module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 199,
"column": 48
} | {
"line": 199,
"column": 59
} | {
"line": 199,
"column": 60
} | [
{
"pp": "τ : ℍ\n⊢ 0 < (-(starRingEnd ℂ) ↑τ).im",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"UpperHalfPlane.coe",
"Real.instZero",
"congrArg",
... | [
"τ : ℍ\n⊢ 0 < τ.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 40
} | {
"line": 222,
"column": 41
} | [
{
"pp": "τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : -ε < r - ‖↑τ‖ ∧ r - ‖↑τ‖ < ε\nhr' : r < 0\nthis✝ : ‖↑τ‖ < ε\nthis : 0 ∈ Metric.ball (↑τ) ε\n⊢ False",
"ppTerm": "?m.1... | [
"τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : -ε < r - ‖↑τ‖ ∧ r - ‖↑τ‖ < ε\nhr' : r < 0\nthis✝ : ‖↑τ‖ < ε\nthis : 0 ∈ Metric.ball (↑τ) ε\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 273,
"column": 22
} | {
"line": 273,
"column": 42
} | {
"line": 274,
"column": 4
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : CharZero K\nn : ℕ\nhn : n ≠ 0\na : K\nm : Matrix (Fin 2) (Fin 2) K\nhg : ↑g = (Matrix.scalar (Fin 2)) a + m\nhm0 : m ≠ 0\nhmsq : m ^ 2 = 0\nthis : a ≠ 0\n⊢ (↑n * a ^ (n - 1)) • m ≠ 0",
"ppTerm": "?m.180",
"assigned": true,
"usedConst... | [] | simp [this, hm0, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 273,
"column": 22
} | {
"line": 273,
"column": 42
} | {
"line": 274,
"column": 4
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : CharZero K\nn : ℕ\nhn : n ≠ 0\na : K\nm : Matrix (Fin 2) (Fin 2) K\nhg : ↑g = (Matrix.scalar (Fin 2)) a + m\nhm0 : m ≠ 0\nhmsq : m ^ 2 = 0\nthis : a ≠ 0\n⊢ (↑n * a ^ (n - 1)) • m ≠ 0",
"ppTerm": "?m.180",
"assigned": true,
"usedConst... | [] | simp [this, hm0, hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 273,
"column": 22
} | {
"line": 273,
"column": 42
} | {
"line": 274,
"column": 4
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : CharZero K\nn : ℕ\nhn : n ≠ 0\na : K\nm : Matrix (Fin 2) (Fin 2) K\nhg : ↑g = (Matrix.scalar (Fin 2)) a + m\nhm0 : m ≠ 0\nhmsq : m ^ 2 = 0\nthis : a ≠ 0\n⊢ (↑n * a ^ (n - 1)) • m ≠ 0",
"ppTerm": "?m.180",
"assigned": true,
"usedConst... | [] | simp [this, hm0, hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 227,
"column": 28
} | {
"line": 227,
"column": 39
} | {
"line": 227,
"column": 40
} | [
{
"pp": "τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\nhr' : 0 ≤ r\n⊢ ‖↑τ‖ ≠ 0",
"ppTerm": "?m.271",
"assigned": true,
"usedConstants": [
... | [
"τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\nhr' : 0 ≤ r\n⊢ ¬↑τ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 43
} | {
"line": 317,
"column": 44
} | [
{
"pp": "z : ℍ\n⊢ ModularGroup.T • z = 1 +ᵥ z",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℍ\n⊢ ModularGroup.T • z = 1 +ᵥ z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 71
} | {
"line": 306,
"column": 8
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\na b : ℝ\ne : X → Y\nhf : ∀ (x : X), f x ∈ Icc a b\nhle : a ≤ b\nhe : IsClosedEmbedding e\ng : Y →ᵇ ℝ\nhgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖\nhge : ⇑g ∘ e = ⇑(f - const X ((a + b) /... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\na b : ℝ\ne : X → Y\nhf : ∀ (x : X), f x ∈ Icc a b\nhle : a ≤ b\nhe : IsClosedEmbedding e\ng : Y →ᵇ ℝ\nhgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖\nhge : ⇑g ∘ e = ⇑(f - const X ((a + b) / 2))\ny : Y\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 15
} | {
"line": 303,
"column": 16
} | [
{
"pp": "case mpr.inl\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nhx : x ≠ 0\nh_det : det (upperRightHom x) = 1 ∨ det (upperRightHom x) = -1\nhg10 : ↑(upperRightHom x) 1 0 = 0\n⊢ ↑(upperRightHom x) 0 0 = ↑(upperRightHom x) 1 1 ∧ ↑(upperRightHom x) 0 1 ≠ 0",
... | [
"case mpr.inl\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nhx : x ≠ 0\nh_det : det (upperRightHom x) = 1 ∨ det (upperRightHom x) = -1\nhg10 : ↑(upperRightHom x) 1 0 = 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 15
} | {
"line": 303,
"column": 16
} | [
{
"pp": "case mpr.inr\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nhx : x ≠ 0\nh_det : det (-upperRightHom x) = 1 ∨ det (-upperRightHom x) = -1\nhg10 : ↑(-upperRightHom x) 1 0 = 0\n⊢ ↑(-upperRightHom x) 0 0 = ↑(-upperRightHom x) 1 1 ∧ ↑(-upperRightHom x) 0 1 ≠ 0... | [
"case mpr.inr\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nx : K\nhx : x ≠ 0\nh_det : det (-upperRightHom x) = 1 ∨ det (-upperRightHom x) = -1\nhg10 : ↑(-upperRightHom x) 1 0 = 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 330,
"column": 30
} | {
"line": 330,
"column": 41
} | {
"line": 330,
"column": 42
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormalSpace Y\ninst✝ : Nonempty X\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\ninhabited_h : Inhabited X\na : ℝ\nha : IsGLB (range ⇑f) a\nhb : IsLUB (range ⇑f) a\nhmem : ∀ (x : X), f x ∈ Icc a a\nhle : a... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormalSpace Y\ninst✝ : Nonempty X\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\ninhabited_h : Inhabited X\na : ℝ\nha : IsGLB (range ⇑f) a\nhb : IsLUB (range ⇑f) a\nhmem : ∀ (x : X), f x ∈ Icc a a\nhle : a ≤ a\n⊢ ∀ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 231,
"column": 28
} | {
"line": 231,
"column": 39
} | {
"line": 231,
"column": 40
} | [
{
"pp": "τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\nhr' : 0 ≤ r\nthis : ↑r / ↑‖↑τ‖ * ↑τ ∈ Metric.ball (↑τ) ε\nξ : ℍ\nhξs : ξ ∈ s\nhξτ : ↑ξ = ↑r / ↑‖↑... | [
