module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 18
} | {
"line": 270,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorn... | [] | exact hmaps hz.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 66
} | {
"line": 457,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun ... | [] | exact (hsc.inter_right I.isClosed_range).image I.continuous_symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 552,
"column": 94
} | {
"line": 559,
"column": 48
} | {
"line": 561,
"column": 0
} | [
{
"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'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H... | [] | by
simp only [LiftPropAt, liftPropWithinAt_iff']
congrm ?_ ∧ ?_
· simp_rw [continuousWithinAt_univ,
(TopologicalSpace.Opens.isOpenEmbedding_of_le hUV).continuousAt_iff]
· apply hG.congr_iff
exact (TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq hUV).fun_comp
(chartAt H' (f (Set.inclus... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 128,
"column": 6
} | {
"line": 130,
"column": 35
} | {
"line": 131,
"column": 4
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : ... | [] | intro y hy
simp only [mfld_simps] at hy
simp only [h, hy, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 128,
"column": 6
} | {
"line": 130,
"column": 35
} | {
"line": 131,
"column": 4
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : ... | [] | intro y hy
simp only [mfld_simps] at hy
simp only [h, hy, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 265,
"column": 2
} | {
"line": 267,
"column": 43
} | {
"line": 269,
"column": 0
} | [
{
"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\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : Normed... | [] | apply contMDiffWithinAt_const.congr_of_eventuallyEq
(eventually_nhdsWithin_of_eventually_nhds <| notMem_mulTSupport_iff_eventuallyEq.mp hx)
(image_eq_one_of_notMem_mulTSupport hx) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 265,
"column": 2
} | {
"line": 267,
"column": 43
} | {
"line": 269,
"column": 0
} | [
{
"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\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : Normed... | [] | apply contMDiffWithinAt_const.congr_of_eventuallyEq
(eventually_nhdsWithin_of_eventually_nhds <| notMem_mulTSupport_iff_eventuallyEq.mp hx)
(image_eq_one_of_notMem_mulTSupport hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 265,
"column": 2
} | {
"line": 267,
"column": 43
} | {
"line": 269,
"column": 0
} | [
{
"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\nE' : Type u_5\ninst✝⁶ : NormedAddCommGroup E'\ninst✝⁵ : Normed... | [] | apply contMDiffWithinAt_const.congr_of_eventuallyEq
(eventually_nhdsWithin_of_eventually_nhds <| notMem_mulTSupport_iff_eventuallyEq.mp hx)
(image_eq_one_of_notMem_mulTSupport hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 65
} | {
"line": 420,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ne : M → H\nh : IsOpenEmbedding e\nn : ℕ∞ω\ninst✝ : Nonempty M\nt... | [
"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\nthis : IsMani... | rw [@contMDiff_iff _ _ _ _ _ _ _ _ _ _ h.singletonChartedSpace] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 447,
"column": 2
} | {
"line": 448,
"column": 57
} | {
"line": 449,
"column": 2
} | [
{
"pp": "case left\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 : ℕ∞ω\ninst✝ : No... | [
"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 : ℕ∞ω\ninst✝ : Nonempty M\nt... | · rw [← h.toOpenPartialHomeomorph_target]
exact (h.toOpenPartialHomeomorph e).continuousOn_symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 53
} | {
"line": 334,
"column": 4
} | [
{
"pp": "case insert\nι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\nin... | [
"case insert\nι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Chart... | simp only [iK, Finset.prod_insert, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 491,
"column": 60
} | {
"line": 494,
"column": 30
} | {
"line": 496,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ∞ω\n⊢ ContMDiffAdd 𝓘(𝕜, E) n E",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prod.normedSpace",
"chartedSpaceSelf_prod",... | [] | by
constructor
rw [← modelWithCornersSelf_prod, chartedSpaceSelf_prod]
exact contDiff_add.contMDiff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 92
} | {
"line": 184,
"column": 2
} | [
{
"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 : ℕ∞ω\nM' : Type u_5\ninst✝² : Topo... | [
"case refine_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\nM' : Type u_5\ninst✝² : T... | refine ⟨s, (f.trans c').open_source.inter ((c.trans e).trans c'.symm).open_source, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 57
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\nM' : Type u_5... | [] | simp only [s, mfld_simps]
rw [← he'] <;> simp only [c, c', hx, hex, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 57
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\nM' : Type u_5... | [] | simp only [s, mfld_simps]
rw [← he'] <;> simp only [c, c', hx, hex, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.VectorBundle.Basic | {
"line": 835,
"column": 6
} | {
"line": 838,
"column": 71
} | {
"line": 839,
"column": 6
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : ... | [
"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : (x : B) → AddCommMonoid (E x)\ninst✝⁴ : (x : B) → Module R (E x)\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace R F\ninst✝¹ : TopologicalSpace B\ninst✝ : (x : B) → TopologicalSpace (E x)\na : VectorPrebun... | have : ContinuousOn (fun x : B × F ↦ a.coordChange he' he x.1 x.2)
