module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 8
} | {
"line": 109,
"column": 4
} | [
{
"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... | [
"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✝ : TopologicalSpace H'\n... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 21
} | {
"line": 247,
"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... | [] | simp [EventuallyEq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.MFDeriv.Defs | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 10
} | {
"line": 162,
"column": 6
} | [
{
"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... | [
"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✝ : TopologicalSpace H'\n... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 837,
"column": 2
} | {
"line": 841,
"column": 53
} | {
"line": 843,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | [] | refine ⟨?_, fun h => h.self_of_nhds⟩
rw [contMDiffAt_iff_contMDiffOn_nhds hn]
rintro ⟨u, hu, h⟩
refine (eventually_mem_nhds_iff.mpr hu).mono fun x' hx' => ?_
exact (h x' <| mem_of_mem_nhds hx').contMDiffAt hx' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 837,
"column": 2
} | {
"line": 841,
"column": 53
} | {
"line": 843,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | [] | refine ⟨?_, fun h => h.self_of_nhds⟩
rw [contMDiffAt_iff_contMDiffOn_nhds hn]
rintro ⟨u, hu, h⟩
refine (eventually_mem_nhds_iff.mpr hu).mono fun x' hx' => ?_
exact (h x' <| mem_of_mem_nhds hx').contMDiffAt hx' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 41
} | {
"line": 135,
"column": 41
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\n⊢ ↑(extChartAt I c).symm '' ... | [
"E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\n⊢ ↑(extChartAt I c).symm '' (ball (↑(ext... | ball_inter_range_eq_ball_inter_target | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 181,
"column": 90
} | {
"line": 185,
"column": 82
} | {
"line": 187,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | by
rw [mdifferentiableWithinAt_iff']
refine and_congr Iff.rfl (exists_congr fun f' => ?_)
rw [inter_comm]
simp only [HasFDerivWithinAt, nhdsWithin_inter, nhdsWithin_extChartAt_target_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 280,
"column": 20
} | {
"line": 280,
"column": 67
} | {
"line": 281,
"column": 8
} | [
{
"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... | [
"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 : TopologicalSpace... | rw [← e.right_inv hx, ← hef (e.symm.mapsTo hx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 78
} | {
"line": 419,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : N... | [
"case e'_13\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | convert! ((mdifferentiableWithinAt_iff_of_mem_source w1 w2).mp h).2.mono _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 453,
"column": 75
} | {
"line": 454,
"column": 75
} | {
"line": 456,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | [] | by
simp [← mdifferentiableOn_univ, mdifferentiableOn_iff, continuousOn_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.VectorBundle.Basic | {
"line": 774,
"column": 2
} | {
"line": 776,
"column": 18
} | {
"line": 778,
"column": 0
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : NontriviallyNormedField R\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace R F\ninst✝ : TopologicalSpace B\nι : Type u_5\nZ : VectorBundleCore R B F ι\ni : ι\nb : B\nhb : b ∈ (Z.localTriv i).baseSet\n⊢ Trivialization.symmL R (Z.localTriv i) b = Z.c... | [] | ext1 v
rw [(Z.localTriv i).symmL_apply hb, (Z.localTriv i).symm_apply]
exacts [rfl, hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.VectorBundle.Basic | {
"line": 774,
"column": 2
} | {
"line": 776,
"column": 18
} | {
"line": 778,
"column": 0
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝³ : NontriviallyNormedField R\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace R F\ninst✝ : TopologicalSpace B\nι : Type u_5\nZ : VectorBundleCore R B F ι\ni : ι\nb : B\nhb : b ∈ (Z.localTriv i).baseSet\n⊢ Trivialization.symmL R (Z.localTriv i) b = Z.c... | [] | ext1 v
rw [(Z.localTriv i).symmL_apply hb, (Z.localTriv i).symm_apply]
exacts [rfl, hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ShrinkingLemma | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 21
} | {
"line": 231,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝ : NormalSpace X\nhs : IsClosed s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s ⊤)\nthis : ∀ (c : Set (PartialRefinement u s ⊤)), IsCha... | [] | exact hv.not_lt hlt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.PartitionOfUnity | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 46
} | {
"line": 229,
"column": 2
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\n⊢ {i | x₀ ∈ tsupport ⇑(ρ i)}.Finite",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real",
"Real.instZero",
"PartitionOfUnity.locall... | [
"ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nt : Set X\nt_in : t ∈ 𝓝 x₀\nht : {i | ((fun i ↦ support ⇑(ρ i)) i ∩ t).Nonempty}.Finite\n⊢ {i | x₀ ∈ tsupport ⇑(ρ i)}.Finite"
] | rcases ρ.locallyFinite x₀ with ⟨t, t_in, ht⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.ShrinkingLemma | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 23
} | {
"line": 362,
"column": 0
} | [
{
