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.IntegralEqImproper | {
"line": 841,
"column": 6
} | {
"line": 841,
"column": 17
} | [
{
"pp": "case inl\ng g' : ℝ → ℝ\na l : ℝ\nhcont : ContinuousWithinAt g (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\ng'pos : ∀ x ∈ Ioi a, 0 ≤ g' x\nhg : Tendsto g atTop (𝓝 l)\nhx : a ∈ Ici a\n⊢ ContinuousWithinAt g (Ici a) a",
"usedConstants": []
}
] | exact hcont | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 841,
"column": 6
} | {
"line": 841,
"column": 17
} | [
{
"pp": "case inl\ng g' : ℝ → ℝ\na l : ℝ\nhcont : ContinuousWithinAt g (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\ng'pos : ∀ x ∈ Ioi a, 0 ≤ g' x\nhg : Tendsto g atTop (𝓝 l)\nhx : a ∈ Ici a\n⊢ ContinuousWithinAt g (Ici a) a",
"usedConstants": []
}
] | exact hcont | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 445,
"column": 4
} | {
"line": 445,
"column": 32
} | [
{
"pp": "case inl.hf\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : a ≤ b\n⊢ Contin... | · rwa [uIcc_of_le hab] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 404,
"column": 2
} | {
"line": 405,
"column": 79
} | [
{
"pp": "case a.inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nhm : ↑0 < ENNReal.ofReal |A.det|\n⊢ {x | (fun δ ↦ ∀ (s : Set E) (f : E → E), Ap... | · filter_upwards
simp only [forall_const, zero_mul, imp_true_iff, zero_le, ENNReal.coe_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 992,
"column": 6
} | {
"line": 992,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhf : Tendsto f atBot (𝓝 m)\nx : ℝ\nhcont : ContinuousWithinAt f (Iic x) x\nhderiv : ∀ x_1 ∈ Iio x, HasDerivAt f (f' x_1) x_1\nf'int : IntegrableOn f' (Iic x) volume\nhx : x ∈ ... | exact hcont | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 992,
"column": 6
} | {
"line": 992,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhf : Tendsto f atBot (𝓝 m)\nx : ℝ\nhcont : ContinuousWithinAt f (Iic x) x\nhderiv : ∀ x_1 ∈ Iio x, HasDerivAt f (f' x_1) x_1\nf'int : IntegrableOn f' (Iic x) volume\nhx : x ∈ ... | exact hcont | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 992,
"column": 6
} | {
"line": 992,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhf : Tendsto f atBot (𝓝 m)\nx : ℝ\nhcont : ContinuousWithinAt f (Iic x) x\nhderiv : ∀ x_1 ∈ Iio x, HasDerivAt f (f' x_1) x_1\nf'int : IntegrableOn f' (Iic x) volume\nhx : x ∈ ... | exact hcont | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1369,
"column": 2
} | {
"line": 1373,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Iic a) volume\nh_zero : Tendsto (u * v) (𝓝[Iic a \\ {a}... | have hderiv : ∀ x ∈ Iio a, HasDerivAt f (u' x * v x + u x * v' x) x := by
intro x hx
apply ((hu x hx).mul (hv x hx)).congr_of_eventuallyEq
filter_upwards [Iio_mem_nhds hx] with x (hx : x < a)
exact Function.update_of_ne (ne_of_lt hx) a' (u * v) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 81
} | [
{
"pp": "case inr\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nU : Set ℂ\nφ : Filter ι\nF : ι → ℂ → E\nf : ℂ → E\ninst✝ : CompleteSpace E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhU : IsOpen U\nhne : φ.NeBot\nK : Set ℂ\nh... | obtain ⟨δ, hδ, hK4, h⟩ := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 237,
"column": 8
} | {
"line": 237,
"column": 23
} | [
{
"pp": "case inr.h.inr\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nx : H\nhx : x ∈ e.source\ns : Set H\nopen_s : IsOpen... | rwa [hs] at x's | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 41
} | [
{
"pp": "case refine_2\nH : Type u_1\nM : Type u_2\nH' : Type u_3\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : TopologicalSpace H'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\ne e' : OpenPartialHomeomorph M H\nP : (H → H') → Set H → H → Prop\ns : Set M\nx... | simp_rw [mem_preimage, OpenPartialHomeomorph.coe_trans_symm, OpenPartialHomeomorph.symm_symm,
Function.comp_apply, e.left_inv hy] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 313,
"column": 2
} | {
"line": 319,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : IsRCLikeNormedField 𝕜\ns : Set E\nhs : Convex ℝ s\n⊢ Convex ℝ s",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"... | letI := h.rclike
letI := NormedSpace.restrictScalars ℝ 𝕜 E
simp only [Convex, StarConvex] at hs ⊢
intro u hu v hv a b ha hb hab
convert! hs hu hv ha hb hab using 2
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_smul (R := ℝ) (A := 𝕜)]
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 313,
"column": 2
} | {
"line": 319,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : IsRCLikeNormedField 𝕜\ns : Set E\nhs : Convex ℝ s\n⊢ Convex ℝ s",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"... | letI := h.rclike
