module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : AddZeroClass β\ninst✝ : ContinuousAdd β\nl : Filter α\nf g : α → β\nhf : l.ZeroAtFilter f\nhg : l.ZeroAtFilter g\n⊢ l.ZeroAtFilter (f + g)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : SubtractionMonoid β\ninst✝ : ContinuousNeg β\nl : Filter α\nf : α → β\nhf : l.ZeroAtFilter f\n⊢ l.ZeroAtFilter (-f)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 51,
"column": 59
} | {
"line": 51,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SMulZeroClass 𝕜 β\ninst✝ : ContinuousConstSMul 𝕜 β\nl : Filter α\nf : α → β\nc : 𝕜\nhf : l.ZeroAtFilter f\n⊢ l.ZeroAtFilter (c • f)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedAddCommGroup β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f + g)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : f =o[l] fun _x ↦ 1\nhg : l.BoundedAtFilter g\n⊢ (f * g) =o[l] fun _x ↦ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"Seminorme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 13
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedRing β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : g =o[l] fun _x ↦ 1\n⊢ (f * g) =o[l] fun _x ↦ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"Seminorme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 135,
"column": 24
} | {
"line": 135,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : SeminormedCommRing 𝕜\ninst✝² : SeminormedRing β\ninst✝¹ : Algebra 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nl : Filter α\nf g : α → β\nhf : f ∈ boundedFilterSubmodule 𝕜 l\nhg : g ∈ boundedFilterSubmodule 𝕜 l\n⊢ f * g ∈ boundedFilterSubmodule 𝕜 l",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 67,
"column": 37
} | {
"line": 67,
"column": 76
} | [
{
"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\nv : ℂ\nhv : v ∈ ball (f z₀) (ε / 2)\nh1 : DiffContOnCl ℂ (fun z ↦ f z - v) (ball z₀ r)\nh2 : ContinuousOn (fun z ↦ ‖f z - v‖) (closedBall z₀ r)\nh3 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | [
{
"pp": "h : ℝ\nhh : 0 < h\nA : ℝ\nz : ℂ\n⊢ -2 * π < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 13
} | [
{
"pp": "h : ℝ\nhh : 0 < h\n⊢ -2 * π < 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 50
} | [
{
"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\n⊢ ∃ ρ > 0, AnalyticOnNhd ℂ f (closedBall z₀ ρ) ∧ ∀ z ∈ closedBall z₀ ρ, z ≠ z₀ → f z ≠ f z₀",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 29
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : h ≠ 0\nhf : Periodic f ↑h\nz : ℂ\nthis : cuspFunction h f (𝕢 h z) = f (invQParam h (𝕢 h z))\nm : ℤ\nhm : invQParam h (𝕢 h z) = z + ↑m * ↑h\n⊢ cuspFunction h f (𝕢 h z) = f z",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Function.Periodic.qParam",
"HMu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 37
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : h ≠ 0\nhf : Periodic f ↑h\nz : ℂ\nhol_z : DifferentiableAt ℂ f z\nq : ℂ := 𝕢 h z\n⊢ HasStrictDerivAt (𝕢 h) (q * (2 * ↑π * I / ↑h)) z",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 51
} | [
{
"pp": "case convert_3\na b t : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (1 + ↑x ^ 2) ^ ↑s\n⊢ ↑t ≠ -1",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Complex.ofReal_neg",
"id",
"Complex.ofReal_one",
"Ne",
"Complex.ofReal",
"Real.inst... | rw [← ofReal_one, ← ofReal_neg, Ne, ofReal_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Periodic | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 46
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nh_zer : I∞.ZeroAtFilter f\n⊢ cuspFunction h f 0 = 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Function.Periodic.invQParam",
"congrArg",
"Function.Periodic.cuspFunction",
"Compl.compl",
"nhdsWithin",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 436,