((e'.baseSet ∩ e.baseSet) ×ˢ univ) :=
isBoundedBilinearMap_apply.continuous.comp_continuousOn
((a.continuousOn_coordChange he' he).prodMap continuousOn_id) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 17
} | {
"line": 567,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [
"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCommGroup E'\ni... | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 40
} | {
"line": 305,
"column": 0
} | [
{
"pp": "case mpr.refine_4\n𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\nn : ℕ∞ω\nM' : Type... | [] | · simp only [c, c', hx', mfld_simps] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 39
} | {
"line": 320,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nF : Type u_6\ninst✝⁴ : NormedAddComm... | [
"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_6\ninst✝⁴ : NormedAddC... | refine ⟨fun h ↦ ⟨?_, ?_⟩, fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 276,
"column": 4
} | {
"line": 280,
"column": 47
} | {
"line": 281,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | [] | simp_rw [mem_aux]
refine ⟨fun _ ↦ ContinuousLinearEquiv.refl 𝕜 F, univ, isOpen_univ, contMDiffOn_const,
contMDiffOn_const, ⟨?_, fun b _hb ↦ rfl⟩⟩
simp only [FiberwiseLinear.openPartialHomeomorph, OpenPartialHomeomorph.refl_partialEquiv,
PartialEquiv.refl_source, univ_prod_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 276,
"column": 4
} | {
"line": 280,
"column": 47
} | {
"line": 281,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | [] | simp_rw [mem_aux]
refine ⟨fun _ ↦ ContinuousLinearEquiv.refl 𝕜 F, univ, isOpen_univ, contMDiffOn_const,
contMDiffOn_const, ⟨?_, fun b _hb ↦ rfl⟩⟩
simp only [FiberwiseLinear.openPartialHomeomorph, OpenPartialHomeomorph.refl_partialEquiv,
PartialEquiv.refl_source, univ_prod_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 502,
"column": 2
} | {
"line": 503,
"column": 40
} | {
"line": 504,
"column": 2
} | [
{
"pp": "n : 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 : TopologicalSpace B\ninst✝¹⁰ : ... | [
"n : 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 : TopologicalSpace B\ninst✝¹⁰ : ChartedSpace... | refine ⟨contMDiffOn_fst.congr fun x hx ↦ e.proj_symm_apply hx,
contMDiffOn_snd.congr fun x hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.ShrinkingLemma | {
"line": 193,
"column": 4
} | {
"line": 194,
"column": 31
} | {
"line": 195,
"column": 2
} | [
{
"pp": "case refine_3.inr\nι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝ : NormalSpace X\nv : PartialRefinement u s ⊤\nhs : IsClosed s\ni : ι\nhi : i ∉ v.carrier\nI : s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ ⊆ v.toFun i\nC : IsClosed (s ∩ ⋂ j, ⋂ (_ : j ≠ i), (v.toFun j)ᶜ)... | [] | · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
exact v.closure_subset hj | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ShrinkingLemma | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 20
} | {
"line": 288,
"column": 6
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ\nhsi : si = s ∩ ... | [
"ι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ\nhsi : si = s ∩ ⋂ i_1, ⋂ (_ ... | rw [hsi] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ShrinkingLemma | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 31
} | {
"line": 319,
"column": 2
} | [
{
"pp": "case refine_3.inr\nι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j... | [] | · rw [update_of_ne (ne_of_mem_of_not_mem hj hi)]
exact v.closure_subset hj | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 33
} | {
"line": 83,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nht : IsSeparable t\nf : α → E\nhf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\n⊢ f =ᵐ[μ] 0",
... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : d.Countable\nhd : t ⊆ closu... | rcases ht with ⟨d, d_count, hd⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 51
} | {
"line": 205,
"column": 0
} | [
{
"pp": "case neg\nι : Type uι\nE : Type uE\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nF : Type uF\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nH : Type uH\ninst✝² : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns ... | [] | · exact contMDiffAt_of_notMem (compl_subset_compl.mpr
(tsupport_smul_subset_left (f i) (g i)) hx) n | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 41
} | {
"line": 204,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\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\nht : MeasurableSet t\nhμt : μ t ≠ ∞\ns : Set α\nhs : MeasurableSet s... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\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\nht : MeasurableSet t\nhμt : μ t ≠ ∞\ns : Set α\nhs : MeasurableSet s\nx✝ : μ (s ... | refine hf_zero (s ∩ t) (hs.inter ht) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 613,
"column": 4
} | {
"line": 613,
"column": 16
} | {
"line": 614,
"column": 4
} | [
{
"pp": "E : Type uE\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℝ E\nH : Type uH\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : IsManifold I ∞ M\ninst✝⁹ : SigmaCompactSpace M... | [
"E : Type uE\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℝ E\nH : Type uH\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : IsManifold I ∞ M\ninst✝⁹ : SigmaCompactSpace M\ninst✝⁸ : T... | intro x_pt _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 378,