letI := NormedSpace.restrictScalars ℝ 𝕜 E
simp only [Convex, StarConvex] at hs ⊢
intro u hu v hv a b ha hb hab
convert! hs hu hv ha hb hab using 2
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_smul (R := ℝ) (A := 𝕜)]
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜), ← @algebraMap_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 29
} | [
{
"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 : Set M\nmono : ∀ ⦃s : Se... | rw [← liftPropOn_univ] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 36
} | [
{
"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... | refine h.mono_of_mem_nhdsWithin ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 335,
"column": 56
} | {
"line": 335,
"column": 81
} | [
{
"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... | (e.extend I).left_inv h2x | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 362,
"column": 68
} | {
"line": 364,
"column": 79
} | [
{
"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\ns : Set M\nx : M\nn : WithTop ℕ∞\nι :... | by
simp only [contMDiffWithinAt_iff, continuousWithinAt_pi, contDiffWithinAt_pi, forall_and,
extChartAt_model_space_eq_id, Function.comp_def, PartialEquiv.refl_coe, id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 443,
"column": 72
} | {
"line": 460,
"column": 70
} | [
{
"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\n⊢... | by
haveI := h.isManifold_singleton (I := I) (n := ω)
rw [@contMDiffOn_iff]
constructor
· rw [← h.toOpenPartialHomeomorph_target]
exact (h.toOpenPartialHomeomorph e).continuousOn_symm
· intro z hz
-- show the function is actually the identity on the range of I ∘ e
apply contDiffOn_id.congr
intr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 253,
"column": 62
} | {
"line": 253,
"column": 87
} | [
{
"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... | (e.extend I).left_inv h2x | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.VectorBundle.Basic | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 58
} | [
{
"pp": "case pos\nR : 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\ne✝ : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb✝ : B\ny : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommM... | · exact (((e.linear R hb).mk' _).inverse (e.symm b) (e.symm_apply_apply_mk hb) fun v ↦
congr_arg Prod.snd <| e.apply_mk_symm hb v).isLinear | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.VectorBundle.Basic | {
"line": 303,
"column": 26
} | {
"line": 303,
"column": 38
} | [
{
"pp": "case fst\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace F E)\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Modu... | e'.coe_fst', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Basic | {
"line": 826,
"column": 26
} | {
"line": 826,
"column": 38
} | [
{
"pp": "case fst\nR : 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'.coe_fst', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 387,
"column": 95
} | {
"line": 398,
"column": 57
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁵ : NormedAddCommGroup EB\ninst✝¹⁴ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹³ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹² : TopologicalSpace B\ninst✝¹¹ : ... | by
have Hmaps : MapsTo Prod.fst (e.target ∩ e'.target) (e.baseSet ∩ e'.baseSet) := fun x hx ↦
⟨e.mem_target.1 hx.1, e'.mem_target.1 hx.2⟩
rw [mapsTo_inter] at Hmaps
-- TODO: drop `congr` https://github.com/leanprover-community/mathlib4/issues/5473
refine (contMDiffOn_fst.prodMk
(contMDiffOn_fst.coordCha... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 79
} | [
{
"pp": "n : 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 : TopologicalSpace... | filter_upwards [hp.continuousWithinAt (e.open_baseSet.mem_nhds he)] with y hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 537,
"column": 2
} | {
"line": 537,
"column": 74
} | [
{
"pp": "case refine_1\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁴ : NormedAddCommGroup EB\ninst✝¹³ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹² : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹¹ : TopologicalSpac... | · exact ((e.contMDiffAt_section_iff (ha' hx)).mp this).contMDiffWithinAt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 669,
"column": 26
} | {
"line": 669,
"column": 38
} | [
{
"pp": "case fst\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝⁹ : NormedAddCommGroup EB\ninst✝⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝⁶ : TopologicalSpace B\ninst... | e'.coe_fst', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 37
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite μ\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s... | ← @Measure.restrict_univ _ _ μ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 59