"column": 18
} | {
"line": 436,
"column": 51
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 46
} | [
{
"pp": "case h\nh : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nh_bd : I∞.BoundedAtFilter f\nc : ℝ\nt : ∀ᶠ (x : ℂ) in 𝓝[≠] 0, DifferentiableAt ℂ (cuspFunction h f) x ∧ ‖cuspFunction h f x‖ ≤ c\nS : Set ℂ\nhS1 : ∀ y ∈ S, y ∈ {0}ᶜ → DifferentiableAt ℂ (cuspFun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 438,
"column": 4
} | {
"line": 438,
"column": 30
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (-cos) (sin x) x",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 67
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nh_bd : I∞.BoundedAtFilter f\nthis : Tendsto (cuspFunction h f) (𝓝[≠] 0) (𝓝 (cuspFunction h f 0))\n⊢ Tendsto f I∞ (𝓝 (cuspFunction h f 0))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 64
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nh_bd : I∞.BoundedAtFilter f\n⊢ (fun z ↦ f z - cuspFunction h f 0) =O[I∞] fun z ↦ rexp (-2 * π * z.im / h)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 132,
"column": 20
} | {
"line": 132,
"column": 80
} | [
{
"pp": "E : 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}\nz : E\nhz : z ∈ sphere 0 1\nt : ℂ\nht : t ∈ ball ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 67
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nh_zer : I∞.ZeroAtFilter f\n⊢ f =O[I∞] fun z ↦ rexp (-2 * π * z.im / h)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 15
} | [
{
"pp": "case h.inl\nh : ℝ\nf : ℂ → ℂ\nhfcts : ContinuousAt (update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h))) 0\na : ℂ\n⊢ update ((a • f) ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder ((a • f) ∘ invQParam h)) 0 =\n (a • update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h))) 0",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 262,
"column": 41
} | {
"line": 262,
"column": 52
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhfcts : ContinuousAt (update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h))) 0\na : ℂ\n⊢ Tendsto (fun x ↦ f (invQParam h x)) (𝓝[≠] 0) (𝓝 ((𝓝[≠] 0).limUnder (f ∘ invQParam h)))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 29
} | [
{
"pp": "E : 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 := ⋯\ngray : E → ℂ → ℂ := ⋯\nr : ℝ\nhr : r > 0\nhgr : ball z₀ r ⊆ {x | AnalyticAt ℂ g x}\nh : ∀ z ∈ sphere 0 1, ∀ᶠ (t : ℂ) in 𝓝 0, gray z t = gray z 0\nz : E\nhz : z ∈ ball ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 148,
"column": 33
} | {
"line": 148,
"column": 59
} | [
{
"pp": "E : 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) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 13
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhfcts : ContinuousAt (cuspFunction h f) 0\n⊢ cuspFunction h (-f) = -cuspFunction h f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 150,
"column": 4
} | {
"line": 151,
"column": 42
} | [
{
"pp": "case pos.h\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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | [
{
"pp": "E : 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}\nz : E\nhz : z ∈ sphere 0 1\nhrz : ¬∀ᶠ (t : ℂ) in ... | simp [gray, ray] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | [
{
"pp": "E : 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}\nz : E\nhz : z ∈ sphere 0 1\nhrz : ¬∀ᶠ (t : ℂ) in ... | simp [gray, ray] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | [
{
"pp": "E : 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}\nz : E\nhz : z ∈ sphere 0 1\nhrz : ¬∀ᶠ (t : ℂ) in ... | simp [gray, ray] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Periodic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 15
} | [
{
