ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SchwartzMap.compCLMOfContinuousLinearEquiv_apply
@[simp] lemma compCLMOfContinuousLinearEquiv_apply (g : D ≃L[ℝ] E) (f : 𝓢(E, F)) :
    compCLMOfContinuousLinearEquiv 𝕜 g f = f ∘ g 
Arguments: (𝕜 : Type u_1) {D : Type u_3} {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (g : D ≃L[ℝ] E) (f : SchwartzMap E F)
Goal: ⇑((SchwartzMap.compCLMOfContinuousLinearEquiv 𝕜 g) f) = ⇑f ∘ ⇑g
-- SchwartzMap.fderivCLM_apply
@[simp]
theorem fderivCLM_apply (f : 𝓢(E, F)) (x : E) : fderivCLM 𝕜 f x = fderiv ℝ f x 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (x : E)
Goal: ((SchwartzMap.fderivCLM 𝕜) f) x = fderiv ℝ (⇑f) x
-- SchwartzMap.derivCLM_apply
@[simp]
theorem derivCLM_apply (f : 𝓢(ℝ, F)) (x : ℝ) : derivCLM 𝕜 f x = deriv f x 
Arguments: (𝕜 : Type u_1) {F : Type u_5} [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap ℝ F) (x : ℝ)
Goal: ((SchwartzMap.derivCLM 𝕜) f) x = deriv (⇑f) x
-- SchwartzMap.pderivCLM_apply
@[simp]
theorem pderivCLM_apply (m : E) (f : 𝓢(E, F)) (x : E) : pderivCLM 𝕜 m f x = fderiv ℝ f x m 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (m : E) (f : SchwartzMap E F) (x : E)
Goal: ((SchwartzMap.pderivCLM 𝕜 m) f) x = (fderiv ℝ (⇑f) x) m
-- SchwartzMap.pderivCLM_eq_lineDeriv
theorem pderivCLM_eq_lineDeriv (m : E) (f : 𝓢(E, F)) (x : E) :
    pderivCLM 𝕜 m f x = lineDeriv ℝ f x m 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (m : E) (f : SchwartzMap E F) (x : E)
Goal: ((SchwartzMap.pderivCLM 𝕜 m) f) x = lineDeriv ℝ (⇑f) x m
-- SchwartzMap.iteratedPDeriv_zero
@[simp]
theorem iteratedPDeriv_zero (m : Fin 0 → E) (f : 𝓢(E, F)) : iteratedPDeriv 𝕜 m f = f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (m : Fin 0 → E) (f : SchwartzMap E F)
Goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = f
-- SchwartzMap.iteratedPDeriv_one
@[simp]
theorem iteratedPDeriv_one (m : Fin 1 → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = pderivCLM 𝕜 (m 0) f 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (m : Fin 1 → E) (f : SchwartzMap E F)
Goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = (SchwartzMap.pderivCLM 𝕜 (m 0)) f
-- SchwartzMap.iteratedPDeriv_succ_left
theorem iteratedPDeriv_succ_left {n : ℕ} (m : Fin (n + 1) → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = pderivCLM 𝕜 (m 0) (iteratedPDeriv 𝕜 (Fin.tail m) f) 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {n : ℕ} (m : Fin (n + 1) → E) (f : SchwartzMap E F)
Goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f = (SchwartzMap.pderivCLM 𝕜 (m 0)) ((SchwartzMap.iteratedPDeriv 𝕜 (Fin.tail m)) f)
-- SchwartzMap.iteratedPDeriv_succ_right
theorem iteratedPDeriv_succ_right {n : ℕ} (m : Fin (n + 1) → E) (f : 𝓢(E, F)) :
    iteratedPDeriv 𝕜 m f = iteratedPDeriv 𝕜 (Fin.init m) (pderivCLM 𝕜 (m (Fin.last n)) f) 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {n : ℕ} (m : Fin (n + 1) → E) (f : SchwartzMap E F)
Goal: (SchwartzMap.iteratedPDeriv 𝕜 m) f =
  (SchwartzMap.iteratedPDeriv 𝕜 (Fin.init m)) ((SchwartzMap.pderivCLM 𝕜 (m (Fin.last n))) f)
-- SchwartzMap.iteratedPDeriv_eq_iteratedFDeriv
theorem iteratedPDeriv_eq_iteratedFDeriv {n : ℕ} {m : Fin n → E} {f : 𝓢(E, F)} {x : E} :
    iteratedPDeriv 𝕜 m f x = iteratedFDeriv ℝ n f x m 
Arguments: (𝕜 : Type u_1) {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] {n : ℕ} {m : Fin n → E} {f : SchwartzMap E F} {x : E}
