module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | [
{
"pp": "case open_pos.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace... | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 38
} | [
{
"pp": "case open_pos.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁸ : MeasurableSpace α\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝⁵ : SFinite ν✝\nX : Type u_4\nY : Type u_5\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace... | exact v_open.measure_pos ν ⟨y, yv⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 873,
"column": 54
} | {
"line": 873,
"column": 65
} | [
{
"pp": "E : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι) (h : x_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 877,
"column": 42
} | {
"line": 877,
"column": 64
} | [
{
"pp": "case bc\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nε : ℝ≥0∞\nhε : 0 < ε\nK : Finset ι\nhK : ∑' (i : { j // j ∉ K }), 2⁻¹ ^ encode ↑i < ε / 2\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ * ↑K.card < ε / 2\nx y : (i : ι) → F i\nhxy : ∀ (x_1 : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ μ' : Measure α\nν ν' : Measure β\ninst✝ : SFinite ν'\nh1 : μ ≪ μ'\nh2 : ν ≪ ν'\ns : Set (α × β)\nhs : MeasurableSet s\nh2s : (fun x ↦ ν' (Prod.mk x ⁻¹' s)) =ᶠ[ae μ'] 0\n⊢ (fun x ↦ ν (Prod.mk x ⁻¹' s)) =ᶠ[ae μ] 0",
... | exact (h2s.filter_mono h1.ae_le).mono fun _ h => h2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 926,
"column": 4
} | {
"line": 926,
"column": 25
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\n⊢ Summable fun i ↦ 2⁻¹ ^ encode i",
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 934,
"column": 2
} | {
"line": 934,
"column": 41
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoMetricSpace (F i)\nx y : (i : ι) → F i\ni : ι\nh : dist x y < 2⁻¹ ^ encode i\n⊢ dist (x i) (y i) ≤ dist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1014,
"column": 2
} | {
"line": 1014,
"column": 83
} | [
{
"pp": "ι : Type u_2\nX : Type u_3\nY : ι → Type u_4\nf : (i : ι) → X → Y i\nseparating_f : Pairwise fun x y ↦ ∃ i, f i x ≠ f i y\n⊢ Injective (embed X Y f)",
"usedConstants": [
"_private.Mathlib.Topology.MetricSpace.PiNat.0.PiCountable.emetricSpace._simp_1",
"Eq.mpr",
"Metric.PiNatEmbed"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 145,
"column": 31
} | {
"line": 145,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\nh : IsComplement S T\nx : G\n⊢ x ∈ S * T",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.GroupTheory.Complement.0.Subgroup.IsComplement.mul_eq._simp_1_1",
"HMul.hMul",
"Monoid.toMulOneClass",
"Membership.mem",
"Exists"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 43
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nT : Set G\nh : IsComplement ∅ T\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 153,
"column": 13
} | {
"line": 153,
"column": 43
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Set G\nh : IsComplement S ∅\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 202,
"column": 65
} | {
"line": 202,
"column": 76
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 210,
"column": 65
} | {
"line": 210,
"column": 76
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑T\nhy : (fun t ↦ g * (↑t)⁻¹ ∈ S) y\n⊢ ?m.180 = ?m.181",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1138,
"column": 6
} | {
"line": 1138,
"column": 95
} | [
{
"pp": "X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Complement | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 46
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\n⊢ IsComplement (↑H) (range f)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Quotient.mk''",
"Set.Elem",
... | isComplement_subgroup_left_iff_bijective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1141,
"column": 4
} | {
"line": 1141,
"column": 15
} | [
{
"pp": "case inr.refine_2\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis✝ : Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 167,
"column": 56
} | {
"line": 167,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nh : ∀ (s : Set β), IsCompact s → IsCompact (f ⁻¹' s)\ns : Set β\nhs : s ∈ cocompact β\nt : Set β\nht : IsCompact t\nhts : tᶜ ⊆ s\n⊢ (f ⁻¹' t)ᶜ ⊆ f ⁻¹' s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\n⊢ ?m.15 ∈ cocompact ?m.13",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 181,
"column": 65
} | {
"line": 181,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\nt : Set α\nht : IsCompact t\nhts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t\n⊢ ⇑f ⁻¹' s ⊆ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Prod | {
"line": 149,
"column": 4
} | {
