ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_egauge_iff
lemma le_egauge_iff : r ≤ egauge 𝕜 s x ↔ ∀ c : 𝕜, x ∈ c • s → r ≤ ‖c‖₊ 
Arguments: {𝕜 : Type u_1} [NNNorm 𝕜] {E : Type u_2} [SMul 𝕜 E] {s : Set E} {x : E} {r : ℝ≥0∞}
Goal: r ≤ egauge 𝕜 s x ↔ ∀ (c : 𝕜), x ∈ c • s → r ≤ ↑‖c‖₊
-- egauge_eq_top
lemma egauge_eq_top : egauge 𝕜 s x = ∞ ↔ ∀ c : 𝕜, x ∉ c • s 
Arguments: {𝕜 : Type u_1} [NNNorm 𝕜] {E : Type u_2} [SMul 𝕜 E] {s : Set E} {x : E}
Goal: egauge 𝕜 s x = ⊤ ↔ ∀ (c : 𝕜), x ∉ c • s
-- egauge_lt_iff
lemma egauge_lt_iff : egauge 𝕜 s x < r ↔ ∃ c : 𝕜, x ∈ c • s ∧ ‖c‖₊ < r 
Arguments: {𝕜 : Type u_1} [NNNorm 𝕜] {E : Type u_2} [SMul 𝕜 E] {s : Set E} {x : E} {r : ℝ≥0∞}
Goal: egauge 𝕜 s x < r ↔ ∃ c, x ∈ c • s ∧ ↑‖c‖₊ < r
-- egauge_zero_left_eq_top
@[simp] lemma egauge_zero_left_eq_top : egauge 𝕜 0 x = ∞ ↔ x ≠ 0 
Arguments: (𝕜 : Type u_1) [NNNorm 𝕜] [Nonempty 𝕜] {E : Type u_2} [Zero E] [SMulZeroClass 𝕜 E] {x : E}
Goal: egauge 𝕜 0 x = ⊤ ↔ x ≠ 0
-- egauge_le_of_smul_mem_of_ne
lemma egauge_le_of_smul_mem_of_ne (h : c • x ∈ s) (hc : c ≠ 0) :
    egauge 𝕜 s x ≤ ↑(‖c‖₊⁻¹ : ℝ≥0) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {c : 𝕜} {s : Set E} {x : E} (h : c • x ∈ s) (hc : c ≠ 0)
Goal: egauge 𝕜 s x ≤ ↑‖c‖₊⁻¹
-- egauge_le_of_smul_mem
lemma egauge_le_of_smul_mem (h : c • x ∈ s) : egauge 𝕜 s x ≤ (‖c‖₊ : ℝ≥0∞)⁻¹ 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {c : 𝕜} {s : Set E} {x : E} (h : c • x ∈ s)
Goal: egauge 𝕜 s x ≤ (↑‖c‖₊)⁻¹
-- mem_of_egauge_lt_one
lemma mem_of_egauge_lt_one (hs : Balanced 𝕜 s) (hx : egauge 𝕜 s x < 1) : x ∈ s 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hs : Balanced 𝕜 s) (hx : egauge 𝕜 s x < 1)
Goal: x ∈ s
-- egauge_zero_right
@[simp]
lemma egauge_zero_right (hs : s.Nonempty) : egauge 𝕜 s 0 = 0 
Arguments: (𝕜 : Type u_1) [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {s : Set E} (hs : s.Nonempty)
Goal: egauge 𝕜 s 0 = 0
-- egauge_zero_zero
@[simp]
lemma egauge_zero_zero : egauge 𝕜 (0 : Set E) 0 = 0 
Arguments: (𝕜 : Type u_1) [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E]
Goal: egauge 𝕜 0 0 = 0
-- egauge_le_one
lemma egauge_le_one (h : x ∈ s) : egauge 𝕜 s x ≤ 1 
Arguments: (𝕜 : Type u_1) [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (h : x ∈ s)
Goal: egauge 𝕜 s x ≤ 1
-- le_egauge_smul_left
lemma le_egauge_smul_left (c : 𝕜) (s : Set E) (x : E) :
    egauge 𝕜 s x / ‖c‖₊ ≤ egauge 𝕜 (c • s) x 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] (c : 𝕜) (s : Set E) (x : E)
Goal: egauge 𝕜 s x / ↑‖c‖₊ ≤ egauge 𝕜 (c • s) x
-- egauge_smul_left
lemma egauge_smul_left (hc : c ≠ 0) (s : Set E) (x : E) :
    egauge 𝕜 (c • s) x = egauge 𝕜 s x / ‖c‖₊ 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {c : 𝕜} (hc : c ≠ 0) (s : Set E) (x : E)
Goal: egauge 𝕜 (c • s) x = egauge 𝕜 s x / ↑‖c‖₊
-- le_egauge_smul_right
lemma le_egauge_smul_right (c : 𝕜) (s : Set E) (x : E) :
    ‖c‖₊ * egauge 𝕜 s x ≤ egauge 𝕜 s (c • x) 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] (c : 𝕜) (s : Set E) (x : E)
Goal: ↑‖c‖₊ * egauge 𝕜 s x ≤ egauge 𝕜 s (c • x)
-- egauge_smul_right
lemma egauge_smul_right (h : c = 0 → s.Nonempty) (x : E) :
    egauge 𝕜 s (c • x) = ‖c‖₊ * egauge 𝕜 s x 
Arguments: {𝕜 : Type u_1} [NormedDivisionRing 𝕜] {E : Type u_3} [AddCommGroup E] [Module 𝕜 E] {c : 𝕜} {s : Set E} (h : c = 0 → s.Nonempty) (x : E)
Goal: egauge 𝕜 s (c • x) = ↑‖c‖₊ * egauge 𝕜 s x
-- div_le_egauge_closedBall
lemma div_le_egauge_closedBall (r : ℝ≥0) (x : E) : ‖x‖₊ / r ≤ egauge 𝕜 (closedBall 0 r) x 
