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.Analysis.Calculus.LagrangeMultipliers | {
"line": 123,
"column": 11
} | {
"line": 123,
"column": 33
} | [
{
"pp": "case refine_2.h\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 26
} | [
{
"pp": "case intro.refine_1\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 142,
"column": 4
} | {
"line": 143,
"column": 23
} | [
{
"pp": "case intro.refine_2\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 34
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[>] a) (𝓝 0)\nhga : Tendsto g (𝓝[>] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) ... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.LHopital | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 34
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) ... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 40
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → ℝ≥0∞\n⊢ ∫⁻ (x : ℝ) in f '' s, g x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal |f' x| * g (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 40
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ IntegrableOn g (f '' s) volume ↔ IntegrableOn (fun x ↦ |f' x| • g (f x)) s volume",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 40
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 26
} | [
{
"pp": "case refine_2\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 19
} | [
{
"pp": "case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 34
} | [
{
"pp": "l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atTop, ∀ y ∈ v, g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\n⊢ Tendsto (fun x ↦... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.LHopital | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 34
} | [
{
"pp": "l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\n⊢ Tendsto (fun x ↦... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 157,
"column": 8
} | {
"line": 157,
"column": 62
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nr : (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 138,
"column": 70
} | {
"line": 138,
"column": 93
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ioi (dist x y)\n⊢ y ∈ Metric.ball x (r a)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 145,
"column": 70
} | {
"line": 145,
"column": 93
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ici (dist x y)\n⊢ y ∈ Metric.closedBall x (r a)",
"usedConstants": [
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 32
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nr : (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 169,
"column": 18
} | {
"line": 169,
"column": 29
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 434,
"column": 8
} | {
"line": 434,
"column": 52
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 440,
"column": 8
} | {
"line": 440,
"column": 52
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i ↦ ∫⁻ (x : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) = volume (f '' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 37
} | [
{
"pp": "case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 88
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\ng : ℝ → F\nH : ¬IntegrableOn g (f '' s) volume\n⊢ ¬Integrable (fun x ↦ f' x • g (f x)) (volume.restrict s)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 375,
"column": 6
} | {
"line": 375,
"column": 37
} | [
{
"pp": "case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 729,
"column": 6
} | {
"line": 729,
"column": 64
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nε : ℝ\nεpos : ε > 0\nL : Tendsto (fun n ↦ ∫ (x : ℝ) in Ici ↑n, ‖f' x‖) atTop (𝓝 (∫ (x : ℝ) in ⋂ n, I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 37
} | [
{
"pp": "case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ deriv f z",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 408,
"column": 6
} | {
"line": 408,
"column": 37
} | [
{
"pp": "case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 336,
"column": 8
} | {
"line": 336,
"column": 90
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ⋯\nε : ℝ\nhε : μ (closedBall... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 37
} | [
{
"pp": "case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ deriv f z",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 13
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : AntitoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (-f' x) = volume (f '' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 37
} | [
{
"pp": "case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 436,
"column": 6
} | {
"line": 436,
"column": 37
} | [
{
"pp": "case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ deriv f z ≤ 0",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 943,
"column": 4
} | {
"line": 943,
"column": 15
