module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1231,
"column": 2
} | {
"line": 1233,
"column": 66
} | {
"line": 1234,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | [
"𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedAddCommGroup ... | have B : mfderivWithin I I'' (g ∘ f) s x = mfderivWithin I I'' (g ∘ f) (s ∩ f ⁻¹' u) x := by
apply MDifferentiableWithinAt.mfderivWithin_mono_of_mem_nhdsWithin _ hxs A
exact hg.comp _ (hf.mono inter_subset_left) inter_subset_right | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 364,
"column": 17
} | {
"line": 364,
"column": 43
} | {
"line": 364,
"column": 43
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ :... | [
"case succ\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddCo... | simp_rw [succ_nsmul, ← ih] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 306,
"column": 89
} | {
"line": 314,
"column": 72
} | {
"line": 316,
"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 g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → Int... | [] | by
rw [← sub_ae_eq_zero]
have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by
intro s hs hμs
rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),
sub_eq_zero.mpr (hfg_eq s hs hμs)]
have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 781,
"column": 8
} | {
"line": 781,
"column": 87
} | {
"line": 782,
"column": 8
} | [
{
"pp": "case pos\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | [
"case pos\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝ :... | simp only [mem_inter_iff, mem_preimage, (chartAt H c).left_inv (hf c Hx)] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 420,
"column": 17
} | {
"line": 427,
"column": 49
} | {
"line": 429,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : CompleteSpace E\nβ : Type u_3\ninst✝⁴ : TopologicalSpace β\ninst✝³ : MeasurableSpace β\ninst✝² : BorelSpace β\ninst✝¹ : SigmaCompactSpace β\ninst✝ : R1Space β\nμ : Measure β\nf : β → E\nhf : LocallyIntegrable f μ\nh'f : ∀ (... | [] | by
rw [← μ.restrict_univ, ← iUnion_closure_compactCovering]
apply (ae_restrict_iUnion_iff _ _).2 (fun n ↦ ?_)
apply ae_eq_zero_of_forall_setIntegral_isCompact_eq_zero
· exact hf.integrableOn_isCompact (isCompact_compactCovering β n).closure
· intro s hs
rw [Measure.restrict_restrict' measurableSet_closure... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 41
} | {
"line": 789,
"column": 4
} | [
{
"pp": "case refine_2\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompac... | [
"case refine_2\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\nin... | apply (g_diff c (chartAt H c x)).comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 37,
"column": 2
} | {
"line": 38,
"column": 27
} | {
"line": 40,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : IntervalIntegrable f volume a (b + c)\nhab : a ≤ b\nhc : 0 ≤ c\n⊢ IntervalIntegrable (fun x ↦ c⁻¹ * (f (x + c) - f x)) volume a b",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real",
"_private.Mathlib.M... | [] | exact hf.comp_add_right c |>.mono_set (by grind [uIcc]) |>.sub (hf.mono_set (by grind [uIcc]))
|>.const_mul (c := c⁻¹) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 86
} | {
"line": 54,
"column": 4
} | [
{
"pp": "case inl\nf : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (Icc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 = c\n⊢ ∫ (x : ℝ) in a..b, slope f x (x + c) ≤ f (b + c) - f a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCo... | [
"case inl\nf : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (Icc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 = c\n⊢ f a ≤ f b"
] | simp only [← hc, add_zero, slope_same, intervalIntegral.integral_zero, sub_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 86
} | {
"line": 230,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng_smooth : Co... | simp_rw [(g_smooth.differentiable (by simp)).differentiableAt.lineDeriv_eq_fderiv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 28
} | {
"line": 124,
"column": 29
} | [
{
"pp": "case left\nf : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop... | [
"case left\nf : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.o... | Filter.EventuallyLE, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 36
} | {
"line": 277,
"column": 6
} | [
{
"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 hs.trans (fun z hz ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 279,
"column": 6
} | {
"line": 283,
"column": 41
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case h₂\nF : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nf : ℝ → M\ng : ℝ → F\nhf :\n Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))\... | [] | rw [mul_comm]
grw [dist_pair_smul]
gcongr
rw [dist_zero_right]
exact hD _ (hnI.left i hi |>.right) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 279,
"column": 6
} | {
"line": 283,
"column": 41
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case h₂\nF : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nf : ℝ → M\ng : ℝ → F\nhf :\n Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))\... | [] | rw [mul_comm]
grw [dist_pair_smul]
gcongr
rw [dist_zero_right]
exact hD _ (hnI.left i hi |>.right) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 274,
"column": 2
} | {
"line": 283,
"column": 41
} | {
"line": 285,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nf : ℝ → M\ng : ℝ → F\nhf :\n Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))\n (𝓝 ... | [] | · simp only [disjWithin, mem_setOf_eq] at hnI
gcongr
· rw [dist_smul₀]
gcongr
exact hC _ (hnI.left i hi |>.left)
· rw [mul_comm]
grw [dist_pair_smul]
gcongr
rw [dist_zero_right]
exact hD _ (hnI.left i hi |>.right) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Taylor | {
"line": 95,
"column": 48
} | {
"line": 95,
"column": 61
} | {
"line": 95,
"column": 62
} | [
{
"pp": "case e_a\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ x : ℝ\n⊢ ((↑((n + 1) * n !))⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =\n ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀",
"ppTer... | [