} | [
{
"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\nh'f : HasCompactSupport f\nhg : Continuous[PseudoMetricSpace... | have K_compact : IsCompact K := IsCompact.cthickening h'f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 67
} | [
{
"pp": "case bound_integrable\nE : 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\nh'f : HasCompactSupport f\nhg : Conti... | · rw [integrable_indicator_iff K_compact.measurableSet]
exact ContinuousOn.integrableOn_compact K_compact (by fun_prop) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 110,
"column": 6
} | {
"line": 119,
"column": 19
} | [
{
"pp": "case tendsto_cd\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx : E\nhx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nc : 𝕜\nhc : 1 < ‖c‖\nd : ℕ → E\nhds : ∀ (n : ℕ), x + d n ∈ s\nhd : ∀ (n... | case tendsto_cd =>
rw [atTop_basis.tendsto_iff (Metric.nhds_basis_ball_pow one_half_pos one_half_lt_one)]
refine fun N _ ↦ ⟨N, trivial, fun n hn ↦ ?_⟩
rw [Set.mem_Ici] at hn
suffices ‖c‖ ^ n * ‖d n‖ < 1 / (2 ^ N) by simpa [norm_smul]
rw [← lt_div_iff₀' (by positivity)]
re... | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 33
} | [
{
"pp": "case mp.inr\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\nhy₀ : y ≠ 0\nc : ℕ → 𝕜\nd : ℕ → E\nhd₀ : Tendsto d atTop (𝓝 0)\nhds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nhcd : Tendsto (fun n ↦ c n • d n) atTop (... | refine ⟨c, d, ?_, hds, hcd⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 75,
"column": 4
} | {
"line": 79,
"column": 51
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\nfU : ∫ (x : ℝ) in b..b + c, f x ≤ c * f (b + c)\n⊢ c * f a ≤ ∫ (x : ℝ) in a..a + c, f x",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Ta... | grw [← intervalIntegral.integral_mono_on (f := fun _ ↦ f a)
(by linarith)
(by simp)
(hf'.mono_set (by grind [uIcc]))
(by intros; apply hf <;> grind [uIcc])] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 312,
"column": 8
} | {
"line": 312,
"column": 37
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nC : ℝ≥0\ninst✝ : FiniteDimensional ℝ E\nf : E → F\nhf : LipschitzWith C f\ns : Set E\nhs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | simpa [norm_rho, hρ] using hv | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Taylor | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 33
} | [
{
"pp": "case h.e'_9\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nk : ℕ\ns t : Set ℝ\nht : UniqueDiffWithinAt ℝ t y\nhs : s ∈ 𝓝[t] y\nhf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y\n⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDer... | rw [iteratedDerivWithin_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | {
"line": 138,
"column": 2
} | {
"line": 141,
"column": 63
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\nf : ℝ → X\nd b y : ℝ\nhdb : d ≤ b\nhf : AbsolutelyContinuousOnInterval f d b\nu : Set (ℝ × ℝ)\nhu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)\nhu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y\nu_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subty... | have u_coe_sum (s : Finset u) (g : ℝ → ℝ → ℝ) :
∑ b ∈ s, (g b.val.1 b.val.2) = ∑ z ∈ u_coe s, (g z.1 z.2) :=
Finset.sum_nbij Subtype.val (by simp [u_coe]) (by simp)
(by simp only [Finset.coe_image, u_coe]; tauto) (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.Taylor | {
"line": 499,
"column": 6
} | {
"line": 499,
"column": 35
} | [
{
"pp": "case h\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis✝¹ : x₀ ≠ x\nn : ℕ\nhf :\n ∀ k ≤ n + 1,\n let u := fun t ↦ (x - t) ^ k / ↑k !;\n let v := fun t ↦ iteratedDerivWithin k f [[x₀, x]] t;\n ∫ (t : ℝ) in x₀..x, u t • deriv v t = u x • v x - u x... | rw [iteratedDerivWithin_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Taylor | {
"line": 563,
"column": 6
} | {
"line": 563,
"column": 35
} | [
{
"pp": "case h\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : ℝ → F\nx x₀ : ℝ\nn : ℕ\nhf : ContDiffOn ℝ (↑(n + 1)) f [[x₀, x]]\nthis✝ : x₀ ≠ x\nthis : UniqueDiffOn ℝ [[x₀, x]]\nk : ℕ\nhk : k ≤ n\na✝³ : ℝ\na✝² : a✝³ ≠ x₀\na✝¹ : a✝³ ≠ x\na✝ : a✝³ ∈ Ι x₀ x\n⊢ ... | rw [iteratedDerivWithin_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 75,
"column": 26
} | {
"line": 75,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * ↑π / 2 = ↑n * ↑π / 2",
"usedConstants": [
"Int.cast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instDivInvMonoid",
"Complex.instMul",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Complex.i... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 75,
"column": 26
} | {
"line": 75,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * ↑π / 2 = ↑n * ↑π / 2",
"usedConstants": [
