module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤... | simpa using h (f x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤... | simpa using h (f x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 525,
"column": 2
} | {
"line": 527,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\n⊢ (↑T.symm).IsPositive",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"LinearEquiv.symm",
"Real",
... | refine ⟨hT.isSymmetric.toLinearMap_symm, fun x ↦ ?_⟩
have := by simpa using hT.2 (T.symm.toLinearMap x)
rwa [← T.symm.coe_toLinearMap, ← hT.isSymmetric.toLinearMap_symm] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 525,
"column": 2
} | {
"line": 527,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsPositive\n⊢ (↑T.symm).IsPositive",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"LinearEquiv.symm",
"Real",
... | refine ⟨hT.isSymmetric.toLinearMap_symm, fun x ↦ ?_⟩
have := by simpa using hT.2 (T.symm.toLinearMap x)
rwa [← T.symm.coe_toLinearMap, ← hT.isSymmetric.toLinearMap_symm] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.WeakOperatorTopology | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : Module 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nα : Type u_4\nl : Filter α\nf : α → E →WOT[𝕜] F\nA : E →WOT[𝕜] F\n⊢ (∀ (x... | exact .symm <| forall_congr' fun _ ↦
Equiv.forall_congr (InnerProductSpace.toDual 𝕜 F) fun _ ↦ Iff.rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 67
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : RKHS 𝕜 H X V\ninst✝¹ : CompleteSpace H\ninst✝ : CompleteSpace V\ns : X →₀... | rw [IsSelfAdjoint, sub_zero, star_finsuppSum, Finsupp.sum_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 285,
"column": 6
} | {
"line": 286,
"column": 17
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix ... | have : @UniformSpace.Completion.coe' (H₀ K) PseudoMetricSpace.toUniformSpace 0 = 0 := rfl
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 285,
"column": 6
} | {
"line": 286,
"column": 17
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix ... | have : @UniformSpace.Completion.coe' (H₀ K) PseudoMetricSpace.toUniformSpace 0 = 0 := rfl
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 63,
"column": 95
} | {
"line": 67,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\n⊢ ⇑(toWeakSpace 𝕜 ... | by
rw [closedConvexHull_eq_closure_convexHull (𝕜 := 𝕜),
((convex_convexHull 𝕜 s).lift ℝ).toWeakSpace_closure _, closedConvexHull_eq_closure_convexHull]
congr
refine LinearMap.image_convexHull (toWeakSpace 𝕜 E).toLinearMap s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 75,
"column": 2
} | {
"line": 82,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | suffices he' : Continuous (toWeakSpace 𝕜 F <| e <| (toWeakSpace 𝕜 E).symm ·) by
have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e
rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 75,
"column": 2
} | {
"line": 82,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | suffices he' : Continuous (toWeakSpace 𝕜 F <| e <| (toWeakSpace 𝕜 E).symm ·) by
have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e
rw [← Set.image_subset_image_iff (toWeakSpace 𝕜 F).injective, h_convex.toWeakSpace_closure 𝕜]
simpa only [Set.image_image, ← hs.toWeakSpace_closure 𝕜, LinearEq... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 51
} | [
{
"pp": "case a\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\na x y : E\n⊢ (⟪x, a⟫ + ((o.areaForm x) a * 0 - 0 * 1)) * (⟪a, y⟫ + ((o.areaForm a) y * 0 - 0 * 1)) -\n (0 + ((o.areaForm x) a * 1 + 0 * 0)) * (0 + ((o.a... | rw [real_inner_comm a x, o.areaForm_swap x a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 51
} | [
{
"pp": "case a\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\na x y : E\n⊢ (⟪x, a⟫ + ((o.areaForm x) a * 0 - 0 * 1)) * (0 + ((o.areaForm a) y * 1 + 0 * 0)) +\n (0 + ((o.areaForm x) a * 1 + 0 * 0)) * (⟪a, y⟫ + ((o.a... | rw [real_inner_comm a x, o.areaForm_swap x a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinInversion | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 98
} | [
{
"pp": "⊢ rexp ∘ Neg.neg '' univ = Ioi 0",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"Real",
"Set.Ioi",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"congrArg",
"Set.univ",
"Function.comp",
"id",
"NonUnitalNonAssocR... | rw [Set.image_comp, Set.image_univ_of_surjective neg_surjective, Set.image_univ, Real.range_exp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinInversion | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 98
} | [
{
"pp": "⊢ rexp ∘ Neg.neg '' univ = Ioi 0",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"Real",
"Set.Ioi",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"congrArg",
"Set.univ",
"Function.comp",
"id",