"case e_a\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ x : ℝ\n⊢ ((↑(n + 1) * ↑n !)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀ =\n ((↑n !)⁻¹ * (↑n + 1)⁻¹ * (x - x₀) ^ (n + 1)) • iteratedDerivWithin (n + 1) f s x₀"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 28
} | {
"line": 146,
"column": 6
} | [
{
"pp": "case e'_3\nX : Type u_1\ninst✝ : PseudoMetricSpace X\nf : ℝ → X\nd b y : ℝ\nhdb : d ≤ b\nhf : AbsolutelyContinuousOnInterval f d b\nu : Set (ℝ × ℝ)\nhu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)\nhu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y\nu_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.... | [
"case e'_3\nX : Type u_1\ninst✝ : PseudoMetricSpace X\nf : ℝ → X\nd b y : ℝ\nhdb : d ≤ b\nhf : AbsolutelyContinuousOnInterval f d b\nu : Set (ℝ × ℝ)\nhu₃ : HasSum (fun z ↦ (↑z).2 - (↑z).1) (b - d)\nhu₄ : HasSum (fun z ↦ dist (f (↑z).1) (f (↑z).2)) y\nu_coe : Finset ↑u → Finset (ℝ × ℝ) := fun s ↦ Finset.image Subtyp... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Taylor | {
"line": 500,
"column": 27
} | {
"line": 500,
"column": 40
} | {
"line": 500,
"column": 41
} | [
{
"pp": "case e'_3.e_a.e_a\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis✝¹ : x₀ ≠ x\nn : ℕ\nih :\n f x - taylorWithinEval f n [[x₀, x]] x₀ x =\n ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t\nhf :\n ∀ k ≤ n + 1,\n ... | [
"case e'_3.e_a.e_a\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nx x₀ : ℝ\nthis✝¹ : x₀ ≠ x\nn : ℕ\nih :\n f x - taylorWithinEval f n [[x₀, x]] x₀ x =\n ∫ (t : ℝ) in x₀..x, ((x - t) ^ n / ↑n !) • iteratedDerivWithin (n + 1) f [[x₀, x]] t\nhf :\n ∀ k ≤ n + 1,\n let u := fun... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 12
} | {
"line": 186,
"column": 2
} | [
{
"pp": "case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nhm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ ra... | [
"case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖\... | clear hm p | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 21
} | {
"line": 95,
"column": 21
} | [
{
"pp": "m : ℤ\nn : ℕ\nhn : Even n\nhm : m ∉ Ico 0 ↑n\na : ℕ\nha : a ∈ Finset.range n\nh : m - ↑a = 0\n⊢ m ∈ Ico 0 ↑n",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"congrArg",
"sub_eq_zero",
"PartialOrder.toPreorder"... | [
"m : ℤ\nn : ℕ\nhn : Even n\nhm : m ∉ Ico 0 ↑n\na : ℕ\nha : a ∈ Finset.range n\nh : m - ↑a = 0\n⊢ ↑a ∈ Ico 0 ↑n"
] | sub_eq_zero.1 h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 187,
"column": 2
} | {
"line": 195,
"column": 27
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l -... | [
"case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ range n, f i - l‖ < ε / 4 / M\nF : ℝ := ∑ i ∈ range B₁, ‖l - s (i + 1)‖\... | suffices ‖(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ < ε / 2 by
calc
_ = ‖l - g z‖ := by rw [norm_sub_rev]
_ = ‖l - g z - (1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i +
(1 - z) * ∑ i ∈ range (max B₁ B₂), (l - s (i + 1)) * z ^ i‖ := by rw [sub_add_cancel _]
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Complex.AbsMax | {
"line": 191,
"column": 52
} | {
"line": 191,
"column": 61
} | {
"line": 192,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\nw : E\nhw : dist z w ≤ r\nhne : z ≠ w\ne : ℂ → E := ⇑(lineMap z w)\nh... | [] | simpa [e] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.AbsMax | {
"line": 191,
"column": 52
} | {
"line": 191,
"column": 61
} | {
"line": 192,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\nw : E\nhw : dist z w ≤ r\nhne : z ≠ w\ne : ℂ → E := ⇑(lineMap z w)\nh... | [] | simpa [e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.AbsMax | {
"line": 191,
"column": 52
} | {
"line": 191,
"column": 61
} | {
"line": 192,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\nw : E\nhw : dist z w ≤ r\nhne : z ≠ w\ne : ℂ → E := ⇑(lineMap z w)\nh... | [] | simpa [e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.AbsMax | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 89
} | {
"line": 343,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : StrictConvexSpace ℝ F\ninst✝ : ProperSpace E\nf : E → F\nr b : ℝ\nh_an : DifferentiableOn ℂ f (ball 0 b)\nhr_nn : 0 ≤ r\nhr_lt : r < b\nhr : ∀ z ∈ ball 0 b,... | [
"E : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : StrictConvexSpace ℝ F\ninst✝ : ProperSpace E\nf : E → F\nr b : ℝ\nh_an : DifferentiableOn ℂ f (ball 0 b)\nhr_nn : 0 ≤ r\nhr_lt : r < b\nhr : ∀ z ∈ ball 0 b, ∃ w ∈ close... | apply eq_const_of_exists_max h_an (closedBall_subset_ball hr_lt hx_mem) (fun z hz ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Convex.Deriv | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 71
} | {
"line": 164,
"column": 4
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"ppTerm": "?m.80",
"assigned": true,
... | [
"D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)"
] | have hxyD' : Ioo x y ⊆ interior D :=
subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 19
} | {
"line": 101,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\n⊢ sin x ^ 2 < x ^ 2",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"PartialOrder.toPreorder",
"Real.instLT",
"Preorder.toLE",
"Eq.mp",
"Ne",
"instOfNatNat",
"LE.le",
"NPow.toPo... | [
"case inr\nx : ℝ\nhx : x ≠ 0\nthis : ∀ {x : ℝ}, x ≠ 0 → 0 < x → sin x ^ 2 < x ^ 2\nhx₀ : x ≤ 0\n⊢ sin x ^ 2 < x ^ 2",
"x✝ x : ℝ\nhx : x ≠ 0\nhx₀ : 0 < x\n⊢ sin x ^ 2 < x ^ 2"
] | wlog! hx₀ : 0 < x | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog!_1 | Mathlib.Tactic.wlog! |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 23
} | {
"line": 160,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 0 < x\nf : ℝ → ℝ := fun t ↦ sin t - (t - t ^ 3 / 6)\nhderiv : ∀ (t : ℝ), deriv f t = cos t - 1 + t ^ 2 / 2\nhmono : StrictMonoOn f (Ici 0)\nh0 : f 0 < f x\n⊢ x - x ^ 3 / 6 < sin x",
"ppTerm": "?m.421",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Special... | [] | grind [Real.sin_zero] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Complex.Schwarz | {
"line": 148,
"column": 57
} | {
"line": 182,
"column": 72
} | {
"line": 184,
"column": 0
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nc z : E\nR₁ R₂ : ℝ\nn : ℕ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)\nhn : (fun x ↦ f x - f c) =o[𝓝 c... | [] | by
-- Note that `0 < R₁`, `0 ≤ R₂`, then discard the trivial case `f z = f c`.
