ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- circle.argPartialEquiv_apply
@[simps (config := .asFn)]
noncomputable def argPartialEquiv : PartialEquiv circle ℝ 
Goal: ↑circle.argPartialEquiv = Complex.arg ∘ Subtype.val
-- circle.argEquiv_apply_coe
@[simps (config := .asFn)]
noncomputable def argEquiv : circle ≃ Ioc (-π) π 
Arguments: (z : ↥circle)
Goal: ↑(circle.argEquiv z) = (↑z).arg
-- circle.argEquiv_symm_apply
@[simps (config := .asFn)]
noncomputable def argEquiv : circle ≃ Ioc (-π) π 
Goal: ⇑circle.argEquiv.symm = ⇑expMapCircle ∘ Subtype.val
-- leftInverse_expMapCircle_arg
theorem leftInverse_expMapCircle_arg : LeftInverse expMapCircle (arg ∘ (↑)) 
Goal: Function.LeftInverse (⇑expMapCircle) (Complex.arg ∘ Subtype.val)
-- invOn_arg_expMapCircle
theorem invOn_arg_expMapCircle : InvOn (arg ∘ (↑)) expMapCircle (Ioc (-π) π) univ 
Goal: Set.InvOn (Complex.arg ∘ Subtype.val) (⇑expMapCircle) (Set.Ioc (-π) π) Set.univ
-- surjOn_expMapCircle_neg_pi_pi
theorem surjOn_expMapCircle_neg_pi_pi : SurjOn expMapCircle (Ioc (-π) π) univ 
Goal: Set.SurjOn (⇑expMapCircle) (Set.Ioc (-π) π) Set.univ
-- expMapCircle_eq_expMapCircle
theorem expMapCircle_eq_expMapCircle {x y : ℝ} :
    expMapCircle x = expMapCircle y ↔ ∃ m : ℤ, x = y + m * (2 * π) 
Arguments: {x y : ℝ}
Goal: expMapCircle x = expMapCircle y ↔ ∃ m, x = y + ↑m * (2 * π)
-- periodic_expMapCircle
theorem periodic_expMapCircle : Periodic expMapCircle (2 * π) 
Goal: Function.Periodic (⇑expMapCircle) (2 * π)
-- expMapCircle_two_pi
@[simp]
theorem expMapCircle_two_pi : expMapCircle (2 * π) = 1 
Goal: expMapCircle (2 * π) = 1
-- expMapCircle_sub_two_pi
theorem expMapCircle_sub_two_pi (x : ℝ) : expMapCircle (x - 2 * π) = expMapCircle x 
Arguments: (x : ℝ)
Goal: expMapCircle (x - 2 * π) = expMapCircle x
-- expMapCircle_add_two_pi
theorem expMapCircle_add_two_pi (x : ℝ) : expMapCircle (x + 2 * π) = expMapCircle x 
Arguments: (x : ℝ)
Goal: expMapCircle (x + 2 * π) = expMapCircle x
-- Real.Angle.expMapCircle_coe
@[simp]
theorem Real.Angle.expMapCircle_coe (x : ℝ) : Real.Angle.expMapCircle x = _root_.expMapCircle x 
Arguments: (x : ℝ)
Goal: (↑x).expMapCircle = expMapCircle x
-- Real.Angle.coe_expMapCircle
theorem Real.Angle.coe_expMapCircle (θ : Real.Angle) :
    (θ.expMapCircle : ℂ) = θ.cos + θ.sin * I 
Arguments: (θ : Real.Angle)
Goal: ↑θ.expMapCircle = ↑θ.cos + ↑θ.sin * Complex.I
-- Real.Angle.expMapCircle_zero
@[simp]
theorem Real.Angle.expMapCircle_zero : Real.Angle.expMapCircle 0 = 1 
Goal: Real.Angle.expMapCircle 0 = 1
-- Real.Angle.expMapCircle_neg
@[simp]
theorem Real.Angle.expMapCircle_neg (θ : Real.Angle) :
    Real.Angle.expMapCircle (-θ) = (Real.Angle.expMapCircle θ)⁻¹ 
Arguments: (θ : Real.Angle)
Goal: (-θ).expMapCircle = θ.expMapCircle⁻¹
-- Real.Angle.expMapCircle_add
@[simp]
theorem Real.Angle.expMapCircle_add (θ₁ θ₂ : Real.Angle) : Real.Angle.expMapCircle (θ₁ + θ₂) =
    Real.Angle.expMapCircle θ₁ * Real.Angle.expMapCircle θ₂ 
Arguments: (θ₁ θ₂ : Real.Angle)
Goal: (θ₁ + θ₂).expMapCircle = θ₁.expMapCircle * θ₂.expMapCircle
-- Real.Angle.arg_expMapCircle
@[simp]
theorem Real.Angle.arg_expMapCircle (θ : Real.Angle) :
    (arg (Real.Angle.expMapCircle θ) : Real.Angle) = θ 
Arguments: (θ : Real.Angle)
Goal: ↑(↑θ.expMapCircle).arg = θ
-- AddCircle.scaled_exp_map_periodic
theorem scaled_exp_map_periodic : Function.Periodic (fun x => expMapCircle (2 * π / T * x)) T 
Arguments: {T : ℝ}
