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.Integral.IntervalIntegral.IntegrationByParts | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 83
} | [
{
"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 [mul_comm] using integrable_deriv_smul_comp_iff_of_deriv_nonneg hf hff' hf' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 83
} | [
{
"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 [mul_comm] using integrable_deriv_smul_comp_iff_of_deriv_nonneg hf hff' hf' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1336,
"column": 2
} | {
"line": 1348,
"column": 67
} | [
{
"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[>] a) (𝓝 a'... | rw [← Ici_diff_left] at h_zero
let f := Function.update (u * v) a a'
have hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x := by
intro x (hx : a < x)
apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
filter_upwards [eventually_ne_nhds hx.ne.symm] with y hy
exact Function.update_of_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1336,
"column": 2
} | {
"line": 1348,
"column": 67
} | [
{
"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[>] a) (𝓝 a'... | rw [← Ici_diff_left] at h_zero
let f := Function.update (u * v) a a'
have hderiv : ∀ x ∈ Ioi a, HasDerivAt f (u' x * v x + u x * v' x) x := by
intro x (hx : a < x)
apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
filter_upwards [eventually_ne_nhds hx.ne.symm] with y hy
exact Function.update_of_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 31
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nhd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\n⊢ Tendsto f (𝓝[≠] c) (𝓝 ((𝓝[≠] c).limUnder f))",
"usedConstants": [
"N... | eventually_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 51
} | [
{
"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\nh'f' : ∀ x ∈ s, (... | exact Tendsto.mono_left this nhdsWithin_le_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 742,
"column": 6
} | {
"line": 742,
"column": 10
} | [
{
"pp": "case h\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 f... | hx2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 832,
"column": 8
} | {
"line": 834,
"column": 36
} | [] | ‖B - A‖ ≤ (min δ δ'' : ℝ≥0) := hB
_ ≤ δ'' := by simp only [le_refl, NNReal.coe_min, min_le_iff, or_true]
_ < δ' := half_lt_self δ'pos | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 866,
"column": 70
} | {
"line": 866,
"column": 83
} | [
{
"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\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 866,
"column": 70
} | {
"line": 866,
"column": 83
} | [
{
"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\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 99
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : SecondCountableTopology H\ns : Set M\nhs : ⋃ x, (chartAt H ↑x).source = univ\nhsc : s.Countable\nthis✝ : ∀ (x : M), SecondCountableTopology ↑(chartAt H x).source\nthis : Encodable ↑s\n... | exact secondCountableTopology_of_countable_cover (fun x : s ↦ (chartAt H (x : M)).open_source) hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 57
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nU : Opens M\nx : ↥U\ne : OpenPartialHomeomorph M H := ⋯\n⊢ ↑e ↑x ∈ (e.subtypeRestr ⋯).target",
"usedConstants": [
"OpenPartialHomeomorph.map_subtype_source",
"mem_chart_source",... | exact e.map_subtype_source ⟨x⟩ (mem_chart_source _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 57
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nU V : Opens M\nhUV : U ≤ V\nx : ↥U\ne : OpenPartialHomeomorph M H := ⋯\n⊢ ↑e ↑x ∈ (e.subtypeRestr ⋯).target",
"usedConstants": [
"OpenPartialHomeomorph.map_subtype_source",
"mem... | exact e.map_subtype_source ⟨x⟩ (mem_chart_source _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 970,
"column": 6
} | {
"line": 976,
"column": 53
} | [
{
"pp": "case inr.refine_2\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, HasFDeri... | intro t g htg
rcases eq_or_ne (μ t) ∞ with (ht | ht)
· simp only [ht, εpos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne,
not_false_iff, _root_.add_top]
have := h t g (htg.mono_num (min_le_left _ _))
rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this
simp o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 970,
"column": 6
} | {
"line": 976,
"column": 53
} | [
{
"pp": "case inr.refine_2\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, HasFDeri... | intro t g htg
rcases eq_or_ne (μ t) ∞ with (ht | ht)
· simp only [ht, εpos.ne', ENNReal.mul_top, ENNReal.coe_eq_zero, le_top, Ne,
not_false_iff, _root_.add_top]
have := h t g (htg.mono_num (min_le_left _ _))
rwa [ENNReal.coe_sub, ENNReal.sub_mul, tsub_le_iff_right] at this