"pp": "case refine_1\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\nhs : IsCompact s\nuo : ∀ (i : ι), IsOpen (u i)\nuf : ∀ x ∈ s, {i | x ∈ u i}.Finite\nus : s ⊆ ⋃ i, u i\nthis✝ : Nonempty (PartialRefinement u s fun w ↦ IsC... | [] | exact hv.not_lt hlt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp | {
"line": 50,
"column": 4
} | {
"line": 52,
"column": 90
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_2\np : ℝ≥0∞\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : MemLp f p μ\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nthis : SeparableSpac... | [] | have h_fs_Lp : ∀ n, MemLp (fs n) p μ :=
SimpleFunc.memLp_approxOn_range hf_meas.measurable hf
exact fun n => (fs n).measure_support_lt_top_of_memLp (h_fs_Lp n) hp_ne_zero hp_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp | {
"line": 50,
"column": 4
} | {
"line": 52,
"column": 90
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_2\np : ℝ≥0∞\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : MemLp f p μ\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nthis : SeparableSpac... | [] | have h_fs_Lp : ∀ n, MemLp (fs n) p μ :=
SimpleFunc.memLp_approxOn_range hf_meas.measurable hf
exact fun n => (fs n).measure_support_lt_top_of_memLp (h_fs_Lp n) hp_ne_zero hp_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 233,
"column": 2
} | {
"line": 241,
"column": 20
} | {
"line": 243,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) i... | [] | rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
⟨u, u_sep, hu⟩
refine ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu
refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt
· intro s hs hμs
exact ContinuousLinearMap.integrable_com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 233,
"column": 2
} | {
"line": 241,
"column": 20
} | {
"line": 243,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫ (x : α) i... | [] | rcases (hf_int_finite t ht hμt.lt_top).aestronglyMeasurable.isSeparable_ae_range with
⟨u, u_sep, hu⟩
refine ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu
refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt
· intro s hs hμs
exact ContinuousLinearMap.integrable_com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.TangentCone.ProperSpace | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 49
} | {
"line": 50,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : ProperSpace E\ns : Set E\nx : E\nhx : AccPt x (𝓟 s)\nu : ℕ → ℝ\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ → E\nhvx : ∀ (n : ℕ), v n ≠ x\nhvu : ∀ (... | [
"𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : ProperSpace E\ns : Set E\nx : E\nhx : AccPt x (𝓟 s)\nu : ℕ → ℝ\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ → E\nhvx : ∀ (n : ℕ), v n ≠ x\nhvu : ∀ (n : ℕ), v n ... | apply IsCompact.tendsto_subseq _ (fun n ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 187,
"column": 25
} | {
"line": 187,
"column": 54
} | {
"line": 188,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC D : ℝ≥0\nf g : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nhg : LipschitzWith D g\nh'g : HasCompactSupport g\nv : E\n... | [] | exact tendsto_nhds_unique A B | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 153,
"column": 12
} | {
"line": 153,
"column": 19
} | {
"line": 153,
"column": 19
} | [
{
"pp": "a b : ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p (uIcc a b)\nhq : MonotoneOn q (uIcc a b)\nhf : BoundedVariationOn (p - q) (uIcc a b)\nh₂ : ∀ᵐ (x : ℝ), x ≠ max a b\nx : ℝ\nhx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ p (uIcc a b) x\nhx₂ : x ∈ uIcc a b → DifferentiableWithinAt ℝ q (uIcc a b) x\nhx₃ : x ≠ max a ... | [
"a b : ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p (uIcc a b)\nhq : MonotoneOn q (uIcc a b)\nhf : BoundedVariationOn (p - q) (uIcc a b)\nh₂ : ∀ᵐ (x : ℝ), x ≠ max a b\nx : ℝ\nhx₁ : x ∈ uIcc a b → DifferentiableWithinAt ℝ p (uIcc a b) x\nhx₂ : x ∈ uIcc a b → DifferentiableWithinAt ℝ q (uIcc a b) x\nhx₃ : x ≠ max a b\nhx₄ : min... | mem_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 37
} | {
"line": 216,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nα : M\nhf : AbsolutelyContinuousOnInterval f a b\nt : ℕ × (ℕ → ℝ × ℝ)\n⊢ ∑ i ∈ Finset.range t.1, dist ((fun x ↦ α • f x) (t.2 i).1) ((fun x ↦ α •... | [] | simp [Finset.mul_sum, dist_smul₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 37
} | {
"line": 216,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nα : M\nhf : AbsolutelyContinuousOnInterval f a b\nt : ℕ × (ℕ → ℝ × ℝ)\n⊢ ∑ i ∈ Finset.range t.1, dist ((fun x ↦ α • f x) (t.2 i).1) ((fun x ↦ α •... | [] | simp [Finset.mul_sum, dist_smul₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 37
} | {
"line": 216,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nα : M\nhf : AbsolutelyContinuousOnInterval f a b\nt : ℕ × (ℕ → ℝ × ℝ)\n⊢ ∑ i ∈ Finset.range t.1, dist ((fun x ↦ α • f x) (t.2 i).1) ((fun x ↦ α •... | [] | simp [Finset.mul_sum, dist_smul₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 48
} | {
"line": 80,
"column": 2
} | [
{
"pp": "case inr\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → F\nd b η : ℝ\nhdb✝ : d ≤ b\nhf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x\nhη : 0 < η\nhdb : d < b\nt : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}\ns : Set ℝ... | [
"case inr\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → F\nd b η : ℝ\nhdb✝ : d ≤ b\nhf : ∀ᵐ (x : ℝ), x ∈ Ioo d b → HasDerivAt f (f' x) x\nhη : 0 < η\nhdb : d < b\nt : Set (ℝ × ℝ) := {z | (d < z.1 ∧ z.1 < z.2 ∧ z.2 < b) ∧ dist (slope f z.1 z.2) (f' z.1) < η}\ns : Set ℝ := {x | x ∈... | simp only [t, subset_def, mem_setOf_eq] at hu₁ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 306,
