module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.FDeriv.Star | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : StarRing 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : StarAddMonoid F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : StarModule 𝕜 F\ninst✝⁴ : ContinuousStar F\nins... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 308,
"column": 34
} | {
"line": 308,
"column": 51
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ... | Finset.sum_sigma, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fin.Parity | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 32
} | [
{
"pp": "n : ℕ\nk : Fin n\nh : Even ↑k\n⊢ Even k",
"usedConstants": [
"Nat.instMulZeroClass",
"NeZero.mk",
"Fin.pos",
"instOfNatNat",
"LT.lt.ne'",
"Nat.instPreorder",
"Nat",
"NeZero",
"OfNat.ofNat",
"MulZeroClass.toZero"
]
}
] | have : NeZero n := ⟨k.pos.ne'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Fin.Parity | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 32
} | [
{
"pp": "n : ℕ\nhn : Odd n\nk : Fin n\n⊢ Even k",
"usedConstants": [
"Nat.instMulZeroClass",
"NeZero.mk",
"Fin.pos",
"instOfNatNat",
"LT.lt.ne'",
"Nat.instPreorder",
"Nat",
"NeZero",
"OfNat.ofNat",
"MulZeroClass.toZero"
]
}
] | have : NeZero n := ⟨k.pos.ne'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 379,
"column": 11
} | {
"line": 379,
"column": 25
} | [
{
"pp": "case hg\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\nf : E → Fu\ns : Set E\nt : Set Fu\nx : E\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 s\nhst : Ma... | Nat.cast_succ, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.Calculus.DifferentialForm.Basic | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nr : WithTop ℕ∞\nω : E → E [⋀^Fin n]→L[𝕜] F\ns : Set E\nx : E\nhω : ContDiffWithinAt 𝕜 r ω s x\nhr :... | exact le_minSmoothness.trans hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 450,
"column": 2
} | {
"line": 452,
"column": 96
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : E\nf : E → F\nf' : E → E →L[ℝ] F\nf'' : E →L[ℝ] E →L[ℝ] F\nhf : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt f (f' y) y\nhx : HasFDerivAt f' f'' x\nv w : E\nε : ℝ\nεpos : ... | exact
Convex.second_derivative_within_at_symmetric (convex_ball x ε) A
(fun y hy => hε (interior_subset hy)) (Metric.mem_ball_self εpos) hx.hasFDerivWithinAt v w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Implicit | {
"line": 230,
"column": 94
} | {
"line": 234,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : CompleteSpace E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : CompleteSpace F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : N... | by
have := φ.hasStrictFDerivAt.to_localInverse.comp (φ.rightFun φ.pt)
((hasStrictFDerivAt_const _ _).prodMk (hasStrictFDerivAt_id _))
convert this
exact this.hasFDerivAt.fderiv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.LHopital | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 35
} | [
{
"pp": "case h\na : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Ioi a, g' x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\na' : ℝ\nhaa' : a < a'\nha' : 0... | simp only [Function.comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 50
} | [
{
"pp": "E : 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∞ := ENNReal.ofReal |A.det|\nHC : IsCompa... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 48
} | [
{
"pp": "E : 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∞ := ENNReal.ofReal |A.det|\nε : ℝ\nhε : ... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1088,
"column": 2
} | {
"line": 1088,
"column": 69
} | [
{
"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 (b * x) = b⁻¹ • ∫ (x : ℝ) in Ioi (b * a), g x",
"usedConstants": [
"instClosedIicTopology",
"Real",
"Set.Ioi",
"Real.lattice",
"MeasurableSet",... | have : ∀ c : ℝ, MeasurableSet (Ioi c) := fun c => measurableSet_Ioi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 69,
"column": 30
} | {
"line": 69,
"column": 50
} | [
{
"pp": "E : 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)\nw : ℂ\nhw : ‖w - z‖ = r\nh : (w - z) ^ 2 = 0\n⊢ 0 = r",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Nor... | sq_eq_zero_iff.mp h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 892,