simp o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1004,
"column": 70
} | {
"line": 1004,
"column": 83
} | [
{
"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\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1004,
"column": 70
} | {
"line": 1004,
"column": 83
} | [
{
"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\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | congr 1; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 30
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nP : (H → H') → Set H → H → Prop\ng : M → M'\ns t : Set M\nx : M\nmono : ... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 275,
"column": 2
} | {
"line": 280,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorn... | refine forall_mem_image.trans <| forall₂_congr fun x hx ↦ ?_
refine (continuousWithinAt_congr_set ?_).trans
(continuousWithinAt_writtenInExtend_iff _ (hs hx) (hmaps hx) hmaps)
rw [← nhdsWithin_eq_iff_eventuallyEq, ← map_extend_nhdsWithin_eq_image_of_subset,
← map_extend_nhdsWithin]
exacts [hs hx, hs hx, h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 275,
"column": 2
} | {
"line": 280,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorn... | refine forall_mem_image.trans <| forall₂_congr fun x hx ↦ ?_
refine (continuousWithinAt_congr_set ?_).trans
(continuousWithinAt_writtenInExtend_iff _ (hs hx) (hmaps hx) hmaps)
rw [← nhdsWithin_eq_iff_eventuallyEq, ← map_extend_nhdsWithin_eq_image_of_subset,
← map_extend_nhdsWithin]
exacts [hs hx, hs hx, h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 34
} | [
{
"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\nI : ModelWithCorners 𝕜 E H\ne e' : OpenPartialHomeomorph M H\n⊢ (extendCoordChange e e').symm... | exact I.extendCoordChange_source | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 594,
"column": 8
} | {
"line": 594,
"column": 36
} | [
{
"pp": "case refine_1\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H → H\ne' : OpenPartialHomeomorph H H\nhe'G : e' ∈ G\nhe'x : x ∈ e'.source\nh : G.IsLocalStructomorphWithinAt f s x\nhx : ↑e' x ∈ ↑e'.symm ⁻¹' s\nhxs : x ∈ s\ne : O... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 608,
"column": 8
} | {
"line": 608,
"column": 36
} | [
{
"pp": "case refine_1\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ns : Set H\nx : H\nf : H → H\ne' : OpenPartialHomeomorph H H\nhe'G : e' ∈ G\na✝ : s ⊆ f ⁻¹' e'.source\nhfx : f x ∈ e'.source\nh : G.IsLocalStructomorphWithinAt f s x\nhx : x ∈ s\ne : Open... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 15
} | [
{
"pp": "case mpr\nH : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\nf : OpenPartialHomeomorph H H\ns : Set H\nx : H\nhx : x ∈ f.source ∪ sᶜ\n⊢ (x ∈ s → ∃ e ∈ G, e.source ⊆ f.source ∧ EqOn (↑f) (↑e) (s ∩ e.source) ∧ x ∈ e.source) →\n G.IsLocalStructomorphWi... | intro hf hx | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 667,
"column": 6
} | {
"line": 667,
"column": 23
} | [
{
"pp": "H₁ : Type u_6\ninst✝⁷ : TopologicalSpace H₁\nH₂ : Type u_7\ninst✝⁶ : TopologicalSpace H₂\nH₃ : Type u_8\ninst✝⁵ : TopologicalSpace H₃\ninst✝⁴ : ChartedSpace H₁ H₂\ninst✝³ : ChartedSpace H₂ H₃\nG₁ : StructureGroupoid H₁\ninst✝² : HasGroupoid H₂ G₁\ninst✝¹ : ClosedUnderRestriction G₁\nG₂ : StructureGroup... | apply G₁.locality | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 34
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : ... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 139,
"column": 18
} | {
"line": 139,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 635,
"column": 76
} | {
"line": 648,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝ : ChartedSpace H M\nx : M\n⊢ (extChartAt I x).target ⊆\n ... | by
intro y hy
rw [mem_closure_iff_nhds]
intro t ht
have A : t ∩ ((extChartAt I x).target ∪ (range I)ᶜ) ∈ 𝓝 y :=
inter_mem ht (extChartAt_target_union_compl_range_mem_nhds_of_mem hy)
have B : y ∈ closure (interior (range I)) := by
apply I.range_subset_closure_interior (extChartAt_target_subset_range x... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 12
} | [
{
"pp": "case right\nn : ℕ∞ω\n𝕜 : 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\nf : OpenPartialHomeomorph H H\nhf : f.source = ∅\nx : E\nhx : x ∈ ↑I.symm ⁻¹' f.target\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 43
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\nn : ℕ∞ω\nf : M →... | simpa using h.comp hf hmaps | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 43
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\nn : ℕ∞ω\nf : M →... | simpa using h.comp hf hmaps | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 43
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\nn : ℕ∞ω\nf : M →... | simpa using h.comp hf hmaps | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 334,
