ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Convex.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt
theorem exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt (hs : Convex ℝ s) {f : E → G}
    (hder : ∀ᶠ y in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x) :
    ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {s : Set E} {x : E} {f' : E → E →L[𝕜] G} (hs : Convex ℝ s) {f : E → G} (hder : ∀ᶠ (y : E) in 𝓝[s] x, HasFDerivWithinAt f (f' y) s y) (hcont : ContinuousWithinAt f' s x)
Goal: ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t
-- Convex.norm_image_sub_le_of_norm_fderivWithin_le
theorem norm_image_sub_le_of_norm_fderivWithin_le (hf : DifferentiableOn 𝕜 f s)
    (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_fderivWithin_le
theorem lipschitzOnWith_of_nnnorm_fderivWithin_le {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s)
    (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {s : Set E} {C : ℝ≥0} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖₊ ≤ C) (hs : Convex ℝ s)
Goal: LipschitzOnWith C f s
-- Convex.norm_image_sub_le_of_norm_fderiv_le
theorem norm_image_sub_le_of_norm_fderiv_le (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
    (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_fderiv_le
theorem lipschitzOnWith_of_nnnorm_fderiv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
    (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {s : Set E} {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x‖₊ ≤ C) (hs : Convex ℝ s)
Goal: LipschitzOnWith C f s
-- lipschitzWith_of_nnnorm_fderiv_le
theorem _root_.lipschitzWith_of_nnnorm_fderiv_le
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
    {C : ℝ≥0} (hf : Differentiable 𝕜 f)
    (bound : ∀ x, ‖fderiv 𝕜 f x‖₊ ≤ C) : LipschitzWith C f 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ (x : E), ‖fderiv 𝕜 f x‖₊ ≤ C)
Goal: LipschitzWith C f
-- Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le'
theorem norm_image_sub_le_of_norm_hasFDerivWithin_le'
    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C)
    (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} {f' : E → E →L[𝕜] G} {φ : E →L[𝕜] G} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖
-- Convex.norm_image_sub_le_of_norm_fderivWithin_le'
theorem norm_image_sub_le_of_norm_fderivWithin_le' (hf : DifferentiableOn 𝕜 f s)
    (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} {φ : E →L[𝕜] G} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖
-- Convex.norm_image_sub_le_of_norm_fderiv_le'
theorem norm_image_sub_le_of_norm_fderiv_le' (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
    (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖ 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {C : ℝ} {s : Set E} {x y : E} {φ : E →L[𝕜] G} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖fderiv 𝕜 f x - φ‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x - φ (y - x)‖ ≤ C * ‖y - x‖
-- Convex.is_const_of_fderivWithin_eq_zero
theorem is_const_of_fderivWithin_eq_zero (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
    (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s) : f x = f y 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → G} {s : Set E} {x y : E} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = 0) (hx : x ∈ s) (hy : y ∈ s)
Goal: f x = f y
-- is_const_of_fderiv_eq_zero
theorem _root_.is_const_of_fderiv_eq_zero
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G}
    (hf : Differentiable 𝕜 f) (hf' : ∀ x, fderiv 𝕜 f x = 0)
    (x y : E) : f x = f y 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f : E → G} (hf : Differentiable 𝕜 f) (hf' : ∀ (x : E), fderiv 𝕜 f x = 0) (x y : E)