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nx : ℝ\nhx : x ∈ Ioi (-a)\nthis : -x ∈ Iic a\n⊢ HasDerivAt g (-f' (-x)) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 950,
"column": 2
} | {
"line": 950,
"column": 17
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nhdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x\nL : Tendsto g atTop (𝓝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 461,
"column": 6
} | {
"line": 461,
"column": 37
} | [
{
"pp": "case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ deriv f z ≤ 0",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 43
} | [
{
"pp": "case neg.h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 441,
"column": 8
} | {
"line": 441,
"column": 38
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 444,
"column": 6
} | {
"line": 444,
"column": 51
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 501,
"column": 69
} | {
"line": 501,
"column": 91
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\n⊢ Integr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\nhg2' : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1081,
"column": 2
} | {
"line": 1081,
"column": 13
} | [
{
"pp": "case pos\nE : Type u_1\nf f' : ℝ → E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x\nhf' : Integrable f' volume\nhf : Integrable f volume\nhE : CompleteSpace E\nA : Tendsto f atBot (𝓝 0)\nB : Tendsto f atTop (𝓝 0)\n⊢ ∫ (x : ℝ), f' x = 0",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u",
"usedConstants": [
"NonUn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 523,
"column": 2
} | {
"line": 523,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nh : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nh' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\n⊢ IntervalIntegrable (fun x ↦ (g ∘ f) x * f' x) volume a b ↔ IntervalIntegrable g volume (f a) (f b)",
"usedConstants": [
"NonUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\n⊢ IntervalIntegrable (fun x ↦ (g ∘ f) x * f' x) volume a b ↔ IntervalIntegrable g volume (f a) (f b)",
"usedConstants": [
"NonUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : Continu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 24
} | [
{
"pp": "a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nhg : ∀ x ∈ [[a, b]], HasDerivAt g (g' (f x)) (f x)\nhf' : ContinuousOn f' [[a, b]]\nhg' : Continuous g'\n⊢ ∫ (x : ℝ) in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1129,
"column": 69
} | {
"line": 1129,
"column": 91
} | [
{
"pp": "f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\n⊢ IntegrableOn (fun x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1130,
"column": 2
} | {
"line": 1130,
"column": 24
} | [
{
"pp": "f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\nhg2' : IntegrableOn (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 48
} | [
{
"pp": "case h0.h\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.QuadraticMap | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : QuadraticMap 𝕜 E F\na b : E\n⊢ HasLineDerivAt 𝕜 (⇑f) (polar (⇑f) a b) a b",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.LineDeriv | {
"line": 29,
"column": 64
} | {
"line": 29,
"column": 75
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nf' : ℝ\nl : Filter E\nh : IsExtrFilter f l a\nhd : HasLineDerivAt ℝ f f' a b\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ IsExtrFilter f l (a + 0 • b)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1174,
"column": 2
} | {
"line": 1174,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\na b : ℝ\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1215,
"column": 2
} | {
"line": 1215,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\np : ℝ\nhp : p ≠ 0\n⊢ IntegrableOn (fun x ↦ x ^ (p - 1) • f (x ^ p)) (Ioi 0) volume ↔ IntegrableOn f (Ioi 0) volume",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Semigroup.toMul",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1229,
"column": 2
} | {
"line": 1229,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nc a : ℝ\nha : 0 < a\n⊢ IntegrableOn (fun x ↦ f (x * a)) (Ioi c) volume ↔ IntegrableOn f (Ioi (c * a)) volume",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"Set.Ioi",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 137,
"column": 40
} | {
"line": 137,
"column": 62
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : Measurab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 19
} | [
{
"pp": "case hx\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1303,
"column": 2
} | {
"line": 1303,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ tsupport v, HasDerivAt u (u' x) x\nhv : ∀ x ∈ tsupport u, HasDerivAt v (v' x) x\nhuv : Integrable (u' * v + u * v') volume\nh_bot : Tendsto (u * v) atBot (𝓝 a')\nh_top : T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1347,
"column": 2
} | {
