module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 51
} | {
"line": 140,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nd : ℕ\nhf : Isometry f\ns : Set X\n⊢ μHE[d] (f '' s) = μHE[d] s",
"ppTerm": "?m.22",
"assigned": true,
... | [] | simp_rw [euclideanHausdorffMeasure_def, Measure.smul_apply]
rw [Isometry.hausdorffMeasure_image hf (by simp)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 51
} | {
"line": 140,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : EMetricSpace Y\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nf : X → Y\nd : ℕ\nhf : Isometry f\ns : Set X\n⊢ μHE[d] (f '' s) = μHE[d] s",
"ppTerm": "?m.22",
"assigned": true,
... | [] | simp_rw [euclideanHausdorffMeasure_def, Measure.smul_apply]
rw [Isometry.hausdorffMeasure_image hf (by simp)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 59
} | {
"line": 370,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\n⊢ dimH (f '' univ) ≤ dimH univ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.univ",
... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, LipschitzOnWith C f t"
] | refine dimH_image_le_of_locally_lipschitzOn fun x _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 91
} | {
"line": 464,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nhle : m₁ ≤ᶠ[𝓝 0] m₂\n⊢ mkMetric m₁ ≤ mkMetric m₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"instHSM... | [] | convert! @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 91
} | {
"line": 464,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nhle : m₁ ≤ᶠ[𝓝 0] m₂\n⊢ mkMetric m₁ ≤ mkMetric m₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"instHSM... | [] | convert! @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 91
} | {
"line": 464,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nm₁ m₂ : ℝ≥0∞ → ℝ≥0∞\nhle : m₁ ≤ᶠ[𝓝 0] m₂\n⊢ mkMetric m₁ ≤ mkMetric m₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"instHSM... | [] | convert! @mkMetric_mono_smul X _ _ _ _ m₂ _ ENNReal.one_ne_top one_ne_zero _ <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 683,
"column": 6
} | {
"line": 683,
"column": 74
} | {
"line": 684,
"column": 4
} | [
{
"pp": "case inl.inl\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f ∅\n⊢ μH[d] (f '' ∅) ≤ ↑0 ^ d * μH[↑r... | [] | simp only [measure_empty, nonpos_iff_eq_zero, mul_zero, image_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 683,
"column": 6
} | {
"line": 683,
"column": 74
} | {
"line": 684,
"column": 4
} | [
{
"pp": "case inl.inl\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f ∅\n⊢ μH[d] (f '' ∅) ≤ ↑0 ^ d * μH[↑r... | [] | simp only [measure_empty, nonpos_iff_eq_zero, mul_zero, image_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 683,
"column": 6
} | {
"line": 683,
"column": 74
} | {
"line": 684,
"column": 4
} | [
{
"pp": "case inl.inl\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f ∅\n⊢ μH[d] (f '' ∅) ≤ ↑0 ^ d * μH[↑r... | [] | simp only [measure_empty, nonpos_iff_eq_zero, mul_zero, image_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 104,
"column": 2
} | {
"line": 113,
"column": 38
} | {
"line": 115,
"column": 0
} | [
{
"pp": "H : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nx : H\nS : ClosedSubmodule ℝ H\n⊢ x ∈ S.symplComp ↔ ∀ y ∈ S, ⟪y, x⟫.im = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Units.val",
"_private.Mathlib.Analysis.InnerProd... | [] | simp only [mem_orthogonal, mem_mapEquiv_iff, scalarSMulCLE_symm_apply, Units.smul_def,
Units.val_inv_eq_inv_val, val_UnitI, inv_I, neg_smul]
constructor
· intro h y hy
have hiy := h (I • y)
simp only [← smul_assoc, smul_eq_mul, I_mul_I, neg_smul, one_smul, neg_neg] at hiy
simpa [inner_real_eq_re_inn... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 104,
"column": 2
} | {
"line": 113,
"column": 38
} | {
"line": 115,
"column": 0
} | [
{
"pp": "H : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nx : H\nS : ClosedSubmodule ℝ H\n⊢ x ∈ S.symplComp ↔ ∀ y ∈ S, ⟪y, x⟫.im = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Units.val",
"_private.Mathlib.Analysis.InnerProd... | [] | simp only [mem_orthogonal, mem_mapEquiv_iff, scalarSMulCLE_symm_apply, Units.smul_def,
Units.val_inv_eq_inv_val, val_UnitI, inv_I, neg_smul]
constructor
· intro h y hy
have hiy := h (I • y)
simp only [← smul_assoc, smul_eq_mul, I_mul_I, neg_smul, one_smul, neg_neg] at hiy
simpa [inner_real_eq_re_inn... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 32
} | {
"line": 753,
"column": 2
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℝ\nhd : 0 ≤ d\nr : 𝕜\nhr : r ≠ 0\ns : Set E\nthis : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH... | [
"𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℝ\nhd : 0 ≤ d\nr : 𝕜\nhr : r ≠ 0\ns : Set E\nthis : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s\n⊢ ‖r‖... | refine le_antisymm (this s) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 90
} | {
"line": 344,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpac... | [] | simp [normDet_eq_zero_iff_ker_ne_bot.mpr hgf, normDet_eq_zero_iff_ker_ne_bot.mpr hg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 347,
"column": 4
} | {
"line": 347,
"column": 88
} | {
"line": 349,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpac... | [] | simp [normDet_eq_zero_iff_ker_ne_bot.mpr hf, normDet_eq_zero_iff_ker_ne_bot.mpr hgf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 217,
"column": 2
} | {
"line": 220,
"column": 42
} | {
"line": 221,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitia... | [
"case refine_2\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitian\nx✝ : Nont... | tfae_have 3 → 1 := fun h ff v ↦ by
rw [Finsupp.sum_comm]
simpa [Finsupp.sum_sum_index, inner_add_right, inner_add_left] using
h (ff.sum fun x T ↦ .single x (T v)) | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 897,
"column": 6
} | {
"line": 897,
"column": 40
} | {
"line": 898,