Arguments: (𝕜 : Type u_1) [NormedField 𝕜] {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (r : ℝ≥0) (x : E)
Goal: ↑‖x‖₊ / ↑r ≤ egauge 𝕜 (Metric.closedBall 0 ↑r) x
-- le_egauge_closedBall_one
lemma le_egauge_closedBall_one (x : E) : ‖x‖₊ ≤ egauge 𝕜 (closedBall 0 1) x 
Arguments: (𝕜 : Type u_1) [NormedField 𝕜] {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ↑‖x‖₊ ≤ egauge 𝕜 (Metric.closedBall 0 1) x
-- div_le_egauge_ball
lemma div_le_egauge_ball (r : ℝ≥0) (x : E) : ‖x‖₊ / r ≤ egauge 𝕜 (ball 0 r) x 
Arguments: (𝕜 : Type u_1) [NormedField 𝕜] {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (r : ℝ≥0) (x : E)
Goal: ↑‖x‖₊ / ↑r ≤ egauge 𝕜 (Metric.ball 0 ↑r) x
-- le_egauge_ball_one
lemma le_egauge_ball_one (x : E) : ‖x‖₊ ≤ egauge 𝕜 (ball 0 1) x 
Arguments: (𝕜 : Type u_1) [NormedField 𝕜] {E : Type u_3} [SeminormedAddCommGroup E] [NormedSpace 𝕜 E] (x : E)
Goal: ↑‖x‖₊ ≤ egauge 𝕜 (Metric.ball 0 1) x
-- egauge_ball_le_of_one_lt_norm
lemma egauge_ball_le_of_one_lt_norm (hc : 1 < ‖c‖) (h₀ : r ≠ 0 ∨ x ≠ 0) :
    egauge 𝕜 (ball 0 r) x ≤ ‖c‖₊ * ‖x‖₊ / r 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} {x : E} {r : ℝ≥0} (hc : 1 < ‖c‖) (h₀ : r ≠ 0 ∨ x ≠ 0)
Goal: egauge 𝕜 (Metric.ball 0 ↑r) x ≤ ↑‖c‖₊ * ↑‖x‖₊ / ↑r
-- egauge_ball_one_le_of_one_lt_norm
lemma egauge_ball_one_le_of_one_lt_norm (hc : 1 < ‖c‖) (x : E) :
    egauge 𝕜 (ball 0 1) x ≤ ‖c‖₊ * ‖x‖₊ 
Arguments: {𝕜 : Type u_1} [NormedField 𝕜] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {c : 𝕜} (hc : 1 < ‖c‖) (x : E)
Goal: egauge 𝕜 (Metric.ball 0 1) x ≤ ↑‖c‖₊ * ↑‖x‖₊
-- convex_halfspace_re_lt
theorem convex_halfspace_re_lt (r : ℝ) : Convex ℝ { c : ℂ | c.re < r } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | c.re < r}
-- convex_halfspace_re_le
theorem convex_halfspace_re_le (r : ℝ) : Convex ℝ { c : ℂ | c.re ≤ r } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | c.re ≤ r}
-- convex_halfspace_re_gt
theorem convex_halfspace_re_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.re } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | r < c.re}
-- convex_halfspace_re_ge
theorem convex_halfspace_re_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.re } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | r ≤ c.re}
-- convex_halfspace_im_lt
theorem convex_halfspace_im_lt (r : ℝ) : Convex ℝ { c : ℂ | c.im < r } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | c.im < r}
-- convex_halfspace_im_le
theorem convex_halfspace_im_le (r : ℝ) : Convex ℝ { c : ℂ | c.im ≤ r } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | c.im ≤ r}
-- convex_halfspace_im_gt
theorem convex_halfspace_im_gt (r : ℝ) : Convex ℝ { c : ℂ | r < c.im } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | r < c.im}
-- convex_halfspace_im_ge
theorem convex_halfspace_im_ge (r : ℝ) : Convex ℝ { c : ℂ | r ≤ c.im } 
Arguments: (r : ℝ)
Goal: Convex ℝ {c | r ≤ c.im}
-- MonotoneOn.convexOn_of_deriv
theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
    (hf'_mono : MonotoneOn (deriv f) (interior D)) : ConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (hf'_mono : MonotoneOn (deriv f) (interior D))
Goal: ConvexOn ℝ D f
-- AntitoneOn.concaveOn_of_deriv
theorem AntitoneOn.concaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D))
    (h_anti : AntitoneOn (deriv f) (interior D)) : ConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (h_anti : AntitoneOn (deriv f) (interior D))
Goal: ConcaveOn ℝ D f