"column": 61
} | {
"line": 306,
"column": 65
} | {
"line": 306,
"column": 65
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\na b : ℝ\nf : ℝ → X\nK : ℝ≥0\nhfK : LipschitzOnWith K f (uIcc a b)\nε : ℝ\nhε : ε > 0\nx✝ : ℕ × (ℕ → ℝ × ℝ)\nn : ℕ\nI : ℕ → ℝ × ℝ\nhnI₁ : (n, I) ∈ disjWithin a b\nhnI₂ : ∑ i ∈ Finset.range (n, I).1, dist ((n, I).2 i).1 ((n, I).2 i).2 < ε / (↑K + 1)\n⊢ ∑ i ∈ Fin... | [
"X : Type u_1\ninst✝ : PseudoMetricSpace X\na b : ℝ\nf : ℝ → X\nK : ℝ≥0\nhfK : LipschitzOnWith K f (uIcc a b)\nε : ℝ\nhε : ε > 0\nx✝ : ℕ × (ℕ → ℝ × ℝ)\nn : ℕ\nI : ℕ → ℝ × ℝ\nhnI₁ : (n, I) ∈ disjWithin a b\nhnI₂ : ∑ i ∈ Finset.range (n, I).1, dist ((n, I).2 i).1 ((n, I).2 i).2 < ε / (↑K + 1)\n⊢ ↑K * ∑ i ∈ Finset.ran... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 56
} | {
"line": 315,
"column": 56
} | [
{
"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⊢ 1 ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"instAddMonoidWithOneENat",
"of_decide_eq_true",
"LinearO... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 56
} | {
"line": 315,
"column": 56
} | [
{
"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⊢ 1 ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"instAddMonoidWithOneENat",
"of_decide_eq_true",
"LinearO... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 56
} | {
"line": 315,
"column": 56
} | [
{
"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⊢ 1 ≠ 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"instAddMonoidWithOneENat",
"of_decide_eq_true",
"LinearO... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 71
} | {
"line": 372,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nC : ℝ≥0\ns : Set E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : FiniteDimensional ℝ F\ninst✝ : μ.I... | [] | filter_upwards [H] with x hx xs
have : f = (A.symm ∘ A) ∘ f := by
simp only [ContinuousLinearEquiv.symm_comp_self, Function.id_comp]
rw [this]
exact A.symm.differentiableAt.comp_differentiableWithinAt x (hx xs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 71
} | {
"line": 372,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nC : ℝ≥0\ns : Set E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : FiniteDimensional ℝ F\ninst✝ : μ.I... | [] | filter_upwards [H] with x hx xs
have : f = (A.symm ∘ A) ∘ f := by
simp only [ContinuousLinearEquiv.symm_comp_self, Function.id_comp]
rw [this]
exact A.symm.differentiableAt.comp_differentiableWithinAt x (hx xs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 371,
"column": 45
} | {
"line": 376,
"column": 44
} | {
"line": 377,
"column": 8
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na✝ b✝ : ℝ\nf : ℝ → F\na b : ℝ\nhf :\n ∀ ε > 0,\n ∃ δ > 0,\n ∀ E ∈ disjWithin a b,\n ∑ i ∈ Finset.range E.1, dist (E.2 i).1 (E.2 i).2 < δ →\n ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2) < ε\nhab₀ : a ≤ b\nhab : a < b\n... | [] | by
convert! this
rw [← Finset.sum_range_sub]
congr; ext i
rw [dist_comm, Real.dist_eq, abs_eq_self.mpr]
linarith [@hp₁ i (i + 1) (by lia)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.TaylorIntegral | {
"line": 47,
"column": 2
} | {
"line": 50,
"column": 22
} | {
"line": 51,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx y : E\nt : 𝕜\nn : ℕ\nhf : ContDiffAt 𝕜 (↑n + 1) f (x + t • y)\n⊢ deriv (fun s ↦ (iteratedFDer... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx y : E\nt : 𝕜\nn : ℕ\nhf : ContDiffAt 𝕜 (↑n + 1) f (x + t • y)\nhf' : DifferentiableAt 𝕜 (iteratedFDeriv ... | have hf' : DifferentiableAt 𝕜 (iteratedFDeriv 𝕜 n f) (x + t • y) := by
apply hf.differentiableAt_iteratedFDeriv
norm_cast
exact lt_add_one n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 23
} | {
"line": 103,
"column": 24
} | [
{
"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 : z ∈ {z | 1 - ε < z.re}\nhzr : z ∈ stolzCone s\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 - ε < z.re\nhzr : z ∈ stolzCone s\n⊢ z ∈ stolzSet M"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 104,
"column": 17
} | {
"line": 104,
"column": 34
} | {
"line": 104,
"column": 35
} | [
{
"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 ∈ {z | |z.im| < s * (1 - z.re)}\n⊢ z ∈ stolzSet M",
"ppTerm": "?h.h.h.h",
"assigned": true,
... | [
"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"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 90
} | {
"line": 555,
"column": 4
} | [
{
"pp": "case inr.hvv'\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\nt : ℝ\nht : t ∈ Ioo (min x₀ x) (max x₀ x)\n⊢ HasDerivA... | [
"case inr.hvv'\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\nt : ℝ\nht : t ∈ Ioo (min x₀ x) (max x₀ x)\n⊢ DifferentiableOn ℝ (f... | refine DifferentiableOn.hasDerivAt (s := uIoo x₀ x) ?_ (by grind [Ioo_mem_nhds, uIoo]) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 134,
"column": 33
} | {
"line": 134,
"column": 46
} | {
"line": 134,
"column": 47
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nz : ℂ\nhz : ‖z‖ < 1\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\nk : Tendsto (fun x ↦ (1 - z) * ∑ x ∈ range x, ∑ i ∈ range x, f x * z ^ i) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))\n⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j... | [