"line": 150,
"column": 24
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhμs : μ s = 0\nhf : Measurable fun z ↦ (z.2 * z.1, z.1⁻¹)\nthis : (map (fun z ↦ (z.2 * z.1,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 23
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Group.LIntegral | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 13
} | [
{
"pp": "G : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ninst✝² : InvolutiveInv G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → ℝ≥0∞\n⊢ ∫⁻ (x : G), f x⁻¹ ∂μ = ∫⁻ (x : G), f x ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 733,
"column": 6
} | {
"line": 733,
"column": 38
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1023,
"column": 11
} | {
"line": 1023,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.prod ν)\n⊢ ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν",
"usedConstants": [
... | ← lintegral_prod_swap f | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 755,
"column": 8
} | {
"line": 755,
"column": 35
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LConvolution | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ 0 ⋆ₘₗ[μ] f = 0",
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiri... | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LConvolution | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ 0 ⋆ₘₗ[μ] f = 0",
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiri... | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LConvolution | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 27
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ f ⋆ₘₗ[μ] 0 = 0",
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiri... | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LConvolution | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 27
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ f ⋆ₘₗ[μ] 0 = 0",
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiri... | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LConvolution | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 25
} | [
{
"pp": "case h.hf\nG : Type u_1\nmG : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : SFinite μ\nf g k : G → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhk : AEMeasurable k μ\nx : G\n⊢ AEMeasurable (Function... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Measure | {
"line": 619,
"column": 2
} | {
"line": 619,
"column": 62
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : BorelSpace G\nμ : Measure G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : μ.Regular\nhμ : μ ≠ 0\ns : Set G\nhs : IsOpen[inst✝⁵] s\n⊢ μ s ≠ 0 ↔ s.Nonempty",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 780,
"column": 8
} | {
"line": 780,
"column": 35
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝¹ : UpgradedIsCompletelyMetrizableSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
"usedConstants": [
"MeasureTheory.withDensity_apply_le",
"MeasureTheory.Measure.withDensity",
"Trans.trans",
"MeasureTheory.Meas... | apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFini... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
"usedConstants": [
"MeasureTheory.withDensity_apply_le",
"MeasureTheory.Measure.withDensity",
"Trans.trans",
"MeasureTheory.Meas... | apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFini... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 29
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : Measurable g\n⊢ μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 44
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (r • f)) s = (r • μ.withDensity f) s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"instHSMul",
"MeasureThe... | rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 51
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nc : ε\nhc : ‖c‖ₑ ≠ ∞\n⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ",
"usedConstants": [
"_private.Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral.0.MeasureTheory.hasFiniteIntegral_c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 255,
"column": 35
} | {
"line": 255,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (-f) μ",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Preorder.toLT",
"Pi.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 263,
"column": 39
} | {
"line": 263,
"column": 80
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ",
"usedConstants": [
"_private.Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral.0.MeasureTheory.HasFiniteIntegral.enorm._si... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 267,
"column": 47
} | {
