module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 21
} | {
"line": 49,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.neg",
"NegZeroClass.toN... | [] | simpa using! hf.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 21
} | {
"line": 49,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.neg",
"NegZeroClass.toN... | [] | simpa using! hf.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | {
"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\nh1 : AnalyticAt ℂ (gray z) 0\nhz : z ∈ sph... | [] | simp [gray, ray] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | {
"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\nh1 : AnalyticAt ℂ (gray z) 0\nhz : z ∈ sph... | [] | simp [gray, ray] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 158,
"column": 32
} | {
"line": 158,
"column": 48
} | {
"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\nh1 : AnalyticAt ℂ (gray z) 0\nhz : z ∈ sph... | [] | simp [gray, ray] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Periodic | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | {
"line": 75,
"column": 14
} | [
{
"pp": "h : ℝ\nhh : 0 < h\nA : ℝ\nz : ℂ\n⊢ -2 * π < 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preo... | [
"h : ℝ\nhh : 0 < h\nA : ℝ\nz : ℂ\n⊢ 0 < π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "h : ℝ\nhh : 0 < h\nz : ℂ\nhz : 0 < z.im\n⊢ ‖𝕢 h z‖ < 1",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"h : ℝ\nhh : 0 < h\nz : ℂ\nhz : 0 < z.im\n⊢ ‖𝕢 h z‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | {
"line": 105,
"column": 14
} | [
{
"pp": "h : ℝ\nhh : 0 < h\n⊢ -2 * π < 0",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | [
"h : ℝ\nhh : 0 < h\n⊢ 0 < π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 29
} | {
"line": 140,
"column": 30
} | [
{
"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",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",... | [
"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⊢ f (z + ↑m * ↑h) = f z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 15
} | {
"line": 256,
"column": 16
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ (Polynomial.X ^ n).natDegree ≠ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Polynomial.natDegree_X",
"HMul.hMul",
"congrArg",
"NormedDivisionRing.toNormMulClass",
"C... | [
"n : ℕ\ninst✝ : NeZero n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 262,
"column": 38
} | {
"line": 262,
"column": 74
} | {
"line": 262,
"column": 75
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nz : { z // z ≠ 0 }\nh : (fun x ↦ x ^ n) 0 = ↑z\n⊢ ↑z = 0",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat",
"Subtype.val",
"Eq"
],
... | [
"n : ℕ\ninst✝ : NeZero n\nz : { z // z ≠ 0 }\nh : (fun x ↦ x ^ n) 0 = ↑z\n⊢ ↑z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 46
} | {
"line": 203,
"column": 47
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nh_zer : I∞.ZeroAtFilter f\n⊢ cuspFunction h f 0 = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Function.Periodic.invQParam",
"congrArg",
"Function.Periodic.cuspFuncti... | [
"h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nh_zer : I∞.ZeroAtFilter f\n⊢ (𝓝[≠] 0).limUnder (f ∘ invQParam h) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 67
} | {
"line": 231,
"column": 68
} | [
{
"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))",
"ppTerm": "?m.59",
"assigned": false,
"usedConst... | [
"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))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 64
} | {
"line": 243,
"column": 4
} | [
{
"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)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 67
} | {
"line": 254,
"column": 4
} | [
{
"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)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnit... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 13
} | {
"line": 270,
"column": 14
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhfcts : ContinuousAt (cuspFunction h f) 0\n⊢ cuspFunction h (-f) = -cuspFunction h f",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"h : ℝ\nf : ℂ → ℂ\nhfcts : ContinuousAt (cuspFunction h f) 0\n⊢ cuspFunction h (-f) = -cuspFunction h f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 77,
"column": 4
} | {
"line": 89,
"column": 15
} | {
"line": 91,
"column": 0
} | [] | [] | ∑ 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.Periodic | {
"line": 285,
"column": 2
} | {
"line": 286,
"column": 9
} | {
"line": 286,
"column": 10
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.instNeg",
"AddGroupWithOne.toAdd... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 13
} | {
"line": 578,
"column": 14
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 13
} | {
"line": 611,
"column": 14