"NonUnitalNonAssocR... | rw [Set.image_comp, Set.image_univ_of_surjective neg_surjective, Set.image_univ, Real.range_exp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinInversion | {
"line": 34,
"column": 2
} | {
"line": 34,
"column": 98
} | [
{
"pp": "⊢ rexp ∘ Neg.neg '' univ = Ioi 0",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"Real",
"Set.Ioi",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"congrArg",
"Set.univ",
"Function.comp",
"id",
"NonUnitalNonAssocR... | rw [Set.image_comp, Set.image_univ_of_surjective neg_surjective, Set.image_univ, Real.range_exp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MellinInversion | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 90
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nσ : ℝ\nf : ℝ → E\nx : ℝ\nhx : 0 < x\nhFf : VerticalIntegrable (mellin f) σ volume\nhfx : ContinuousAt f x\ng : ℝ → E := fun u ↦ rexp (-σ * u) • f (rexp (-u))\nhf : Integrable g volume\nh2π : 2 * π ≠ 0\n⊢ Int... | simpa [mellin_eq_fourier, mul_div_cancel_right₀ _ h2π] using hFf.comp_mul_right' h2π | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.MellinTransform | {
"line": 369,
"column": 6
} | {
"line": 369,
"column": 66
} | [
{
"pp": "case hbc.inr\nE : 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 ... | rwa [sub_re, sub_le_iff_le_add, ← sub_le_iff_le_add'] at hz' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 86
} | [
{
"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... | ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 393,
"column": 40
} | {
"line": 393,
"column": 59
} | [
{
"pp": "case h\ng : ℝ → ℝ\nt : ℝ\nhg0 : Tendsto g atTop (𝓝 0)\nhg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)\nx : ℝ\nhg1 : -1 ≤ g x\n⊢ ↑x * Complex.log (1 + ↑(g x)) = ↑x * Complex.log ↑(1 + g x)",
"usedConstants": [
"Eq.mpr",
"Complex.log",
"Real",
"HMul.hMul",
"congrArg",... | Complex.ofReal_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.LogBounds | {
"line": 413,
"column": 41
} | {
"line": 413,
"column": 60
} | [
{
"pp": "case h\ng : ℝ → ℝ\nt : ℝ\nhg0 : Tendsto g atTop (𝓝 0)\nhg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)\nx : ℝ\nhg1 : -1 ≤ g x\n⊢ (1 + ↑(g x)) ^ ↑x = ↑(1 + g x) ^ ↑x",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Complex.instPow",
"Complex.ofReal_add",
"id"... | Complex.ofReal_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 64
} | [
{
"pp": "f : ℝ → ℝ\nx : ℝ\nn : ℕ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhn : n ≠ 0\nhx : 0 < x\nhx' : x ≤ 1\nhn' : 0 < ↑n\nthis : f ↑n + x * log ↑n = (1 - x) * f ↑n + x * f (↑n + 1)\n⊢ f (↑n + x) ≤ f ↑n + x * log ↑n",
"usedConstants": [
"Mathlib.Tactic.Ri... | rw [this, (by ring : (n : ℝ) + x = (1 - x) * n + x * (n + 1))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.AsymptoticCone | {
"line": 38,
"column": 32
} | {
"line": 41,
"column": 45
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Bornology.IsBounded s\n⊢ asymptoticCone ℝ s ⊆ {0}",
"usedConstants": [
"Real",
"NormedSpace.toModule",
"Classical.byContradicti... | by
intro v h
by_contra! hv
exact h (asymptoticNhds_le_cobounded hv hs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 75
} | [
{
"pp": "case refine_1.inl\nf : ℝ → ℝ\nx : ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nthis : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))\nh✝ : x < 1\n⊢ ↑⌈x - 1⌉₊ < x",
"usedConstants": [
"Iff.mpr",
"Real... | rwa [Nat.ceil_eq_zero.mpr (by linarith : x - 1 ≤ 0), Nat.cast_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 75
} | [
{
"pp": "case refine_1.inl\nf : ℝ → ℝ\nx : ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nthis : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))\nh✝ : x < 1\n⊢ ↑⌈x - 1⌉₊ < x",
"usedConstants": [
"Iff.mpr",
"Real... | rwa [Nat.ceil_eq_zero.mpr (by linarith : x - 1 ≤ 0), Nat.cast_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 75
} | [
{
"pp": "case refine_1.inl\nf : ℝ → ℝ\nx : ℝ\nhf_conv : ConvexOn ℝ (Ioi 0) f\nhf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y\nhx : 0 < x\nthis : ∀ (m : ℕ), ↑m < x → x ≤ ↑m + 1 → Tendsto (logGammaSeq x) atTop (𝓝 (f x - f 1))\nh✝ : x < 1\n⊢ ↑⌈x - 1⌉₊ < x",
"usedConstants": [
"Iff.mpr",
"Real... | rwa [Nat.ceil_eq_zero.mpr (by linarith : x - 1 ≤ 0), Nat.cast_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 234,
"column": 59
} | {
"line": 235,
"column": 74
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nn : ℕ\n⊢ s.GammaSeq n = ↑n ^ s * s.betaIntegral (↑n + 1)",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Complex.commRing",
"Monoid.toMulOneClass",
"congrArg",
"mul_div_assoc",
"Complex.instPow",
"Complex.instDivIn... | by
rw [GammaSeq, betaIntegral_eval_nat_add_one_right hs n, ← mul_div_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 21
} | [
{
