ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- measurableEmbedding_const_vadd
@[to_additive]
theorem _root_.measurableEmbedding_const_smul (c : G) : MeasurableEmbedding (c • · : α → α) 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [AddGroup G] [AddAction G α] [MeasurableVAdd G α] (c : G)
Goal: MeasurableEmbedding fun x => c +ᵥ x
-- MeasurableEquiv.symm_smul
@[to_additive (attr := simp)]
theorem symm_smul (c : G) : (smul c : α ≃ᵐ α).symm = smul c⁻¹ 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [Group G] [MulAction G α] [MeasurableSMul G α] (c : G)
Goal: (MeasurableEquiv.smul c).symm = MeasurableEquiv.smul c⁻¹
-- MeasurableEquiv.symm_vadd
@[to_additive (attr := simp)]
theorem symm_smul (c : G) : (smul c : α ≃ᵐ α).symm = smul c⁻¹ 
Arguments: {G : Type u_1} {α : Type u_3} [MeasurableSpace G] [MeasurableSpace α] [AddGroup G] [AddAction G α] [MeasurableVAdd G α] (c : G)
Goal: (MeasurableEquiv.vadd c).symm = MeasurableEquiv.vadd (-c)
-- MeasurableEquiv.coe_smul₀
@[simp] lemma coe_smul₀ {c : G₀} (hc : c ≠ 0) : ⇑(smul₀ c hc : α ≃ᵐ α) = (c • ·) 
Arguments: {G₀ : Type u_2} {α : Type u_3} [MeasurableSpace G₀] [MeasurableSpace α] [GroupWithZero G₀] [MulAction G₀ α] [MeasurableSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: ⇑(MeasurableEquiv.smul₀ c hc) = fun x => c • x
-- MeasurableEquiv.symm_smul₀
@[simp]
theorem symm_smul₀ {c : G₀} (hc : c ≠ 0) :
    (smul₀ c hc : α ≃ᵐ α).symm = smul₀ c⁻¹ (inv_ne_zero hc) 
Arguments: {G₀ : Type u_2} {α : Type u_3} [MeasurableSpace G₀] [MeasurableSpace α] [GroupWithZero G₀] [MulAction G₀ α] [MeasurableSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: (MeasurableEquiv.smul₀ c hc).symm = MeasurableEquiv.smul₀ c⁻¹ ⋯
-- measurableEmbedding_const_smul₀
theorem _root_.measurableEmbedding_const_smul₀ {c : G₀} (hc : c ≠ 0) :
    MeasurableEmbedding (c • · : α → α) 
Arguments: {G₀ : Type u_2} {α : Type u_3} [MeasurableSpace G₀] [MeasurableSpace α] [GroupWithZero G₀] [MulAction G₀ α] [MeasurableSMul G₀ α] {c : G₀} (hc : c ≠ 0)
Goal: MeasurableEmbedding fun x => c • x
-- MeasurableEquiv.coe_mulLeft
@[to_additive (attr := simp)]
theorem coe_mulLeft (g : G) : ⇑(mulLeft g) = (g * ·) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: ⇑(MeasurableEquiv.mulLeft g) = fun x => g * x
-- MeasurableEquiv.coe_addLeft
@[to_additive (attr := simp)]
theorem coe_mulLeft (g : G) : ⇑(mulLeft g) = (g * ·) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: ⇑(MeasurableEquiv.addLeft g) = fun x => g + x
-- MeasurableEquiv.symm_mulLeft
@[to_additive (attr := simp)]
theorem symm_mulLeft (g : G) : (mulLeft g).symm = mulLeft g⁻¹ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: (MeasurableEquiv.mulLeft g).symm = MeasurableEquiv.mulLeft g⁻¹
-- MeasurableEquiv.symm_addLeft
@[to_additive (attr := simp)]
theorem symm_mulLeft (g : G) : (mulLeft g).symm = mulLeft g⁻¹ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: (MeasurableEquiv.addLeft g).symm = MeasurableEquiv.addLeft (-g)
-- MeasurableEquiv.toEquiv_addLeft
@[to_additive (attr := simp)]
theorem toEquiv_mulLeft (g : G) : (mulLeft g).toEquiv = Equiv.mulLeft g 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: (MeasurableEquiv.addLeft g).toEquiv = Equiv.addLeft g