Goal: ((SchwartzMap.iteratedPDeriv 𝕜 m) f) x = (iteratedFDeriv ℝ n (⇑f) x) m
-- SchwartzMap.integrable_pow_mul_iteratedFDeriv
lemma integrable_pow_mul_iteratedFDeriv (f : 𝓢(D, V))
    (k n : ℕ) : Integrable (fun x ↦ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖) μ 
Arguments: {D : Type u_3} {V : Type u_7} [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace D] [BorelSpace D] [SecondCountableTopology D] (μ : MeasureTheory.Measure D) [hμ : μ.HasTemperateGrowth] (f : SchwartzMap D V) (k n : ℕ)
Goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖) μ
-- SchwartzMap.integrable_pow_mul
lemma integrable_pow_mul (f : 𝓢(D, V))
    (k : ℕ) : Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ 
Arguments: {D : Type u_3} {V : Type u_7} [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace D] [BorelSpace D] [SecondCountableTopology D] (μ : MeasureTheory.Measure D) [hμ : μ.HasTemperateGrowth] (f : SchwartzMap D V) (k : ℕ)
Goal: MeasureTheory.Integrable (fun x => ‖x‖ ^ k * ‖f x‖) μ
-- SchwartzMap.integral_pow_mul_iteratedFDeriv_le
lemma integral_pow_mul_iteratedFDeriv_le (f : 𝓢(D, V)) (k n : ℕ) :
    ∫ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ∂μ ≤ 2 ^ μ.integrablePower *
      (∫ x, (1 + ‖x‖) ^ (- (μ.integrablePower : ℝ)) ∂μ) *
        (SchwartzMap.seminorm 𝕜 0 n f + SchwartzMap.seminorm 𝕜 (k + μ.integrablePower) n f) 
Arguments: (𝕜 : Type u_1) {D : Type u_3} {V : Type u_7} [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedSpace 𝕜 V] [MeasurableSpace D] (μ : MeasureTheory.Measure D) [hμ : μ.HasTemperateGrowth] (f : SchwartzMap D V) (k n : ℕ)
Goal: ∫ (x : D), ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖ ∂μ ≤
  (2 ^ μ.integrablePower * ∫ (x : D), (1 + ‖x‖) ^ (-↑μ.integrablePower) ∂μ) *
    ((SchwartzMap.seminorm 𝕜 0 n) f + (SchwartzMap.seminorm 𝕜 (k + μ.integrablePower) n) f)
-- SchwartzMap.integrable
lemma integrable (f : 𝓢(D, V)) : Integrable f μ 
Arguments: {D : Type u_3} {V : Type u_7} [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace D] [BorelSpace D] [SecondCountableTopology D] {μ : MeasureTheory.Measure D} [hμ : μ.HasTemperateGrowth] (f : SchwartzMap D V)
Goal: MeasureTheory.Integrable (⇑f) μ
-- SchwartzMap.integralCLM_apply
@[simp]
lemma integralCLM_apply (f : 𝓢(D, V)) : integralCLM 𝕜 μ f = ∫ x, f x ∂μ 
Arguments: (𝕜 : Type u_1) {D : Type u_3} {V : Type u_7} [RCLike 𝕜] [NormedAddCommGroup D] [NormedSpace ℝ D] [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedSpace 𝕜 V] [MeasurableSpace D] [BorelSpace D] [SecondCountableTopology D] {μ : MeasureTheory.Measure D} [hμ : μ.HasTemperateGrowth] (f : SchwartzMap D V)
Goal: (SchwartzMap.integralCLM 𝕜 μ) f = ∫ (x : D), f x ∂μ
-- SchwartzMap.toBoundedContinuousFunction_apply
@[simp]
theorem toBoundedContinuousFunction_apply (f : 𝓢(E, F)) (x : E) :
    f.toBoundedContinuousFunction x = f x 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] (f : SchwartzMap E F) (x : E)
Goal: f.toBoundedContinuousFunction x = f x
-- SchwartzMap.toBoundedContinuousFunctionCLM_apply
@[simp]
theorem toBoundedContinuousFunctionCLM_apply (f : 𝓢(E, F)) (x : E) :
    toBoundedContinuousFunctionCLM 𝕜 E F f x = f x 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (x : E)
Goal: ((SchwartzMap.toBoundedContinuousFunctionCLM 𝕜 E F) f) x = f x
-- SchwartzMap.delta_apply