"column": 4
} | {
"line": 892,
"column": 48
} | [
{
"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... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 129,
"column": 35
} | {
"line": 129,
"column": 65
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nG : StructureGroupoid H\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ maximalAtlas M G\nhe' : e' ∈ maximalAtlas M G\nx : H\nhx : x ∈ (e.symm ≫ₕ e').source\nf : OpenPartialHomeomorph M H := chartA... | rw [trans_of_set']; apply refl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 129,
"column": 35
} | {
"line": 129,
"column": 65
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nG : StructureGroupoid H\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ maximalAtlas M G\nhe' : e' ∈ maximalAtlas M G\nx : H\nhx : x ∈ (e.symm ≫ₕ e').source\nf : OpenPartialHomeomorph M H := chartA... | rw [trans_of_set']; apply refl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 41
} | [
{
"pp": "case h.e'_5\nH : Type u_1\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne : OpenPartialHomeomorph H H\nhe :\n ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\nx : H\nxu : x ∈ e.source\ns : Set H\ns_open : IsOpen s\nxs : x ∈ s\nhs ... | dsimp [OpenPartialHomeomorph.restr] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 43
} | [
{
"pp": "case h.e'_5\nH : Type u_1\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne : OpenPartialHomeomorph H H\nhe :\n ∀ x ∈ e.source, ∃ s, IsOpen s ∧ x ∈ s ∧ e.restr s ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\nx : H\nxu : x ∈ e.target\ns : Set H\ns_open : IsOpen s\nxs : ↑e.symm x ... | dsimp [OpenPartialHomeomorph.restr] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1045,
"column": 4
} | {
"line": 1045,
"column": 48
} | [
{
"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... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 33
} | [
{
"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\ns : Set E\nhs : IsOpen s\n⊢ IsOpen... | rw [← extend_source f (I := I)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 419,
"column": 95
} | {
"line": 420,
"column": 63
} | [
{
"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\nx : H\ns : Set H\n⊢ map (↑I.symm) (𝓝[↑I '' s] ↑I x) = 𝓝[s] x",
"usedConstants": [
"Eq.mpr",
... | by
rw [← I.map_nhdsWithin_eq, map_map, I.symm_comp_self, map_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 535,
"column": 19
} | {
"line": 541,
"column": 56
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nι : Type v\ninst✝³ : Fintype ι\nE : ι → Type w\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\nH : ι → Type u'\ninst✝ : (i : ι) → TopologicalSpace (H i)\nI : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)\n⊢ if h : IsRC... | by
rw [PartialEquiv.pi_apply, Set.range_piMap]
split_ifs with h
· letI := h.rclike
letI := fun i ↦ NormedSpace.restrictScalars ℝ 𝕜 (E i)
exact convex_pi fun i _hi ↦ (I i).convex_range
· simp [range_eq_univ_of_not_isRCLikeNormedField, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 766,
"column": 2
} | {
"line": 767,
"column": 58
} | [
{
"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✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAdd... | rw [← (extChartAt I x).image_source_inter_eq', ← map_extChartAt_nhdsWithin_eq_image,
← map_extChartAt_nhdsWithin, nhdsWithin_inter_of_mem'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 977,
"column": 6
} | {
"line": 978,
"column": 34
} | [
{
"pp": "case inr.inl.inl\n𝕜 : 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 : WithTop ℕ∞\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' ... | · rw [hef, he'f', f.lift_openEmbedding_trans f' IsOpenEmbedding.inl]
exact hM.compatible hf hf' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 59
} | [
{
"pp": "case refine_1\n𝕜 : 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\nF : Type u_8\ninst✝⁴ ... | · exact (extChartAt I p.1).right_inv ⟨hy.1.1.1, hy.1.2.1⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 21
} | [
{