"pp": "case h.inr\nh : ℝ\nf g : ℂ → ℂ\nhfcts : ContinuousAt (cuspFunction h f) 0\nhgcts : ContinuousAt (cuspFunction h g) 0\n⊢ update ((f + g) ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder ((f + g) ∘ invQParam h)) 0 =\n (update (f ∘ invQParam h) 0 ((𝓝[≠] 0).limUnder (f ∘ invQParam h)) +\n update (g ∘ invQPa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 15
} | [
{
"pp": "case neg.h\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}\nz : E\nhz : z ∈ sphere 0 1\nhrz : ¬∀ᶠ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 9
} | [
{
"pp": "h : ℝ\nf g : ℂ → ℂ\nhfcts : ContinuousAt (cuspFunction h f) 0\nhgcts : ContinuousAt (cuspFunction h g) 0\n⊢ cuspFunction h (f - g) = cuspFunction h f - cuspFunction h g",
"usedConstants": [
"Eq.mpr",
"Pi.instNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Function.Per... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 440,
"column": 4
} | {
"line": 448,
"column": 58
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | calc
(∫ 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... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 440,
"column": 4
} | {
"line": 448,
"column": 58
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | calc
(∫ 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... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 440,
"column": 4
} | {
"line": 448,
"column": 58
} | [
{
"pp": "case refine_3\na b : ℝ\nn : ℕ\nC : ℝ := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nhu : ∀ x ∈ [[a, b]], HasDerivAt (fun y ↦ sin y ^ (n + 1)) (↑(n + 1) * cos x * sin x ^ n) x\nhv : ∀ x ∈ [[a, b]], HasDerivAt (-cos) (sin x) x\nH :\n ∫ (x : ℝ) i... | calc
(∫ 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... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 77,
"column": 4
} | {
"line": 89,
"column": 15
} | [] | ∑ x ∈ s, weight x • (z - x) = conj (∑ x ∈ s, P.rootMultiplicity x • (1 / (z - x))) := by
simp only [map_sum, weight, derivRootWeight, if_neg hzP]
refine Finset.sum_congr rfl fun x hx ↦ ?_
have : z - x ≠ 0 := by
rw [sub_ne_zero]
rintro rfl
simp_all [s]
simp [← Complex.conj... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 15
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ (Polynomial.X ^ n).natDegree ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Polynomial.natDegree_X",
"HMul.hMul",
"congrArg",
"NormedDivisionRing.toNormMulClass",
"Complex.instNormedField",
"Nat.instMulOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 262,
"column": 38
} | {
"line": 262,
"column": 74
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nz : { z // z ≠ 0 }\nh : (fun x ↦ x ^ n) 0 = ↑z\n⊢ ↑z = 0",
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat",
"Subtype.val",
"Eq"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 510,
"column": 6
} | {
"line": 510,
"column": 57
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Positivity | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 64
} | [
{
"pp": "f : ℂ → ℂ\nhf : Differentiable ℂ f\nc : ℂ\nh : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f c\nz : ℂ\nhz : 0 ≤ z - c\n⊢ ‖↑(z - c).re‖ < (↑(z - c).re).re + 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Positivity | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 67
} | [
{
"pp": "f : ℂ → ℂ\nc : ℂ\nhf : Differentiable ℂ f\nh : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f c\nz : ℂ\nhz : z ≤ c\nn : ℕ\nhn : n ≠ 0\n⊢ 0 ≤ iteratedDeriv n (fun z ↦ f (-z)) (-c)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 13
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.TaylorSeries | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 50
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nr : ℝ\nhf : DifferentiableOn ℂ f (Metric.ball c r)\nz : ℂ\nhz : z ∈ Metric.ball c r\nr' : NNReal\nhr' : ↑r' < r\nhr'₀ : 0 < ↑r'\nhzr' : z ∈ Metric.ball c ↑r'\n⊢ z - c ∈ Metric.ball 0 ↑r'",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.TaylorSeries | {
