ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- EReal.coe_ne_bot
@[simp]
theorem coe_ne_bot (x : ℝ) : (x : EReal) ≠ ⊥ 
Arguments: (x : ℝ)
Goal: ↑x ≠ ⊥
-- EReal.bot_ne_coe
@[simp]
theorem bot_ne_coe (x : ℝ) : (⊥ : EReal) ≠ x 
Arguments: (x : ℝ)
Goal: ⊥ ≠ ↑x
-- EReal.coe_lt_top
@[simp]
theorem coe_lt_top (x : ℝ) : (x : EReal) < ⊤ 
Arguments: (x : ℝ)
Goal: ↑x < ⊤
-- EReal.coe_ne_top
@[simp]
theorem coe_ne_top (x : ℝ) : (x : EReal) ≠ ⊤ 
Arguments: (x : ℝ)
Goal: ↑x ≠ ⊤
-- EReal.top_ne_coe
@[simp]
theorem top_ne_coe (x : ℝ) : (⊤ : EReal) ≠ x 
Arguments: (x : ℝ)
Goal: ⊤ ≠ ↑x
-- EReal.bot_lt_zero
@[simp]
theorem bot_lt_zero : (⊥ : EReal) < 0 
Goal: ⊥ < 0
-- EReal.bot_ne_zero
@[simp]
theorem bot_ne_zero : (⊥ : EReal) ≠ 0 
Goal: ⊥ ≠ 0
-- EReal.zero_ne_bot
@[simp]
theorem zero_ne_bot : (0 : EReal) ≠ ⊥ 
Goal: 0 ≠ ⊥
-- EReal.zero_lt_top
@[simp]
theorem zero_lt_top : (0 : EReal) < ⊤ 
Goal: 0 < ⊤
-- EReal.zero_ne_top
@[simp]
theorem zero_ne_top : (0 : EReal) ≠ ⊤ 
Goal: 0 ≠ ⊤
-- EReal.top_ne_zero
@[simp]
theorem top_ne_zero : (⊤ : EReal) ≠ 0 
Goal: ⊤ ≠ 0
-- EReal.range_coe
theorem range_coe : range Real.toEReal = {⊥, ⊤}ᶜ 
Goal: Set.range Real.toEReal = {⊥, ⊤}ᶜ
-- EReal.range_coe_eq_Ioo
theorem range_coe_eq_Ioo : range Real.toEReal = Ioo ⊥ ⊤ 
Goal: Set.range Real.toEReal = Set.Ioo ⊥ ⊤
-- EReal.coe_add
@[simp, norm_cast]
theorem coe_add (x y : ℝ) : (↑(x + y) : EReal) = x + y 
Arguments: (x y : ℝ)
Goal: ↑(x + y) = ↑x + ↑y
-- EReal.coe_nsmul
@[norm_cast]
theorem coe_nsmul (n : ℕ) (x : ℝ) : (↑(n • x) : EReal) = n • (x : EReal) 
Arguments: (n : ℕ) (x : ℝ)
Goal: ↑(n • x) = n • ↑x
-- EReal.coe_eq_zero
@[simp, norm_cast]
theorem coe_eq_zero {x : ℝ} : (x : EReal) = 0 ↔ x = 0 
Arguments: {x : ℝ}
Goal: ↑x = 0 ↔ x = 0
-- EReal.coe_eq_one
@[simp, norm_cast]
theorem coe_eq_one {x : ℝ} : (x : EReal) = 1 ↔ x = 1 
Arguments: {x : ℝ}
Goal: ↑x = 1 ↔ x = 1
-- EReal.coe_ne_zero
theorem coe_ne_zero {x : ℝ} : (x : EReal) ≠ 0 ↔ x ≠ 0 
Arguments: {x : ℝ}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- EReal.coe_ne_one
theorem coe_ne_one {x : ℝ} : (x : EReal) ≠ 1 ↔ x ≠ 1 
Arguments: {x : ℝ}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- EReal.coe_nonneg
@[simp, norm_cast]
protected theorem coe_nonneg {x : ℝ} : (0 : EReal) ≤ x ↔ 0 ≤ x 
Arguments: {x : ℝ}
Goal: 0 ≤ ↑x ↔ 0 ≤ x
-- EReal.coe_nonpos
@[simp, norm_cast]
protected theorem coe_nonpos {x : ℝ} : (x : EReal) ≤ 0 ↔ x ≤ 0 
Arguments: {x : ℝ}
Goal: ↑x ≤ 0 ↔ x ≤ 0
-- EReal.coe_pos
@[simp, norm_cast]
protected theorem coe_pos {x : ℝ} : (0 : EReal) < x ↔ 0 < x 
Arguments: {x : ℝ}
Goal: 0 < ↑x ↔ 0 < x
-- EReal.coe_neg'
@[simp, norm_cast]
protected theorem coe_neg' {x : ℝ} : (x : EReal) < 0 ↔ x < 0 
Arguments: {x : ℝ}
Goal: ↑x < 0 ↔ x < 0
-- EReal.toReal_le_toReal
theorem toReal_le_toReal {x y : EReal} (h : x ≤ y) (hx : x ≠ ⊥) (hy : y ≠ ⊤) :
    x.toReal ≤ y.toReal 
Arguments: {x y : EReal} (h : x ≤ y) (hx : x ≠ ⊥) (hy : y ≠ ⊤)
Goal: x.toReal ≤ y.toReal
-- EReal.coe_toReal
theorem coe_toReal {x : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) : (x.toReal : EReal) = x 
