module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 34
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case mp.inr\nx : ℝ\nn : ℤ\nh : -(π / 2) + ↑n * (2 * π) = x\n⊢ ∃ k, π / 2 + ↑k * π = x",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAd... | [] | exact ⟨-1 + n * 2, by grind⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 34
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case mp.inr\nx : ℝ\nn : ℤ\nh : -(π / 2) + ↑n * (2 * π) = x\n⊢ ∃ k, π / 2 + ↑k * π = x",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAd... | [] | exact ⟨-1 + n * 2, by grind⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 34
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case mp.inr\nx : ℝ\nn : ℤ\nh : -(π / 2) + ↑n * (2 * π) = x\n⊢ ∃ k, π / 2 + ↑k * π = x",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAd... | [] | exact ⟨-1 + n * 2, by grind⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Deriv | {
"line": 272,
"column": 16
} | {
"line": 272,
"column": 33
} | {
"line": 272,
"column": 33
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x\n⊢ ∀ x ∈ interior (interior D), 0 < deriv (deriv f) x",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"Real.dense... | [
"D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf'' : ∀ x ∈ interior D, 0 < deriv^[2] f x\n⊢ ∀ x ∈ interior D, 0 < deriv (deriv f) x"
] | interior_interior | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 285,
"column": 16
} | {
"line": 285,
"column": 33
} | {
"line": 285,
"column": 33
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf'' : ∀ x ∈ interior D, deriv^[2] f x < 0\n⊢ ∀ x ∈ interior (interior D), deriv (deriv f) x < 0",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"Real.dense... | [
"D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf'' : ∀ x ∈ interior D, deriv^[2] f x < 0\n⊢ ∀ x ∈ interior D, deriv (deriv f) x < 0"
] | interior_interior | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 272,
"column": 29
} | {
"line": 272,
"column": 43
} | {
"line": 272,
"column": 44
} | [
{
"pp": "x : ℝ\nk : ℤ\nh : arctan x = (2 * ↑k + 1) * π / 2\nlb : -1 * (π / 2) < (2 * ↑k + 1) * π / 2\nub : arctan x < π / 2\n⊢ False",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Int.cast",
"MulOne.toOne",
"Real",
"Preorder.toLT",
"instHDiv",
"NonUnit... | [
"x : ℝ\nk : ℤ\nh : arctan x = (2 * ↑k + 1) * π / 2\nlb : -1 * (π / 2) < (2 * ↑k + 1) * (π / 2)\nub : arctan x < π / 2\n⊢ False"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 273,
"column": 28
} | {
"line": 273,
"column": 42
} | {
"line": 273,
"column": 43
} | [
{
"pp": "x : ℝ\nk : ℤ\nh : arctan x = (2 * ↑k + 1) * π / 2\nlb : -1 < 2 * ↑k + 1\nub : (2 * ↑k + 1) * π / 2 < 1 * (π / 2)\n⊢ False",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Int.cast",
"MulOne.toOne",
"Real",
"Preorder.toLT",
"instHDiv",
"Real.pi"... | [
"x : ℝ\nk : ℤ\nh : arctan x = (2 * ↑k + 1) * π / 2\nlb : -1 < 2 * ↑k + 1\nub : (2 * ↑k + 1) * (π / 2) < 1 * (π / 2)\n⊢ False"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 89,
"column": 74
} | {
"line": 90,
"column": 57
} | {
"line": 92,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : -(π / 2) ≤ x\nhx₀ : x ≤ 0\n⊢ sin x ≤ 2 / π * x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNo... | [] | by
simpa using mul_le_sin (neg_nonneg.2 hx₀) (neg_le.2 hx) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 46
} | {
"line": 336,
"column": 47
} | [
{
"pp": "⊢ 4 * arctan 5⁻¹ - arctan 239⁻¹ = π / 4",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Tactic.Ring.Common.mul_congr",
"HMul.hMul",
"Real.ar... | [
"⊢ 2 * (2 * arctan 5⁻¹) - arctan 239⁻¹ = π / 4"
] | show 4 * arctan _ = 2 * (2 * _) by ring, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 50
} | {
"line": 108,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx : V\n⊢ arccos (⟪0, x⟫ / (‖0‖ * ‖x‖)) = π / 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"instHDiv",
... | [] | rw [inner_zero_left, zero_div, Real.arccos_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Deriv | {
"line": 693,
"column": 2
} | {
"line": 706,
"column": 71
} | {
"line": 708,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx : ℝ\nhf : ConvexOn ℝ S f\nhx : x ∈ interior S\nhf_ld : derivWithin f (Iio x) x ≤ 0\nhf_rd : 0 ≤ derivWithin f (Ioi x) x\n⊢ IsMinOn f S x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
... | [] | intro y hy
rcases lt_trichotomy x y with hxy | h_eq | hyx
· suffices 0 ≤ slope f x y by
simp only [slope_def_field, div_nonneg_iff, sub_nonneg, tsub_le_iff_right, zero_add,
not_le.mpr hxy, and_false, or_false] at this
exact this.1
exact hf_rd.trans <| rightDeriv_le_slope_of_mem_interior hf h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Deriv | {
"line": 693,
"column": 2
} | {
"line": 706,
"column": 71
} | {
"line": 708,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx : ℝ\nhf : ConvexOn ℝ S f\nhx : x ∈ interior S\nhf_ld : derivWithin f (Iio x) x ≤ 0\nhf_rd : 0 ≤ derivWithin f (Ioi x) x\n⊢ IsMinOn f S x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
... | [] | intro y hy
rcases lt_trichotomy x y with hxy | h_eq | hyx
· suffices 0 ≤ slope f x y by
simp only [slope_def_field, div_nonneg_iff, sub_nonneg, tsub_le_iff_right, zero_add,
not_le.mpr hxy, and_false, or_false] at this
exact this.1
exact hf_rd.trans <| rightDeriv_le_slope_of_mem_interior hf h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 78
} | {
"line": 78,
"column": 0
} | [
{
"pp": "f : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\nx : ℂ\nhx : x ∈ ball 0 R\n⊢ f x / (2 * ↑M - f x) ∈ closedBall (f 0 / (2 * ↑M - f 0)) 1",
"ppTerm": "?m.110",
"assigned": true,
"usedCo... | [] | · simpa [hf₂] using
div_le_one_of_le₀ (norm_le_norm_two_mul_sub hM (hf₁ hx)) (by positivity) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Schwarz | {
"line": 74,
"column": 4
} | {
"line": 77,
"column": 36
} | {
"line": 78,
"column": 4
} | [
{
"pp": "case inr\nf : ℂ → ℂ\nc z : ℂ\nR₁ R₂ : ℝ\nn : ℕ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nthis :\n ∀ {R₁ : ℝ},\n DifferentiableOn ℂ f (ball c R₁) →\n MapsT... | [
"case inr\nf : ℂ → ℂ\nc z : ℂ\nR₁ R₂ : ℝ\nn : ℕ\nhd : DifferentiableOn ℂ f (ball c R₁)\nh_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nthis :\n ∀ {R₁ : ℝ},\n DifferentiableOn ℂ f (ball c R₁) →\n MapsTo f (ball c ... | suffices ∀ᶠ r in 𝓝[<] R₁, ‖f z - f c‖ ≤ R₂ * (‖z - c‖ / r) ^ (n + 1) by
refine ge_of_tendsto ?_ this
refine ContinuousAt.continuousWithinAt ?_