"line": 58,
"column": 2
} | {
"line": 60,
"column": 9
} | [
{
"pp": "case h.e'_5.h.h.e'_6\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nr : ℝ\nhf : DifferentiableOn ℂ f (Metric.ball c r)\nz : ℂ\nhz : z ∈ Metric.ball c r\nr' : NNReal\nhr' : ↑r' < r\nhr'₀ : 0 < ↑r'\nhzr' : z ∈ Metric.ball c ↑r'\nhz' : z ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 13
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (cos a ^ 2 - cos b ^ 2) / 2",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"False",
"Real",
"instHDiv",
"HMul.hMul",
"Real.cos",
"Real.instRCLike"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 17
} | [
{
"pp": "case hdR\nρ w : ℂ\nR : ℝ\nhR : 0 < R\nhρ : ‖ρ‖ = R\nhw : ‖w‖ < R\nr : ℕ → ℝ\nhr_lt : ∀ (n : ℕ), r n < R\nhr_tendsto : Tendsto r atTop (𝓝 R)\nbound : ℝ → ℝ := ⋯\nn : ℕ\nhn : (R + ‖w‖) / 2 ≤ r n\nθ : ℝ\nh : ¬‖circleMap 0 R θ - ρ‖ = 0\n⊢ 0 < ‖circleMap 0 R θ - ρ‖",
"usedConstants": [
"AddGroup.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 29
} | [
{
"pp": "case h\nρ w : ℂ\nR : ℝ\nhR : 0 < R\nhρ : ‖ρ‖ = R\nhw : ‖w‖ < R\nr : ℕ → ℝ\nhr_lt : ∀ (n : ℕ), r n < R\nhr_tendsto : Tendsto r atTop (𝓝 R)\nbound : ℝ → ℝ := ⋯\nn : ℕ\nhn : (R + ‖w‖) / 2 ≤ r n\nh_bound : ∀ {θ : ℝ}, ‖herglotzLogIntegrand w ρ (circleMap 0 (r n) θ)‖ ≤ bound θ ∨ ‖circleMap 0 R θ - ρ‖ = 0\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 54
} | [
{
"pp": "case inr.refine_1\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : ¬0 ∉ U\na : ℂ\nha : a ∉ U\nf : ℂ → ℂ\nhf_inj : Injective f\nhf_dense : ¬Dense (f '' (-a +ᵥ U))\nhdf : ∀ z ∈ -a +ᵥ U, deriv f z ≠ 0\n⊢ ¬Dense ((f ∘ fun x ↦ -a + x) '' U)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 60,
"column": 6
} | {
"line": 60,
"column": 59
} | [
{
"pp": "case inr.refine_2\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : ¬0 ∉ U\na : ℂ\nha : a ∉ U\nf : ℂ → ℂ\nhf_inj : Injective f\nhf_dense : ¬Dense (f '' (-a +ᵥ U))\nhdf : ∀ z ∈ -a +ᵥ U, deriv f z ≠ 0\nz : ℂ\nhz : z ∈ U\n⊢ deriv (f ∘ fun x ↦ -a + x) z ≠ 0",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 57
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nz : ℂ\nhz : z ∈ U\nhfz : f z = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 17
} | [
{
"pp": "case hf\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nz : ℂ\nhz : z ∈ U\n⊢ HasStrictDerivAt ?f (2 * f z) (f z)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 17
} | [
{
"pp": "case hf'\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nz : ℂ\nhz : z ∈ U\n⊢ 2 * f z ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 77
} | [
{
"pp": "case refine_1\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nhdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z\n⊢ ¬Dense (f '' U)",
"usedConstants": [
"Eq.mpr",... | simp only [Dense, not_forall, mem_closure_iff_frequently, not_frequently] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 89,
"column": 39
} | {
"line": 89,
"column": 50
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nhdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z\nx : ℂ\nhx : x ∈ U\n⊢ (2 * f x)⁻¹ ≠ 0",
"usedConstants": [
"Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 31
} | [
{
"pp": "w ρ : ℂ\nR : ℝ\nhρ : ‖ρ‖ = R\nhw : ‖w‖ < R\nhR : 0 < R\nr : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)\nhr_lt : ∀ (n : ℕ), r n < R\nhr_pos : ∀ (n : ℕ), 0 < r n\nhr_tendsto : Tendsto r atTop (𝓝 R)\nDCT :\n Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop\n (𝓝 (circleAverage ... | unfold herglotzLogIntegrand | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 34
} | [
{