Arguments: {x : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥)
Goal: ↑x.toReal = x
-- EReal.le_coe_toReal
theorem le_coe_toReal {x : EReal} (h : x ≠ ⊤) : x ≤ x.toReal 
Arguments: {x : EReal} (h : x ≠ ⊤)
Goal: x ≤ ↑x.toReal
-- EReal.coe_toReal_le
theorem coe_toReal_le {x : EReal} (h : x ≠ ⊥) : ↑x.toReal ≤ x 
Arguments: {x : EReal} (h : x ≠ ⊥)
Goal: ↑x.toReal ≤ x
-- EReal.eq_top_iff_forall_lt
theorem eq_top_iff_forall_lt (x : EReal) : x = ⊤ ↔ ∀ y : ℝ, (y : EReal) < x 
Arguments: (x : EReal)
Goal: x = ⊤ ↔ ∀ (y : ℝ), ↑y < x
-- EReal.eq_bot_iff_forall_lt
theorem eq_bot_iff_forall_lt (x : EReal) : x = ⊥ ↔ ∀ y : ℝ, x < (y : EReal) 
Arguments: (x : EReal)
Goal: x = ⊥ ↔ ∀ (y : ℝ), x < ↑y
-- EReal.exists_between_coe_real
lemma exists_between_coe_real {x z : EReal} (h : x < z) : ∃ y : ℝ, x < y ∧ y < z 
Arguments: {x z : EReal} (h : x < z)
Goal: ∃ y, x < ↑y ∧ ↑y < z
-- EReal.image_coe_Icc
@[simp]
lemma image_coe_Icc (x y : ℝ) : Real.toEReal '' Icc x y = Icc ↑x ↑y 
Arguments: (x y : ℝ)
Goal: Real.toEReal '' Set.Icc x y = Set.Icc ↑x ↑y
-- EReal.image_coe_Ico
@[simp]
lemma image_coe_Ico (x y : ℝ) : Real.toEReal '' Ico x y = Ico ↑x ↑y 
Arguments: (x y : ℝ)
Goal: Real.toEReal '' Set.Ico x y = Set.Ico ↑x ↑y
-- EReal.image_coe_Ici
@[simp]
lemma image_coe_Ici (x : ℝ) : Real.toEReal '' Ici x = Ico ↑x ⊤ 
Arguments: (x : ℝ)
Goal: Real.toEReal '' Set.Ici x = Set.Ico ↑x ⊤
-- EReal.image_coe_Ioc
@[simp]
lemma image_coe_Ioc (x y : ℝ) : Real.toEReal '' Ioc x y = Ioc ↑x ↑y 
Arguments: (x y : ℝ)
Goal: Real.toEReal '' Set.Ioc x y = Set.Ioc ↑x ↑y
-- EReal.image_coe_Ioo
@[simp]
lemma image_coe_Ioo (x y : ℝ) : Real.toEReal '' Ioo x y = Ioo ↑x ↑y 
Arguments: (x y : ℝ)
Goal: Real.toEReal '' Set.Ioo x y = Set.Ioo ↑x ↑y
-- EReal.image_coe_Ioi
@[simp]
lemma image_coe_Ioi (x : ℝ) : Real.toEReal '' Ioi x = Ioo ↑x ⊤ 
Arguments: (x : ℝ)
Goal: Real.toEReal '' Set.Ioi x = Set.Ioo ↑x ⊤
-- EReal.image_coe_Iic
@[simp]
lemma image_coe_Iic (x : ℝ) : Real.toEReal '' Iic x = Ioc ⊥ ↑x 
Arguments: (x : ℝ)
Goal: Real.toEReal '' Set.Iic x = Set.Ioc ⊥ ↑x
-- EReal.image_coe_Iio
@[simp]
lemma image_coe_Iio (x : ℝ) : Real.toEReal '' Iio x = Ioo ⊥ ↑x 
Arguments: (x : ℝ)
Goal: Real.toEReal '' Set.Iio x = Set.Ioo ⊥ ↑x
-- EReal.preimage_coe_Ici
@[simp]
lemma preimage_coe_Ici (x : ℝ) : Real.toEReal ⁻¹' Ici x = Ici x 
Arguments: (x : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ici ↑x = Set.Ici x
-- EReal.preimage_coe_Ioi
@[simp]
lemma preimage_coe_Ioi (x : ℝ) : Real.toEReal ⁻¹' Ioi x = Ioi x 
Arguments: (x : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioi ↑x = Set.Ioi x
-- EReal.preimage_coe_Ioi_bot
@[simp]
lemma preimage_coe_Ioi_bot : Real.toEReal ⁻¹' Ioi ⊥ = univ 
Goal: Real.toEReal ⁻¹' Set.Ioi ⊥ = Set.univ
-- EReal.preimage_coe_Iic
@[simp]
lemma preimage_coe_Iic (y : ℝ) : Real.toEReal ⁻¹' Iic y = Iic y 
Arguments: (y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Iic ↑y = Set.Iic y
-- EReal.preimage_coe_Iio
@[simp]
lemma preimage_coe_Iio (y : ℝ) : Real.toEReal ⁻¹' Iio y = Iio y 