have hR₁ : 0 < R₁ := nonempty_ball.mp ⟨_, hz⟩
have hR₂ : 0 ≤ R₂ := nonempty_closedBall.mp ⟨_, h_maps hz⟩
rcases eq_or_ne (f z) (f c) with heq | hfne
· trans 0 <;> [simp [heq]; positivity]
have hne : z ≠ c := ne_of_apply_ne _ hfne... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 342,
"column": 24
} | {
"line": 342,
"column": 38
} | {
"line": 342,
"column": 39
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nι : Type u_3\nf : ι → 𝕜 → 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nhs :\n (∀ i ∈ s, MeromorphicOn (f i) U) →\n (∀ i ∈ s, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤) → divisor (∏ i ∈ s, f i) U = ∑ i ∈ s, divisor (f i) U\nh₁f : ∀ i... | [
"case insert\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nι : Type u_3\nf : ι → 𝕜 → 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nhs :\n (∀ i ∈ s, MeromorphicOn (f i) U) →\n (∀ i ∈ s, ∀ z ∈ U, meromorphicOrderAt (f i) z ≠ ⊤) → divisor (∏ i ∈ s, f i) U = ∑ i ∈ s, divisor (f i) U\nh₁f : ∀ i ∈ insert a ... | sum_insert ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 39
} | {
"line": 209,
"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\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x",
"ppTerm": "?pos✝"... | [] | simp_all [← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 39
} | {
"line": 209,
"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\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x",
"ppTerm": "?pos✝"... | [] | simp_all [← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 39
} | {
"line": 209,
"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\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x",
"ppTerm": "?pos✝"... | [] | simp_all [← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 194,
"column": 25
} | {
"line": 194,
"column": 48
} | {
"line": 195,
"column": 4
} | [
{
"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\nho : 0 < meromorphicOrderAt f x\nhf : MeromorphicAt f x\nn : ℤ\nh'o : meromorphicOrderAt f x = ↑n\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nhg : ∀ᶠ (... | [] | simpa [h'o] using ho.le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Meromorphic.Order | {
"line": 194,
"column": 25
} | {
"line": 194,
"column": 48
} | {
"line": 195,
"column": 4
} | [
{
"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\nho : 0 < meromorphicOrderAt f x\nhf : MeromorphicAt f x\nn : ℤ\nh'o : meromorphicOrderAt f x = ↑n\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nhg : ∀ᶠ (... | [] | simpa [h'o] using ho.le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 194,
"column": 25
} | {
"line": 194,
"column": 48
} | {
"line": 195,
"column": 4
} | [
{
"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\nho : 0 < meromorphicOrderAt f x\nhf : MeromorphicAt f x\nn : ℤ\nh'o : meromorphicOrderAt f x = ↑n\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nhg : ∀ᶠ (... | [] | simpa [h'o] using ho.le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 509,
"column": 12
} | {
"line": 509,
"column": 60
} | {
"line": 510,
"column": 12
} | [
{
"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 z : 𝕜\nhz : z = x\nh₀f : MeromorphicNFAt f x\nn : ℤ\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝 x] (fun x_1 ↦ x_1 - x) ^ 0 • g\nthis : meromor... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx z : 𝕜\nhz : z = x\nh₀f : MeromorphicNFAt f x\nn : ℤ\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nthis : meromorphicOrderAt f x = ↑n\nh₃f : meromorphicOrderAt f x = 0\nhn... | simp only [zpow_zero, one_smul, ne_eq] at h₃g h₂ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Meromorphic.Order | {
"line": 336,
"column": 42
} | {
"line": 336,
"column": 64
} | {
"line": 337,
"column": 4
} | [
{
"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\nh : MeromorphicAt f x\nh' : ContinuousAt f x\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ≠ 0\nthis : 0 ≤ meromorphicOrderAt f x\nn : ℕ\nho : meromorphicOrde... | [] | by simpa using hz.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 372,
"column": 48
} | {
"line": 372,
"column": 67
} | {
"line": 372,
"column": 68
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (closedBall 0... | Finset.prod_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Isometry | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 33
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case h.refine_2\nf : ℂ ≃ₗᵢ[ℝ] ℂ\na : Circle := ⟨f 1, ⋯⟩\nthis : (f.trans (rotation a).symm) 1 = 1\nh₂ : f.trans (rotation a).symm = conjLIE\n⊢ f = conjLIE.trans (rotation a)",
"ppTerm": "?h.refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"R... | [] | exact eq_mul_of_inv_mul_eq h₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.Isometry | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 33
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case h.refine_2\nf : ℂ ≃ₗᵢ[ℝ] ℂ\na : Circle := ⟨f 1, ⋯⟩\nthis : (f.trans (rotation a).symm) 1 = 1\nh₂ : f.trans (rotation a).symm = conjLIE\n⊢ f = conjLIE.trans (rotation a)",
"ppTerm": "?h.refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"R... | [] | exact eq_mul_of_inv_mul_eq h₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Isometry | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 33
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case h.refine_2\nf : ℂ ≃ₗᵢ[ℝ] ℂ\na : Circle := ⟨f 1, ⋯⟩\nthis : (f.trans (rotation a).symm) 1 = 1\nh₂ : f.trans (rotation a).symm = conjLIE\n⊢ f = conjLIE.trans (rotation a)",
"ppTerm": "?h.refine_2",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"R... | [] | exact eq_mul_of_inv_mul_eq h₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 710,
"column": 6
} | {
"line": 710,
"column": 88
} | {
"line": 711,
"column": 4
} | [
{