"τ : ℍ\nU : Set ℝ\nhU : U ∈ map (fun τ ↦ ‖↑τ‖) (𝓝 τ)\ns : Set ℍ\nhs' : (fun τ ↦ ‖↑τ‖) '' s ⊆ U\nε : ℝ\nhεpos : ε > 0\nhεs : Metric.ball (↑τ) ε ⊆ UpperHalfPlane.coe '' s\nr : ℝ\nhr : r ∈ Metric.ball ‖↑τ‖ ε\nhr' : 0 ≤ r\nthis : ↑r / ↑‖↑τ‖ * ↑τ ∈ Metric.ball (↑τ) ε\nξ : ℍ\nhξs : ξ ∈ s\nhξτ : ↑ξ = ↑r / ↑‖↑τ‖ * ↑τ\n⊢ ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 39
} | {
"line": 89,
"column": 40
} | [
{
"pp": "⊢ Tendsto UpperHalfPlane.coe atImInfty (comap Complex.im atTop)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"UpperHalfPlane.coe",
"congrArg",
"Complex.im",
"UpperHalfPlane.atImInfty",
"Function.comp",
"id",
... | [
"⊢ Tendsto im (comap im atTop) atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 38
} | {
"line": 97,
"column": 39
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : ↑g 1 0 = 0\n⊢ 0 < |↑g 0 0 / ↑g 1 1|",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Preorder.toLT",
"instHDiv",
... | [
"g : GL (Fin 2) ℝ\nhg : ↑g 1 0 = 0\n⊢ ¬↑g 0 0 = 0 ∧ ¬↑g 1 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 361,
"column": 31
} | {
"line": 361,
"column": 63
} | {
"line": 361,
"column": 64
} | [
{
"pp": "z : ℍ\n⊢ ↑√z.im ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Complex.instZero",
"Real.instLT",
"id",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction.0.UpperHalfPl... | [
"z : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 19
} | {
"line": 419,
"column": 20
} | [
{
"pp": "a b : SL(2, ℤ)\ni j : Fin 2\nh : ∀ (i j : Fin 2), ↑↑(coe a) i j = ↑↑(coe b) i j\n⊢ ↑a i j = ↑b i j",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : SL(2, ℤ)\ni j : Fin 2\nh : ∀ (i j : Fin 2), ↑↑(coe a) i j = ↑↑(coe b) i j\n⊢ ↑a i j = ↑b i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 13
} | {
"line": 475,
"column": 14
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\n⊢ (g • z).im = z.im / Complex.normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunLike",
"in... | [
"g : SL(2, ℤ)\nz : ℍ\n⊢ (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g) • z).im =\n z.im / Complex.normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | {
"line": 69,
"column": 2
} | [
{
"pp": "n : ℕ∞ω\nf : ℍ → ℂ\nτ : ℍ\n⊢ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n f τ ↔ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n (f ∘ ↑ofComplex) ↑τ",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommR... | [
"case refine_1\nn : ℕ∞ω\nf : ℍ → ℂ\nτ : ℍ\nhf : ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n f τ\n⊢ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n (f ∘ ↑ofComplex) ↑τ",
"case refine_2\nn : ℕ∞ω\nf : ℍ → ℂ\nτ : ℍ\nhf : ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n (f ∘ ↑ofComplex) ↑τ\n⊢ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n f τ"
] | refine ⟨fun hf ↦ ?_, fun hf ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 57
} | {
"line": 70,
"column": 58
} | [
{
"pp": "case refine_2\nn : ℕ∞ω\nf : ℍ → ℂ\nτ : ℍ\nhf : ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n (f ∘ ↑ofComplex) ↑τ\n⊢ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n f τ",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nn : ℕ∞ω\nf : ℍ → ℂ\nτ : ℍ\nhf : ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n (f ∘ ↑ofComplex) ↑τ\n⊢ ContMDiffAt 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n f τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 35
} | {
"line": 76,
"column": 2
} | [
{
"pp": "f : ℍ → ℂ\nτ : ℍ\n⊢ MDiffAt f τ ↔ MDiffAt (f ∘ ↑ofComplex) ↑τ",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"chartedSpaceSelf",
"Complex.instNormed... | [
"case refine_1\nf : ℍ → ℂ\nτ : ℍ\nhf : MDiffAt f τ\n⊢ MDiffAt (f ∘ ↑ofComplex) ↑τ",
"case refine_2\nf : ℍ → ℂ\nτ : ℍ\nhf : MDiffAt (f ∘ ↑ofComplex) ↑τ\n⊢ MDiffAt f τ"
] | refine ⟨fun hf ↦ ?_, fun hf ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 57
} | {
"line": 77,
"column": 58
} | [
{
"pp": "case refine_2\nf : ℍ → ℂ\nτ : ℍ\nhf : MDiffAt (f ∘ ↑ofComplex) ↑τ\n⊢ MDiffAt f τ",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nf : ℍ → ℂ\nτ : ℍ\nhf : MDiffAt (f ∘ ↑ofComplex) ↑τ\n⊢ MDiffAt f τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 14
} | [
{
"pp": "n : ℕ∞ω\ng : GL (Fin 2) ℝ\n⊢ ContMDiff 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n fun τ ↦ (denom g ↑τ)⁻¹",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ∞ω\ng : GL (Fin 2) ℝ\n⊢ ContMDiff 𝓘(ℂ, ℂ) 𝓘(ℂ, ℂ) n fun τ ↦ (denom g ↑τ)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 52
} | {
"line": 217,
"column": 53
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_6\nK : Type u_7\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra R L\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K L\nx : L\nhx : IsIntegral R x\nh : ¬FiniteDimensional K L\n⊢ IsIntegral R ((norm K) x)",
"ppTerm":... | [
"case neg\nR : Type u_1\ninst✝⁶ : CommRing R\nL : Type u_6\nK : Type u_7\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra R L\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K L\nx : L\nhx : IsIntegral R x\nh : ¬FiniteDimensional K L\n⊢ IsIntegral R 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 15
} | {
"line": 134,
"column": 16
} | [
{
"pp": "f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ a ∈ {τ}ᶜ → f a ≠ 0",
"ppTerm": "?m.221",
"... | [
"f : ℍ → ℂ\nhf : DifferentiableOn ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nτ : ℍ\nthis : AnalyticOnNhd ℂ (f ∘ ↑ofComplex) {z | 0 < z.im}\nw : ℍ\nhτ : ∀ᶠ (a : ℍ) in 𝓝 τ, ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\na : ℍ\nha : ↑a ∈ {↑τ}ᶜ → (f ∘ ↑ofComplex) ↑a ≠ 0\n⊢ ¬a = τ → ¬f a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 137,