Arguments: (y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Iio ↑y = Set.Iio y
-- EReal.preimage_coe_Iio_top
@[simp]
lemma preimage_coe_Iio_top : Real.toEReal ⁻¹' Iio ⊤ = univ 
Goal: Real.toEReal ⁻¹' Set.Iio ⊤ = Set.univ
-- EReal.preimage_coe_Icc
@[simp]
lemma preimage_coe_Icc (x y : ℝ) : Real.toEReal ⁻¹' Icc x y = Icc x y 
Arguments: (x y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Icc ↑x ↑y = Set.Icc x y
-- EReal.preimage_coe_Ico
@[simp]
lemma preimage_coe_Ico (x y : ℝ) : Real.toEReal ⁻¹' Ico x y = Ico x y 
Arguments: (x y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ico ↑x ↑y = Set.Ico x y
-- EReal.preimage_coe_Ioc
@[simp]
lemma preimage_coe_Ioc (x y : ℝ) : Real.toEReal ⁻¹' Ioc x y = Ioc x y 
Arguments: (x y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioc ↑x ↑y = Set.Ioc x y
-- EReal.preimage_coe_Ioo
@[simp]
lemma preimage_coe_Ioo (x y : ℝ) : Real.toEReal ⁻¹' Ioo x y = Ioo x y 
Arguments: (x y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioo ↑x ↑y = Set.Ioo x y
-- EReal.preimage_coe_Ico_top
@[simp]
lemma preimage_coe_Ico_top (x : ℝ) : Real.toEReal ⁻¹' Ico x ⊤ = Ici x 
Arguments: (x : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ico ↑x ⊤ = Set.Ici x
-- EReal.preimage_coe_Ioo_top
@[simp]
lemma preimage_coe_Ioo_top (x : ℝ) : Real.toEReal ⁻¹' Ioo x ⊤ = Ioi x 
Arguments: (x : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioo ↑x ⊤ = Set.Ioi x
-- EReal.preimage_coe_Ioc_bot
@[simp]
lemma preimage_coe_Ioc_bot (y : ℝ) : Real.toEReal ⁻¹' Ioc ⊥ y = Iic y 
Arguments: (y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioc ⊥ ↑y = Set.Iic y
-- EReal.preimage_coe_Ioo_bot
@[simp]
lemma preimage_coe_Ioo_bot (y : ℝ) : Real.toEReal ⁻¹' Ioo ⊥ y = Iio y 
Arguments: (y : ℝ)
Goal: Real.toEReal ⁻¹' Set.Ioo ⊥ ↑y = Set.Iio y
-- EReal.preimage_coe_Ioo_bot_top
@[simp]
lemma preimage_coe_Ioo_bot_top : Real.toEReal ⁻¹' Ioo ⊥ ⊤ = univ 
Goal: Real.toEReal ⁻¹' Set.Ioo ⊥ ⊤ = Set.univ
-- EReal.toReal_coe_ennreal
@[simp]
theorem toReal_coe_ennreal : ∀ {x : ℝ≥0∞}, toReal (x : EReal) = ENNReal.toReal x
  
Arguments: {x : ℝ≥0∞}
Goal: (↑x).toReal = x.toReal
-- EReal.coe_ennreal_ofReal
@[simp]
theorem coe_ennreal_ofReal {x : ℝ} : (ENNReal.ofReal x : EReal) = max x 0 
Arguments: {x : ℝ}
Goal: ↑(ENNReal.ofReal x) = ↑(max x 0)
-- EReal.coe_nnreal_eq_coe_real
theorem coe_nnreal_eq_coe_real (x : ℝ≥0) : ((x : ℝ≥0∞) : EReal) = (x : ℝ) 
Arguments: (x : ℝ≥0)
Goal: ↑↑x = ↑↑x
-- EReal.coe_ennreal_zero
@[simp, norm_cast]
theorem coe_ennreal_zero : ((0 : ℝ≥0∞) : EReal) = 0 
Goal: ↑0 = 0
-- EReal.coe_ennreal_one
@[simp, norm_cast]
theorem coe_ennreal_one : ((1 : ℝ≥0∞) : EReal) = 1 
Goal: ↑1 = 1
-- EReal.coe_ennreal_top
@[simp, norm_cast]
theorem coe_ennreal_top : ((⊤ : ℝ≥0∞) : EReal) = ⊤ 
Goal: ↑⊤ = ⊤
-- EReal.coe_ennreal_strictMono
theorem coe_ennreal_strictMono : StrictMono ((↑) : ℝ≥0∞ → EReal) 
Goal: StrictMono ENNReal.toEReal
-- EReal.coe_ennreal_injective
theorem coe_ennreal_injective : Injective ((↑) : ℝ≥0∞ → EReal) 
Goal: Function.Injective ENNReal.toEReal
-- EReal.coe_ennreal_eq_top_iff
@[simp]