"pp": "case h.e'_6\n⊢ Ioi 0 ∩ Ici 2 = Ici 2",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Set.Ici",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLat... | rw [inter_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 400,
"column": 24
} | {
"line": 400,
"column": 59
} | [
{
"pp": "⊢ √π / √π = 1",
"usedConstants": [
"Iff.mpr",
"Real",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"Real.pi",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",
"Real.instLT",... | div_self (sqrt_ne_zero'.mpr pi_pos) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 285,
"column": 2
} | {
"line": 286,
"column": 82
} | [
{
"pp": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAd... | simp_rw [← asymptoticCone_submodule, mem_asymptoticCone_iff, ← asymptoticNhds_vadd_pure v p,
vadd_pure, frequently_map, SetLike.mem_coe, s.vadd_mem_iff_mem_direction _ hp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 19
} | [
{
"pp": "case pos\ns : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\nh_im : s.im = 0\nthis : s = ↑s.re\nn : ℕ\n⊢ s.re ≠ -↑n",
"usedConstants": []
}
] | specialize hs n | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 131,
"column": 10
} | {
"line": 131,
"column": 66
} | [
{
"pp": "case h.e'_3.h.e'_4.h.e'_10\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex R P n\nhr : s.Regular\ni j : Fin (n + 1)\nhij : i ≠ j\nhn : n ≠ 0\nhi : ¬i = 1\nx ... | Equiv.swap_apply_of_ne_of_ne (by simp [hn]) (Ne.symm hi) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Ring.Ultra | {
"line": 72,
"column": 22
} | {
"line": 75,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedRing R\ninst✝¹ : NormOneClass R\ninst✝ : IsUltrametricDist R\nz : ℤ\n⊢ ‖↑z‖₊ ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Int.cast_natCast",
"AddGroupWithOne.toAddGroup",
... | by
cases z <;>
simpa only [Int.ofNat_eq_natCast, Int.cast_natCast, Int.cast_negSucc, Nat.cast_one, nnnorm_neg]
using nnnorm_natCast_le_one _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"... | simpa using H.trans' zero_lt_one | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"... | simpa using H.trans' zero_lt_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 61,
"column": 19
} | {
"line": 61,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R), ‖x‖ ≤ 1 → ‖x + 1‖ ≤ 1\nx : R\nH : 1 < ‖x‖\n⊢ x ≠ 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"... | simpa using H.trans' zero_lt_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 96,
"column": 37
} | {
"line": 96,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (x : R) (m : ℕ), ‖x + 1‖ ^ m ≤ (m + 1) • max 1 (‖x‖ ^ m)\nx : R\na : ℝ\nha : max 1 ‖x‖ < a\nha' : 1 < a\nm : ℕ\nhm : (m + 1) • max 1 ‖x‖ ^ m < a ^ m\n⊢ 1 ∈ {x | 0 ≤ x}",
"usedConstants": [
"Real",
"Real.instZeroLEOneClass",
"Partia... | by simp [zero_le_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 107,
"column": 2
} | {
"line": 134,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\n⊢ IsUltrametricDist R",
"usedConstants": [
"Nat.cast_comm",
"one_pow",
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | refine isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm (fun x m ↦ ?_)
-- we first use our hypothesis about the norm of naturals to have that multiplication by
-- naturals keeps the norm small
replace h (x : R) (n : ℕ) : ‖n • x‖ ≤ ‖x‖ := by
rw [nsmul_eq_mul, norm_mul]
rcases (norm_nonneg x).... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 107,
"column": 2
} | {
"line": 134,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nh : ∀ (n : ℕ), ‖↑n‖ ≤ 1\n⊢ IsUltrametricDist R",
"usedConstants": [
"Nat.cast_comm",
"one_pow",
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | refine isUltrametricDist_of_forall_pow_norm_le_nsmul_pow_max_one_norm (fun x m ↦ ?_)
-- we first use our hypothesis about the norm of naturals to have that multiplication by
-- naturals keeps the norm small
replace h (x : R) (n : ℕ) : ‖n • x‖ ≤ ‖x‖ := by
rw [nsmul_eq_mul, norm_mul]
rcases (norm_nonneg x).... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 41
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : AddCommGroup α\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : DenselyOrdered α\ninst✝¹ : AddLeftMono α\nf : Filter ι\ninst✝ : f.NeBot\nu v : ι → α\nh₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₃ : IsBounded... | have h := isCoboundedUnder_le_add h₄ h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Order.LiminfLimsup | {
"line": 46,
"column": 52
} | {
"line": 53,
"column": 45
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝⁴ : AddCommGroup α\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : DenselyOrdered α\ninst✝¹ : AddLeftMono α\nf : Filter ι\ninst✝ : f.NeBot\nu v : ι → α\nh₁ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₃ : IsBounded... | by
have h := isCoboundedUnder_le_add h₄ h₂ -- These `have` tactic improve performance.