"column": 2
} | {
"line": 389,
"column": 89
} | {
"line": 391,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ =... | [] | suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)
have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
→ ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
intro t t_meas ht
have I : ∫ x, f x ∂μ = 0 := by rw [← setIntegral_u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 378,
"column": 2
} | {
"line": 389,
"column": 89
} | {
"line": 391,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\nμ : Measure β\nf : β → E\nhf : Integrable f μ\nh'f : ∀ (s : Set β), IsClosed[inst✝²] s → ∫ (x : β) in s, f x ∂μ =... | [] | suffices ∀ s, MeasurableSet s → ∫ x in s, f x ∂μ = 0 from
hf.ae_eq_zero_of_forall_setIntegral_eq_zero (fun s hs _ ↦ this s hs)
have A : ∀ (t : Set β), MeasurableSet t → ∫ (x : β) in t, f x ∂μ = 0
→ ∫ (x : β) in tᶜ, f x ∂μ = 0 := by
intro t t_meas ht
have I : ∫ x, f x ∂μ = 0 := by rw [← setIntegral_u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 817,
"column": 45
} | {
"line": 817,
"column": 59
} | {
"line": 817,
"column": 59
} | [
{
"pp": "E : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝... | [] | simpa using xs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 817,
"column": 45
} | {
"line": 817,
"column": 59
} | {
"line": 817,
"column": 59
} | [
{
"pp": "E : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝... | [] | simpa using xs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 817,
"column": 45
} | {
"line": 817,
"column": 59
} | {
"line": 817,
"column": 59
} | [
{
"pp": "E : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝... | [] | simpa using xs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 55
} | {
"line": 289,
"column": 2
} | [
{
"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... | [
"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.toTopologicalSpace] s\nL : E... | apply Metric.mem_nhds_iff.2 ⟨r, r_pos, fun v hv ↦ ?_⟩ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Taylor | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 82
} | {
"line": 143,
"column": 2
} | [
{
"pp": "t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) (-(↑n + 1) * (-t + x) ^ n) t",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZeroClass.... | [
"t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * (-t + x) ^ n * -1) t"
] | rw [← neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ← mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 96,
"column": 44
} | {
"line": 96,
"column": 73
} | {
"line": 97,
"column": 4
} | [
{
"pp": "s x y : ℝ\nhx₀ : 0 < x\nhx₁ : x < 1 / (1 + s ^ 2)\nhy : |y| < s * x\nH : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2\n⊢ 2 * √(1 + s ^ 2) * (x / 2) < (2 * √(1 + s ^ 2) + 1) * (x / 2)",
"ppTerm": "?m.882",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
... | [] | by gcongr; exact lt_add_one _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 314,
"column": 44
} | {
"line": 316,
"column": 41
} | {
"line": 318,
"column": 0
} | [
{
"pp": "a b : ℝ\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nhf : ContDiffOn ℝ 1 f (uIcc a b)\n⊢ AbsolutelyContinuousOnInterval f a b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"instDecidableNot",
... | [] | by
obtain ⟨K, hK⟩ := hf.exists_lipschitzOnWith (by decide) (convex_Icc _ _) isCompact_Icc
exact hK.absolutelyContinuousOnInterval | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 103,
"column": 37
} | {
"line": 103,
"column": 46
} | {
"line": 103,
"column": 47
} | [
{
"pp": "case h.h.h.h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nhM : 0 < M\nhε : 0 < ε\nH : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))\nz : ℂ\nhzl : 1 - z.re < ε\nhzr : z ∈ stolzCone s\n⊢ z ∈ stolzSet M",
"ppTerm": "?h.h.h.h",
"assigned": true,
"usedConstants":... | [
"case h.h.h.h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nhM : 0 < M\nhε : 0 < ε\nH : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))\nz : ℂ\nhzl : re 1 - z.re < ε\nhzr : z ∈ stolzCone s\n⊢ z ∈ stolzSet M"
] | ← one_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 104,
"column": 35
} | {
"line": 104,
"column": 44
} | {
"line": 104,
"column": 45
} | [
{
"pp": "case h.h.h.h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nhM : 0 < M\nhε : 0 < ε\nH : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))\nz : ℂ\nhzl : (1 - z).re < ε\nhzr : |z.im| < s * (1 - z.re)\n⊢ z ∈ stolzSet M",
"ppTerm": "?h.h.h.h",
"assigned": true,
"usedC... | [
"case h.h.h.h\ns : ℝ\nhs : 0 < s\nM ε : ℝ\nhM : 0 < M\nhε : 0 < ε\nH : ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))\nz : ℂ\nhzl : (1 - z).re < ε\nhzr : |z.im| < s * (re 1 - z.re)\n⊢ z ∈ stolzSet M"
] | ← one_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 231,
"column": 12
} | {
"line": 231,
"column": 56
} | {
"line": 231,
"column": 57
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖\... | [
"f : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖\nz : ℂ\nzn :... | tsum_geometric_of_lt_one (by positivity) zn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 87,
"column": 50
} | {
"line": 87,
"column": 88
} | {
"line": 87,
"column": 88
} | [
{
"pp": "case inl\nx : ℝ\nhx_gt : -(π / 2) < x\nhx_lt : ¬π / 2 ≤ x\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : 0 ≤ r\n⊢ 1 * (π / 2) ≤ (2 * ↑r + 1) * (π / 2)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"NonAssocSe... | [