theorem coe_ennreal_eq_top_iff {x : ℝ≥0∞} : (x : EReal) = ⊤ ↔ x = ⊤ 
Arguments: {x : ℝ≥0∞}
Goal: ↑x = ⊤ ↔ x = ⊤
-- EReal.coe_nnreal_ne_top
theorem coe_nnreal_ne_top (x : ℝ≥0) : ((x : ℝ≥0∞) : EReal) ≠ ⊤ 
Arguments: (x : ℝ≥0)
Goal: ↑↑x ≠ ⊤
-- EReal.coe_nnreal_lt_top
@[simp]
theorem coe_nnreal_lt_top (x : ℝ≥0) : ((x : ℝ≥0∞) : EReal) < ⊤ 
Arguments: (x : ℝ≥0)
Goal: ↑↑x < ⊤
-- EReal.coe_ennreal_le_coe_ennreal_iff
@[simp, norm_cast]
theorem coe_ennreal_le_coe_ennreal_iff {x y : ℝ≥0∞} : (x : EReal) ≤ (y : EReal) ↔ x ≤ y 
Arguments: {x y : ℝ≥0∞}
Goal: ↑x ≤ ↑y ↔ x ≤ y
-- EReal.coe_ennreal_lt_coe_ennreal_iff
@[simp, norm_cast]
theorem coe_ennreal_lt_coe_ennreal_iff {x y : ℝ≥0∞} : (x : EReal) < (y : EReal) ↔ x < y 
Arguments: {x y : ℝ≥0∞}
Goal: ↑x < ↑y ↔ x < y
-- EReal.coe_ennreal_eq_coe_ennreal_iff
@[simp, norm_cast]
theorem coe_ennreal_eq_coe_ennreal_iff {x y : ℝ≥0∞} : (x : EReal) = (y : EReal) ↔ x = y 
Arguments: {x y : ℝ≥0∞}
Goal: ↑x = ↑y ↔ x = y
-- EReal.coe_ennreal_ne_coe_ennreal_iff
theorem coe_ennreal_ne_coe_ennreal_iff {x y : ℝ≥0∞} : (x : EReal) ≠ (y : EReal) ↔ x ≠ y 
Arguments: {x y : ℝ≥0∞}
Goal: ↑x ≠ ↑y ↔ x ≠ y
-- EReal.coe_ennreal_eq_zero
@[simp, norm_cast]
theorem coe_ennreal_eq_zero {x : ℝ≥0∞} : (x : EReal) = 0 ↔ x = 0 
Arguments: {x : ℝ≥0∞}
Goal: ↑x = 0 ↔ x = 0
-- EReal.coe_ennreal_eq_one
@[simp, norm_cast]
theorem coe_ennreal_eq_one {x : ℝ≥0∞} : (x : EReal) = 1 ↔ x = 1 
Arguments: {x : ℝ≥0∞}
Goal: ↑x = 1 ↔ x = 1
-- EReal.coe_ennreal_ne_zero
@[norm_cast]
theorem coe_ennreal_ne_zero {x : ℝ≥0∞} : (x : EReal) ≠ 0 ↔ x ≠ 0 
Arguments: {x : ℝ≥0∞}
Goal: ↑x ≠ 0 ↔ x ≠ 0
-- EReal.coe_ennreal_ne_one
@[norm_cast]
theorem coe_ennreal_ne_one {x : ℝ≥0∞} : (x : EReal) ≠ 1 ↔ x ≠ 1 
Arguments: {x : ℝ≥0∞}
Goal: ↑x ≠ 1 ↔ x ≠ 1
-- EReal.coe_ennreal_nonneg
theorem coe_ennreal_nonneg (x : ℝ≥0∞) : (0 : EReal) ≤ x 
Arguments: (x : ℝ≥0∞)
Goal: 0 ≤ ↑x
-- EReal.range_coe_ennreal
@[simp] theorem range_coe_ennreal : range ((↑) : ℝ≥0∞ → EReal) = Set.Ici 0 
Goal: Set.range ENNReal.toEReal = Set.Ici 0
-- EReal.coe_ennreal_pos
@[simp, norm_cast]
theorem coe_ennreal_pos {x : ℝ≥0∞} : (0 : EReal) < x ↔ 0 < x 
Arguments: {x : ℝ≥0∞}
Goal: 0 < ↑x ↔ 0 < x
-- EReal.bot_lt_coe_ennreal
@[simp]
theorem bot_lt_coe_ennreal (x : ℝ≥0∞) : (⊥ : EReal) < x 
Arguments: (x : ℝ≥0∞)
Goal: ⊥ < ↑x
-- EReal.coe_ennreal_ne_bot
@[simp]
theorem coe_ennreal_ne_bot (x : ℝ≥0∞) : (x : EReal) ≠ ⊥ 
Arguments: (x : ℝ≥0∞)
Goal: ↑x ≠ ⊥
-- EReal.coe_ennreal_add
@[simp, norm_cast]
theorem coe_ennreal_add (x y : ENNReal) : ((x + y : ℝ≥0∞) : EReal) = x + y 
Arguments: (x y : ℝ≥0∞)
Goal: ↑(x + y) = ↑x + ↑y
-- _private.Mathlib.Data.Real.EReal.0.EReal.coe_ennreal_top_mul
private theorem coe_ennreal_top_mul (x : ℝ≥0) : ((⊤ * x : ℝ≥0∞) : EReal) = ⊤ * x 
Arguments: (x : ℝ≥0)
Goal: ↑(⊤ * ↑x) = ⊤ * ↑↑x
-- EReal.coe_ennreal_mul
@[simp, norm_cast]
theorem coe_ennreal_mul : ∀ x y : ℝ≥0∞, ((x * y : ℝ≥0∞) : EReal) = (x : EReal) * y
  
Arguments: (x y : ℝ≥0∞)
Goal: ↑(x * y) = ↑x * ↑y
-- EReal.coe_ennreal_nsmul
@[norm_cast]
theorem coe_ennreal_nsmul (n : ℕ) (x : ℝ≥0∞) : (↑(n • x) : EReal) = n • (x : EReal) 