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, sin x * cos x = (cos a ^ 2 - cos b ^ 2) / 2",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"False",
"Real.partialOrder",
"Real",
... | [
"a b : ℝ\n⊢ sin b ^ 2 - sin a ^ 2 = cos a ^ 2 - cos b ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 31
} | {
"line": 183,
"column": 4
} | [
{
"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 ... | [
"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 (herglotzLog... | unfold herglotzLogIntegrand | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Analysis.Complex.Positivity | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 67
} | {
"line": 83,
"column": 68
} | [
{
"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)",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.m... | [
"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 ≤ (-1) ^ n * iteratedDeriv n f c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.TaylorSeries | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 50
} | {
"line": 53,
"column": 51
} | [
{
"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'",
... | [
"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‖ < ↑r'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.TaylorSeries | {
"line": 58,
"column": 2
} | {
"line": 60,
"column": 9
} | {
"line": 60,
"column": 10
} | [
{
"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'\nhz' : z - c ∈ Metric.eball 0 ↑... | [
"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'\nhz' : z - c ∈ Metric.eball 0 ↑r'\nH : HasS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 34
} | {
"line": 188,
"column": 35
} | [
{
"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 ... | [
"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 (herglotzLog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 60,
"column": 6
} | {
"line": 60,
"column": 59
} | {
"line": 60,
"column": 60
} | [
{
"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",
"ppTerm": "?inr.r... | [
"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 (-a + z) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 57
} | {
"line": 69,
"column": 58
} | [
{
"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",
"ppTerm": "?m.254",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 18
} | [
{
"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)",
"ppTerm": "?hf",
"assigned": false,
"usedC... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 17
} | {
"line": 77,
"column": 18
} | [
{
"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",
"ppTerm": "?hf'",
"assigned": true,
"usedConstants": [
"Nor... | [
"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⊢ ¬f z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 77
} | {
"line": 83,
"column": 4
} | [
{
"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)",
"ppTerm": "?refine_1",
"assign... | [
"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⊢ ∃ x, ∀ᶠ (x : ℂ) in 𝓝 x, x ∉ f '' U"
] | 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
} | {
"line": 89,
"column": 51
} | [
{
"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",
"ppTerm": "?m.384",
"assig... | [
"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⊢ ¬f x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 45
} | {
"line": 103,
"column": 46
} | [
{
"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",
"ppTerm": "?refin... | [
"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⊢ ¬f z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 80
} | {
"line": 125,
"column": 81
} | [
{
"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",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 126,
"column": 47
} | {
"line": 126,
"column": 58
} | {
"line": 126,
"column": 59
} | [
{
"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",
"ppTerm": "?m.113",
"assigned": true,
"usedCon... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 32
} | {
"line": 133,
"column": 33
} | [
{
"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‖",
"ppTerm": "?m.183",
... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 31
} | {
"line": 134,
"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⊢ 0 < ‖f z - x‖",
"ppTerm": "?m.182",
... | [
"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⊢ ¬f z = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 56
} | {
"line": 137,
"column": 57
} | [
{
"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 - ... | [
"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 - x)) z = (fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 32
} | {
"line": 366,
"column": 33
} | [
{
"pp": "c : 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⊢ f z = 0... | [
"c : 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⊢ f z = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 15
} | {
"line": 402,
"column": 16
} | [
{
"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 < ... | [