"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\nz : ↑U\nhz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}\nx✝¹ : Set ↑U\nx✝ : ↑U\n| meromorphicOrderAt f ↑x✝ = ⊤",
"ppTerm": "?m.288",
... | [
"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\nz : ↑U\nhz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}\nx✝¹ : Set ↑U\nx✝ : ↑U\n| ∃ t, (∀ y ∈ t, y ∈ {↑x✝}ᶜ → f y = 0) ∧ IsOpen[PseudoMetricSpace.toU... | rw [meromorphicOrderAt_eq_top_iff, eventually_nhdsWithin_iff, eventually_nhds_iff] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.Complex.ExponentialBounds | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 23
} | {
"line": 56,
"column": 2
} | [
{
"pp": "⊢ ↑3 ≤ rexp 1 + 1 / 2",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real.instNNRatCast",
"le_refl",
"Real",
"NNRatCast.toOfScientific",
"instHDiv",
"covariant_swap_add_of_covariant_add",
"add_le_add",
"Nat.instAt... | [
"⊢ ↑3 ≤ 2.7182818283 + 1 / 2"
] | grw [← exp_one_gt_d9] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 197,
"column": 4
} | {
"line": 198,
"column": 50
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nz : ℂ\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz0 : z.re = 0\n⊢ (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re) * ‖f z‖ ≤ 1",
"ppTerm": "?inl",
"assigne... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nz : ℂ\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nhz0 : z.re = 0\n⊢ ‖f z‖ ≤ ε + sSupNormIm f 0"
] | simp only [hz0, zero_sub, Real.rpow_neg_one, neg_zero, Real.rpow_zero, mul_one,
inv_mul_le_iff₀ (sSupNormIm_eps_pos f hε 0)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 25
} | {
"line": 134,
"column": 26
} | [
{
"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))\nhB :\n ... | add_sub_sub_cancel, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Homotopy.Lifting | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 85
} | {
"line": 463,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\ne₀ e₁ : E\nγ₀ γ₁ : Path e₀ e₁\n⊢ (γ₀.map ⋯).Homotopic (γ₁.map ⋯) → γ₀.Homotopic γ₁",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.in... | [] | exact (cov.homotopicRel_iff_comp ⟨0, .inl rfl, γ₀.source.trans γ₁.source.symm⟩).mpr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Homotopy.Lifting | {
"line": 472,
"column": 2
} | {
"line": 483,
"column": 43
} | {
"line": 485,
"column": 0
} | [
{
"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 : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\n⊢ ∃! F, F a₀ = e₀ ∧ p ∘... | [] | refine cov.isLocalHomeomorph.existsUnique_continuousMap_lifts f a₀ e₀ he (fun γ γ_0 ↦ ?_)
fun γ γ' Γ Γ' γ_0 γ'_0 Γ_0 Γ'_0 Γ_lifts Γ'_lifts γγ'1 ↦ ?_
· simpa [and_comm] using cov.exists_path_lifts (f.comp γ) e₀ (by simp [γ_0, he])
let pγ : Path a₀ (γ 1) := ⟨γ, γ_0, rfl⟩
let pγ' : Path a₀ (γ 1) := ⟨γ', γ'_0, γγ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.Lifting | {
"line": 472,
"column": 2
} | {
"line": 483,
"column": 43
} | {
"line": 485,
"column": 0
} | [
{
"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 : SimplyConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\n⊢ ∃! F, F a₀ = e₀ ∧ p ∘... | [] | refine cov.isLocalHomeomorph.existsUnique_continuousMap_lifts f a₀ e₀ he (fun γ γ_0 ↦ ?_)
fun γ γ' Γ Γ' γ_0 γ'_0 Γ_0 Γ'_0 Γ_lifts Γ'_lifts γγ'1 ↦ ?_
· simpa [and_comm] using cov.exists_path_lifts (f.comp γ) e₀ (by simp [γ_0, he])
let pγ : Path a₀ (γ 1) := ⟨γ, γ_0, rfl⟩
let pγ' : Path a₀ (γ 1) := ⟨γ', γ'_0, γγ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 87,
"column": 23
} | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 32
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\ninst✝ : NullSingletonClass μ\nhf : ContinuousOn f [[a, b]]\nhμfin : μ (Ι a b) ≠ ⊤\nhμ0 : μ (Ι a b) ≠ 0\nhint : IntegrableOn f (Ι a b) μ\n⊢ a ≠ b",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Real",
"Eq"
],
"usedFVars": [
... | [
"f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\ninst✝ : NullSingletonClass μ\nhf : ContinuousOn f [[a, b]]\nhμfin : μ (Ι a b) ≠ ⊤\nhμ0 : μ (Ι a b) ≠ 0\nhint : IntegrableOn f (Ι a b) μ\nhab : a = b\n⊢ False"
] | intro hab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 87,
"column": 23
} | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 32
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\ninst✝ : NullSingletonClass μ\nhf : ContinuousOn f [[a, b]]\nhμfin : μ (Ι a b) ≠ ⊤\nhμ0 : μ (Ι a b) ≠ 0\nhint : IntegrableOn f (Ι a b) μ\n⊢ a ≠ b",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Real",
"Eq"
],
"usedFVars": [
... | [
"f : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\ninst✝ : NullSingletonClass μ\nhf : ContinuousOn f [[a, b]]\nhμfin : μ (Ι a b) ≠ ⊤\nhμ0 : μ (Ι a b) ≠ 0\nhint : IntegrableOn f (Ι a b) μ\nhab : a = b\n⊢ False"
] | intro hab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Complex.Harmonic.Analytic | {
"line": 41,
"column": 2
} | {
"line": 42,
"column": 28
} | {
"line": 43,
"column": 2
} | [
{
"pp": "f : 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 (⇑ofRealCLM) ((fderiv ℝ f x) 1) ∘SL fderiv ℝ (fun x ↦ (fderiv ℝ f x)... | [
"f : 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 (fun x ↦ (fderiv ℝ f x) 1) x) I) - I * ↑((fderiv ℝ (fun x ↦ (fderiv ℝ f x) I) ... | simp only [ContinuousLinearMap.fderiv, sub_apply, ContinuousLinearMap.comp_apply, ofRealCLM_apply,
smul_apply, smul_eq_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 230,
"column": 27
} | {
"line": 230,
"column": 59
} | {
"line": 230,
"column": 59
} | [
{
"pp": "f : ℝ → ℝ\nc r R : ℝ\nh₁f : ContinuousOn f (Ioc r R)\nhR : r < R\nh₂f : EqOn f (fun x ↦ c) (Ioo r R)\n⊢ Ioc r R ∈ 𝓝[<] R",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"Real.par... | [