fun_prop (disch := positivity) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 35
} | {
"line": 82,
"column": 36
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑↑n ≠ 0\n⊢ IsCoveringMap fun x ↦ ⟨↑x ^ ↑n, ⋯⟩",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | [
"case e'_5\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑↑n ≠ 0\nx✝ : { x // x ≠ 0 }\n⊢ ↑x✝ ^ ↑n = ↑x✝ ^ n",
"case inl.convert_3\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑↑n ≠ 0\n⊢ ProperSpace 𝕜",
"case inl.convert_4\n�... | convert! isCoveringMap_npow n _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 109,
"column": 11
} | {
"line": 109,
"column": 27
} | {
"line": 110,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑n ≠ 0\nsurj : Function.Surjective fun x ↦ x ^ n\nthis✝ : NeZero n\nthis : IsQuotientMap ({0}ᶜ.restrictPreimage fun x ↦ x ^ n)\ne : 𝕜ˣ ≃ₜ { g // g ≠ 0 } := ⋯\nx✝ : 𝕜\n⊢ x✝ ≠ 0 ↔ x✝ ∈ (fun x ↦ x ^ n) ⁻¹' {0}ᶜ",
... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 51
} | {
"line": 116,
"column": 51
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ↑n ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"NormedField.toField",
"Field.toDivisionRing",
"_private.Mathlib.Analysis.Complex.CoveringMap.0.Complex.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 51
} | {
"line": 116,
"column": 51
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ↑n ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"NormedField.toField",
"Field.toDivisionRing",
"_private.Mathlib.Analysis.Complex.CoveringMap.0.Complex.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 51
} | {
"line": 116,
"column": 51
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ↑n ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"NormedField.toField",
"Field.toDivisionRing",
"_private.Mathlib.Analysis.Complex.CoveringMap.0.Complex.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Divisor | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 40
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nU : Set 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nV : Set 𝕜\nhf : MeromorphicOn f U\nhV : V ⊆ U\nx : 𝕜\nhx : x ∈ V\n⊢ (if x ∈ V then (divisor f U) x else 0) = (divisor f V) x",
"ppTerm": "?p... | [] | simp [hf, hx, hf.mono_set hV, hV hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 67
} | {
"line": 82,
"column": 4
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nx : 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\nh₁f : MeromorphicAt f x\nh₃ : ¬meromorphicOrderA... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nx : 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\nh₁f : MeromorphicAt f x\nh₃ : ¬meromorphicOrderAt f x = ⊤\nh'₁ : Analy... | filter_upwards [h, h'₃, self_mem_nhdsWithin] with y h₁y h₂y h₃y | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 44
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\nh : f =ᶠ[𝓝[≠] x] g\nt₀ : meromorphicOrderAt f x = meromorphicOrderAt g x\n⊢ f =ᶠ[𝓝 x]... | [
"case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\ng : 𝕜 → E\nhf : MeromorphicNFAt f x\nhg : MeromorphicNFAt g x\nh : f =ᶠ[𝓝[≠] x] g\nt₀ : meromorphicOrderAt f x = meromorphicOrderAt g x\ncs : meromorphicOrderAt... | by_cases cs : meromorphicOrderAt f x = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 66
} | {
"line": 277,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_3\ns : Finset ι\nf : ι → 𝕜 → 𝕜\nh₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x\nh₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton\nτ : ι\nh₁τ : τ ∈ s\nh₂τ : τ ∉ {σ ∈ s | f σ x = 0}\n⊢ 0 ≤ meromorphicOrderAt (f τ) x",
"ppTerm": "?m.63",
"a... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_3\ns : Finset ι\nf : ι → 𝕜 → 𝕜\nh₁f : ∀ i ∈ s, MeromorphicNFAt (f i) x\nh₂f : {σ | σ ∈ s ∧ f σ x = 0}.Subsingleton\nτ : ι\nh₁τ : τ ∈ s\nh₂τ : τ ∉ {σ ∈ s | f σ x = 0}\n⊢ f τ x ≠ 0"
] | apply ((h₁f τ h₁τ).meromorphicOrderAt_eq_zero_iff.2 _).symm.le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 39
} | {
"line": 209,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x",
"ppTerm": "?pos✝"... | [
"case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (-f) x = -meromorphicTrailingCoeffAt f x"
] | · simp_all [← meromorphicOrderAt_neg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 258,
"column": 2
} | {
"line": 260,
"column": 64
} | {
"line": 261,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf₁ f₂ : 𝕜 → E\nhf₂ : MeromorphicAt f₂ x\nhf₁ : MeromorphicAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\ng₂ : 𝕜 → E\nh₁g₂ : AnalyticAt 𝕜 g₂ x\nh₂g₂ : g₂... | [
"case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf₁ f₂ : 𝕜 → E\nhf₂ : MeromorphicAt f₂ x\nhf₁ : MeromorphicAt f₁ x\nn₂ : ℤ\nhn₂ : ↑n₂ = meromorphicOrderAt f₂ x\ng₂ : 𝕜 → E\nh₁g₂ : AnalyticAt 𝕜 g₂ x\nh₂g₂ : g₂ x ≠ 0\nh₃g₂... | rw [h₁g₁.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE h₂g₁ h₃g₁,
τ₁.meromorphicTrailingCoeffAt_of_ne_zero_of_eq_nhdsNE τ₂ τ₀, sub_self, add_eq_left,
smul_eq_zero, zero_zpow _ (sub_ne_zero.2 (ne_of_lt h).symm)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 98
} | {
"line": 170,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nho : meromorphicOrderAt f x < 0\nhf : MeromorphicAt f x\nm : ℤ\nhm : ↑m = meromorphicOrderAt f x\nm_neg : m < 0\ng : 𝕜 → E\ng_an : AnalyticAt 𝕜 g x\ngx : g x ... | [] | exact (tendsto_norm_cobounded_atTop.comp (tendsto_zpow_nhdsNE_zero_cobounded m_neg)).comp this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 470,
"column": 2
} | {
"line": 473,
"column": 59
} | {
"line": 474,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (f ^ n) x = meromorphicTrailingCoeffAt f x ^ n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"zero... | [
"case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\n⊢ meromorphicTrailingCoeffAt (f ^ n) x = meromorphicTrailingCoeffAt f x ^ n"
] | · by_cases h₃ : n = 0
· simp only [h₃, zpow_zero]
apply analyticAt_const.meromorphicTrailingCoeffAt_of_ne_zero (ne_zero_of_eq_one rfl)
· simp_all [meromorphicOrderAt_zpow h₁, zero_zpow n h₃] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Meromorphic.Order | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 67
} | {
"line": 290,
"column": 4
} | [
{
"pp": "case coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\na✝ : ℕ\nhn : analyticOrderAt f x = ↑a✝\n⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ a✝ • g z"... | [
