[simp]
theorem inv_omega : ω⁻¹ = ε 
Goal: Hyperreal.omega⁻¹ = Hyperreal.epsilon
-- Hyperreal.inv_epsilon
@[simp]
theorem inv_epsilon : ε⁻¹ = ω 
Goal: Hyperreal.epsilon⁻¹ = Hyperreal.omega
-- Hyperreal.omega_pos
theorem omega_pos : 0 < ω 
Goal: 0 < Hyperreal.omega
-- Hyperreal.epsilon_pos
theorem epsilon_pos : 0 < ε 
Goal: 0 < Hyperreal.epsilon
-- Hyperreal.epsilon_ne_zero
theorem epsilon_ne_zero : ε ≠ 0 
Goal: Hyperreal.epsilon ≠ 0
-- Hyperreal.omega_ne_zero
theorem omega_ne_zero : ω ≠ 0 
Goal: Hyperreal.omega ≠ 0
-- Hyperreal.epsilon_mul_omega
theorem epsilon_mul_omega : ε * ω = 1 
Goal: Hyperreal.epsilon * Hyperreal.omega = 1
-- Hyperreal.lt_of_tendsto_zero_of_pos
theorem lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :
    ∀ {r : ℝ}, 0 < r → ofSeq f < (r : ℝ*) 
Arguments: {f : ℕ → ℝ} (hf : Filter.Tendsto f Filter.atTop (𝓝 0)) {r : ℝ}
Goal: 0 < r → Hyperreal.ofSeq f < ↑r
-- Hyperreal.neg_lt_of_tendsto_zero_of_pos
theorem neg_lt_of_tendsto_zero_of_pos {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :
    ∀ {r : ℝ}, 0 < r → (-r : ℝ*) < ofSeq f 
Arguments: {f : ℕ → ℝ} (hf : Filter.Tendsto f Filter.atTop (𝓝 0)) {r : ℝ}
Goal: 0 < r → -↑r < Hyperreal.ofSeq f
-- Hyperreal.gt_of_tendsto_zero_of_neg
theorem gt_of_tendsto_zero_of_neg {f : ℕ → ℝ} (hf : Tendsto f atTop (𝓝 0)) :
    ∀ {r : ℝ}, r < 0 → (r : ℝ*) < ofSeq f 
Arguments: {f : ℕ → ℝ} (hf : Filter.Tendsto f Filter.atTop (𝓝 0)) {r : ℝ}
Goal: r < 0 → ↑r < Hyperreal.ofSeq f
-- Hyperreal.epsilon_lt_pos
theorem epsilon_lt_pos (x : ℝ) : 0 < x → ε < x 
Arguments: (x : ℝ)
Goal: 0 < x → Hyperreal.epsilon < ↑x
-- Hyperreal.isSt_ofSeq_iff_tendsto
theorem isSt_ofSeq_iff_tendsto {f : ℕ → ℝ} {r : ℝ} :
    IsSt (ofSeq f) r ↔ Tendsto f (hyperfilter ℕ) (𝓝 r) 
Arguments: {f : ℕ → ℝ} {r : ℝ}
Goal: (Hyperreal.ofSeq f).IsSt r ↔ Filter.Tendsto f (↑(Filter.hyperfilter ℕ)) (𝓝 r)
-- Hyperreal.isSt_iff_tendsto
theorem isSt_iff_tendsto {x : ℝ*} {r : ℝ} : IsSt x r ↔ x.Tendsto (𝓝 r) 
Arguments: {x : ℝ*} {r : ℝ}
Goal: x.IsSt r ↔ Filter.Germ.Tendsto x (𝓝 r)
-- Hyperreal.isSt_of_tendsto
theorem isSt_of_tendsto {f : ℕ → ℝ} {r : ℝ} (hf : Tendsto f atTop (𝓝 r)) : IsSt (ofSeq f) r 
Arguments: {f : ℕ → ℝ} {r : ℝ} (hf : Filter.Tendsto f Filter.atTop (𝓝 r))
Goal: (Hyperreal.ofSeq f).IsSt r
-- Hyperreal.IsSt.lt
protected theorem IsSt.lt {x y : ℝ*} {r s : ℝ} (hxr : IsSt x r) (hys : IsSt y s) (hrs : r < s) :
    x < y 
Arguments: {x y : ℝ*} {r s : ℝ} (hxr : x.IsSt r) (hys : y.IsSt s) (hrs : r < s)
Goal: x < y
-- Hyperreal.IsSt.unique