"f : ℝ → ℝ\nc r R : ℝ\nh₁f : ContinuousOn f (Ioc r R)\nhR : r < R\nh₂f : EqOn f (fun x ↦ c) (Ioo r R)\n⊢ ∃ l ∈ Iio R, Ioo l R ⊆ Ioc r R"
] | mem_nhdsLT_iff_exists_Ioo_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Harmonic.Constructions | {
"line": 81,
"column": 9
} | {
"line": 102,
"column": 45
} | {
"line": 104,
"column": 0
} | [] | [] | Real.log ∘ normSq ∘ g
_ =ᶠ[𝓝 z] reCLM ∘ ofRealCLM ∘ Real.log ∘ normSq ∘ g := by aesop
_ =ᶠ[𝓝 z] reCLM ∘ log ∘ ((conjCLE ∘ g) * g) := by
filter_upwards with x
simp only [Function.comp_apply, ofRealCLM_apply, Pi.mul_apply, conjCLE_apply]
rw [ofReal_log, normSq_eq_conj_mul_self]
exact nor... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 20
} | {
"line": 141,
"column": 20
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ 0 < deriv^[2] (fun x ↦ x * log x) x",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"Real.instInv",
... | [
"x : ℝ\nhx : 0 < x\n⊢ 0 < x⁻¹"
] | deriv2_mul_log | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 195,
"column": 12
} | {
"line": 195,
"column": 15
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case mp\nx : ℝ\nh : DifferentiableAt ℝ (fun x ↦ -x * log x) x\n⊢ x ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq"
],
"usedFVars": [
"x"
],
"usedGoals": [
{
... | [
"case mp\nx : ℝ\nh : DifferentiableAt ℝ (fun x ↦ -x * log x) x\neq0 : x = 0\n⊢ False"
] | eq0 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 38
} | {
"line": 56,
"column": 4
} | [
{
"pp": "case e'_9\na b r : ℝ\nh : -1 < r\nc : ℝ\nhc : 0 ≤ c\nx : ℝ\nhx : x ∈ Ioo 0 c\n⊢ x ^ r = (r + 1) * x ^ (r + 1 - 1) / (r + 1)",
"ppTerm": "?e'_9",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr... | [] | simp [(by linarith : r + 1 ≠ 0)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 676,
"column": 40
} | {
"line": 676,
"column": 69
} | {
"line": 677,
"column": 4
} | [
{
"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' ... | [] | rwa [norm_zero, norm_pos_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 57
} | {
"line": 144,
"column": 4
} | [
{
"pp": "a b : ℝ\nr : ℂ\nh : -1 < r.re\nc : ℝ\nhc : 0 ≤ c\n⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Complex.instNormedAddCommGroup",
... | [
"a b : ℝ\nr : ℂ\nh : -1 < r.re\nc : ℝ\nhc : 0 ≤ c\n⊢ IntervalIntegrable (fun t ↦ ‖↑t ^ r‖) volume 0 c",
"a b : ℝ\nr : ℂ\nh : -1 < r.re\nc : ℝ\nhc : 0 ≤ c\n⊢ AEStronglyMeasurable (fun x ↦ ↑x ^ r) (volume.restrict (Ι 0 c))"
] | rw [← IntervalIntegrable.intervalIntegrable_norm_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 160,
"column": 4
} | {
"line": 171,
"column": 8
} | {
"line": 173,
"column": 0
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : -1 < r.re\nthis : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c\nc : ℝ\nhc : c ≤ 0\n⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnit... | [] | rw [IntervalIntegrable.iff_comp_neg, neg_zero]
have m := (this (-c) (by linarith)).const_mul (Complex.exp (π * Complex.I * r))
rw [intervalIntegrable_iff, uIoc_of_le (by linarith : 0 ≤ -c)] at m ⊢
refine m.congr_fun (fun x hx => ?_) measurableSet_Ioc
#adaptation_note /-- 2026-05-17(kmill) added `dsimp o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrability.Basic | {
"line": 160,
"column": 4
} | {
"line": 171,
"column": 8
} | {
"line": 173,
"column": 0
} | [
{
"pp": "case inr\na b : ℝ\nr : ℂ\nh : -1 < r.re\nthis : ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c\nc : ℝ\nhc : c ≤ 0\n⊢ IntervalIntegrable (fun x ↦ ↑x ^ r) volume 0 c",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NonUnit... | [] | rw [IntervalIntegrable.iff_comp_neg, neg_zero]
have m := (this (-c) (by linarith)).const_mul (Complex.exp (π * Complex.I * r))
rw [intervalIntegrable_iff, uIoc_of_le (by linarith : 0 ≤ -c)] at m ⊢
refine m.congr_fun (fun x hx => ?_) measurableSet_Ioc
#adaptation_note /-- 2026-05-17(kmill) added `dsimp o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 81
} | {
"line": 250,
"column": 2
} | [
{
"pp": "a b : ℝ\nc : ℂ\nhc : c ≠ 0\nx : ℝ\n⊢ HasDerivAt (fun x ↦ c * ↑x) c x",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HasDerivAt.comp_ofReal",
"NormedRing.toRing",
"No... | [] | simpa only [mul_one] using! ((hasDerivAt_id (x : ℂ)).const_mul _).comp_ofReal | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Complex.Periodic | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 59
} | {
"line": 181,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : h ≠ 0\nhf : Periodic f ↑h\nz : ℂ\nq : ℂ := 𝕢 h z\nqdiff : HasStrictDerivAt (𝕢 h) (q * (2 * ↑π * I / ↑h)) z\ndiff_ne : q * (2 * ↑π * I / ↑h) ≠ 0\nhol_z :\n DifferentiableAt ℂ f\n (HasStrictFDerivAt.localInverse (𝕢 h)\n ((ContinuousLinearEquiv.unitsEquivAut ℂ) (Units.mk... | [] | exact (hol_z.comp q diff_L).congr_of_eventuallyEq hF.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Monotone.Union | {
"line": 78,
"column": 93
} | {
"line": 102,
"column": 22
} | {
"line": 104,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\n⊢ MonotoneOn f (s ∪ t)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
... | [] | by
have A : ∀ x, x ∈ s ∪ t → x ≤ c → x ∈ s := by
intro x hx hxc
cases hx
· assumption
rcases eq_or_lt_of_le hxc with (rfl | h'x)
· exact hs.1
exact (lt_irrefl _ (h'x.trans_le (ht.2 (by assumption)))).elim
have B : ∀ x, x ∈ s ∪ t → c ≤ x → x ∈ t := by
intro x hx hxc
match hx with
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.IsoIoo | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 66
} | {
"line": 32,
"column": 4
} | [
{
"pp": "case refine_1\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsStrictOrderedRing k\nx : k\n⊢ |x / (1 + |x|)| < 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",... | [
"case refine_1\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsStrictOrderedRing k\nx : k\nH : 0 < 1 + |x|\n⊢ |x / (1 + |x|)| < 1"