"case coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\na✝ : ℕ\nhn : analyticOrderAt f x = ↑a✝\ng : 𝕜 → E\nh1 : AnalyticAt 𝕜 g x\nh2 : g x ≠ 0\nh3 : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = (z - x) ^ a✝ • g... | rcases hf.analyticOrderAt_eq_natCast.mp hn with ⟨g, h1, h2, h3⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 21
} | {
"line": 86,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\n⊢ AnalyticAt 𝕜 ((fun x ↦ x - u) ^ d u) x",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"AddGroupWithOne.toAddGroup",
"HSub.hSub",... | [
"case pos\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\nh₂ : x = u\n⊢ AnalyticAt 𝕜 ((fun x ↦ x - u) ^ d u) x",
"case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : 0 ≤ d x\nu : 𝕜\nh₂ : ¬x = u\n⊢ AnalyticAt 𝕜 ((fun x ↦ x - u) ^ d u)... | by_cases h₂ : x = u | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 583,
"column": 4
} | {
"line": 586,
"column": 24
} | {
"line": 587,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nh₁f : MeromorphicNFOn f U\nh₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤\nu : 𝕜\nhu : u ∈ U ∩ f ⁻¹' {0}\n⊢ u ∈ Function.support ⇑(MeromorphicOn.di... | [] | simp_all only [ne_eq, Subtype.forall, Set.mem_inter_iff, Set.mem_preimage,
Set.mem_singleton_iff, Function.mem_support, h₁f.meromorphicOn, MeromorphicOn.divisor_apply,
WithTop.untop₀_eq_zero, (h₁f hu.1).meromorphicOrderAt_eq_zero_iff, not_true_eq_false, or_self,
not_false_eq_true] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 583,
"column": 4
} | {
"line": 586,
"column": 24
} | {
"line": 587,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nh₁f : MeromorphicNFOn f U\nh₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤\nu : 𝕜\nhu : u ∈ U ∩ f ⁻¹' {0}\n⊢ u ∈ Function.support ⇑(MeromorphicOn.di... | [] | simp_all only [ne_eq, Subtype.forall, Set.mem_inter_iff, Set.mem_preimage,
Set.mem_singleton_iff, Function.mem_support, h₁f.meromorphicOn, MeromorphicOn.divisor_apply,
WithTop.untop₀_eq_zero, (h₁f hu.1).meromorphicOrderAt_eq_zero_iff, not_true_eq_false, or_self,
not_false_eq_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 583,
"column": 4
} | {
"line": 586,
"column": 24
} | {
"line": 587,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nh₁f : MeromorphicNFOn f U\nh₂f : ∀ (u : ↑U), meromorphicOrderAt f ↑u ≠ ⊤\nu : 𝕜\nhu : u ∈ U ∩ f ⁻¹' {0}\n⊢ u ∈ Function.support ⇑(MeromorphicOn.di... | [] | simp_all only [ne_eq, Subtype.forall, Set.mem_inter_iff, Set.mem_preimage,
Set.mem_singleton_iff, Function.mem_support, h₁f.meromorphicOn, MeromorphicOn.divisor_apply,
WithTop.untop₀_eq_zero, (h₁f hu.1).meromorphicOrderAt_eq_zero_iff, not_true_eq_false, or_self,
not_false_eq_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 23
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case pos\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fun x ↦... | [
"case pos\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, inst✝²] fun x ↦ g (x, a)\nq... | · exact g'_a t0 htn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Homotopy.Lifting | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 32
} | {
"line": 182,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\nhomeo : IsLocalHomeomorph p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nuniq :\n ∀ (γ γ' :... | [
"case refine_1\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace A\np : E → X\nhomeo : IsLocalHomeomorph p\ninst✝¹ : PathConnectedSpace A\ninst✝ : LocallyPathConnectedSpace A\nf : C(A, X)\na₀ : A\ne₀ : E\nhe : p e₀ = f a₀\nuniq :\n ∀ (γ γ... | refine ⟨⟨F, continuous_iff_continuousAt.mpr fun a ↦ ?_⟩, ⟨?_, funext this⟩, fun F' ⟨F'_0, hpF'⟩ ↦
DFunLike.ext _ _ fun a ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Conformal | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 38
} | {
"line": 81,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖map 1‖⁻¹ • (↑ℝ ↑map) (x • 1)‖ = ‖x‖",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants... | [
"case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ E\nmap : ℂ →L[ℂ] E\nnonzero : map ≠ 0\nminor₁ : ‖map 1‖ ≠ 0\nx : ℂ\nthis : x = x • 1\n⊢ ‖‖map 1‖⁻¹ • ↑map (x • 1)‖ = ‖x‖"
] | rw [LinearMap.coe_restrictScalars] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 859,
"column": 13
} | {
"line": 859,
"column": 28
} | {
"line": 859,
"column": 28
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhf : MeromorphicAt f (g x)\nhg : AnalyticAt 𝕜 g x\nhg_nc : ¬EventuallyConst g (𝓝 x)\nhf' : meromorphicOrderAt f (g x) = ⊤\n⊢ meromorphi... | [
"case inl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhf : MeromorphicAt f (g x)\nhg : AnalyticAt 𝕜 g x\nhg_nc : ¬EventuallyConst g (𝓝 x)\nhf' : meromorphicOrderAt f (g x) = ⊤\n⊢ meromorphicOrderAt (f ... | WithTop.top_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 83,
"column": 77
} | {
"line": 94,
"column": 83
} | {
"line": 96,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\na : ℝ\nf g : ℂ → E\nl : Filter ℂ\nhBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)\nhBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)\n⊢ ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z ↦ expR (B * ‖z‖ ^ c)",
"ppTerm"... | [] | by
have : ∀ {c₁ c₂ B₁ B₂ : ℝ}, c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ →
(fun z : ℂ => expR (B₁ * ‖z‖ ^ c₁)) =O[cobounded ℂ ⊓ l]
fun z => expR (B₂ * ‖z‖ ^ c₂) := fun hc hB₀ hB ↦ .of_norm_eventuallyLE <| by
filter_upwards [(eventually_cobounded_le_norm 1).filter_mono inf_le_left] with z hz
simp only [Real.nor... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Hadamard | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 31
} | {
"line": 270,
"column": 6
} | [
{
"pp": "case pos.inl\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nz : ℂ\nhz : z ∈ verticalStrip 0 1\nh0 : sSupNormIm f 0 = 0\n⊢ ¬1 - z = 0",
"ppTerm": "?pos.inl✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"AddGroupWithOne.toAddGroup... | [
"case pos.inl\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nz : ℂ\nhz : z ∈ verticalStrip 0 1\nh0 : sSupNormIm f 0 = 0\n⊢ ¬z = 1"
] | rw [sub_eq_zero, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 284,
"column": 2
} | {
"line": 288,
"column": 56
} | {
"line": 290,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nx y z : X\ne : E\nhpe : x = p e\nγ : Path x y\nγ' : Path y z\n⊢ cov.liftPath (↑(γ.trans γ')) e ⋯ =\n ↑({ toContinuousMap := cov.liftPath (↑γ) e ⋯, source' := ⋯, target' := ⋯ }.trans... | [] | refine .symm <| (cov.eq_liftPath_iff' _).mpr ⟨funext fun _ ↦ ?_, by simp⟩