"column": 59
} | {
"line": 137,
"column": 70
} | {
"line": 137,
"column": 71
} | [
{
"pp": "f g : ℍ → ℂ\nhf : MDiff f\nhg : MDiff g\nhfg : f * g = 0\n⊢ ∀ (x : ℍ), f x = 0 ∨ g x = 0",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f g : ℍ → ℂ\nhf : MDiff f\nhg : MDiff g\nhfg : f * g = 0\n⊢ ∀ (x : ℍ), f x = 0 ∨ g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 144,
"column": 64
} | {
"line": 144,
"column": 75
} | {
"line": 144,
"column": 76
} | [
{
"pp": "ι : Type u_1\nf : ι → ℍ → ℂ\ns : Finset ι\nhf : ∀ i ∈ s, MDiff (f i)\nh0 : ∏ i ∈ s, f i = 0\n⊢ ∀ (x : ℍ), ∏ i ∈ s, f i x = 0",
"ppTerm": "?m.82",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nf : ι → ℍ → ℂ\ns : Finset ι\nhf : ∀ i ∈ s, MDiff (f i)\nh0 : ∏ i ∈ s, f i = 0\n⊢ ∀ (x : ℍ), ∏ i ∈ s, f i x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 23
} | {
"line": 165,
"column": 24
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\n⊢ HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Units.val",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCo... | [
"g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\n⊢ HasDerivAt (HMul.hMul ↑(↑g 1 0)) ↑(↑g 1 0) ↑τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 54
} | {
"line": 167,
"column": 55
} | [
{
"pp": "g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\nthis : HasDerivAt ((fun x ↦ x ^ k) ∘ denom g) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * ↑(↑g 1 0) * denom g ↑τ ^ (k - 1)) ↑τ",
"ppTerm": "?m.109",
"assigned": true,
... | [
"g : GL (Fin 2) ℝ\nk : ℤ\nτ : ℍ\nhd : HasDerivAt (fun x ↦ denom g x) ↑(↑g 1 0) ↑τ\nthis : HasDerivAt ((fun x ↦ x ^ k) ∘ denom g) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ\n⊢ HasDerivAt (fun z ↦ denom g z ^ k) (↑k * denom g ↑τ ^ (k - 1) * ↑(↑g 1 0)) ↑τ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 178,
"column": 4
} | {
"line": 180,
"column": 52
} | {
"line": 181,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nτ : ℍ\nthis : HasStrictDerivAt (num g / denom g) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ\n⊢ HasStrictDerivAt (fun z ↦ ↑(g • ↑ofComplex z)) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | refine this.congr_of_eventuallyEq ?_
rw [← isOpenEmbedding_coe.map_nhds_eq, eventuallyEq_map]
simp [Function.comp_def, coe_smul_of_det_pos hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Manifold | {
"line": 178,
"column": 4
} | {
"line": 180,
"column": 52
} | {
"line": 181,
"column": 2
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : 0 < (↑g).det\nτ : ℍ\nthis : HasStrictDerivAt (num g / denom g) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ\n⊢ HasStrictDerivAt (fun z ↦ ↑(g • ↑ofComplex z)) (↑(↑g).det / denom g ↑τ ^ 2) ↑τ",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | refine this.congr_of_eventuallyEq ?_
rw [← isOpenEmbedding_coe.map_nhds_eq, eventuallyEq_map]
simp [Function.comp_def, coe_smul_of_det_pos hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 26
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case pos\nL : Type u_6\nK : Type u_7\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nh : FiniteDimensional K L\nF : IntermediateField K L := K⟮x⟯\n⊢ (norm K) x = (norm K) (gen K x) ^ finrank (↥K⟮x⟯) L",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"case pos\nL : Type u_6\nK : Type u_7\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nh : FiniteDimensional K L\nF : IntermediateField K L := K⟮x⟯\n⊢ (norm K) ↑(gen K x) = (norm K) (gen K x) ^ finrank (↥K⟮x⟯) L"
] | nth_rw 1 [← coe_gen K x] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 43
} | {
"line": 161,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc x y : P\nhx : x ≠ c\nhy : y ≠ c\nR : ℝ\n⊢ dist ((R / ‖x -ᵥ c‖) ^ 2 • (x -ᵥ c)) ((R / ‖y -ᵥ c‖) ^ 2 • (y -ᵥ c)) = R ^ 2 / (‖x -ᵥ c‖ * ‖y -ᵥ c‖) * dist x y",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc x y : P\nhx : x ≠ c\nhy : y ≠ c\nR : ℝ\n⊢ dist ((R / ‖x -ᵥ c‖) ^ 2 • (x -ᵥ c)) ((R / ‖y -ᵥ c‖) ^ 2 • (y -ᵥ c)) = R ^ 2 / (‖x -ᵥ c‖ * ‖y -ᵥ c‖) * dist x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | {
"line": 250,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc : P\nR : ℝ\nhR : R ≠ 0\nhdist : Tendsto (dist c) (𝓝[≠] c) (𝓝[>] 0)\nhratio : Tendsto (fun x ↦ dist c (inversion c R x)) (𝓝[≠] c) atTop\n⊢ Tendsto (fun x ↦... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nc : P\nR : ℝ\nhR : R ≠ 0\nhdist : Tendsto (dist c) (𝓝[≠] c) (𝓝[>] 0)\nhratio : Tendsto (fun x ↦ dist c (inversion c R x)) (𝓝[≠] c) atTop\n⊢ Tendsto (inversion c R) (𝓝[... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 94
} | {
"line": 119,
"column": 95
} | [
{
"pp": "z✝ w✝ : ℍ\nr : ℝ\nz w : ℍ\nh : dist z w = 0\n⊢ z = w",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z✝ w✝ : ℍ\nr : ℝ\nz w : ℍ\nh : dist z w = 0\n⊢ z = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 408,
"column": 48
} | {
"line": 408,
"column": 76
} | {
"line": 408,
"column": 77
} | [
{
"pp": "x : ℝ\n⊢ 0 < sinh x ↔ 0 < x",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ 0 < sinh x ↔ 0 < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 411,
"column": 51
} | {
"line": 411,
"column": 79
} | {
"line": 411,
"column": 80
} | [
{
"pp": "x : ℝ\n⊢ sinh x ≤ 0 ↔ x ≤ 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ sinh x ≤ 0 ↔ x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 414,
"column": 48
} | {
"line": 414,
"column": 76
} | {
"line": 414,
"column": 77
} | [
{
"pp": "x : ℝ\n⊢ sinh x < 0 ↔ x < 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ sinh x < 0 ↔ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 417,