] | have H : 0 < 1 + |x| := (abs_nonneg x).trans_lt (lt_one_add _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 63,
"column": 2
} | {
"line": 64,
"column": 24
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nh : I = 0\n⊢ (map ℂ 𝕜)\n (↑√((‖(map 𝕜 ℂ) a‖ + ((map 𝕜 ℂ) a).re) / 2) +\n (if 0 ≤ ((map 𝕜 ℂ) a).im then 1 else -1) * ↑√((‖(map 𝕜 ℂ) a‖ - ((map 𝕜 ℂ) a).re) / 2) * Complex.I) =\n ↑√((‖a‖ + re a) / 2) + (if 0 ≤ im a then 1 else -1) * ... | [
"case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nh : im I = 1\n⊢ (map ℂ 𝕜)\n (↑√((‖(map 𝕜 ℂ) a‖ + ((map 𝕜 ℂ) a).re) / 2) +\n (if 0 ≤ ((map 𝕜 ℂ) a).im then 1 else -1) * ↑√((‖(map 𝕜 ℂ) a‖ - ((map 𝕜 ℂ) a).re) / 2) * Complex.I) =\n ↑√((‖a‖ + re a) / 2) + (if 0 ≤ im a then 1 else -1) * ↑√((‖a‖ -... | · rw [← re_add_im a]
simp [h, im_eq_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 42
} | {
"line": 123,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nha : 0 ≤ a\nh : im I = 1\n⊢ sqrt (-a) = I * sqrt a",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"RCLike.sqrt_eq_ite",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"HMul.hMul",
"SemilinearMapClass.... | [
"case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\nha : 0 ≤ a\nh : im I = 1\n⊢ Complex.I * ((complexRingEquiv h) a).sqrt = (complexRingEquiv h) (I * sqrt a)"
] | rw [sqrt_eq_ite, dif_pos h, RingEquiv.symm_apply_eq, map_neg,
Complex.sqrt_neg_of_nonneg (by simpa)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 43
} | {
"line": 152,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nh : im I = 1\n⊢ ((complexRingEquiv h) (-I)).sqrt =\n (complexRingEquiv h) ↑√2⁻¹ * (complexRingEquiv h) (1 + I) * (complexRingEquiv h) (-I)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"RCLike.one_re",
"add_mul",
"... | [] | simp [h, mul_assoc, add_comm, Complex.sqrt_neg_I, neg_mul, mul_add, add_mul, mul_sub,
mul_comm Complex.I, ← sub_eq_add_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 67
} | {
"line": 134,
"column": 2
} | [
{
"pp": "case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + ↑(-↑N * ⌊z.re / ↑N⌋)| ≤ ↑N",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq... | [
"case h\nz : ℍ\nN : ℕ\nhn : 0 < N\nn : ℤ := ⌊z.re / ↑N⌋\nh : (↑(↑N * -n) +ᵥ z).re = ↑(-↑N * ⌊z.re / ↑N⌋) + z.re\n⊢ |z.re + -(↑N * ↑⌊z.re / ↑N⌋)| ≤ ↑N"
] | simp only [neg_mul, Int.cast_neg, Int.cast_mul, Int.cast_natCast] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 33
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case pos\nw : ℂ\nhw : 0 < w.im\n⊢ ↑ofComplex w = { coe := w, coe_im_pos := hw }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.mk",
"UpperHalfPlane.ofComplex_apply"
],
"usedFVars": [
"w",
"hw"
],
"usedGoals": []
}
] | [] | exact ofComplex_apply ⟨w, hw⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 33
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case pos\nw : ℂ\nhw : 0 < w.im\n⊢ ↑ofComplex w = { coe := w, coe_im_pos := hw }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.mk",
"UpperHalfPlane.ofComplex_apply"
],
"usedFVars": [
"w",
"hw"
],
"usedGoals": []
}
] | [] | exact ofComplex_apply ⟨w, hw⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 33
} | {
"line": 157,
"column": 2
} | [
{
"pp": "case pos\nw : ℂ\nhw : 0 < w.im\n⊢ ↑ofComplex w = { coe := w, coe_im_pos := hw }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.mk",
"UpperHalfPlane.ofComplex_apply"
],
"usedFVars": [
"w",
"hw"
],
"usedGoals": []
}
] | [] | exact ofComplex_apply ⟨w, hw⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.TietzeExtension | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 29
} | {
"line": 487,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : C(X, ℝ)\nt : Set ℝ\ne : X → Y\nhs : t.OrdConnected\nhf : ∀ (x : X), f x ∈ t\nhne : t.Nonempty\nhe : IsClosedEmbedding e\nh : ℝ ≃o ↑(Ioo (-1) 1)\nF : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h ∘ ⇑... | [
"case refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : C(X, ℝ)\nt : Set ℝ\ne : X → Y\nhs : t.OrdConnected\nhf : ∀ (x : X), f x ∈ t\nhne : t.Nonempty\nhe : IsClosedEmbedding e\nh : ℝ ≃o ↑(Ioo (-1) 1)\nF : X →ᵇ ℝ := { toFun := Subtype.val ∘ ⇑h ... | refine ⟨g, fun y => ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 24
} | {
"line": 82,
"column": 2
} | [
{
"pp": "case inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nx : P\n⊢ inversion x (dist x x) x = x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MetricSpace.toPseudoMetricSpace",
... | [
"case inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc x : P\nhne : x ≠ c\n⊢ inversion c (dist x c) x = x"
] | · apply inversion_self | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 114,
"column": 4
} | {
"line": 116,
"column": 16
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc : P\nR : ℝ\nhR : R ≠ 0\nx : P\nhne : x ≠ c\n⊢ inversion c R (inversion c R x) = x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants"... | [
"case inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc : P\nR : ℝ\nhR : R ≠ 0\nx : P\nhne : x ≠ c\n⊢ R ^ 2 ≠ 0"
] | rw [inversion, dist_inversion_center, inversion_vsub_center, smul_smul, ← mul_pow,
div_mul_div_comm, div_mul_cancel₀ _ (dist_ne_zero.2 hne), ← sq, div_self, one_pow, one_smul,
vsub_vadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 784,
"column": 2
} | {
"line": 785,
"column": 74
} | {
"line": 787,
"column": 0
} | [
{
"pp": "⊢ logDeriv cosh = tanh",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Semiring.toModule",
"RCLike.toNormedAlgebra",
"Real.denselyNormedField",
"... | [] | ext
rw [logDeriv, Real.deriv_cosh, Pi.div_apply, Real.tanh_eq_sinh_div_cosh] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 784,
"column": 2
} | {
"line": 785,
"column": 74
} | {
"line": 787,