simp only [ContinuousMap.coe_coe, Function.comp_apply, Path.trans_apply]; split_ifs
· exact congr_fun (cov.liftPath_lifts γ e (γ.source.trans hpe)) _
· refine congr_fun (cov.liftPath_lifts γ' _ ?_) _
simpa using congr($(cov.liftPath_lif... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.Lifting | {
"line": 284,
"column": 2
} | {
"line": 288,
"column": 56
} | {
"line": 290,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nx y z : X\ne : E\nhpe : x = p e\nγ : Path x y\nγ' : Path y z\n⊢ cov.liftPath (↑(γ.trans γ')) e ⋯ =\n ↑({ toContinuousMap := cov.liftPath (↑γ) e ⋯, source' := ⋯, target' := ⋯ }.trans... | [] | refine .symm <| (cov.eq_liftPath_iff' _).mpr ⟨funext fun _ ↦ ?_, by simp⟩
simp only [ContinuousMap.coe_coe, Function.comp_apply, Path.trans_apply]; split_ifs
· exact congr_fun (cov.liftPath_lifts γ e (γ.source.trans hpe)) _
· refine congr_fun (cov.liftPath_lifts γ' _ ?_) _
simpa using congr($(cov.liftPath_lif... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 229,
"column": 4
} | {
"line": 231,
"column": 77
} | {
"line": 232,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nc : ℂ\nr : ℝ\nf : ℂ → E\ninst✝ : CompleteSpace E\nf_cont : ContinuousOn f (ball c r)\nz : ℂ\nhz : z ∈ ball c r\nthis : (fun w ↦ ∫ (y : ℝ) in z.im..w.im, f (↑w.re + ↑y * I) - f z) =o[𝓝 z] fun w ↦ w - z\nw : ℂ\nhw : w ∈ ball z (r - d... | [] | exact (intervalIntegral.integral_sub
((f_cont.mono (mem_ball_of_map_im_aux₂ hw)).comp (by fun_prop)
(mapsTo_image _ _)).intervalIntegrable intervalIntegrable_const).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.DerivNotation | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 50
} | {
"line": 334,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_6\nV₁ : Type u_8\nV₂ : Type u_9\nV₃ : Type u_10\ninst✝¹⁵ : LineDeriv E V₁ V₂\ninst✝¹⁴ : LineDeriv E V₂ V₃\ninst✝¹³ : AddCommGroup V₁\ninst✝¹² : AddCommGroup V₂\ninst✝¹¹ : AddCommGroup V₃\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional... | [] | simp [InnerProductSpace.canonicalCovariantTensor_eq_sum E v,
tensorLineDerivTwo_eq_lineDerivOp_lineDerivOp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Distribution.DerivNotation | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 50
} | {
"line": 334,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_6\nV₁ : Type u_8\nV₂ : Type u_9\nV₃ : Type u_10\ninst✝¹⁵ : LineDeriv E V₁ V₂\ninst✝¹⁴ : LineDeriv E V₂ V₃\ninst✝¹³ : AddCommGroup V₁\ninst✝¹² : AddCommGroup V₂\ninst✝¹¹ : AddCommGroup V₃\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional... | [] | simp [InnerProductSpace.canonicalCovariantTensor_eq_sum E v,
tensorLineDerivTwo_eq_lineDerivOp_lineDerivOp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.DerivNotation | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 50
} | {
"line": 334,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_6\nV₁ : Type u_8\nV₂ : Type u_9\nV₃ : Type u_10\ninst✝¹⁵ : LineDeriv E V₁ V₂\ninst✝¹⁴ : LineDeriv E V₂ V₃\ninst✝¹³ : AddCommGroup V₁\ninst✝¹² : AddCommGroup V₂\ninst✝¹¹ : AddCommGroup V₃\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional... | [] | simp [InnerProductSpace.canonicalCovariantTensor_eq_sum E v,
tensorLineDerivTwo_eq_lineDerivOp_lineDerivOp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 516,
"column": 6
} | {
"line": 518,
"column": 19
} | {
"line": 520,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz : ℂ\na b : ℝ\nhz : z ∈ verticalClosedStrip 0 1\nhd : DiffContOnCl ℂ f (verticalStrip 0 1)\nhB : BddAbove (norm ∘ f '' verticalClosedStrip 0 1)\nha : ∀ z ∈ re ⁻¹' {0}, ‖f z‖ ≤ a\nhb : ∀ z ∈ re ⁻¹' {1}, ‖f z‖ ≤ b\nthis : ‖... | [] | · use ‖(f 1)‖, 1
simp only [mem_preimage, one_re, mem_singleton_iff, comp_apply,
and_self] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Hadamard | {
"line": 542,
"column": 34
} | {
"line": 542,
"column": 48
} | {
"line": 542,
"column": 49
} | [
{
"pp": "z : ℂ\nl u : ℝ\nhul : l < u\nhz : z ∈ re ⁻¹' Icc l u\n⊢ l * (u - l) / ((u - l) * (u - l)) ≤ z.re * (u - l) / ((u - l) * (u - l)) ∧\n z.re * (u - l) / ((u - l) * (u - l)) ≤ 1 + l * (u - l) / ((u - l) * (u - l))",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"z : ℂ\nl u : ℝ\nhul : l < u\nhz : z ∈ re ⁻¹' Icc l u\n⊢ l * ((u - l) / ((u - l) * (u - l))) ≤ z.re * ((u - l) / ((u - l) * (u - l))) ∧\n z.re * ((u - l) / ((u - l) * (u - l))) ≤ 1 + l * ((u - l) / ((u - l) * (u - l)))"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Complex.Hadamard | {
"line": 575,
"column": 83
} | {
"line": 575,
"column": 97
} | {
"line": 576,
"column": 4
} | [
{
"pp": "l u : ℝ\nhul : l < u\nz : ℂ\n⊢ z.re * (u - l) / ((u - l) * (u - l)) + z.im * 0 / ((u - l) * (u - l)) - l / (u - l) = (z.re - l) / (u - l)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
... | [
"l u : ℝ\nhul : l < u\nz : ℂ\n⊢ z.re * ((u - l) / ((u - l) * (u - l))) + z.im * 0 / ((u - l) * (u - l)) - l / (u - l) = (z.re - l) / (u - l)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 300,
"column": 2
} | {
"line": 303,
"column": 60
} | {
"line": 305,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\ninst✝ : CompleteSpace E\nh₁ : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nh₂ : IsConservativeOn f univ\n⊢ IsExactOn f univ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
... | [] | use (wedgeIntegral 0 · f)
intro z _
have h₃ : IsConservativeOn f (ball 0 (‖z‖ + 1)) := h₂.mono (subset_univ _)
exact h₃.hasDerivAt_wedgeIntegral (by fun_prop) (by aesop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 300,
"column": 2
} | {
"line": 303,
"column": 60
} | {
"line": 305,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\ninst✝ : CompleteSpace E\nh₁ : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nh₂ : IsConservativeOn f univ\n⊢ IsExactOn f univ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
... | [] | use (wedgeIntegral 0 · f)
intro z _
have h₃ : IsConservativeOn f (ball 0 (‖z‖ + 1)) := h₂.mono (subset_univ _)
exact h₃.hasDerivAt_wedgeIntegral (by fun_prop) (by aesop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalAverage | {
"line": 67,
"column": 27
} | {
"line": 67,
"column": 62
} | {
"line": 67,
"column": 63
} | [
{
"pp": "a b : ℝ\nf₁ f₂ : ℝ → ℝ\nhf : f₁ =ᶠ[Filter.codiscreteWithin (Ι a b)] f₂\n⊢ (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f₁ x = ⨍ (x : ℝ) in a..b, f₂ x",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"... | [