"column": 53
} | {
"line": 334,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : ChartedSpace H M\nE' : Type u_5\ninst✝⁵ : NormedAddCo... | contDiffWithinAtProp_self_source, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 982,
"column": 35
} | {
"line": 982,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 982,
"column": 35
} | {
"line": 982,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 982,
"column": 35
} | {
"line": 982,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 988,
"column": 35
} | {
"line": 988,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 988,
"column": 35
} | {
"line": 988,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 988,
"column": 35
} | {
"line": 988,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_5\ninst✝¹ : Topo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 391,
"column": 2
} | {
"line": 403,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalS... | intro x
rw [contMDiffAt_iff]
refine ⟨continuous_inl.continuousAt, ?_⟩
-- In extended charts, .inl equals the identity (on the chart sources).
apply contDiffWithinAt_id.congr_of_eventuallyEq; swap
· simp [ChartedSpace.sum_chartAt_inl, Sum.inl_injective.extend_apply (chartAt _ x)]
set C := chartAt H x with hC... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 391,
"column": 2
} | {
"line": 403,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalS... | intro x
rw [contMDiffAt_iff]
refine ⟨continuous_inl.continuousAt, ?_⟩
-- In extended charts, .inl equals the identity (on the chart sources).
apply contDiffWithinAt_id.congr_of_eventuallyEq; swap
· simp [ChartedSpace.sum_chartAt_inl, Sum.inl_injective.extend_apply (chartAt _ x)]
set C := chartAt H x with hC... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 456,
"column": 6
} | {
"line": 456,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ne : M → H\nh : IsOpenEmbedding e\nn : ℕ∞ω\ninst✝ : Nonempty M\nt... | rw [ModelWithCorners.symm, ← mem_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 186,
"column": 8
} | {
"line": 186,
"column": 36
} | [
{
"pp": "case ht\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : ... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 197,
"column": 20
} | {
"line": 197,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : Topologic... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Algebra.SMul | {
"line": 170,
"column": 20
} | {
"line": 173,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹⁴ : TopologicalSpace H\nE : Type u_3\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nH' : Type u_4\ninst✝¹¹ : TopologicalSpace H'\nE' : Type u_5\ninst✝¹⁰ : NormedAddCommGroup E'\ninst✝⁹ : ... | by
have h : ContMDiff (𝓘(𝕜).prod 𝓘(𝕜, E)) 𝓘(𝕜, 𝕜 × E) n (@id (𝕜 × E)) := by
rw [contMDiff_prod_module_iff, ← contMDiff_prod_iff]; exact contMDiff_id
exact contDiff_smul.contMDiff.comp h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nn : ℕ∞ω\nx : M\ny : E\nhy : y ∈ (extCh... | convert! PartialEquiv.right_inv (extChartAt I x) hy | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 34
} | [
{
"pp": "case mpr.refine_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\ninst✝² : ... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 34
} | [
{
"pp": "case mpr.refine_2\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\ninst✝² : ... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | apply HasSubset.Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 545,
"column": 31
} | {
"line": 545,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simp only [mfderiv, h, if_neg, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 545,
"column": 31
} | {
"line": 545,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simp only [mfderiv, h, if_neg, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 545,
"column": 31
} | {
"line": 545,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | simp only [mfderiv, h, if_neg, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 573,
"column": 34
} | {
"line": 575,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | by
simp only [← mdifferentiableWithinAt_univ, ← mfderivWithin_univ] at hf ⊢
exact mdifferentiableWithinAt_of_mfderivWithin_injective hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.VectorBundle.Basic | {
"line": 460,
"column": 4
} | {
"line": 460,
"column": 23
} | [
{
"pp": "case fst\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpac... | refine e.coe_fst ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 643,
"column": 2
} | {
"line": 643,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | refine ⟨h.1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.VectorBundle.Basic | {
"line": 734,
"column": 2
} | {
"line": 734,
"column": 10
} | [