"column": 4
} | [
{
"pp": "ι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\ns : Set (ι → ℝ)\nx✝ : ediam s ≤ ∞\n⊢ volume s ≤ ediam s ^ Fintype.card ι",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Real.volume_pi_le_diam_pow"
],
"usedFVars": [
"ι",
"inst✝",
... | [] | exact Real.volume_pi_le_diam_pow s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 1018,
"column": 32
} | {
"line": 1018,
"column": 49
} | {
"line": 1018,
"column": 50
} | [
{
"pp": "⊢ Measure.map ⇑(MeasurableEquiv.piFinTwo fun x ↦ ℝ) μH[2] =\n Measure.map ⇑(MeasurableEquiv.piFinTwo fun x ↦ ℝ) μH[↑(Fintype.card (Fin 2))]",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Eq.mpr",
"emetricSpacePi",
"Real... | [
"⊢ Measure.map ⇑(MeasurableEquiv.piFinTwo fun x ↦ ℝ) μH[2] = Measure.map ⇑(MeasurableEquiv.piFinTwo fun x ↦ ℝ) μH[↑2]"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 115,
"column": 53
} | {
"line": 115,
"column": 68
} | {
"line": 115,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [] | simpa using hc1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 115,
"column": 53
} | {
"line": 115,
"column": 68
} | {
"line": 115,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [] | simpa using hc1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 115,
"column": 53
} | {
"line": 115,
"column": 68
} | {
"line": 115,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [] | simpa using hc1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 255,
"column": 92
} | {
"line": 257,
"column": 58
} | {
"line": 259,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\n⊢ o.rightAngleRotation.symm = o.rightAngleRotation.trans (LinearIsometryEquiv.neg ℝ)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Lin... | [] | by
rw [rightAngleRotation]
exact LinearIsometryEquiv.toLinearIsometry_injective rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MellinInversion | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 9
} | {
"line": 49,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nx : ℝ\ns : ℂ\nf : E\n⊢ cexp (-↑x * (1 + (s - 1))) • f = cexp (-s * ↑x) • f",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_z... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.MellinInversion | {
"line": 53,
"column": 63
} | {
"line": 56,
"column": 26
} | {
"line": 57,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\n⊢ mellin f s = ∫ (u : ℝ), cexp (-s * ↑u) • f (rexp (-u))",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"mellin.eq_1",
"Eq.mpr",
"NegZer... | [] | by
rw [mellin, ← rexp_neg_image_aux, integral_image_eq_integral_abs_deriv_smul
MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux]
simp [rexp_cexp_aux] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MellinInversion | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 15
} | {
"line": 67,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\nu : ℝ\n⊢ cexp (-(↑s.im * I) * ↑u) • rexp (-s.re * u) • f (rexp (-u)) =\n cexp (-↑s.im * ↑u * I) • rexp (-s.re * u) • f (rexp (-u))",
"ppTerm": "?m.258",
"assigned": true,
"usedConstants": [
"Mat... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.MellinInversion | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 11
} | {
"line": 89,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nσ : ℝ\nf : ℂ → E\nx : ℝ\nhx : 0 < x\nhx0 : ↑x ≠ 0\nx✝ : ℝ\n⊢ cexp (↑(Real.log x) * -(2 * ↑π * ↑x✝ * I)) • f (↑σ + 2 * ↑π * ↑x✝ * I) =\n cexp (2 * ↑π * (↑x✝ * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑x✝ * I)",
"ppTerm": "?m.340... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.MellinTransform | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 37
} | {
"line": 233,
"column": 2
} | [
{
"pp": "b : ℝ\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\nhf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\ns : ℝ\nhs : b < s\n⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Normed... | [
"b : ℝ\nf : ℝ → ℝ\nhfc : AEStronglyMeasurable f (volume.restrict (Ioi 0))\nhf : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\ns : ℝ\nhs : b < s\nd : ℝ\nleft✝ : d > 0\nhd' : IsBigOWith d (𝓝[>] 0) f fun x ↦ x ^ (-b)\n⊢ ∃ c, 0 < c ∧ IntegrableOn (fun t ↦ t ^ (s - 1) * f t) (Ioc 0 c) volume"
] | obtain ⟨d, _, hd'⟩ := hf.exists_pos | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Gamma.Deriv | {
"line": 79,
"column": 28
} | {
"line": 79,
"column": 40
} | {
"line": 80,
"column": 4
} | [
{
"pp": "s✝ : ℂ\nhs✝ : ∀ (m : ℕ), s✝ ≠ -↑m\nn : ℕ\nIH : ∀ (s : ℂ), -↑n < s.re → (∀ (m : ℕ), s ≠ -↑m) → DifferentiableAt ℂ Gamma s\ns : ℂ\nhsre : -↑(n + 1) < s.re\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ s ≠ 0",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunct... | [] | grind [hs 0] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 11
} | {
"line": 109,
"column": 2
} | [
{
"pp": "case e'_2\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 :\n ∀ x ∈ uIcc 0 (π / 2),\n HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ Complex.sin (2 * z * ↑x) * ↑(sin x) * ↑(cos x) ^ (n - 1) = ↑(sin x)... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 27
} | {
"line": 87,
"column": 28
} | [
{
"pp": "s t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ... | [
"s t a b : ℝ\nhs : 0 < s\nht : 0 < t\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nf : ℝ → ℝ → ℝ → ℝ := fun c u x ↦ rexp (-c * x) * x ^ (c * (u - 1))\ne : (1 / a).HolderConjugate (1 / b)\nhab' : b = 1 - a\nhst : 0 < a * s + b * t\nposf : ∀ (c u x : ℝ), x ∈ Ioi 0 → 0 ≤ f c u x\nposf' : ∀ (c u : ℝ), ∀ᵐ (x : ℝ) ∂volume.re... | ENNReal.div_self A B, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MellinTransform | {
"line": 389,
"column": 4
} | {
"line": 392,
"column": 10
} | {
"line": 393,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na b : ℝ\nf : ℝ → E\ns : ℂ\nhfc : LocallyIntegrableOn f (Ioi 0) volume\nhf_top : f =O[atTop] fun x ↦ x ^ (-a)\nhs_top : s.re < a\nhf_bot : f =O[𝓝[>] 0] fun x ↦ x ^ (-b)\nhs_bot : b < s.re\nF : ℂ → ℝ → E := fun z t ↦ ↑t ^ (z - 1) • f t\nF' : ℂ → ... | have u1 : HasDerivAt (fun z : ℂ => (t : ℂ) ^ (z - 1)) (t ^ (y - 1) * log t) y := by