"f : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nz : ℂ\nhz : ‖z‖ < 1\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\nk : Tendsto (fun x ↦ (1 - z) * ∑ x ∈ Ico 0 x, ∑ x_1 ∈ Ico 0 x, f x * z ^ x_1) atTop (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))\n⊢ Tendsto (fun n ↦ (1 - z) * ∑ i ∈ range n, (l - ∑ j ∈ range... | range_eq_Ico, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 108,
"column": 32
} | {
"line": 108,
"column": 38
} | {
"line": 108,
"column": 38
} | [
{
"pp": "m : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\nx : ℝ\nhx : 0 < x\n⊢ Even 2",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNa... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 108,
"column": 32
} | {
"line": 108,
"column": 38
} | {
"line": 108,
"column": 38
} | [
{
"pp": "m : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\nx : ℝ\nhx : 0 < x\n⊢ Even 2",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNa... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 108,
"column": 32
} | {
"line": 108,
"column": 38
} | {
"line": 108,
"column": 38
} | [
{
"pp": "m : ℤ\nhm₀ : m ≠ 0\nhm₁ : m ≠ 1\nx : ℝ\nhx : 0 < x\n⊢ Even 2",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Nat.instDecidablePredEven",
"Bool.true",
"Nat",
"Even",
"Bool",
"instAddNa... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 29
} | {
"line": 36,
"column": 30
} | [
{
"pp": "θ : ℂ\n⊢ cexp (θ * I) + cexp (-θ * I) = 0 ↔ cexp (2 * θ * I) = -1",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMo... | [
"θ : ℂ\n⊢ cexp (θ * I) = -cexp (-θ * I) ↔ cexp (2 * θ * I) = -1"
] | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 36
} | {
"line": 104,
"column": 2
} | [
{
"pp": "x✝ x : ℝ\nhx : x ≠ 0\nhx₀ : 0 < x\n⊢ sin x ^ 2 < x ^ 2",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Real.instLT",
"Preorder.toLE",
"instOfNatNat",
"LE.le",
"NPow.toPow",
"... | [
"case inl\nx✝ x : ℝ\nhx : x ≠ 0\nhx₀ : 0 < x\nhxπ : x ≤ 1\n⊢ sin x ^ 2 < x ^ 2",
"case inr\nx✝ x : ℝ\nhx : x ≠ 0\nhx₀ : 0 < x\nhxπ : 1 < x\n⊢ sin x ^ 2 < x ^ 2"
] | rcases le_or_gt x 1 with hxπ | hxπ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 70
} | {
"line": 137,
"column": 0
} | [
{
"pp": "x : ℝ\nhx₀ : -(π / 2) ≤ x\nhx : x ≤ 0\n⊢ 1 + 2 / π * x ≤ cos x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using one_sub_mul_le_cos (x := -x) (by linarith) (by linarith) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 70
} | {
"line": 137,
"column": 0
} | [
{
"pp": "x : ℝ\nhx₀ : -(π / 2) ≤ x\nhx : x ≤ 0\n⊢ 1 + 2 / π * x ≤ cos x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using one_sub_mul_le_cos (x := -x) (by linarith) (by linarith) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 70
} | {
"line": 137,
"column": 0
} | [
{
"pp": "x : ℝ\nhx₀ : -(π / 2) ≤ x\nhx : x ≤ 0\n⊢ 1 + 2 / π * x ≤ cos x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | simpa using one_sub_mul_le_cos (x := -x) (by linarith) (by linarith) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Deriv | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 83
} | {
"line": 446,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx : ℝ\nhfc : ConvexOn ℝ S f\nhxs : x ∈ interior S\na b : ℝ\nhxab : x ∈ Ioo a b\nhabs : Ioo a b ⊆ S\nh : Ioo x b ⊆ {y | y ∈ S ∧ x < y}\nh_Ioo : Tendsto (slope f x) (𝓝[>] x) (𝓝 (sInf (slope f x '' Ioo x b)))\ny : ℝ\nhyS : y ∈ S\nhxy : x < y\nz : ℝ\nhxz : x < z\nhzy : z < min b y\n... | [] | exact ⟨z, ⟨hxz, hzy.trans_le (min_le_left _ _)⟩, hzy.le.trans (min_le_right _ _)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 12
} | {
"line": 51,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ angle (c • x) (c • y) = angle x y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ arccos (⟪c • x, c • y⟫ / (‖c • x‖ * ‖c • y‖)) = angle x y"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 51,
"column": 13
} | {
"line": 51,
"column": 19
} | {
"line": 51,
"column": 20
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ arccos (⟪c • x, c • y⟫ / (‖c • x‖ * ‖c • y‖)) = angle x y",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProd... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nc : ℝ\nhc : c ≠ 0\nx y : V\nthis : c * c ≠ 0\n⊢ arccos (⟪c • x, c • y⟫ / (‖c • x‖ * ‖c • y‖)) = arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 12
} | {
"line": 58,
"column": 13
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E →ₗᵢ[ℝ] F\nu v : E\n⊢ angle (f u) (f v) = angle u v",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearIsometry",... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E →ₗᵢ[ℝ] F\nu v : E\n⊢ arccos (⟪f u, f v⟫ / (‖f u‖ * ‖f v‖)) = angle u v"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 19
} | {
"line": 58,
"column": 20
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E →ₗᵢ[ℝ] F\nu v : E\n⊢ arccos (⟪f u, f v⟫ / (‖f u‖ * ‖f v‖)) = angle u v",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E →ₗᵢ[ℝ] F\nu v : E\n⊢ arccos (⟪f u, f v⟫ / (‖f u‖ * ‖f v‖)) = arccos (⟪u, v⟫ / (‖u‖ * ‖v‖))"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 192,
"column": 6
} | {
"line": 192,
"column": 12
} | {
"line": 192,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ angle x y = 0 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"in... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = 0 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 199,
"column": 6
} | {
"line": 199,
"column": 12
} | {
"line": 199,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ angle x y = π ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHS... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = π ↔ x ≠ 0 ∧ ∃ r < 0, y = r • x"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 12