-- MeasurableEquiv.toEquiv_mulLeft
@[to_additive (attr := simp)]
theorem toEquiv_mulLeft (g : G) : (mulLeft g).toEquiv = Equiv.mulLeft g 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: (MeasurableEquiv.mulLeft g).toEquiv = Equiv.mulLeft g
-- measurableEmbedding_mulLeft
@[to_additive]
theorem _root_.measurableEmbedding_mulLeft (g : G) : MeasurableEmbedding (g * ·) 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: MeasurableEmbedding fun x => g * x
-- measurableEmbedding_addLeft
@[to_additive]
theorem _root_.measurableEmbedding_mulLeft (g : G) : MeasurableEmbedding (g * ·) 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: MeasurableEmbedding fun x => g + x
-- measurableEmbedding_mulRight
@[to_additive]
theorem _root_.measurableEmbedding_mulRight (g : G) : MeasurableEmbedding fun x => x * g 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: MeasurableEmbedding fun x => x * g
-- measurableEmbedding_addRight
@[to_additive]
theorem _root_.measurableEmbedding_mulRight (g : G) : MeasurableEmbedding fun x => x * g 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: MeasurableEmbedding fun x => x + g
-- MeasurableEquiv.coe_mulRight
@[to_additive (attr := simp)]
theorem coe_mulRight (g : G) : ⇑(mulRight g) = fun x => x * g 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: ⇑(MeasurableEquiv.mulRight g) = fun x => x * g
-- MeasurableEquiv.coe_addRight
@[to_additive (attr := simp)]
theorem coe_mulRight (g : G) : ⇑(mulRight g) = fun x => x * g 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: ⇑(MeasurableEquiv.addRight g) = fun x => x + g
-- MeasurableEquiv.symm_mulRight
@[to_additive (attr := simp)]
theorem symm_mulRight (g : G) : (mulRight g).symm = mulRight g⁻¹ 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: (MeasurableEquiv.mulRight g).symm = MeasurableEquiv.mulRight g⁻¹
-- MeasurableEquiv.symm_addRight
@[to_additive (attr := simp)]
theorem symm_mulRight (g : G) : (mulRight g).symm = mulRight g⁻¹ 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: (MeasurableEquiv.addRight g).symm = MeasurableEquiv.addRight (-g)
-- MeasurableEquiv.toEquiv_addRight
@[to_additive (attr := simp)]
theorem toEquiv_mulRight (g : G) : (mulRight g).toEquiv = Equiv.mulRight g 
Arguments: {G : Type u_1} [MeasurableSpace G] [AddGroup G] [MeasurableAdd G] (g : G)
Goal: (MeasurableEquiv.addRight g).toEquiv = Equiv.addRight g
-- MeasurableEquiv.toEquiv_mulRight
@[to_additive (attr := simp)]
theorem toEquiv_mulRight (g : G) : (mulRight g).toEquiv = Equiv.mulRight g 
Arguments: {G : Type u_1} [MeasurableSpace G] [Group G] [MeasurableMul G] (g : G)
Goal: (MeasurableEquiv.mulRight g).toEquiv = Equiv.mulRight g
-- measurableEmbedding_mulLeft₀
theorem _root_.measurableEmbedding_mulLeft₀ {g : G₀} (hg : g ≠ 0) :
    MeasurableEmbedding (g * ·) 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: MeasurableEmbedding fun x => g * x
-- MeasurableEquiv.coe_mulLeft₀
@[simp] lemma coe_mulLeft₀ {g : G₀} (hg : g ≠ 0) : ⇑(mulLeft₀ g hg) = (g * ·) 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: ⇑(MeasurableEquiv.mulLeft₀ g hg) = fun x => g * x
-- MeasurableEquiv.symm_mulLeft₀
@[simp]
theorem symm_mulLeft₀ {g : G₀} (hg : g ≠ 0) :