theorem IsSt.unique {x : ℝ*} {r s : ℝ} (hr : IsSt x r) (hs : IsSt x s) : r = s 
Arguments: {x : ℝ*} {r s : ℝ} (hr : x.IsSt r) (hs : x.IsSt s)
Goal: r = s
-- Hyperreal.IsSt.st_eq
theorem IsSt.st_eq {x : ℝ*} {r : ℝ} (hxr : IsSt x r) : st x = r 
Arguments: {x : ℝ*} {r : ℝ} (hxr : x.IsSt r)
Goal: x.st = r
-- Hyperreal.IsSt.not_infinite
theorem IsSt.not_infinite {x : ℝ*} {r : ℝ} (h : IsSt x r) : ¬Infinite x 
Arguments: {x : ℝ*} {r : ℝ} (h : x.IsSt r)
Goal: ¬x.Infinite
-- Hyperreal.not_infinite_of_exists_st
theorem not_infinite_of_exists_st {x : ℝ*} : (∃ r : ℝ, IsSt x r) → ¬Infinite x 
Arguments: {x : ℝ*}
Goal: (∃ r, x.IsSt r) → ¬x.Infinite
-- Hyperreal.Infinite.st_eq
theorem Infinite.st_eq {x : ℝ*} (hi : Infinite x) : st x = 0 
Arguments: {x : ℝ*} (hi : x.Infinite)
Goal: x.st = 0
-- Hyperreal.isSt_sSup
theorem isSt_sSup {x : ℝ*} (hni : ¬Infinite x) : IsSt x (sSup { y : ℝ | (y : ℝ*) < x }) 
Arguments: {x : ℝ*} (hni : ¬x.Infinite)
Goal: x.IsSt (sSup {y | ↑y < x})
-- Hyperreal.exists_st_of_not_infinite
theorem exists_st_of_not_infinite {x : ℝ*} (hni : ¬Infinite x) : ∃ r : ℝ, IsSt x r 
Arguments: {x : ℝ*} (hni : ¬x.Infinite)
Goal: ∃ r, x.IsSt r
-- Hyperreal.st_eq_sSup
theorem st_eq_sSup {x : ℝ*} : st x = sSup { y : ℝ | (y : ℝ*) < x } 
Arguments: {x : ℝ*}
Goal: x.st = sSup {y | ↑y < x}
-- Hyperreal.exists_st_iff_not_infinite
theorem exists_st_iff_not_infinite {x : ℝ*} : (∃ r : ℝ, IsSt x r) ↔ ¬Infinite x 
Arguments: {x : ℝ*}
Goal: (∃ r, x.IsSt r) ↔ ¬x.Infinite
-- Hyperreal.infinite_iff_not_exists_st
theorem infinite_iff_not_exists_st {x : ℝ*} : Infinite x ↔ ¬∃ r : ℝ, IsSt x r 
Arguments: {x : ℝ*}
Goal: x.Infinite ↔ ¬∃ r, x.IsSt r
-- Hyperreal.IsSt.isSt_st
theorem IsSt.isSt_st {x : ℝ*} {r : ℝ} (hxr : IsSt x r) : IsSt x (st x) 
Arguments: {x : ℝ*} {r : ℝ} (hxr : x.IsSt r)
Goal: x.IsSt x.st
-- Hyperreal.isSt_st_of_exists_st
theorem isSt_st_of_exists_st {x : ℝ*} (hx : ∃ r : ℝ, IsSt x r) : IsSt x (st x) 
Arguments: {x : ℝ*} (hx : ∃ r, x.IsSt r)
Goal: x.IsSt x.st
-- Hyperreal.isSt_st'
theorem isSt_st' {x : ℝ*} (hx : ¬Infinite x) : IsSt x (st x) 
Arguments: {x : ℝ*} (hx : ¬x.Infinite)
Goal: x.IsSt x.st
-- Hyperreal.isSt_st
theorem isSt_st {x : ℝ*} (hx : st x ≠ 0) : IsSt x (st x) 
Arguments: {x : ℝ*} (hx : x.st ≠ 0)
Goal: x.IsSt x.st
-- Hyperreal.isSt_refl_real
theorem isSt_refl_real (r : ℝ) : IsSt r r 
Arguments: (r : ℝ)
Goal: (↑r).IsSt r
-- Hyperreal.st_id_real
theorem st_id_real (r : ℝ) : st r = r 
Arguments: (r : ℝ)
Goal: (↑r).st = r
-- Hyperreal.eq_of_isSt_real
theorem eq_of_isSt_real {r s : ℝ} : IsSt r s → r = s 
Arguments: {r s : ℝ}