convert! ((hasDerivAt_id' y).sub_const 1).const_cpow (Or.inl ht') using 1
rw [ofReal_log (le_of_lt ht)]
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 412,
"column": 2
} | {
"line": 412,
"column": 9
} | {
"line": 414,
"column": 0
} | [
{
"pp": "s : ℝ\nhs : s ∈ Ioi 0\nh1 : √π ≠ 0\nh2 : Γ (s / 2) ≠ 0\nh3 : Γ (s / 2 + 1 / 2) ≠ 0\nh4 : 2 ^ (s - 1) ≠ 0\n⊢ log (Γ (s / 2)) + log (Γ (s / 2 + 1 / 2)) + (s - 1) * log 2 - log √π =\n (fun s ↦ log (Γ (s / 2)) + log (Γ (s / 2 + 1 / 2)) + s * log 2 - (log 2 + log √π)) s",
"ppTerm": "?m.231",
"ass... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 437,
"column": 83
} | {
"line": 442,
"column": 36
} | {
"line": 444,
"column": 0
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ s.doublingGamma = Γ s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real.Gamma_pos_of_pos",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"MulZer... | [] | by
refine
eq_Gamma_of_log_convex doublingGamma_log_convex_Ioi
(fun {y} hy => doublingGamma_add_one y hy.ne') (fun {y} hy => ?_) doublingGamma_one hs
apply_rules [mul_pos, Gamma_pos_of_pos, add_pos, inv_pos_of_pos, rpow_pos_of_pos, two_pos,
one_pos, sqrt_pos_of_pos pi_pos] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 29
} | {
"line": 86,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P\nhp' : ∀ ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nhs : s.Nonempty\nfs : ↑s → Finset ι\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np' : P\nhp' : ∀ (i : ↑s), p'... | simp_rw [← hw i, fsx, wx] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Affine.Ceva | {
"line": 49,
"column": 78
} | {
"line": 58,
"column": 11
} | {
"line": 60,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle 𝕜 P\np : Fin 3 → P\np' : P\nhp0 : ∀ (i : Fin 3), p i ≠ t.points (i + 2)\nhp : ∀ (i : Fin 3), p i ∈ line[𝕜... | [] | by
have aux (i) : dist (p i) (t.points (i + 2)) ≠ 0 := by simpa using hp0 i
have key := prod_dist_eq_prod_dist_of_mem_line_of_mem_line hp hp'
rw [Fin.prod_univ_three] at key ⊢
rw [Fin.prod_univ_three] at key
have := aux 0
have := aux 1
have := aux 2
field_simp
exact key | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 405,
"column": 4
} | {
"line": 406,
"column": 30
} | {
"line": 407,
"column": 4
} | [
{
"pp": "case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : -z * ↑π = ↑k * ↑π\n⊢ Gamma z * Gamma (1 - z) = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Int.cast",
"False",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
... | [
"case pos\nz : ℂ\npi_ne : ↑π ≠ 0\nk : ℤ\nhk : z = -↑k\n⊢ Gamma z * Gamma (1 - z) = 0"
] | rw [mul_eq_mul_right_iff, eq_false (ofReal_ne_zero.mpr pi_pos.ne'), or_false,
neg_eq_iff_eq_neg] at hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Factorization | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 30
} | {
"line": 56,
"column": 2
} | [
{
"pp": "f : ℝ[X]\nn : ℕ\nhf : f.IsMonicOfDegree (n + 1)\nf₁ : ℝ[X]\nhm : f₁.Monic\nhirr : Irreducible f₁\nf₂ : ℝ[X]\nhf₂ : f = f₁ * f₂\n⊢ f₁.IsMonicOfDegree 1 ∨ f₁.IsMonicOfDegree 2",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Mathlib.Tactic.IntervalCases.of_lt_... | [
"f : ℝ[X]\nn : ℕ\nhf : f.IsMonicOfDegree (n + 1)\nf₁ : ℝ[X]\nhm : f₁.Monic\nhirr : Irreducible f₁\nf₂ : ℝ[X]\nhf₂ : f = f₁ * f₂\nhelp : ∀ {P : ℕ → Prop} {m : ℕ}, 0 < m → m ≤ 2 → P m → P 1 ∨ P 2\n⊢ f₁.IsMonicOfDegree 1 ∨ f₁.IsMonicOfDegree 2"
] | have help {P : ℕ → Prop} {m : ℕ} (hm₀ : 0 < m) (hm₂ : m ≤ 2) (h : P m) : P 1 ∨ P 2 := by
interval_cases m <;> tauto | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 41
} | {
"line": 156,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ ∃ z, ∀ (x_1 : 𝕜), ‖x - (algebraMap 𝕜 F) z‖ ≤ ‖x - (algeb... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : SeminormedRing F\ninst✝¹ : NormedAlgebra 𝕜 F\ninst✝ : NormOneClass F\nx : F\nthis : Tendsto (fun x_1 ↦ ‖x - (algebraMap 𝕜 F) x_1‖) (cobounded 𝕜) atTop\n⊢ Bornology.IsBounded {x_1 | ‖x - (algebraMap 𝕜 F) x_1‖ ≤ ‖x - (algebra... | refine (show Continuous fun z : 𝕜 ↦ ‖x - algebraMap 𝕜 F z‖ by fun_prop)
|>.exists_forall_le_of_isBounded 0 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Quaternion | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 22
} | {
"line": 157,
"column": 2
} | [
{
"pp": "x : ℍ\n⊢ ‖WithLp.toLp 2 ((equivTuple ℝ) x)‖ = ‖x‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"WithLp",
"PiLp.instNorm",
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real",
"QuaternionAlgeb... | [
"x : ℍ\n⊢ √(⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 0, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 0⟫ +\n ⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 1, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 1⟫ +\n ⟪(WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 2, (WithLp.toLp 2 ((equivTuple ℝ) x)).ofLp 2⟫ +\n ⟪(Wi... | rw [norm_eq_sqrt_real_inner, norm_eq_sqrt_real_inner, inner_self, normSq_def', PiLp.inner_apply,
Fin.sum_univ_four] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 11
} | {
"line": 58,
"column": 0
} | [
{
"pp": "case calc_2\nq : ℍ\nhq : q.re = 0\nn : ℕ\nhq2 : q ^ 2 = -↑(normSq q)\nk : ℝ := ↑(2 * n)!\n⊢ ↑(k⁻¹ * (↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n))) = ↑(↑(Int.negSucc 0 ^ n) * ‖q‖ ^ (2 * n) * k⁻¹)",
"ppTerm": "?calc_2",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_le... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 32
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ ‖x + 1‖ ≤ max ‖x‖ 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.m... | [] | · suffices ‖x + 1‖ ≤ ‖x‖ from this.trans (le_max_left _ _)
rw [← div_le_one (by positivity), ← norm_div, add_div,
div_self (by simpa using H.trans' zero_lt_one), add_comm]
apply h