Arguments: (n : ℕ) (x : ℝ≥0∞)
Goal: ↑(n • x) = n • ↑x
-- EReal.coe_coe_eq_natCast
theorem coe_coe_eq_natCast (n : ℕ) : (n : ℝ) = (n : EReal) 
Arguments: (n : ℕ)
Goal: ↑↑n = ↑n
-- EReal.natCast_ne_bot
theorem natCast_ne_bot (n : ℕ) : (n : EReal) ≠ ⊥ 
Arguments: (n : ℕ)
Goal: ↑n ≠ ⊥
-- EReal.natCast_ne_top
theorem natCast_ne_top (n : ℕ) : (n : EReal) ≠ ⊤ 
Arguments: (n : ℕ)
Goal: ↑n ≠ ⊤
-- EReal.natCast_eq_iff
@[simp, norm_cast]
theorem natCast_eq_iff {m n : ℕ} : (m : EReal) = (n : EReal) ↔ m = n 
Arguments: {m n : ℕ}
Goal: ↑m = ↑n ↔ m = n
-- EReal.natCast_ne_iff
theorem natCast_ne_iff {m n : ℕ} : (m : EReal) ≠ (n : EReal) ↔ m ≠ n 
Arguments: {m n : ℕ}
Goal: ↑m ≠ ↑n ↔ m ≠ n
-- EReal.natCast_le_iff
@[simp, norm_cast]
theorem natCast_le_iff {m n : ℕ} : (m : EReal) ≤ (n : EReal) ↔ m ≤ n 
Arguments: {m n : ℕ}
Goal: ↑m ≤ ↑n ↔ m ≤ n
-- EReal.natCast_lt_iff
@[simp, norm_cast]
theorem natCast_lt_iff {m n : ℕ} : (m : EReal) < (n : EReal) ↔ m < n 
Arguments: {m n : ℕ}
Goal: ↑m < ↑n ↔ m < n
-- EReal.natCast_mul
@[simp, norm_cast]
theorem natCast_mul (m n : ℕ) :
    (m * n : ℕ) = (m : EReal) * (n : EReal) 
Arguments: (m n : ℕ)
Goal: ↑(m * n) = ↑m * ↑n
-- EReal.exists_rat_btwn_of_lt
theorem exists_rat_btwn_of_lt :
    ∀ {a b : EReal}, a < b → ∃ x : ℚ, a < (x : ℝ) ∧ ((x : ℝ) : EReal) < b
  
Arguments: {a b : EReal}
Goal: a < b → ∃ x, a < ↑↑x ∧ ↑↑x < b
-- EReal.lt_iff_exists_rat_btwn
theorem lt_iff_exists_rat_btwn {a b : EReal} :
    a < b ↔ ∃ x : ℚ, a < (x : ℝ) ∧ ((x : ℝ) : EReal) < b 
Arguments: {a b : EReal}
Goal: a < b ↔ ∃ x, a < ↑↑x ∧ ↑↑x < b
-- EReal.lt_iff_exists_real_btwn
theorem lt_iff_exists_real_btwn {a b : EReal} : a < b ↔ ∃ x : ℝ, a < x ∧ (x : EReal) < b 
Arguments: {a b : EReal}
Goal: a < b ↔ ∃ x, a < ↑x ∧ ↑x < b
-- EReal.add_bot
@[simp]
theorem add_bot (x : EReal) : x + ⊥ = ⊥ 
Arguments: (x : EReal)
Goal: x + ⊥ = ⊥
-- EReal.bot_add
@[simp]
theorem bot_add (x : EReal) : ⊥ + x = ⊥ 
Arguments: (x : EReal)
Goal: ⊥ + x = ⊥
-- EReal.add_eq_bot_iff
@[simp]
theorem add_eq_bot_iff {x y : EReal} : x + y = ⊥ ↔ x = ⊥ ∨ y = ⊥ 
Arguments: {x y : EReal}
Goal: x + y = ⊥ ↔ x = ⊥ ∨ y = ⊥
-- EReal.bot_lt_add_iff
@[simp]
theorem bot_lt_add_iff {x y : EReal} : ⊥ < x + y ↔ ⊥ < x ∧ ⊥ < y 
Arguments: {x y : EReal}
Goal: ⊥ < x + y ↔ ⊥ < x ∧ ⊥ < y
-- EReal.top_add_top
@[simp]
theorem top_add_top : (⊤ : EReal) + ⊤ = ⊤ 
Goal: ⊤ + ⊤ = ⊤
-- EReal.top_add_coe
@[simp]
theorem top_add_coe (x : ℝ) : (⊤ : EReal) + x = ⊤ 
Arguments: (x : ℝ)
Goal: ⊤ + ↑x = ⊤
-- EReal.top_add_of_ne_bot
@[simp]
theorem top_add_of_ne_bot {x : EReal} (h : x ≠ ⊥) : ⊤ + x = ⊤ 
Arguments: {x : EReal} (h : x ≠ ⊥)
Goal: ⊤ + x = ⊤
-- EReal.top_add_iff_ne_bot
theorem top_add_iff_ne_bot {x : EReal} : ⊤ + x = ⊤ ↔ x ≠ ⊥ 
Arguments: {x : EReal}
Goal: ⊤ + x = ⊤ ↔ x ≠ ⊥
-- EReal.add_top_of_ne_bot
@[simp]
theorem add_top_of_ne_bot {x : EReal} (h : x ≠ ⊥) : x + ⊤ = ⊤ 
Arguments: {x : EReal} (h : x ≠ ⊥)
Goal: x + ⊤ = ⊤
-- EReal.add_top_iff_ne_bot
theorem add_top_iff_ne_bot {x : EReal} : x + ⊤ = ⊤ ↔ x ≠ ⊥ 
Arguments: {x : EReal}