Goal: (↑r).IsSt s → r = s
-- Hyperreal.isSt_real_iff_eq
theorem isSt_real_iff_eq {r s : ℝ} : IsSt r s ↔ r = s 
Arguments: {r s : ℝ}
Goal: (↑r).IsSt s ↔ r = s
-- Hyperreal.isSt_symm_real
theorem isSt_symm_real {r s : ℝ} : IsSt r s ↔ IsSt s r 
Arguments: {r s : ℝ}
Goal: (↑r).IsSt s ↔ (↑s).IsSt r
-- Hyperreal.isSt_trans_real
theorem isSt_trans_real {r s t : ℝ} : IsSt r s → IsSt s t → IsSt r t 
Arguments: {r s t : ℝ}
Goal: (↑r).IsSt s → (↑s).IsSt t → (↑r).IsSt t
-- Hyperreal.isSt_inj_real
theorem isSt_inj_real {r₁ r₂ s : ℝ} (h1 : IsSt r₁ s) (h2 : IsSt r₂ s) : r₁ = r₂ 
Arguments: {r₁ r₂ s : ℝ} (h1 : (↑r₁).IsSt s) (h2 : (↑r₂).IsSt s)
Goal: r₁ = r₂
-- Hyperreal.isSt_iff_abs_sub_lt_delta
theorem isSt_iff_abs_sub_lt_delta {x : ℝ*} {r : ℝ} : IsSt x r ↔ ∀ δ : ℝ, 0 < δ → |x - ↑r| < δ 
Arguments: {x : ℝ*} {r : ℝ}
Goal: x.IsSt r ↔ ∀ (δ : ℝ), 0 < δ → |x - ↑r| < ↑δ
-- Hyperreal.IsSt.map
theorem IsSt.map {x : ℝ*} {r : ℝ} (hxr : IsSt x r) {f : ℝ → ℝ} (hf : ContinuousAt f r) :
    IsSt (x.map f) (f r) 
Arguments: {x : ℝ*} {r : ℝ} (hxr : x.IsSt r) {f : ℝ → ℝ} (hf : ContinuousAt f r)
Goal: Hyperreal.IsSt (Filter.Germ.map f x) (f r)
-- Hyperreal.IsSt.map₂
theorem IsSt.map₂ {x y : ℝ*} {r s : ℝ} (hxr : IsSt x r) (hys : IsSt y s) {f : ℝ → ℝ → ℝ}
    (hf : ContinuousAt (Function.uncurry f) (r, s)) : IsSt (x.map₂ f y) (f r s) 
Arguments: {x y : ℝ*} {r s : ℝ} (hxr : x.IsSt r) (hys : y.IsSt s) {f : ℝ → ℝ → ℝ} (hf : ContinuousAt (Function.uncurry f) (r, s))
Goal: Hyperreal.IsSt (Filter.Germ.map₂ f x y) (f r s)
-- Hyperreal.IsSt.add
theorem IsSt.add {x y : ℝ*} {r s : ℝ} (hxr : IsSt x r) (hys : IsSt y s) :
    IsSt (x + y) (r + s) 
Arguments: {x y : ℝ*} {r s : ℝ} (hxr : x.IsSt r) (hys : y.IsSt s)
Goal: (x + y).IsSt (r + s)
-- Hyperreal.IsSt.neg
theorem IsSt.neg {x : ℝ*} {r : ℝ} (hxr : IsSt x r) : IsSt (-x) (-r) 
Arguments: {x : ℝ*} {r : ℝ} (hxr : x.IsSt r)
Goal: (-x).IsSt (-r)
-- Hyperreal.IsSt.sub
theorem IsSt.sub {x y : ℝ*} {r s : ℝ} (hxr : IsSt x r) (hys : IsSt y s) : IsSt (x - y) (r - s) 
Arguments: {x y : ℝ*} {r s : ℝ} (hxr : x.IsSt r) (hys : y.IsSt s)
Goal: (x - y).IsSt (r - s)
-- Hyperreal.IsSt.le
theorem IsSt.le {x y : ℝ*} {r s : ℝ} (hrx : IsSt x r) (hsy : IsSt y s) (hxy : x ≤ y) : r ≤ s 
Arguments: {x y : ℝ*} {r s : ℝ} (hrx : x.IsSt r) (hsy : y.IsSt s) (hxy : x ≤ y)
Goal: r ≤ s
-- Hyperreal.st_le_of_le
theorem st_le_of_le {x y : ℝ*} (hix : ¬Infinite x) (hiy : ¬Infinite y) : x ≤ y → st x ≤ st y 
Arguments: {x y : ℝ*} (hix : ¬x.Infinite) (hiy : ¬y.Infinite)
Goal: x ≤ y → x.st ≤ y.st
-- Hyperreal.lt_of_st_lt