"line": 267,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun a ↦ ‖f a‖) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Pre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 50
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : μ ({x | f x ≠ 0} ∩ s) = 0\nt✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)\nA : s ⊆ t✝ ∪ {x | f x = 0}\ng : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᶠ[ae μ] g\nt : {x | f x = 0} =ᶠ[ae (μ.withDensity f)] {x | g x = 0}\n⊢ (μ... | rw [measure_congr t, withDensity_congr_ae hfg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 333,
"column": 37
} | {
"line": 333,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nf' : α → ℝ≥0\nhf'_m : Measurable f'\nhf'_ae : f =ᶠ[ae μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : ↑(f a) ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 410,
"column": 4
} | {
"line": 410,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nF : ℕ → α → β\nf : α → β\nbound : α → ℝ\nF_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ\nbound_hasFiniteIntegral : HasFiniteIntegral bound μ\nh_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a\nh_... | refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 363,
"column": 2
} | {
"line": 365,
"column": 66
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ (dirac a).withDensity f = f a • dirac a",
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HMul.hMul",
... | ext s hs
classical
simp [withDensity_apply f hs, setLIntegral_dirac, Set.indicator] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 363,
"column": 2
} | {
"line": 365,
"column": 66
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ (dirac a).withDensity f = f a • dirac a",
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HMul.hMul",
... | ext s hs
classical
simp [withDensity_apply f hs, setLIntegral_dirac, Set.indicator] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 441,
"column": 65
} | {
"line": 444,
"column": 19
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nhg : AEMeasurable g (μ.withDensity f)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ",
"usedConstants": [
"Eq.mpr",
"Measure... | by
rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul₀' hf.restrict]
rw [← restrict_withDensity hs]
exact hg.restrict | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 66
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : HasFiniteIntegral f μ\nx : α\n⊢ ‖min (f x) 0‖ ≤ ‖f x‖",
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 57
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_7\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\nh : HasFiniteIntegral (↑c • f) μ\n⊢ HasFiniteIntegral f μ",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 523,
"column": 2
} | {
"line": 523,
"column": 37
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nh : HasFiniteIntegral f μ\ns : Set α\n⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Sqrt | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 29
} | [
{
"pp": "case mp\nx y : ℝ\nh : 0 ≤ y\n⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 27
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ≤ |z.re| + |z.im|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Sqrt | {
"line": 266,
"column": 56
} | {
"line": 266,
"column": 67
} | [
{
"pp": "x : ℝ\nh : 0 ≤ x\n⊢ √x = 0 ↔ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | [
{
"pp": "z : ℂ\n⊢ |z.im| < ‖z‖ ↔ z.re ≠ 0",
"usedConstants": [
"Norm.norm",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"Complex.im",
"Real.instLT",
"Complex.instNorm",
"id",
"Real.instAddGroup",
"Ne",
"Complex.re",
"Iff",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Sqrt | {
"line": 426,
"column": 63
} | {
"line": 430,
"column": 20
} | [
{
"pp": "x : ℝ\nh : -1 ≤ x\n⊢ √(1 + x) ≤ 1 + x / 2",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | by
refine sqrt_le_iff.mpr ⟨by linarith, ?_⟩
calc 1 + x
_ ≤ 1 + x + (x / 2) ^ 2 := le_add_of_nonneg_right <| sq_nonneg _
_ = _ := by ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Sqrt | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 17
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Sqrt | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 86
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"O... | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Real.Sqrt | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 86
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"O... | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Sqrt | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 86
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Eq.mp",