have h' := isBoundedUnder_le_add h₃ h₁
rw [add_comm] at h h'
refine add_le_of_forall_lt fun a a_u b b_v ↦ (le_limsup_iff h h').2 fun c c_ab ↦ ?_
refine ((frequently_lt_of_lt_limsup h₂ a_u).and_eventually
(eventually_lt_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 69
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nf_ne_zero : f ≠ 0\n⊢ seminormFromBounded' f 1 ≤ 1",
"usedConstants": [
"Real",
"AddGroupWithOne.toAddMonoidWithOne",
"seminormFromBounded_one",
"Re... | · exact le_of_eq (seminormFromBounded_one f_ne_zero f_nonneg f_mul) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 483,
"column": 2
} | {
"line": 498,
"column": 72
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦... | have h_mul : smoothingFun μ x ^ a * smoothingFun μ y ^ b + ε ≤
max (smoothingFun μ x) (smoothingFun μ y) + ε := by
rw [max_def]
split_ifs with h
· rw [add_le_add_iff_right]
apply le_trans (mul_le_mul_of_nonneg_right
(rpow_le_rpow (smoothingFun_nonneg μ hμ1 _) h a_in.1)
(rpow_nonn... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 511,
"column": 2
} | {
"line": 512,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦ n - mu ... | apply (ciInf_le (smoothingSeminormSeq_bddBelow μ _)
⟨ψ N, (hψ_mono.lt_iff_lt.mpr N.pos).pos⟩).trans (hN.le.trans' _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 281,
"column": 20
} | {
"line": 281,
"column": 44
} | [
{
"pp": "K : 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‖ < ... | ← max_eq_left_of_lt h_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.SemiNormedGrp | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 42
} | [
{
"pp": "V W : SemiNormedGrp\ni : V ≅ W\nh1 : (Hom.hom i.hom).NormNoninc\nh2 : (Hom.hom i.inv).NormNoninc\n⊢ Isometry ⇑(ConcreteCategory.hom i.hom)",
"usedConstants": [
"NormedAddGroupHom",
"AddMonoidHomClass.isometry_of_norm",
"CategoryTheory.ConcreteCategory.hom",
"SemiNormedGrp.in... | apply AddMonoidHomClass.isometry_of_norm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 105,
"column": 57
} | {
"line": 105,
"column": 90
} | [
{
"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... | rw [← add_assoc, sum_range_succ'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 105,
"column": 57
} | {
"line": 105,
"column": 90
} | [
{
"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... | rw [← add_assoc, sum_range_succ'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 105,
"column": 57
} | {
"line": 105,
"column": 90
} | [
{
"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... | rw [← add_assoc, sum_range_succ'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 555,
"column": 93
} | {
"line": 559,
"column": 85
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh_fin : FiniteDimensional K L\nhn : Normal K L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf_ext : ∀ (x : K), f ((algebraMap K L) x) = ↑‖x‖₊\nhf_iso : ∀ (σ : Gal(L/K)) (x : L), f x = ... | by
have h_sup : (⨆ σ : Gal(L/K), f (σ x)) = f x := by
rw [← @ciSup_const _ Gal(L/K) _ _ (f x)]
exact iSup_congr fun σ ↦ by rw [hf_iso σ x]
rw [spectralNorm_eq_iSup_of_finiteDimensional_normal K L hf_pm hf_na hf_ext, h_sup] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 32
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound ... | rw [Metric.tendsto_nhds] at h2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 939,
"column": 2
} | {
"line": 939,
"column": 38
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nh_fin : FiniteDimensional K L\nthis✝ : NormedAddCommGroup L := normedAddCommGroup K L\nthis : NormedSpac... | exact FiniteDimensional.complete K L | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 143,
"column": 2
} | {
"line": 155,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nt : Finset ι\nf : ι → R\n⊢ ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ Real.exp (∑ i ∈ t, ‖f i‖) - 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing"... | induction t using Finset.induction_on with
| empty => simp
| insert x t hx IH =>
rw [Finset.prod_insert hx, Finset.sum_insert hx, Real.exp_add,
show (1 + f x) * ∏ i ∈ t, (1 + f i) - 1 =
(∏ i ∈ t, (1 + f i) - 1) + f x * ∏ x ∈ t, (1 + f x) by ring]
refine (norm_add_le_of_le IH (norm_mul_le _ _))... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 143,
"column": 2
} | {
"line": 155,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nt : Finset ι\nf : ι → R\n⊢ ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ Real.exp (∑ i ∈ t, ‖f i‖) - 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing"... | induction t using Finset.induction_on with
| empty => simp
| insert x t hx IH =>
rw [Finset.prod_insert hx, Finset.sum_insert hx, Real.exp_add,