theorem lt_of_st_lt {x y : ℝ*} (hix : ¬Infinite x) (hiy : ¬Infinite y) : st x < st y → x < y 
Arguments: {x y : ℝ*} (hix : ¬x.Infinite) (hiy : ¬y.Infinite)
Goal: x.st < y.st → x < y
-- Hyperreal.infinitePos_def
theorem infinitePos_def {x : ℝ*} : InfinitePos x ↔ ∀ r : ℝ, ↑r < x 
Arguments: {x : ℝ*}
Goal: x.InfinitePos ↔ ∀ (r : ℝ), ↑r < x
-- Hyperreal.infiniteNeg_def
theorem infiniteNeg_def {x : ℝ*} : InfiniteNeg x ↔ ∀ r : ℝ, x < r 
Arguments: {x : ℝ*}
Goal: x.InfiniteNeg ↔ ∀ (r : ℝ), x < ↑r
-- Hyperreal.InfinitePos.pos
theorem InfinitePos.pos {x : ℝ*} (hip : InfinitePos x) : 0 < x 
Arguments: {x : ℝ*} (hip : x.InfinitePos)
Goal: 0 < x
-- Hyperreal.InfiniteNeg.lt_zero
theorem InfiniteNeg.lt_zero {x : ℝ*} : InfiniteNeg x → x < 0 
Arguments: {x : ℝ*}
Goal: x.InfiniteNeg → x < 0
-- Hyperreal.Infinite.ne_zero
theorem Infinite.ne_zero {x : ℝ*} (hI : Infinite x) : x ≠ 0 
Arguments: {x : ℝ*} (hI : x.Infinite)
Goal: x ≠ 0
-- Hyperreal.not_infinite_zero
theorem not_infinite_zero : ¬Infinite 0 
Goal: ¬Hyperreal.Infinite 0
-- Hyperreal.InfiniteNeg.not_infinitePos
theorem InfiniteNeg.not_infinitePos {x : ℝ*} : InfiniteNeg x → ¬InfinitePos x 
Arguments: {x : ℝ*}
Goal: x.InfiniteNeg → ¬x.InfinitePos
-- Hyperreal.InfinitePos.not_infiniteNeg
theorem InfinitePos.not_infiniteNeg {x : ℝ*} (hp : InfinitePos x) : ¬InfiniteNeg x 
Arguments: {x : ℝ*} (hp : x.InfinitePos)
Goal: ¬x.InfiniteNeg
-- Hyperreal.InfinitePos.neg
theorem InfinitePos.neg {x : ℝ*} : InfinitePos x → InfiniteNeg (-x) 
Arguments: {x : ℝ*}
Goal: x.InfinitePos → (-x).InfiniteNeg
-- Hyperreal.InfiniteNeg.neg
theorem InfiniteNeg.neg {x : ℝ*} : InfiniteNeg x → InfinitePos (-x) 
Arguments: {x : ℝ*}
Goal: x.InfiniteNeg → (-x).InfinitePos
-- Hyperreal.infiniteNeg_neg
@[simp] theorem infiniteNeg_neg {x : ℝ*} : InfiniteNeg (-x) ↔ InfinitePos x 
Arguments: {x : ℝ*}
Goal: (-x).InfiniteNeg ↔ x.InfinitePos
-- Hyperreal.infinitePos_neg
@[simp] theorem infinitePos_neg {x : ℝ*} : InfinitePos (-x) ↔ InfiniteNeg x 
Arguments: {x : ℝ*}
Goal: (-x).InfinitePos ↔ x.InfiniteNeg
-- Hyperreal.infinite_neg
@[simp] theorem infinite_neg {x : ℝ*} : Infinite (-x) ↔ Infinite x 
Arguments: {x : ℝ*}
Goal: (-x).Infinite ↔ x.Infinite
-- Hyperreal.Infinitesimal.not_infinite
nonrec theorem Infinitesimal.not_infinite {x : ℝ*} (h : Infinitesimal x) : ¬Infinite x 
Arguments: {x : ℝ*} (h : x.Infinitesimal)
Goal: ¬x.Infinite
-- Hyperreal.Infinite.not_infinitesimal
theorem Infinite.not_infinitesimal {x : ℝ*} (h : Infinite x) : ¬Infinitesimal x 
Arguments: {x : ℝ*} (h : x.Infinite)
Goal: ¬x.Infinitesimal