"line": 1347,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Ioi a) volume\nh_zero : Tendsto (u * v) (𝓝[Ici a \\ {a}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1359,
"column": 2
} | {
"line": 1359,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Ioi a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1378,
"column": 2
} | {
"line": 1378,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Iic a) volume\nh_zero : Tendsto (u * v) (𝓝[Iic a \\ {a}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1390,
"column": 2
} | {
"line": 1390,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Iic a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 39
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : Measurab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 64
} | [
{
"pp": "case pos.inr.hf'g\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.Polynomial | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 70
} | [
{
"pp": "p : ℝ[X]\nx : ℝ\na✝ : x ∈ p.roots.toFinset ∪ (derivative p).roots.toFinset\nhx₂ : x ∉ (derivative p).roots.toFinset\n⊢ Multiset.count x (derivative p).roots = 0",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Real",
"Polynomial.roots",
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.Polynomial | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 87
} | [
{
"pp": "F : Type u_1\ninst✝¹ : CommRing F\ninst✝ : Algebra F ℝ\np : F[X]\n⊢ Fintype.card ↑(p.rootSet ℝ) ≤ Fintype.card ↑((derivative p).rootSet ℝ) + 1",
"usedConstants": [
"Multiset.toFinset",
"Polynomial.derivative",
"Eq.mpr",
"Real",
"Semiring.toModule",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 64
} | [
{
"pp": "case pos.inr.hfg'\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 64
} | [
{
"pp": "case pos.inr.hfg\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 630,
"column": 69
} | {
"line": 632,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nR : ℝ\nhs : s ⊆ c... | by
gcongr
exact (hδ (A _)).2 _ (ht _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 43
} | [
{
"pp": "case e_a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nr : ℝ\nf g : ℂ → E\nhr : 0 < r\nhf : ContinuousOn f (sphere z r)\nhg : ContinuousOn g (sphere z r)\nh1 : ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)\n⊢ ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 55
} | [
{
"pp": "case h\nE : Type u_1\nι : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nK : Set ℂ\nδ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf : ℂ → E\nhF : ∀ ε > 0, ∀ᶠ (n : ι) in φ, ∀ x ∈ cthickening δ K, dist (f x) (F n x) < ε\nhδ : 0 < δ\nhFn : ∀ᶠ (n : ι) in φ, ContinuousOn (F n) (cthickening δ K)\nhn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 95
} | [
{
"pp": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhs : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nhc : ContinuousAt f c\nx : ℂ\nhx : x ∈ s\nhne : x ≠ c\n⊢ DifferentiableWithinAt ℂ f s x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 46
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhc : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\nF : ℂ → E := fun z ↦ (z - c) • f z\nH : Tendsto (fun x ↦ (x - c)⁻¹⁻¹ • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 71
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nhd : ∀ᶠ (x : ℂ) in 𝓝 c, x ∈ {c}ᶜ → DifferentiableAt ℂ f x\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\nthis : DifferentiableOn ℂ f ({z | z ≠ c → DifferentiableAt ℂ f z} \\ {c})\n... | exact continuousAt_update_same.1 (H.differentiableAt hd).continuousAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝ : UniformSpace α\n⊢ HasProdUniformlyOn f g s ↔ TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop s",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, HasProdUniformlyOn f g t\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\nhs : IsOpen[inst✝¹] s\ninst✝ : LocallyCompactSpace β\nh : ∀ K ⊆ s, IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ K ⊆ s, IsCompact K → TendstoUniforml... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝ : UniformSpace α\n⊢ HasProdUniformly f g ↔ TendstoUniformly (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Finset",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 17
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU : Set ℂ\nhU : IsOpen U\nc w₀ : ℂ\nR : ℝ\nf : ℂ → E\nhc : closedBall c R ⊆ U\nhf : DifferentiableOn ℂ f U\nhw₀ : w₀ ∈ ball c R\nhf' : DifferentiableOn ℂ (dslope f w₀) U\nh0 : (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ (x : β), ∃ t ∈ 𝓝 x, HasProdUniformlyOn f g t\n⊢ ∀ (x : β), ∃ t ∈ 𝓝 x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : LocallyCompactSpace β\nh : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ (K : Set β), IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : ℕ → β → α\nh : HasProdLocallyUniformly f g\n⊢ TendstoLocallyUniformly (fun x1 x2 ↦ ∏ i ∈ Finset.range x1, f i x2) g atTop",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 893,