-- StrictMonoOn.exists_slope_lt_deriv_aux
theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
    ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a 
Arguments: {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Set.Icc x y)) (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Set.Ioo x y)) (h : ∀ w ∈ Set.Ioo x y, deriv f w ≠ 0)
Goal: ∃ a ∈ Set.Ioo x y, (f y - f x) / (y - x) < deriv f a
-- StrictMonoOn.exists_slope_lt_deriv
theorem StrictMonoOn.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
    ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a 
Arguments: {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Set.Icc x y)) (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Set.Ioo x y))
Goal: ∃ a ∈ Set.Ioo x y, (f y - f x) / (y - x) < deriv f a
-- StrictMonoOn.exists_deriv_lt_slope_aux
theorem StrictMonoOn.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
    ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) 
Arguments: {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Set.Icc x y)) (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Set.Ioo x y)) (h : ∀ w ∈ Set.Ioo x y, deriv f w ≠ 0)
Goal: ∃ a ∈ Set.Ioo x y, deriv f a < (f y - f x) / (y - x)
-- StrictMonoOn.exists_deriv_lt_slope
theorem StrictMonoOn.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
    (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) :
    ∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) 
Arguments: {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Set.Icc x y)) (hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Set.Ioo x y))
Goal: ∃ a ∈ Set.Ioo x y, deriv f a < (f y - f x) / (y - x)
-- StrictMonoOn.strictConvexOn_of_deriv
theorem StrictMonoOn.strictConvexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : StrictMonoOn (deriv f) (interior D)) : StrictConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : StrictMonoOn (deriv f) (interior D))
Goal: StrictConvexOn ℝ D f
-- StrictAntiOn.strictConcaveOn_of_deriv
theorem StrictAntiOn.strictConcaveOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (h_anti : StrictAntiOn (deriv f) (interior D)) :
    StrictConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (h_anti : StrictAntiOn (deriv f) (interior D))
Goal: StrictConcaveOn ℝ D f
-- Monotone.convexOn_univ_of_deriv
theorem Monotone.convexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
    (hf'_mono : Monotone (deriv f)) : ConvexOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf'_mono : Monotone (deriv f))
Goal: ConvexOn ℝ Set.univ f
-- Antitone.concaveOn_univ_of_deriv
theorem Antitone.concaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Differentiable ℝ f)
    (hf'_anti : Antitone (deriv f)) : ConcaveOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Differentiable ℝ f) (hf'_anti : Antitone (deriv f))
Goal: ConcaveOn ℝ Set.univ f
-- StrictMono.strictConvexOn_univ_of_deriv
theorem StrictMono.strictConvexOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continuous f)
    (hf'_mono : StrictMono (deriv f)) : StrictConvexOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Continuous f) (hf'_mono : StrictMono (deriv f))
Goal: StrictConvexOn ℝ Set.univ f
-- StrictAnti.strictConcaveOn_univ_of_deriv
theorem StrictAnti.strictConcaveOn_univ_of_deriv {f : ℝ → ℝ} (hf : Continuous f)
    (hf'_anti : StrictAnti (deriv f)) : StrictConcaveOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Continuous f) (hf'_anti : StrictAnti (deriv f))