Goal: Function.Periodic (fun x => expMapCircle (2 * π / T * x)) T
-- AddCircle.toCircle_apply_mk
theorem toCircle_apply_mk (x : ℝ) : @toCircle T x = expMapCircle (2 * π / T * x) 
Arguments: {T : ℝ} (x : ℝ)
Goal: AddCircle.toCircle ↑x = expMapCircle (2 * π / T * x)
-- AddCircle.toCircle_add
theorem toCircle_add (x : AddCircle T) (y : AddCircle T) :
    @toCircle T (x + y) = toCircle x * toCircle y 
Arguments: {T : ℝ} (x y : AddCircle T)
Goal: (x + y).toCircle = x.toCircle * y.toCircle
-- AddCircle.continuous_toCircle
theorem continuous_toCircle : Continuous (@toCircle T) 
Arguments: {T : ℝ}
Goal: Continuous AddCircle.toCircle
-- AddCircle.injective_toCircle
theorem injective_toCircle (hT : T ≠ 0) : Function.Injective (@toCircle T) 
Arguments: {T : ℝ} (hT : T ≠ 0)
Goal: Function.Injective AddCircle.toCircle
-- AddCircle.homeomorphCircle'_apply
@[simps] noncomputable def homeomorphCircle' : AddCircle (2 * π) ≃ₜ circle 
Arguments: (θ : Real.Angle)
Goal: AddCircle.homeomorphCircle' θ = θ.expMapCircle
-- AddCircle.homeomorphCircle'_symm_apply
@[simps] noncomputable def homeomorphCircle' : AddCircle (2 * π) ≃ₜ circle 
Arguments: (x : ↥circle)
Goal: AddCircle.homeomorphCircle'.symm x = ↑(↑x).arg
-- AddCircle.homeomorphCircle'_apply_mk
theorem homeomorphCircle'_apply_mk (x : ℝ) : homeomorphCircle' x = expMapCircle x 
Arguments: (x : ℝ)
Goal: AddCircle.homeomorphCircle' ↑x = expMapCircle x
-- AddCircle.homeomorphCircle_apply
theorem homeomorphCircle_apply (hT : T ≠ 0) (x : AddCircle T) :
    homeomorphCircle hT x = toCircle x 
Arguments: {T : ℝ} (hT : T ≠ 0) (x : AddCircle T)
Goal: (AddCircle.homeomorphCircle hT) x = x.toCircle
-- isLocalHomeomorph_expMapCircle
lemma isLocalHomeomorph_expMapCircle : IsLocalHomeomorph expMapCircle 
Goal: IsLocalHomeomorph ⇑expMapCircle
-- ZMod.toCircle_intCast
lemma toCircle_intCast (j : ℤ) :
    toCircle (j : ZMod N) = exp (2 * π * I * j / N) 
Arguments: {N : ℕ} [NeZero N] (j : ℤ)
Goal: ↑(ZMod.toCircle ↑j) = cexp (2 * ↑π * Complex.I * ↑j / ↑N)
-- ZMod.toCircle_natCast
lemma toCircle_natCast (j : ℕ) :
    toCircle (j : ZMod N) = exp (2 * π * I * j / N) 
Arguments: {N : ℕ} [NeZero N] (j : ℕ)
Goal: ↑(ZMod.toCircle ↑j) = cexp (2 * ↑π * Complex.I * ↑j / ↑N)
-- ZMod.toCircle_apply
lemma toCircle_apply (j : ZMod N) :
    toCircle j = exp (2 * π * I * j.val / N) 
Arguments: {N : ℕ} [NeZero N] (j : ZMod N)
Goal: ↑(ZMod.toCircle j) = cexp (2 * ↑π * Complex.I * ↑j.val / ↑N)
-- ZMod.stdAddChar_coe
lemma stdAddChar_coe (j : ℤ) :
    stdAddChar (j : ZMod N) = exp (2 * π * I * j / N) 
Arguments: {N : ℕ} [NeZero N] (j : ℤ)
Goal: ZMod.stdAddChar ↑j = cexp (2 * ↑π * Complex.I * ↑j / ↑N)
-- ZMod.stdAddChar_apply
lemma stdAddChar_apply (j : ZMod N) : stdAddChar j = ↑(toCircle j) 
Arguments: {N : ℕ} [NeZero N] (j : ZMod N)
Goal: ZMod.stdAddChar j = ↑(ZMod.toCircle j)
-- Subadditive.lim_le_div
theorem lim_le_div (hbdd : BddBelow (range fun n => u n / n)) {n : ℕ} (hn : n ≠ 0) :
    h.lim ≤ u n / n 
Arguments: {u : ℕ → ℝ} (h : Subadditive u) (hbdd : BddBelow (Set.range fun n => u n / ↑n)) {n : ℕ} (hn : n ≠ 0)
Goal: h.lim ≤ u n / ↑n
-- Subadditive.apply_mul_add_le
theorem apply_mul_add_le (k n r) : u (k * n + r) ≤ k * u n + u r 
Arguments: {u : ℕ → ℝ} (h : Subadditive u) (k n r : ℕ)
Goal: u (k * n + r) ≤ ↑k * u n + u r
-- Subadditive.eventually_div_lt_of_div_lt
theorem eventually_div_lt_of_div_lt {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / n < L) :