@[simp]
theorem delta_apply (x₀ : E) (f : 𝓢(E, F)) : delta 𝕜 F x₀ f = f x₀ 
Arguments: (𝕜 : Type u_1) {E : Type u_4} (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (x₀ : E) (f : SchwartzMap E F)
Goal: (SchwartzMap.delta 𝕜 F x₀) f = f x₀
-- SchwartzMap.integralCLM_dirac_eq_delta
@[simp]
theorem integralCLM_dirac_eq_delta (x : E) : integralCLM 𝕜 (dirac x) = delta 𝕜 F x 
Arguments: (𝕜 : Type u_1) {E : Type u_4} (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] [CompleteSpace F] (x : E)
Goal: SchwartzMap.integralCLM 𝕜 (MeasureTheory.Measure.dirac x) = SchwartzMap.delta 𝕜 F x
-- SchwartzMap.toZeroAtInfty_apply
@[simp] theorem toZeroAtInfty_apply (f : 𝓢(E, F)) (x : E) : f.toZeroAtInfty x = f x 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [ProperSpace E] (f : SchwartzMap E F) (x : E)
Goal: f.toZeroAtInfty x = f x
-- SchwartzMap.toZeroAtInfty_toBCF
@[simp] theorem toZeroAtInfty_toBCF (f : 𝓢(E, F)) :
    f.toZeroAtInfty.toBCF = f.toBoundedContinuousFunction 
Arguments: {E : Type u_4} {F : Type u_5} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [ProperSpace E] (f : SchwartzMap E F)
Goal: f.toZeroAtInfty.toBCF = f.toBoundedContinuousFunction
-- SchwartzMap.toZeroAtInftyCLM_apply
@[simp] theorem toZeroAtInftyCLM_apply (f : 𝓢(E, F)) (x : E) : toZeroAtInftyCLM 𝕜 E F f x = f x 
Arguments: (𝕜 : Type u_1) (E : Type u_4) (F : Type u_5) [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [ProperSpace E] [RCLike 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (f : SchwartzMap E F) (x : E)
Goal: ((SchwartzMap.toZeroAtInftyCLM 𝕜 E F) f) x = f x
-- mem_convexJoin
theorem mem_convexJoin : x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} {x : E}
Goal: x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b
-- convexJoin_comm
theorem convexJoin_comm (s t : Set E) : convexJoin 𝕜 s t = convexJoin 𝕜 t s 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s t : Set E)
Goal: convexJoin 𝕜 s t = convexJoin 𝕜 t s
-- convexJoin_mono
theorem convexJoin_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : convexJoin 𝕜 s₁ t₁ ⊆ convexJoin 𝕜 s₂ t₂ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s₁ s₂ t₁ t₂ : Set E} (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂)
Goal: convexJoin 𝕜 s₁ t₁ ⊆ convexJoin 𝕜 s₂ t₂
-- convexJoin_mono_left
theorem convexJoin_mono_left (hs : s₁ ⊆ s₂) : convexJoin 𝕜 s₁ t ⊆ convexJoin 𝕜 s₂ t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {t s₁ s₂ : Set E} (hs : s₁ ⊆ s₂)
Goal: convexJoin 𝕜 s₁ t ⊆ convexJoin 𝕜 s₂ t
-- convexJoin_mono_right
theorem convexJoin_mono_right (ht : t₁ ⊆ t₂) : convexJoin 𝕜 s t₁ ⊆ convexJoin 𝕜 s t₂ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t₁ t₂ : Set E} (ht : t₁ ⊆ t₂)
Goal: convexJoin 𝕜 s t₁ ⊆ convexJoin 𝕜 s t₂
-- convexJoin_empty_left
@[simp]
theorem convexJoin_empty_left (t : Set E) : convexJoin 𝕜 ∅ t = ∅ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (t : Set E)
Goal: convexJoin 𝕜 ∅ t = ∅
-- convexJoin_empty_right
@[simp]
theorem convexJoin_empty_right (s : Set E) : convexJoin 𝕜 s ∅ = ∅ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E)
Goal: convexJoin 𝕜 s ∅ = ∅
-- convexJoin_singleton_left
@[simp]
theorem convexJoin_singleton_left (t : Set E) (x : E) :
    convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (t : Set E) (x : E)
Goal: convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y