Goal: f x = f y
-- Convex.eqOn_of_fderivWithin_eq
theorem eqOn_of_fderivWithin_eq (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s)
    (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s)
    (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x) :
    s.EqOn f g 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f g : E → G} {s : Set E} {x : E} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (hg : DifferentiableOn 𝕜 g s) (hs' : UniqueDiffOn 𝕜 s) (hf' : ∀ x ∈ s, fderivWithin 𝕜 f s x = fderivWithin 𝕜 g s x) (hx : x ∈ s) (hfgx : f x = g x)
Goal: Set.EqOn f g s
-- eq_of_fderiv_eq
theorem _root_.eq_of_fderiv_eq
    {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G}
    (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g)
    (hf' : ∀ x, fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x) : f = g 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {E : Type u_5} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {f g : E → G} (hf : Differentiable 𝕜 f) (hg : Differentiable 𝕜 g) (hf' : ∀ (x : E), fderiv 𝕜 f x = fderiv 𝕜 g x) (x : E) (hfgx : f x = g x)
Goal: f = g
-- Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
theorem norm_image_sub_le_of_norm_hasDerivWithin_le {C : ℝ}
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s)
    (xs : x ∈ s) (ys : y ∈ s) : ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f f' : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} {C : ℝ} (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le
theorem lipschitzOnWith_of_nnnorm_hasDerivWithin_le {C : ℝ≥0} (hs : Convex ℝ s)
    (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C) :
    LipschitzOnWith C f s 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f f' : 𝕜 → G} {s : Set 𝕜} {C : ℝ≥0} (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) (bound : ∀ x ∈ s, ‖f' x‖₊ ≤ C)
Goal: LipschitzOnWith C f s
-- Convex.norm_image_sub_le_of_norm_derivWithin_le
theorem norm_image_sub_le_of_norm_derivWithin_le {C : ℝ} (hf : DifferentiableOn 𝕜 f s)
    (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} {C : ℝ} (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_derivWithin_le
theorem lipschitzOnWith_of_nnnorm_derivWithin_le {C : ℝ≥0} (hs : Convex ℝ s)
    (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C) :
    LipschitzOnWith C f s 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} {s : Set 𝕜} {C : ℝ≥0} (hs : Convex ℝ s) (hf : DifferentiableOn 𝕜 f s) (bound : ∀ x ∈ s, ‖derivWithin f s x‖₊ ≤ C)
Goal: LipschitzOnWith C f s
-- Convex.norm_image_sub_le_of_norm_deriv_le
theorem norm_image_sub_le_of_norm_deriv_le {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
    (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ‖f y - f x‖ ≤ C * ‖y - x‖ 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} {s : Set 𝕜} {x y : 𝕜} {C : ℝ} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖ ≤ C) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ‖f y - f x‖ ≤ C * ‖y - x‖
-- Convex.lipschitzOnWith_of_nnnorm_deriv_le
theorem lipschitzOnWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x)
    (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s) : LipschitzOnWith C f s 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} {s : Set 𝕜} {C : ℝ≥0} (hf : ∀ x ∈ s, DifferentiableAt 𝕜 f x) (bound : ∀ x ∈ s, ‖deriv f x‖₊ ≤ C) (hs : Convex ℝ s)
Goal: LipschitzOnWith C f s
-- lipschitzWith_of_nnnorm_deriv_le
theorem _root_.lipschitzWith_of_nnnorm_deriv_le {C : ℝ≥0} (hf : Differentiable 𝕜 f)
    (bound : ∀ x, ‖deriv f x‖₊ ≤ C) : LipschitzWith C f 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} {C : ℝ≥0} (hf : Differentiable 𝕜 f) (bound : ∀ (x : 𝕜), ‖deriv f x‖₊ ≤ C)