"pp": "case right\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\ne : M → H\nh : IsOpenEmbedding e\nn : WithTop ℕ∞\nin... | apply I.right_inv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Operator.Prod | {
"line": 162,
"column": 23
} | {
"line": 167,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : Nontrivial F\n⊢ ‖snd 𝕜 E F‖ = 1",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSu... | by
refine le_antisymm (norm_snd_le ..) ?_
let ⟨f, hf⟩ := exists_ne (0 : F)
have : ‖f‖ ≤ _ * max ‖(0 : E)‖ ‖f‖ := (snd 𝕜 E F).le_opNorm (0, f)
rw [norm_zero, max_eq_right (norm_nonneg f)] at this
rwa [← mul_le_mul_iff_of_pos_right (norm_pos_iff.mpr hf), one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 109,
"column": 52
} | {
"line": 112,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : WithTop ℕ∞\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\n... | by
constructor
rw [contMDiff_zero_iff]
exact continuous_inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : WithTop ℕ∞\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\n... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : WithTop ℕ∞\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\n... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : WithTop ℕ∞\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\n... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 175,
"column": 37
} | {
"line": 175,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : WithTop ℕ∞\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\n... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 63
} | [
{
"pp": "𝕜 : 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 : WithTop ℕ∞\ninst✝² : IsManifold I... | have hy₁ : f y ∈ c'.source := by simp only [hy, mfld_simps] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.ContMDiff.NormedSpace | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nE' : Type u_5\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace 𝕜 E'\nn : WithTop ℕ∞\nf : E → E'\ns : Set E\nx : E\n⊢ ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiff... | simp +contextual only [ContMDiffWithinAt, liftPropWithinAt_iff',
ContDiffWithinAtProp, iff_def, mfld_simps] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.VectorBundle.Basic | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 19
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ni... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.VectorBundle.Basic | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 44
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ni... | rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.VectorBundle.Basic | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 44
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ni... | rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Paracompact | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 94
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\npow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k\nhpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m\nh2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n\nι : Type u_1\ns : ι → Set α\nho : ∀ (a : ι), IsOpen (s a)\nhcov : ∀ (x : α), ∃ i, x ∈ s i\nw✝ : LinearOrder ι\nwf : WellFounded... | refine ⟨ℕ × ι, fun ni => D ni.1 ni.2, fun _ => Dopen _ _, ?_, ?_, fun ni => ⟨ni.2, HDS _ _⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.EMetricSpace.Paracompact | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\npow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k\nhpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m\nh2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n\nι : Type u_1\ns : ι → Set α\nho : ∀ (a : ι), IsOpen (s a)\nhcov : ∀ (x : α), ∃ i, x ∈ s i\nw✝ : LinearOrder ι\nwf : WellFounded... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 165,
"column": 7
} | {
"line": 165,
"column": 32
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nt : Set α\nhf : IntegrableOn f t μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ (s ∩ t) < ∞ → 0 ≤ ∫ (x : α) in s ∩ t, f x ∂μ\ns : Set α\nhs : MeasurableSet s\nh's : (μ.restrict t) s < ∞\n⊢ μ (s ∩ t) < ∞",
"usedConstants": [
"Mea... | Measure.restrict_apply hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 189,