-- convexJoin_singleton_right
@[simp]
theorem convexJoin_singleton_right (s : Set E) (y : E) :
    convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E) (y : E)
Goal: convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y
-- convexJoin_singletons
theorem convexJoin_singletons (x : E) : convexJoin 𝕜 {x} {y} = segment 𝕜 x y 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {y : E} (x : E)
Goal: convexJoin 𝕜 {x} {y} = segment 𝕜 x y
-- convexJoin_union_left
@[simp]
theorem convexJoin_union_left (s₁ s₂ t : Set E) :
    convexJoin 𝕜 (s₁ ∪ s₂) t = convexJoin 𝕜 s₁ t ∪ convexJoin 𝕜 s₂ t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s₁ s₂ t : Set E)
Goal: convexJoin 𝕜 (s₁ ∪ s₂) t = convexJoin 𝕜 s₁ t ∪ convexJoin 𝕜 s₂ t
-- convexJoin_union_right
@[simp]
theorem convexJoin_union_right (s t₁ t₂ : Set E) :
    convexJoin 𝕜 s (t₁ ∪ t₂) = convexJoin 𝕜 s t₁ ∪ convexJoin 𝕜 s t₂ 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s t₁ t₂ : Set E)
Goal: convexJoin 𝕜 s (t₁ ∪ t₂) = convexJoin 𝕜 s t₁ ∪ convexJoin 𝕜 s t₂
-- convexJoin_iUnion_left
@[simp]
theorem convexJoin_iUnion_left (s : ι → Set E) (t : Set E) :
    convexJoin 𝕜 (⋃ i, s i) t = ⋃ i, convexJoin 𝕜 (s i) t 
Arguments: {ι : Sort u_1} {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : ι → Set E) (t : Set E)
Goal: convexJoin 𝕜 (⋃ i, s i) t = ⋃ i, convexJoin 𝕜 (s i) t
-- convexJoin_iUnion_right
@[simp]
theorem convexJoin_iUnion_right (s : Set E) (t : ι → Set E) :
    convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i) 
Arguments: {ι : Sort u_1} {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s : Set E) (t : ι → Set E)
Goal: convexJoin 𝕜 s (⋃ i, t i) = ⋃ i, convexJoin 𝕜 s (t i)
-- segment_subset_convexJoin
theorem segment_subset_convexJoin (hx : x ∈ s) (hy : y ∈ t) : segment 𝕜 x y ⊆ convexJoin 𝕜 s t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} {x y : E} (hx : x ∈ s) (hy : y ∈ t)
Goal: segment 𝕜 x y ⊆ convexJoin 𝕜 s t
-- subset_convexJoin_left
theorem subset_convexJoin_left (h : t.Nonempty) : s ⊆ convexJoin 𝕜 s t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} (h : t.Nonempty)
Goal: s ⊆ convexJoin 𝕜 s t
-- subset_convexJoin_right
theorem subset_convexJoin_right (h : s.Nonempty) : t ⊆ convexJoin 𝕜 s t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t : Set E} (h : s.Nonempty)
Goal: t ⊆ convexJoin 𝕜 s t
-- convexJoin_subset
theorem convexJoin_subset (hs : s ⊆ u) (ht : t ⊆ u) (hu : Convex 𝕜 u) : convexJoin 𝕜 s t ⊆ u 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t u : Set E} (hs : s ⊆ u) (ht : t ⊆ u) (hu : Convex 𝕜 u)
Goal: convexJoin 𝕜 s t ⊆ u
-- convexJoin_subset_convexHull
theorem convexJoin_subset_convexHull (s t : Set E) : convexJoin 𝕜 s t ⊆ convexHull 𝕜 (s ∪ t) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] (s t : Set E)
Goal: convexJoin 𝕜 s t ⊆ (convexHull 𝕜) (s ∪ t)
-- convexJoin_assoc_aux
theorem convexJoin_assoc_aux (s t u : Set E) :
    convexJoin 𝕜 (convexJoin 𝕜 s t) u ⊆ convexJoin 𝕜 s (convexJoin 𝕜 t u) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s t u : Set E)
Goal: convexJoin 𝕜 (convexJoin 𝕜 s t) u ⊆ convexJoin 𝕜 s (convexJoin 𝕜 t u)
-- convexJoin_assoc
theorem convexJoin_assoc (s t u : Set E) :
    convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 s (convexJoin 𝕜 t u) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s t u : Set E)