} | {
"line": 230,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ⟪x, y⟫ = -(‖x‖ * ‖y‖)\nh₁ : ‖x‖ * ‖y‖ ≠ 0\n⊢ angle x y = π",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instH... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ⟪x, y⟫ = -(‖x‖ * ‖y‖)\nh₁ : ‖x‖ * ‖y‖ ≠ 0\n⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = π"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 12
} | {
"line": 238,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ⟪x, y⟫ = ‖x‖ * ‖y‖\nh₁ : ‖x‖ * ‖y‖ ≠ 0\n⊢ angle x y = 0",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : ⟪x, y⟫ = ‖x‖ * ‖y‖\nh₁ : ‖x‖ * ‖y‖ ≠ 0\n⊢ arccos (⟪x, y⟫ / (‖x‖ * ‖y‖)) = 0"
] | angle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 235,
"column": 42
} | {
"line": 238,
"column": 45
} | {
"line": 240,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ⟪x, y⟫ = ‖x‖ * ‖y‖ ↔ angle x y = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr... | [] | by
refine ⟨fun h => ?_, inner_eq_mul_norm_of_angle_eq_zero⟩
have h₁ : ‖x‖ * ‖y‖ ≠ 0 := (mul_pos (norm_pos_iff.mpr hx) (norm_pos_iff.mpr hy)).ne'
rw [angle, h, div_self h₁, Real.arccos_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.UnitDisc.Basic | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"pp": "z : 𝔻\n⊢ ‖↑z‖ ≠ ‖1‖",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real",
"eq_false",
"congrArg",
"Complex.instNormedField",
"NormOneClass.norm_one",
"Complex.instNorm",
"NormedDivisionRing.to_normOneCl... | [] | simp [z.norm_ne_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UnitDisc.Basic | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"pp": "z : 𝔻\n⊢ ‖↑z‖ ≠ ‖1‖",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real",
"eq_false",
"congrArg",
"Complex.instNormedField",
"NormOneClass.norm_one",
"Complex.instNorm",
"NormedDivisionRing.to_normOneCl... | [] | simp [z.norm_ne_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UnitDisc.Basic | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"pp": "z : 𝔻\n⊢ ‖↑z‖ ≠ ‖1‖",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real",
"eq_false",
"congrArg",
"Complex.instNormedField",
"NormOneClass.norm_one",
"Complex.instNorm",
"NormedDivisionRing.to_normOneCl... | [] | simp [z.norm_ne_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 80,
"column": 2
} | {
"line": 84,
"column": 37
} | {
"line": 87,
"column": 2
} | [
{
"pp": "case refine_2\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fu... | [
"case refine_2\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fun x ↦ g (x, ... | have : Set.Icc (t n) (t (n + 1)) ×ˢ {a} ⊆ f ⁻¹' (q e).target := by
rintro ⟨t0, a'⟩ ⟨ht, ha⟩
rw [Set.mem_singleton_iff] at ha; dsimp only at ha
rw [← g_lifts, hpq e, ha]
exact (q e).map_source (h_sub ht) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 190,
"column": 6
} | {
"line": 196,
"column": 19
} | {
"line": 197,
"column": 2
} | [
{
"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\nx : 𝕜\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\nh : f =ᶠ[𝓝[≠] x] g\nt₀ : meromorphicOrderAt f x = meromorphicOrderAt g x\ncs : ¬merom... | [] | apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE h
let h₁f := cs
rw [hf.meromorphicOrderAt_eq_zero_iff] at h₁f
let h₁g := cs
rw [t₀, hg.meromorphicOrderAt_eq_zero_iff] at h₁g
simp only [not_not] at *
rw [h₁f, h₁g] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 190,
"column": 6
} | {
"line": 196,
"column": 19
} | {
"line": 197,
"column": 2
} | [
{
"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\nx : 𝕜\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\nh : f =ᶠ[𝓝[≠] x] g\nt₀ : meromorphicOrderAt f x = meromorphicOrderAt g x\ncs : ¬merom... | [] | apply eventuallyEq_nhds_of_eventuallyEq_nhdsNE h
let h₁f := cs
rw [hf.meromorphicOrderAt_eq_zero_iff] at h₁f
let h₁g := cs
rw [t₀, hg.meromorphicOrderAt_eq_zero_iff] at h₁g
simp only [not_not] at *
rw [h₁f, h₁g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 21
} | {
"line": 364,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nf g : 𝕜 → 𝕜\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\nhg : MeromorphicNFAt g x\nhor : g x ≠ 0 ∨ f x ≠ 0\n⊢ MeromorphicNFAt (f / g) x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instH... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nf g : 𝕜 → 𝕜\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\nhg : MeromorphicNFAt g x\nhor : g x ≠ 0 ∨ f x ≠ 0\n⊢ MeromorphicNFAt (f * g⁻¹) x"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 57
} | {
"line": 307,
"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\nx : 𝕜\nf₁ f₂ : 𝕜 → E\nhf₁ : MeromorphicAt f₁ x\nhf₂ : MeromorphicAt f₂ x\nh₁ : meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x\nh₂ : meromorphicTrailingCoeffAt f₁ x +... | [
"𝕜 : 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\nhf₁ : MeromorphicAt f₁ x\nhf₂ : MeromorphicAt f₂ x\nh₁ : meromorphicOrderAt f₁ x = meromorphicOrderAt f₂ x\nh₂ : meromorphicTrailingCoeffAt f₁ x + meromorphicTrailingCo... | filter_upwards [meromorphicOrderAt_eq_top_iff.1 h₁f₁] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 50
} | {
"line": 183,
"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\nhf : MeromorphicAt f x\nho : meromorphicOrderAt f x = 0\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nhg : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ 0 • g ... | [] | filter_upwards [hg] with y hy using by simp [hy] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 52