"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 < Real.log (R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Exp | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 63
} | {
"line": 45,
"column": 64
} | [
{
"pp": "τ : ℍ\n⊢ ‖cexp (2 * ↑π * Complex.I * ↑τ)‖ < 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Complex.mul_re",
"Real.pi",
"H... | [
"τ : ℍ\n⊢ 0 < 2 * π * τ.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 23
} | {
"line": 410,
"column": 24
} | [
{
"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 (... | [
"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 (closedBall c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 130,
"column": 17
} | {
"line": 130,
"column": 52
} | {
"line": 130,
"column": 53
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing R\nhR : ∀ (r : Rˣ), ∃ k, k ^ Fintype.card n = r\ng : GL n R\nr : Rˣ\nhr : r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹\n⊢ r ^ Fintype.card n * GeneralLinearGroup.det g = 1",
"ppTerm": "?m.81",
"assigne... | [
"n : Type u_1\nR : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing R\nhR : ∀ (r : Rˣ), ∃ k, k ^ Fintype.card n = r\ng : GL n R\nr : Rˣ\nhr : r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹\n⊢ r ^ Fintype.card n = (GeneralLinearGroup.det g)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 30
} | {
"line": 147,
"column": 31
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Nonempty n\nh : Function.Surjective ⇑toPGL\nr : Rˣ\nA : GL n R\nhA : GeneralLinearGroup.det A = r\nX : SpecialLinearGroup n R\nhX : toPGL ↑X = mk A\n⊢ ∃ u, toGL X * (GeneralLinearGroup.scalar n) u = A",... | [
"n : Type u_1\nR : Type u_2\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Nonempty n\nh : Function.Surjective ⇑toPGL\nr : Rˣ\nA : GL n R\nhA : GeneralLinearGroup.det A = r\nX : SpecialLinearGroup n R\nhX : toPGL ↑X = mk A\n⊢ ∃ u, toGL X * (GeneralLinearGroup.scalar n) u = A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 432,
"column": 37
} | {
"line": 432,
"column": 62
} | {
"line": 432,
"column": 63
} | [
{
"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|))... | [
"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|)) u) * Real.l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 148,
"column": 15
} | {
"line": 148,
"column": 31
} | {
"line": 148,
"column": 32
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Nonempty n\nh : Function.Surjective ⇑toPGL\nr : Rˣ\nA : GL n R\nhA : GeneralLinearGroup.det A = r\nX : SpecialLinearGroup n R\nhX : toPGL ↑X = mk A\nu : Rˣ\nhu : toGL X * (GeneralLinearGroup.scalar n) u... | [
"n : Type u_1\nR : Type u_2\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing R\ninst✝ : Nonempty n\nh : Function.Surjective ⇑toPGL\nr : Rˣ\nA : GL n R\nhA : GeneralLinearGroup.det A = r\nX : SpecialLinearGroup n R\nhX : toPGL ↑X = mk A\nu : Rˣ\nhu : toGL X * (GeneralLinearGroup.scalar n) u = A\n⊢ u ^ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 44
} | {
"line": 162,
"column": 45
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : Nonempty n\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : IsAlgClosed F\nr : Fˣ\nx : F\nhx : (X ^ Fintype.card n - C ↑r).IsRoot x\nhx' : x ≠ 0\n⊢ { val := x, inv := x⁻¹, val_inv := ⋯, inv_val := ⋯ } ^ Fintyp... | [
"n : Type u_1\nR : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : CommRing R\ninst✝² : Nonempty n\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : IsAlgClosed F\nr : Fˣ\nx : F\nhx : (X ^ Fintype.card n - C ↑r).IsRoot x\nhx' : x ≠ 0\n⊢ x ^ Fintype.card n = ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 433,
"column": 12
} | {
"line": 433,
"column": 75
} | {
"line": 433,
"column": 76
} | [
{
"pp": "case 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 (closedBa... | [
"case 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 (closedBall c |R|)) u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 52
} | {
"line": 66,
"column": 53
} | [
{
"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 ... | [
"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 : x ∈ Metric... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Projective | {
"line": 170,
"column": 25
} | {
"line": 170,
"column": 36
} | {
"line": 170,
"column": 37
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing R\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : IsAlgClosed F\nh : ¬Nonempty n\n⊢ IsEmpty n",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing R\nF : Type u_3\ninst✝¹ : Field F\ninst✝ : IsAlgClosed F\nh : ¬Nonempty n\n⊢ IsEmpty n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Tietze | {
"line": 72,
"column": 31
} | {
"line": 72,
"column": 84
} | {
"line": 72,
"column": 85
} | [
{
"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",
"ppTerm": "?m.17... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 33
} | {
"line": 197,
"column": 34
} | [
{