{
"pp": "case h\nR : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : NontriviallyNormedField R\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace R F\ninst✝ : TopologicalSpace B\nι : Type u_5\nZ : VectorBundleCore R B F ι\ni : ι\nb : B\nhb : b ∈ (Z.localTriv i).baseSet\nv : Z.Fiber b\n⊢ (Trivialization.continu... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.VectorBundle.Basic | {
"line": 896,
"column": 6
} | {
"line": 896,
"column": 60
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : ... | apply linear_trivializationOfMemPretrivializationAtlas | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 752,
"column": 58
} | {
"line": 752,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | hasMFDerivWithinAt_insert | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 10
} | [
{
"pp": "case h\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²¹ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝²⁰ : NormedAddCommGroup EB\ninst✝¹⁹ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁸ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁷ : Topologi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 928,
"column": 59
} | {
"line": 933,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | by
by_cases hx : MDifferentiableWithinAt I I' f s x
· simp only [mfderivWithin, hx, (mdifferentiableWithinAt_congr_set' y h).1 hx, ↓reduceIte]
apply fderivWithin_congr_set' (extChartAt I x x)
exact preimage_extChartAt_eventuallyEq_compl_singleton y h
· simp [mfderivWithin, hx, ← mdifferentiableWithinAt_co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1095,
"column": 66
} | {
"line": 1097,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | by
refine TotalSpace.ext (h p.1 hp) ?_
rw [tangentMapWithin, h p.1 hp, tangentMapWithin, mfderivWithin_congr h (h _ hp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1141,
"column": 4
} | {
"line": 1141,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | simp only [mfld_simps] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1142,
"column": 4
} | {
"line": 1142,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | have : f (((chartAt H x).symm : H → M) (I.symm y)) ∈ u := hst hy.1.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 261,
"column": 2
} | {
"line": 265,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | have : {x | t x y ≠ 0} ⊆ {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ U).Nonempty} := by
intro x hx
rw [Set.mem_setOf] at hx ⊢
use y
simpa using ⟨hx, mem_of_mem_nhds hu⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1262,
"column": 82
} | {
"line": 1263,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | by
subst hy; exact mfderiv_comp x hg hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.PartitionOfUnity | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 84
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : NormalSpace X\np : (X → ℝ) → Prop\nh01 :\n ∀ (s t : Set X),\n IsClosed[inst✝¹] s →\n IsClosed[inst✝¹] t → Disjoint s t → ∃ f, p ⇑f ∧ EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Icc 0 1\nhs : IsClosed[inst✝¹] s\nU : ι ... | exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.PartitionOfUnity | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 84
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : T2Space X\np : (X → ℝ) → Prop\nh01 :\n ∀ (s t : Set X),\n IsClosed[inst✝²] s →\n IsCompact t → Disjoint s t → ∃ f, p ⇑f ∧ EqOn (⇑f) 0 s ∧ EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Icc 0 1\nhs : IsCom... | exact ⟨i, ((hf1 i).mono subset_closure).eventuallyEq_of_mem ((hWo i).mem_nhds hi)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ShrinkingLemma | {
"line": 326,
"column": 8
} | {
"line": 327,
"column": 32
} | [
{
"pp": "case neg.left\nι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := ⋯\nhsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 =... | intro hji
exact False.elim (h hji) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ShrinkingLemma | {
"line": 326,
"column": 8
} | {
"line": 327,
"column": 32
} | [
{
"pp": "case neg.left\nι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := ⋯\nhsi : si = s ∩ ⋂ i_1, ⋂ (_ : ¬i_1 =... | intro hji
exact False.elim (h hji) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 132,
"column": 25
} | {
"line": 132,
"column": 38
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\n⊢ ∀ᵐ (x : α) ∂μ, 0 x ≤ f x",
"usedConstants": [
"MeasureTheory.ae",
"Real.instLE",
"Real",
"MeasureTheory.Mea... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 380,
"column": 2
} | {
"line": 384,
"column": 52
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ =... | have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
→ ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