"column": 8
} | {
"line": 189,
"column": 33
} | [
{
"pp": "case refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet... | Measure.restrict_apply hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 33
} | [
{
"pp": "case refine_2\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : AEFinStronglyMeasurable f μ\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nt : Set α := hf.sigmaFiniteSet... | Measure.restrict_apply hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 69
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) i... | suffices f =ᵐ[μ.restrict t] 0 from
ae_of_ae_restrict_of_ae_restrict_compl _ this hf.ae_eq_zero_compl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 294,
"column": 8
} | {
"line": 294,
"column": 33
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ... | Measure.restrict_apply hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 381,
"column": 2
} | {
"line": 383,
"column": 59
} | [
{
"pp": "case refine_2\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nU : M → Set M\ninst✝¹ : T2Space M\ni... | · refine (mem_iUnion.1 <| hsV hx).imp fun i hi => ?_
exact ((f i).updateRIn _ _).eventuallyEq_one_of_dist_lt
((f i).support_subset_source <| hVf _ hi) (hr i hi).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 298,
"column": 8
} | {
"line": 298,
"column": 33
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ... | Measure.restrict_apply hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 48
} | [
{
"pp": "case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhm : m ≤ m0\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), Measurabl... | exact hf_int_finite _ (hs.inter ht_meas) hμs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : CompleteSpace F\nH : Type u_3\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_4\ninst✝⁵ : Topolo... | let U : Opens M := ⟨U, hU⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 96,
"column": 4
} | {
"line": 130,
"column": 71
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx y : E\n⊢ y ∈ tangentConeAt 𝕜 s x →\n ∃ c d,\n Tendsto (fun x ↦ ‖c x‖) atTop atTop ∧ (∀ᶠ (n : ℕ) in atTop, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) ... | rcases eq_or_ne y 0 with rfl | hy₀
· rw [zero_mem_tangentConeAt_iff]
intro hx
obtain ⟨c, hc⟩ := NormedField.exists_lt_norm 𝕜 1
have (n : ℕ) : ∃ d : E, x + d ∈ s ∧ ‖d‖ < (1 / (2 * ‖c‖)) ^ n := by
rw [Metric.mem_closure_iff] at hx
rcases hx ((1 / (2 * ‖c‖)) ^ n) (by positivity) with... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 96,
"column": 4
} | {
"line": 130,
"column": 71
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx y : E\n⊢ y ∈ tangentConeAt 𝕜 s x →\n ∃ c d,\n Tendsto (fun x ↦ ‖c x‖) atTop atTop ∧ (∀ᶠ (n : ℕ) in atTop, x + d n ∈ s) ∧ Tendsto (fun n ↦ c n • d n) ... | rcases eq_or_ne y 0 with rfl | hy₀
· rw [zero_mem_tangentConeAt_iff]
intro hx
obtain ⟨c, hc⟩ := NormedField.exists_lt_norm 𝕜 1
have (n : ℕ) : ∃ d : E, x + d ∈ s ∧ ‖d‖ < (1 / (2 * ‖c‖)) ^ n := by
rw [Metric.mem_closure_iff] at hx
rcases hx ((1 / (2 * ‖c‖)) ^ n) (by positivity) with... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 326,
"column": 41
} | {
"line": 326,
"column": 64
} | [
{
"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 : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\ns : Set E\ns_count : s.Countable\ns_dense : Dense s\n⊢ sphere ... | rw [s_dense.closure_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 71
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nC : ℝ≥0\ns : Set E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : FiniteDimensional ℝ F\nins... | exact A.symm.differentiableAt.comp_differentiableWithinAt x (hx xs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Taylor | {
"line": 231,
"column": 65
} | {
"line": 231,
"column": 74
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx₀ x : ℝ\ns : Set ℝ\nf : ℝ → E\nn : ℕ\nthis :\n ∀ (i : ℕ) {c : ℝ} {c' : E}, HasDerivAt (fun x ↦ (c * (x - x₀) ^ i) • c') ((c * (↑i * (x - x₀) ^ (i - 1) * 1)) • c') x\n⊢ ∑ k ∈ Finset.range (n + 1),\n ((↑(k + 1)!)⁻¹ * (↑(k + 1)... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 148,
"column": 9
} | {
"line": 148,
"column": 18