simp [inv_le_one_iff₀, H.le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 41
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 = ‖x‖\n⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NonAssocSemiring.toAddCommMonoid... | [] | simp only [← hx, mul_zero, le_refl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 41
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 = ‖x‖\n⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NonAssocSemiring.toAddCommMonoid... | [] | simp only [← hx, mul_zero, le_refl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 41
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\nx✝ : R\nm : ℕ\nx : R\nn : ℕ\nhx : 0 = ‖x‖\n⊢ ‖↑n‖ * ‖x‖ ≤ ‖x‖",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NonAssocSemiring.toAddCommMonoid... | [] | simp only [← hx, mul_zero, le_refl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 115,
"column": 78
} | {
"line": 115,
"column": 91
} | {
"line": 117,
"column": 0
} | [
{
"pp": "q : ℍ\n⊢ (exp q).re = exp q.re * Real.cos ‖q - ↑q.re‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Norm.norm",
"Quaternion.instDistribMulAction",
"NegZeroClass.toNeg",
"Real",
"instHSMul",
"instHDiv",
"Semiring.... | [] | simp [exp_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 115,
"column": 78
} | {
"line": 115,
"column": 91
} | {
"line": 117,
"column": 0
} | [
{
"pp": "q : ℍ\n⊢ (exp q).re = exp q.re * Real.cos ‖q - ↑q.re‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Norm.norm",
"Quaternion.instDistribMulAction",
"NegZeroClass.toNeg",
"Real",
"instHSMul",
"instHDiv",
"Semiring.... | [] | simp [exp_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.QuaternionExponential | {
"line": 115,
"column": 78
} | {
"line": 115,
"column": 91
} | {
"line": 117,
"column": 0
} | [
{
"pp": "q : ℍ\n⊢ (exp q).re = exp q.re * Real.cos ‖q - ↑q.re‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Quaternion.coe",
"Norm.norm",
"Quaternion.instDistribMulAction",
"NegZeroClass.toNeg",
"Real",
"instHSMul",
"instHDiv",
"Semiring.... | [] | simp [exp_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 84,
"column": 4
} | {
"line": 85,
"column": 24
} | {
"line": 86,
"column": 4
} | [
{
"pp": "case hg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nL : ℝ\nhL : 0 ≤ L\nε : ℝ\nhε : 0 < ε\nm1 : ℕ\nhm1 : 0 < m1\nx : R\nhx : μ x ≠ 0\nh_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)\n⊢ Tendsto (fun x_1 ↦ (μ x ^ (x_1 % m1)) ^ (1 * 1 / ↑x_1)) atTop (𝓝 1)",
"ppTerm": "?hg",
"assig... | [
"case hg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nL : ℝ\nhL : 0 ≤ L\nε : ℝ\nhε : 0 < ε\nm1 : ℕ\nhm1 : 0 < m1\nx : R\nhx : μ x ≠ 0\nh_exp : Tendsto (fun n ↦ ↑(n % m1) / ↑n) atTop (𝓝 0)\n⊢ Tendsto (fun x_1 ↦ μ x ^ (↑(x_1 % m1) / ↑x_1)) atTop (𝓝 (μ x ^ 0))"
] | simp_rw [mul_one, ← rpow_natCast, ← rpow_mul (apply_nonneg μ x), ← mul_div_assoc, mul_one,
← rpow_zero (μ x)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Unbundled.IsPowMulFaithful | {
"line": 48,
"column": 2
} | {
"line": 50,
"column": 82
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case h\nF : Type u_1\nα : Type u_2\ninst✝³ : Ring α\ninst✝² : FunLike F α ℝ\ninst✝¹ : RingSeminormClass F α ℝ\nβ : Type u_3\ninst✝ : Ring β\nnα : F\nnβ : β → ℝ\nhβ : IsPowMul nβ\nf : α →+* β\nx : α\nC : ℝ\nhC0 : 0 < C\nhC : ∀ (x : α), nβ (f x) ≤ C * nα x\nhlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) at... | [
"case h\nF : Type u_1\nα : Type u_2\ninst✝³ : Ring α\ninst✝² : FunLike F α ℝ\ninst✝¹ : RingSeminormClass F α ℝ\nβ : Type u_3\ninst✝ : Ring β\nnα : F\nnβ : β → ℝ\nhβ : IsPowMul nβ\nf : α →+* β\nx : α\nC : ℝ\nhC0 : 0 < C\nhC : ∀ (x : α), nβ (f x) ≤ C * nα x\nhlim : Tendsto (fun n ↦ C ^ (1 / ↑n) * nα x) atTop (𝓝 (nα ... | have h : (C ^ (1 / n : ℝ)) ^ n = C := by
have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr (ne_of_gt hn)
rw [← rpow_natCast, ← rpow_mul hC0.le, one_div, inv_mul_cancel₀ hn0, rpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 214,
"column": 8
} | {
"line": 215,
"column": 62
} | {
"line": 216,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\nn : ℕ\nhn : ¬n = 0\n⊢ seminormFromConst_seq c f x n = f x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff... | [] | simp only [seminormFromConst_seq, hx (c ^ n), hpm _ (Nat.one_le_iff_ne_zero.mpr hn),
mul_div_assoc, div_self (pow_ne_zero n hc), mul_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 214,
"column": 8
} | {
"line": 215,
"column": 62
} | {
"line": 216,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\nn : ℕ\nhn : ¬n = 0\n⊢ seminormFromConst_seq c f x n = f x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff... | [] | simp only [seminormFromConst_seq, hx (c ^ n), hpm _ (Nat.one_le_iff_ne_zero.mpr hn),
mul_div_assoc, div_self (pow_ne_zero n hc), mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 214,
"column": 8
} | {
"line": 215,
"column": 62
} | {
"line": 216,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\nn : ℕ\nhn : ¬n = 0\n⊢ seminormFromConst_seq c f x n = f x",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff... | [] | simp only [seminormFromConst_seq, hx (c ^ n), hpm _ (Nat.one_le_iff_ne_zero.mpr hn),
mul_div_assoc, div_self (pow_ne_zero n hc), mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 11
} | {
"line": 261,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nn : ℕ\n⊢ f (c * x * c ^ n) / f c ^ n = f c * (f (x * c ^ (n + 1)) / f c ^ (n + 1))",
... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nn : ℕ\n⊢ f (c * c ^ n * x) * (f c)⁻¹ ^ n = f (c * c ^ n * x) * f c * (f c)⁻¹ * (f c)⁻¹ ^ n"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 122,
"column": 2
} | {
"line": 239,
"column": 21
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"PNat.val",
"le_max_right",
"Iff.mpr",
"AddGroup.toS... | [] | let L := iInf fun n : PNat => μ (x ^ (n : ℕ)) ^ (1 / (n : ℝ))
have hL0 : 0 ≤ L := le_ciInf fun x ↦ by positivity
rw [Metric.tendsto_atTop]
intro ε hε