    (mulLeft₀ g hg).symm = mulLeft₀ g⁻¹ (inv_ne_zero hg) 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: (MeasurableEquiv.mulLeft₀ g hg).symm = MeasurableEquiv.mulLeft₀ g⁻¹ ⋯
-- MeasurableEquiv.toEquiv_mulLeft₀
@[simp]
theorem toEquiv_mulLeft₀ {g : G₀} (hg : g ≠ 0) : (mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: (MeasurableEquiv.mulLeft₀ g hg).toEquiv = Equiv.mulLeft₀ g hg
-- measurableEmbedding_mulRight₀
theorem _root_.measurableEmbedding_mulRight₀ {g : G₀} (hg : g ≠ 0) :
    MeasurableEmbedding fun x => x * g 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: MeasurableEmbedding fun x => x * g
-- MeasurableEquiv.coe_mulRight₀
@[simp]
theorem coe_mulRight₀ {g : G₀} (hg : g ≠ 0) : ⇑(mulRight₀ g hg) = fun x => x * g 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: ⇑(MeasurableEquiv.mulRight₀ g hg) = fun x => x * g
-- MeasurableEquiv.symm_mulRight₀
@[simp]
theorem symm_mulRight₀ {g : G₀} (hg : g ≠ 0) :
    (mulRight₀ g hg).symm = mulRight₀ g⁻¹ (inv_ne_zero hg) 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: (MeasurableEquiv.mulRight₀ g hg).symm = MeasurableEquiv.mulRight₀ g⁻¹ ⋯
-- MeasurableEquiv.toEquiv_mulRight₀
@[simp]
theorem toEquiv_mulRight₀ {g : G₀} (hg : g ≠ 0) : (mulRight₀ g hg).toEquiv = Equiv.mulRight₀ g hg 
Arguments: {G₀ : Type u_2} [MeasurableSpace G₀] [GroupWithZero G₀] [MeasurableMul G₀] {g : G₀} (hg : g ≠ 0)
Goal: (MeasurableEquiv.mulRight₀ g hg).toEquiv = Equiv.mulRight₀ g hg
-- MeasurableEquiv.inv_toEquiv
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
Arguments: (G : Type u_4) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G]
Goal: (MeasurableEquiv.inv G).toEquiv = Equiv.inv G
-- MeasurableEquiv.neg_apply
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
Arguments: (G : Type u_4) [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G]
Goal: ⇑(MeasurableEquiv.neg G) = Neg.neg
-- MeasurableEquiv.neg_toEquiv
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
Arguments: (G : Type u_4) [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G]
Goal: (MeasurableEquiv.neg G).toEquiv = Equiv.neg G
-- MeasurableEquiv.inv_apply
@[to_additive (attr := simps! (config := .asFn) toEquiv apply)
    "Negation as a measurable automorphism of an additive group."]
def inv (G) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] : G ≃ᵐ G 
Arguments: (G : Type u_4) [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G]
Goal: ⇑(MeasurableEquiv.inv G) = Inv.inv
-- MeasurableEquiv.symm_inv
@[to_additive (attr := simp)]
theorem symm_inv {G} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] :
    (inv G).symm = inv G 
Arguments: {G : Type u_4} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G]
Goal: (MeasurableEquiv.inv G).symm = MeasurableEquiv.inv G
-- MeasurableEquiv.symm_neg
@[to_additive (attr := simp)]
theorem symm_inv {G} [MeasurableSpace G] [InvolutiveInv G] [MeasurableInv G] :
    (inv G).symm = inv G 
Arguments: {G : Type u_4} [MeasurableSpace G] [InvolutiveNeg G] [MeasurableNeg G]
Goal: (MeasurableEquiv.neg G).symm = MeasurableEquiv.neg G
-- MeasureTheory.OuterMeasure.IsMetric.finset_iUnion_of_pairwise_separated
theorem finset_iUnion_of_pairwise_separated (hm : IsMetric μ) {I : Finset ι} {s : ι → Set X}
    (hI : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → IsMetricSeparated (s i) (s j)) :