Goal: convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 s (convexJoin 𝕜 t u)
-- convexJoin_left_comm
theorem convexJoin_left_comm (s t u : Set E) :
    convexJoin 𝕜 s (convexJoin 𝕜 t u) = convexJoin 𝕜 t (convexJoin 𝕜 s u) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s t u : Set E)
Goal: convexJoin 𝕜 s (convexJoin 𝕜 t u) = convexJoin 𝕜 t (convexJoin 𝕜 s u)
-- convexJoin_right_comm
theorem convexJoin_right_comm (s t u : Set E) :
    convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 (convexJoin 𝕜 s u) t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s t u : Set E)
Goal: convexJoin 𝕜 (convexJoin 𝕜 s t) u = convexJoin 𝕜 (convexJoin 𝕜 s u) t
-- convexJoin_convexJoin_convexJoin_comm
theorem convexJoin_convexJoin_convexJoin_comm (s t u v : Set E) :
    convexJoin 𝕜 (convexJoin 𝕜 s t) (convexJoin 𝕜 u v) =
      convexJoin 𝕜 (convexJoin 𝕜 s u) (convexJoin 𝕜 t v) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (s t u v : Set E)
Goal: convexJoin 𝕜 (convexJoin 𝕜 s t) (convexJoin 𝕜 u v) = convexJoin 𝕜 (convexJoin 𝕜 s u) (convexJoin 𝕜 t v)
-- Convex.convexJoin
protected theorem Convex.convexJoin (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) :
    Convex 𝕜 (convexJoin 𝕜 s t) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Convex 𝕜 s) (ht : Convex 𝕜 t)
Goal: Convex 𝕜 (convexJoin 𝕜 s t)
-- Convex.convexHull_union
protected theorem Convex.convexHull_union (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hs₀ : s.Nonempty)
    (ht₀ : t.Nonempty) : convexHull 𝕜 (s ∪ t) = convexJoin 𝕜 s t 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hs₀ : s.Nonempty) (ht₀ : t.Nonempty)
Goal: (convexHull 𝕜) (s ∪ t) = convexJoin 𝕜 s t
-- convexHull_union
theorem convexHull_union (hs : s.Nonempty) (ht : t.Nonempty) :
    convexHull 𝕜 (s ∪ t) = convexJoin 𝕜 (convexHull 𝕜 s) (convexHull 𝕜 t) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s t : Set E} (hs : s.Nonempty) (ht : t.Nonempty)
Goal: (convexHull 𝕜) (s ∪ t) = convexJoin 𝕜 ((convexHull 𝕜) s) ((convexHull 𝕜) t)
-- convexHull_insert
theorem convexHull_insert (hs : s.Nonempty) :
    convexHull 𝕜 (insert x s) = convexJoin 𝕜 {x} (convexHull 𝕜 s) 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} {x : E} (hs : s.Nonempty)
Goal: (convexHull 𝕜) (insert x s) = convexJoin 𝕜 {x} ((convexHull 𝕜) s)
-- convexJoin_segments
theorem convexJoin_segments (a b c d : E) :
    convexJoin 𝕜 (segment 𝕜 a b) (segment 𝕜 c d) = convexHull 𝕜 {a, b, c, d} 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c d : E)
Goal: convexJoin 𝕜 (segment 𝕜 a b) (segment 𝕜 c d) = (convexHull 𝕜) {a, b, c, d}
-- convexJoin_segment_singleton
theorem convexJoin_segment_singleton (a b c : E) :
    convexJoin 𝕜 (segment 𝕜 a b) {c} = convexHull 𝕜 {a, b, c} 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: convexJoin 𝕜 (segment 𝕜 a b) {c} = (convexHull 𝕜) {a, b, c}
-- convexJoin_singleton_segment
theorem convexJoin_singleton_segment (a b c : E) :
    convexJoin 𝕜 {a} (segment 𝕜 b c) = convexHull 𝕜 {a, b, c} 
Arguments: {𝕜 : Type u_2} {E : Type u_3} [LinearOrderedField 𝕜] [AddCommGroup E] [Module 𝕜 E] (a b c : E)
Goal: convexJoin 𝕜 {a} (segment 𝕜 b c) = (convexHull 𝕜) {a, b, c}
-- InnerProductSpace.smul_left
class InnerProductSpace (𝕜 : Type*) (E : Type*) [RCLike 𝕜] [NormedAddCommGroup E] extends