"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... | [
"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))\nhc₂ : IsClos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 49
} | {
"line": 200,
"column": 50
} | [
{
"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... | [
"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✝¹] (⇑e '' ⇑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 23
} | {
"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",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
... | [] | 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
} | {
"line": 52,
"column": 14
} | [
{
"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",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 40
} | {
"line": 59,
"column": 41
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : m.IsParabolic\n⊢ (-m).discr = 0",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Fintype.card_fin",
"NonU... | [
"case right\nR : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : m.IsParabolic\n⊢ m.trace ^ 2 - 4 * m.det = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | {
"line": 62,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : (-m).IsParabolic\n⊢ m.IsParabolic",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nm : Matrix (Fin 2) (Fin 2) R\nh : (-m).IsParabolic\n⊢ m.IsParabolic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 44
} | {
"line": 56,
"column": 45
} | [
{
"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",
"ppTerm": "?m.82",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 35
} | {
"line": 116,
"column": 2
} | [
{
"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",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Matrix.scala... | [
"case mpr\nK : Type u_1\ninst✝¹ : Field K\nm : Matrix (Fin 2) (Fin 2) K\ninst✝ : NeZero 2\n⊢ (∃ a n, m = (scalar (Fin 2)) a + n ∧ n ≠ 0 ∧ n ^ 2 = 0) → m.IsParabolic"
] | · 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.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 120,
"column": 6
} | {
"line": 121,
"column": 29
} | {
"line": 121,
"column": 30
} | [
{
"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",
"ppTerm": "?mpr.lef... | [
"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⊢ b - a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 46
} | {
"line": 159,
"column": 47
} | [
{
"pp": "g h : GL (Fin 2) ℝ\nz : ℍ\n⊢ denom (g * h) ↑z = (σ h) (denom g ↑(smulAux h z)) * denom h ↑z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ContinuousAlgEquivClass.toAlgEquivClass",
"Real"... | [
"g h : GL (Fin 2) ℝ\nz : ℍ\n⊢ ↑((↑g * ↑h) 1 0) * ↑z + ↑((↑g * ↑h) 1 1) =\n (↑(↑g 1 0) * (num h ↑z / (↑(↑h 1 0) * ↑z + ↑(↑h 1 1))) + ↑(↑g 1 1)) * (↑(↑h 1 0) * ↑z + ↑(↑h 1 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 79
} | {
"line": 124,
"column": 80
} | [
{
"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",
"ppTerm": "?m.260",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"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⊢ ¬2 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 15
} | {
"line": 158,
"column": 16
} | [
{
"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",
"ppTerm": "?inr",
"assigned": false,
"usedCons... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Preorder R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ ((↑g)⁻¹ * m * ↑g).IsHyperbolic ↔ m.IsHyperbolic",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Preorder R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ ((↑g)⁻¹ * m * ↑g).IsHyperbolic ↔ m.IsHyperbolic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Preorder R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ ((↑g)⁻¹ * m * ↑g).IsElliptic ↔ m.IsElliptic",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Preorder R\nm : Matrix (Fin 2) (Fin 2) R\ng : GL (Fin 2) R\n⊢ ((↑g)⁻¹ * m * ↑g).IsElliptic ↔ m.IsElliptic"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 17
} | {
"line": 163,
"column": 18
} | [
{
"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",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
"Real.instLE",
"Real",
... | [
"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⊢ (↑g).discr ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.TietzeExtension | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 71
} | {
"line": 302,
"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": 238,
"column": 30
} | {
"line": 238,
"column": 41
} | {
"line": 238,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\n⊢ ↑g 0 1 = 0",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\n⊢ ↑g 0 1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 239,
"column": 30
} | {
"line": 239,
"column": 41
} | {
"line": 239,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\nhb : ↑g 0 1 = 0\n⊢ ↑g 1 0 = 0",
"ppTerm": "?m.82",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\nhb : ↑g 0 1 = 0\n⊢ ↑g 1 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 240,