"a b : ℝ\nf₁ f₂ : ℝ → ℝ\nhf : f₁ =ᶠ[Filter.codiscreteWithin (Ι a b)] f₂\n⊢ (b - a)⁻¹ • ∫ (x : ℝ) in a..b, f₂ x = ⨍ (x : ℝ) in a..b, f₂ x"
] | integral_congr_codiscreteWithin hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Harmonic.MeanValue | {
"line": 53,
"column": 2
} | {
"line": 66,
"column": 39
} | {
"line": 67,
"column": 0
} | [
{
"pp": "f : ℂ → ℝ\nc : ℂ\nR : ℝ\nh₁f : HarmonicContOnCl f (ball c |R|)\n⊢ circleAverage f c R = f c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
"In... | [] | by_cases hR : R = 0
· simp_all
have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by
refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)
intro x hx
rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]
exact hx.2
rw [← circleAverage_abs_radius]
apply H.eq_of_eqOn_Ioo (by aesop)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Harmonic.MeanValue | {
"line": 53,
"column": 2
} | {
"line": 66,
"column": 39
} | {
"line": 67,
"column": 0
} | [
{
"pp": "f : ℂ → ℝ\nc : ℂ\nR : ℝ\nh₁f : HarmonicContOnCl f (ball c |R|)\n⊢ circleAverage f c R = f c",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
"In... | [] | by_cases hR : R = 0
· simp_all
have H : ContinuousOn (circleAverage f c) (Set.Ioc 0 |R|) := by
refine (h₁f.2.mono ?_).circleAverage (fun z hz ↦ hz.1.le)
intro x hx
rw [closure_ball _ (by aesop), mem_closedBall_iff_norm]
exact hx.2
rw [← circleAverage_abs_radius]
apply H.eq_of_eqOn_Ioo (by aesop)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleAverage | {
"line": 180,
"column": 8
} | {
"line": 180,
"column": 72
} | {
"line": 180,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\n⊢ ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 (-θ)) = ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 θ)",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"HMul... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℂ → E\n⊢ ∫ (x : ℝ) in -(2 * π)..-0, f (circleMap 0 1 x) = ∫ (θ : ℝ) in 0..2 * π, f (circleMap 0 1 θ)"
] | intervalIntegral.integral_comp_neg (fun w ↦ f (circleMap 0 1 w)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 230,
"column": 20
} | {
"line": 230,
"column": 52
} | {
"line": 232,
"column": 0
} | [
{
"pp": "x y : ℝ\nz : ℍ\n⊢ (x + y) +ᵥ z = x +ᵥ y +ᵥ z",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real",
"UpperHalfPlane.instAddActionReal._proof_1",
"AddMonoid.toAddSemigroup",
"UpperHalfPlane.coe",
"Real.instZero",
"Real.instAddMonoid",
"con... | [] | by simp [HVAdd.hVAdd, add_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc | {
"line": 72,
"column": 4
} | {
"line": 74,
"column": 28
} | {
"line": 75,
"column": 4
} | [
{
"pp": "case inl\nx : ℝ\nhx : x < 0\n⊢ sin x * x⁻¹ * -x ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZe... | [
"case inl\nx : ℝ\nhx : x < 0\n⊢ 0 < -x"
] | · ring_nf
rw [mul_assoc, mul_inv_cancel₀ hx.ne, mul_one, neg_le]
exact neg_one_le_sin x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 129,
"column": 8
} | {
"line": 129,
"column": 11
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case pos\nx : ℝ\nhx : x = 0\n⊢ deriv (deriv fun x ↦ x * log x) x = x⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"Real.... | [
"case pos\nx : ℝ\nhx : x = 0\n⊢ deriv (deriv fun x ↦ x * log x) 0 = 0⁻¹"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 297,
"column": 62
} | {
"line": 297,
"column": 76
} | {
"line": 297,
"column": 76
} | [
{
"pp": "case inl\nb : ℝ\nh : b < 0\n⊢ -(-b * log (-b) - -b) = b * log b - b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"HMul.hMul",
"Real.log_neg_eq_log",
"congrArg",
... | [
"case inl\nb : ℝ\nh : b < 0\n⊢ -(-b * log b - -b) = b * log b - b"
] | log_neg_eq_log | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 54,
"column": 50
} | {
"line": 78,
"column": 81
} | {
"line": 80,
"column": 0
} | [
{
"pp": "f : ℂ → ℂ\nz₀ : ℂ\nε r : ℝ\nh : DiffContOnCl ℂ f (ball z₀ r)\nhr : 0 < r\nhf : ∀ z ∈ sphere z₀ r, ε ≤ ‖f z - f z₀‖\nhz₀ : ∃ᶠ (z : ℂ) in 𝓝 z₀, f z ≠ f z₀\n⊢ ball (f z₀) (ε / 2) ⊆ f '' closedBall z₀ r",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealC... | [] | by
/- This is a direct application of the maximum principle. Pick `v` close to `f z₀`, and look at
the function `fun z ↦ ‖f z - v‖`: it is bounded below on the circle, and takes a small value
at `z₀` so it is not constant on the disk, which implies that its infimum is equal to `0` and
hence that `v` is in... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 16
} | {
"line": 102,
"column": 2
} | [
{
"pp": "f : ℂ → ℂ\nz₀ : ℂ\nhf : AnalyticAt ℂ f z₀\nh : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀\nR : ℝ\nhR : 0 < R\nh1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀\nh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z\nρ : ℝ\nhρ : ρ > 0\nh4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀\nh3 : DiffContOnCl ℂ f (ball z₀ ρ)\n⊢ ∃ i, 0 < i ∧ ... | [
"f : ℂ → ℂ\nz₀ : ℂ\nhf : AnalyticAt ℂ f z₀\nh : ¬∀ᶠ (z : ℂ) in 𝓝 z₀, f z = f z₀\nR : ℝ\nhR : 0 < R\nh1 : ∀ᶠ (z : ℂ) in 𝓝[≠] z₀, f z ≠ f z₀\nh2 : ∀ᶠ (z : ℂ) in 𝓝 z₀, AnalyticAt ℂ f z\nρ : ℝ\nhρ : ρ > 0\nh4 : ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀\nh3 : DiffContOnCl ℂ f (ball z₀ ρ)\nr : ℝ := min ρ R\n⊢ ∃ i, 0 ... | let r := ρ ⊓ R | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 462,
"column": 6
} | {
"line": 469,
"column": 58
} | {
"line": 470,
"column": 2
} | [] | [] | (∫ x in a..b, sin x ^ (n + 2)) = ∫ x in a..b, sin x ^ (n + 1) * sin x := by
simp only [_root_.pow_succ]
_ = C + (↑n + 1) * ∫ x in a..b, cos x ^ 2 * sin x ^ n := by simp [H, h, sq]; ring
_ = C + (↑n + 1) * ∫ x in a..b, sin x ^ n - sin x ^ (n + 2) := by
simp [cos_sq', sub_mul, ← pow_add, add_c... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 55
} | {
"line": 150,
"column": 4
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nh1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ ... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\ng : E → ℂ\nz₀ : E\nhg : AnalyticAt ℂ g z₀\nray : E → ℂ → E := fun z t ↦ z₀ + t • z\ngray : E → ℂ → ℂ := fun z ↦ g ∘ ray z\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nh1 : ∀ z ∈ sphere 0 1, AnalyticOnNhd ℂ (gray z) (ba... | replace h4 : ↑‖z - z₀‖ ∈ ball (0 : ℂ) r := by simpa | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Complex.Periodic | {
"line": 69,
"column": 24
} | {
"line": 69,
"column": 38
} | {
"line": 69,
"column": 39
} | [
{