} | [
{
"pp": "f : 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 :\n Tends... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 42
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b C x : ℝ\nn : ℕ\nhab : a ≤ b\nhf : ContDiffOn ℝ (↑n + 1) f (Icc a b)\nhx : x ∈ Icc a b\nhC : ∀ y ∈ Icc a b, ‖iteratedDerivWithin (n + 1) f (Icc a b) y‖ ≤ C\n⊢ ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (... | rcases eq_or_lt_of_le hab with (rfl | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Taylor | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 42
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nn : ℕ\nhab : a ≤ b\nhf : ContDiffOn ℝ (↑n + 1) f (Icc a b)\n⊢ ∃ C, ∀ x ∈ Icc a b, ‖f x - taylorWithinEval f n (Icc a b) a x‖ ≤ C * (x - a) ^ (n + 1)",
"usedConstants": [
"Real.partialOrder",
"Real"... | rcases eq_or_lt_of_le hab with (rfl | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.AbsMax | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 51
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm : IsMaxOn (norm ∘ f) U c\nV : Set E := U ∩ {z | IsMaxO... | have hVne : (U ∩ V).Nonempty := ⟨c, hcU, hcU, hm⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 81,
"column": 19
} | {
"line": 81,
"column": 27
} | [
{
"pp": "case succ\nm : ℤ\nn : ℕ\nihn : 0 ≤ ∏ k ∈ Finset.range (2 * n), (m - ↑k)\n⊢ 0 ≤ ∏ k ∈ Finset.range (2 * (n + 1)), (m - ↑k)",
"usedConstants": [
"Int.instCommMonoid",
"Distrib.leftDistribClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"con... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 125,
"column": 35
} | {
"line": 125,
"column": 44
} | [
{
"pp": "case inr\nx : ℝ\nhx : x ≤ 0\n⊢ deriv (fun x ↦ √x * log x) x = (2 + log x) / (2 * 0)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.toMul",... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 46
} | [
{
"pp": "x : ℝ\n⊢ deriv^[2] (fun x ↦ √x * log x) x = -log x / (4 * √x ^ 3)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"HMul.hMul",
"Real.denselyNormedField",
"congrArg",
"deriv",
"Real.instDivInvMonoid",
... | simp only [Nat.iterate, deriv_sqrt_mul_log'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 138,
"column": 59
} | {
"line": 138,
"column": 68
} | [
{
"pp": "case inl\nx : ℝ\nhx : x ≤ 0\n⊢ deriv (fun x ↦ (2 + log x) / (2 * √x)) x = -log x / (4 * 0)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 141,
"column": 35
} | {
"line": 141,
"column": 44
} | [
{
"pp": "case inl\nx✝ : ℝ\nhx✝ : x✝ ≤ 0\nx : ℝ\nhx : x ∈ Iic 0\n⊢ (2 + log x) / (2 * 0) = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZero",
"congrArg",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 67,
"column": 73
} | {
"line": 67,
"column": 82
} | [
{
"pp": "θ : ℂ\n⊢ 2 * (sin θ * cos θ) = 2 * 0 ↔ ∃ k, ↑k * ↑π / 2 = θ",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"Complex.cos",
"MulZeroClass.toMul",
"congrArg",
"C... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 93,
"column": 8
} | {
"line": 94,
"column": 72
} | [
{
"pp": "case h₁\nx y : ℂ\n⊢ sin ((x - y) / 2) = 0 ↔ ∃ k, y = 2 * ↑k * ↑π + x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Mathlib.Tactic.Fi... | simp [field, sin_eq_zero_iff, eq_sub_iff_add_eq',
sub_eq_iff_eq_add, mul_comm (2 : ℂ), mul_right_comm _ (2 : ℂ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 93,
"column": 8
} | {
"line": 94,
"column": 72
} | [
{
"pp": "case h₂\nx y : ℂ\n⊢ sin ((x + y) / 2) = 0 ↔ ∃ k, y = 2 * ↑k * ↑π - x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Mathlib.Tactic.Fi... | simp [field, sin_eq_zero_iff, eq_sub_iff_add_eq',
sub_eq_iff_eq_add, mul_comm (2 : ℂ), mul_right_comm _ (2 : ℂ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 57
} | [
{
"pp": "case refine_1\nx : ℝ\nh : 0 < x\n⊢ tan (π / 2 - arcsin x) = √(1 - x ^ 2) / x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Real.pi",
"DivisionCommMonoid.toDivisionMonoid",
"Real.arcsin",
"congrArg",
"Real.instInv",
... | rw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticRw_mod_cast____1 | Lean.Parser.Tactic.tacticRw_mod_cast___ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 57
} | [
{
"pp": "case refine_1\nx : ℝ\nh : 0 < x\n⊢ tan (π / 2 - arcsin x) = √(1 - x ^ 2) / x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Real.pi",