/- For each `ε > 0`, we can find a positive natural number `m1` such that
`μ x ^ (1 / m1) < L + ε/2`. -/
obtain ⟨m1, hm1⟩ := smoothingSeminormSeq_exists_pnat ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 122,
"column": 2
} | {
"line": 239,
"column": 21
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"PNat.val",
"le_max_right",
"Iff.mpr",
"AddGroup.toS... | [] | let L := iInf fun n : PNat => μ (x ^ (n : ℕ)) ^ (1 / (n : ℝ))
have hL0 : 0 ≤ L := le_ciInf fun x ↦ by positivity
rw [Metric.tendsto_atTop]
intro ε hε
/- For each `ε > 0`, we can find a positive natural number `m1` such that
`μ x ^ (1 / m1) < L + ε/2`. -/
obtain ⟨m1, hm1⟩ := smoothingSeminormSeq_exists_pnat ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 86
} | {
"line": 62,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝² : NormedAddCommGroup G\ninst✝¹ : CompleteSpace G\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC ε : ℝ\nhC : 0 < C\nhε : 0 < ε\nhyp : f.SurjectiveOnWith K C\nh : H\nh_in : h ∈ K.topologicalClosure\nhyp_h : ¬h = 0\nb : ℕ → ℝ := fun i ↦ (1... | [
"G : Type u_1\ninst✝² : NormedAddCommGroup G\ninst✝¹ : CompleteSpace G\nH : Type u_2\ninst✝ : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : AddSubgroup H\nC ε : ℝ\nhC : 0 < C\nhε : 0 < ε\nhyp : f.SurjectiveOnWith K C\nh : H\nh_in : h ∈ K.topologicalClosure\nhyp_h : ¬h = 0\nb : ℕ → ℝ := ⋯\nb_pos : ∀ (i : ℕ), ... | apply NormedAddCommGroup.cauchy_series_of_le_geometric'' (by simp) one_half_lt_one | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 259,
"column": 4
} | {
"line": 278,
"column": 51
} | {
"line": 279,
"column": 4
} | [
{
"pp": "case neg\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ¬f x ≤ spectralValue p\nhn_lt : ∀ n < p.natDegree, ‖p.c... | [
"case neg\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\np : K[X]\nhp : p.Monic\nx : L\nhx : (aeval x) p = 0\nhx0 : ¬f x = 0\nh_ge : ¬f x ≤ spectralValue p\nhn_lt : ∀ n < p.natDegree, ‖p.coeff n‖ < f ... | have h_lt : f ((Finset.range p.natDegree).sum fun i : ℕ ↦ p.coeff i • x ^ i) <
f (x ^ p.natDegree) := by
have hn' (n : ℕ) (hn : n < p.natDegree) : f (p.coeff n • x ^ n) < f (x ^ p.natDegree) := by
by_cases hn0 : n = 0
· rw [hn0, pow_zero, map_smul_eq_mul, hf_pm _ (succ_le_iff.mpr h_deg),
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 600,
"column": 28
} | {
"line": 600,
"column": 79
} | {
"line": 601,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : ∀ (y : R), μ (x * y) = μ x * μ y\nn : ℕ\nhn : 1 ≤ n\nhx0 : ¬μ x = 0\n⊢ μ 1 = 1",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"NonUnitalComm... | [] | rw [← mul_right_inj' hx0, ← hx 1, mul_one, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 17
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case h_below\nM : Type u_1\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : NontrivialTopology M\n⊢ ∀ N ≥ 0, (∀ (x : M), ‖normedMk x‖ ≤ N * ‖x‖) → 1 ≤ N",
"ppTerm": "?h_below",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"NormedAddGroupHom",
... | [
"case h_below\nM : Type u_1\ninst✝¹ : SeminormedAddCommGroup M\ninst✝ : NontrivialTopology M\nN : ℝ\na✝ : N ≥ 0\nhle : ∀ (x : M), ‖normedMk x‖ ≤ N * ‖x‖\n⊢ 1 ≤ N"
] | intro N _ hle | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 381,
"column": 8
} | {
"line": 381,
"column": 48
} | {
"line": 381,
"column": 48
} | [
{
"pp": "case neg.inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nP : ℕ → X →L[𝕜] X\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\ni j : ℕ\nx : X\nh : ¬i = j\nh' : i < j\n⊢ (P (i + 1)) x - (P (min i (j + 1))) x - ... | [
"case neg.inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nP : ℕ → X →L[𝕜] X\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\ni j : ℕ\nx : X\nh : ¬i = j\nh' : i < j\n⊢ (P (i + 1)) x - (P i) x - ((P i.succ) x - (P i) x) =... | min_eq_left_of_lt (Nat.lt_succ_of_lt h') | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 26
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case intro\nι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nhs : IsClosed s\nhs' : BddBelow s\nval✝ : Fintype ι\nf : ℕ → ι → ℝ\nx : ι → ℝ\nhf : ∀ (n : ℕ), f n ∈ ↑(upperClosure s)\nhx : Filter.Tendsto f Filter.atTop (nhds x)\n⊢ x ∈ ↑(upperClosure s)",
"ppTerm": "?intro",
"assigned": true,
"... | [
"case intro\nι : Type u_2\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nhs : IsClosed s\nhs' : BddBelow s\nval✝ : Fintype ι\nf : ℕ → ι → ℝ\nx : ι → ℝ\nhx : Filter.Tendsto f Filter.atTop (nhds x)\ng : ℕ → ι → ℝ\nhg : ∀ (n : ℕ), g n ∈ s\nhgf : ∀ (n : ℕ), g n ≤ f n\n⊢ x ∈ ↑(upperClosure s)"
] | choose g hg hgf using hf | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Analysis.ODE.Gronwall | {
"line": 87,
"column": 2
} | {
"line": 91,
"column": 12
} | {
"line": 93,
"column": 0
} | [
{
"pp": "δ K x : ℝ\n⊢ Continuous fun ε ↦ gronwallBound δ K ε x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Continuous.comp'",
"Eq.mpr",
"Real",
"Continuous",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSeparatelyContinuous... | [] | by_cases hK : K = 0
· simp only [gronwallBound_K0, hK]
fun_prop
· simp only [gronwallBound_of_K_ne_0 hK]
fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Gronwall | {
"line": 87,
"column": 2
} | {
"line": 91,
"column": 12
} | {
"line": 93,
"column": 0
} | [
{
"pp": "δ K x : ℝ\n⊢ Continuous fun ε ↦ gronwallBound δ K ε x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Continuous.comp'",
"Eq.mpr",
"Real",
"Continuous",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSeparatelyContinuous... | [] | by_cases hK : K = 0
· simp only [gronwallBound_K0, hK]
fun_prop
· simp only [gronwallBound_of_K_ne_0 hK]
fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 51,
"column": 4
} | {
"line": 63,
"column": 65
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn k : ℕ\nhk : n₀ ≤ k\nih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c... | [] | have hck : 0 ≤ c k := hc k hk