    ∀ᶠ p in atTop, u p / p < L 
Arguments: {u : ℕ → ℝ} (h : Subadditive u) {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / ↑n < L)
Goal: ∀ᶠ (p : ℕ) in Filter.atTop, u p / ↑p < L
-- Subadditive.tendsto_lim
theorem tendsto_lim (hbdd : BddBelow (range fun n => u n / n)) :
    Tendsto (fun n => u n / n) atTop (𝓝 h.lim) 
Arguments: {u : ℕ → ℝ} (h : Subadditive u) (hbdd : BddBelow (Set.range fun n => u n / ↑n))
Goal: Filter.Tendsto (fun n => u n / ↑n) Filter.atTop (𝓝 h.lim)
-- Complex.isOpenMap_exp
theorem isOpenMap_exp : IsOpenMap exp 
Goal: IsOpenMap Complex.exp
-- Complex.hasStrictDerivAt_log
theorem hasStrictDerivAt_log {x : ℂ} (h : x ∈ slitPlane) : HasStrictDerivAt log x⁻¹ x 
Arguments: {x : ℂ} (h : x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt Complex.log x⁻¹ x
-- Complex.hasDerivAt_log
lemma hasDerivAt_log {z : ℂ} (hz : z ∈ slitPlane) : HasDerivAt log z⁻¹ z 
Arguments: {z : ℂ} (hz : z ∈ Complex.slitPlane)
Goal: HasDerivAt Complex.log z⁻¹ z
-- Complex.differentiableAt_log
lemma differentiableAt_log {z : ℂ} (hz : z ∈ slitPlane) : DifferentiableAt ℂ log z 
Arguments: {z : ℂ} (hz : z ∈ Complex.slitPlane)
Goal: DifferentiableAt ℂ Complex.log z
-- Complex.hasStrictFDerivAt_log_real
theorem hasStrictFDerivAt_log_real {x : ℂ} (h : x ∈ slitPlane) :
    HasStrictFDerivAt log (x⁻¹ • (1 : ℂ →L[ℝ] ℂ)) x 
Arguments: {x : ℂ} (h : x ∈ Complex.slitPlane)
Goal: HasStrictFDerivAt Complex.log (x⁻¹ • 1) x
-- Complex.contDiffAt_log
theorem contDiffAt_log {x : ℂ} (h : x ∈ slitPlane) {n : ℕ∞} : ContDiffAt ℂ n log x 
Arguments: {x : ℂ} (h : x ∈ Complex.slitPlane) {n : ℕ∞}
Goal: ContDiffAt ℂ n Complex.log x
-- HasStrictFDerivAt.clog
theorem HasStrictFDerivAt.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : HasStrictFDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasStrictFDerivAt (fun t => log (f t)) ((f x)⁻¹ • f') x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : HasStrictFDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasStrictFDerivAt (fun t => Complex.log (f t)) ((f x)⁻¹ • f') x
-- HasStrictDerivAt.clog
theorem HasStrictDerivAt.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasStrictDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasStrictDerivAt (fun t => log (f t)) (f' / f x) x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasStrictDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt (fun t => Complex.log (f t)) (f' / f x) x
-- HasStrictDerivAt.clog_real
theorem HasStrictDerivAt.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : HasStrictDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasStrictDerivAt (fun t => log (f t)) (f' / f x) x 
Arguments: {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : HasStrictDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasStrictDerivAt (fun t => Complex.log (f t)) (f' / f x) x
-- HasFDerivAt.clog
theorem HasFDerivAt.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : HasFDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasFDerivAt (fun t => log (f t)) ((f x)⁻¹ • f') x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {x : E} (h₁ : HasFDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasFDerivAt (fun t => Complex.log (f t)) ((f x)⁻¹ • f') x
-- HasDerivAt.clog
theorem HasDerivAt.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasDerivAt (fun t => log (f t)) (f' / f x) x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasDerivAt (fun t => Complex.log (f t)) (f' / f x) x