show (1 + f x) * ∏ i ∈ t, (1 + f i) - 1 =
(∏ i ∈ t, (1 + f i) - 1) + f x * ∏ x ∈ t, (1 + f x) by ring]
refine (norm_add_le_of_le IH (norm_mul_le _ _))... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 143,
"column": 2
} | {
"line": 155,
"column": 84
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nt : Finset ι\nf : ι → R\n⊢ ‖∏ i ∈ t, (1 + f i) - 1‖ ≤ Real.exp (∑ i ∈ t, ‖f i‖) - 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing"... | induction t using Finset.induction_on with
| empty => simp
| insert x t hx IH =>
rw [Finset.prod_insert hx, Finset.sum_insert hx, Real.exp_add,
show (1 + f x) * ∏ i ∈ t, (1 + f i) - 1 =
(∏ i ∈ t, (1 + f i) - 1) + f x * ∏ x ∈ t, (1 + f x) by ring]
refine (norm_add_le_of_le IH (norm_mul_le _ _))... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 39
} | [
{
"pp": "𝕜 : 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\nhrank : ∀ (n : ℕ), Module.finrank 𝕜 ↥(↑(P n)).range = n\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\nn : ℕ\nU : Submodule 𝕜 X := (↑(succSu... | exact Nat.add_right_cancel h_dim.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.ODE.Basic | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 92
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ : ℝ\n⊢ IsIntegralCurveAt γ v t₀ ↔ ∃ ε > 0, IsIntegralCurveOn γ v (Metric.ball t₀ ε)",
"usedConstants": [
"Eq.mpr",
"IsIntegralCurveAt.eq_1",
"IsIntegralCurveOn._proof_1",
"HasD... | rw [IsIntegralCurveAt, Metric.eventually_nhds_iff_ball]
congrm ∃ ε > 0, ∀ (y : ℝ) (hy : y ∈ Metric.ball t₀ ε), ?_
exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.Basic | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 92
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ : ℝ\n⊢ IsIntegralCurveAt γ v t₀ ↔ ∃ ε > 0, IsIntegralCurveOn γ v (Metric.ball t₀ ε)",
"usedConstants": [
"Eq.mpr",
"IsIntegralCurveAt.eq_1",
"IsIntegralCurveOn._proof_1",
"HasD... | rw [IsIntegralCurveAt, Metric.eventually_nhds_iff_ball]
congrm ∃ ε > 0, ∀ (y : ℝ) (hy : y ∈ Metric.ball t₀ ε), ?_
exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 266,
"column": 38
} | {
"line": 266,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\ng : α → α\nx y : α\nhx : IsFixedPt f x\nhy : IsFixedPt g y\nC : ℝ\nhfg : ∀ (z : α), dist (f z) (g z) ≤ C\n⊢ dist y (f y) / (1 - ↑K) = dist (f y) (g y) / (1 - ↑K)",
"usedConstants": [
"Eq.mpr",
"Real",
... | by rw [hy.eq, dist_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.Transform | {
"line": 40,
"column": 18
} | {
"line": 40,
"column": 57
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\nhγ : IsIntegralCurveOn γ v s\ndt t : ℝ\nht : t ∈ -dt +ᵥ s\n⊢ HasDerivWithinAt (γ ∘ fun x ↦ x + dt) (v (t + dt) ((γ ∘ fun x ↦ x + dt) t)) (-dt +ᵥ s) t",
"usedConstants": [
"NormedCommRing... | hasDerivWithinAt_iff_hasFDerivWithinAt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 77,
"column": 83
} | {
"line": 79,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ Tendsto (fun x ↦ eval x P) atTop atBot ↔ 0 < P.degree ∧ P.leadingCoeff ≤ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtraction... | by
simp only [← tendsto_neg_atTop_iff, ← eval_neg, tendsto_atTop_iff_leadingCoeff_nonneg,
degree_neg, leadingCoeff_neg, neg_nonneg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.Gronwall | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 18
} | [
{
"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 : ℝ → E\na b t₀ : ℝ\nhv : ∀ t ∈ Ioo a b, LipschitzOnWith K (v t) (s t)\nht : t₀ ∈ Ioo a b\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t\nhfs : ∀ t ∈ Ioo a b... | apply EqOn.union | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Polynomial.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 21
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhP : ¬P = 0\n⊢ (fun x ↦ eval x P / eval x Q) ~[atTop] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)",
"usedConstants": [
... | by_cases hQ : Q = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Polynomial.Basic | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 21
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhP : ¬P = 0\n⊢ (fun x ↦ eval x P / eval x Q) ~[atBot] fun x ↦ P.leadingCoeff / Q.leadingCoeff * x ^ (↑P.natDegree - ↑Q.natDegree)",
"usedConstants": [
... | by_cases hQ : Q = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Polynomial.CauchyBound | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝ : NormedDivisionRing K\np : K[X]\nhp : p ≠ 0\na : K\nh : ‖p.leadingCoeff‖₊ * ‖a‖₊ ^ p.natDegree ≤ ‖∑ i ∈ range p.natDegree, p.coeff i * a ^ i‖₊\n⊢ ‖a‖₊ ^ p.natDegree ≤ (p.cauchyBound - 1) * ∑ x ∈ range p.natDegree, ‖a‖₊ ^ x",