"pp": "h : ℝ\nhh : h ≠ 0\nz : ℂ\nm : ℤ\nhm : log (cexp (2 * ↑π * I * z / ↑h)) = 2 * ↑π * I * z / ↑h + ↑m * (2 * ↑π * I)\n⊢ ↑h / (2 * ↑π * I) * (2 * ↑π * I * z / ↑h + ↑m * (2 * ↑π * I)) = z + ↑m * ↑h",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"... | [
"h : ℝ\nhh : h ≠ 0\nz : ℂ\nm : ℤ\nhm : log (cexp (2 * ↑π * I * z / ↑h)) = 2 * ↑π * I * z / ↑h + ↑m * (2 * ↑π * I)\n⊢ ↑h / (2 * ↑π * I) * (2 * ↑π * I * (z / ↑h) + ↑m * (2 * ↑π * I)) = z + ↑m * ↑h"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 99
} | {
"line": 516,
"column": 0
} | [
{
"pp": "n : ℕ\nH : ∀ x ∈ Set.Icc 0 π, sin x ^ (n + 1) ≤ sin x ^ n :=\n fun x h ↦ pow_le_pow_of_le_one (sin_nonneg_of_mem_Icc h) (sin_le_one x) (Nat.le_add_right n 1)\n⊢ ∫ (x : ℝ) in 0..π, sin x ^ (n + 1) ≤ ∫ (x : ℝ) in 0..π, sin x ^ n",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
... | [] | refine integral_mono_on pi_pos.le ?_ ?_ H <;> exact (continuous_sin.pow _).intervalIntegrable 0 π | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 600,
"column": 6
} | {
"line": 601,
"column": 30
} | {
"line": 602,
"column": 6
} | [
{
"pp": "a b : ℝ\nm n : ℕ\nhc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m\n⊢ -∫ (x : ℝ) in b..a, sin x ^ (2 * m + 1) * cos x ^ n = ∫ (x : ℝ) in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n",
"ppTerm": "?m.250",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormed... | [
"a b : ℝ\nm n : ℕ\nhc : Continuous fun u ↦ u ^ n * (1 - u ^ 2) ^ m\n⊢ ∫ (x : ℝ) in b..a, (sin x * sin x) ^ m * sin x * cos x ^ n =\n ∫ (x : ℝ) in b..a, (1 - cos x * cos x) ^ m * sin x * cos x ^ n"
] | simp only [_root_.pow_succ, pow_mul, _root_.pow_zero, one_mul, mul_neg, neg_mul,
integral_neg, neg_inj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.TaylorSeries | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 68
} | {
"line": 90,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nr : ENNReal\nhf : DifferentiableOn ℂ f (Metric.eball c r)\nz : ℂ\nhz : z ∈ Metric.eball c r\n⊢ HasSum (fun n ↦ (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z)",
"ppTerm": "?m.47",
... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nr : ENNReal\nhf : DifferentiableOn ℂ f (Metric.eball c r)\nz : ℂ\nhz : z ∈ Metric.eball c r\nr' : ENNReal\nhzr' : edist c z < r'\nhr' : r' < r\n⊢ HasSum (fun n ↦ (↑n !)⁻¹ • (z - c) ^ n • iteratedDeriv ... | obtain ⟨r', hzr', hr'⟩ := exists_between (Metric.mem_eball'.mp hz) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 250,
"column": 13
} | {
"line": 250,
"column": 27
} | {
"line": 250,
"column": 28
} | [
{
"pp": "R : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\nu : ℂ\nhu : u ∈ h.toFinset\n⊢ circleAverage (fun x ↦ ↑(D u) * log ‖x - u‖) c R = ↑(D u) * log R",
"ppTerm": "?m.248",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"... | [
"R : ℝ\nc : ℂ\nD : Function.locallyFinsuppWithin (closedBall c |R|) ℤ\nh : D.support.Finite\nu : ℂ\nhu : u ∈ h.toFinset\n⊢ circleAverage (fun x ↦ ↑(D u) • log ‖x - u‖) c R = ↑(D u) • log R"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 41
} | {
"line": 36,
"column": 2
} | [
{
"pp": "case pos\na : ℂ\nh : 0 ≤ a.im\n⊢ a ^ 2⁻¹ = ↑(a ^ 2⁻¹).re + (if 0 ≤ a.im then 1 else -1) * ↑√((‖a‖ - a.re) / 2) * I",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
... | [
"case neg\na : ℂ\nh : a.im < 0\n⊢ a ^ 2⁻¹ = ↑(a ^ 2⁻¹).re + (if 0 ≤ a.im then 1 else -1) * ↑√((‖a‖ - a.re) / 2) * I"
] | · simp [← cpow_inv_two_im_eq_sqrt h, h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Tietze | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 10
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nthis✝¹ : NormedSpace ℝ E\nthis✝ : IsScalarTower ℝ 𝕜 E\ng : E → E := fun x ↦ ‖x‖⁻¹ • x\nthis : Continuous ((Metric.closedBall 0 1).piecewise id g)\n⊢ { t... | [] | · ext x
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 47
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inl\ng g' : GL (Fin 2) ℝ\nz : ℂ\nh : (↑g).det < 0\n⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ\n else Complex.conjCAE)\n z =\n (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE)\n ((... | [
"case inl.inl\ng g' : GL (Fin 2) ℝ\nz : ℂ\nh : (↑g).det < 0\nh' : (↑g').det < 0\n⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ\n else Complex.conjCAE)\n z =\n (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCA... | rcases g'.det_ne_zero.lt_or_gt with (h' | h') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 47
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inr\ng g' : GL (Fin 2) ℝ\nz : ℂ\nh : 0 < (↑g).det\n⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ\n else Complex.conjCAE)\n z =\n (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCAE)\n ((... | [
"case inr.inl\ng g' : GL (Fin 2) ℝ\nz : ℂ\nh : 0 < (↑g).det\nh' : (↑g').det < 0\n⊢ (if 0 < ↑(GeneralLinearGroup.det g) * ↑(GeneralLinearGroup.det g') then ContinuousAlgEquiv.refl ℝ ℂ\n else Complex.conjCAE)\n z =\n (if 0 < ↑(GeneralLinearGroup.det g) then ContinuousAlgEquiv.refl ℝ ℂ else Complex.conjCA... | rcases g'.det_ne_zero.lt_or_gt with (h' | h') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 94
} | {
"line": 323,
"column": 2
} | [
{
"pp": "a b : ℝ\nha : a ≠ 0\nhc : ↑⟨!![a, b; 0, a⁻¹], ⋯⟩ 1 0 = 0\nz : ℂ\nhz : 0 < z.im\n⊢ ↑(⟨!![a, b; 0, a⁻¹], ⋯⟩ • { coe := z, coe_im_pos := hz }) =\n ↑(((fun x ↦ b * a +ᵥ x) ∘ fun x ↦ ⟨a * a, ⋯⟩ • x) { coe := z, coe_im_pos := hz })",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
... | [
"a b : ℝ\nha : a ≠ 0\nhc : ↑⟨!![a, b; 0, a⁻¹], ⋯⟩ 1 0 = 0\nz : ℂ\nhz : 0 < z.im\n⊢ ↑a * z * ↑a + ↑b * ↑a = ↑b * ↑a + ↑a * ↑a * z"
] | suffices ↑a * z * a + b * a = b * a + a * a * z by simpa [specialLinearGroup_apply, add_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 392,
"column": 49
} | {
"line": 392,
"column": 67
} | {
"line": 394,
"column": 0
} | [
{
"pp": "τ : ℂ\n⊢ denom J τ = 1",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Units.val",
"Matrix.GeneralLinearGroup.val_mkOfDetNeZero",
"Real",
"Equiv.instEquivLike",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Matrix",
"MulZeroClass.... | [] | by simp [J, denom] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 64
} | {
"line": 71,
"column": 4
} | [
{
"pp": "S : Type u_2\nm : Type u_5\ninst✝² : CommRing S\nM : Matrix m m S\ninst✝¹ : DecidableEq m\nk : m\ninst✝ : Fintype m\ni j : m\nlt : (fun x ↦ x = k) j < (fun x ↦ x = k) i\n⊢ (M * auxMat M k) i j = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Partia... | [