have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =
∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by
rw [sum_Ico_succ_top hk, mul_sum, Ico_self, prod_empty, mul_one]
refine congr_arg (· + b k) (sum_congr rfl fun j hj ↦ ?_)
r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 51,
"column": 4
} | {
"line": 63,
"column": 65
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn k : ℕ\nhk : n₀ ≤ k\nih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c... | [] | have hck : 0 ≤ c k := hc k hk
have heq : c k * ∑ j ∈ Ico n₀ k, b j * ∏ i ∈ Ico (j + 1) k, c i + b k =
∑ j ∈ Ico n₀ (k + 1), b j * ∏ i ∈ Ico (j + 1) (k + 1), c i := by
rw [sum_Ico_succ_top hk, mul_sum, Ico_self, prod_empty, mul_one]
refine congr_arg (· + b k) (sum_congr rfl fun j hj ↦ ?_)
r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Gronwall | {
"line": 174,
"column": 2
} | {
"line": 181,
"column": 22
} | {
"line": 183,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g f' g' : ℝ → E\na b εf εg δ : ℝ\nhv : ∀ t ∈ Ico a b, LipschitzOnWith K (v t) (s t)\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : ∀ t ∈ Ic... | [] | simp only [dist_eq_norm] at ha ⊢
have h_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t :=
fun t ht => (hf' t ht).sub (hg' t ht)
apply norm_le_gronwallBound_of_norm_deriv_right_le (hf.fun_sub hg) h_deriv ha
intro t ht
have := dist_triangle4_right (f' t) (g' t) (v t (f t)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Gronwall | {
"line": 174,
"column": 2
} | {
"line": 181,
"column": 22
} | {
"line": 183,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g f' g' : ℝ → E\na b εf εg δ : ℝ\nhv : ∀ t ∈ Ico a b, LipschitzOnWith K (v t) (s t)\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ico a b, HasDerivWithinAt f (f' t) (Ici t) t\nf_bound : ∀ t ∈ Ic... | [] | simp only [dist_eq_norm] at ha ⊢
have h_deriv : ∀ t ∈ Ico a b, HasDerivWithinAt (fun t => f t - g t) (f' t - g' t) (Ici t) t :=
fun t ht => (hf' t ht).sub (hg' t ht)
apply norm_le_gronwallBound_of_norm_deriv_right_le (hf.fun_sub hg) h_deriv ha
intro t ht
have := dist_triangle4_right (f' t) (g' t) (v t (f t)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 44
} | {
"line": 66,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ dt : ℝ\n⊢ IsIntegralCurveAt (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (t₀ - dt) ↔ IsIntegralCurveAt γ v t₀",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ dt : ℝ\n⊢ (∃ ε > 0, IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) (Metric.ball (t₀ - dt) ε)) ↔\n ∃ ε > 0, IsIntegralCurveOn γ v (Metric.ball t₀ ε)"
] | simp_rw [isIntegralCurveAt_iff_exists_pos] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 36
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nn : ℕ\nhf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Ic... | [] | exact contDiffWithinAt_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 38
} | {
"line": 588,
"column": 0
} | [
{
"pp": "case neg.inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\ntmin tmax : ℝ\nn : ℕ∞\nhf : ContDiffOn ℝ (↑n) (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ∀ t ∈ Icc tmin tmax, HasDerivWithinAt α (f t (α t)) (Icc tmin tmax) t\n... | [] | exact contDiffWithinAt_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Polynomial.Order | {
"line": 88,
"column": 33
} | {
"line": 88,
"column": 98
} | {
"line": 88,
"column": 98
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\ny : ℝ\nhy : (P.comp (-X)).IsRoot y\n⊢ P.IsRoot (-y)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"NegZeroClass.toNeg",
"Real",
"Polynomial.instNeg",
... | [] | rwa [IsRoot.def, eval_comp, eval_neg, eval_X, ← IsRoot.def] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Polynomial.Order | {
"line": 88,
"column": 33
} | {
"line": 88,
"column": 98
} | {
"line": 88,
"column": 98
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\ny : ℝ\nhy : (P.comp (-X)).IsRoot y\n⊢ P.IsRoot (-y)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"NegZeroClass.toNeg",
"Real",
"Polynomial.instNeg",
... | [] | rwa [IsRoot.def, eval_comp, eval_neg, eval_X, ← IsRoot.def] at hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.Order | {
"line": 88,
"column": 33
} | {
"line": 88,
"column": 98
} | {
"line": 88,
"column": 98
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\ny : ℝ\nhy : (P.comp (-X)).IsRoot y\n⊢ P.IsRoot (-y)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"NegZeroClass.toNeg",
"Real",
"Polynomial.instNeg",
... | [] | rwa [IsRoot.def, eval_comp, eval_neg, eval_X, ← IsRoot.def] at hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Order | {
"line": 91,
"column": 52
} | {
"line": 91,
"column": 73
} | {
"line": 91,
"column": 73
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\n⊢ (P.comp (-X)).natDegree = P.natDegree",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.... | [] | simp [natDegree_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Polynomial.Order | {
"line": 91,
"column": 52
} | {
"line": 91,
"column": 73
} | {
"line": 91,
"column": 73
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\n⊢ (P.comp (-X)).natDegree = P.natDegree",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.... | [] | simp [natDegree_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.Order | {
"line": 91,
"column": 52
} | {
"line": 91,
"column": 73
} | {
"line": 91,
"column": 73
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\n⊢ (P.comp (-X)).natDegree = P.natDegree",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.... | [] | simp [natDegree_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 52
} | {
"line": 129,
"column": 2
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nh_eq_zero : ∀ (x : R), v x = 0 → x = 0\nhc : 0 < c\n⊢ gaussNorm v c p = 0 ↔ p = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants... | [