"O... | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Sqrt | {
"line": 494,
"column": 2
} | {
"line": 494,
"column": 17
} | [
{
"pp": "ι : Type u_2\nf g : ι → ℝ\ns : Finset ι\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\n⊢ ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 277,
"column": 23
} | {
"line": 277,
"column": 34
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nx✝ : ℝ\nε0 : x✝ > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ (↑f n).re) j - (fun n ↦ (↑f n).re) i| ≤ ‖↑f j - ↑f i‖",
"usedConstants": [
"Norm.norm",
"Real",
"Real.lattice",
"AddGroupWithOne.toAddGroup",
"abs",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 62
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nε : ℝ\nε0 : ε > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ (↑f n).im) j - (fun n ↦ (↑f n).im) i| < ε",
"usedConstants": [
"Norm.norm",
"Real",
"Preorder.toLT",
"Real.lattice",
"AddGroupWithOne.toAddGroup",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 15
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nε : ℝ\nε0 : ε > 0\nx✝ : ℕ\nH :\n ∀ j ≥ x✝,\n |↑(⟨fun n ↦ (↑f n).re, ⋯⟩ - CauSeq.const abs (CauSeq.lim ⟨fun n ↦ (↑f n).re, ⋯⟩)) j| < ε / 2 ∧\n |↑(⟨fun n ↦ (↑f n).im, ⋯⟩ - CauSeq.const abs (CauSeq.lim ⟨fun n ↦ (↑f n).im, ⋯⟩)) j| < ε / 2\nj : ℕ\nij : j ≥ x✝\nH₁ : |↑(⟨fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 377,
"column": 51
} | {
"line": 377,
"column": 62
} | [
{
"pp": "z : ℂ\nhz : z ∈ Metric.sphere 0 1\n⊢ ‖z‖ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : ‖x‖ ≤ 1\n⊢ normSq (-↑x + I * ↑√(1 - x ^ 2)) = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 45
} | [
{
"pp": "z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)",
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"and_true",
"Real.instZero",
"congrArg",
"sub_zero",
"Complex.im",
"Real.instSub",
"Nat.instAtLeast... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Order | {
"line": 83,
"column": 30
} | {
"line": 83,
"column": 45
} | [
{
"pp": "z : ℂ\nh : z.re = 0\n⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"and_true",
"Real.instZero",
"congrArg",
"true_or",
"Complex.im",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.MonoidHom | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 77
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\nD : Type u_5\ninst✝⁷ : Mul A\ninst✝⁶ : Mul B\ninst✝⁵ : Mul C\ninst✝⁴ : Mul D\ninst✝³ : Star A\ninst✝² : Star B\ninst✝¹ : Star C\ninst✝ : Star D\ne : A ≃⋆* B\nb : B\n⊢ e.symm.toFun (star b) = star (e.symm.toFun b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 81
} | [
{
"pp": "V : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nK : ℝ≥0\nh : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 150,
"column": 49
} | {
"line": 150,
"column": 89
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nh : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖\nx y : V₁\n⊢ dist x y ≤ ↑K * dist (f x) (f y)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 20
} | [
{
"pp": "case pos\nV₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC C' : ℝ\nh : f.SurjectiveOnWith K C\nH : C ≤ C'\ng : V₁\nk_in : f g ∈ K\nhg : ‖g‖ ≤ C * ‖f g‖\nHg : ‖f g‖ = 0\n⊢ ‖g‖ ≤ C' * ‖f g‖",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 254,
"column": 36
} | {
"line": 254,
"column": 80
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nhf : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 714,
"column": 4
} | {
"line": 714,
"column": 25
} | [
{
"pp": "case refine_1\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : f.NormNoninc\nv : V\n⊢ ‖f v‖ ≤ 1 * ‖v‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"NormedAddGroupHom",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 715,
"column": 4
} | {
"line": 715,
"column": 15
} | [
{
"pp": "case refine_2\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : ‖f‖ ≤ 1\nv : V\n⊢ ‖f v‖ ≤ ‖v‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 726,
"column": 17
} | {
"line": 726,
"column": 28
} | [
{
"pp": "V₁ : Type u_3\nV₂ : Type u_4\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nh : (-f).NormNoninc\nx : V₁\n⊢ ‖f x‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 57,
"column": 51
} | {
"line": 57,
"column": 62
} | [
{
"pp": "E : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : StarAddMonoid E\ninst✝ : NormedStarGroup E\nx : E\n⊢ ‖x‖ ≤ ‖x⋆‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 108,
"column": 10
} | {
"line": 108,
"column": 41
} | [
{
"pp": "case inr\nE : Type u_2\ninst✝¹ : NonUnitalNormedRing E\ninst✝ : StarRing E\nh : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"NonUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 17
