module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.FrullaniIntegral | {
"line": 224,
"column": 47
} | {
"line": 224,
"column": 65
} | {
"line": 224,
"column": 65
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nL R : E\ninst✝ : CompleteSpace E\nhf : LocallyIntegrableOn f (Ioi 0) volume\nha : 0 < a\nhb : 0 < b\nhL : Tendsto f (𝓝[>] 0) (𝓝 L)\nhR : Tendsto f atTop (𝓝 R)\ng : ℝ → E := fun x ↦ x⁻¹ • (f (a * x) - f (b * x)... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nL R : E\ninst✝ : CompleteSpace E\nhf : LocallyIntegrableOn f (Ioi 0) volume\nha : 0 < a\nhb : 0 < b\nhL : Tendsto f (𝓝[>] 0) (𝓝 L)\nhR : Tendsto f atTop (𝓝 R)\ng : ℝ → E := fun x ↦ x⁻¹ • (f (a * x) - f (b * x))\nhint : In... | mem_nhdsWithin_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 14
} | [
{
"pp": "case e_a\np t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\n⊢ x ^ (p - 1) * x = x ^ p",
"ppTerm": "?e_a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case e_a\np t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\n⊢ x ^ (p - 1) * x = x ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 162,
"column": 14
} | {
"line": 162,
"column": 46
} | {
"line": 162,
"column": 47
} | [
{
"pp": "p t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nx : ℝ\nhx : x ∈ Ici 0\ny : ℝ\nhy : y ∈ Ici 0\nhxy : x ≤ y\nh : x = 0\n⊢ p.rpowIntegrand₀₁ t x ≤ p.rpowIntegrand₀₁ t y",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
... | [
"p t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nx : ℝ\nhx : x ∈ Ici 0\ny : ℝ\nhy : y ∈ Ici 0\nhxy : x ≤ y\nh : x = 0\n⊢ 0 ≤ t ^ p * (t⁻¹ - (t + y)⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 169,
"column": 2
} | {
"line": 177,
"column": 87
} | {
"line": 179,
"column": 0
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\ns : Set ℝ\nhs : s ⊆ Ici 0\n⊢ ContinuousOn (Function.uncurry p.rpowIntegrand₀₁) (Ioi 0 ×ˢ s)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRi... | [] | let g : ℝ × ℝ → ℝ := fun q => q.1 ^ (p - 1) * q.2 / (q.1 + q.2)
refine ContinuousOn.congr (f := g) ?_ fun q => ?_
· simp only [g]
refine ContinuousOn.mul ?_ ?_
· refine ContinuousOn.mul ?_ (by fun_prop)
exact ContinuousOn.rpow_const (by fun_prop) (by grind)
· exact ContinuousOn.inv₀ (by fun_prop) ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 169,
"column": 2
} | {
"line": 177,
"column": 87
} | {
"line": 179,
"column": 0
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\ns : Set ℝ\nhs : s ⊆ Ici 0\n⊢ ContinuousOn (Function.uncurry p.rpowIntegrand₀₁) (Ioi 0 ×ˢ s)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRi... | [] | let g : ℝ × ℝ → ℝ := fun q => q.1 ^ (p - 1) * q.2 / (q.1 + q.2)
refine ContinuousOn.congr (f := g) ?_ fun q => ?_
· simp only [g]
refine ContinuousOn.mul ?_ ?_
· refine ContinuousOn.mul ?_ (by fun_prop)
exact ContinuousOn.rpow_const (by fun_prop) (by grind)
· exact ContinuousOn.inv₀ (by fun_prop) ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.EulerMascheroni | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 16
} | [
{
"pp": "this : Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)\n⊢ Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)\n⊢ Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 17
} | {
"line": 294,
"column": 18
} | [
{
"pp": "case hff'\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ∀ x_1 ∈ Ioi 0, HasDerivWithinAt (fun x_2 ↦ x * x_2) x (Ioi x_1) x_1",
"ppTerm": "?... | [
"case hff'\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ∀ (x_1 : ℝ), 0 < x_1 → HasDerivWithinAt (fun x_2 ↦ x * x_2) x (Ici x_1) x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 45
} | {
"line": 295,
"column": 46
} | [
{
"pp": "case hg_cont\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ContinuousOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ioi 0)",
... | [
"case hg_cont\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ContinuousOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 45
} | {
"line": 296,
"column": 46
} | [
{
"pp": "case hg1\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ IntegrableOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ici 0) volume"... | [
"case hg1\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ IntegrableOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) (Ici 0) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 57,
"column": 25
} | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 37
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ s ≠ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat"
],
"usedFVars": [
"s"
],
"usedGoals": [
{
"new"... | [
"s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 469,
"column": 10
} | {
"line": 472,
"column": 18
} | {
"line": 473,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | [] | refine cfcₙ_smul (R := ℝ) (t ^ ((p : ℝ) - 1)) _ a ?_
refine ContinuousOn.mono ?_ hspec
have := continuousOn_rpowIntegrand₀₁_Ici hp zero_lt_one
fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 469,
"column": 10
} | {
"line": 472,
"column": 18
} | {
"line": 473,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | [] | refine cfcₙ_smul (R := ℝ) (t ^ ((p : ℝ) - 1)) _ a ?_
refine ContinuousOn.mono ?_ hspec
have := continuousOn_rpowIntegrand₀₁_Ici hp zero_lt_one
fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 459,
"column": 89
} | {
"line": 477,
"column": 51
} | {
"line": 479,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | [] | by
have hspec : quasispectrum ℝ a ⊆ Ici 0 := by grind