"R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nh_eq_zero : ∀ (x : R), v x = 0 → x = 0\nhc : 0 < c\n⊢ PowerSeries.gaussNorm (⇑v) c ↑p = 0 ↔ p = 0"
] | rw [← gaussNorm_coe_powerSeries _ _ (le_of_lt hc)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 185,
"column": 91
} | {
"line": 191,
"column": 60
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\n⊢ f (a + b) = max (f a) (f b)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
... | [] | by
wlog hab : f a > f b generalizing a b with H
· simpa [add_comm, max_comm] using (H hne.symm ((not_lt.mp hab).lt_of_ne hne))
apply le_antisymm (na a b)
rcases le_max_iff.mp (na (a + b) (-b)) with h | h
· simpa [max_eq_left (le_of_lt hab)] using h
· exact absurd h (not_le.mpr (by simpa [Neg b] using hab)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 234,
"column": 10
} | {
"line": 235,
"column": 51
} | {
"line": 236,
"column": 10
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nf g : MvPowerSeries σ R\nvMulEq : ∀ (a b : R), v (a * b) = v a * v b\nvna : IsNonarchimedean v\nvNeg : ∀ (a : R), v a = v (-a)\nhbfg : HasGaussNorm v c (f * g)\ni₀ j₀ : σ →₀ ℕ\nhi₀ : (v ((coeff i₀) f) * i₀.prod fu... | [
"R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝¹ : Ring R\ninst✝ : DecidableEq σ\nf g : MvPowerSeries σ R\nvMulEq : ∀ (a b : R), v (a * b) = v a * v b\nvna : IsNonarchimedean v\nvNeg : ∀ (a : R), v a = v (-a)\nhbfg : HasGaussNorm v c (f * g)\ni₀ j₀ : σ →₀ ℕ\nhi₀ : (v ((coeff i₀) f) * i₀.prod fun x1 x2 ↦ c ... | have hprod : (i₀ + j₀).prod (c · ^ ·) = i₀.prod (c · ^ ·) * j₀.prod (c · ^ ·) := by
simp [Finsupp.prod_add_index', pow_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 321,
"column": 4
} | {
"line": 322,
"column": 83
} | {
"line": 323,
"column": 2
} | [
{
"pp": "case refine_2\np : ℂ[X]\nthis✝ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\n⊢ InjOn (circleMap 0 1) {a | a ∈ uIoc 0 (2 * π) ∧ eval (circleMap 0 1 a) p = 0}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddG... | [] | · grw [setOf_and, inter_subset_left]
exact injOn_circleMap_of_abs_sub_le one_ne_zero (by simp [abs_of_pos pi_pos]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 339,
"column": 12
} | {
"line": 339,
"column": 19
} | {
"line": 339,
"column": 19
} | [
{
"pp": "p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p... | [
"p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p‖) = ‖eval (... | sqrt_sq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.OfDigits | {
"line": 136,
"column": 22
} | {
"line": 136,
"column": 53
} | {
"line": 136,
"column": 53
} | [
{
"pp": "x : ℝ\nb : ℕ\ninst✝ : NeZero b\nhx : x ∈ Set.Ico 0 1\nn : ℕ\nthis✝ : 0 < b\nthis : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊\nh_le : ↑b ^ n * x < ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i + 1\n⊢ ↑b ^ n * x - ↑b ^ n * (↑b ^ n)⁻¹ ≤ ↑b ^ n * ∑ i ∈ Finset.... | [
"x : ℝ\nb : ℕ\ninst✝ : NeZero b\nhx : x ∈ Set.Ico 0 1\nn : ℕ\nthis✝ : 0 < b\nthis : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊\nh_le : ↑b ^ n * x < ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i + 1\n⊢ ↑b ^ n * x - 1 ≤ ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digit... | mul_inv_cancel₀ (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 69,
"column": 20
} | {
"line": 69,
"column": 29
} | {
"line": 69,
"column": 30
} | [
{
"pp": "n : ℕ\nm : ℤ\nhnpos : 0 < n\nN : ℤ\nD : ℕ\nP : D ≠ 0\nC : N.natAbs.Coprime D\nhxr : ↑((N /. ↑D) ^ n) = ↑m\nhv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y\nc1 : ↑↑D ≠ 0\nc2 : ↑↑D ^ n ≠ 0\n⊢ False",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Int.cast",
... | [
"n : ℕ\nm : ℤ\nhnpos : 0 < n\nN : ℤ\nD : ℕ\nP : D ≠ 0\nC : N.natAbs.Coprime D\nhxr : ↑(N /. ↑D) ^ n = ↑m\nhv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y\nc1 : ↑↑D ≠ 0\nc2 : ↑↑D ^ n ≠ 0\n⊢ False"
] | cast_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.OfDigits | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 79
} | {
"line": 154,
"column": 2
} | [
{
"pp": "x : ℝ\nb : ℕ\ninst✝ : NeZero b\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ Filter.Tendsto (fun i ↦ x - (↑b)⁻¹ ^ i) Filter.atTop (nhds x)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instZero",
... | [
"case e'_5\nx : ℝ\nb : ℕ\ninst✝ : NeZero b\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ x = x - 0",
"case convert_3\nx : ℝ\nb : ℕ\ninst✝ : NeZero b\nhb : 1 < b\nhx : x ∈ Set.Ico 0 1\n⊢ |(↑b)⁻¹| < 1"
] | convert! tendsto_const_nhds.sub (tendsto_pow_atTop_nhds_zero_of_abs_lt_one _) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.Real.Irrational | {
"line": 338,
"column": 15
} | {
"line": 338,
"column": 28
} | {
"line": 340,
"column": 0
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ Irrational x⁻¹⁻¹",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"Real.instInv",
"InvolutiveInv.toInv",
"id",
"DivisionMonoid.toInvolutive... | [] | rwa [inv_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.Real.Irrational | {
"line": 338,
"column": 15
} | {
"line": 338,
"column": 28
} | {
"line": 340,
"column": 0
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ Irrational x⁻¹⁻¹",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"Real.instInv",
"InvolutiveInv.toInv",
"id",
"DivisionMonoid.toInvolutive... | [] | rwa [inv_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Real.Irrational | {
"line": 338,
"column": 15
} | {
"line": 338,
"column": 28
} | {
"line": 340,
"column": 0
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ Irrational x⁻¹⁻¹",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"Real.instInv",
"InvolutiveInv.toInv",
"id",
"DivisionMonoid.toInvolutive... | [] | rwa [inv_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 397,
"column": 8
} | {
"line": 397,
"column": 20
} | {
"line": 397,
"column": 20
} | [
{
"pp": "x : ℝ\nn : ℕ\nh : Irrational (x ^ ↑n)\n⊢ Irrational x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Real",
"congrArg",
"Real.instDivInvMonoid",
"DivInvMonoid.toZPow",