-- Hyperreal.InfinitePos.not_infinitesimal
theorem InfinitePos.not_infinitesimal {x : ℝ*} (h : InfinitePos x) : ¬Infinitesimal x 
Arguments: {x : ℝ*} (h : x.InfinitePos)
Goal: ¬x.Infinitesimal
-- Hyperreal.InfiniteNeg.not_infinitesimal
theorem InfiniteNeg.not_infinitesimal {x : ℝ*} (h : InfiniteNeg x) : ¬Infinitesimal x 
Arguments: {x : ℝ*} (h : x.InfiniteNeg)
Goal: ¬x.Infinitesimal
-- Hyperreal.infinitePos_iff_infinite_and_pos
theorem infinitePos_iff_infinite_and_pos {x : ℝ*} : InfinitePos x ↔ Infinite x ∧ 0 < x 
Arguments: {x : ℝ*}
Goal: x.InfinitePos ↔ x.Infinite ∧ 0 < x
-- Hyperreal.infiniteNeg_iff_infinite_and_neg
theorem infiniteNeg_iff_infinite_and_neg {x : ℝ*} : InfiniteNeg x ↔ Infinite x ∧ x < 0 
Arguments: {x : ℝ*}
Goal: x.InfiniteNeg ↔ x.Infinite ∧ x < 0
-- Hyperreal.infinitePos_iff_infinite_of_nonneg
theorem infinitePos_iff_infinite_of_nonneg {x : ℝ*} (hp : 0 ≤ x) : InfinitePos x ↔ Infinite x 
Arguments: {x : ℝ*} (hp : 0 ≤ x)
Goal: x.InfinitePos ↔ x.Infinite
-- Hyperreal.infinitePos_iff_infinite_of_pos
theorem infinitePos_iff_infinite_of_pos {x : ℝ*} (hp : 0 < x) : InfinitePos x ↔ Infinite x 
Arguments: {x : ℝ*} (hp : 0 < x)
Goal: x.InfinitePos ↔ x.Infinite
-- Hyperreal.infiniteNeg_iff_infinite_of_neg
theorem infiniteNeg_iff_infinite_of_neg {x : ℝ*} (hn : x < 0) : InfiniteNeg x ↔ Infinite x 
Arguments: {x : ℝ*} (hn : x < 0)
Goal: x.InfiniteNeg ↔ x.Infinite
-- Hyperreal.infinitePos_abs_iff_infinite_abs
theorem infinitePos_abs_iff_infinite_abs {x : ℝ*} : InfinitePos |x| ↔ Infinite |x| 
Arguments: {x : ℝ*}
Goal: |x|.InfinitePos ↔ |x|.Infinite
-- Hyperreal.infinite_abs_iff
@[simp] theorem infinite_abs_iff {x : ℝ*} : Infinite |x| ↔ Infinite x 
Arguments: {x : ℝ*}
Goal: |x|.Infinite ↔ x.Infinite
-- Hyperreal.infinitePos_abs_iff_infinite
@[simp] theorem infinitePos_abs_iff_infinite {x : ℝ*} : InfinitePos |x| ↔ Infinite x 
Arguments: {x : ℝ*}
Goal: |x|.InfinitePos ↔ x.Infinite
-- Hyperreal.infinite_iff_abs_lt_abs
theorem infinite_iff_abs_lt_abs {x : ℝ*} : Infinite x ↔ ∀ r : ℝ, (|r| : ℝ*) < |x| 
Arguments: {x : ℝ*}
Goal: x.Infinite ↔ ∀ (r : ℝ), |↑r| < |x|
-- Hyperreal.infinitePos_add_not_infiniteNeg
theorem infinitePos_add_not_infiniteNeg {x y : ℝ*} :
    InfinitePos x → ¬InfiniteNeg y → InfinitePos (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfinitePos → ¬y.InfiniteNeg → (x + y).InfinitePos
-- Hyperreal.not_infiniteNeg_add_infinitePos
theorem not_infiniteNeg_add_infinitePos {x y : ℝ*} :
    ¬InfiniteNeg x → InfinitePos y → InfinitePos (x + y) 
Arguments: {x y : ℝ*}