"DivisionCommMonoid.toDivisionMonoid",
"Real.arcsin",
"congrArg",
"Real.instInv",
... | rw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 57
} | [
{
"pp": "case refine_1\nx : ℝ\nh : 0 < x\n⊢ tan (π / 2 - arcsin x) = √(1 - x ^ 2) / x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Real.pi",
"DivisionCommMonoid.toDivisionMonoid",
"Real.arcsin",
"congrArg",
"Real.instInv",
... | rw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Deriv | {
"line": 512,
"column": 2
} | {
"line": 513,
"column": 52
} | [
{
"pp": "case inr\nS : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nx : ℝ\nhxs : x ∈ interior S\ny : ℝ\nhys : y ∈ interior S\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (fun x ↦ derivWithin f (Iio x) x) x ≤ (fun x ↦ derivWithin f (Iio x) x) y",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | simp_rw [hfc.leftDeriv_eq_sSup_slope_of_mem_interior hxs,
hfc.leftDeriv_eq_sSup_slope_of_mem_interior hys] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Convex.Deriv | {
"line": 792,
"column": 11
} | {
"line": 792,
"column": 29
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConvexOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\nu : ℝ\nhxu : x < u\nhuy : u < y\nhu : u ∈ S\nthis : (f x - f y) / (x - y) < (f u - f y) / (u - y)\n⊢ slope f x y < f'",
"usedConstants": [
"Real",
"P... | ← slope_def_field, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 32
} | [
{
"pp": "x✝ x : ℝ\nhx : |x| ≤ π / 2\nhx₀ : 0 ≤ x\n⊢ 2 / π * |x| ≤ |sin x|",
"usedConstants": [
"Real.instLE",
"Real",
"instHDiv",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Eq.mp",
"... | rw [abs_of_nonneg hx₀] at hx ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 45
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ √(1 - (⟪x, y⟫ / (‖x‖ * ‖y‖)) ^ 2) * √((‖x‖ * ‖y‖) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring.toA... | rw [← Real.sqrt_mul' _ (by positivity), sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 34
} | [
{
"pp": "case inr\nx : ℝ\nh1 : 0 ≤ x\nh2 : x < π / 2\nh1' : 0 < x\n⊢ x ≤ tan x",
"usedConstants": [
"Real",
"le_of_lt",
"Real.lt_tan",
"Real.instPreorder",
"Real.tan"
]
}
] | · exact le_of_lt (lt_tan h1' h2) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.LocallyFinsupp | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 66
} | [
{
"pp": "case h.h.h\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\ninst✝ : Zero Y\nf : X → Y\nz : X\nt : Set X\nht : t ∈ 𝓝 z\n⊢ {i | ({↑i} ∩ t).Nonempty}.Finite ↔ (t ∩ support f).Finite",
"usedConstants": [
"Set.ext",
"Function.mem_support._simp_1",
"Iff.of_eq",
"congrArg... | have aux1 : t ∩ f.support = {i : f.support | ↑i ∈ t} := by aesop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.LocallyFinsupp | {
"line": 397,
"column": 6
} | {
"line": 397,
"column": 66
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeSup Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\n⊢ ∃ t ∈ 𝓝 z, (t ∩ Function.support fun z ↦ D₁ z ⊔ D₂ z).Finite",
"usedConstant... | obtain ⟨t₂, ht₂⟩ := D₂.supportLocallyFiniteWithinDomain z hz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 401,
"column": 60
} | {
"line": 401,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nz₀ x : 𝕜\nhx : x ≠ z₀\nhu : x ∈ U\n⊢ (fun x ↦ x - z₀) x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, z - z₀ = (z - x) ^ 0 • (fun x ↦ x - z₀) z",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"False",
... | simp [sub_ne_zero_of_ne hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 401,
"column": 60
} | {
"line": 401,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nz₀ x : 𝕜\nhx : x ≠ z₀\nhu : x ∈ U\n⊢ (fun x ↦ x - z₀) x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, z - z₀ = (z - x) ^ 0 • (fun x ↦ x - z₀) z",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"False",
... | simp [sub_ne_zero_of_ne hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 401,
"column": 60
} | {
"line": 401,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nz₀ x : 𝕜\nhx : x ≠ z₀\nhu : x ∈ U\n⊢ (fun x ↦ x - z₀) x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, z - z₀ = (z - x) ^ 0 • (fun x ↦ x - z₀) z",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"False",