"Eq.mp",
"DivInvMonoid.toMonoid",
"Int",
"... | [
"x : ℝ\nn : ℕ\nh : Irrational (x ^ n)\n⊢ Irrational x"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 39,
"column": 32
} | {
"line": 39,
"column": 55
} | {
"line": 39,
"column": 56
} | [
{
"pp": "z : ℂ\nh₂ : z ≠ -I\nh₁ : -1 = z * I\n⊢ z = I",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
... | [
"z : ℂ\nh₂ : z ≠ -I\nh₁ : -1 / I = z\n⊢ z = I"
] | ← div_eq_iff I_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 43,
"column": 21
} | {
"line": 43,
"column": 44
} | {
"line": 43,
"column": 45
} | [
{
"pp": "z : ℂ\nh₁ : z ≠ I\nz₁ : 1 + z * I ≠ 0\nh₂ : 1 = z * I\n⊢ z = -I",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"MulZeroClass.toMul",
"congrArg",
"... | [
"z : ℂ\nh₁ : z ≠ I\nz₁ : 1 + z * I ≠ 0\nh₂ : 1 / I = z\n⊢ z = -I"
] | ← div_eq_iff I_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 958,
"column": 2
} | {
"line": 958,
"column": 25
} | {
"line": 959,
"column": 2
} | [
{
"pp": "x y : ℝ*\nhx : x.InfinitePos\nhy₁ : ¬y.Infinitesimal\nhy₂ : 0 < y\nr : ℝ\nhy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)\n⊢ ↑r < x * y",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"NegZeroClass.toNeg",
"Real",
"Preorder.toLT",
"HMul... | [
"x y : ℝ*\nhx : x.InfinitePos\nhy₁ : ¬y.Infinitesimal\nhy₂ : 0 < y\nr : ℝ\nhy₁' : ∃ x, ¬(0 < x → -↑x < y ∧ y < ↑x)\nr₁ : ℝ\nhy₁'' : ¬(0 < r₁ → -↑r₁ < y ∧ y < ↑r₁)\n⊢ ↑r < x * y"
] | let ⟨r₁, hy₁''⟩ := hy₁' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 11
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case succ.refine_3\nn : ℕ\nIH :\n ∏ i ∈ range n, (2 * ↑i + 2) / (2 * ↑i + 1) * ((2 * ↑i + 2) / (2 * ↑i + 3)) =\n 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)! ^ 2 * (2 * ↑n + 1))\n⊢ 2 ^ (4 * n) * (↑n ! ^ 0 * ↑n ! * ↑n ! * ↑n ! * ↑n !) * ((2 * ↑n + 2) * (2 * ↑n + 2)) *\n (((2 * ↑n + 1 + 1) * ((2 * ↑n + 1... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 88,
"column": 36
} | {
"line": 88,
"column": 58
} | {
"line": 88,
"column": 58
} | [
{
"pp": "k : ℕ\n⊢ (2 * ↑k + 1) / (2 * ↑k + 2) ≤ W k / (π / 2)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"DivisionCommMonoid.toDiv... | [
"k : ℕ\n⊢ (2 * ↑k + 1) / (2 * ↑k + 2) ≤ (π / 2)⁻¹ * W k"
] | div_eq_inv_mul (W k) _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Artanh | {
"line": 56,
"column": 50
} | {
"line": 57,
"column": 79
} | {
"line": 59,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc (-1) 1\n⊢ artanh x = 1 / 2 * log ((1 + x) / (1 - x))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.log_sqrt",
"Real.partialOrder",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"GroupWith... | [] | by
rw [artanh, log_sqrt <| div_nonneg (by grind) (by grind), one_div_mul_eq_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 84,
"column": 29
} | {
"line": 84,
"column": 36
} | {
"line": 86,
"column": 0
} | [
{
"pp": "p : ℝ\n⊢ binEntropy (1 - (2⁻¹ + p)) = binEntropy (2⁻¹ - p)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidW... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 58
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case refine_1\nn : ℕ\n⊢ (sinPoly (n + 1)).natDegree ≤ n + 2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"add_le_add_iff_left._simp_1",
"Nat.instIsOrderedAddMonoid",
"_private.Mathlib.Analysis.Real.Pi.Irrational.0.sinPoly",
"instIsLeftCancelAddOfAddL... | [] | exact (sinPoly_natDegree_le (n + 1)).trans (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 58
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case refine_1\nn : ℕ\n⊢ (sinPoly (n + 1)).natDegree ≤ n + 2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"add_le_add_iff_left._simp_1",
"Nat.instIsOrderedAddMonoid",
"_private.Mathlib.Analysis.Real.Pi.Irrational.0.sinPoly",
"instIsLeftCancelAddOfAddL... | [] | exact (sinPoly_natDegree_le (n + 1)).trans (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 58
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case refine_1\nn : ℕ\n⊢ (sinPoly (n + 1)).natDegree ≤ n + 2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"add_le_add_iff_left._simp_1",
"Nat.instIsOrderedAddMonoid",
"_private.Mathlib.Analysis.Real.Pi.Irrational.0.sinPoly",
"instIsLeftCancelAddOfAddL... | [] | exact (sinPoly_natDegree_le (n + 1)).trans (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 309,
"column": 2
} | {
"line": 310,
"column": 38
} | {
"line": 312,
"column": 0
} | [
{
"pp": "q : ℕ\ntendstoBot : Tendsto (fun p ↦ log (↑q - 1) + log (1 - p) - log p) (𝓝[<] 1) atBot\na✝¹ : ℝ\na✝ : a✝¹ ∈ Ioo (1 - 2⁻¹) 1\n⊢ a✝¹ ≠ 1",
"ppTerm": "?m.173",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.... | [] | · simp_all only [mem_Ioo, ne_eq]
linarith [two_inv_lt_one (α := ℝ)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 326,
"column": 2
} | {
"line": 327,
"column": 38
} | {
"line": 329,
"column": 0
} | [
{
"pp": "q : ℕ\ntendstoTop : Tendsto (fun p ↦ log (↑q - 1) + log (1 - p) - log p) (𝓝[>] 0) atTop\na✝¹ : ℝ\na✝ : a✝¹ ∈ Ioo 0 2⁻¹\n⊢ a✝¹ ≠ 1",
"ppTerm": "?m.166",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common... | [] | · simp_all only [mem_Ioo, ne_eq]
linarith [two_inv_lt_one (α := ℝ)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 411,
"column": 8
} | {
"line": 411,
"column": 15
} | {
"line": 412,
"column": 8
} | [
{
"pp": "case a\nq : ℕ\nqLe2 : 2 ≤ q\np1 : ℝ\nhp1 : p1 ∈ Icc (1 - 1 / ↑q) 1\np2 : ℝ\nhp2 : p2 ∈ Icc (1 - 1 / ↑q) 1\np1le2 : p1 < p2\np : ℝ\nthis : 2 ≤ ↑q\nqinv_lt_1 : (↑q)⁻¹ < 1\nzero_lt_1_sub_p : 0 < 1 - p\nhp : 1 - (↑q)⁻¹ < p ∧ p < 1\nqpos : 0 < ↑q\n⊢ (↑q - 1) * (1 - p) < p",
"ppTerm": "?a✝",
"assigne... | [
"case a\nq : ℕ\nqLe2 : 2 ≤ q\np1 : ℝ\nhp1 : p1 ∈ Icc (1 - 1 / ↑q) 1\np2 : ℝ\nhp2 : p2 ∈ Icc (1 - 1 / ↑q) 1\np1le2 : p1 < p2\np : ℝ\nthis : 2 ≤ ↑q\nqinv_lt_1 : (↑q)⁻¹ < 1\nzero_lt_1_sub_p : 0 < 1 - p\nhp : 1 - (↑q)⁻¹ < p ∧ p < 1\nqpos : 0 < ↑q\n⊢ -1 + ↑q - ↑q * p + p < p"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.