"usedConstants": [
"AddGroup.toSubtractionMonoi... | have pld : ‖p.leadingCoeff‖₊ ≠ 0 := by simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 86,
"column": 25
} | {
"line": 89,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV :... | by
change Continuous Tₗ.symm.toLinearMap
suffices T'.toLinearMap = Tₗ.symm from this ▸ T'.continuous
simp [LinearMap.ext_iff, ← Tₗ.injective.eq_iff, T', this, hT, hd, Tₗ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 50
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuou... | by_cases! Subsingleton W
· use { toLinearEquiv := 0, norm_map' _ := by simp [Subsingleton.eq_zero] }
exact ext fun _ ↦ Subsingleton.allEq _ _
simpa using congr(f $(Subsingleton.allEq 0 1)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 164,
"column": 4
} | {
"line": 167,
"column": 50
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuou... | by_cases! Subsingleton W
· use { toLinearEquiv := 0, norm_map' _ := by simp [Subsingleton.eq_zero] }
exact ext fun _ ↦ Subsingleton.allEq _ _
simpa using congr(f $(Subsingleton.allEq 0 1)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 526,
"column": 57
} | {
"line": 544,
"column": 36
} | [
{
"pp": "E : 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 ∈ Icc tmin tma... | by
by_cases hlt : tmin < tmax
· have (t) (ht : t ∈ Icc tmin tmax) :=
hasDerivWithinAt_picard_Icc ht₀ hf.continuousOn hα hmem x₀ ht
induction n with
| zero =>
simp only [Nat.cast_zero, contDiffOn_zero] at *
exact HasDerivWithinAt.continuousOn this
| succ n hn =>
simp only [Nat.cas... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 63
} | [
{
"pp": "case pos\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\nhmem... | have ht₀ : t₀ ∈ Icc tmin tmax := ⟨by linarith, by linarith⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 31
} | [
{
"pp": "case a\nR : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\nhpq : p + q ≠ 0\ni : ℕ\na✝ : i ∈ (p + q).support\n⊢ v (p.coeff i) * c ^ i... | exact le_gaussNorm v _ hc i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 31
} | [
{
"pp": "case a\nR : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\nhpq : p + q ≠ 0\ni : ℕ\na✝ : i ∈ (p + q).support\n⊢ v (q.coeff i) * c ^ i... | exact le_gaussNorm v _ hc i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 258,
"column": 12
} | {
"line": 258,
"column": 16
} | [
{
"pp": "c✝ : ℝ\nR✝ : Type u_3\ninst✝¹ : Ring R✝\nv✝ : AbsoluteValue R✝ ℝ\nc : ℝ\nR : Type u_3\ninst✝ : Ring R\nv : AbsoluteValue R ℝ\nhna : IsNonarchimedean ⇑v\nhc : 0 < c\np q : R[X]\nhc0 : 0 ≤ c\ni : ℕ\nhi_p : gaussNorm v c p = v (p.coeff i) * c ^ i\nhlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p\n... | hj_q | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 34,
"column": 14
} | {
"line": 34,
"column": 37
} | [
{
"pp": "case h\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"usedConstants": [
"Real.instIsOrderedRing",
"pow_pos",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.semi... | apply pow_pos; norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 34,
"column": 14
} | {
"line": 34,
"column": 37
} | [
{
"pp": "case h\nn : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"usedConstants": [
"Real.instIsOrderedRing",
"pow_pos",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.semi... | apply pow_pos; norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 34,
"column": 14
} | {
"line": 34,
"column": 37
} | [
{
"pp": "n : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"usedConstants": [
"Real.instIsOrderedRing",
"pow_pos",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.semiring",
... | apply pow_pos; norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 34,
"column": 14
} | {
"line": 34,
"column": 37
} | [
{
"pp": "n : ℕ\n⊢ 0 < 2 ^ (n + 2)",
"usedConstants": [
"Real.instIsOrderedRing",
"pow_pos",
"Real.partialOrder",
"Real",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.semiring",
... | apply pow_pos; norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Pi.Bounds | {
"line": 54,
"column": 57
} | {
"line": 54,
"column": 80
} | [
{
"pp": "case e_a\nn : ℕ\nthis : π < (√(2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)\n⊢ 2 ^ (n * 3 + 2) * (2 ^ (2 * n))⁻¹ = 2 ^ (n + 2)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"DivInvMonoid.toInv",
"GroupWithZero.toDivision... | mul_inv_eq_iff_eq_mul₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 331,
"column": 2
} | {
"line": 349,
"column": 93
} | [
{
"pp": "case neg\np : 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 (circleMa... | calc exp (⨍ (θ : ℝ) in 0..(2 * π), log ‖p.eval (circleMap 0 1 θ)‖)