-- HasDerivAt.clog_real
theorem HasDerivAt.clog_real {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : HasDerivAt f f' x)
    (h₂ : f x ∈ slitPlane) : HasDerivAt (fun t => log (f t)) (f' / f x) x 
Arguments: {f : ℝ → ℂ} {x : ℝ} {f' : ℂ} (h₁ : HasDerivAt f f' x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasDerivAt (fun t => Complex.log (f t)) (f' / f x) x
-- DifferentiableAt.clog
theorem DifferentiableAt.clog {f : E → ℂ} {x : E} (h₁ : DifferentiableAt ℂ f x)
    (h₂ : f x ∈ slitPlane) : DifferentiableAt ℂ (fun t => log (f t)) x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {x : E} (h₁ : DifferentiableAt ℂ f x) (h₂ : f x ∈ Complex.slitPlane)
Goal: DifferentiableAt ℂ (fun t => Complex.log (f t)) x
-- HasFDerivWithinAt.clog
theorem HasFDerivWithinAt.clog {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : Set E} {x : E}
    (h₁ : HasFDerivWithinAt f f' s x) (h₂ : f x ∈ slitPlane) :
    HasFDerivWithinAt (fun t => log (f t)) ((f x)⁻¹ • f') s x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {f' : E →L[ℂ] ℂ} {s : Set E} {x : E} (h₁ : HasFDerivWithinAt f f' s x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasFDerivWithinAt (fun t => Complex.log (f t)) ((f x)⁻¹ • f') s x
-- HasDerivWithinAt.clog
theorem HasDerivWithinAt.clog {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (h₁ : HasDerivWithinAt f f' s x)
    (h₂ : f x ∈ slitPlane) : HasDerivWithinAt (fun t => log (f t)) (f' / f x) s x 
Arguments: {f : ℂ → ℂ} {f' x : ℂ} {s : Set ℂ} (h₁ : HasDerivWithinAt f f' s x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasDerivWithinAt (fun t => Complex.log (f t)) (f' / f x) s x
-- HasDerivWithinAt.clog_real
theorem HasDerivWithinAt.clog_real {f : ℝ → ℂ} {s : Set ℝ} {x : ℝ} {f' : ℂ}
    (h₁ : HasDerivWithinAt f f' s x) (h₂ : f x ∈ slitPlane) :
    HasDerivWithinAt (fun t => log (f t)) (f' / f x) s x 
Arguments: {f : ℝ → ℂ} {s : Set ℝ} {x : ℝ} {f' : ℂ} (h₁ : HasDerivWithinAt f f' s x) (h₂ : f x ∈ Complex.slitPlane)
Goal: HasDerivWithinAt (fun t => Complex.log (f t)) (f' / f x) s x
-- DifferentiableWithinAt.clog
theorem DifferentiableWithinAt.clog {f : E → ℂ} {s : Set E} {x : E}
    (h₁ : DifferentiableWithinAt ℂ f s x) (h₂ : f x ∈ slitPlane) :
    DifferentiableWithinAt ℂ (fun t => log (f t)) s x 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} {x : E} (h₁ : DifferentiableWithinAt ℂ f s x) (h₂ : f x ∈ Complex.slitPlane)
Goal: DifferentiableWithinAt ℂ (fun t => Complex.log (f t)) s x
-- DifferentiableOn.clog
theorem DifferentiableOn.clog {f : E → ℂ} {s : Set E} (h₁ : DifferentiableOn ℂ f s)
    (h₂ : ∀ x ∈ s, f x ∈ slitPlane) : DifferentiableOn ℂ (fun t => log (f t)) s 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} {s : Set E} (h₁ : DifferentiableOn ℂ f s) (h₂ : ∀ x ∈ s, f x ∈ Complex.slitPlane)
Goal: DifferentiableOn ℂ (fun t => Complex.log (f t)) s
-- Differentiable.clog
theorem Differentiable.clog {f : E → ℂ} (h₁ : Differentiable ℂ f)
    (h₂ : ∀ x, f x ∈ slitPlane) : Differentiable ℂ fun t => log (f t) 
Arguments: {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : E → ℂ} (h₁ : Differentiable ℂ f) (h₂ : ∀ (x : E), f x ∈ Complex.slitPlane)
Goal: Differentiable ℂ fun t => Complex.log (f t)
-- Complex.deriv_log_comp_eq_logDeriv
lemma Complex.deriv_log_comp_eq_logDeriv {f : ℂ → ℂ} {x : ℂ} (h₁ : DifferentiableAt ℂ f x)