"Int.cast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instDivInvMonoid",
"Complex.instMul",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Complex.i... | use n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 75,
"column": 26
} | {
"line": 75,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * ↑π / 2 = ↑n * ↑π / 2",
"usedConstants": [
"Int.cast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instDivInvMonoid",
"Complex.instMul",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Complex.i... | use n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 67,
"column": 26
} | {
"line": 67,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * π / 2 = ↑n * π / 2",
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Nat.instNeZeroSucc",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 67,
"column": 26
} | {
"line": 67,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * π / 2 = ↑n * π / 2",
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Nat.instNeZeroSucc",
... | use n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 67,
"column": 26
} | {
"line": 67,
"column": 31
} | [
{
"pp": "n : ℤ\n⊢ ∃ k, ↑k * π / 2 = ↑n * π / 2",
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"HDiv.hDiv",
"instOfNatNat",
"Int",
"Nat.instNeZeroSucc",
... | use n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Deriv | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 43
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nhfd : DifferentiableOn ℝ f S\nx : ℝ\nhx : x ∈ S\ny : ℝ\nhy : y ∈ S\nhxy : x ≤ y\n⊢ derivWithin f S x ≤ derivWithin f S y",
"usedConstants": [
"Real.partialOrder",
"Real",
"eq_or_lt_of_le"
]
}
] | rcases eq_or_lt_of_le hxy with rfl | hxy' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.Deriv | {
"line": 686,
"column": 2
} | {
"line": 686,
"column": 43
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nhfd : ∀ x ∈ S, DifferentiableAt ℝ f x\nx : ℝ\nhx : x ∈ S\ny : ℝ\nhy : y ∈ S\nhxy : x ≤ y\n⊢ deriv f x ≤ deriv f y",
"usedConstants": [
"Real.partialOrder",
"Real",
"eq_or_lt_of_le"
]
}
] | rcases eq_or_lt_of_le hxy with rfl | hxy' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.Deriv | {
"line": 942,
"column": 2
} | {
"line": 942,
"column": 43
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : ConcaveOn ℝ S f\nhfd : DifferentiableOn ℝ f S\nx : ℝ\nhx : x ∈ S\ny : ℝ\nhy : y ∈ S\nhxy : x ≤ y\n⊢ derivWithin f S y ≤ derivWithin f S x",
"usedConstants": [
"Real.partialOrder",
"Real",
"eq_or_lt_of_le"
]
}
] | rcases eq_or_lt_of_le hxy with rfl | hxy' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 323,
"column": 30
} | {
"line": 323,
"column": 59
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ cos (angle x y) = 1 ∨ cos (angle x y) = -1 ↔ angle x y = 0 ∨ angle x y = π",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.instZero",
"Real.cos",
"congrArg",
"Inn... | cos_eq_one_iff_angle_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Schwarz | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 57
} | [
{
"pp": "case ho.refine_2\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n +... | rw [mem_compl_singleton_iff, ← sub_ne_zero] at hw | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 46,
"column": 4
} | {
"line": 48,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nU✝ : Set 𝕜\nz : 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\n⊢ ∀ z ∈ U,\n ∃ t ∈ 𝓝 z,\n (t ∩ Function.support fun z ↦ if MeromorphicOn f U ∧ z ∈ U then (meromorphicOrderAt f z).untop₀ else... | simp_all only [Function.support_subset_iff, ne_eq, ite_eq_right_iff, WithTop.untop₀_eq_zero,
and_imp, Classical.not_imp, not_or, implies_true,
← supportDiscreteWithin_iff_locallyFiniteWithin] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 81,
"column": 8
} | {
"line": 81,
"column": 84
} | [
{
"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\nh : AnalyticAt 𝕜 f x\nh₂f : ¬analyticOrderAt f x = ⊤\nthis : analyticOrderAt f x ≠ ⊤\n⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - ... | rw [← ENat.coe_toNat_eq_self, eq_comm, h.analyticOrderAt_eq_natCast] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 64
} | [
{
"pp": "case mpr\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\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\n⊢ f =ᶠ[𝓝 x] g → f =ᶠ[𝓝[≠] x] g",
"usedConstants": [
"NormedCommRing.toSemi... | exact (Filter.EventuallyEq.filter_mono · nhdsWithin_le_nhds) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 64
} | [
{
"pp": "case mpr\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\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\n⊢ f =ᶠ[𝓝 x] g → f =ᶠ[𝓝[≠] x] g",
"usedConstants": [
"NormedCommRing.toSemi... | exact (Filter.EventuallyEq.filter_mono · nhdsWithin_le_nhds) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 64