"case inl\nx : ℝ\nhx_gt : -(π / 2) < x\nhx_lt : ¬π / 2 ≤ x\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : 0 ≤ r\n⊢ 1 ≤ 2 * ↑r + 1"
] | mul_le_mul_iff_left₀ (half_pos pi_pos) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 44
} | {
"line": 92,
"column": 44
} | [
{
"pp": "case inr\nx : ℝ\nhx_gt : ¬x ≤ -(π / 2)\nhx_lt : x < π / 2\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : r < 0\n⊢ (2 * ↑r + 1) * (π / 2) ≤ -1 * (π / 2)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"NonAssocS... | [
"case inr\nx : ℝ\nhx_gt : ¬x ≤ -(π / 2)\nhx_lt : x < π / 2\nr : ℤ\nhxr_eq : x = (2 * ↑r + 1) * π / 2\nh : r < 0\n⊢ 2 * ↑r + 1 ≤ -1"
] | mul_le_mul_iff_left₀ (half_pos pi_pos) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 22
} | {
"line": 250,
"column": 0
} | [
{
"pp": "case hx\nx : ℝ\nh : 0 ≤ x\n⊢ 0 ≤ x ^ 2",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"Even.pow_nonneg",
"PartialOrder.toPreorder",
"Nat.instAtLea... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 290,
"column": 25
} | {
"line": 290,
"column": 38
} | {
"line": 290,
"column": 39
} | [
{
"pp": "case hx₁\nx y : ℝ\nh : x * y < 1\n⊢ -arctan x + -arctan y < π / 2",
"ppTerm": "?hx₁",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHDiv",
"Real.pi",
"Real.arctan",
"congrArg",
"Real.instDivInvMonoid",
... | [
"case hx₁\nx y : ℝ\nh : x * y < 1\n⊢ arctan (-x) + -arctan y < π / 2"
] | ← arctan_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 302,
"column": 61
} | {
"line": 302,
"column": 74
} | {
"line": 302,
"column": 75
} | [
{
"pp": "x y : ℝ\nh : 1 < x * y\nhx : 0 < x\nhy : 0 < y\nk : π + -arctan ((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹)) = arctan x + arctan y\n⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
... | [
"x y : ℝ\nh : 1 < x * y\nhx : 0 < x\nhy : 0 < y\nk : π + arctan (-((x⁻¹ + y⁻¹) / (1 - x⁻¹ * y⁻¹))) = arctan x + arctan y\n⊢ arctan x + arctan y = arctan ((x + y) / (1 - x * y)) + π"
] | ← arctan_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 65
} | {
"line": 990,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhfd : HasDerivAt f f' x\n⊢ slope f x y < f'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.part... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivAt hx hy hxy hfd.neg) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Deriv | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 65
} | {
"line": 990,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhfd : HasDerivAt f f' x\n⊢ slope f x y < f'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.part... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivAt hx hy hxy hfd.neg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Deriv | {
"line": 987,
"column": 2
} | {
"line": 988,
"column": 65
} | {
"line": 990,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhfd : HasDerivAt f f' x\n⊢ slope f x y < f'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.part... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivAt hx hy hxy hfd.neg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinsupp | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 23
} | {
"line": 112,
"column": 23
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\nW : Set X\ninst✝ : Zero Y\nf : X → Y\nh : LocallyFiniteSupport f\nhW : IsCompact W\nthis : {i | ({↑i} ∩ W).Nonempty}.Finite\nlem : ∀ {α : Type u_1} (s t : Set α), Subtype.val '' {i | ({↑i} ∩ t).Nonempty} = t ∩ s\n⊢ (W ∩ support f).Finite",
"p... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nY : Type u_2\nW : Set X\ninst✝ : Zero Y\nf : X → Y\nh : LocallyFiniteSupport f\nhW : IsCompact W\nthis : {i | ({↑i} ∩ W).Nonempty}.Finite\nlem : ∀ {α : Type u_1} (s t : Set α), Subtype.val '' {i | ({↑i} ∩ t).Nonempty} = t ∩ s\n⊢ (Subtype.val '' {i | ({↑i} ∩ W).Nonempty}).... | ← lem f.support W | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 268,
"column": 45
} | {
"line": 270,
"column": 35
} | {
"line": 272,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nU : Set 𝕜\nhf₁ : MeromorphicOn f₁ U\nhf₂ : MeromorphicOn f₂ U\n⊢ (divisor f₁ U)⁻ ⊔ (divisor f₂ U)⁻ ≤ (divisor f₁ U)⁻ + (divisor f₂ U)⁻",
"ppTerm": "?m.112",
... | [] | by
by_cases h : (divisor f₁ U)⁻ ≤ (divisor f₂ U)⁻
<;> simp_all [negPart_nonneg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 82
} | {
"line": 149,
"column": 0
} | [
{
"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\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x ≠ ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicO... | [] | simpa [h₁g.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g h₃g] using h₂g | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 173,
"column": 4
} | {
"line": 175,
"column": 26
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : DecidableEq 𝕜\nx y : 𝕜\nh : ¬x = y\n⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGr... | [] | simp_all only [ite_false]
apply AnalyticAt.meromorphicTrailingCoeffAt_of_ne_zero (by fun_prop)
simp_all [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 173,
"column": 4
} | {
"line": 175,
"column": 26
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : DecidableEq 𝕜\nx y : 𝕜\nh : ¬x = y\n⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x = if x = y then 1 else x - y",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGr... | [] | simp_all only [ite_false]
apply AnalyticAt.meromorphicTrailingCoeffAt_of_ne_zero (by fun_prop)
simp_all [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 214,
"column": 4
} | {
"line": 215,