Goal: ¬x.InfiniteNeg → y.InfinitePos → (x + y).InfinitePos
-- Hyperreal.infiniteNeg_add_not_infinitePos
theorem infiniteNeg_add_not_infinitePos {x y : ℝ*} :
    InfiniteNeg x → ¬InfinitePos y → InfiniteNeg (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfiniteNeg → ¬y.InfinitePos → (x + y).InfiniteNeg
-- Hyperreal.not_infinitePos_add_infiniteNeg
theorem not_infinitePos_add_infiniteNeg {x y : ℝ*} :
    ¬InfinitePos x → InfiniteNeg y → InfiniteNeg (x + y) 
Arguments: {x y : ℝ*}
Goal: ¬x.InfinitePos → y.InfiniteNeg → (x + y).InfiniteNeg
-- Hyperreal.infinitePos_add_infinitePos
theorem infinitePos_add_infinitePos {x y : ℝ*} :
    InfinitePos x → InfinitePos y → InfinitePos (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfinitePos → y.InfinitePos → (x + y).InfinitePos
-- Hyperreal.infiniteNeg_add_infiniteNeg
theorem infiniteNeg_add_infiniteNeg {x y : ℝ*} :
    InfiniteNeg x → InfiniteNeg y → InfiniteNeg (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfiniteNeg → y.InfiniteNeg → (x + y).InfiniteNeg
-- Hyperreal.infinitePos_add_not_infinite
theorem infinitePos_add_not_infinite {x y : ℝ*} :
    InfinitePos x → ¬Infinite y → InfinitePos (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfinitePos → ¬y.Infinite → (x + y).InfinitePos
-- Hyperreal.infiniteNeg_add_not_infinite
theorem infiniteNeg_add_not_infinite {x y : ℝ*} :
    InfiniteNeg x → ¬Infinite y → InfiniteNeg (x + y) 
Arguments: {x y : ℝ*}
Goal: x.InfiniteNeg → ¬y.Infinite → (x + y).InfiniteNeg
-- Hyperreal.infinitePos_of_tendsto_top
theorem infinitePos_of_tendsto_top {f : ℕ → ℝ} (hf : Tendsto f atTop atTop) :
    InfinitePos (ofSeq f) 
Arguments: {f : ℕ → ℝ} (hf : Filter.Tendsto f Filter.atTop Filter.atTop)
Goal: (Hyperreal.ofSeq f).InfinitePos
-- Hyperreal.infiniteNeg_of_tendsto_bot
theorem infiniteNeg_of_tendsto_bot {f : ℕ → ℝ} (hf : Tendsto f atTop atBot) :
    InfiniteNeg (ofSeq f) 
Arguments: {f : ℕ → ℝ} (hf : Filter.Tendsto f Filter.atTop Filter.atBot)
Goal: (Hyperreal.ofSeq f).InfiniteNeg
-- Hyperreal.not_infinite_neg
theorem not_infinite_neg {x : ℝ*} : ¬Infinite x → ¬Infinite (-x) 
Arguments: {x : ℝ*}
Goal: ¬x.Infinite → ¬(-x).Infinite
-- Hyperreal.not_infinite_add
theorem not_infinite_add {x y : ℝ*} (hx : ¬Infinite x) (hy : ¬Infinite y) : ¬Infinite (x + y) 
Arguments: {x y : ℝ*} (hx : ¬x.Infinite) (hy : ¬y.Infinite)
Goal: ¬(x + y).Infinite
-- Hyperreal.not_infinite_iff_exist_lt_gt
theorem not_infinite_iff_exist_lt_gt {x : ℝ*} : ¬Infinite x ↔ ∃ r s : ℝ, (r : ℝ*) < x ∧ x < s 
Arguments: {x : ℝ*}