} | [
{
"pp": "case mpr\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\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\n⊢ f =ᶠ[𝓝 x] g → f =ᶠ[𝓝[≠] x] g",
"usedConstants": [
"NormedCommRing.toSemi... | exact (Filter.EventuallyEq.filter_mono · nhdsWithin_le_nhds) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 96,
"column": 2
} | {
"line": 99,
"column": 11
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nx : 𝕜\nn : ℤ\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ n • g z\nh₄ : MeromorphicAt f x\n⊢ meromorphicTrailingCoeffAt f x = g x",
... | have : meromorphicOrderAt f x = n := by
simp only [meromorphicOrderAt_eq_int_iff h₄, ne_eq]
use g, h₁g, h₂g
exact h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 309,
"column": 37
} | {
"line": 313,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_3\nf : ι → 𝕜 → 𝕜\nh₁f : ∀ (i : ι), MeromorphicNFAt (f i) x\nh₂f : {σ | f σ x = 0}.Subsingleton\n⊢ MeromorphicNFAt (∏ᶠ (i : ι), f i) x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"MulOne.toOne... | by
by_cases h₃f : Function.HasFiniteMulSupport f
· simp_rw [finprod_eq_prod f h₃f]
exact meromorphicNFAt_prod (by aesop) (fun _ _ _ _ ↦ by aesop)
· exact finprod_of_not_hasFiniteMulSupport h₃f ▸ analyticAt_const.meromorphicNFAt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 233,
"column": 4
} | {
"line": 234,
"column": 62
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf₁ f₂ : 𝕜 → E\nhf₂ : MeromorphicAt f₂ x\nh : meromorphicOrderAt f₁ x < meromorphicOrderAt f₂ x\nhf₁ : ¬MeromorphicAt f₁ x\n⊢ meromorphicTrailingCoeffAt (f₁ + f₂)... | have : ¬MeromorphicAt (f₁ + f₂) x := by
rwa [add_comm, hf₂.meromorphicAt_add_iff_meromorphicAt₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.Order | {
"line": 498,
"column": 2
} | {
"line": 499,
"column": 46
} | [
{
"pp": "case neg\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\n⊢ meromorphicO... | rw [← WithTop.coe_untop₀_of_ne_top h, ← WithTop.coe_mul,
meromorphicOrderAt_eq_int_iff (hf.zpow n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nx : 𝕜\nhf₁ : MeromorphicAt f₁ x\nhf₂ : MeromorphicAt f₂ x\nh₂f₂ : ¬meromorphicOrderAt f₂ x = ⊤\nn₁ : ℤ\nhn₁ : ↑n₁ = meromorphicOrderAt f₁ x\nmeromorphicO... | lift meromorphicOrderAt f₂ x to ℤ using h₂f₂ with n₂ hn₂ | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 37
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nz : 𝕜\nd : 𝕜 → ℤ\nhd : (support d).Finite\n⊢ meromorphicOrderAt (∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ d u) z ≠ ⊤",
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"NormedRing.toRing",
"AddGroupWithOne.toAddG... | · simp [meromorphicOrderAt_eq d hd] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Order | {
"line": 673,
"column": 45
} | {
"line": 673,
"column": 59
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nhz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}ᶜ\nh : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y... | not_eventually | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Order | {
"line": 816,
"column": 4
} | {
"line": 816,
"column": 29
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhf : MeromorphicAt f (g x)\nhg : AnalyticAt 𝕜 g x\nhg_nc : ¬EventuallyConst g (𝓝 x)\nhf' : meromorphicOrderAt f (g x) = ⊤\n⊢ meromorphi... | rw [hf', WithTop.top_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 351,
"column": 6
} | {
"line": 351,
"column": 31
} | [
{
"pp": "case h.mp\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\nh : ∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e\n⊢ ∃ x, x.re = l ∧ f x = e",
"usedConstants": []
}
] | obtain ⟨z, hz₁, hz₂⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.Hadamard | {
"line": 355,
"column": 6
} | {
"line": 355,
"column": 31
} | [
{
"pp": "case h.mpr\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\nh : ∃ x, x.re = l ∧ f x = e\n⊢ ∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e",
"usedConstants": []
}
] | obtain ⟨z, hz₁, hz₂⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.Hadamard | {
"line": 375,
"column": 6
} | {
"line": 375,
"column": 31
} | [
{
"pp": "case h.mp\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\nh : ∃ x, x.re = 1 ∧ f (↑l + x * (↑u - ↑l)) = e\n⊢ ∃ x, x.re = u ∧ f x = e",
"usedConstants": []
}
] | obtain ⟨z, hz₁, hz₂⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.Hadamard | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 31
} | [
{
"pp": "case h.mpr\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\nh : ∃ x, x.re = u ∧ f x = e\n⊢ ∃ x, x.re = 1 ∧ f (↑l + x * (↑u - ↑l)) = e",
"usedConstants": []
}