"column": 39
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case neg.h\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\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ ... | [] | filter_upwards [h₃g] with a ha
simp [ha, ← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 214,
"column": 4
} | {
"line": 215,
"column": 39
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case neg.h\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\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ ... | [] | filter_upwards [h₃g] with a ha
simp [ha, ← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 484,
"column": 6
} | {
"line": 520,
"column": 15
} | {
"line": 521,
"column": 4
} | [
{
"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 z = f z",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [] | rw [hz]
simp only [toMeromorphicNFAt, hf.meromorphicAt, WithTop.coe_zero, ne_eq]
have h₀f := hf
rcases hf with h₁f | h₁f
· simpa [meromorphicOrderAt_eq_top_iff.2 (h₁f.filter_mono nhdsWithin_le_nhds)]
using h₁f.eq_of_nhds.symm
· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h₁f
rw [Filt... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 484,
"column": 6
} | {
"line": 520,
"column": 15
} | {
"line": 521,
"column": 4
} | [
{
"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 z = f z",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [] | rw [hz]
simp only [toMeromorphicNFAt, hf.meromorphicAt, WithTop.coe_zero, ne_eq]
have h₀f := hf
rcases hf with h₁f | h₁f
· simpa [meromorphicOrderAt_eq_top_iff.2 (h₁f.filter_mono nhdsWithin_le_nhds)]
using h₁f.eq_of_nhds.symm
· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h₁f
rw [Filt... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 578,
"column": 4
} | {
"line": 580,
"column": 8
} | {
"line": 581,
"column": 2
} | [
{
"pp": "case pos\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\nh₂f₁ : meromorphicOrderAt f₁ x = ⊤\n⊢ min (meromorphicOrderAt f₁ x) (meromorphicOrderAt f₂ x) ... | [] | rw [h₂f₁, min_top_left, meromorphicOrderAt_congr]
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂f₁]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 578,
"column": 4
} | {
"line": 580,
"column": 8
} | {
"line": 581,
"column": 2
} | [
{
"pp": "case pos\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\nh₂f₁ : meromorphicOrderAt f₁ x = ⊤\n⊢ min (meromorphicOrderAt f₁ x) (meromorphicOrderAt f₂ x) ... | [] | rw [h₂f₁, min_top_left, meromorphicOrderAt_congr]
filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₂f₁]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 14
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\nh₂ : x = u\n⊢ AnalyticAt 𝕜 ((fun x ↦ x - u) ^ d u) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedRing.toRing... | [] | apply AnalyticAt.fun_zpow_nonneg (by fun_prop)
rwa [← h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 86,
"column": 4
} | {
"line": 87,
"column": 14
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\nh₂ : x = u\n⊢ AnalyticAt 𝕜 ((fun x ↦ x - u) ^ d u) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedRing.toRing... | [] | apply AnalyticAt.fun_zpow_nonneg (by fun_prop)
rwa [← h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 603,
"column": 37
} | {
"line": 603,
"column": 82
} | {
"line": 603,
"column": 83
} | [
{
"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\nn₁ : ℤ\nhn₁ : ↑n₁ = meromorphicOrderAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\nmero... | [
"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\nn₁ : ℤ\nhn₁ : ↑n₁ = meromorphicOrderAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\nmeromorphicOrder... | meromorphicOrderAt_smul t₀ h₁g.meromorphicAt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Order | {
"line": 678,
"column": 6
} | {
"line": 678,
"column": 28
} | {
"line": 679,
"column": 6
} | [
{
"pp": "case left.inr\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 ∈ ... | [
"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\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 ∈ {↑z}ᶜ → f y... | use Subtype.val ⁻¹' t' | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Meromorphic.Order | {
"line": 698,
"column": 4
} | {
"line": 698,
"column": 26
} | {
"line": 699,
"column": 4
} | [
{
"pp": "case right\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\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalS... | [
"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\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : ↑U\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ → f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'\nh₃t' :... | use Subtype.val ⁻¹' t' | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 327,
"column": 49
} | {
"line": 327,
"column": 65
} | {
"line": 327,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nh₁f : MeromorphicOn f U\nh₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤\nh₃f : (divisor f U).support.Finite\nφ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (d... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nh₁f : MeromorphicOn f U\nh₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤\nh₃f : (divisor f U).support.Finite\nφ : 𝕜 → 𝕜 := ∏ᶠ (u : 𝕜), (fun x ↦ x - u) ^ (divisor f U) ... | WithTop.coe_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Order | {
"line": 769,
"column": 2
} | {
"line": 769,
"column": 58
} | {
"line": 770,
"column": 2
} | [
{