Goal: ¬x.Infinite ↔ ∃ r s, ↑r < x ∧ x < ↑s
-- Hyperreal.not_infinite_real
theorem not_infinite_real (r : ℝ) : ¬Infinite r 
Arguments: (r : ℝ)
Goal: ¬(↑r).Infinite
-- Hyperreal.Infinite.ne_real
theorem Infinite.ne_real {x : ℝ*} : Infinite x → ∀ r : ℝ, x ≠ r 
Arguments: {x : ℝ*}
Goal: x.Infinite → ∀ (r : ℝ), x ≠ ↑r
-- Hyperreal.IsSt.mul
theorem IsSt.mul {x y : ℝ*} {r s : ℝ} (hxr : IsSt x r) (hys : IsSt y s) : IsSt (x * y) (r * s) 
Arguments: {x y : ℝ*} {r s : ℝ} (hxr : x.IsSt r) (hys : y.IsSt s)
Goal: (x * y).IsSt (r * s)
-- Hyperreal.not_infinite_mul
theorem not_infinite_mul {x y : ℝ*} (hx : ¬Infinite x) (hy : ¬Infinite y) : ¬Infinite (x * y) 
Arguments: {x y : ℝ*} (hx : ¬x.Infinite) (hy : ¬y.Infinite)
Goal: ¬(x * y).Infinite
-- Hyperreal.st_add
theorem st_add {x y : ℝ*} (hx : ¬Infinite x) (hy : ¬Infinite y) : st (x + y) = st x + st y 
Arguments: {x y : ℝ*} (hx : ¬x.Infinite) (hy : ¬y.Infinite)
Goal: (x + y).st = x.st + y.st
-- Hyperreal.st_neg
theorem st_neg (x : ℝ*) : st (-x) = -st x 
Arguments: (x : ℝ*)
Goal: (-x).st = -x.st
-- Hyperreal.st_mul
theorem st_mul {x y : ℝ*} (hx : ¬Infinite x) (hy : ¬Infinite y) : st (x * y) = st x * st y 
Arguments: {x y : ℝ*} (hx : ¬x.Infinite) (hy : ¬y.Infinite)
Goal: (x * y).st = x.st * y.st
-- Hyperreal.infinitesimal_def
theorem infinitesimal_def {x : ℝ*} : Infinitesimal x ↔ ∀ r : ℝ, 0 < r → -(r : ℝ*) < x ∧ x < r 
Arguments: {x : ℝ*}
Goal: x.Infinitesimal ↔ ∀ (r : ℝ), 0 < r → -↑r < x ∧ x < ↑r
-- Hyperreal.lt_of_pos_of_infinitesimal
theorem lt_of_pos_of_infinitesimal {x : ℝ*} : Infinitesimal x → ∀ r : ℝ, 0 < r → x < r 
Arguments: {x : ℝ*}
Goal: x.Infinitesimal → ∀ (r : ℝ), 0 < r → x < ↑r
-- Hyperreal.lt_neg_of_pos_of_infinitesimal
theorem lt_neg_of_pos_of_infinitesimal {x : ℝ*} : Infinitesimal x → ∀ r : ℝ, 0 < r → -↑r < x 
Arguments: {x : ℝ*}
Goal: x.Infinitesimal → ∀ (r : ℝ), 0 < r → -↑r < x
-- Hyperreal.gt_of_neg_of_infinitesimal
theorem gt_of_neg_of_infinitesimal {x : ℝ*} (hi : Infinitesimal x) (r : ℝ) (hr : r < 0) : ↑r < x 
Arguments: {x : ℝ*} (hi : x.Infinitesimal) (r : ℝ) (hr : r < 0)
Goal: ↑r < x
-- Hyperreal.abs_lt_real_iff_infinitesimal
theorem abs_lt_real_iff_infinitesimal {x : ℝ*} : Infinitesimal x ↔ ∀ r : ℝ, r ≠ 0 → |x| < |↑r| 
Arguments: {x : ℝ*}
Goal: x.Infinitesimal ↔ ∀ (r : ℝ), r ≠ 0 → |x| < |↑r|
-- Hyperreal.infinitesimal_zero
theorem infinitesimal_zero : Infinitesimal 0 
Goal: Hyperreal.Infinitesimal 0
-- Hyperreal.Infinitesimal.eq_zero
theorem Infinitesimal.eq_zero {r : ℝ} : Infinitesimal r → r = 0 
Arguments: {r : ℝ}
Goal: (↑r).Infinitesimal → r = 0