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 30
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : IsSelfAdjoint x\n⊢ ‖x * x‖ = ‖x‖ ^ 2",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 44
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\n⊢ x * x⋆ = 0 ↔ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 190,
"column": 27
} | {
"line": 190,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\nι : Type u_4\nR₁ : Type u_5\nR₂ : Type u_6\nR : ι → Type u_7\ninst✝⁹ : NonUnitalNormedRing R₁\ninst✝⁸ : StarRing R₁\ninst✝⁷ : CStarRing R₁\ninst✝⁶ : NonUnitalNormedRing R₂\ninst✝⁵ : StarRing R₂\ninst✝⁴ : CStarRing R₂\ninst✝³ : (i : ι) → NonUnitalNormedRing (R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nA : E\nU : ↥(unitary E)\n⊢ ‖A * ↑U‖ = ‖A‖",
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"HMul.hMul",
"Ring.toNonAssocRing",
"Monoid.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Unitary | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 29
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : StarMul G\na b : G\n⊢ a⁻¹ * b ∈ unitary G ↔ a * star a = b * star b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Unitary | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 33
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁶ : Monoid R\ninst✝⁵ : StarMul R\ninst✝⁴ : Monoid S\ninst✝³ : StarMul S\nF : Type u_5\ninst✝² : FunLike F R S\ninst✝¹ : StarHomClass F R S\ninst✝ : MonoidHomClass F R S\nf : F\nr : R\nhr : star r * r = 1 ∧ r * star r = 1\n⊢ f r ∈ unitary S",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Unitary | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 13
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : StarMul A\na : A\nU : ↥(unitary A)\n⊢ spectrum R (star ↑U * a * ↑U) = spectrum R a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 18
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalSpace M₂\nσ : R →+* R₂\nL : SummationFilte... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 141,
"column": 14
} | {
"line": 141,
"column": 67
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 142,
"column": 15
} | {
"line": 142,
"column": 63
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : ... | simp only [tsum_bot hL, eq_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 17
} | [
{
"pp": "case neg.refine_2\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 43
} | [
{
"pp": "⊢ Continuous ⇑normSq",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 13
} | [
{
"pp": "z : ℂ\n⊢ ‖equivRealProd z‖ ≤ 1 * ‖z‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Equiv.instEquivLike",
"HMul.hMul",
"Real.lattice",
"abs",
"congrArg",
"Real.instSemilatticeSup",
"Complex.im",
"Prod.toNor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | [
{
"pp": "⊢ LipschitzWith 1 ⇑equivRealProd",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 54
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ≤ ↑(NNReal.sqrt 2) * ‖equivRealProdLm z‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Prod.seminormedAddGroup",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 9
} | [
{
"pp": "⊢ Tendsto (⇑normSq) (cocompact ℂ) atTop",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 37
} | [
{
"pp": "f : ℂ →+* ℂ\nhf : Continuous ⇑f\n⊢ f = RingHom.id ℂ ∨ f = starRingEnd ℂ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"RingHom",
"id",
"RingHom.instFunLike",
"Complex.instCommSemiring",
"congr",
"Complex.instStarRing",
"Or",
"_private.Mathlib... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 525,
"column": 15
} | {
"line": 525,
"column": 70
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nx : ℝ\nh : HasSum (fun x ↦ ↑(f x)) (↑x) L\n⊢ HasSum f x L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 530,
"column": 15
} | {
"line": 530,
"column": 70
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nh : Summable (fun x ↦ ↑(f x)) L\n⊢ Summable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 30
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → 𝕜\nc : 𝕜\nh₁ : HasSum (fun x ↦ re (f x)) (re c) L\nh₂ : HasSum (fun x ↦ im (f x)) (im c) L\n⊢ HasSum f c L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 654,
"column": 2
} | {
"line": 654,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ -↑x ∈ slitPlane ↔ x < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 31
} | [
{
"pp": "n : ℕ\n⊢ ↑n ∈ slitPlane ↔ n ≠ 0",
"usedConstants": [
"Membership.mem",
"id",
"Ne",
"instOfNatNat",
"Complex.instNatCast",
"Nat.cast",
"Iff",
"Nat",
"Complex",
"OfNat.ofNat",
"Set.instMembership",
"Complex.slitPlane",
"Set... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.