    (h₂ : f x ∈ Complex.slitPlane) : deriv (Complex.log ∘ f) x = logDeriv f x 
Arguments: {f : ℂ → ℂ} {x : ℂ} (h₁ : DifferentiableAt ℂ f x) (h₂ : f x ∈ Complex.slitPlane)
Goal: deriv (Complex.log ∘ f) x = logDeriv f x
-- norm_indicator_eq_indicator_norm
theorem norm_indicator_eq_indicator_norm : ‖indicator s f a‖ = indicator s (fun a => ‖f a‖) a 
Arguments: {α : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] {s : Set α} (f : α → E) (a : α)
Goal: ‖s.indicator f a‖ = s.indicator (fun a => ‖f a‖) a
-- nnnorm_indicator_eq_indicator_nnnorm
theorem nnnorm_indicator_eq_indicator_nnnorm :
    ‖indicator s f a‖₊ = indicator s (fun a => ‖f a‖₊) a 
Arguments: {α : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] {s : Set α} (f : α → E) (a : α)
Goal: ‖s.indicator f a‖₊ = s.indicator (fun a => ‖f a‖₊) a
-- norm_indicator_le_of_subset
theorem norm_indicator_le_of_subset (h : s ⊆ t) (f : α → E) (a : α) :
    ‖indicator s f a‖ ≤ ‖indicator t f a‖ 
Arguments: {α : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] {s t : Set α} (h : s ⊆ t) (f : α → E) (a : α)
Goal: ‖s.indicator f a‖ ≤ ‖t.indicator f a‖
-- indicator_norm_le_norm_self
theorem indicator_norm_le_norm_self : indicator s (fun a => ‖f a‖) a ≤ ‖f a‖ 
Arguments: {α : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] {s : Set α} (f : α → E) (a : α)
Goal: s.indicator (fun a => ‖f a‖) a ≤ ‖f a‖
-- norm_indicator_le_norm_self
theorem norm_indicator_le_norm_self : ‖indicator s f a‖ ≤ ‖f a‖ 
Arguments: {α : Type u_1} {E : Type u_2} [SeminormedAddCommGroup E] {s : Set α} (f : α → E) (a : α)
Goal: ‖s.indicator f a‖ ≤ ‖f a‖
-- Real.geom_mean_le_arith_mean_weighted
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) :
    ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i)
Goal: ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i
-- Real.geom_mean_le_arith_mean
theorem geom_mean_le_arith_mean {ι : Type*} (s : Finset ι) (w : ι → ℝ) (z : ι → ℝ)
    (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 ≤ z i) :
    (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹  ≤  (∑ i ∈ s, w i * z i) / (∑ i ∈ s, w i) 
Arguments: {ι : Type u_1} (s : Finset ι) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 ≤ z i)
Goal: (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / ∑ i ∈ s, w i
-- Real.geom_mean_weighted_of_constant
theorem geom_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
    ∏ i ∈ s, z i ^ w i = x 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x)
Goal: ∏ i ∈ s, z i ^ w i = x
-- Real.arith_mean_weighted_of_constant
theorem arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i ∈ s, w i = 1)
    (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) : ∑ i ∈ s, w i * z i = x 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (x : ℝ) (hw' : ∑ i ∈ s, w i = 1) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x)
Goal: ∑ i ∈ s, w i * z i = x
-- Real.geom_mean_eq_arith_mean_weighted_of_constant
theorem geom_mean_eq_arith_mean_weighted_of_constant (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
    (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x) :
    ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (x : ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (hx : ∀ i ∈ s, w i ≠ 0 → z i = x)
Goal: ∏ i ∈ s, z i ^ w i = ∑ i ∈ s, w i * z i
-- NNReal.geom_mean_le_arith_mean_weighted
theorem geom_mean_le_arith_mean_weighted (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1) :