"pp": "w ρ : ℂ\nR : ℝ\nhρ : ‖ρ‖ = R\nhw : ‖w‖ < R\nhR : 0 < R\nr : ℕ → ℝ := fun n ↦ R - (R - ‖w‖) / (↑n + 2)\nhr_lt : ∀ (n : ℕ), r n < R\nhr_pos : ∀ (n : ℕ), 0 < r n\nhr_tendsto : Tendsto r atTop (𝓝 R)\nDCT :\n Tendsto (fun n ↦ circleAverage (herglotzLogIntegrand w ρ) 0 (r n)) atTop\n (𝓝 (circleAverage ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 45
} | [
{
"pp": "case refine_2\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nhdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z\nz : ℂ\nhz : z ∈ U\n⊢ deriv f z ≠ 0",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 80
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nf : ℂ → ℂ\nhf_inj : Injective f\nhfd : ¬Dense (f '' U)\nhdf : ∀ z ∈ U, deriv f z ≠ 0\n⊢ ∃ x ε, 0 < ε ∧ ∀ a ∈ U, ε < dist (f a) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 126,
"column": 47
} | {
"line": 126,
"column": 58
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nf : ℂ → ℂ\nhf_inj : Injective f\nhfd : ¬Dense (f '' U)\nhdf : ∀ z ∈ U, deriv f z ≠ 0\nx : ℂ\nε : ℝ\nhε₀ : 0 < ε\nhε : ∀ a ∈ U, ε < dist (f a) x\nz : ℂ\nhz : z ∈ U\n⊢ f z ≠ x",
"usedConstants": [
"id",
"Ne",
"Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 32
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nf : ℂ → ℂ\nhf_inj : Injective f\nhfd : ¬Dense (f '' U)\nhdf : ∀ z ∈ U, deriv f z ≠ 0\nx : ℂ\nε : ℝ\nhε₀ : 0 < ε\nhε : ∀ a ∈ U, ε < dist (f a) x\nhfx : ∀ z ∈ U, f z ≠ x\nz : ℂ\nhz : z ∈ U\n⊢ ε < ‖f z - x‖",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 31
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nf : ℂ → ℂ\nhf_inj : Injective f\nhfd : ¬Dense (f '' U)\nhdf : ∀ z ∈ U, deriv f z ≠ 0\nx : ℂ\nε : ℝ\nhε₀ : 0 < ε\nhε : ∀ a ∈ U, ε < dist (f a) x\nhfx : ∀ z ∈ U, f z ≠ x\nz : ℂ\nhz : z ∈ U\n⊢ 0 < ‖f z - x‖",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 56
} | [
{
"pp": "U : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nf : ℂ → ℂ\nhf_inj : Injective f\nhfd : ¬Dense (f '' U)\nhdf : ∀ z ∈ U, deriv f z ≠ 0\nx : ℂ\nε : ℝ\nhε₀ : 0 < ε\nhε : ∀ a ∈ U, ε < dist (f a) x\nhfx : ∀ z ∈ U, f z ≠ x\nz : ℂ\nhz : z ∈ U\nw : ℂ\nhw : w ∈ U\nheq : (fun z ↦ ↑ε / (f z - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 338,
"column": 8
} | {
"line": 338,
"column": 24
} | [
{
"pp": "case h\nc : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := ⋯\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[codiscr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 338,
"column": 8
} | {
"line": 338,
"column": 64
} | [
{
"pp": "case h\nc : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := ⋯\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[codiscr... | simpa [hR] using fun _ ⟨h, _⟩ ↦ ball_subset_closedBall h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 32
} | [
{
"pp": "case h\nc : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f✝ : ¬∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₂f : ∀ (u : ↑(closedBall c |R|)), meromorphicOrderAt f ↑u = ⊤\nthis : divisor f CB = 0\nz : ℂ\nh₁z : MeromorphicNFAt f z\nh₂z : z ∈ CB\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Exp | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 63
} | [
{
"pp": "τ : ℍ\n⊢ ‖cexp (2 * ↑π * Complex.I * ↑τ)‖ < 1",
"usedConstants": [
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Complex.mul_re",
"Real.pi",
"HMul.hMul",
"Left.neg_neg_iff._simp_1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 15
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\nthis : (∑ᶠ (a : ℂ), ↑((divisor f (closedBall c |r|)) a)) * Real.log (R / r) ≤ Real.log (M / ‖f c‖)\n⊢ 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 23
} | [
{
"pp": "case h₂f.h.h.inl\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 80,
"column": 22
} | {
"line": 80,
"column": 33
} | [
{