Goal: x + ⊤ = ⊤ ↔ x ≠ ⊥
-- EReal.add_pos
theorem add_pos {a b : EReal} (ha : 0 < a) (hb : 0 < b) : 0 < a + b 
Arguments: {a b : EReal} (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a + b
-- EReal.coe_add_top
@[simp]
theorem coe_add_top (x : ℝ) : (x : EReal) + ⊤ = ⊤ 
Arguments: (x : ℝ)
Goal: ↑x + ⊤ = ⊤
-- EReal.toReal_add
theorem toReal_add {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :
    toReal (x + y) = toReal x + toReal y 
Arguments: {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥)
Goal: (x + y).toReal = x.toReal + y.toReal
-- EReal.addLECancellable_coe
theorem addLECancellable_coe (x : ℝ) : AddLECancellable (x : EReal)
  
Arguments: (x : ℝ)
Goal: AddLECancellable ↑x
-- EReal.add_lt_add_right_coe
theorem add_lt_add_right_coe {x y : EReal} (h : x < y) (z : ℝ) : x + z < y + z 
Arguments: {x y : EReal} (h : x < y) (z : ℝ)
Goal: x + ↑z < y + ↑z
-- EReal.add_lt_add_left_coe
theorem add_lt_add_left_coe {x y : EReal} (h : x < y) (z : ℝ) : (z : EReal) + x < z + y 
Arguments: {x y : EReal} (h : x < y) (z : ℝ)
Goal: ↑z + x < ↑z + y
-- EReal.add_lt_add
theorem add_lt_add {x y z t : EReal} (h1 : x < y) (h2 : z < t) : x + z < y + t 
Arguments: {x y z t : EReal} (h1 : x < y) (h2 : z < t)
Goal: x + z < y + t
-- EReal.add_lt_add_of_lt_of_le'
theorem add_lt_add_of_lt_of_le' {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hbot : t ≠ ⊥)
    (htop : t = ⊤ → z = ⊤ → x = ⊥) : x + z < y + t 
Arguments: {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hbot : t ≠ ⊥) (htop : t = ⊤ → z = ⊤ → x = ⊥)
Goal: x + z < y + t
-- EReal.add_lt_add_of_lt_of_le
theorem add_lt_add_of_lt_of_le {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥)
    (ht : t ≠ ⊤) : x + z < y + t 
Arguments: {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥) (ht : t ≠ ⊤)
Goal: x + z < y + t
-- EReal.add_lt_top
theorem add_lt_top {x y : EReal} (hx : x ≠ ⊤) (hy : y ≠ ⊤) : x + y < ⊤ 
Arguments: {x y : EReal} (hx : x ≠ ⊤) (hy : y ≠ ⊤)
Goal: x + y < ⊤
-- EReal.neg_top
@[simp]
theorem neg_top : -(⊤ : EReal) = ⊥ 
Goal: -⊤ = ⊥
-- EReal.neg_bot
@[simp]
theorem neg_bot : -(⊥ : EReal) = ⊤ 
Goal: -⊥ = ⊤
-- EReal.coe_neg
@[simp, norm_cast] theorem coe_neg (x : ℝ) : (↑(-x) : EReal) = -↑x 
Arguments: (x : ℝ)
Goal: ↑(-x) = -↑x
-- EReal.coe_sub
@[simp, norm_cast] theorem coe_sub (x y : ℝ) : (↑(x - y) : EReal) = x - y 
Arguments: (x y : ℝ)
Goal: ↑(x - y) = ↑x - ↑y
-- EReal.coe_zsmul
@[norm_cast]
theorem coe_zsmul (n : ℤ) (x : ℝ) : (↑(n • x) : EReal) = n • (x : EReal) 
Arguments: (n : ℤ) (x : ℝ)
Goal: ↑(n • x) = n • ↑x
-- EReal.toReal_neg
@[simp]
theorem toReal_neg : ∀ {a : EReal}, toReal (-a) = -toReal a
  
Arguments: {a : EReal}
Goal: (-a).toReal = -a.toReal
-- EReal.neg_eq_top_iff
@[simp]
theorem neg_eq_top_iff {x : EReal} : -x = ⊤ ↔ x = ⊥ 
Arguments: {x : EReal}
Goal: -x = ⊤ ↔ x = ⊥
-- EReal.neg_eq_bot_iff
@[simp]
theorem neg_eq_bot_iff {x : EReal} : -x = ⊥ ↔ x = ⊤ 
Arguments: {x : EReal}
Goal: -x = ⊥ ↔ x = ⊤
-- EReal.neg_eq_zero_iff
@[simp]
theorem neg_eq_zero_iff {x : EReal} : -x = 0 ↔ x = 0 
Arguments: {x : EReal}