Goal: StrictConcaveOn ℝ Set.univ f
-- convexOn_of_deriv2_nonneg
theorem convexOn_of_deriv2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
    (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
    (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x) : ConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D)) (hf''_nonneg : ∀ x ∈ interior D, 0 ≤ deriv^[2] f x)
Goal: ConvexOn ℝ D f
-- concaveOn_of_deriv2_nonpos
theorem concaveOn_of_deriv2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D)
    (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D))
    (hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (interior D)) (hf'' : DifferentiableOn ℝ (deriv f) (interior D)) (hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0)
Goal: ConcaveOn ℝ D f
-- convexOn_of_hasDerivWithinAt2_nonneg
lemma convexOn_of_hasDerivWithinAt2_nonneg {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
    (hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x) : ConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x) (hf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x)
Goal: ConvexOn ℝ D f
-- concaveOn_of_hasDerivWithinAt2_nonpos
lemma concaveOn_of_hasDerivWithinAt2_nonpos {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x)
    (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x)
    (hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0) : ConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f f' f'' : ℝ → ℝ} (hf : ContinuousOn f D) (hf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x) (hf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x) (hf''₀ : ∀ x ∈ interior D, f'' x ≤ 0)
Goal: ConcaveOn ℝ D f
-- strictConvexOn_of_deriv2_pos
theorem strictConvexOn_of_deriv2_pos {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < (deriv^[2] f) x) :
    StrictConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x)
Goal: StrictConvexOn ℝ D f
-- strictConcaveOn_of_deriv2_neg
theorem strictConcaveOn_of_deriv2_neg {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0) :
    StrictConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0)
Goal: StrictConcaveOn ℝ D f
-- convexOn_of_deriv2_nonneg'
theorem convexOn_of_deriv2_nonneg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
    (hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f) x) : ConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D) (hf''_nonneg : ∀ x ∈ D, 0 ≤ deriv^[2] f x)
Goal: ConvexOn ℝ D f
-- concaveOn_of_deriv2_nonpos'
theorem concaveOn_of_deriv2_nonpos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D)
    (hf''_nonpos : ∀ x ∈ D, deriv^[2] f x ≤ 0) : ConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf' : DifferentiableOn ℝ f D) (hf'' : DifferentiableOn ℝ (deriv f) D) (hf''_nonpos : ∀ x ∈ D, deriv^[2] f x ≤ 0)
Goal: ConcaveOn ℝ D f
-- strictConvexOn_of_deriv2_pos'
theorem strictConvexOn_of_deriv2_pos' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, 0 < deriv^[2] f x)
Goal: StrictConvexOn ℝ D f
-- strictConcaveOn_of_deriv2_neg'
theorem strictConcaveOn_of_deriv2_neg' {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
    (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, deriv^[2] f x < 0) : StrictConcaveOn ℝ D f 
Arguments: {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ} (hf : ContinuousOn f D) (hf'' : ∀ x ∈ D, deriv^[2] f x < 0)
Goal: StrictConcaveOn ℝ D f
-- convexOn_univ_of_deriv2_nonneg
theorem convexOn_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f) x) :
    ConvexOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf' : Differentiable ℝ f) (hf'' : Differentiable ℝ (deriv f)) (hf''_nonneg : ∀ (x : ℝ), 0 ≤ deriv^[2] f x)
Goal: ConvexOn ℝ Set.univ f
-- concaveOn_univ_of_deriv2_nonpos
theorem concaveOn_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : Differentiable ℝ f)
    (hf'' : Differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, deriv^[2] f x ≤ 0) :
    ConcaveOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf' : Differentiable ℝ f) (hf'' : Differentiable ℝ (deriv f)) (hf''_nonpos : ∀ (x : ℝ), deriv^[2] f x ≤ 0)
Goal: ConcaveOn ℝ Set.univ f
-- strictConvexOn_univ_of_deriv2_pos
theorem strictConvexOn_univ_of_deriv2_pos {f : ℝ → ℝ} (hf : Continuous f)
    (hf'' : ∀ x, 0 < (deriv^[2] f) x) : StrictConvexOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Continuous f) (hf'' : ∀ (x : ℝ), 0 < deriv^[2] f x)
Goal: StrictConvexOn ℝ Set.univ f
-- strictConcaveOn_univ_of_deriv2_neg
theorem strictConcaveOn_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : Continuous f)
    (hf'' : ∀ x, deriv^[2] f x < 0) : StrictConcaveOn ℝ univ f 
Arguments: {f : ℝ → ℝ} (hf : Continuous f) (hf'' : ∀ (x : ℝ), deriv^[2] f x < 0)
Goal: StrictConcaveOn ℝ Set.univ f
-- ConvexOn.slope_mono
lemma ConvexOn.slope_mono (hfc : ConvexOn 𝕜 s f) (hx : x ∈ s) : MonotoneOn (slope f x) (s \ {x}) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} {x : 𝕜} (hfc : ConvexOn 𝕜 s f) (hx : x ∈ s)
Goal: MonotoneOn (slope f x) (s \ {x})
-- ConcaveOn.slope_anti
lemma ConcaveOn.slope_anti (hfc : ConcaveOn 𝕜 s f) (hx : x ∈ s) :
    AntitoneOn (slope f x) (s \ {x}) 
Arguments: {𝕜 : Type u_1} [LinearOrderedField 𝕜] {s : Set 𝕜} {f : 𝕜 → 𝕜} {x : 𝕜} (hfc : ConcaveOn 𝕜 s f) (hx : x ∈ s)
Goal: AntitoneOn (slope f x) (s \ {x})
-- ConvexOn.le_slope_of_hasDerivWithinAt_Ioi
lemma le_slope_of_hasDerivWithinAt_Ioi (hfc : ConvexOn ℝ S f)
    (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Ioi x) x) :
    f' ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Set.Ioi x) x)
Goal: f' ≤ slope f x y
-- ConvexOn.right_deriv_le_slope
lemma right_deriv_le_slope (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableWithinAt ℝ f (Ioi x) x) :
    derivWithin f (Ioi x) x ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableWithinAt ℝ f (Set.Ioi x) x)
Goal: derivWithin f (Set.Ioi x) x ≤ slope f x y
-- ConvexOn.le_slope_of_hasDerivWithinAt
lemma le_slope_of_hasDerivWithinAt (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hf' : HasDerivWithinAt f f' S x) :
    f' ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' S x)
Goal: f' ≤ slope f x y
-- ConvexOn.derivWithin_le_slope
lemma derivWithin_le_slope (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableWithinAt ℝ f S x) :
    derivWithin f S x ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableWithinAt ℝ f S x)
Goal: derivWithin f S x ≤ slope f x y
-- ConvexOn.le_slope_of_hasDerivAt
lemma le_slope_of_hasDerivAt (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (ha : HasDerivAt f f' x) :
    f' ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (ha : HasDerivAt f f' x)
Goal: f' ≤ slope f x y
-- ConvexOn.deriv_le_slope
lemma deriv_le_slope (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableAt ℝ f x) :
    deriv f x ≤ slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableAt ℝ f x)
Goal: deriv f x ≤ slope f x y
-- ConvexOn.slope_le_of_hasDerivWithinAt_Iio