... | simp [sub_ne_zero_of_ne hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 66
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nU : Set X\nY : Type u_2\ninst✝¹ : SemilatticeInf Y\ninst✝ : Zero Y\nD₁ D₂ : locallyFinsuppWithin U Y\nz : X\nhz : z ∈ U\nt₁ : Set X\nht₁ : t₁ ∈ 𝓝 z ∧ (t₁ ∩ D₁.support).Finite\n⊢ ∃ t ∈ 𝓝 z, (t ∩ Function.support fun z ↦ D₁ z ⊓ D₂ z).Finite",
"usedConstant... | obtain ⟨t₂, ht₂⟩ := D₂.supportLocallyFiniteWithinDomain z hz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Meromorphic.Order | {
"line": 107,
"column": 9
} | {
"line": 107,
"column": 19
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nn : ℤ\nhf : MeromorphicAt f x\nh : ∀ᶠ (z : 𝕜) in 𝓝 x, (z - x) ^ Exists.choose hf • f z = 0\nx✝ : ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[... | hfz_eq hz, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 62
} | [
{
"pp": "case mpr.inr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : MeromorphicNFAt f x\nh : f x ≠ 0\nn : ℤ\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nthis : n = 0\nh₃g : f =ᶠ[𝓝 x] 1 • g\n⊢ meromorp... | apply (meromorphicOrderAt_eq_int_iff hf.meromorphicAt).2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 21
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\nh₂ : ¬x = u\n⊢ x - u ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NormedRing.toRing",
"Ring.toNonAssocRing",
"congrArg",
"sub_ne_zero",
... | rwa [sub_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 96
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nx : 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nhprod : MeromorphicNFAt (g • f) x\n⊢ f =ᶠ[𝓝 x] g⁻¹ • g • f",
"usedConstants": [
"Filter.instMemb... | filter_upwards [h₁g.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.mpr h₂g)] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 135,
"column": 31
} | {
"line": 135,
"column": 44
} | [
{
"pp": "R : ℝ\nw z : ℂ\nh₂z : z ≠ w\nhR : 0 < R\nhzw : z - w ≠ 0\nhw : ‖w‖ < R\nh₁z : ‖z‖ ≤ R\nthis : ‖w‖ * ‖z‖ < R * R\n⊢ ‖(starRingEnd ℂ) w * z‖ < ‖↑R ^ 2‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.lattice",
"RCLike.norm_conj",
"Ring.... | by simpa [sq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : toMeromorphicNFAt f x = f\n⊢ MeromorphicNFAt f x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"toMeromorphicNFAt",
"M... | rw [hf.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 78
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : MeromorphicNFAt f x\nz : 𝕜\nhz : z = x\n⊢ toMeromorphicNFAt f x x = f x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminor... | simp only [toMeromorphicNFAt, hf.meromorphicAt, WithTop.coe_zero, ne_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Meromorphic.Order | {
"line": 428,
"column": 4
} | {
"line": 430,
"column": 75
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_3\nf : ι → 𝕜 → 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (∀ i ∈ s, MeromorphicAt (f i) x) → meromorphicOrderAt (∏ i ∈ s, f i) x = ∑ i ∈ s, meromorphicOrderAt (f i) x\nhf : ∀ i ∈ insert a s, MeromorphicAt (f i) x\n⊢ mero... | rw [Finset.sum_insert ha, Finset.prod_insert ha, meromorphicOrderAt_mul
(hf a (Finset.mem_insert_self a s))
(MeromorphicAt.prod (fun i hi ↦ hf i (Finset.mem_insert_of_mem hi)))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 27
} | [
{
"pp": "case h.e_a\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\nhf : MeromorphicAt f x\nn : ℤ\nhn : ¬n = 0\nh : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\ny : 𝕜\nhy :... | rw [mul_comm, zpow_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 120,
"column": 2
} | {
"line": 125,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℤ\nhn : ↑n ≠ 0\nsurj : Function.Surjective fun x ↦ x ^ n\n⊢ IsQuotientCoveringMap (fun x ↦ x ^ n) ↥(zpowGroupHom n).ker",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
"AddGroup.toS... | obtain ⟨n, rfl | rfl⟩ := n.eq_nat_or_neg
· exact isQuotientCoveringMap_npow n (by aesop) (by simpa using surj)
rw [show (zpowGroupHom (α := 𝕜ˣ) (-n)).ker = (powMonoidHom n).ker by ext; simp]