"column": 4
} | {
"line": 893,
"column": 98
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhs : MeasurableSet s\nh's : μ s ≠ ∞\nhf' : ∀ x ∈ s, HasFDerivWit... | exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 82
} | [
{
"pp": "case refine_1\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyConnectedSpace H\ne : M → OpenPartialHomeomorph M H := chartAt H\nx : M\n⊢ (𝓝 x).HasBasis ((fun x s ↦ (IsOpen[inst✝³] s ∧ ↑(e x) x ∈ s ∧ IsConnected s) ∧ s ⊆ (e ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 81
} | [
{
"pp": "case inr.h\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : e ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nE : e ∈ {e | e.source = ∅}\n⊢ e.symm ∈ {e | e.source = ∅}",
"usedConstants": [
"Eq.mpr",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 225,
"column": 10
} | {
"line": 225,
"column": 26
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nx : H\nhx : x ∈ e.source\ns : Set H\nopen_s : IsOpen[inst✝] s\nxs : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 504,
"column": 24
} | {
"line": 504,
"column": 50
} | [
{
"pp": "case inl\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M\n⊢ Sum.inl x ∈ ((ChartedSpace.chartAt x).lift_openEmbedding... | lift_openEmbedding_source, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 508,
"column": 24
} | {
"line": 508,
"column": 50
} | [
{
"pp": "case inr\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M'\n⊢ Sum.inr x ∈ ((ChartedSpace.chartAt x).lift_openEmbeddin... | lift_openEmbedding_source, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 963,
"column": 8
} | {
"line": 963,
"column": 89
} | [
{
"pp": "case ha₀\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasFDerivWithinAt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 43
} | [
{
"pp": "case inr.h.inl\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nx : H\nhx : x ∈ e.source\ns : Set H\nopen_s : IsOpen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 44
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne : PartialEquiv M H\nhe : e ∈ c.atlas\nE : e.target ∩ ↑e.symm ⁻¹' e.source = e.target\n⊢ IsOpen[inst✝] e.target",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 666,
"column": 8
} | {
"line": 666,
"column": 35
} | [
{
"pp": "H : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne✝ e : PartialEquiv M H\nhe : e ∈ c.atlas\nthis : TopologicalSpace M := c.toTopologicalSpace\nt : Set M\ne' : PartialEquiv M H\ne'_atlas : e' ∈ c.atlas\ns : Set H\ns_open : IsOp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 673,
"column": 6
} | {
"line": 673,
"column": 87
} | [
{
"pp": "case h\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝ : TopologicalSpace H\nc : ChartedSpaceCore H M\ne✝ e : PartialEquiv M H\nhe : e ∈ c.atlas\nthis✝ : TopologicalSpace M := c.toTopologicalSpace\nt : Set M\ne' : PartialEquiv M H\ne'_atlas : e' ∈ c.atlas\ns : Set H\ns_op... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 412,
"column": 4
} | {
"line": 412,
"column": 33
} | [
{
"pp": "H : Type u_1\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nh : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {e | ∃ s, ∃ (h : IsOpen[inst✝] s), e ≈ ofSet s h}\nx : H\nhx : x ∈ e.source\ns : Set H\nhs : IsOpen[inst✝] s\nhxs : x ∈ s\ns' : Set H\nhs' : IsOpen[inst✝] s'\nhes' : e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 97,
"column": 90
} | {
"line": 98,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n⊢ (f.extend I).target = ↑(f.extend... | by
rw [f.extend_target', ← f.image_source_eq_target, ← image_comp, f.extend_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1046,
"column": 4
} | {
"line": 1046,
"column": 98
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhs : MeasurableSet s\nh's : μ s ≠ ∞\nhf' : ∀ x ∈ s, HasFDerivWit... | exact ENNReal.Tendsto.const_mul (ENNReal.tendsto_coe.2 tendsto_id) (Or.inr ENNReal.coe_ne_top) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 39
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\ninst✝⁴ : TopologicalSpace H'\ninst✝³ : TopologicalSpace M'\ninst✝² : ChartedSpace H' M'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne : OpenPartialHome... | exact this.comp_continuousWithinAt h1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 201,
"column": 24
} | {
"line": 203,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝ : TopologicalSpace H\nφ : PartialEquiv H E\nhsource : φ.source = univ\nhtarget : φ.target = univ\nhcont : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.to... | by
have : range φ = φ.target := by rw [← φ.image_source_eq_target, hsource, image_univ.symm]
simp [this, htarget] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nh : ¬IsRCLikeNormedField 𝕜\n⊢ range ↑I = univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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