Goal: LipschitzWith C f
-- is_const_of_deriv_eq_zero
theorem _root_.is_const_of_deriv_eq_zero (hf : Differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0)
    (x y : 𝕜) : f x = f y 
Arguments: {𝕜 : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : 𝕜 → G} (hf : Differentiable 𝕜 f) (hf' : ∀ (x : 𝕜), deriv f x = 0) (x y : 𝕜)
Goal: f x = f y
-- exists_ratio_hasDerivAt_eq_ratio_slope
theorem exists_ratio_hasDerivAt_eq_ratio_slope :
    ∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c 
Arguments: (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (g g' : ℝ → ℝ) (hgc : ContinuousOn g (Set.Icc a b)) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x)
Goal: ∃ c ∈ Set.Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c
-- exists_ratio_hasDerivAt_eq_ratio_slope'
theorem exists_ratio_hasDerivAt_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
    (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga))
    (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) :
    ∃ c ∈ Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c 
Arguments: (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (g g' : ℝ → ℝ) {lfa lga lfb lgb : ℝ} (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x) (hgg' : ∀ x ∈ Set.Ioo a b, HasDerivAt g (g' x) x) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Filter.Tendsto g (𝓝[<] b) (𝓝 lgb))
Goal: ∃ c ∈ Set.Ioo a b, (lgb - lga) * f' c = (lfb - lfa) * g' c
-- exists_hasDerivAt_eq_slope
theorem exists_hasDerivAt_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a) 
Arguments: (f f' : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hff' : ∀ x ∈ Set.Ioo a b, HasDerivAt f (f' x) x)
Goal: ∃ c ∈ Set.Ioo a b, f' c = (f b - f a) / (b - a)
-- exists_ratio_deriv_eq_ratio_slope
theorem exists_ratio_deriv_eq_ratio_slope :
    ∃ c ∈ Ioo a b, (g b - g a) * deriv f c = (f b - f a) * deriv g c 
Arguments: (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfd : DifferentiableOn ℝ f (Set.Ioo a b)) (g : ℝ → ℝ) (hgc : ContinuousOn g (Set.Icc a b)) (hgd : DifferentiableOn ℝ g (Set.Ioo a b))
Goal: ∃ c ∈ Set.Ioo a b, (g b - g a) * deriv f c = (f b - f a) * deriv g c
-- exists_ratio_deriv_eq_ratio_slope'
theorem exists_ratio_deriv_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
    (hdf : DifferentiableOn ℝ f <| Ioo a b) (hdg : DifferentiableOn ℝ g <| Ioo a b)
    (hfa : Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Tendsto g (𝓝[>] a) (𝓝 lga))
    (hfb : Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Tendsto g (𝓝[<] b) (𝓝 lgb)) :
    ∃ c ∈ Ioo a b, (lgb - lga) * deriv f c = (lfb - lfa) * deriv g c 
Arguments: (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (g : ℝ → ℝ) {lfa lga lfb lgb : ℝ} (hdf : DifferentiableOn ℝ f (Set.Ioo a b)) (hdg : DifferentiableOn ℝ g (Set.Ioo a b)) (hfa : Filter.Tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : Filter.Tendsto g (𝓝[>] a) (𝓝 lga)) (hfb : Filter.Tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : Filter.Tendsto g (𝓝[<] b) (𝓝 lgb))
Goal: ∃ c ∈ Set.Ioo a b, (lgb - lga) * deriv f c = (lfb - lfa) * deriv g c
-- exists_deriv_eq_slope
theorem exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) 
Arguments: (f : ℝ → ℝ) {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Set.Icc a b)) (hfd : DifferentiableOn ℝ f (Set.Ioo a b))
Goal: ∃ c ∈ Set.Ioo a b, deriv f c = (f b - f a) / (b - a)
-- Convex.mul_sub_lt_image_sub_of_lt_deriv
theorem Convex.mul_sub_lt_image_sub_of_lt_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C}
    (hf'_gt : ∀ x ∈ interior D, C < deriv f x) :
    ∀ᵉ (x ∈ D) (y ∈ D), x < y → C * (y - x) < f y - f x 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C : ℝ} (hf'_gt : ∀ x ∈ interior D, C < deriv f x) (x : ℝ)