"column": 34
} | {
"line": 240,
"column": 59
} | {
"line": 240,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\nhb : ↑g 0 1 = 0\nhc : ↑g 1 0 = 0\n⊢ ↑g 1 1 = ↑g 0 0",
"ppTerm": "?m.116",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\ng : GL (Fin 2) R\nhP : C (↑g 1 0) * X ^ 2 + C (↑g 1 1 - ↑g 0 0) * X - C (↑g 0 1) = 0\nhb : ↑g 0 1 = 0\nhc : ↑g 1 0 = 0\n⊢ ↑g 1 1 = ↑g 0 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 43
} | {
"line": 315,
"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": 326,
"column": 30
} | {
"line": 326,
"column": 41
} | {
"line": 326,
"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.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 250,
"column": 47
} | {
"line": 250,
"column": 87
} | {
"line": 250,
"column": 88
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : g.IsParabolic\n⊢ (↑g).trace ^ 2 = 4 * (↑g).det",
"ppTerm": "?m.51",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : g.IsParabolic\n⊢ (↑g).trace ^ 2 = 4 * (↑g).det"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 194,
"column": 8
} | {
"line": 196,
"column": 16
} | {
"line": 197,
"column": 6
} | [
{
"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",
"ppTerm": "?mp.inl.inl",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | 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
} | {
"line": 197,
"column": 6
} | [
{
"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",
"ppTerm": "?mp.inl.inl",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.glAction",
... | [] | 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 |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.FinTwo | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 27
} | {
"line": 254,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : g.IsParabolic\nthis : (↑g).trace ^ 2 = 4 * (↑g).det\n⊢ (↑g).parabolicEigenvalue ≠ 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | [
"K : Type u_2\ninst✝¹ : Field K\ng : GL (Fin 2) K\ninst✝ : NeZero 2\nhg : g.IsParabolic\nthis : (↑g).trace ^ 2 = 4 * (↑g).det\n⊢ ¬2 = 0 ∧ ¬(↑g).det = 0"
] | rw [parabolicEigenvalue, div_ne_zero_iff, eq_true_intro (two_ne_zero' K), and_true,
Ne, ← sq_eq_zero_iff, this, show (4 : K) = 2 ^ 2 by norm_num, mul_eq_zero,
sq_eq_zero_iff, not_or] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 357,
"column": 31
} | {
"line": 357,
"column": 63
} | {
"line": 357,
"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": 415,
"column": 2
} | {
"line": 415,
"column": 19
} | {
"line": 415,
"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": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | {
"line": 471,
"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.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": 160,
"column": 31
} | {
"line": 160,
"column": 42
} | {
"line": 160,
"column": 43
} | [
{
"pp": "a : ℍ\nhw : ¬0 < (↑a).im\n⊢ a.im ≤ 0",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℍ\nhw : ¬0 < (↑a).im\n⊢ a.im ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 39
} | {
"line": 88,
"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": 96,
"column": 2
} | {
"line": 96,
"column": 38
} | {
"line": 96,
"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.Topology | {
"line": 191,
"column": 48
} | {
"line": 191,
"column": 59
} | {
"line": 191,
"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.Geometry.Manifold.MFDeriv.FDeriv | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 74
} | {
"line": 57,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace 𝕜 E'\nf : E → E'\ns : Set E\nx : E\nf' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)\n⊢ HasMF... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : NormedSpace 𝕜 E'\nf : E → E'\ns : Set E\nx : E\nf' : TangentSpace 𝓘(𝕜, E) x →L[𝕜] TangentSpace 𝓘(𝕜, E') (f x)\n⊢ HasFDerivWithinAt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 40
} | {
"line": 214,
"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.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 219,
"column": 28
} | {
"line": 219,
"column": 39
} | {
"line": 219,
"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.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.Analysis.Complex.UpperHalfPlane.Topology | {
"line": 223,
"column": 28
} | {
"line": 223,
"column": 39
} | {
"line": 223,
"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.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.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.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.Geometry.Euclidean.Inversion.Basic | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 43
} | {
"line": 160,
"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.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 |
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