convert (isQuotientCoveringMap_npow n (by aesop) _).homeomorph_comp (.inv 𝕜ˣ) using 1
· ext; simp
convert inv_involut... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 120,
"column": 2
} | {
"line": 125,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℤ\nhn : ↑n ≠ 0\nsurj : Function.Surjective fun x ↦ x ^ n\n⊢ IsQuotientCoveringMap (fun x ↦ x ^ n) ↥(zpowGroupHom n).ker",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
"AddGroup.toS... | obtain ⟨n, rfl | rfl⟩ := n.eq_nat_or_neg
· exact isQuotientCoveringMap_npow n (by aesop) (by simpa using surj)
rw [show (zpowGroupHom (α := 𝕜ˣ) (-n)).ker = (powMonoidHom n).ker by ext; simp]
convert (isQuotientCoveringMap_npow n (by aesop) _).homeomorph_comp (.inv 𝕜ˣ) using 1
· ext; simp
convert inv_involut... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 92
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nhB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B\nz : ℂ\nhset : z ∈ verticalClosedStrip 0 1\n⊢ ‖((↑ε + ↑(sSupNormIm f 0)) ^ (z - 1) * (↑ε + ↑(sSupNormIm f 1)) ^ (-z)) • f z‖ ≤\n ma... | specialize hB (‖f z‖) (by simpa [image_congr, mem_image, comp_apply] using ⟨z, hset, rfl⟩) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Complex.Hadamard | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 29
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : 0 < ε\nz : ℂ\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz : z ∈ verticalClosedStrip 0 1\nBF : ℝ\nhBF : ∀ a ∈ verticalClosedStrip 0 1, ‖F f ε a‖ ≤ B... | rw [Asymptotics.isBigO_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 151,
"column": 2
} | {
"line": 165,
"column": 66
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - ... | obtain ⟨δ, δ₀, hδ⟩ :
∃ δ : ℝ,
δ < 0 ∧ ∀ ⦃w⦄, im w ∈ Icc (a - b) (a + b) → ‖g ε w‖ ≤ expR (δ * expR (d * |re w|)) := by
refine
⟨ε * Real.cos (d * b),
mul_neg_of_neg_of_pos ε₀
(Real.cos_pos_of_mem_Ioo <| abs_lt.1 <| (abs_of_pos (mul_pos hd₀ hb)).symm ▸ hb'),
fun w hw => ?_⟩
... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 81
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | refine fun w hw => (hδ <| hw.by_cases ?_ ?_).trans (Real.exp_le_one_iff.2 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 19
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC✝ : ℝ\nf : ℂ → E\nz : ℂ\nC : ℝ\nhC₀ : 0 < C\na b : ℝ\nhza : a - b < z.im\nhle_a : ∀ (z : ℂ), z.im = a - b → ‖f z‖ ≤ C\nhzb : z.im < a + b\nhle_b : ∀ (z : ℂ), z.im = a + b → ‖f z‖ ≤ C\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo (a - b) (a + ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 66,
"column": 57
} | {
"line": 67,
"column": 85
} | [
{
"pp": "a b : ℝ\nf₁ f₂ : ℝ → ℝ\nhf : f₁ =ᶠ[Filter.codiscreteWithin (Ι a b)] f₂\n⊢ ⨍ (x : ℝ) in a..b, f₁ x = ⨍ (x : ℝ) in a..b, f₂ x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"Real.instRCLike",
"congrArg",
"Real.instInv",... | by
rw [interval_average_eq, integral_congr_codiscreteWithin hf, ← interval_average_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 686,
"column": 6
} | {
"line": 686,
"column": 56
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : Tendsto (fun x ↦ f ↑x) atTop (𝓝 0)\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhle : ∀ (C' ... | simpa [max_eq_right h.le] using hle _ hmax _ hz.le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 743,
"column": 28
} | {
"line": 743,
"column": 33
} | [
{
"pp": "case h.refine_3\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz : ℂ\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz :... | I_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 743,
"column": 34
} | {
"line": 743,
"column": 43
} | [
{
"pp": "case h.refine_3\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz : ℂ\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz :... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 743,
"column": 44
} | {
"line": 743,
"column": 53
} | [
{
"pp": "case h.refine_3\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nz : ℂ\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ ‖f ↑x‖\nhim : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\nhz :... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
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