"S : Type u_2\nm : Type u_5\ninst✝² : CommRing S\nM : Matrix m m S\ninst✝¹ : DecidableEq m\nk : m\ninst✝ : Fintype m\ni j : m\nlt : i = k ∧ ¬j = k\n⊢ (M * auxMat M k) i j = 0"
] | simp_rw [lt_iff_not_ge, le_Prop_eq, Classical.not_imp] at lt | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 132,
"column": 75
} | {
"line": 132,
"column": 93
} | {
"line": 133,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ ((compRingEquiv m n R) (f.mapMatrix M)).det * (f (M k k)).det ^ (Fi... | [
"R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ ((compRingEquiv m n R) (M.map ⇑f)).det * (f (M k k)).det ^ (Fintype.card m - 1)... | f.mapMatrix_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 294,
"column": 2
} | {
"line": 296,
"column": 66
} | {
"line": 298,
"column": 0
} | [
{
"pp": "case mpr\nz : ℍ\nr : ℝ\nw : ℂ\n⊢ w ∈ ball (↑(z.center r)) (z.im * Real.sinh r) → w ∈ UpperHalfPlane.coe '' ball z r",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.center",
"Iff.mpr",
"NormedCommRing.toSeminormedCommRing",
"Metric.ball_subset... | [] | · intro hw
lift w to ℍ using im_pos_of_dist_center_le (ball_subset_closedBall hw)
exact mem_image_of_mem _ (dist_lt_iff_dist_coe_center_lt.2 hw) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 314,
"column": 6
} | {
"line": 314,
"column": 19
} | {
"line": 314,
"column": 19
} | [
{
"pp": "a y₁ y₂ : ℝ\n⊢ dist { coe := { re := a, im := rexp y₁ }, coe_im_pos := ⋯ } { coe := { re := a, im := rexp y₂ }, coe_im_pos := ⋯ } =\n dist y₁ y₂",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"UpperHalfPlane.dist_of_re_eq",... | [
"a y₁ y₂ : ℝ\n⊢ dist (log { coe := { re := a, im := rexp y₁ }, coe_im_pos := ⋯ }.im)\n (log { coe := { re := a, im := rexp y₂ }, coe_im_pos := ⋯ }.im) =\n dist y₁ y₂",
"a y₁ y₂ : ℝ\n⊢ { coe := { re := a, im := rexp y₁ }, coe_im_pos := ⋯ }.re = { coe := { re := a, im := rexp y₂ }, coe_im_pos := ⋯ }.re"
] | dist_of_re_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 48
} | {
"line": 90,
"column": 2
} | [
{
"pp": "K : Set ℍ\nhK : IsCompact K\n⊢ ∃ A, ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 0 0 ^ 2 + ↑g 0 1 ^ 2 ≤ A",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Real",
"Matrix.SpecialLinearGroup",
"Equiv.instEquivLike",
"HMul.hMul",
"Real.instZero",
"Real.instAdd... | [
"K : Set ℍ\nhK : IsCompact K\nS : SL(2, ℝ) := ⟨!![0, -1; 1, 0], ⋯⟩\n⊢ ∃ A, ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 0 0 ^ 2 + ↑g 0 1 ^ 2 ≤ A"
] | let S : SL(2, ℝ) := ⟨!![0, -1; 1, 0], by simp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 107,
"column": 2
} | {
"line": 116,
"column": 35
} | {
"line": 118,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nα : Type u_2\ns : Finset α\nf : α → ℂ → E\nr : ℝ\nhf : ∀ a ∈ s, Meromorphic (f a)\nhr : 1 ≤ r\n⊢ characteristic (∑ a ∈ s, f a) ⊤ r ≤ (∑ a ∈ s, characteristic (f a) ⊤) r + log ↑s.card",
"ppTerm": "?m.43",
"assigned": true,
... | [] | simp only [characteristic, Pi.add_apply, Finset.sum_apply]
calc proximity (∑ a ∈ s, f a) ⊤ r + logCounting (∑ a ∈ s, f a) ⊤ r
_ ≤ ((∑ a ∈ s, proximity (f a) ⊤) r) + log s.card + (∑ a ∈ s, (logCounting (f a) ⊤)) r := by
gcongr
· apply proximity_sum_top_le s f hf r
· apply logCounting_sum_top_le s f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 107,
"column": 2
} | {
"line": 116,
"column": 35
} | {
"line": 118,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nα : Type u_2\ns : Finset α\nf : α → ℂ → E\nr : ℝ\nhf : ∀ a ∈ s, Meromorphic (f a)\nhr : 1 ≤ r\n⊢ characteristic (∑ a ∈ s, f a) ⊤ r ≤ (∑ a ∈ s, characteristic (f a) ⊤) r + log ↑s.card",
"ppTerm": "?m.43",
"assigned": true,
... | [] | simp only [characteristic, Pi.add_apply, Finset.sum_apply]
calc proximity (∑ a ∈ s, f a) ⊤ r + logCounting (∑ a ∈ s, f a) ⊤ r
_ ≤ ((∑ a ∈ s, proximity (f a) ⊤) r) + log s.card + (∑ a ∈ s, (logCounting (f a) ⊤)) r := by
gcongr
· apply proximity_sum_top_le s f hf r
· apply logCounting_sum_top_le s f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.ConstantSpeed | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 32
} | {
"line": 108,
"column": 8
} | [
{
"pp": "case mp\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofR... | [
"case mp.inl\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l ... | rintro ⟨ws | wt, zw, wy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 142,
"column": 4
} | {
"line": 145,
"column": 9
} | {
"line": 147,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\ne : E\nr : ℝ\nn : ℤ\nhr : ‖e‖ ≤ r\nhe : ¬0 = e\n⊢ ↑((single e n) e) * log (r * ‖e‖⁻¹) + ↑((single e n) 0) * log r = ↑n * (log r - log ‖e‖)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConsta... | [] | simp only [single_apply, he, reduceIte, Int.cast_zero, zero_mul, add_zero,
log_mul (ne_of_lt (lt_of_lt_of_le (norm_pos_iff.mpr (he ·.symm)) hr)).symm
(inv_ne_zero (norm_ne_zero_iff.mpr (he ·.symm))), log_inv]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 142,
"column": 4
} | {
"line": 145,
"column": 9
} | {
"line": 147,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\ne : E\nr : ℝ\nn : ℤ\nhr : ‖e‖ ≤ r\nhe : ¬0 = e\n⊢ ↑((single e n) e) * log (r * ‖e‖⁻¹) + ↑((single e n) 0) * log r = ↑n * (log r - log ‖e‖)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConsta... | [] | simp only [single_apply, he, reduceIte, Int.cast_zero, zero_mul, add_zero,
log_mul (ne_of_lt (lt_of_lt_of_le (norm_pos_iff.mpr (he ·.symm)) hr)).symm
(inv_ne_zero (norm_ne_zero_iff.mpr (he ·.symm))), log_inv]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.AmpleSet | {
"line": 61,
"column": 79
} | {
"line": 61,
"column": 94
} | {
"line": 61,
"column": 94
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\na✝ : x ∈ univ\n⊢ (convexHull ℝ) univ = univ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"ChainCompletePartialOrder.instOfCompleteLattice"... | [
"F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx : F\na✝ : x ∈ univ\n⊢ univ = univ"
] | convexHull_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 32
} | {
"line": 115,
"column": 8
} | [
{
"pp": "case mp\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofR... | [
"case mp.inl\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l ... | rintro ⟨ws | wt, zw, wy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.ConstantSpeed | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 32
} | {
"line": 132,
"column": 8
} | [
{
"pp": "case mp\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofR... | [