"pp": "case h\nX₁ : Type u₁\ninst✝⁴ : TopologicalSpace X₁\nX : Type u\ninst✝³ : TopologicalSpace X\ninst✝² : NormalSpace X\ne : X₁ → X\nY : Type v\ninst✝¹ : TopologicalSpace Y\ninst✝ : TietzeExtension Y\nhe : IsClosedEmbedding e\nf : C(X₁, Y)\ne' : X₁ ≃ₜ ↑(Set.range e) := ⋯.toHomeomorph\ng : C(X, Y)\nhg : res... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 52
} | [
{
"pp": "case pos\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)\nx : E\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 33
} | [
{
"pp": "case inr.refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < ‖f‖\nhf3 : -‖f‖ / 3 < ‖f‖ / 3\nhc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 49
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\nh3 : 0 < 3\nh23 : 0 < 2 / 3\nhf : 0 < ‖f‖\nhf3 : -‖f‖ / 3 < ‖f‖ / 3\nhc₁ : IsClosed[inst✝¹] (⇑e '' ⇑f ⁻¹' Iic (-‖f‖ / 3))\nhc₂ : IsClosed[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 72,
"column": 31
} | {
"line": 72,
"column": 84
} | [
{
"pp": "𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nthis✝ : NormedSpace ℝ E\nthis : IsScalarTower ℝ 𝕜 E\ng : E → E := fun x ↦ ‖x‖⁻¹ • x\nx : E\nhx : x ∈ frontier (Metric.closedBall 0 1)\n⊢ ‖x‖ = 1",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 432,
"column": 37
} | {
"line": 432,
"column": 62
} | [
{
"pp": "c : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (closedBall c |R|))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 433,
"column": 12
} | {
"line": 433,
"column": 75
} | [
{
"pp": "case neg.hbc.hxy\nc : ℂ\nr R M : ℝ\nf : ℂ → ℂ\nr_pos : 0 < |r|\nr_lt_R : |r| < |R|\nhM : 1 ≤ M\nh₁f : AnalyticOnNhd ℂ f (closedBall c |R|)\nh₂f : f c ≠ 0\nf_bound : ∀ z ∈ sphere c |R|, ‖f z‖ ≤ M\nhrR : 1 < |R / r|\njensen :\n circleAverage (fun x ↦ Real.log ‖f x‖) c R =\n ∑ᶠ (u : ℂ), ↑((divisor f (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 118,
"column": 51
} | {
"line": 118,
"column": 62
} | [
{
"pp": "X : Type u\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : NormalSpace X\ns : Set X\nhs : IsClosed s\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nf : ↑s →ᵇ E\nhf : ¬‖f‖ = 0\nthis : TietzeExtension ↑(Metric.closedBall 0 ‖f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 125,
"column": 29
} | {
"line": 125,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : NormalSpace X\ns : Set X\nhs : IsClosed s\n𝕜 : Type v\ninst✝³ : RCLike 𝕜\nE : Type w\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nf : ↑s →ᵇ E\nhf : ¬‖f‖ = 0\nthis : TietzeExtension ↑(Metric.closedBall 0 ‖f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 71
} | [
{
"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) /... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 45
} | [
{
"pp": "case refine_1\nX : 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 - con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 326,
"column": 30
} | {
"line": 326,
"column": 41
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ (¬∃ y, (↑g * diagonal fun x ↦ y) = ↑g * m) ∧ m.discr = 0 ↔ (¬∃ y, (diagonal fun x ↦ y) = m) ∧ m.discr = 0",
"usedConstants": [
"Units.val",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hM... | Units.mul_right_inj | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ ((↑g)⁻¹ * m * ↑g).IsParabolic ↔ m.IsParabolic",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 40
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : m.IsParabolic\n⊢ (-m).discr = 0",
"usedConstants": [
"one_pow",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Fintype.card_fin",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : (-m).IsParabolic\n⊢ m.IsParabolic",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 44
} | [
{