lemma slope_le_of_hasDerivWithinAt_Iio (hfc : ConvexOn ℝ S f)
    (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Iio y) y) :
    slope f x y ≤ f' 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Set.Iio y) y)
Goal: slope f x y ≤ f'
-- ConvexOn.slope_le_left_deriv
lemma slope_le_left_deriv (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableWithinAt ℝ f (Iio y) y) :
    slope f x y ≤ derivWithin f (Iio y) y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableWithinAt ℝ f (Set.Iio y) y)
Goal: slope f x y ≤ derivWithin f (Set.Iio y) y
-- ConvexOn.slope_le_of_hasDerivWithinAt
lemma slope_le_of_hasDerivWithinAt (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hf' : HasDerivWithinAt f f' S y) :
    slope f x y ≤ f' 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' S y)
Goal: slope f x y ≤ f'
-- ConvexOn.slope_le_derivWithin
lemma slope_le_derivWithin (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableWithinAt ℝ f S y) :
    slope f x y ≤ derivWithin f S y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableWithinAt ℝ f S y)
Goal: slope f x y ≤ derivWithin f S y
-- ConvexOn.slope_le_of_hasDerivAt
lemma slope_le_of_hasDerivAt (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hf' : HasDerivAt f f' y) :
    slope f x y ≤ f' 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivAt f f' y)
Goal: slope f x y ≤ f'
-- ConvexOn.slope_le_deriv
lemma slope_le_deriv (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableAt ℝ f y) :
    slope f x y ≤ deriv f y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : ConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableAt ℝ f y)
Goal: slope f x y ≤ deriv f y
-- ConvexOn.monotoneOn_derivWithin
lemma monotoneOn_derivWithin (hfc : ConvexOn ℝ S f) (hfd : DifferentiableOn ℝ f S) :
    MonotoneOn (derivWithin f S) S 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} (hfc : ConvexOn ℝ S f) (hfd : DifferentiableOn ℝ f S)
Goal: MonotoneOn (derivWithin f S) S
-- ConvexOn.monotoneOn_deriv
theorem monotoneOn_deriv (hfc : ConvexOn ℝ S f) (hfd : ∀ x ∈ S, DifferentiableAt ℝ f x) :
    MonotoneOn (deriv f) S 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} (hfc : ConvexOn ℝ S f) (hfd : ∀ x ∈ S, DifferentiableAt ℝ f x)
Goal: MonotoneOn (deriv f) S
-- StrictConvexOn.lt_slope_of_hasDerivWithinAt_Ioi
lemma lt_slope_of_hasDerivWithinAt_Ioi (hfc : StrictConvexOn ℝ S f)
    (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Ioi x) x) :
    f' < slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : StrictConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' (Set.Ioi x) x)
Goal: f' < slope f x y
-- StrictConvexOn.right_deriv_lt_slope
lemma right_deriv_lt_slope (hfc : StrictConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y)
    (hfd : DifferentiableWithinAt ℝ f (Ioi x) x) :
    derivWithin f (Ioi x) x < slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y : ℝ} (hfc : StrictConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hfd : DifferentiableWithinAt ℝ f (Set.Ioi x) x)
Goal: derivWithin f (Set.Ioi x) x < slope f x y
-- StrictConvexOn.lt_slope_of_hasDerivWithinAt
lemma lt_slope_of_hasDerivWithinAt (hfc : StrictConvexOn ℝ S f)
    (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' S x) :
    f' < slope f x y 
Arguments: {S : Set ℝ} {f : ℝ → ℝ} {x y f' : ℝ} (hfc : StrictConvexOn ℝ S f) (hx : x ∈ S) (hy : y ∈ S) (hxy : x < y) (hf' : HasDerivWithinAt f f' S x)
Goal: f' < slope f x y