} | {
"line": 203,
"column": 0
} | [
{
"pp": "case coe\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\nho : 0 < meromorphicOrderAt f x\nhf : MeromorphicAt f x\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nn : ℕ\nh'o : meromorphicOrderAt f x = ↑↑n... | [] | filter_upwards [hg] with y hy using by simp [hy] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 83
} | {
"line": 398,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ meromorphicOrderAt (fun x_1 ↦ x_1 - x) x = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.h... | [] | rw [← WithTop.coe_one, ← meromorphicOrderAt_zpow_id_sub_const (𝕜 := 𝕜), zpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 83
} | {
"line": 398,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ meromorphicOrderAt (fun x_1 ↦ x_1 - x) x = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.h... | [] | rw [← WithTop.coe_one, ← meromorphicOrderAt_zpow_id_sub_const (𝕜 := 𝕜), zpow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 83
} | {
"line": 398,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ meromorphicOrderAt (fun x_1 ↦ x_1 - x) x = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.h... | [] | rw [← WithTop.coe_one, ← meromorphicOrderAt_zpow_id_sub_const (𝕜 := 𝕜), zpow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 310,
"column": 8
} | {
"line": 310,
"column": 49
} | {
"line": 310,
"column": 50
} | [
{
"pp": "case refine_2\n𝕜 : 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 ... | [
"case refine_2\n𝕜 : 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) ^... | ← (hg hu).meromorphicOrderAt_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 254,
"column": 72
} | {
"line": 254,
"column": 90
} | {
"line": 255,
"column": 4
} | [
{
"pp": "case left\nR : ℝ\nF : locallyFinsuppWithin (ball 0 R) ℤ\nz : ℂ\nhz : z ∈ ball 0 R\na : ℂ\nha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}\nb : ℂ\nhb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}\n⊢ a ∈ ball 0 R",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"NormedCommRing... | [] | (by_contra; aesop) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 254,
"column": 72
} | {
"line": 254,
"column": 90
} | {
"line": 255,
"column": 4
} | [
{
"pp": "case right\nR : ℝ\nF : locallyFinsuppWithin (ball 0 R) ℤ\nz : ℂ\nhz : z ∈ ball 0 R\na : ℂ\nha : a ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}\nb : ℂ\nhb : b ∈ {σ | (canonicalFactor R σ ^ F σ) z = 0}\n⊢ b ∈ ball 0 R",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"NormedCommRi... | [] | (by_contra; aesop) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 386,
"column": 88
} | {
"line": 393,
"column": 57
} | {
"line": 395,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nx : 𝕜\nf g : 𝕜 → E\nD : 𝕜 → ℤ\nhD : HasFiniteSupport D\nh₁x : x ∈ U\nh₂x : AccPt x (𝓟 U)\nhf : MeromorphicAt f x\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh : f =ᶠ[c... | [] | by
have t₀ : MeromorphicAt (∏ᶠ u, (· - u) ^ D u) x :=
(FactorizedRational.meromorphicNFOn D U).meromorphicOn x h₁x
rw [meromorphicTrailingCoeffAt_congr_nhdsNE
(hf.eventuallyEq_nhdsNE_of_eventuallyEq_codiscreteWithin (by fun_prop) h₁x h₂x h),
t₀.meromorphicTrailingCoeffAt_smul h₁g.meromorphicAt,
h₁... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Homotopy.Lifting | {
"line": 160,
"column": 9
} | {
"line": 160,
"column": 22
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\nhomeo : IsLocalHomeomorph p\nsep : IsSeparatedMap p\nγ₀ γ₁ : C(↑I, X)\nγ : γ₀.HomotopyRel γ₁ {0, 1}\nΓ : ↑I → C(↑I, E)\nΓ_lifts : ∀ (t s : ↑I), p ((Γ t) s) = γ (t, s)\nΓ_0 : ∀ (t : ↑I), (Γ t) ... | [] | apply Γ_lifts | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Homotopy.Lifting | {
"line": 193,
"column": 48
} | {
"line": 193,
"column": 64
} | {
"line": 193,
"column": 64
} | [
{
"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\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nΓ : (γ : C(↑I, A)) → γ 0 = a₀ → C(↑I, E)\nΓ_0 : ∀ (γ : C(↑I, A)) (a : γ 0 = a... | [] | exact hUp (hγ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Homotopy.Lifting | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 73
} | {
"line": 235,
"column": 4
} | [
{
"pp": "case zero\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x : X), IsOp... | [
"case zero\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x : X), IsOpen[inst✝¹] (... | refine ⟨fun _ ↦ e, continuous_const.continuousOn, fun t ht ↦ ?_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.ExponentialBounds | {
"line": 60,
"column": 2
} | {
"line": 61,
"column": 12
} | {
"line": 62,
"column": 2
} | [
{
"pp": "⊢ rexp 1 < 0.36787944116⁻¹",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NNRat.divNat",
"Real.instNNRatCast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"lt_of_le_of_lt",
"Real.partialOrder",
"Real",
"NNRatCast.toOfScientific",
"i... | [
"⊢ 0 < 0.36787944116"
] | · refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Order | {
"line": 859,
"column": 4
} | {
"line": 863,
"column": 82
} | {
"line": 865,
"column": 2
} | [
{
"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]
· rw [meromorphicOrderAt_eq_top_iff] at hf' ⊢
rw [Function.comp_def, ← eventually_map (P := (f · = 0))]
exact EventuallyEq.filter_mono hf' (hg.map_nhdsNE hg_nc)
· simp [(show AnalyticAt 𝕜 (g · - g x) x by fun_prop).analyticOrderAt_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 859,
"column": 4
} | {
"line": 863,
"column": 82