    (∏ i ∈ s, z i ^ (w i : ℝ)) ≤ ∑ i ∈ s, w i * z i 
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ≥0) (hw' : ∑ i ∈ s, w i = 1)
Goal: ∏ i ∈ s, z i ^ ↑(w i) ≤ ∑ i ∈ s, w i * z i
-- NNReal.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean2_weighted (w₁ w₂ p₁ p₂ : ℝ≥0) :
    w₁ + w₂ = 1 → p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ 
Arguments: (w₁ w₂ p₁ p₂ : ℝ≥0)
Goal: w₁ + w₂ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ ≤ w₁ * p₁ + w₂ * p₂
-- NNReal.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean3_weighted (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0) :
    w₁ + w₂ + w₃ = 1 →
      p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ 
Arguments: (w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0)
Goal: w₁ + w₂ + w₃ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃
-- NNReal.geom_mean_le_arith_mean4_weighted
theorem geom_mean_le_arith_mean4_weighted (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0) :
    w₁ + w₂ + w₃ + w₄ = 1 →
      p₁ ^ (w₁ : ℝ) * p₂ ^ (w₂ : ℝ) * p₃ ^ (w₃ : ℝ) * p₄ ^ (w₄ : ℝ) ≤
        w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ 
Arguments: (w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0)
Goal: w₁ + w₂ + w₃ + w₄ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ * p₄ ^ ↑w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄
-- Real.geom_mean_le_arith_mean2_weighted
theorem geom_mean_le_arith_mean2_weighted {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
    (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1) : p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂ 
Arguments: {w₁ w₂ p₁ p₂ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hw : w₁ + w₂ = 1)
Goal: p₁ ^ w₁ * p₂ ^ w₂ ≤ w₁ * p₁ + w₂ * p₂
-- Real.geom_mean_le_arith_mean3_weighted
theorem geom_mean_le_arith_mean3_weighted {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂)
    (hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1) :
    p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ 
Arguments: {w₁ w₂ w₃ p₁ p₂ p₃ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hw : w₁ + w₂ + w₃ = 1)
Goal: p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃
-- Real.geom_mean_le_arith_mean4_weighted
theorem geom_mean_le_arith_mean4_weighted {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁)
    (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃)
    (hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1) :
    p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄ 
Arguments: {w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ} (hw₁ : 0 ≤ w₁) (hw₂ : 0 ≤ w₂) (hw₃ : 0 ≤ w₃) (hw₄ : 0 ≤ w₄) (hp₁ : 0 ≤ p₁) (hp₂ : 0 ≤ p₂) (hp₃ : 0 ≤ p₃) (hp₄ : 0 ≤ p₄) (hw : w₁ + w₂ + w₃ + w₄ = 1)
Goal: p₁ ^ w₁ * p₂ ^ w₂ * p₃ ^ w₃ * p₄ ^ w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄
-- Real.harm_mean_le_geom_mean_weighted
theorem harm_mean_le_geom_mean_weighted (w z : ι → ℝ) (hs : s.Nonempty) (hw : ∀ i ∈ s, 0 < w i)
    (hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) :
    (∑ i in s, w i / z i)⁻¹ ≤ ∏ i in s, z i ^ w i  
Arguments: {ι : Type u} (s : Finset ι) (w z : ι → ℝ) (hs : s.Nonempty) (hw : ∀ i ∈ s, 0 < w i) (hw' : ∑ i ∈ s, w i = 1) (hz : ∀ i ∈ s, 0 < z i)
Goal: (∑ i ∈ s, w i / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ w i
-- Real.harm_mean_le_geom_mean