≤ ⨍ (θ : ℝ) in 0..(2 * π), exp (log ‖p.eval (circleMap 0 1 θ)‖) := by
-- First Jensen's inequality invocation
refine convexOn_exp.map_average_le continuousOn_exp isClosed_univ (by simp) ?_ ?_
· rw [Set.uIoc_of_le (by positivi... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 50,
"column": 16
} | {
"line": 50,
"column": 17
} | [
{
"pp": "θ : ℝ\n⊢ θ * I 0 θ = 2 * sin θ",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Real.cos",
"Real.instR... | I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 7
} | [
{
"pp": "θ : ℝ\nn : ℕ\n⊢ I (n + 1) θ * θ ^ 2 =\n -(2 * 2 * ((↑n + 1) * (0 ^ n * cos θ))) + 2 * (↑n + 1) * (2 * ↑n + 1) * I n θ - 4 * (↑n + 1) * ↑n * I (n - 1) θ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"HMul.hMul",
"Real.instZero",
"Rea... | I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 56
} | [
{
"pp": "case h₀\nx : ℝ\n⊢ ¬Real.arctan x = π / 2",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"Real.arctan",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"HDiv.hDiv",
"Ne",
"instOfNatNat",
... | · rw [← ne_eq]; exact (Real.arctan_lt_pi_div_two _).ne | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 96,
"column": 72
} | {
"line": 96,
"column": 93
} | [
{
"pp": "z : ℂ\nhz : ‖z‖ < 1\n⊢ 0 < 1 + z.re ∨ 1 + z = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Real.instZero",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"AddMonoid.to... | ← neg_lt_iff_pos_add' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 13
} | [
{
"pp": "n : ℕ\nx : ℝ\nhx : -1 < x ∧ x ≤ 1\n⊢ |1 - x ^ 2| ≤ 1",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"abs_le",
"Real.instSub",
"PartialOrder.toPreorder",
"... | rw [abs_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 135,
"column": 17
} | {
"line": 135,
"column": 32
} | [
{
"pp": "case h.inl\nl : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\nz : ℂ\nhre : ↑n * ‖Real.log z.re‖ ≤ ‖z.re‖\nhim : |z.im| ^ n ≤ Real.exp z.re\nh₁ : 1 < z.re\nhle : |z.im| ≤ z.re\n⊢ ↑n * ‖Real.log (max z.re |z.im|)‖ ≤ ‖z.re‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSemi... | max_eq_left hle | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 22
} | [
{
"pp": "| Differentiable ℂ fun s ↦ s.Gammaℝ⁻¹",
"usedConstants": [
"Differentiable",
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"Complex.Gammaℝ",
"Complex.instNormedField",
"PseudoMetricSpace.toUniformSpace",
"NormedField.toField",
"Eq.rec",
... | enter [2, s] | Lean.Elab.Tactic.Conv.evalEnter | Lean.Parser.Tactic.Conv.enter |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 89
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\nb : ℂ\nf : ℝ → ℂ := fun x ↦ cexp (-↑π * a * ↑x ^ 2 + 2 * ↑π * b * ↑x)\nhCf : Continuous f\nhFf : 𝓕 f = fun x ↦ 1 / a ^ (1 / 2) * cexp (-↑π / a * (↑x + I * b) ^ 2)\nh1 : 0 < (↑π * a).re\nh2 : 0 < (↑π / a).re\nf_bd : f =O[cocompact ℝ] fun x ↦ |x| ^ (-2)\nthis : ∀ (x : ℝ), -↑π / a *... | conv => enter [2, x]; rw [Complex.exp_add, ← mul_assoc _ _ (Complex.exp _), mul_comm] | Lean.Elab.Tactic.Conv.evalConv | Lean.Parser.Tactic.Conv.conv |
Mathlib.Data.Int.Log | {
"line": 197,
"column": 2
} | {
"line": 202,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhr : r ≤ 0\n⊢ clog b r = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroCla... | rw [clog, if_neg (hr.trans_lt zero_lt_one).not_ge, neg_eq_zero, Int.natCast_eq_zero,
Nat.log_eq_zero_iff]
rcases le_or_gt b 1 with hb | hb
· exact Or.inr hb
· refine Or.inl (lt_of_le_of_lt ?_ hb)
exact Nat.floor_le_one_of_le_one ((inv_nonpos.2 hr).trans zero_le_one) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Log | {
"line": 197,
"column": 2
} | {
"line": 202,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhr : r ≤ 0\n⊢ clog b r = 0",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroCla... | rw [clog, if_neg (hr.trans_lt zero_lt_one).not_ge, neg_eq_zero, Int.natCast_eq_zero,
Nat.log_eq_zero_iff]
rcases le_or_gt b 1 with hb | hb
· exact Or.inr hb
· refine Or.inl (lt_of_le_of_lt ?_ hb)
exact Nat.floor_le_one_of_le_one ((inv_nonpos.2 hr).trans zero_le_one) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 29
} | [
{
"pp": "case inr\nb x : ℝ\nhb : 1 < b\nhx✝ : 0 ≤ x\nhx : 0 < x\n⊢ logb b x ≤ 0 ↔ x ≤ 1",
"usedConstants": [
"Real.logb_nonpos_iff"
]
}
] | exact logb_nonpos_iff hb hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 38
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.partialOrder",
... | · have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb
apply le_antisymm
· rw [← Int.zpow_le_iff_le_log hb hr, ← rpow_intCast b]
refine le_of_le_of_eq ?_ (rpow_logb (zero_lt_one.trans hb1') hb1'.ne' hr)
exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _)