    μ (⋃ i ∈ I, s i) = ∑ i ∈ I, μ (s i) 
Arguments: {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {μ : MeasureTheory.OuterMeasure X} (hm : μ.IsMetric) {I : Finset ι} {s : ι → Set X} (hI : ∀ i ∈ I, ∀ j ∈ I, i ≠ j → IsMetricSeparated (s i) (s j))
Goal: μ (⋃ i ∈ I, s i) = ∑ i ∈ I, μ (s i)
-- MeasureTheory.OuterMeasure.IsMetric.borel_le_caratheodory
theorem borel_le_caratheodory (hm : IsMetric μ) : borel X ≤ μ.caratheodory 
Arguments: {X : Type u_2} [EMetricSpace X] {μ : MeasureTheory.OuterMeasure X} (hm : μ.IsMetric)
Goal: borel X ≤ μ.caratheodory
-- MeasureTheory.OuterMeasure.IsMetric.le_caratheodory
theorem le_caratheodory [MeasurableSpace X] [BorelSpace X] (hm : IsMetric μ) :
    ‹MeasurableSpace X› ≤ μ.caratheodory 
Arguments: {X : Type u_2} [EMetricSpace X] {μ : MeasureTheory.OuterMeasure X} [MeasurableSpace X] [BorelSpace X] (hm : μ.IsMetric)
Goal: inst✝ ≤ μ.caratheodory
-- MeasureTheory.OuterMeasure.mkMetric'.le_pre
theorem le_pre : μ ≤ pre m r ↔ ∀ s : Set X, diam s ≤ r → μ s ≤ m s 
Arguments: {X : Type u_2} [EMetricSpace X] {m : Set X → ℝ≥0∞} {r : ℝ≥0∞} {μ : MeasureTheory.OuterMeasure X}
Goal: μ ≤ MeasureTheory.OuterMeasure.mkMetric'.pre m r ↔ ∀ (s : Set X), EMetric.diam s ≤ r → μ s ≤ m s
-- MeasureTheory.OuterMeasure.mkMetric'.pre_le
theorem pre_le (hs : diam s ≤ r) : pre m r s ≤ m s 
Arguments: {X : Type u_2} [EMetricSpace X] {m : Set X → ℝ≥0∞} {r : ℝ≥0∞} {s : Set X} (hs : EMetric.diam s ≤ r)
Goal: (MeasureTheory.OuterMeasure.mkMetric'.pre m r) s ≤ m s
-- MeasureTheory.OuterMeasure.mkMetric'.mono_pre
theorem mono_pre (m : Set X → ℝ≥0∞) {r r' : ℝ≥0∞} (h : r ≤ r') : pre m r' ≤ pre m r 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞) {r r' : ℝ≥0∞} (h : r ≤ r')
Goal: MeasureTheory.OuterMeasure.mkMetric'.pre m r' ≤ MeasureTheory.OuterMeasure.mkMetric'.pre m r
-- MeasureTheory.OuterMeasure.mkMetric'.mono_pre_nat
theorem mono_pre_nat (m : Set X → ℝ≥0∞) : Monotone fun k : ℕ => pre m k⁻¹ 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞)
Goal: Monotone fun k => MeasureTheory.OuterMeasure.mkMetric'.pre m (↑k)⁻¹
-- MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre
theorem tendsto_pre (m : Set X → ℝ≥0∞) (s : Set X) :
    Tendsto (fun r => pre m r s) (𝓝[>] 0) (𝓝 <| mkMetric' m s) 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞) (s : Set X)
Goal: Filter.Tendsto (fun r => (MeasureTheory.OuterMeasure.mkMetric'.pre m r) s) (𝓝[>] 0)
  (𝓝 ((MeasureTheory.OuterMeasure.mkMetric' m) s))
-- MeasureTheory.OuterMeasure.mkMetric'.tendsto_pre_nat
theorem tendsto_pre_nat (m : Set X → ℝ≥0∞) (s : Set X) :
    Tendsto (fun n : ℕ => pre m n⁻¹ s) atTop (𝓝 <| mkMetric' m s) 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞) (s : Set X)
Goal: Filter.Tendsto (fun n => (MeasureTheory.OuterMeasure.mkMetric'.pre m (↑n)⁻¹) s) Filter.atTop
  (𝓝 ((MeasureTheory.OuterMeasure.mkMetric' m) s))
-- MeasureTheory.OuterMeasure.mkMetric'.eq_iSup_nat
theorem eq_iSup_nat (m : Set X → ℝ≥0∞) : mkMetric' m = ⨆ n : ℕ, mkMetric'.pre m n⁻¹ 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞)
Goal: MeasureTheory.OuterMeasure.mkMetric' m = ⨆ n, MeasureTheory.OuterMeasure.mkMetric'.pre m (↑n)⁻¹
-- MeasureTheory.OuterMeasure.mkMetric'.trim_pre
theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
    (hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞) (hcl : ∀ (s : Set X), m (closure s) = m s) (r : ℝ≥0∞)
Goal: (MeasureTheory.OuterMeasure.mkMetric'.pre m r).trim = MeasureTheory.OuterMeasure.mkMetric'.pre m r
-- MeasureTheory.OuterMeasure.mkMetric'_isMetric
theorem mkMetric'_isMetric (m : Set X → ℝ≥0∞) : (mkMetric' m).IsMetric 
Arguments: {X : Type u_2} [EMetricSpace X] (m : Set X → ℝ≥0∞)
Goal: (MeasureTheory.OuterMeasure.mkMetric' m).IsMetric
-- MeasureTheory.OuterMeasure.mkMetric_mono_smul
theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0)
    (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : OuterMeasure X) ≤ c • mkMetric m₂ 
Arguments: {X : Type u_2} [EMetricSpace X] {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ⊤) (h0 : c ≠ 0) (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂)
Goal: MeasureTheory.OuterMeasure.mkMetric m₁ ≤ c • MeasureTheory.OuterMeasure.mkMetric m₂
-- MeasureTheory.OuterMeasure.mkMetric_top
@[simp]
theorem mkMetric_top : (mkMetric (fun _ => ∞ : ℝ≥0∞ → ℝ≥0∞) : OuterMeasure X) = ⊤ 
Arguments: {X : Type u_2} [EMetricSpace X]
Goal: (MeasureTheory.OuterMeasure.mkMetric fun x => ⊤) = ⊤
-- MeasureTheory.OuterMeasure.mkMetric_mono
theorem mkMetric_mono {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} (hle : m₁ ≤ᶠ[𝓝[≥] 0] m₂) :
    (mkMetric m₁ : OuterMeasure X) ≤ mkMetric m₂ 
Arguments: {X : Type u_2} [EMetricSpace X] {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} (hle : m₁ ≤ᶠ[𝓝[≥] 0] m₂)
Goal: MeasureTheory.OuterMeasure.mkMetric m₁ ≤ MeasureTheory.OuterMeasure.mkMetric m₂
-- MeasureTheory.OuterMeasure.isometry_comap_mkMetric
theorem isometry_comap_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f)
    (H : Monotone m ∨ Surjective f) : comap f (mkMetric m) = mkMetric m 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f) (H : Monotone m ∨ Function.Surjective f)
Goal: (MeasureTheory.OuterMeasure.comap f) (MeasureTheory.OuterMeasure.mkMetric m) = MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.mkMetric_smul
theorem mkMetric_smul (m : ℝ≥0∞ → ℝ≥0∞) {c : ℝ≥0∞} (hc : c ≠ ∞) (hc' : c ≠ 0) :
    (mkMetric (c • m) : OuterMeasure X) = c • mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] (m : ℝ≥0∞ → ℝ≥0∞) {c : ℝ≥0∞} (hc : c ≠ ⊤) (hc' : c ≠ 0)
Goal: MeasureTheory.OuterMeasure.mkMetric (c • m) = c • MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.mkMetric_nnreal_smul
theorem mkMetric_nnreal_smul (m : ℝ≥0∞ → ℝ≥0∞) {c : ℝ≥0} (hc : c ≠ 0) :
    (mkMetric (c • m) : OuterMeasure X) = c • mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] (m : ℝ≥0∞ → ℝ≥0∞) {c : ℝ≥0} (hc : c ≠ 0)
Goal: MeasureTheory.OuterMeasure.mkMetric (c • m) = c • MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.isometry_map_mkMetric
theorem isometry_map_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f)
    (H : Monotone m ∨ Surjective f) : map f (mkMetric m) = restrict (range f) (mkMetric m) 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f) (H : Monotone m ∨ Function.Surjective f)
Goal: (MeasureTheory.OuterMeasure.map f) (MeasureTheory.OuterMeasure.mkMetric m) =
  (MeasureTheory.OuterMeasure.restrict (Set.range f)) (MeasureTheory.OuterMeasure.mkMetric m)