"case mp.inl\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nt : Set ℝ\nhfs : ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → x ≤ y → eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (↑l * (y - x))\nhft : ∀ ⦃x : ℝ⦄, x ∈ t → ∀ ⦃y : ℝ⦄, y ∈ t → x ≤ y → eVariationOn f (t ∩ Icc x y) = ENNReal.ofReal (↑l ... | rintro ⟨ws | wt, zw, wy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 48
} | {
"line": 442,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ locallyFinsuppWithin.logCounting (divisor (f₁ + f₂) univ)⁻ r ≤\n (loc... | [
"𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf₁ f₂ : 𝕜 → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ locallyFinsuppWithin.logCounting (divisor (f₁ + f₂) univ)⁻ r ≤\n locallyFinsuppWi... | ← locallyFinsuppWithin.logCounting.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 506,
"column": 4
} | {
"line": 506,
"column": 46
} | {
"line": 506,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor f₁ un... | [
"𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor f₁ univ + divisor... | ← locallyFinsuppWithin.logCounting.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 46
} | {
"line": 536,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor f₁ un... | [
"𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor f₁ univ + divisor... | ← locallyFinsuppWithin.logCounting.map_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 79,
"column": 56
} | {
"line": 79,
"column": 70
} | {
"line": 79,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na... | [
"𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nx : E\na : ℝ\nhx : x... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 95,
"column": 48
} | {
"line": 95,
"column": 62
} | {
"line": 95,
"column": 63
} | [
{
"pp": "s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\ny : ℝ\n⊢ y * l 1 = l y",
"ppTerm": "?m.120",
"ass... | [
"s : Set ℝ\nf : ℝ → ℝ\nx a : ℝ\nhx : x ∈ s\nhax : a < f x\nhsc : IsClosed s\nhfc : LowerSemicontinuousOn f s\nhf : ConvexOn ℝ s f\nl : ℝ →L[ℝ] ℝ\nc' : ℝ\nhlc'_le : s.restrict (⇑re ∘ ⇑l) + const (↑s) c' ≤ s.restrict f\nhlc'_eq : re (l x) + c' = a\ny : ℝ\n⊢ y • l 1 = l y"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Hall.Finite | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 24
} | {
"line": 175,
"column": 2
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnio... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n.succ\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ ... | rw [Nat.add_one] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Approximation | {
"line": 131,
"column": 72
} | {
"line": 148,
"column": 37
} | {
"line": 150,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ :... | [] | by
by_cases! hs : s.Nonempty
· let 𝓕 := {f | f ≤ s.restrict φ ∧
∃ (l : E →L[𝕜] 𝕜) (c : ℝ), f = s.restrict (re ∘ l) + const s c}
have hl : IsLUB 𝓕 (s.restrict φ) := by
refine (hφcv.sSup_affine_eq (𝕜 := 𝕜) hsc hφc) ▸ isLUB_csSup ?_ ?_
· obtain ⟨l, c, hlc⟩ := exists_affine_le_of_lt (𝕜 := �... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Between | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 25
} | {
"line": 208,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nV' : Type u_3\nP : Type u_4\nP' : Type u_5\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx y z : P\nf : P ≃ᵃ[R] P'\nthis : Function.Inj... | [] | apply this.sbtw_map_iff | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Convex.Between | {
"line": 569,
"column": 88
} | {
"line": 572,
"column": 29
} | {
"line": 574,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Sbtw R x y z\n⊢ Sbtw R w y z",
"pp... | [] | by
rw [wbtw_comm] at *
rw [sbtw_comm] at *
exact h₁.trans_sbtw_left h₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 24
} | {
"line": 291,
"column": 4
} | [
{
"pp": "case inr\nJ : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nhFn : ∀ (j : J), Nonempty (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\nhs : s.Nonempty\nj : J\nji : j ⟶ i\n⊢ ∃ x, ∀ (i_1 : j ... | [
"case inr\nJ : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni : J\ns : Set (F.obj i)\ninst✝ : IsCofilteredOrEmpty J\nhFn : ∀ (j : J), Nonempty (F.obj j)\nFsur : ∀ ⦃i j : J⦄ (f : i ⟶ j), Function.Surjective ⇑(ConcreteCategory.hom (F.map f))\nj : J\nji : j ⟶ i\ny : F.obj i\nys : y ∈ s\n⊢ ∃ x, ∀ (i_1 : j ⟶ i)... | obtain ⟨y, ys⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.DoublyStochasticMatrix | {
"line": 150,
"column": 76
} | {
"line": 168,
"column": 79
} | {
"line": 170,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 ≤ s\n⊢ (∃ M' ∈ doublyStochastic R n, M = s • M') ↔\n (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n)... | [] | by
classical
constructor
case mp =>
rintro ⟨M', hM', rfl⟩
rw [mem_doublyStochastic_iff_sum] at hM'
simp only [Matrix.smul_apply, smul_eq_mul, ← mul_sum]
exact ⟨fun i j => mul_nonneg hs (hM'.1 _ _), by simp [hM']⟩
rcases eq_or_lt_of_le hs with rfl | hs
case inl =>
simp only [zero_smul, exis... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 80,
"column": 28
} | {
"line": 80,
"column": 44
} | {
"line": 80,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e... | [] | by convert! ksum | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Continuous | {
"line": 37,
"column": 41
} | {
"line": 37,
"column": 57
} | {
"line": 37,
"column": 57
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhxy : x ≠ y\nhx₀... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhxy : x ≠ y\nhx₀r : ball x₀ ... | norm_sub_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Independent | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 53
} | {
"line": 74,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → E\nhc : ConvexIndependent 𝕜 p\ni j : ι\nhij : p i = p j\n⊢ p i ∈ (convexHull 𝕜) (p '' {j})",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants... | [
"𝕜 : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → E\nhc : ConvexIndependent 𝕜 p\ni j : ι\nhij : p i = p j\n⊢ p j ∈ {p j}"
] | rw [hij, Set.image_singleton, convexHull_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Continuous | {
"line": 85,
"column": 75
} | {
"line": 85,
"column": 99
} | {
"line": 86,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhr : r' < r\nhf' : IsBounded (f '' ball x₀ r)\nx✝ : ℝ\n⊢ x✝ ∈ (-f) '' ball x₀ r ↔ x✝ ∈ -f '' ball x₀ r",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": ... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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