} | {
"line": 865,
"column": 2
} | [
{
"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]
· rw [meromorphicOrderAt_eq_top_iff] at hf' ⊢
rw [Function.comp_def, ← eventually_map (P := (f · = 0))]
exact EventuallyEq.filter_mono hf' (hg.map_nhdsNE hg_nc)
· simp [(show AnalyticAt 𝕜 (g · - g x) x by fun_prop).analyticOrderAt_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 254,
"column": 2
} | {
"line": 255,
"column": 78
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case succ.refine_4\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x :... | [
"case succ.refine_5\nE : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x : X), IsOpen[... | · rw [Function.comp_apply]; split_ifs with h
exacts [eqOn ⟨hs.1, h⟩, q.proj_symm_apply' (t_sub ⟨le_of_not_ge h, hs.2⟩)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Homotopy.Lifting | {
"line": 508,
"column": 6
} | {
"line": 508,
"column": 20
} | {
"line": 508,
"column": 21
} | [
{
"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 : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (Fund... | [
"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 : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nle : (FundamentalGroup... | Path.map_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 41
} | {
"line": 189,
"column": 4
} | [
{
"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 + ... | [
"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 + b))\nhab : a... | refine Real.tendsto_exp_atBot.comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 18
} | {
"line": 75,
"column": 0
} | [
{
"pp": "E : 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\nf : E → F\nx : E\nh : HarmonicAt f x\n⊢ ∀ᶠ (y : E) in 𝓝 x, HarmonicAt f y",
"ppTerm": "?m.28",
"assigned": true,
... | [] | filter_upwards [h.1.eventually (by simp), h.2.eventually_nhds] with a h₁a h₂a
exact ⟨h₁a, h₂a⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Harmonic.Basic | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 18
} | {
"line": 75,
"column": 0
} | [
{
"pp": "E : 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\nf : E → F\nx : E\nh : HarmonicAt f x\n⊢ ∀ᶠ (y : E) in 𝓝 x, HarmonicAt f y",
"ppTerm": "?m.28",
"assigned": true,
... | [] | filter_upwards [h.1.eventually (by simp), h.2.eventually_nhds] with a h₁a h₂a
exact ⟨h₁a, h₂a⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 26
} | {
"line": 359,
"column": 2
} | [
{
"pp": "case inr\nE : 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... | [
"case inr\nE : 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 ... | change ‖(f ∘ exp) ζ‖ ≤ C | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Complex.MeanValue | {
"line": 50,
"column": 4
} | {
"line": 57,
"column": 9
} | {
"line": 58,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc w : ℂ\ns : Set ℂ\nhs : s.Countable\nh₁f : ContinuousOn f (closedBall c |R|)\nh₂f : ∀ z ∈ ball c |R| \\ s, DifferentiableAt ℂ f z\nhw : w ∈ ball c |R|\nhR : 0 < |R|\n⊢ circleAverage (fun z... | [] | simp only [circleAverage_eq_circleIntegral hR.ne', mul_inv_rev, inv_I, neg_mul, neg_smul,
neg_inj, ne_eq, mul_eq_zero, I_ne_zero, inv_eq_zero, ofReal_eq_zero, pi_ne_zero,
OfNat.ofNat_ne_zero, or_self, not_false_eq_true, smul_right_inj]
apply circleIntegral.integral_congr hR.le
intro z hz
match_s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.MeanValue | {
"line": 50,
"column": 4
} | {
"line": 57,
"column": 9
} | {
"line": 58,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ\nc w : ℂ\ns : Set ℂ\nhs : s.Countable\nh₁f : ContinuousOn f (closedBall c |R|)\nh₂f : ∀ z ∈ ball c |R| \\ s, DifferentiableAt ℂ f z\nhw : w ∈ ball c |R|\nhR : 0 < |R|\n⊢ circleAverage (fun z... | [] | simp only [circleAverage_eq_circleIntegral hR.ne', mul_inv_rev, inv_I, neg_mul, neg_smul,
neg_inj, ne_eq, mul_eq_zero, I_ne_zero, inv_eq_zero, ofReal_eq_zero, pi_ne_zero,
OfNat.ofNat_ne_zero, or_self, not_false_eq_true, smul_right_inj]
apply circleIntegral.integral_congr hR.le
intro z hz
match_s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Poisson | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 21
} | {
"line": 83,
"column": 2
} | [
{
"pp": "φ θ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) / (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))).re ≤ (R + r) / (R - r)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
... | [
"φ θ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\n⊢ ((↑R * cexp (↑θ * I) + ↑r * cexp (↑φ * I)) * (↑R * cexp (↑θ * I) - ↑r * cexp (↑φ * I))⁻¹).re ≤ (R + r) / (R - r)"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 68
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ sin x / x ≤ |x|⁻¹",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real.partialOrder",
"Real.in... | [
"case inl\nx : ℝ\nhx : x < 0\n⊢ sin x * x⁻¹ * -x ≤ 1",
"case inl\nx : ℝ\nhx : x < 0\n⊢ 0 < -x"
] | rw [abs_of_nonpos hx.le, ← one_div, le_div_iff₀, div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 72,
"column": 26
} | {
"line": 72,
"column": 46
} | {
"line": 72,
"column": 46
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ HasDerivAt (fun x ↦ x * log x) (deriv (fun x ↦ x * log x) x) x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toM... | [
"x : ℝ\nhx : x ≠ 0\n⊢ DifferentiableAt ℝ (fun x ↦ x * log x) x"
] | hasDerivAt_deriv_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 21
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case pos\nx : ℝ\nhx : x = 0\n⊢ deriv (deriv fun x ↦ x * log x) x = x⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"DivInvMonoid.toInv",