"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\nU : Set 𝕜\nh : MeromorphicOn f U\nhx : x ∈ U\nthis : ∀ᶠ (y : 𝕜) in 𝓝[U \\ {x}] x, ContinuousAt f y\n⊢ ∀ᶠ (y : 𝕜) in 𝓝[U \\ {x}] x, AnalyticAt 𝕜 f y",
... | [
"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\nU : Set 𝕜\nh : MeromorphicOn f U\nhx : x ∈ U\nthis : ∀ᶠ (y : 𝕜) in 𝓝[U \\ {x}] x, ContinuousAt f y\ny : 𝕜\nhy : ContinuousAt f y\nh'y : y ∈ U \\ {x}\n⊢ AnalyticAt 𝕜 f ... | filter_upwards [this, self_mem_nhdsWithin] with y hy h'y | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 775,
"column": 2
} | {
"line": 775,
"column": 29
} | {
"line": 776,
"column": 2
} | [
{
"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\nU : Set 𝕜\nh : MeromorphicOn f U\nhx : x ∈ U\nthis : {x}ᶜ = U \\ {x} ∪ Uᶜ\n⊢ ∀ᶠ (y : 𝕜) in 𝓝[≠] x, AnalyticAt 𝕜 f y ∨ y ∈ Uᶜ",
"ppTerm": "?m.50",
"a... | [
"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\nU : Set 𝕜\nh : MeromorphicOn f U\nhx : x ∈ U\nthis : {x}ᶜ = U \\ {x} ∪ Uᶜ\n⊢ ∀ᶠ (y : 𝕜) in 𝓝[U \\ {x}] x ⊔ 𝓝[Uᶜ] x, AnalyticAt 𝕜 f y ∨ y ∈ Uᶜ"
] | rw [this, nhdsWithin_union] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 67
} | {
"line": 787,
"column": 2
} | [
{
"pp": "𝕜 : 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\nx : 𝕜\nhx : x ∈ U\n⊢ Disjoint (𝓝[≠] x) (𝓟 (U \\ {x | AnalyticAt 𝕜 f x}))",
"ppTerm": "?m.30",
"assigned": true,
"use... | [
"𝕜 : 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\nx : 𝕜\nhx : x ∈ U\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[≠] x, x ∈ (U \\ {x | AnalyticAt 𝕜 f x})ᶜ"
] | rw [Filter.disjoint_principal_right, ← Filter.eventually_mem_set] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 859,
"column": 6
} | {
"line": 859,
"column": 22
} | {
"line": 859,
"column": 22
} | [
{
"pp": "𝕜 : 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)\nr : ℤ := (meromorphicOrderAt f (g x)).untop₀\nhr : meromorph... | [
"𝕜 : 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)\nr : ℤ := (meromorphicOrderAt f (g x)).untop₀\nhr : meromorphicOrderAt f ... | WithTop.coe_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Conformal | {
"line": 235,
"column": 31
} | {
"line": 235,
"column": 62
} | {
"line": 235,
"column": 62
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nx : ℂ\ns : Set ℂ\nhs : UniqueDiffWithinAt ℝ s x\nh : DifferentiableWithinAt ℂ f s x\n⊢ (fderivWithin ℂ f s x) (I • 1) = (I • 1) • (fderivWithin ℂ f s x) 1",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nx : ℂ\ns : Set ℂ\nhs : UniqueDiffWithinAt ℝ s x\nh : DifferentiableWithinAt ℂ f s x\n⊢ I • (fderivWithin ℂ f s x) 1 = (I • 1) • (fderivWithin ℂ f s x) 1"
] | (fderivWithin ℂ f s x).map_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 429,
"column": 2
} | {
"line": 434,
"column": 25
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case refine_2\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nγ γ' : C(↑I,... | [
"case e'_2\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nγ γ' : C(↑I, A)\nΓ Γ' : C(↑I... | convert!
cov.liftPath_apply_one_eq_of_homotopicRel
(ContinuousMap.HomotopicRel.comp_continuousMap (SimplyConnectedSpace.paths_homotopic pγ pγ')
f)
e₀ (by simp [he]) (by simp [he]) <;>
rw [eq_liftPath_iff'] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Complex.Hadamard | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 40
} | {
"line": 170,
"column": 6
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nz : ℂ\nhB : ‖f z‖ ≤ B\nhset : z ∈ verticalClosedStrip 0 1\nhM0_one : 1 ≤ ε + sSupNormIm f 0\n⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ ≤ max 1 ((ε + sSupNormIm f 0) ^ (-1))",
"ppTerm": "?p... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nz : ℂ\nhB : ‖f z‖ ≤ B\nhset : z ∈ verticalClosedStrip 0 1\nhM0_one : 1 ≤ ε + sSupNormIm f 0\n⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ ≤ 1"
] | apply le_trans _ (le_max_left _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.Hadamard | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 40
} | {
"line": 181,
"column": 6
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nz : ℂ\nhB : ‖f z‖ ≤ B\nhset : z ∈ verticalClosedStrip 0 1\nhM1_one : 1 ≤ ε + sSupNormIm f 1\n⊢ ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ ≤ max 1 ((ε + sSupNormIm f 1) ^ (-1))",
"ppTerm": "?pos✝... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nz : ℂ\nhB : ‖f z‖ ≤ B\nhset : z ∈ verticalClosedStrip 0 1\nhM1_one : 1 ≤ ε + sSupNormIm f 1\n⊢ ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ ≤ 1"
] | apply le_trans _ (le_max_left _ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Homotopy.Lifting | {
"line": 459,
"column": 18
} | {
"line": 459,
"column": 44
} | {
"line": 459,
"column": 45
} | [
{
"pp": "E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (FundamentalGroup.map f ... | [
"E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ncov : IsCoveringMap p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (FundamentalGroup.map f a₀).range ≤ ... | ← eq.2 ⟨.reflTransSymm _⟩, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 495,
"column": 22
} | {
"line": 495,
"column": 67
} | {
"line": 495,
"column": 67
} | [
{
"pp": "E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\ninst✝¹ : SimplyConnectedSpace A\ninst✝ : LocPathConnectedSpace A\ns : Set X\ncov : IsCoveringMapOn p s\na₀ : A\nf : C(A, ↑s)\nhs : ∀ (a : A), ({ toFun := Subtype.v... | [] | by rw [Set.mem_preimage, hF'₂]; exact (f a).2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 120,
"column": 31
} | {
"line": 120,
"column": 53