-- MeasureTheory.OuterMeasure.isometryEquiv_comap_mkMetric
theorem isometryEquiv_comap_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (f : X ≃ᵢ Y) :
    comap f (mkMetric m) = mkMetric m 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ℝ≥0∞ → ℝ≥0∞) (f : X ≃ᵢ Y)
Goal: (MeasureTheory.OuterMeasure.comap ⇑f) (MeasureTheory.OuterMeasure.mkMetric m) = MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.isometryEquiv_map_mkMetric
theorem isometryEquiv_map_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (f : X ≃ᵢ Y) :
    map f (mkMetric m) = mkMetric m 
Arguments: {X : Type u_2} {Y : Type u_3} [EMetricSpace X] [EMetricSpace Y] (m : ℝ≥0∞ → ℝ≥0∞) (f : X ≃ᵢ Y)
Goal: (MeasureTheory.OuterMeasure.map ⇑f) (MeasureTheory.OuterMeasure.mkMetric m) = MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.trim_mkMetric
theorem trim_mkMetric [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) :
    (mkMetric m : OuterMeasure X).trim = mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞)
Goal: (MeasureTheory.OuterMeasure.mkMetric m).trim = MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.le_mkMetric
theorem le_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (μ : OuterMeasure X) (r : ℝ≥0∞) (h0 : 0 < r)
    (hr : ∀ s, diam s ≤ r → μ s ≤ m (diam s)) : μ ≤ mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] (m : ℝ≥0∞ → ℝ≥0∞) (μ : MeasureTheory.OuterMeasure X) (r : ℝ≥0∞) (h0 : 0 < r) (hr : ∀ (s : Set X), EMetric.diam s ≤ r → μ s ≤ m (EMetric.diam s))
Goal: μ ≤ MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.Measure.mkMetric'_toOuterMeasure
@[simp]
theorem mkMetric'_toOuterMeasure (m : Set X → ℝ≥0∞) :
    (mkMetric' m).toOuterMeasure = (OuterMeasure.mkMetric' m).trim 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : Set X → ℝ≥0∞)
Goal: (MeasureTheory.Measure.mkMetric' m).toOuterMeasure = (MeasureTheory.OuterMeasure.mkMetric' m).trim
-- MeasureTheory.Measure.mkMetric_toOuterMeasure
@[simp]
theorem mkMetric_toOuterMeasure (m : ℝ≥0∞ → ℝ≥0∞) :
    (mkMetric m : Measure X).toOuterMeasure = OuterMeasure.mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞)
Goal: (MeasureTheory.Measure.mkMetric m).toOuterMeasure = MeasureTheory.OuterMeasure.mkMetric m
-- MeasureTheory.OuterMeasure.coe_mkMetric
theorem OuterMeasure.coe_mkMetric [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) :
    ⇑(OuterMeasure.mkMetric m : OuterMeasure X) = Measure.mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞)
Goal: ⇑(MeasureTheory.OuterMeasure.mkMetric m) = ⇑(MeasureTheory.Measure.mkMetric m)
-- MeasureTheory.Measure.mkMetric_mono_smul
theorem mkMetric_mono_smul {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ∞) (h0 : c ≠ 0)
    (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂) : (mkMetric m₁ : Measure X) ≤ c • mkMetric m₂ 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} {c : ℝ≥0∞} (hc : c ≠ ⊤) (h0 : c ≠ 0) (hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂)
Goal: MeasureTheory.Measure.mkMetric m₁ ≤ c • MeasureTheory.Measure.mkMetric m₂
-- MeasureTheory.Measure.mkMetric_top
@[simp]
theorem mkMetric_top : (mkMetric (fun _ => ∞ : ℝ≥0∞ → ℝ≥0∞) : Measure X) = ⊤ 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X]