"Semiring.toModule",
"HMul.hMul",
"GroupWithZe... | [
"case pos\nx : ℝ\nhx : x = 0\n⊢ deriv (deriv fun x ↦ x * log x) 0 = 0"
] | rw [hx, inv_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 137,
"column": 89
} | {
"line": 142,
"column": 12
} | {
"line": 144,
"column": 0
} | [
{
"pp": "⊢ StrictConvexOn ℝ (Set.Ici 0) fun x ↦ x * log x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Log.NegMulLog.0.Real.strictConvexOn_mul_log._simp_1_1",
"Real.instIsOrderedRing",
"Eq.mpr",
"IsOrderedModule.toPosSM... | [] | by
refine strictConvexOn_of_deriv2_pos (convex_Ici 0) (continuous_mul_log.continuousOn) ?_
intro x hx
simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
rw [deriv2_mul_log]
positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 47
} | {
"line": 57,
"column": 4
} | [
{
"pp": "a b r : ℝ\nh : -1 < r\nc : ℝ\nhc : 0 ≤ c\nhderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x\n⊢ IntegrableOn (fun x ↦ x ^ r) (Ioc 0 c) volume",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"instHDiv",
"int... | [
"a b r : ℝ\nh : -1 < r\nc : ℝ\nhc : 0 ≤ c\nhderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x\n⊢ ∀ x ∈ Ioo 0 c, 0 ≤ x ^ r",
"a b r : ℝ\nh : -1 < r\nc : ℝ\nhc : 0 ≤ c\nhderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x ↦ x ^ (r + 1) / (r + 1)) (x ^ r) x\n⊢ ContinuousOn (fun x ↦ x ^ (r + 1) / (r + ... | apply integrableOn_deriv_of_nonneg _ hderiv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 192,
"column": 89
} | {
"line": 204,
"column": 35
} | {
"line": 206,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ DifferentiableAt ℝ negMulLog x ↔ x ≠ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"False",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"CommRing.toNonUni... | [] | by
constructor
· unfold negMulLog
intro h eq0
simp only [neg_mul, differentiableAt_fun_neg_iff, eq0] at h
exact not_DifferentiableAt_log_mul_zero h
· intro hx
have : x ∈ ({0} : Set ℝ)ᶜ := by
simp_all only [ne_eq, Set.mem_compl_iff, Set.mem_singleton_iff, not_false_eq_true]
have := differ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 213,
"column": 28
} | {
"line": 213,
"column": 48
} | {
"line": 213,
"column": 48
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ HasDerivAt negMulLog (deriv negMulLog x) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"Norme... | [
"x : ℝ\nhx : x ≠ 0\n⊢ DifferentiableAt ℝ negMulLog x"
] | hasDerivAt_deriv_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 94,
"column": 2
} | {
"line": 97,
"column": 85
} | {
"line": 99,
"column": 0
} | [
{
"pp": "⊢ ¬ContinuousAt (fun x ↦ log (log x)) 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toMod... | [] | suffices Tendsto (fun x ↦ log (log x)) (nhdsWithin 1 {1}ᶜ) (cobounded ℝ) from
not_continuousAt_of_tendsto this nhdsWithin_le_nhds (disjoint_nhds_cobounded _)
exact (tendsto_log_nhdsNE_zero.mono_right atBot_le_cobounded).comp <| log_one ▸
HasDerivAt.tendsto_nhdsNE (by simpa using hasDerivAt_log one_ne_zero) on... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 94,
"column": 2
} | {
"line": 97,
"column": 85
} | {
"line": 99,
"column": 0
} | [
{
"pp": "⊢ ¬ContinuousAt (fun x ↦ log (log x)) 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toMod... | [] | suffices Tendsto (fun x ↦ log (log x)) (nhdsWithin 1 {1}ᶜ) (cobounded ℝ) from
not_continuousAt_of_tendsto this nhdsWithin_le_nhds (disjoint_nhds_cobounded _)
exact (tendsto_log_nhdsNE_zero.mono_right atBot_le_cobounded).comp <| log_one ▸
HasDerivAt.tendsto_nhdsNE (by simpa using hasDerivAt_log one_ne_zero) on... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 65
} | {
"line": 131,
"column": 4
} | [
{
"pp": "case pos\nx : ℝ\nhx : x = 0\n⊢ (fun x ↦ 0) =ᶠ[nhds x] log⁺",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.posLog",
"FloorRing.toFloorSemiring",
"Real.instZero",
"Real.instZeroLEOneClass",
"NeZero.charZer... | [
"x : ℝ\nhx : x = 0\ny : ℝ\nhy : y ∈ Metric.ball x 1\n⊢ 0 = log⁺ y"
] | filter_upwards [Metric.ball_mem_nhds _ zero_lt_one] with y hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 162,
"column": 2
} | {
"line": 169,
"column": 68
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\n⊢ log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Real.instLE",
"Real",
"Trans.trans",
"Finset.prod_insert",
"HMul.hMul",
"Real.po... | [] | induction s using Finset.induction with
| empty => simp [posLog]
| insert a s ha hs =>
calc log⁺ (∏ t ∈ insert a s, f t)
_ = log⁺ (f a * ∏ t ∈ s, f t) := by rw [Finset.prod_insert ha]
_ ≤ log⁺ (f a) + log⁺ (∏ t ∈ s, f t) := posLog_mul
_ ≤ log⁺ (f a) + ∑ t ∈ s, log⁺ (f t) := add_le_add (by rfl) hs
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 162,
"column": 2
} | {
"line": 169,
"column": 68
} | {
"line": 171,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\n⊢ log⁺ (∏ t ∈ s, f t) ≤ ∑ t ∈ s, log⁺ (f t)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Real.instLE",
"Real",
"Trans.trans",
"Finset.prod_insert",
"HMul.hMul",
"Real.po... | [] | induction s using Finset.induction with
| empty => simp [posLog]
| insert a s ha hs =>
calc log⁺ (∏ t ∈ insert a s, f t)
_ = log⁺ (f a * ∏ t ∈ s, f t) := by rw [Finset.prod_insert ha]
_ ≤ log⁺ (f a) + log⁺ (∏ t ∈ s, f t) := posLog_mul
_ ≤ log⁺ (f a) + ∑ t ∈ s, log⁺ (f t) := add_le_add (by rfl) hs
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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