· rw [Int.le_floor, le_logb_iff_rpow_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 321,
"column": 45
} | {
"line": 325,
"column": 60
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\n⊢ ∫ (t : ℝ) in Ioi 1, p.rpowIntegrand₀₁ t 1 ≤ ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1",
"usedConstants": [
"MeasureTheory.ae",
"Real.integrableOn_rpowIntegrand₀₁_Ioi",
"InnerProductSpace.toNormedSpace",
"instClosedIicTopology",
"Real.instStar... | by
refine setIntegral_mono_set (integrableOn_rpowIntegrand₀₁_Ioi hp zero_le_one) ?_ ?_
· refine ae_restrict_of_forall_mem measurableSet_Ioi fun t ht => ?_
exact rpowIntegrand₀₁_nonneg hp.1 (le_of_lt ht) zero_le_one
· exact .of_forall <| Set.Ioi_subset_Ioi zero_le_one | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 96
} | [
{
"pp": "⊢ StrictAntiOn (fun x ↦ x * log x) (Icc 0 (rexp (-1)))",
"usedConstants": [
"IsOrderedModule.toPosSMulMono",
"Real.partialOrder",
"Real",
"Real.continuous_mul_log",
"Semiring.toModule",
"Continuous.continuousOn",
"HMul.hMul",
"Real.instZero",
"I... | refine strictAntiOn_of_deriv_neg (convex_Icc ..) continuous_mul_log.continuousOn fun x hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 41,
"column": 2
} | {
"line": 42,
"column": 90
} | [
{
"pp": "⊢ ¬ContinuousAt (fun x ↦ (log x)⁻¹) 1",
"usedConstants": [
"Real",
"not_continuousAt_of_tendsto",
"PseudoMetricSpace.toBornology",
"Real.denselyNormedField",
"Real.instInv",
"Compl.compl",
"nhdsWithin",
"Metric.disjoint_nhds_cobounded",
"PseudoM... | suffices Tendsto (fun x ↦ (log x)⁻¹) (nhdsWithin 1 {1}ᶜ) (Bornology.cobounded ℝ) from
not_continuousAt_of_tendsto this nhdsWithin_le_nhds (Metric.disjoint_nhds_cobounded _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 7
} | [
{
"pp": "n a b : ℕ\nhb : b ≠ 0\n⊢ a ≤ (a ^ (n + 1) / b ^ n + n * b) / (n + 1)",
"usedConstants": [
"Iff.mpr",
"zero_le",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [Nat.le_div_iff_mul_le (by positivity), Nat.mul_comm,
← Nat.add_mul_div_right _ _ (by positivity),
Nat.le_div_iff_mul_le (by positivity)]
have := (Commute.all (b : ℤ) (a - b)).pow_add_mul_le_add_pow_of_sq_nonneg
(by positivity) (sq_nonneg _) (sq_nonneg _) (by grind) (n + 1)
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 7
} | [
{
"pp": "n a b : ℕ\nhb : b ≠ 0\n⊢ a ≤ (a ^ (n + 1) / b ^ n + n * b) / (n + 1)",
"usedConstants": [
"Iff.mpr",
"zero_le",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [Nat.le_div_iff_mul_le (by positivity), Nat.mul_comm,
← Nat.add_mul_div_right _ _ (by positivity),
Nat.le_div_iff_mul_le (by positivity)]
have := (Commute.all (b : ℤ) (a - b)).pow_add_mul_le_add_pow_of_sq_nonneg
(by positivity) (sq_nonneg _) (sq_nonneg _) (by grind) (n + 1)
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 49
} | [
{
"pp": "n a b : ℕ\nhb : b ≠ 0\n⊢ a < ((a / b ^ n + n * b) / (n + 1) + 1) ^ (n + 1)",
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Nat.instMonoid",
"Exists",
"HDiv.hDiv",
"instMulNat",
"instOfNatNat",
"LE.le",
"instLENat",
"Monoid.toPow",
"in... | have ⟨c, hc1, hc2⟩ := nthRoot.always_exists n a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 230,
"column": 4
} | {
"line": 233,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nx : ↑(sphere 0 1)\nthis : Nontrivial E\nε : ℝ\nhε : 0 < ε\nhε₂ : ε ≤ 2\n⊢ ↑(toSphereBallBound (dim E) ε) *... | have hdim : Module.finrank ℝ E ≠ 0 := Module.finrank_pos.ne'
have : min (ENNReal.ofReal ε) 2 = ENNReal.ofReal ε := by simpa
simp (disch := positivity) [μ.addHaar_ball_of_pos (r := ε / 4), ENNReal.ofReal_div_of_pos,
toSphereBallBound, mul_assoc, ENNReal.ofNNReal_toNNReal, this, hdim, hε] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 230,
"column": 4
} | {
"line": 233,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nx : ↑(sphere 0 1)\nthis : Nontrivial E\nε : ℝ\nhε : 0 < ε\nhε₂ : ε ≤ 2\n⊢ ↑(toSphereBallBound (dim E) ε) *... | have hdim : Module.finrank ℝ E ≠ 0 := Module.finrank_pos.ne'
have : min (ENNReal.ofReal ε) 2 = ENNReal.ofReal ε := by simpa
simp (disch := positivity) [μ.addHaar_ball_of_pos (r := ε / 4), ENNReal.ofReal_div_of_pos,
toSphereBallBound, mul_assoc, ENNReal.ofNNReal_toNNReal, this, hdim, hε] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff | {
"line": 68,
"column": 4
} | {
"line": 70,
"column": 8
} | [
{
"pp": "case inr\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f [[a, b]]\nhab : b < a\n⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"N... | simp only [uIcc_of_ge hab.le] at h
rw [integral_symm, integral_deriv_of_contDiffOn_Icc h hab.le]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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