Goal: (MeasureTheory.Measure.mkMetric fun x => ⊤) = ⊤
-- MeasureTheory.Measure.mkMetric_mono
theorem mkMetric_mono {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} (hle : m₁ ≤ᶠ[𝓝[≥] 0] m₂) :
    (mkMetric m₁ : Measure X) ≤ mkMetric m₂ 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {m₁ m₂ : ℝ≥0∞ → ℝ≥0∞} (hle : m₁ ≤ᶠ[𝓝[≥] 0] m₂)
Goal: MeasureTheory.Measure.mkMetric m₁ ≤ MeasureTheory.Measure.mkMetric m₂
-- MeasureTheory.Measure.mkMetric_apply
theorem mkMetric_apply (m : ℝ≥0∞ → ℝ≥0∞) (s : Set X) :
    mkMetric m s =
      ⨆ (r : ℝ≥0∞) (_ : 0 < r),
        ⨅ (t : ℕ → Set X) (_ : s ⊆ iUnion t) (_ : ∀ n, diam (t n) ≤ r),
          ∑' n, ⨆ _ : (t n).Nonempty, m (diam (t n)) 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) (s : Set X)
Goal: (MeasureTheory.Measure.mkMetric m) s =
  ⨆ r,
    ⨆ (_ : 0 < r),
      ⨅ t,
        ⨅ (_ : s ⊆ Set.iUnion t),
          ⨅ (_ : ∀ (n : ℕ), EMetric.diam (t n) ≤ r), ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), m (EMetric.diam (t n))
-- MeasureTheory.Measure.le_mkMetric
theorem le_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (μ : Measure X) (ε : ℝ≥0∞) (h₀ : 0 < ε)
    (h : ∀ s : Set X, diam s ≤ ε → μ s ≤ m (diam s)) : μ ≤ mkMetric m 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] (m : ℝ≥0∞ → ℝ≥0∞) (μ : MeasureTheory.Measure X) (ε : ℝ≥0∞) (h₀ : 0 < ε) (h : ∀ (s : Set X), EMetric.diam s ≤ ε → μ s ≤ m (EMetric.diam s))
Goal: μ ≤ MeasureTheory.Measure.mkMetric m
-- MeasureTheory.Measure.mkMetric_le_liminf_tsum
theorem mkMetric_le_liminf_tsum {β : Type*} {ι : β → Type*} [∀ n, Countable (ι n)] (s : Set X)
    {l : Filter β} (r : β → ℝ≥0∞) (hr : Tendsto r l (𝓝 0)) (t : ∀ n : β, ι n → Set X)
    (ht : ∀ᶠ n in l, ∀ i, diam (t n i) ≤ r n) (hst : ∀ᶠ n in l, s ⊆ ⋃ i, t n i) (m : ℝ≥0∞ → ℝ≥0∞) :
    mkMetric m s ≤ liminf (fun n => ∑' i, m (diam (t n i))) l 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {β : Type u_4} {ι : β → Type u_5} [∀ (n : β), Countable (ι n)] (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Filter.Tendsto r l (𝓝 0)) (t : (n : β) → ι n → Set X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), EMetric.diam (t n i) ≤ r n) (hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i) (m : ℝ≥0∞ → ℝ≥0∞)
Goal: (MeasureTheory.Measure.mkMetric m) s ≤ Filter.liminf (fun n => ∑' (i : ι n), m (EMetric.diam (t n i))) l
-- MeasureTheory.Measure.mkMetric_le_liminf_sum
theorem mkMetric_le_liminf_sum {β : Type*} {ι : β → Type*} [hι : ∀ n, Fintype (ι n)] (s : Set X)
    {l : Filter β} (r : β → ℝ≥0∞) (hr : Tendsto r l (𝓝 0)) (t : ∀ n : β, ι n → Set X)
    (ht : ∀ᶠ n in l, ∀ i, diam (t n i) ≤ r n) (hst : ∀ᶠ n in l, s ⊆ ⋃ i, t n i) (m : ℝ≥0∞ → ℝ≥0∞) :
    mkMetric m s ≤ liminf (fun n => ∑ i, m (diam (t n i))) l 
Arguments: {X : Type u_2} [EMetricSpace X] [MeasurableSpace X] [BorelSpace X] {β : Type u_4} {ι : β → Type u_5} [hι : (n : β) → Fintype (ι n)] (s : Set X) {l : Filter β} (r : β → ℝ≥0∞) (hr : Filter.Tendsto r l (𝓝 0)) (t : (n : β) → ι n → Set X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), EMetric.diam (t n i) ≤ r n) (hst : ∀ᶠ (n : β) in l, s ⊆ ⋃ i, t n i) (m : ℝ≥0∞ → ℝ≥0∞)
Goal: (MeasureTheory.Measure.mkMetric m) s ≤ Filter.liminf (fun n => ∑ i : ι n, m (EMetric.diam (t n i))) l