theorem harm_mean_le_geom_mean {ι : Type*} (s : Finset ι) (hs : s.Nonempty) (w : ι → ℝ)
    (z : ι → ℝ) (hw : ∀ i ∈ s, 0 < w i) (hw' : 0 < ∑ i in s, w i) (hz : ∀ i ∈ s, 0 < z i) :
    (∑ i in s, w i) / (∑ i in s, w i / z i) ≤ (∏ i in s, z i ^ w i) ^ (∑ i in s, w i)⁻¹ 
Arguments: {ι : Type u_1} (s : Finset ι) (hs : s.Nonempty) (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 < w i) (hw' : 0 < ∑ i ∈ s, w i) (hz : ∀ i ∈ s, 0 < z i)
Goal: (∑ i ∈ s, w i) / ∑ i ∈ s, w i / z i ≤ (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹
-- Real.young_inequality_of_nonneg
theorem young_inequality_of_nonneg {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
    (hpq : p.IsConjExponent q) : a * b ≤ a ^ p / p + b ^ q / q 
Arguments: {a b p q : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hpq : p.IsConjExponent q)
Goal: a * b ≤ a ^ p / p + b ^ q / q
-- Real.young_inequality
theorem young_inequality (a b : ℝ) {p q : ℝ} (hpq : p.IsConjExponent q) :
    a * b ≤ |a| ^ p / p + |b| ^ q / q 
Arguments: (a b : ℝ) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: a * b ≤ |a| ^ p / p + |b| ^ q / q
-- NNReal.young_inequality
theorem young_inequality (a b : ℝ≥0) {p q : ℝ≥0} (hpq : p.IsConjExponent q) :
    a * b ≤ a ^ (p : ℝ) / p + b ^ (q : ℝ) / q 
Arguments: (a b : ℝ≥0) {p q : ℝ≥0} (hpq : p.IsConjExponent q)
Goal: a * b ≤ a ^ ↑p / p + b ^ ↑q / q
-- NNReal.young_inequality_real
theorem young_inequality_real (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) :
    a * b ≤ a ^ p / Real.toNNReal p + b ^ q / Real.toNNReal q 
Arguments: (a b : ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: a * b ≤ a ^ p / p.toNNReal + b ^ q / q.toNNReal
-- ENNReal.young_inequality
theorem young_inequality (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjExponent q) :
    a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q 
Arguments: (a b : ℝ≥0∞) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: a * b ≤ a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q
-- _private.Mathlib.Analysis.MeanInequalities.0.NNReal.inner_le_Lp_mul_Lp_of_norm_le_one
private theorem inner_le_Lp_mul_Lp_of_norm_le_one (f g : ι → ℝ≥0) {p q : ℝ}
    (hpq : p.IsConjExponent q) (hf : ∑ i ∈ s, f i ^ p ≤ 1) (hg : ∑ i ∈ s, g i ^ q ≤ 1) :
    ∑ i ∈ s, f i * g i ≤ 1 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∑ i ∈ s, f i ^ p ≤ 1) (hg : ∑ i ∈ s, g i ^ q ≤ 1)
Goal: ∑ i ∈ s, f i * g i ≤ 1
-- _private.Mathlib.Analysis.MeanInequalities.0.NNReal.inner_le_Lp_mul_Lp_of_norm_eq_zero
private theorem inner_le_Lp_mul_Lp_of_norm_eq_zero (f g : ι → ℝ≥0) {p q : ℝ}
    (hpq : p.IsConjExponent q) (hf : ∑ i ∈ s, f i ^ p = 0) :
    ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) (hf : ∑ i ∈ s, f i ^ p = 0)
Goal: ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
-- NNReal.inner_le_Lp_mul_Lq
theorem inner_le_Lp_mul_Lq (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q) :
    ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) 
Arguments: {ι : Type u} (s : Finset ι) (f g : ι → ℝ≥0) {p q : ℝ} (hpq : p.IsConjExponent q)
Goal: ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
-- NNReal.inner_le_weight_mul_Lp
lemma inner_le_weight_mul_Lp (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0) :
    ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ 
Arguments: {ι : Type u} (s : Finset ι) {p : ℝ} (hp : 1 ≤ p) (w f : ι → ℝ≥0)
Goal: ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