"column": 51
} | {
"line": 417,
"column": 79
} | {
"line": 417,
"column": 80
} | [
{
"pp": "x : ℝ\n⊢ 0 ≤ sinh x ↔ 0 ≤ x",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ 0 ≤ sinh x ↔ 0 ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 50
} | {
"line": 223,
"column": 51
} | [
{
"pp": "case hab\nz w : ℍ\n⊢ dist { re := 0, im := z.im } { re := 0, im := w.im } ≤ dist ↑z ↑w",
"ppTerm": "?hab",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.partialOrder",
"Real",
"Real.lattice",
"C... | [
"case hab\nz w : ℍ\n⊢ |z.im - w.im| ≤ dist ↑z ↑w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Matrix | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 29
} | {
"line": 154,
"column": 30
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nA : Type u_4\ninst✝ : CommRing A\nx : GL n A\n⊢ x ∈ Set.range ⇑toGL ↔ x ∈ ⇑GeneralLinearGroup.det ⁻¹' {1}",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MulOne.toOne",
"Matri... | [
"n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nA : Type u_4\ninst✝ : CommRing A\nx : GL n A\n⊢ (∃ y, ↑y = ↑x) ↔ (↑x).det = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.Matrix | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 48
} | {
"line": 158,
"column": 49
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T0Space R\n⊢ IsClosed[Units.instTopologicalSpaceUnits] (Set.range ⇑toGL)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : TopologicalSpace R\ninst✝¹ : IsTopologicalRing R\ninst✝ : T0Space R\n⊢ IsClosed[Units.instTopologicalSpaceUnits] (⇑GeneralLinearGroup.det ⁻¹' {1})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 32
} | {
"line": 152,
"column": 33
} | [
{
"pp": "X : Type u_1\nx : OnePoint X\n⊢ x ≠ ∞ ↔ ∃ y, ↑y = x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"OnePoint.some",
"Exists",
"Ne",
"OnePoint.rec",
"Iff",
"Eq",
"OnePoint"
],
"usedFVars": [
"X",
... | [
"case infty\nX : Type u_1\n⊢ ∞ ≠ ∞ ↔ ∃ y, ↑y = ∞",
"case coe\nX : Type u_1\nx✝ : X\n⊢ ↑x✝ ≠ ∞ ↔ ∃ y, ↑y = ↑x✝"
] | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 157,
"column": 74
} | {
"line": 158,
"column": 58
} | {
"line": 160,
"column": 0
} | [
{
"pp": "X : Type u_1\nx : OnePoint X\n⊢ x ∉ range some ↔ x = ∞",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OnePoint.infty",
"congrArg",
"Compl.compl",
"Iff.rfl",
"OnePoint.compl_range_coe",
"OnePoint.some",
"Membership.mem",
... | [] | by
rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 29
} | {
"line": 208,
"column": 30
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ\nhs : IsOpen[inst✝] (some ⁻¹' s)\nhmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ\nht : IsOpen[inst✝] (some ⁻¹' t)\nhms' : ∞ ∈ s\nhmt' : ∞ ∈ t\n⊢ IsCompact (some ⁻¹' (s ∩ t))ᶜ",
"ppTerm": "?m... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ns t : Set (OnePoint X)\nhms : ∞ ∈ s → IsCompact (some ⁻¹' s)ᶜ\nhs : IsOpen[inst✝] (some ⁻¹' s)\nhmt : ∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ\nht : IsOpen[inst✝] (some ⁻¹' t)\nhms' : ∞ ∈ s\nhmt' : ∞ ∈ t\n⊢ IsCompact ((some ⁻¹' s)ᶜ ∪ (some ⁻¹' t)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 50
} | {
"line": 245,
"column": 51
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ IsOpen[instTopologicalSpace] (some '' s) ↔ IsOpen[inst✝] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OnePoint.infty_notMem_image_coe",
"OnePoint.some",
"id",
"One... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ IsOpen[inst✝] (some ⁻¹' some '' s) ↔ IsOpen[inst✝] s"
] | isOpen_iff_of_notMem infty_notMem_image_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 32
} | {
"line": 472,
"column": 33
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : OnePoint X\n⊢ Inseparable x y ↔ x = ∞ ∧ y = ∞ ∨ ∃ x', x = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"OnePoint.some",
"Exists",
"OnePoint.rec",
... | [
"case infty\nX : Type u_1\ninst✝ : TopologicalSpace X\ny : OnePoint X\n⊢ Inseparable ∞ y ↔ ∞ = ∞ ∧ y = ∞ ∨ ∃ x', ∞ = ↑x' ∧ ∃ y', y = ↑y' ∧ Inseparable x' y'",
"case coe\nX : Type u_1\ninst✝ : TopologicalSpace X\ny : OnePoint X\nx✝ : X\n⊢ Inseparable (↑x✝) y ↔ ↑x✝ = ∞ ∧ y = ∞ ∨ ∃ x', ↑x✝ = ↑x' ∧ ∃ y', y = ↑y' ∧ In... | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 32
} | {
"line": 531,
"column": 33
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝¹ : WeaklyLocallyCompactSpace X\ninst✝ : R1Space X\nkey : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)\nx y : OnePoint X\n⊢ x ⤳ y ∨ Disjoint (𝓝 x) (𝓝 y)",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
... | [
"case infty\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝¹ : WeaklyLocallyCompactSpace X\ninst✝ : R1Space X\nkey : ∀ (z : X), Disjoint (𝓝 ↑z) (𝓝 ∞)\ny : OnePoint X\n⊢ ∞ ⤳ y ∨ Disjoint (𝓝 ∞) (𝓝 y)",
"case coe\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ns : ... | induction x using OnePoint.rec | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 554,
"column": 2
} | {
"line": 555,
"column": 47
} | {
"line": 555,
"column": 48
} | [
{
"pp": "case h\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : Infinite X\ninst✝ : DiscreteTopology X\ninhabited_h : Inhabited X\n⊢ ¬Tendsto (⇑CofiniteTopology.of.symm) (𝓝 (CofiniteTopology.of ↑default))\n (𝓝 (CofiniteTopology.of.symm (CofiniteTopology.of ↑default)))",
"ppTerm": "?h",
"assi... | [
"case h\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : Infinite X\ninst✝ : DiscreteTopology X\ninhabited_h : Inhabited X\n⊢ {x | ¬x = CofiniteTopology.of ↑default}.Infinite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 570,