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))\nhle_a : ∀ (z : ℂ), z.im = a → ‖f z‖... | [] | exact hle_a _ hza.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 120,
"column": 31
} | {
"line": 120,
"column": 53
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))\nhle_a : ∀ (z : ℂ), z.im = a → ‖f z‖... | [] | exact hle_a _ hza.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 120,
"column": 31
} | {
"line": 120,
"column": 53
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b C : ℝ\nf : ℂ → E\nz : ℂ\nhfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b)\nhB : ∃ c < π / (b - a), ∃ B, f =O[comap (abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)] fun z ↦ expR (B * expR (c * |z.re|))\nhle_a : ∀ (z : ℂ), z.im = a → ‖f z‖... | [] | exact hle_a _ hza.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 239,
"column": 2
} | {
"line": 242,
"column": 89
} | {
"line": 243,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nε : ℝ\nε_pos : 0 < ε\nthis : ∀ᶠ (x : ℂ) in 𝓝 z, ‖f x - f z‖ ≤ ε\n⊢ ∀ᶠ (x : ℂ) in 𝓝 z, ‖∫ (y : ℝ) in z.im..x.im, f (↑x.... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nε : ℝ\nε_pos : 0 < ε\nthis : ∀ᶠ (w : ℂ) in 𝓝 z, ∀ y ∈ Ι z.im w.im, ‖f (↑w.re + ↑y * I) - f z‖ ≤ ε\n⊢ ∀ᶠ (x : ℂ) in 𝓝 z, ‖∫ (y : ℝ)... | replace this : ∀ᶠ w in 𝓝 z, ∀ y ∈ Ι z.im w.im, ‖f (w.re + y * I) - f z‖ ≤ ε := by
rw [Metric.nhds_basis_closedBall.eventually_iff] at this ⊢
obtain ⟨i, i_pos, hi⟩ := this
exact ⟨i, i_pos, fun w w_in_ball y y_in_I ↦ hi (mem_closedBall_aux w_in_ball y_in_I)⟩ | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 41
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case e_a\nf : ℂ → ℝ\nx : ℂ\nhf : HarmonicAt f x\nthis :\n (fun z ↦ ↑((fderiv ℝ f z) 1) - I * ↑((fderiv ℝ f z) I)) =\n (⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) 1) - I • ⇑ofRealCLM ∘ fun x ↦ (fderiv ℝ f x) I\nh₁f : ContDiffAt ℝ 2 f x\n⊢ ((fderiv ℝ (fderiv ℝ f) x) I) 1 = ((fderiv ℝ (fderiv ℝ f) x) 1) I",
... | [] | apply h₁f.isSymmSndFDerivAt (by simp) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 55
} | {
"line": 361,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nζ : ℂ\nh... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nC : ℝ\nf : ℂ → E\nhd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0)\nhB : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C\nhim : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * I)‖ ≤ C\nζ : ℂ\nhζ : ζ.im ∈ I... | rw [mem_reProdIm, exp_re, exp_im, mem_Ioi, mem_Ioi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 26
} | {
"line": 89,
"column": 2
} | [
{
"pp": "φ θ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\nh_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2\nh_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)\n⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) * (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))⁻¹).re ≤ (R + ... | [
"φ θ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\nh_cos : R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ) ≥ (R - r) ^ 2\nh_subst : (R ^ 2 - r ^ 2) / (R ^ 2 + r ^ 2 - 2 * R * r * Real.cos (θ - φ)) ≤ (R + r) / (R - r)\n⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) * (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))⁻¹).re =\n (R ^ 2 - r ^... | convert! h_subst using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 70
} | {
"line": 74,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ DifferentiableAt ℝ (fun x ↦ x * log x) x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Semiring.toModule",
"NormedRing.toRing",
"HMul.hMul",
... | [
"x : ℝ\nhx : x ≠ 0\n⊢ {0}ᶜ ∈ 𝓝 x"
] | refine DifferentiableOn.differentiableAt differentiableOn_mul_log ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 56
} | {
"line": 210,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ deriv negMulLog x = -log x - 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"congrArg",
"deriv",
"Real... | [
"x : ℝ\nhx : x ≠ 0\n⊢ -(log x + 1) = -log x - 1"
] | rw [negMulLog_eq_neg, deriv.fun_neg, deriv_mul_log hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 34
} | {
"line": 151,
"column": 0
} | [
{
"pp": "case neg\nx y : ℝ\nha : ¬x = 0\nhb : ¬y = 0\n⊢ max (0 + 0) (log x + log y) ≤ max 0 (log x) + max 0 (log y)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real",
"max_add_add_le_max_add_max",
"Real.instZero",
"Real.instAddMonoid",
"covariant_swap_add... | [] | exact max_add_add_le_max_add_max | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 166,
"column": 44
} | {
"line": 166,
"column": 65
} | {
"line": 166,
"column": 65
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\na : α\ns : Finset α\nha : a ∉ s\nhs : log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)\n⊢ log⁺ (∏ t ∈ insert a s, f t) = log⁺ (f a * ∏ t ∈ s, f t)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Finset.prod_insert",
"HMu... | [
"α : Type u_1\nf : α → ℝ\na : α\ns : Finset α\nha : a ∉ s\nhs : log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)\n⊢ log⁺ (f a * ∏ x ∈ s, f x) = log⁺ (f a * ∏ t ∈ s, f t)"
] | Finset.prod_insert ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 80
} | {
"line": 193,
"column": 4
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ log⁺ (∑ t ∈ s, |f t|) ≤ log⁺ (∑ t ∈ s, |f t_max|)",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoi... | [
"α : Type u_1\ns : Finset α\nf : α → ℝ\nhs : s.Nonempty\nt_max : α\nht_max : t_max ∈ s ∧ ∀ x' ∈ s, |f x'| ≤ |f t_max|\n⊢ ∑ t ∈ s, |f t| ≤ ∑ t ∈ s, |f t_max|"