Goal: x ∈ D → ∀ y ∈ D, x < y → C * (y - x) < f y - f x
-- mul_sub_lt_image_sub_of_lt_deriv
theorem mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
    (hf'_gt : ∀ x, C < deriv f x) ⦃x y⦄ (hxy : x < y) : C * (y - x) < f y - f x 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C : ℝ} (hf'_gt : ∀ (x : ℝ), C < deriv f x) ⦃x y : ℝ⦄ (hxy : x < y)
Goal: C * (y - x) < f y - f x
-- Convex.mul_sub_le_image_sub_of_le_deriv
theorem Convex.mul_sub_le_image_sub_of_le_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C}
    (hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x) :
    ∀ᵉ (x ∈ D) (y ∈ D), x ≤ y → C * (y - x) ≤ f y - f x 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C : ℝ} (hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x) (x : ℝ)
Goal: x ∈ D → ∀ y ∈ D, x ≤ y → C * (y - x) ≤ f y - f x
-- mul_sub_le_image_sub_of_le_deriv
theorem mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
    (hf'_ge : ∀ x, C ≤ deriv f x) ⦃x y⦄ (hxy : x ≤ y) : C * (y - x) ≤ f y - f x 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C : ℝ} (hf'_ge : ∀ (x : ℝ), C ≤ deriv f x) ⦃x y : ℝ⦄ (hxy : x ≤ y)
Goal: C * (y - x) ≤ f y - f x
-- Convex.image_sub_lt_mul_sub_of_deriv_lt
theorem Convex.image_sub_lt_mul_sub_of_deriv_lt {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C}
    (lt_hf' : ∀ x ∈ interior D, deriv f x < C) (x : ℝ) (hx : x ∈ D) (y : ℝ) (hy : y ∈ D)
    (hxy : x < y) : f y - f x < C * (y - x) 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C : ℝ} (lt_hf' : ∀ x ∈ interior D, deriv f x < C) (x : ℝ) (hx : x ∈ D) (y : ℝ) (hy : y ∈ D) (hxy : x < y)
Goal: f y - f x < C * (y - x)
-- image_sub_lt_mul_sub_of_deriv_lt
theorem image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
    (lt_hf' : ∀ x, deriv f x < C) ⦃x y⦄ (hxy : x < y) : f y - f x < C * (y - x) 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C : ℝ} (lt_hf' : ∀ (x : ℝ), deriv f x < C) ⦃x y : ℝ⦄ (hxy : x < y)
Goal: f y - f x < C * (y - x)
-- Convex.image_sub_le_mul_sub_of_deriv_le
theorem Convex.image_sub_le_mul_sub_of_deriv_le {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C}
    (le_hf' : ∀ x ∈ interior D, deriv f x ≤ C) (x : ℝ) (hx : x ∈ D) (y : ℝ) (hy : y ∈ D)
    (hxy : x ≤ y) : f y - f x ≤ C * (y - x) 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) {C : ℝ} (le_hf' : ∀ x ∈ interior D, deriv f x ≤ C) (x : ℝ) (hx : x ∈ D) (y : ℝ) (hy : y ∈ D) (hxy : x ≤ y)
Goal: f y - f x ≤ C * (y - x)
-- image_sub_le_mul_sub_of_deriv_le
theorem image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C}
    (le_hf' : ∀ x, deriv f x ≤ C) ⦃x y⦄ (hxy : x ≤ y) : f y - f x ≤ C * (y - x) 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) {C : ℝ} (le_hf' : ∀ (x : ℝ), deriv f x ≤ C) ⦃x y : ℝ⦄ (hxy : x ≤ y)
Goal: f y - f x ≤ C * (y - x)
-- strictMonoOn_of_deriv_pos
theorem strictMonoOn_of_deriv_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x) : StrictMonoOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x)
Goal: StrictMonoOn f D
-- strictMono_of_deriv_pos
theorem strictMono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) : StrictMono f 
Arguments: {f : ℝ → ℝ} (hf' : ∀ (x : ℝ), 0 < deriv f x)
Goal: StrictMono f
-- strictMonoOn_of_hasDerivWithinAt_pos
lemma strictMonoOn_of_hasDerivWithinAt_pos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'₀ : ∀ x ∈ interior D, 0 < f' x) : StrictMonoOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'₀ : ∀ x ∈ interior D, 0 < f' x)
Goal: StrictMonoOn f D
-- strictMono_of_hasDerivAt_pos
lemma strictMono_of_hasDerivAt_pos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