intro t t_meas ht
have I : ∫ x, f x ∂μ = 0 := by rw [← setIntegral_univ]; exact h'f _ isClosed_univ
simpa [ht, I] using integral_add_compl t_meas hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 405,
"column": 60
} | {
"line": 405,
"column": 74
} | [
{
"pp": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : CompleteSpace E\nβ : Type u_3\ninst✝⁴ : TopologicalSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\ninst✝¹ : SigmaCompactSpace β\ninst✝ : R1Space β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 82
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝... | exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 82
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝... | exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 82
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝... | exact hf'.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 108,
"column": 47
} | {
"line": 125,
"column": 75
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf g : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nhg : Integrable g μ\nv : E\n⊢ Tendsto (fun t ↦ ∫ (x : E), t⁻... | by
apply tendsto_integral_filter_of_dominated_convergence (fun x ↦ (C * ‖v‖) * ‖g x‖)
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeasurable_const.smul
apply AEStronglyMeasurable.sub _ hf.continuous.measurable.aestronglyMeasurable
apply AEMeasura... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 778,
"column": 8
} | {
"line": 778,
"column": 70
} | [
{
"pp": "case pos\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | suffices g c (chartAt H c x) = 0 by simp only [this, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 186,
"column": 75
} | {
"line": 187,
"column": 54
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC D : ℝ≥0\nf g : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nhg : LipschitzWith D g\nh'g : HasCompactSupport g\nv : E\n... | by
simp only [S1] at A; exact tendsto_nhds_unique A B | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 46,
"column": 49
} | {
"line": 46,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ici (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i ≤ b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 46,
"column": 49
} | {
"line": 46,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ici (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i ≤ b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 46,
"column": 49
} | {
"line": 46,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ici (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ico (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i ≤ b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 58,
"column": 49
} | {
"line": 58,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ioi (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ioc (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i < b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 58,
"column": 49
} | {
"line": 58,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ioi (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ioc (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i < b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 58,
"column": 49
} | {
"line": 58,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\na : α\nhf : ∀ i ∈ Ici a, f a ≤ f i\nb : β\nhb : b ∈ Ioi (f a)\nh2f : b ∉ ⋃ i ∈ Ici a, Ioc (f i) (f (succ i))\n⊢ ∀ (n : α) (hmn : a ≤ n), (fun i x ↦ f i < b) n hmn → ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 222,
"column": 2
} | {
"line": 224,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\n⊢ uniformity X = comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"HEq.refl",
"Finset",
"Nat.inst... | refine Filter.HasBasis.eq_of_same_basis Metric.uniformity_basis_dist ?_
convert! hasBasis_totalLengthFilter.comap _
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 222,
"column": 2
} | {
"line": 224,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\n⊢ uniformity X = comap (fun x ↦ (1, fun x_1 ↦ x)) totalLengthFilter",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"HEq.refl",
"Finset",
"Nat.inst... | refine Filter.HasBasis.eq_of_same_basis Metric.uniformity_basis_dist ?_
convert! hasBasis_totalLengthFilter.comap _
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 82
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ) ∂volume.restrict (Icc a b), Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (Icc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in Icc a b, ‖G n x‖ₑ) atTop ... | exact (intervalIntegrable_iff_integrableOn_Icc_of_le hab).mpr integrable_f_deriv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 89,
"column": 12
} | {
"line": 89,
"column": 38
} | [