"column": 0
} | [
{
"pp": "⊢ logDeriv cosh = tanh",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Semiring.toModule",
"RCLike.toNormedAlgebra",
"Real.denselyNormedField",
"... | [] | ext
rw [logDeriv, Real.deriv_cosh, Pi.div_apply, Real.tanh_eq_sinh_div_cosh] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 50
} | {
"line": 166,
"column": 2
} | [
{
"pp": "case zero\nn : Type u_4\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\nR : Type u_1\nS : Type u_2\nm : Type u_5\ninst✝³ : CommRing R\ninst✝² : CommRing S\nM : Matrix m m S\ninst✝¹ : DecidableEq m\ninst✝ : Fintype m\nf : S →+* Matrix n n R\nl : IsEmpty m\n⊢ (f M.det).det = ((comp m m n n R) (M.map ⇑f)).de... | [] | simp_rw [Matrix.det_isEmpty, map_one, det_one] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 65
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case succ\nn : Type u_4\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\nl✝ : ℕ\nih :\n ∀ {R : Type u_1} {S : Type u_2} {m : Type u_5} [inst : CommRing R] [inst_1 : CommRing S] (M : Matrix m m S)\n [inst_2 : DecidableEq m] [inst_3 : Fintype m] (f : S →+* Matrix n n R),\n Fintype.card m = l✝ → (f M.d... | [
"case succ\nn : Type u_4\ninst✝⁵ : DecidableEq n\ninst✝⁴ : Fintype n\nl✝ : ℕ\nih :\n ∀ {R : Type u_1} {S : Type u_2} {m : Type u_5} [inst : CommRing R] [inst_1 : CommRing S] (M : Matrix m m S)\n [inst_2 : DecidableEq m] [inst_3 : Fintype m] (f : S →+* Matrix n n R),\n Fintype.card m = l✝ → (f M.det).det = ((... | have ⟨k⟩ := Fintype.card_pos_iff.mp (Nat.lt_of_sub_eq_succ l) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 39
} | {
"line": 90,
"column": 40
} | [
{
"pp": "z w : ℍ\nr : ℝ\n⊢ dist z w = r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"instHDiv",
"HMul.hMul",
"Group... | [
"z w : ℍ\nr : ℝ\n⊢ dist z w / 2 = r / 2 ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2)"
] | ← div_left_inj' (two_ne_zero' ℝ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 67
} | {
"line": 92,
"column": 0
} | [
{
"pp": "z w : ℍ\nr : ℝ\n⊢ dist z w = r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"instHDiv",
"HMul.hMul",
"Group... | [] | rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 67
} | {
"line": 92,
"column": 0
} | [
{
"pp": "z w : ℍ\nr : ℝ\n⊢ dist z w = r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"instHDiv",
"HMul.hMul",
"Group... | [] | rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 67
} | {
"line": 92,
"column": 0
} | [
{
"pp": "z w : ℍ\nr : ℝ\n⊢ dist z w = r ↔ dist ↑z ↑w / (2 * √(z.im * w.im)) = sinh (r / 2)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"instHDiv",
"HMul.hMul",
"Group... | [] | rw [← div_left_inj' (two_ne_zero' ℝ), ← sinh_inj, sinh_half_dist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 244,
"column": 4
} | {
"line": 245,
"column": 68
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℂ\nh : 0 < (↑g).det\n⊢ (↑↑(if 0 < ↑(Matrix.GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) ∘SL\n ContinuousLinearMap.restrictScalars ℝ\n (ContinuousLinearMap.toSpanSingleton ℂ (↑↑(Matrix.GeneralLinearGroup.det g) / denom g... | [] | simp [ContinuousLinearMap.det, h, LinearMap.det_restrictScalars,
Algebra.norm_complex_eq, Complex.normSq_eq_norm_sq, ← pow_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 244,
"column": 4
} | {
"line": 245,
"column": 68
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℂ\nh : 0 < (↑g).det\n⊢ (↑↑(if 0 < ↑(Matrix.GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) ∘SL\n ContinuousLinearMap.restrictScalars ℝ\n (ContinuousLinearMap.toSpanSingleton ℂ (↑↑(Matrix.GeneralLinearGroup.det g) / denom g... | [] | simp [ContinuousLinearMap.det, h, LinearMap.det_restrictScalars,
Algebra.norm_complex_eq, Complex.normSq_eq_norm_sq, ← pow_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 244,
"column": 4
} | {
"line": 245,
"column": 68
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℂ\nh : 0 < (↑g).det\n⊢ (↑↑(if 0 < ↑(Matrix.GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE) ∘SL\n ContinuousLinearMap.restrictScalars ℝ\n (ContinuousLinearMap.toSpanSingleton ℂ (↑↑(Matrix.GeneralLinearGroup.det g) / denom g... | [] | simp [ContinuousLinearMap.det, h, LinearMap.det_restrictScalars,
Algebra.norm_complex_eq, Complex.normSq_eq_norm_sq, ← pow_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.StoneCech | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 26
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\nu : Ultrafilter (Ultrafilter α)\nx : Ultrafilter α\n⊢ (∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s ↦ {u | s ∈ u}) → i ∈ ↑u) → ∀ s ∈ x, {v | s ∈ v} ∈ u",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"setOf",
"... | [
"case mpr\nα : Type u\nu : Ultrafilter (Ultrafilter α)\nx : Ultrafilter α\n⊢ (∀ s ∈ x, {v | s ∈ v} ∈ u) → ∀ (i : Set (Ultrafilter α)), (x ∈ i ∧ i ∈ range fun s ↦ {u | s ∈ u}) → i ∈ ↑u"
] | · intro h a ha
exact h _ ⟨ha, a, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 54
} | {
"line": 269,
"column": 6
} | [
{
"pp": "case refine_2\nz✝ w : ℍ\nr : ℝ\nthis : MetricSpace ℍ := metricSpaceAux\nz : ℍ\nR : ℝ\nhR : 0 < R\nh₁ : 1 < R / z.im + 1\nh₀ : 0 < R / z.im + 1\n⊢ ball z (log (R / z.im + 1)) ⊆ UpperHalfPlane.coe ⁻¹' ball (↑z) R",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NormedCommR... | [
"case refine_2\nz✝ w✝ : ℍ\nr : ℝ\nthis : MetricSpace ℍ := metricSpaceAux\nz : ℍ\nR : ℝ\nhR : 0 < R\nh₁ : 1 < R / z.im + 1\nh₀ : 0 < R / z.im + 1\nw : ℍ\nhw : w ∈ ball z (log (R / z.im + 1))\n⊢ z.im * (rexp (dist w z) - 1) < R"
] | refine fun w hw => (dist_coe_le w z).trans_lt ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.ProperAction.CompactlyGenerated | {
"line": 79,
"column": 77
} | {
"line": 81,
"column": 63
} | {