    (hf' : ∀ x, 0 < f' x) : StrictMono f 
Arguments: {f f' : ℝ → ℝ} (hf : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : ∀ (x : ℝ), 0 < f' x)
Goal: StrictMono f
-- monotoneOn_of_deriv_nonneg
theorem monotoneOn_of_deriv_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
    (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) : MonotoneOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x)
Goal: MonotoneOn f D
-- monotone_of_deriv_nonneg
theorem monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, 0 ≤ deriv f x) :
    Monotone f 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ (x : ℝ), 0 ≤ deriv f x)
Goal: Monotone f
-- monotoneOn_of_hasDerivWithinAt_nonneg
lemma monotoneOn_of_hasDerivWithinAt_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'₀ : ∀ x ∈ interior D, 0 ≤ f' x) : MonotoneOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'₀ : ∀ x ∈ interior D, 0 ≤ f' x)
Goal: MonotoneOn f D
-- monotone_of_hasDerivAt_nonneg
lemma monotone_of_hasDerivAt_nonneg {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
    (hf' : 0 ≤ f') : Monotone f 
Arguments: {f f' : ℝ → ℝ} (hf : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : 0 ≤ f')
Goal: Monotone f
-- strictAntiOn_of_deriv_neg
theorem strictAntiOn_of_deriv_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, deriv f x < 0) : StrictAntiOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, deriv f x < 0)
Goal: StrictAntiOn f D
-- strictAnti_of_deriv_neg
theorem strictAnti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) : StrictAnti f 
Arguments: {f : ℝ → ℝ} (hf' : ∀ (x : ℝ), deriv f x < 0)
Goal: StrictAnti f
-- strictAntiOn_of_hasDerivWithinAt_neg
lemma strictAntiOn_of_hasDerivWithinAt_neg {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'₀ : ∀ x ∈ interior D, f' x < 0) : StrictAntiOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'₀ : ∀ x ∈ interior D, f' x < 0)
Goal: StrictAntiOn f D
-- strictAnti_of_hasDerivAt_neg
lemma strictAnti_of_hasDerivAt_neg {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
    (hf' : ∀ x, f' x < 0) : StrictAnti f 
Arguments: {f f' : ℝ → ℝ} (hf : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : ∀ (x : ℝ), f' x < 0)
Goal: StrictAnti f
-- antitoneOn_of_deriv_nonpos
theorem antitoneOn_of_deriv_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
    (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) : AntitoneOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0)
Goal: AntitoneOn f D
-- antitone_of_deriv_nonpos
theorem antitone_of_deriv_nonpos {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ x, deriv f x ≤ 0) :
    Antitone f 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf' : ∀ (x : ℝ), deriv f x ≤ 0)
Goal: Antitone f
-- antitoneOn_of_hasDerivWithinAt_nonpos
lemma antitoneOn_of_hasDerivWithinAt_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'₀ : ∀ x ∈ interior D, f' x ≤ 0) : AntitoneOn f D 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'₀ : ∀ x ∈ interior D, f' x ≤ 0)
Goal: AntitoneOn f D
-- antitone_of_hasDerivAt_nonpos
lemma antitone_of_hasDerivAt_nonpos {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x)
    (hf' : f' ≤ 0) : Antitone f 
Arguments: {f f' : ℝ → ℝ} (hf : ∀ (x : ℝ), HasDerivAt f (f' x) x) (hf' : f' ≤ 0)
Goal: Antitone f
-- domain_mvt
theorem domain_mvt {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ}
    (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
    ∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : E → ℝ} {s : Set E} {x y : E} {f' : E → E →L[ℝ] ℝ} (hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hs : Convex ℝ s) (xs : x ∈ s) (ys : y ∈ s)
Goal: ∃ z ∈ segment ℝ x y, f y - f x = (f' z) (y - x)