{
"pp": "s x y : ℝ\nhx₀ : 0 < x\nhx₁ : x < 1 / (1 + s ^ 2)\nhy : |y| < s * x\n⊢ 1 / (1 + s ^ 2) ≤ 1",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"instHDiv",
"HMul.hMul",
"Group... | div_le_one (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 138,
"column": 8
} | {
"line": 139,
"column": 75
} | [
{
"pp": "case h\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nz : ℂ\nhz : ‖z‖ < 1\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\nk :\n Tendsto (fun x ↦ (1 - z) * ∑ x_1 ∈ range x, (∑ i ∈ Ico (x_1 + 1) x, f i) * z ^ x_1) atTop\n (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))\nn : ℕ\n| (1 - z) * ∑ x... | sum_congr (g := fun j ↦ (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j)
rfl (fun j hj ↦ by congr 1; exact sum_Ico_eq_sub _ (mem_range.mp hj)) | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 70
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\nhf : ContDiff ℝ (↑n) f\n⊢ (fun x ↦ f x - taylorWithinEval f n univ x₀ x) =o[𝓝 x₀] fun x ↦ (x - x₀) ^ n",
"usedConstants": [
"taylorWithinEval",
"InnerProductSpace.toNormedSpace",
"Real... | simpa using taylor_isLittleO convex_univ (mem_univ x₀) hf.contDiffOn | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Taylor | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 70
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\nhf : ContDiff ℝ (↑n) f\n⊢ (fun x ↦ f x - taylorWithinEval f n univ x₀ x) =o[𝓝 x₀] fun x ↦ (x - x₀) ^ n",
"usedConstants": [
"taylorWithinEval",
"InnerProductSpace.toNormedSpace",
"Real... | simpa using taylor_isLittleO convex_univ (mem_univ x₀) hf.contDiffOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Taylor | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 70
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\nhf : ContDiff ℝ (↑n) f\n⊢ (fun x ↦ f x - taylorWithinEval f n univ x₀ x) =o[𝓝 x₀] fun x ↦ (x - x₀) ^ n",
"usedConstants": [
"taylorWithinEval",
"InnerProductSpace.toNormedSpace",
"Real... | simpa using taylor_isLittleO convex_univ (mem_univ x₀) hf.contDiffOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 147,
"column": 21
} | {
"line": 147,
"column": 39
} | [
{
"pp": "case h.h\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nz : ℂ\nhz : ‖z‖ < 1\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\nk :\n Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop\n (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))\nthis ... | sub_add_sub_cancel | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 474,
"column": 8
} | {
"line": 474,
"column": 12
} | [
{
"pp": "case inr.zero\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis : x₀ ≠ x\nhf : ∫ (t : ℝ) in x₀..x, deriv (fun t ↦ f t) t = f x - f x₀\n⊢ f x - f x₀ = ∫ (t : ℝ) in x₀..x, derivWithin f [[x₀, x]] t",
"usedConstants": [
"Eq.mpr",
"Real",
... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 208,
"column": 38
} | {
"line": 208,
"column": 49
} | [
{
"pp": "case h₁.hbc\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l ... | ← norm_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 208,
"column": 38
} | {
"line": 208,
"column": 49
} | [
{
"pp": "case h₂.hbc\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l ... | ← norm_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 71
} | [
{
"pp": "case succ.inr\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : ↑k < m\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"usedConstants": [
"mul_nonneg",
"Int.instIsStrictOrderedRing",
"IsOrderedRing.toPosMulMono",
"instConditionallyCompleteLinearOrder",
"co... | exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 71
} | [
{
"pp": "case succ.inr\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : ↑k < m\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"usedConstants": [
"mul_nonneg",
"Int.instIsStrictOrderedRing",
"IsOrderedRing.toPosMulMono",
"instConditionallyCompleteLinearOrder",
"co... | exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 71
} | [
{
"pp": "case succ.inr\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)\nk : ℕ\nhmk : ↑k < m\n⊢ 0 ≤ (m - ↑k) * (m - ↑(k + 1))",
"usedConstants": [
"mul_nonneg",
"Int.instIsStrictOrderedRing",
"IsOrderedRing.toPosMulMono",
"instConditionallyCompleteLinearOrder",
"co... | exact mul_nonneg (sub_nonneg_of_le hmk.le) (sub_nonneg_of_le hmk) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 27
} | [
{
"pp": "case hf''.«0»\nx : ℝ\nhx : 0 < x\nhm₀ : (Nat.castEmbedding.trans (addLeftEmbedding 0)) 0 ≠ 0\nhm₁ : (Nat.castEmbedding.trans (addLeftEmbedding 0)) 0 ≠ 1\n⊢ False",
"usedConstants": [
"addLeftEmbedding",
"CharP.cast_eq_zero",
"False",
"Int.instLocallyFiniteOrder._proof_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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