] | apply monotoneOn_posLog (by simp [Finset.sum_nonneg]) (by simp [mul_nonneg]) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 676,
"column": 56
} | {
"line": 676,
"column": 68
} | {
"line": 676,
"column": 68
} | [
{
"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' ... | [
"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' : ℝ), (∀ (x ... | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 77
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case hderiv\na b : ℝ\nr : ℂ\nhr : r + 1 ≠ 0\nhab : ¬0 ∉ [[a, b]]\nh : -1 < r.re\nc : ℝ\n⊢ ∀ x ∈ Set.Ioo (min 0 c) (max 0 c), HasDerivWithinAt (fun c ↦ ↑c ^ (r + 1) / (r + 1)) (↑x ^ r) (Set.Ioi x) x",
"ppTerm": "?hderiv",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealC... | [
"case hderiv.refine_1\na b : ℝ\nr : ℂ\nhr : r + 1 ≠ 0\nhab : ¬0 ∉ [[a, b]]\nh : -1 < r.re\nc x : ℝ\nhx : x ∈ Set.Ioo (min 0 c) (max 0 c)\n⊢ x ≠ 0",
"case hderiv.refine_2\na b : ℝ\nr : ℂ\nhr : r + 1 ≠ 0\nhab : ¬0 ∉ [[a, b]]\nh : -1 < r.re\nc x : ℝ\nhx : x ∈ Set.Ioo (min 0 c) (max 0 c)\n⊢ r ≠ -1"
] | refine fun x hx => (hasDerivAt_ofReal_cpow_const' ?_ ?_).hasDerivWithinAt | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 162,
"column": 2
} | {
"line": 168,
"column": 10
} | {
"line": 170,
"column": 0
} | [
{
"pp": "a : ℂ\nh : 1 < ‖a‖\n⊢ circleAverage (fun x ↦ log ‖x - a‖) 0 1 = log ‖a‖",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"SeminormedAddGroup.t... | [] | rw [HarmonicOnNhd.circleAverage_eq, zero_sub, norm_neg]
intro x hx
apply AnalyticAt.harmonicAt_log_norm (by fun_prop)
rw [sub_ne_zero]
by_contra!
simp_all only [abs_one, Metric.mem_closedBall, dist_zero_right]
linarith | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 162,
"column": 2
} | {
"line": 168,
"column": 10
} | {
"line": 170,
"column": 0
} | [
{
"pp": "a : ℂ\nh : 1 < ‖a‖\n⊢ circleAverage (fun x ↦ log ‖x - a‖) 0 1 = log ‖a‖",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"SeminormedAddGroup.t... | [] | rw [HarmonicOnNhd.circleAverage_eq, zero_sub, norm_neg]
intro x hx
apply AnalyticAt.harmonicAt_log_norm (by fun_prop)
rw [sub_ne_zero]
by_contra!
simp_all only [abs_one, Metric.mem_closedBall, dist_zero_right]
linarith | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 286,
"column": 4
} | {
"line": 287,
"column": 61
} | {
"line": 288,
"column": 2
} | [
{
"pp": "case ha\nb : ℝ\nht : 0 < b\n⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instPow",
"Real.partialOrder",
"Real",
... | [] | simpa [mul_comm] using! ((tendsto_log_mul_rpow_nhdsGT_zero zero_lt_one).sub
(tendsto_nhdsWithin_of_tendsto_nhds Filter.tendsto_id)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 286,
"column": 4
} | {
"line": 287,
"column": 61
} | {
"line": 288,
"column": 2
} | [
{
"pp": "case ha\nb : ℝ\nht : 0 < b\n⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instPow",
"Real.partialOrder",
"Real",
... | [] | simpa [mul_comm] using! ((tendsto_log_mul_rpow_nhdsGT_zero zero_lt_one).sub
(tendsto_nhdsWithin_of_tendsto_nhds Filter.tendsto_id)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 286,
"column": 4
} | {
"line": 287,
"column": 61
} | {
"line": 288,
"column": 2
} | [
{
"pp": "case ha\nb : ℝ\nht : 0 < b\n⊢ Filter.Tendsto (fun x ↦ x * log x - x) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0)",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instPow",
"Real.partialOrder",
"Real",
... | [] | simpa [mul_comm] using! ((tendsto_log_mul_rpow_nhdsGT_zero zero_lt_one).sub
(tendsto_nhdsWithin_of_tendsto_nhds Filter.tendsto_id)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 42
} | {
"line": 410,
"column": 42
} | [
{
"pp": "case e'_2\na b t : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s\n⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceReal... | [
"case e'_2\na b t : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s\n⊢ ∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t"
] | ← intervalIntegral.integral_ofReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 78
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nz : E\nh1 : AnalyticAt ℂ (gray z) 0\nhz... | [
"case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nz : E\nh1 : AnalyticAt ℂ (gray z) 0\nhz : z ∈ spher... | have h7 := h1.eventually_constant_or_nhds_le_map_nhds_aux.resolve_left hrz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 93
} | {
"line": 171,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nU : Set E\ng : E → ℂ\nhg : AnalyticOnNhd ℂ g U\nhU : IsPreconnected U\nh : ∃ z₀ ∈ U, ∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀\n⊢ (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s → IsOp... | [
"case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nU : Set E\ng : E → ℂ\nhg : AnalyticOnNhd ℂ g U\nhU : IsPreconnected U\nh : ¬∃ z₀ ∈ U, ∀ᶠ (z : E) in 𝓝 z₀, g z = g z₀\n⊢ (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s → IsOpen (g '' s)... | · obtain ⟨z₀, hz₀, h⟩ := h
exact Or.inl ⟨g z₀, hg.eqOn_of_preconnected_of_eventuallyEq analyticOnNhd_const hU hz₀ h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 41
} | {
"line": 825,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ... | [] | exact isBigO_sub_exp_rpow hfexp hgexp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 41
} | {
"line": 825,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ... | [] | exact isBigO_sub_exp_rpow hfexp hgexp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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