Goal: -x = 0 ↔ x = 0
-- EReal.neg_strictAnti
theorem neg_strictAnti : StrictAnti (- · : EReal → EReal) 
Goal: StrictAnti fun x => -x
-- EReal.neg_le_neg_iff
@[simp] theorem neg_le_neg_iff {a b : EReal} : -a ≤ -b ↔ b ≤ a 
Arguments: {a b : EReal}
Goal: -a ≤ -b ↔ b ≤ a
-- EReal.neg_lt_neg_iff
@[simp] theorem neg_lt_neg_iff {a b : EReal} : -a < -b ↔ b < a 
Arguments: {a b : EReal}
Goal: -a < -b ↔ b < a
-- EReal.neg_le
protected theorem neg_le {a b : EReal} : -a ≤ b ↔ -b ≤ a 
Arguments: {a b : EReal}
Goal: -a ≤ b ↔ -b ≤ a
-- EReal.neg_le_of_neg_le
protected theorem neg_le_of_neg_le {a b : EReal} (h : -a ≤ b) : -b ≤ a 
Arguments: {a b : EReal} (h : -a ≤ b)
Goal: -b ≤ a
-- EReal.le_neg_of_le_neg
theorem le_neg_of_le_neg {a b : EReal} (h : a ≤ -b) : b ≤ -a 
Arguments: {a b : EReal} (h : a ≤ -b)
Goal: b ≤ -a
-- EReal.neg_lt_iff_neg_lt
theorem neg_lt_iff_neg_lt {a b : EReal} : -a < b ↔ -b < a 
Arguments: {a b : EReal}
Goal: -a < b ↔ -b < a
-- EReal.neg_lt_of_neg_lt
theorem neg_lt_of_neg_lt {a b : EReal} (h : -a < b) : -b < a 
Arguments: {a b : EReal} (h : -a < b)
Goal: -b < a
-- EReal.neg_add
lemma neg_add {x y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊤) (h2 : x ≠ ⊤ ∨ y ≠ ⊥) :
    - (x + y) = - x - y 
Arguments: {x y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊤) (h2 : x ≠ ⊤ ∨ y ≠ ⊥)
Goal: -(x + y) = -x - y
-- EReal.neg_sub
lemma neg_sub {x y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊥) (h2 : x ≠ ⊤ ∨ y ≠ ⊤) :
    - (x - y) = - x + y 
Arguments: {x y : EReal} (h1 : x ≠ ⊥ ∨ y ≠ ⊥) (h2 : x ≠ ⊤ ∨ y ≠ ⊤)
Goal: -(x - y) = -x + y
-- EReal.bot_sub
@[simp]
theorem bot_sub (x : EReal) : ⊥ - x = ⊥ 
Arguments: (x : EReal)
Goal: ⊥ - x = ⊥
-- EReal.sub_top
@[simp]
theorem sub_top (x : EReal) : x - ⊤ = ⊥ 
Arguments: (x : EReal)
Goal: x - ⊤ = ⊥
-- EReal.top_sub_bot
@[simp]
theorem top_sub_bot : (⊤ : EReal) - ⊥ = ⊤ 
Goal: ⊤ - ⊥ = ⊤
-- EReal.top_sub_coe
@[simp]
theorem top_sub_coe (x : ℝ) : (⊤ : EReal) - x = ⊤ 
Arguments: (x : ℝ)
Goal: ⊤ - ↑x = ⊤
-- EReal.coe_sub_bot
@[simp]
theorem coe_sub_bot (x : ℝ) : (x : EReal) - ⊥ = ⊤ 
Arguments: (x : ℝ)
Goal: ↑x - ⊥ = ⊤
-- EReal.sub_le_sub
theorem sub_le_sub {x y z t : EReal} (h : x ≤ y) (h' : t ≤ z) : x - z ≤ y - t 
Arguments: {x y z t : EReal} (h : x ≤ y) (h' : t ≤ z)
Goal: x - z ≤ y - t
-- EReal.sub_lt_sub_of_lt_of_le
theorem sub_lt_sub_of_lt_of_le {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥)
    (ht : t ≠ ⊤) : x - t < y - z 
Arguments: {x y z t : EReal} (h : x < y) (h' : z ≤ t) (hz : z ≠ ⊥) (ht : t ≠ ⊤)
Goal: x - t < y - z
-- EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal
theorem coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal (x : ℝ) :
    (x : EReal) = Real.toNNReal x - Real.toNNReal (-x) 
Arguments: (x : ℝ)
Goal: ↑x = ↑↑x.toNNReal - ↑↑(-x).toNNReal
-- EReal.toReal_sub
theorem toReal_sub {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥) :
    toReal (x - y) = toReal x - toReal y 
Arguments: {x y : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) (hy : y ≠ ⊤) (h'y : y ≠ ⊥)