] | obtain ⟨z, hz₁, hz₂⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.Conformal | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 41
} | [
{
"pp": "case mpr\ng : ℂ →L[ℝ] ℂ\n⊢ ((∃ map, restrictScalars ℝ map = g) ∨ ∃ map, restrictScalars ℝ map = g ∘SL ↑conjCLE) ∧ g ≠ 0 → IsConformalMap g",
"usedConstants": [
"ContinuousLinearMap.comp",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | rintro ⟨⟨map, rfl⟩ | ⟨map, hmap⟩, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 146,
"column": 4
} | {
"line": 148,
"column": 8
} | [
{
"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 le_of_tendsto (Tendsto.mono_left ?_ nhdsWithin_le_nhds) this
apply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 146,
"column": 4
} | {
"line": 148,
"column": 8
} | [
{
"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 le_of_tendsto (Tendsto.mono_left ?_ nhdsWithin_le_nhds) this
apply ((continuous_ofReal.mul continuous_const).cexp.smul continuous_const).norm.tendsto'
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case right\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf₁ f₂ : E → F\nx : E\nh₁ : HarmonicAt f₁ x\nh₂ : HarmonicAt f₂ x\n⊢ Δ (f₁ - f₂) =ᶠ[𝓝 x] 0",
"usedConstants"... | · filter_upwards [h₁.1.laplacian_sub_nhds h₂.1, h₁.2, h₂.2]
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 42
} | [
{
"pp": "case right.hf'.h\nf : ℂ → ℝ\nz : ℂ\nR : ℝ\nhf : HarmonicOnNhd f (ball z R)\nhR : ¬R ≤ 0\ng : ℂ → ℂ := (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nhg : DifferentiableOn ℂ g (ball z R)\nF : ℂ → ℂ\nhF : F z = ↑(f z) ∧ ∀ x ∈ ball z R, HasDerivAt F (g x) x\nh₁F : Dif... | simp [HasDerivAt.deriv (hF.2 y hy), g] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 258,
"column": 4
} | {
"line": 260,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : CompleteSpace E\na : E\nhf : ∀ x ∈ sphere c |R|, f x = a\n| (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) = a",
"usedConstants": [
"Real",
"instHSMul",
"MeasureTheory.M... | left; arg 2; arg 1
intro θ
rw [hf (circleMap c R θ) (circleMap_mem_sphere' c R θ)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 258,
"column": 4
} | {
"line": 260,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℂ → E\nc : ℂ\nR : ℝ\ninst✝ : CompleteSpace E\na : E\nhf : ∀ x ∈ sphere c |R|, f x = a\n| (2 * π)⁻¹ • ∫ (θ : ℝ) in 0..2 * π, f (circleMap c R θ) = a",
"usedConstants": [
"Real",
"instHSMul",
"MeasureTheory.M... | left; arg 2; arg 1
intro θ
rw [hf (circleMap c R θ) (circleMap_mem_sphere' c R θ)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.InnerProductSpace.Laplacian | {
"line": 391,
"column": 42
} | {
"line": 393,
"column": 84
} | [
{
"pp": "E : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : E → F\nx : E\ns : Set E\nl : F →L[ℝ] G\nh : ContD... | by
filter_upwards [(h.eventually (by simp)).filter_mono (nhdsWithin_mono _ (Set.subset_insert ..)),
eventually_mem_nhdsWithin] with a h₁a using h₁a.laplacianWithin_CLM_comp_left hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Poisson | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 28
} | [
{
"pp": "w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\n⊢ ((z - c + (w - c)) / (z - c - (w - c))).re ≤ (‖z - c‖ + ‖w - c‖) / (‖z - c‖ - ‖w - c‖)",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero... | by_cases h₁w : ‖w - c‖ = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Complex.Poisson | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 28
} | [
{
"pp": "w c z : ℂ\nη₂ : z - c ≠ 0\nhz : z ∈ sphere c ‖z - c‖\nhw : w ∈ ball c ‖z - c‖\nη₀ : 0 < ‖z - c‖\n⊢ (‖z - c‖ - ‖w - c‖) / (‖z - c‖ + ‖w - c‖) ≤ ((z - c + (w - c)) / (z - c - (w - c))).re",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero... | by_cases h₁w : ‖w - c‖ = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 397,
"column": 2
} | {
"line": 402,
"column": 83
} | [
{
"pp": "case hint\na b : ℝ\nt : ℂ\nht : t ≠ -1\nthis : t + 1 ≠ 0\n⊢ IntervalIntegrable (fun y ↦ ↑y * (1 + ↑y ^ 2) ^ t) MeasureTheory.volume a b",
"usedConstants": [
"Continuous.comp'",
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | · apply Continuous.intervalIntegrable
refine continuous_ofReal.mul ?_
apply Continuous.cpow (by fun_prop) continuous_const
intro a
norm_cast
exact ofReal_mem_slitPlane.2 <| add_pos_of_pos_of_nonneg one_pos <| sq_nonneg a | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Periodic | {
"line": 214,
"column": 17
} | {
"line": 214,
"column": 20
} | [
{
"pp": "case h\nh : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nh_bd : I∞.BoundedAtFilter f\nc : ℝ\nt : ∀ᶠ (x : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (cuspFunction h f) x ∧ ‖cuspFunction h f x‖ ≤ c\nS : Set ℂ\nhS1 : ∀ y ∈ S, y ∈ {0}ᶜ → DifferentiableAt ℂ (cuspFun... | hq2 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 61