  NormedSpace 𝕜 E, Inner 𝕜 E where
    norm_sq_eq_inner : ∀ x : E, ‖x‖ ^ 2 = re (inner x x)
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {E : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [self : InnerProductSpace 𝕜 E] (x y : E) (r : 𝕜)
Goal: ⟪r • x, y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪x, y⟫_𝕜
-- InnerProductSpace.norm_sq_eq_inner
class InnerProductSpace (𝕜 : Type*) (E : Type*) [RCLike 𝕜] [NormedAddCommGroup E] extends
  NormedSpace 𝕜 E, Inner 𝕜 E where
    norm_sq_eq_inner : ∀ x : E, ‖x‖ ^ 2 = re (inner x x)
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {E : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [self : InnerProductSpace 𝕜 E] (x : E)
Goal: ‖x‖ ^ 2 = RCLike.re ⟪x, x⟫_𝕜
-- InnerProductSpace.conj_symm
class InnerProductSpace (𝕜 : Type*) (E : Type*) [RCLike 𝕜] [NormedAddCommGroup E] extends
  NormedSpace 𝕜 E, Inner 𝕜 E where
    norm_sq_eq_inner : ∀ x : E, ‖x‖ ^ 2 = re (inner x x)
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {E : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [self : InnerProductSpace 𝕜 E] (x y : E)
Goal: (starRingEnd 𝕜) ⟪y, x⟫_𝕜 = ⟪x, y⟫_𝕜
-- InnerProductSpace.add_left
class InnerProductSpace (𝕜 : Type*) (E : Type*) [RCLike 𝕜] [NormedAddCommGroup E] extends
  NormedSpace 𝕜 E, Inner 𝕜 E where
    norm_sq_eq_inner : ∀ x : E, ‖x‖ ^ 2 = re (inner x x)
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {E : Type u_5} [RCLike 𝕜] [NormedAddCommGroup E] [self : InnerProductSpace 𝕜 E] (x y z : E)
Goal: ⟪x + y, z⟫_𝕜 = ⟪x, z⟫_𝕜 + ⟪y, z⟫_𝕜
-- InnerProductSpace.Core.smul_left
-- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore
structure InnerProductSpace.Core (𝕜 : Type*) (F : Type*) [RCLike 𝕜] [AddCommGroup F]
  [Module 𝕜 F] extends Inner 𝕜 F where
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    nonneg_re : ∀ x, 0 ≤ re (inner x x)
    definite : ∀ x, inner x x = 0 → x = 0
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {F : Type u_5} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] (self : InnerProductSpace.Core 𝕜 F) (x y : F) (r : 𝕜)
Goal: ⟪r • x, y⟫_𝕜 = (starRingEnd 𝕜) r * ⟪x, y⟫_𝕜
-- InnerProductSpace.Core.add_left
-- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore
structure InnerProductSpace.Core (𝕜 : Type*) (F : Type*) [RCLike 𝕜] [AddCommGroup F]
  [Module 𝕜 F] extends Inner 𝕜 F where
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    nonneg_re : ∀ x, 0 ≤ re (inner x x)
    definite : ∀ x, inner x x = 0 → x = 0
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {F : Type u_5} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] (self : InnerProductSpace.Core 𝕜 F) (x y z : F)
Goal: ⟪x + y, z⟫_𝕜 = ⟪x, z⟫_𝕜 + ⟪y, z⟫_𝕜
-- InnerProductSpace.Core.nonneg_re
-- @[nolint HasNonemptyInstance] porting note: I don't think we have this linter anymore
structure InnerProductSpace.Core (𝕜 : Type*) (F : Type*) [RCLike 𝕜] [AddCommGroup F]
  [Module 𝕜 F] extends Inner 𝕜 F where
    conj_symm : ∀ x y, conj (inner y x) = inner x y
    nonneg_re : ∀ x, 0 ≤ re (inner x x)
    definite : ∀ x, inner x x = 0 → x = 0
    add_left : ∀ x y z, inner (x + y) z = inner x z + inner y z
    smul_left : ∀ x y r, inner (r • x) y = conj r * inner x y
Arguments: {𝕜 : Type u_4} {F : Type u_5} [RCLike 𝕜] [AddCommGroup F] [Module 𝕜 F] (self : InnerProductSpace.Core 𝕜 F) (x : F)
Goal: 0 ≤ RCLike.re ⟪x, x⟫_𝕜