"column": 6
} | {
"line": 570,
"column": 64
} | {
"line": 572,
"column": 0
} | [
{
"pp": "case coe\nX✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ns : Set (OnePoint X✝)\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\ny : OnePoint X\nx✝¹ : y ∈ univ\nval : X\nx✝ : ↑val ∈ univ\nhxy : ↑val ≠ y\n⊢ IsOpen[inst✝¹] (some ⁻¹' {↑val})",
"ppTerm": "?coe",
"a... | [] | exacts [isOpen_discrete _, (Option.some_ne_none val).symm] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 587,
"column": 28
} | {
"line": 587,
"column": 39
} | {
"line": 587,
"column": 40
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nN : Set Y\nhN : N ∈ 𝓝 y\nU : Set Y\nhU₁ : U ⊆ N\nhU₂ : IsOpen[inst✝²] U\nhU... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nN : Set Y\nhN : N ∈ 𝓝 y\nU : Set Y\nhU₁ : U ⊆ N\nhU₂ : IsOpen[inst✝²] U\nhU₃ : y ∈ U\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 597,
"column": 34
} | {
"line": 597,
"column": 50
} | {
"line": 597,
"column": 51
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\n⊢ f p ≠ y",
"ppTerm"... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\n⊢ ¬f p = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 598,
"column": 10
} | {
"line": 598,
"column": 26
} | {
"line": 598,
"column": 27
} | [
{
"pp": "case coe\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\nhp : f p ≠ y\n... | [
"case coe\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\np : X\nhp : f p ≠ y\n⊢ ⋯.choose =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 603,
"column": 10
} | {
"line": 603,
"column": 26
} | {
"line": 603,
"column": 27
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\nhq : q ≠ y\nhq... | [
"case inr\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\nhq : q ≠ y\nhq' : q ∈ rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 599,
"column": 27
} | {
"line": 603,
"column": 42
} | {
"line": 603,
"column": 43
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ns : Set (OnePoint X)\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactSpace Y\ny : Y\nf : X → Y\nhf : IsEmbedding f\nhy : range f = {y}ᶜ\n_i : T2Space X\nthis : Tendsto f (coclosedCompact X) (𝓝 y)\nq : Y\n⊢ (fun p ↦ p.elim y f) (... | [] | by
rcases eq_or_ne q y with rfl | hq
· simp
· have hq' : q ∈ range f := by simpa [hy]
simpa [hq] using hq'.choose_spec | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 300,
"column": 2
} | {
"line": 302,
"column": 45
} | {
"line": 304,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyGeneratedSpace X\ns : Set X\nhs : ∀ ⦃K : Set X⦄, IsCompact K → IsOpen[inst✝¹] (s ∩ K)\n⊢ IsOpen[inst✝¹] s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"congrArg",
"Set.prei... | [] | refine isOpen' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 300,
"column": 2
} | {
"line": 302,
"column": 45
} | {
"line": 304,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyGeneratedSpace X\ns : Set X\nhs : ∀ ⦃K : Set X⦄, IsCompact K → IsOpen[inst✝¹] (s ∩ K)\n⊢ IsOpen[inst✝¹] s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"congrArg",
"Set.prei... | [] | refine isOpen' fun K _ _ _ f hf ↦ ?_
rw [← Set.preimage_inter_range]
exact (hs (isCompact_range hf)).preimage hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 335,
"column": 6
} | {
"line": 335,
"column": 21
} | {
"line": 335,
"column": 22
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nh : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed[inst✝¹] (s ∩ K)) → IsClosed[inst✝¹] s\ns : Set X\nhs :\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] [T2Space K] (f : K → X),\n Continuous[_, inst✝¹] f → IsClosed (f... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nh : ∀ (s : Set X), (∀ (K : Set X), IsCompact K → IsClosed[inst✝¹] (s ∩ K)) → IsClosed[inst✝¹] s\ns : Set X\nhs :\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] [T2Space K] (f : K → X),\n Continuous[_, inst✝¹] f → IsClosed (f ⁻¹' s)\nK :... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 31
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : a = ⊤\nr : ℝ\n⊢ 0 r ≤ proximity f a r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"R... | [
"case pos\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : a = ⊤\nr : ℝ\n⊢ 0 ≤ circleAverage (fun x ↦ log⁺ ‖f x‖) 0 r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 30
} | {
"line": 149,
"column": 31
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : ¬a = ⊤\nr : ℝ\n⊢ 0 r ≤ proximity f a r",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",... | [
"case neg\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\na : WithTop E\nh : ¬a = ⊤\nr : ℝ\n⊢ 0 ≤ circleAverage (fun x ↦ log⁺ ‖f x - a.untop₀‖⁻¹) 0 r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 51
} | {
"line": 85,
"column": 52
} | [
{
"pp": "case inr\nK : Set ℍ\nhK : IsCompact K\nhKne : K.Nonempty\nδ : ℝ\nhδ : δ > 0\ng : SL(2, ℝ)\nhg : g • I ∈ K\nhδK : Complex.normSq (denom ((Matrix.SpecialLinearGroup.mapGL ℝ) g) ↑I) ≤ 1 / δ\n⊢ ↑g 1 0 ^ 2 + ↑g 1 1 ^ 2 ≤ 1 / δ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"case inr\nK : Set ℍ\nhK : IsCompact K\nhKne : K.Nonempty\nδ : ℝ\nhδ : δ > 0\ng : SL(2, ℝ)\nhg : g • I ∈ K\nhδK : Complex.normSq (denom ((Matrix.SpecialLinearGroup.mapGL ℝ) g) ↑I) ≤ 1 / δ\n⊢ ↑g 1 0 * ↑g 1 0 + ↑g 1 1 * ↑g 1 1 ≤ δ⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.Basic | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | {
"line": 202,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\n⊢ proximity (f₁ + f₂) ⊤ ≤ proximity f₁ ⊤ + proximity f₂ ⊤ + fun x ↦ log 2",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\n⊢ proximity (f₁ + f₂) ⊤ ≤ proximity f₁ ⊤ + proximity f₂ ⊤ + fun x ↦ log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.ProperAction | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 30
} | {
"line": 101,
"column": 2
} | [
{
"pp": "K : Set ℍ\nhK : IsCompact K\n⊢ IsCompact ((fun g ↦ g • I) ⁻¹' K)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"Matrix",
"Matrix.SpecialLinearGro... | [
"K : Set ℍ\nhK : IsCompact K\nA : ℝ\nhA : ∀ (g : SL(2, ℝ)), g • I ∈ K → ↑g 0 0 ^ 2 + ↑g 0 1 ^ 2 ≤ A\n⊢ IsCompact ((fun g ↦ g • I) ⁻¹' K)"
] | obtain ⟨A, hA⟩ := absq_le hK | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 138,
"column": 16
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Meromorphic",... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\n⊢ Meromorphic f₁ ∧ Meromorphic f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.CharacteristicFunction | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\nh_meromorphic : ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)\n⊢ characteristic (f₁ + f₂) ⊤ r ≤ characteristic f₁ ⊤ r + characteristic f₂ ⊤ r + log 2",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf₁ f₂ : ℂ → E\nr : ℝ\nh₁f₁ : Meromorphic f₁\nh₁f₂ : Meromorphic f₂\nhr : 1 ≤ r\nh_meromorphic : ∀ a ∈ Finset.univ, Meromorphic (![f₁, f₂] a)\n⊢ characteristic (f₁ + f₂) ⊤ r ≤ characteristic f₁ ⊤ r + characteristic f₂ ⊤ r + log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 113,
"column": 16
} | {
"line": 113,
"column": 27
} | {
"line": 113,
"column": 28
} | [
{
"pp": "f : ℂ → ℂ\nh : Meromorphic f\nR : ℝ\n⊢ ‖(characteristic f ⊤ - characteristic f⁻¹ ⊤) R‖ ≤ max |log ‖f 0‖| |log ‖meromorphicTrailingCoeffAt f 0‖| * ‖1 R‖",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | [
"f : ℂ → ℂ\nh : Meromorphic f\nR : ℝ\n⊢ |characteristic f ⊤ R - characteristic f⁻¹ ⊤ R| ≤ |log ‖f 0‖| ∨\n |characteristic f ⊤ R - characteristic f⁻¹ ⊤ R| ≤ |log ‖meromorphicTrailingCoeffAt f 0‖|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 147,
"column": 6
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nr : ℝ\nh✝ : Meromorphic f\nh₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r\nh₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r\nθ : ℂ\nhθ : θ ∈ sphere 0 |r|\nh : 0 ≤ log⁺ ‖f θ‖ - log⁺ ‖f θ - a₀‖\n⊢ |log... | [
"case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nr : ℝ\nh✝ : Meromorphic f\nh₁f : CircleIntegrable (fun x ↦ log⁺ ‖f x‖) 0 r\nh₂f : CircleIntegrable (fun x ↦ log⁺ ‖f x - a₀‖) 0 r\nθ : ℂ\nhθ : θ ∈ sphere 0 |r|\nh : 0 ≤ log⁺ ‖f θ‖ - log⁺ ‖f θ - a₀‖\n⊢ log⁺ ‖f θ‖ ≤ log... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.FirstMainTheorem | {
"line": 164,
"column": 16
} | {
"line": 164,
"column": 27
} | {
"line": 164,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nh : Meromorphic f\nR : ℝ\n⊢ ‖(characteristic f ⊤ - characteristic (fun x ↦ f x - a₀) ⊤) R‖ ≤ (log⁺ ‖a₀‖ + log 2) * ‖1 R‖",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\na₀ : E\nf : ℂ → E\nh : Meromorphic f\nR : ℝ\n⊢ |characteristic f ⊤ R - characteristic (fun x ↦ f x - a₀) ⊤ R| ≤ log⁺ ‖a₀‖ + log 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 60
} | {
"line": 56,
"column": 61
} | [
{
"pp": "f : ℂ → ℂ\nR β : ℝ\n⊢ IntervalIntegrable (fun x ↦ cartanKernel f R x β) volume 0 (2 * π)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"HMul.hMul",
"Real.instZero",
"IntervalIntegrable",
"ValueDistribution.Cartan.cartanKerne... | [
"f : ℂ → ℂ\nR β : ℝ\n⊢ IntervalIntegrable (fun x ↦ log ‖f (circleMap 0 R β) - circleMap 0 1 x‖) volume 0 (2 * π)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 30
} | {
"line": 89,
"column": 31
} | [
{
"pp": "case neg.inr.inl\nf : ℂ → ℂ\nh : ¬¬MeromorphicAt f 0\nhzero : meromorphicOrderAt f 0 = 0\n⊢ CircleIntegrable (fun x ↦ log ‖meromorphicTrailingCoeffAt f 0 - x‖) 0 1",
"ppTerm": "?neg.inr.inl✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"case neg.inr.inl\nf : ℂ → ℂ\nh : ¬¬MeromorphicAt f 0\nhzero : meromorphicOrderAt f 0 = 0\n⊢ CircleIntegrable (fun x ↦ log ‖x - meromorphicTrailingCoeffAt f 0‖) 0 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 13
} | {
"line": 157,
"column": 14
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : ¬R = 0\nx✝ : ℂ\n⊢ logCounting f (↑x✝) R =\n (((fun a ↦ circleAverage (fun x ↦ log ‖f x - a‖) 0 ?neg.convert_1✝) + fun x ↦ logCounting f ⊤ R) - fun a ↦\n log ‖meromorphicTrailingCoeffAt (fun x ↦ ?neg.convert_2✝ x - a) 0‖)\n x✝",
"ppTerm": "?m.... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : ¬R = 0\nx✝ : ℂ\n⊢ logCounting f (↑x✝) R =\n circleAverage (fun x ↦ log ‖f x - x✝‖) 0 ?neg.convert_1✝ + logCounting f ⊤ R -\n log ‖meromorphicTrailingCoeffAt (fun x ↦ ?neg.convert_2✝ x - x✝) 0‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 67,
"column": 2
} | {
"line": 69,
"column": 53
} | {
"line": 71,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nhs : s.Subsingleton\nl : ℝ≥0\n⊢ HasConstantSpeedOnWith f s l",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"sub_self",
"ENNReal.ofReal",
"congrArg"... | [] | rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ConstantSpeed | {
"line": 67,
"column": 2
} | {
"line": 69,
"column": 53
} | {
"line": 71,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nhs : s.Subsingleton\nl : ℝ≥0\n⊢ HasConstantSpeedOnWith f s l",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"sub_self",
"ENNReal.ofReal",
"congrArg"... | [] | rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 40