} | [
{
"pp": "f : ℂ → ℂ\nU : Set ℂ\nc₀ : ℝ\nh₁f : AnalyticOnNhd ℂ f U\nh₂f : ∀ x ∈ U, (f x).re = c₀\nh₁U : IsOpen U\nh₂U : IsConnected U\n⊢ ∃ c, ∀ x ∈ U, f x = ↑c₀ + ↑c * I",
"usedConstants": [
"AnalyticOnNhd.eq_const_of_re_eq_const"
]
}
] | obtain ⟨cc, hcc⟩ := eq_const_of_re_eq_const h₁f h₂f h₁U h₂U | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 485,
"column": 2
} | {
"line": 487,
"column": 71
} | [
{
"pp": "n : ℕ\n⊢ 0 < ∫ (x : ℝ) in 0..π, sin x ^ n",
"usedConstants": [
"Nat.even_or_odd'"
]
}
] | rcases even_or_odd' n with ⟨k, rfl | rfl⟩ <;>
simp only [integral_sin_pow_even, integral_sin_pow_odd] <;>
refine mul_pos (by simp [pi_pos]) (prod_pos fun n _ => div_pos ?_ ?_) | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 618,
"column": 62
} | {
"line": 618,
"column": 78
} | [
{
"pp": "a b : ℝ\nh1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4\nh2 : Continuous fun x ↦ cos (2 * x) ^ 2\nx✝ : ℝ\n⊢ cos x✝ * sin x✝ = sin (2 * x✝) / 2",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Real.cos",
"congrArg",
"Real.instD... | rw [sin_two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.SummableUniformlyOn | {
"line": 28,
"column": 4
} | {
"line": 28,
"column": 18
} | [
{
"pp": "case refine_2\nι : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ι → ℂ → E\ns : Set ℂ\nhs : IsOpen s\nhf2 : ∀ (n : ι), ∀ r ∈ s, DifferentiableAt ℂ (f n) r\ng : ℂ → E\nhg : HasSumLocallyUniformlyOn f g s\nhc : DifferentiableOn ℂ g s\n⊢ Diff... | apply hc.congr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.TietzeExtension | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 57
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < ‖f‖\nhf3 : -‖f‖ / 3 < ‖f‖ / 3\nhc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))\n⊢ ∃ g, ‖g... | have hc₂ : IsClosed (e '' f ⁻¹' Ici (‖f‖ / 3)) :=
he.isClosedMap _ (isClosed_Ici.preimage f.continuous) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 421,
"column": 6
} | {
"line": 446,
"column": 23
} | [
{
"pp": "case refine_3\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (clo... | by_cases h1 : u ∈ closedBall c |R|
· by_cases h2 : u ∈ closedBall c |r|
· --In the smaller ball: the divisors agree and we bound the log factor
simp only [(h₁f.mono (closedBall_subset_closedBall r_lt_R.le)), h2,
AnalyticOnNhd.divisor_apply, h₁f, h1]
by_cases! h3 : u = c --N... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 421,
"column": 6
} | {
"line": 446,
"column": 23
} | [
{
"pp": "case refine_3\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (clo... | by_cases h1 : u ∈ closedBall c |R|
· by_cases h2 : u ∈ closedBall c |r|
· --In the smaller ball: the divisors agree and we bound the log factor
simp only [(h₁f.mono (closedBall_subset_closedBall r_lt_R.le)), h2,
AnalyticOnNhd.divisor_apply, h₁f, h1]
by_cases! h3 : u = c --N... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.TietzeExtension | {
"line": 476,
"column": 4
} | {
"line": 476,
"column": 28
} | [
{
"pp": "case out'\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : C(X, ℝ)\nt : Set ℝ\ne : X → Y\nhs : t.OrdConnected\nhf : ∀ (x : X), f x ∈ t\nhne : t.Nonempty\nhe : IsClosedEmbedding e\nh : ℝ ≃o ↑(Ioo (-1) 1)\nF : X →ᵇ ℝ := { toFun := Subtype.v... | rw [← h.image_Icc] at hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 33
} | [
{
"pp": "case coe\ng h : GL (Fin 2) ℝ\nz : ℍ\n⊢ num (g * h) ((σ g) ((σ h) ↑z)) / denom (g * h) ((σ g) ((σ h) ↑z)) =\n num g (num h ((σ g) ((σ h) ↑z)) / denom h ((σ g) ((σ h) ↑z))) /\n denom g (num h ((σ g) ((σ h) ↑z)) / denom h ((σ g) ((σ h) ↑z)))",
"usedConstants": [
"NormedCommRing.toSeminor... | generalize hu : σ g (σ h z) = u | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 75
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ eval 0 (f.polyToMatrix (cornerAddX M k).det).det = (f M.det).det",
... | rw [← coe_evalRingHom, RingHom.map_det, ← RingHom.comp_apply,
evalRingHom_mapMatrix_comp_polyToMatrix, f.comp_apply, RingHom.map_det] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Arsinh | {
"line": 70,
"column": 52
} | {
"line": 70,
"column": 64
} | [
{
"pp": "case h\nx : ℝ\n⊢ (√(1 + x ^ 2) + x) * (√(1 + x ^ 2) - x) = 1",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"MulOne.toOne",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"DivisionCommMonoi... | ← sq_sub_sq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 162,
"column": 2
} | {
"line": 176,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ (f M.det).det = ((comp m m n n R) (M.map ⇑f)).det",
"usedConstants": [... | induction l : Fintype.card m generalizing R S m with
| zero =>
rw [Fintype.card_eq_zero_iff] at l
simp_rw [Matrix.det_isEmpty, map_one, det_one]
| succ l ih =>