"line": 82,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MulAction G X\ninst✝² : CompactlyGeneratedSpace (X × X)\ninst✝¹ : T2Space X\ninst✝ : ContinuousSMul G X\nh : ∀ {U V : Set X}, IsCompact U → IsCompact V → IsCompact {g | (g • U ∩ V).Nonempty}... | [] | by
apply this.of_isClosed_subset (hK.isClosed.preimage <| by fun_prop)
exact Set.preimage_mono Set.subset_fst_image_prod_snd_image | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 57
} | {
"line": 209,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ\nhs : IsOpen[inst✝] (some ⁻¹' s)\nhmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ\nht : IsOpen[inst✝] (some ⁻¹' t)\nhms' : ∞ ∈ s\nhmt' : ∞ ∈ t\n⊢ IsCompact (some ⁻¹' (s ∩ t))ᶜ",
"ppTerm": "?m... | [] | simpa [compl_inter] using (hms hms').union (hmt hmt') | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 157,
"column": 2
} | {
"line": 159,
"column": 27
} | {
"line": 161,
"column": 0
} | [
{
"pp": "X : Type w\ntX : TopologicalSpace X\ninst✝ : UCompactlyGeneratedSpace X\ns : Set X\nhs : ∀ (S : CompHaus) (f : C(↑S.toTop, X)), IsClosed (⇑f ⁻¹' s)\n⊢ IsClosed[tX] s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.compactlyGenerated.match_1"... | [] | rw [eq_compactlyGenerated (X := X), TopologicalSpace.compactlyGenerated, isClosed_coinduced,
isClosed_sigma_iff]
exact fun ⟨S, f⟩ ↦ hs S f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 157,
"column": 2
} | {
"line": 159,
"column": 27
} | {
"line": 161,
"column": 0
} | [
{
"pp": "X : Type w\ntX : TopologicalSpace X\ninst✝ : UCompactlyGeneratedSpace X\ns : Set X\nhs : ∀ (S : CompHaus) (f : C(↑S.toTop, X)), IsClosed (⇑f ⁻¹' s)\n⊢ IsClosed[tX] s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.compactlyGenerated.match_1"... | [] | rw [eq_compactlyGenerated (X := X), TopologicalSpace.compactlyGenerated, isClosed_coinduced,
isClosed_sigma_iff]
exact fun ⟨S, f⟩ ↦ hs S f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 70,
"column": 76
} | {
"line": 71,
"column": 63
} | {
"line": 72,
"column": 2
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nhf : Meromorphic f\nhR : R ≠ 0\n⊢ (characteristic f ⊤ - characteristic f⁻¹ ⊤) R =\n circleAverage (fun x ↦ log ‖f x‖) 0 R - Function.locallyFinsuppWithin.logCounting (divisor f Set.univ) R",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | by
rw [characteristic_sub_characteristic_inv hf, Pi.sub_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ConstantSpeed | {
"line": 216,
"column": 37
} | {
"line": 216,
"column": 84
} | {
"line": 217,
"column": 4
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nφ : ℝ → ℝ\nφm : MonotoneOn φ (Icc 0 s)\nφst : φ '' Icc 0 s = Icc 0 t\nhfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)\nhf : HasUnitSpeedOn f (φ '' Icc 0 s)\nx✝ : ℝ\n⊢ 0 ∈ φ '' Icc 0 s",
"ppTerm": "?m.149",
"assigned"... | [] | simp only [φst, ht, mem_Icc, le_refl, and_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.ConstantSpeed | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 72
} | {
"line": 221,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nφ : ℝ → ℝ\nφm : MonotoneOn φ (Icc 0 s)\nφst : φ '' Icc 0 s = Icc 0 t\nhfφ : HasUnitSpeedOn (f ∘ φ) (Icc 0 s)\nhf : HasUnitSpeedOn f (φ '' Icc 0 s)\nx✝ x : ℝ\nxs : x ∈ Icc 0 s\nhx : φ x = 0\nthis : MapsTo φ (Icc 0 s)... | [] | exact (mem_Icc.mp (@this 0 (by rw [mem_Icc]; exact ⟨le_rfl, hs⟩))).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 86
} | {
"line": 360,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"dite_cond_eq... | [] | by_cases h : e = ⊤ <;>
simpa [logCounting, h] using locallyFinsuppWithin.logCounting_mono (by positivity) | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 86
} | {
"line": 360,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"dite_cond_eq... | [] | by_cases h : e = ⊤ <;>
simpa [logCounting, h] using locallyFinsuppWithin.logCounting_mono (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 86
} | {
"line": 360,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"dite_cond_eq... | [] | by_cases h : e = ⊤ <;>
simpa [logCounting, h] using locallyFinsuppWithin.logCounting_mono (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.BetweenList | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 41
} | {
"line": 129,
"column": 6
} | [
{
"pp": "case cons.refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nx✝ : (Pairwise (Sbtw R head) tail ∧ Triplewise (Sbtw R) tail) ∧ ∀ (a : P), head... | [
"case cons.refine_2.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nx✝ : (Pairwise (Sbtw R head) tail ∧ Triplewise (Sbtw R) tail) ∧ ∀ (a : P), head ::... | refine ⟨fun a ha' ↦ ?_, fun a ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Approximation | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 24
} | {
"line": 91,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpac... | [] | · simp [u, c, smul_re] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Between | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 33
} | {
"line": 398,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝¹¹ : Ring R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module R V\ninst✝⁷ : AddTorsor V P\ninst✝⁶ : ZeroLEOneClass R\nR' : Type u_6\ninst✝⁵ : Ring R'\ninst✝⁴ : PartialOrder R'\ninst✝³ : Module R' V\ninst✝² : Module R' R\ninst✝¹ : IsScalar... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝¹¹ : Ring R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : AddCommGroup V\ninst✝⁸ : Module R V\ninst✝⁷ : AddTorsor V P\ninst✝⁶ : ZeroLEOneClass R\nR' : Type u_6\ninst✝⁵ : Ring R'\ninst✝⁴ : PartialOrder R'\ninst✝³ : Module R' V\ninst✝² : Module R' R\ninst✝¹ : IsScalarTower R' R V... | rintro p ⟨a, ⟨⟨ha₀, ha₁⟩, rfl⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
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