} | {
"line": 131,
"column": 41
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nthis : Measurable f\n⊢ Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (uIoc 0 (2 * π))).prod (volume.restrict (uIoc 0 (2 * π))))",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Norme... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nthis : Measurable f\n⊢ Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 9
} | {
"line": 145,
"column": 10
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ IntegrableOn (fun x ↦ ∫ (α : ℝ) in 0..2 * π, cartanKernel f R α x) (Ioc 0 (2 * π)) volume",
"ppTerm": "?m.91",
"assign... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ Integrable (fun x ↦ ∫ (α : ℝ) in Ioc 0 (2 * π), log ‖f (circleMap 0 R x) - circleMap 0 1 α‖ ∂volume)\n (volume.restrict (Ioc 0 (2 * π))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 104,
"column": 4
} | {
"line": 120,
"column": 10
} | {
"line": 122,
"column": 0
} | [
{
"pp": "E✝ : Type u_1\ninst✝² : NormedAddCommGroup E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD₁ D₂ : locallyFinsupp E ℤ\n⊢ (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) (D₁ + D₂)) z) * log (r * ‖z‖⁻¹) + ↑((D₁ + D₂) 0) * log r) =\n (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * lo... | [] | simp only [map_add, coe_add, Pi.add_apply, Int.cast_add]
ext r
have {A B C D : ℝ} : A + B + (C + D) = A + C + (B + D) := by ring
rw [Pi.add_apply, this]
congr 1
· have h₁s : ((D₁.toClosedBall r).support ∪ (D₂.toClosedBall r).support).Finite := by
apply Set.finite_union.2
constructor
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.Proximity.IntegralPresentation | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 9
} | {
"line": 157,
"column": 10
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ IntegrableOn (fun x ↦ ∫ (β : ℝ) in 0..2 * π, cartanKernel f R x β) (Ioc 0 (2 * π)) volume",
"ppTerm": "?m.91",
"assign... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nh_int :\n Integrable (fun p ↦ cartanKernel f R p.1 p.2)\n ((volume.restrict (Ioc 0 (2 * π))).prod (volume.restrict (Ioc 0 (2 * π))))\n⊢ Integrable (fun x ↦ ∫ (β : ℝ) in Ioc 0 (2 * π), log ‖f (circleMap 0 R β) - circleMap 0 1 x‖ ∂volume)\n (volume.restrict (Ioc 0 (2 * π))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 104,
"column": 4
} | {
"line": 120,
"column": 10
} | {
"line": 122,
"column": 0
} | [
{
"pp": "E✝ : Type u_1\ninst✝² : NormedAddCommGroup E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD₁ D₂ : locallyFinsupp E ℤ\n⊢ (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) (D₁ + D₂)) z) * log (r * ‖z‖⁻¹) + ↑((D₁ + D₂) 0) * log r) =\n (fun r ↦ ∑ᶠ (z : E), ↑(((toClosedBall r) D₁) z) * lo... | [] | simp only [map_add, coe_add, Pi.add_apply, Int.cast_add]
ext r
have {A B C D : ℝ} : A + B + (C + D) = A + C + (B + D) := by ring
rw [Pi.add_apply, this]
congr 1
· have h₁s : ((D₁.toClosedBall r).support ∪ (D₂.toClosedBall r).support).Finite := by
apply Set.finite_union.2
constructor
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 79,
"column": 49
} | {
"line": 79,
"column": 66
} | {
"line": 79,
"column": 67
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\nha : a > 0\nb c : ℝ\nhc : ∀ (b : ℝ), c ≤ b → ‖1 b‖ ≤ a * ‖logCounting (single e 1) b‖\nℓ : ℝ := 1 + max ‖e‖ (max |b| |c|)\nh₁ℓ : c ≤ ℓ\nh₂ℓ : 1... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\nha : a > 0\nb c : ℝ\nhc : ∀ (b : ℝ), c ≤ b → ‖1 b‖ ≤ a * ‖logCounting (single e 1) b‖\nℓ : ℝ := 1 + max ‖e‖ (max |b| |c|)\nh₁ℓ : c ≤ ℓ\nh₂ℓ : 1 ≤ ℓ\n⊢ 1 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Lindelof | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 71
} | {
"line": 83,
"column": 72
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous ... | [
"X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous f\nh𝓕 : ∀ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 27
} | {
"line": 216,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\nb : ℝ\nhb : b ∈ Ioi ‖e‖\nhab : a ≤ b\n⊢ a ∈ Ioi 0",
"ppTerm": "?m.178",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\nb : ℝ\nhb : b ∈ Ioi ‖e‖\nhab : a ≤ b\n⊢ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 213,
"column": 2
} | {
"line": 216,
"column": 55
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\n⊢ MonotoneOn (logCounting (D - single e 1)) (Ioi ‖e‖)",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup"... | [] | · intro a ha b hb hab
apply logCounting_mono _ _ ((norm_nonneg e).trans_lt hb) hab
· simp [hD]
· simpa [mem_Ioi] using (norm_nonneg e).trans_lt ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 226,
"column": 54
} | {
"line": 226,
"column": 65
} | {
"line": 226,
"column": 66
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\nthis : ∀ (z : E), 0 ≤ ↑(((toClosedBall r) f) z) * log (r * ‖z‖⁻¹)\n⊢ 0 ≤ ↑(f 0)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Int.cast_no... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\nthis : ∀ (z : E), 0 ≤ ↑(((toClosedBall r) f) z) * log (r * ‖z‖⁻¹)\n⊢ 0 ≤ f 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 112,
"column": 4
} | {
"line": 117,
"column": 89
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case inl.inr\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) = ENNRea... | [
"case inl.inr\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... | have : (s ∪ t) ∩ Icc z y = s ∩ Icc z x ∪ t ∩ Icc x y := by
ext w; constructor
· rintro ⟨ws | wt, zw, wy⟩
exacts [Or.inl ⟨ws, zw, hs.2 ws⟩, Or.inr ⟨wt, ht.2 wt, wy⟩]
· rintro (⟨ws, zw, wx⟩ | ⟨wt, xw, wy⟩)