have ⟨k⟩ := Fintype.card_pos_iff.mp (Nat.lt_of_sub_eq_succ l)
let f' := f.polyToMatrix
let M' := cornerAddX M k
have : (f' M'.det).d... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 162,
"column": 2
} | {
"line": 176,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ (f M.det).det = ((comp m m n n R) (M.map ⇑f)).det",
"usedConstants": [... | induction l : Fintype.card m generalizing R S m with
| zero =>
rw [Fintype.card_eq_zero_iff] at l
simp_rw [Matrix.det_isEmpty, map_one, det_one]
| succ l ih =>
have ⟨k⟩ := Fintype.card_pos_iff.mp (Nat.lt_of_sub_eq_succ l)
let f' := f.polyToMatrix
let M' := cornerAddX M k
have : (f' M'.det).d... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 162,
"column": 2
} | {
"line": 176,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ (f M.det).det = ((comp m m n n R) (M.map ⇑f)).det",
"usedConstants": [... | induction l : Fintype.card m generalizing R S m with
| zero =>
rw [Fintype.card_eq_zero_iff] at l
simp_rw [Matrix.det_isEmpty, map_one, det_one]
| succ l ih =>
have ⟨k⟩ := Fintype.card_pos_iff.mp (Nat.lt_of_sub_eq_succ l)
let f' := f.polyToMatrix
let M' := cornerAddX M k
have : (f' M'.det).d... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 52,
"column": 2
} | {
"line": 55,
"column": 8
} | [
{
"pp": "z w : ℍ\n⊢ (2 ^ 2 * (z.im * w.im) + dist ↑z ↑w ^ 2) / (2 ^ 2 * (z.im * w.im)) =\n dist (↑z) ((starRingEnd ℂ) ↑w) ^ 2 / (2 ^ 2 * (z.im * w.im))",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | · congr 1
simp only [Complex.dist_eq, Complex.sq_norm, Complex.normSq_sub, Complex.normSq_conj,
Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 78
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)\nthis : IsOpen[inst✝] (some ⁻¹' ⋃₀ S)\ns : Set (OnePoint X)\nhsS : s ∈ S\nhs : ∞ ∈ s\n⊢ (some ⁻¹' ⋃₀ S)ᶜ ⊆ (some ⁻¹' s)ᶜ",
"usedConstants"... | exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 32
} | [
{
"pp": "X : Type w\nY : Type x\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\ninst✝ : SequentialSpace X\ns : Set X\nh : ∀ (S : CompHaus) (f : C(↑S.toTop, X)), IsClosed (⇑f ⁻¹' s)\nu : ℕ → X\np : X\nhu : ∀ (n : ℕ), u n ∈ s\nhup : Tendsto u atTop (𝓝 p)\ng : ULift.{u, 0} (OnePoint ℕ) → X := ⇑(continuousMapMk... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 132,
"column": 2
} | {
"line": 133,
"column": 60
} | [
{
"pp": "case h.hf₂\nf : ℂ → ℂ\nh₁f : Meromorphic f\nR : ℝ\n⊢ CircleIntegrable (fun x ↦ log⁺ ‖f x‖⁻¹) 0 R",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"_private.Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic.0.ValueDistribution.proxim... | · simp_rw [← norm_inv]
apply h₁f.inv.meromorphicOn.circleIntegrable_posLog_norm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 55
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nα : Type u_2\ns : Finset α\nf : α → ℂ → E\nhf : ∀ a ∈ s, Meromorphic (f a)\nr : ℝ\nh₂f : ∀ i ∈ s, CircleIntegrable (fun x ↦ log⁺ ‖f i x‖) 0 r\n⊢ circleAverage (fun x ↦ ∑ c ∈ s, log⁺ ‖f c x‖ + log ↑s.card) 0 r =\n ∑ c ∈ s, circleAv... | nth_rw 2 [← circleAverage_const (log s.card) 0 r] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 148,
"column": 84
} | {
"line": 150,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\n⊢ characteristic (f₁ + f₂) ⊤ ≤ᶠ[atTop] characteristic f₁ ⊤ + characteristic f₂ ⊤ + fun x ↦ log 2",
"usedConstants": [
"Real.instLE",
"Real",
"AddCommG... | by
filter_upwards [Filter.eventually_ge_atTop 1] with r hr
using characteristic_add_top_le h₁f₁ h₁f₂ hr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ConstantSpeed | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 42
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nφ : ℝ → ℝ\nφm : MonotoneOn φ (Icc 0 s)\nφst : φ '' Icc 0 s = Icc 0 t\nhfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)\nhf : HasUnitSpeedOn f (φ '' Icc 0 s)\nx : ℝ\nxs : x ∈ Icc 0 s\nhx : φ x = 0\n⊢ 0 ≤ φ 0",
"usedConstan... | have := φst ▸ mapsTo_image φ (Icc 0 s) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 204,
"column": 2
} | {
"line": 208,
"column": 37
} | [
{
"pp": "case hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\n⊢ StrictMonoOn (logCounting (single e 1)) (Ioi ‖e‖)",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Funct... | · intro a ha b hb hab
rw [mem_Ioi] at ha hb
rw [logCounting_single_eq_log_sub_const ha.le, logCounting_single_eq_log_sub_const hb.le]
gcongr
exact (norm_nonneg e).trans_lt ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 505,
"column": 6
} | {
"line": 505,
"column": 94
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor (fun ... | divisor_mul h₁f₁.meromorphicOn h₁f₂.meromorphicOn (fun z _ ↦ h₂f₁ z) (fun z _ ↦ h₂f₂ z), | Lean.Elab.Tactic.evalRewriteSeq | null |
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