Goal: (x - y).toReal = x.toReal - y.toReal
-- EReal.add_sub_cancel_right
lemma add_sub_cancel_right {a : EReal} {b : Real} : a + b - b = a 
Arguments: {a : EReal} {b : ℝ}
Goal: a + ↑b - ↑b = a
-- EReal.top_mul_top
@[simp] lemma top_mul_top : (⊤ : EReal) * ⊤ = ⊤ 
Goal: ⊤ * ⊤ = ⊤
-- EReal.top_mul_bot
@[simp] lemma top_mul_bot : (⊤ : EReal) * ⊥ = ⊥ 
Goal: ⊤ * ⊥ = ⊥
-- EReal.bot_mul_top
@[simp] lemma bot_mul_top : (⊥ : EReal) * ⊤ = ⊥ 
Goal: ⊥ * ⊤ = ⊥
-- EReal.bot_mul_bot
@[simp] lemma bot_mul_bot : (⊥ : EReal) * ⊥ = ⊤ 
Goal: ⊥ * ⊥ = ⊤
-- EReal.coe_mul_top_of_pos
lemma coe_mul_top_of_pos {x : ℝ} (h : 0 < x) : (x : EReal) * ⊤ = ⊤ 
Arguments: {x : ℝ} (h : 0 < x)
Goal: ↑x * ⊤ = ⊤
-- EReal.coe_mul_top_of_neg
lemma coe_mul_top_of_neg {x : ℝ} (h : x < 0) : (x : EReal) * ⊤ = ⊥ 
Arguments: {x : ℝ} (h : x < 0)
Goal: ↑x * ⊤ = ⊥
-- EReal.top_mul_coe_of_pos
lemma top_mul_coe_of_pos {x : ℝ} (h : 0 < x) : (⊤ : EReal) * x = ⊤ 
Arguments: {x : ℝ} (h : 0 < x)
Goal: ⊤ * ↑x = ⊤
-- EReal.top_mul_coe_of_neg
lemma top_mul_coe_of_neg {x : ℝ} (h : x < 0) : (⊤ : EReal) * x = ⊥ 
Arguments: {x : ℝ} (h : x < 0)
Goal: ⊤ * ↑x = ⊥
-- EReal.mul_top_of_pos
lemma mul_top_of_pos : ∀ {x : EReal}, 0 < x → x * ⊤ = ⊤
  
Arguments: {x : EReal}
Goal: 0 < x → x * ⊤ = ⊤
-- EReal.mul_top_of_neg
lemma mul_top_of_neg : ∀ {x : EReal}, x < 0 → x * ⊤ = ⊥
  
Arguments: {x : EReal}
Goal: x < 0 → x * ⊤ = ⊥
-- EReal.top_mul_of_pos
lemma top_mul_of_pos {x : EReal} (h : 0 < x) : ⊤ * x = ⊤ 
Arguments: {x : EReal} (h : 0 < x)
Goal: ⊤ * x = ⊤
-- EReal.mul_pos
lemma mul_pos {a b : EReal} (ha : 0 < a) (hb : 0 < b) : 0 < a * b 
Arguments: {a b : EReal} (ha : 0 < a) (hb : 0 < b)
Goal: 0 < a * b
-- EReal.top_mul_of_neg
lemma top_mul_of_neg {x : EReal} (h : x < 0) : ⊤ * x = ⊥ 
Arguments: {x : EReal} (h : x < 0)
Goal: ⊤ * x = ⊥
-- EReal.coe_mul_bot_of_pos
lemma coe_mul_bot_of_pos {x : ℝ} (h : 0 < x) : (x : EReal) * ⊥ = ⊥ 
Arguments: {x : ℝ} (h : 0 < x)
Goal: ↑x * ⊥ = ⊥
-- EReal.coe_mul_bot_of_neg
lemma coe_mul_bot_of_neg {x : ℝ} (h : x < 0) : (x : EReal) * ⊥ = ⊤ 
Arguments: {x : ℝ} (h : x < 0)
Goal: ↑x * ⊥ = ⊤
-- EReal.bot_mul_coe_of_pos
lemma bot_mul_coe_of_pos {x : ℝ} (h : 0 < x) : (⊥ : EReal) * x = ⊥ 
Arguments: {x : ℝ} (h : 0 < x)
Goal: ⊥ * ↑x = ⊥
-- EReal.bot_mul_coe_of_neg
lemma bot_mul_coe_of_neg {x : ℝ} (h : x < 0) : (⊥ : EReal) * x = ⊤ 
Arguments: {x : ℝ} (h : x < 0)
Goal: ⊥ * ↑x = ⊤
-- EReal.mul_bot_of_pos
lemma mul_bot_of_pos : ∀ {x : EReal}, 0 < x → x * ⊥ = ⊥
  
Arguments: {x : EReal}
Goal: 0 < x → x * ⊥ = ⊥
-- EReal.mul_bot_of_neg
lemma mul_bot_of_neg : ∀ {x : EReal}, x < 0 → x * ⊥ = ⊤
  
Arguments: {x : EReal}
Goal: x < 0 → x * ⊥ = ⊤
-- EReal.bot_mul_of_pos
lemma bot_mul_of_pos {x : EReal} (h : 0 < x) : ⊥ * x = ⊥ 
Arguments: {x : EReal} (h : 0 < x)
Goal: ⊥ * x = ⊥
-- EReal.bot_mul_of_neg
lemma bot_mul_of_neg {x : EReal} (h : x < 0) : ⊥ * x = ⊤ 
Arguments: {x : EReal} (h : x < 0)
Goal: ⊥ * x = ⊤
-- EReal.toReal_mul
lemma toReal_mul {x y : EReal} : toReal (x * y) = toReal x * toReal y 
Arguments: {x y : EReal}
Goal: (x * y).toReal = x.toReal * y.toReal