exacts [⟨Or.inl ws, zw, wx.trans (ht.2 yt)⟩, ⟨Or.inr wt, (hs.2 zs).trans xw, wy⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 23
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case pos.refine_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ ↑(((toClosedBall r) f) a)",
"ppTerm": "?pos.refine_1✝",
"assigned": true,
"usedCon... | [
"case pos.refine_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 86
} | {
"line": 233,
"column": 87
} | [
{
"pp": "case pos.refine_2\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 1 ≤ r * ‖a‖⁻¹",
"ppTerm": "?pos.refine_2✝",
"assigned": true,
"usedConstants": [
... | [
"case pos.refine_2\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ ‖a‖ ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 14
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf₁ f₂ : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f₂ - f₁\nhr : 1 ≤ r\n⊢ 0 ≤ logCounting f₂ r - logCounting f₁ r",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf₁ f₂ : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f₂ - f₁\nhr : 1 ≤ r\n⊢ logCounting f₁ r ≤ logCounting f₂ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 39
} | {
"line": 277,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 39
} | {
"line": 277,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 39
} | {
"line": 277,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf g : 𝕜 → E\na : WithTop E\na₀ : E\nh : a = ⊤\n⊢ ℝ → ℝ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | exact (divisor f univ)⁻.logCounting | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 137,
"column": 36
} | {
"line": 137,
"column": 47
} | {
"line": 137,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub : ↑t ⊆ su... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub : ↑t ⊆ support D\nhtc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 32
} | {
"line": 139,
"column": 33
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub... | [
"case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub : ↑t ⊆ supp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 32
} | {
"line": 360,
"column": 33
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : e = ⊤\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [
"case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : e = ⊤\n⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor f univ)⁻) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 32
} | {
"line": 360,
"column": 33
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : ¬e = ⊤\n⊢ MonotoneOn (logCounting f e) (Ioi 0)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants":... | [
"case neg\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ne : WithTop E\nh : ¬e = ⊤\n⊢ MonotoneOn (locallyFinsuppWithin.logCounting (divisor (fun x ↦ f x - e.untop₀) univ)⁺) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 368,
"column": 4
} | {
"line": 369,
"column": 43
} | {
"line": 370,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 368,
"column": 4
} | {
"line": 369,
"column": 43
} | {
"line": 370,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 368,
"column": 4
} | {
"line": 369,
"column": 43
} | {
"line": 370,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nf : 𝕜 → E\ne : WithTop E\nhr : 1 ≤ r\nh : e = ⊤\n⊢ 0 ≤ logCounting f e r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [] | simp [logCounting, h, locallyFinsuppWithin.logCounting_nonneg
(negPart_nonneg (divisor f univ)) hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 20
} | [
{
"pp": "case a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub :... | [
"case a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nh : 0 ≤ D\nhO : logCounting D =O[atTop] log\nC : ℝ\nhC : ∀ᶠ (x : ℝ) in atTop, ‖logCounting D x‖ ≤ C * ‖log x‖\nN : ℕ\nhCN : max C 0 < ↑N\nhCN' : C < ↑N\nhInf : (support D).Infinite\nt : Finset E\nhtsub : ↑t ⊆ suppor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 590,
"column": 4
} | {
"line": 590,
"column": 19
} | {
"line": 592,
"column": 0
} | [
{
"pp": "case e_a.hf\nR : ℝ\nf : ℂ → ℂ\nh : Meromorphic f\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall 0 |R|)\n⊢ MeromorphicOn f (closedBall 0 |R|)",
"ppTerm": "?e_a.hf✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁f"
],
"usedGoals": []
},
{
"pp": "case e_a.... | [] | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.ConstantSpeed | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 19
} | {
"line": 166,
"column": 20
} | [
{
"pp": "case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nh : eVariationOn f s = 0\nx : ℝ\nx✝¹ : x ∈ s\ny : ℝ\nx✝ : y ∈ s\n⊢ eVariationOn f (s ∩ Icc x y) = 0",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nh : eVariationOn f s = 0\nx : ℝ\nx✝¹ : x ∈ s\ny : ℝ\nx✝ : y ∈ s\n⊢ eVariationOn f (s ∩ Icc x y) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hall.Finite | {
"line": 62,
"column": 33
} | {
"line": 62,
"column": 50
} | {
"line": 62,
"column": 51
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nx : ι\na : α\ns' : Finset ↑{x' | x' ≠ x}\nthis : DecidableEq ι\nha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)\nhe : s'.Nonempty\nh : image (fun z ↦ ↑z) s' = univ\n⊢ False"... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nx : ι\na : α\ns' : Finset ↑{x' | x' ≠ x}\nthis : DecidableEq ι\nha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)\nhe : s'.Nonempty\nh : image (fun z ↦ ↑z) s' = univ\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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