"pp": "g : GL (Fin 2) ℝ\nz : ℍ\nhtrace : ↑g 0 0 = -↑g 1 1\nhc : ↑g 1 0 = 0\nh₀ : ↑g 1 1 = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℍ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nthis : ∀ {g : GL (Fin 2) ℝ}, 0 < (↑g).det → ∀ (hell : g.IsElliptic), 0 < ↑g 1 0 → (g • z = z ↔ z = fixedPt g hell)\nhc : ↑g 1 0 < 0\n⊢ g • z = z ↔ z = fixedPt g hell",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 49
} | [
{
"pp": "h : IsCompact univ\n⊢ IsCompact (Complex.im ⁻¹' Ioi 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 37
} | [
{
"pp": "h : IsCompact univ\nthis : IsCompact (Complex.im ⁻¹' Ioi 0)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 17
} | [
{
"pp": "g✝ : GL (Fin 2) ℝ\nz : ℍ\ng : GL (Fin 2) ℝ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nhc : 0 < ↑g 1 0\n⊢ 0 ≤ -(↑g).discr",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"Matrix",
"ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 160,
"column": 31
} | {
"line": 160,
"column": 42
} | [
{
"pp": "a : ℍ\nhw : ¬0 < (↑a).im\n⊢ a.im ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 35
} | [
{
"pp": "case mp\nK : Type u_1\ninst✝¹ : Field K\nm : Matrix (Fin 2) (Fin 2) K\ninst✝ : NeZero 2\n⊢ m.IsParabolic → ∃ a n, m = (scalar (Fin 2)) a + n ∧ n ≠ 0 ∧ n ^ 2 = 0",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Matrix.scalar",
"Matrix.add",
"AddGroupWith... | · exact fun hm ↦ ⟨_, _, (add_sub_cancel ..).symm, sub_ne_zero.mpr fun h ↦ hm.1 ⟨_, h.symm⟩,
hm.sub_eigenvalue_sq_eq_zero⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 191,
"column": 48
} | {
"line": 191,
"column": 59
} | [
{
"pp": "τ : ℍ\n⊢ 0 < (-(starRingEnd ℂ) ↑τ).im",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"UpperHalfPlane.coe",
"Real.instZero",
"congrArg",
"CommSemiring.toSemiring",
"Complex.im",
"Real.instLT",
"RingH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 29
} | [
{
"pp": "case mpr.left\nK : Type u_1\ninst✝¹ : Field K\nm : Matrix (Fin 2) (Fin 2) K\ninst✝ : NeZero 2\na : K\nn : Matrix (Fin 2) (Fin 2) K\nhm : m - (scalar (Fin 2)) a = n\nhn0 : n ≠ 0\nhnsq : n ^ 2 = 0\nx✝ : m ∈ Set.range ⇑(scalar (Fin 2))\nb : K\nhb : (scalar (Fin 2)) b = m\n⊢ n = 0",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nm : Matrix (Fin 2) (Fin 2) K\ninst✝ : NeZero 2\na : K\nn : Matrix (Fin 2) (Fin 2) K\nhm : m = (scalar (Fin 2)) a + n\nhn0 : n ≠ 0\nhnsq : n ^ 2 = 0\nthis : m.discr = 0 ∨ 4 = 0\n⊢ 4 ≠ 0",
"usedConstants": [
"Eq.mpr",
"False",
"IsDomain.to_noZeroDivis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 214,
"column": 4
} | {
"line": 214,
"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 : -ε < r - ‖↑τ‖ ∧ r - ‖↑τ‖ < ε\nhr' : r < 0\nthis✝ : ‖↑τ‖ < ε\nthis : 0 ∈ Metric.ball (↑τ) ε\n⊢ False",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 219,
"column": 28
} | {
"line": 219,
"column": 39
} | [
{
"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",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 194,
"column": 8
} | {
"line": 196,
"column": 16
} | [
{
"pp": "case mp.inl.inl\ng : GL (Fin 2) ℝ\nhg : ∀ (z : ℍ), g • z = z\nhgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)\nhlt : (↑g).det < 0\nha : (↑g).trace = 0\nhb : ↑g 0 1 = ↑g 1 0\nhc : ↑g 1 0 = 0\n⊢ False",
"usedConstants": [
"UpperHalfPlane.glAction",
"Units.val",
"GroupWithZero.toMonoidWithZ... | specialize hg ⟨1 + .I, by simp⟩
rw [gl_smul_eq_self_iff_re_eq ha hc] at hg
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 194,
"column": 8
} | {
"line": 196,
"column": 16
} | [
{
"pp": "case mp.inl.inl\ng : GL (Fin 2) ℝ\nhg : ∀ (z : ℍ), g • z = z\nhgc : g ∉ Subgroup.center (GL (Fin 2) ℝ)\nhlt : (↑g).det < 0\nha : (↑g).trace = 0\nhb : ↑g 0 1 = ↑g 1 0\nhc : ↑g 1 0 = 0\n⊢ False",
"usedConstants": [
"UpperHalfPlane.glAction",
"Units.val",
"GroupWithZero.toMonoidWithZ... | specialize hg ⟨1 + .I, by simp⟩
rw [gl_smul_eq_self_iff_re_eq ha hc] at hg
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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