have h_mapsTo : MapsTo (t⁻¹ • · : ℝ → ℝ) (Ici 0) (Ici 0) := by
intro x hx
simp only [mem_Ici, smul_eq_mul] at hx ⊢
positivity
calc _ = cfcₙ (fun x => t ^ ((p : ℝ) - 1) * (rpowIntegrand₀₁ p 1 (t⁻¹ • x))) a := by
refine cfcₙ_congr ?_
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 31
} | [
{
"pp": "⊢ HasDerivAt Gamma (-γ) 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt Gamma (-γ) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 34
} | {
"line": 42,
"column": 35
} | [
{
"pp": "a : ℝ\nha : a < 0\nb s : ℝ\nthis : (fun x ↦ rexp (a * x ^ 2 + b * x)) =o[atTop] fun x ↦ rexp (-x)\n⊢ (fun x ↦ rexp (-x)) =o[atTop] fun x ↦ x ^ s",
"ppTerm": "?m.114",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℝ\nha : a < 0\nb s : ℝ\nthis : (fun x ↦ rexp (a * x ^ 2 + b * x)) =o[atTop] fun x ↦ rexp (-x)\n⊢ (fun x ↦ rexp (-x)) =o[atTop] fun x ↦ x ^ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 30
} | {
"line": 182,
"column": 31
} | [
{
"pp": "⊢ HasDerivAt Gamma (-↑γ) 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt Gamma (-↑γ) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 24
} | [
{
"pp": "case refine_2\nthis : HasDerivAt Gamma ↑(-√π * (γ + 2 * Real.log 2)) ↑(1 / 2)\n⊢ HasDerivAt Gamma (-↑√π * (↑γ + 2 * log 2)) (1 / 2)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NormedCommRing.toSeminormedComm... | [
"case refine_2\nthis : HasDerivAt Gamma ↑(-√π * (γ + 2 * Real.log 2)) ↑(1 / 2)\n⊢ HasDerivAt Gamma (-(↑√π * (↑γ + 2 * log 2))) 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 49
} | {
"line": 121,
"column": 50
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\n⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℂ\nha : 0 < a.re\n⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 20
} | [
{
"pp": "a : ℝ\nha : 0 < a\n⊢ ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
... | [
"a : ℝ\nha : 0 < a\n⊢ ∑' (a_1 : ℤ), cexp (-↑π * ↑a * ↑a_1 ^ 2) = 1 / ↑a ^ (1 / 2) * ∑' (a_1 : ℤ), cexp (-↑π / ↑a * ↑a_1 ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 239,
"column": 63
} | {
"line": 240,
"column": 43
} | {
"line": 242,
"column": 0
} | [
{
"pp": "b x : ℝ\nhb : 1 < b\nhx : 0 < x\n⊢ logb b x ≤ 0 ↔ x ≤ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.... | [] | by
rw [← not_lt, logb_pos_iff hb hx, not_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 610,
"column": 6
} | {
"line": 615,
"column": 26
} | {
"line": 616,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\na : A\nha : 0 ≤ a\nb : A\nhb : 0 ≤ b\nhab : a ≤ b\n⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)",
"... | [] | gcongr
unfold rpowIntegrand₀₁
simp only [Real.one_rpow, one_mul, inv_one]
refine CFC.monotoneOn_one_sub_one_add_inv_real
(?_ : 0 ≤ t⁻¹ • a) (?_ : 0 ≤ t⁻¹ • b) (by gcongr)
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 610,
"column": 6
} | {
"line": 615,
"column": 26
} | {
"line": 616,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\na : A\nha : 0 ≤ a\nb : A\nhb : 0 ≤ b\nhab : a ≤ b\n⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)",
"... | [] | gcongr
unfold rpowIntegrand₀₁
simp only [Real.one_rpow, one_mul, inv_one]
refine CFC.monotoneOn_one_sub_one_add_inv_real
(?_ : 0 ≤ t⁻¹ • a) (?_ : 0 ≤ t⁻¹ • b) (by gcongr)
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 35,
"column": 33
} | {
"line": 35,
"column": 44
} | {
"line": 35,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : x ∈ interior (Ici (rexp (-1)))\n⊢ rexp (-1) < x",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ∈ interior (Ici (rexp (-1)))\n⊢ rexp (-1) < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 53
} | {
"line": 370,
"column": 4
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"FloorRing.toFloorSemiring",
"Real.instZeroLEOneC... | [
"case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r"
] | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 53
} | {
"line": 388,
"column": 4
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌈logb (↑b) r⌉ = Int.clog b r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"FloorRing.toFloorSemiring",
"Real.instZeroLEOne... | [
"case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ ⌈logb (↑b) r⌉ = Int.clog b r"
] | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 36
} | {
"line": 39,
"column": 37
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nf : X →ᵇ ℝ≥0\nx : X\n⊢ ↑(f x) ≤ edist 0 f",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nf : X →ᵇ ℝ≥0\nx : X\n⊢ ↑(f x) ≤ edist 0 f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 514,
"column": 4
} | {
"line": 514,
"column": 15
} | {
"line": 514,
"column": 16
} | [
{
"pp": "b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / id x) atTop (𝓝 0)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"NormedDivisionRing.toNormedRing",
"PseudoMetricSpace.toUniformSpace",
"NormedDivisionRing.toDivisionRing",
"n... | [
"b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / x) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 15
} | {
"line": 535,
"column": 16
} | [
{
"pp": "case inl\n⊢ (fun x ↦ log (0 * x)) =O[atTop] log",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"MulZeroClass.zero_mul",
"Asymptotics.IsBigO",
"Real.semiring",
"id",
... | [
"case inl\n⊢ (fun x ↦ 0) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 24
} | {
"line": 543,
"column": 25
} | [
{
"pp": "c : ℝ\n⊢ (fun x ↦ log (x * c)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.t... | [
"c : ℝ\n⊢ (fun x ↦ log (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 546,
"column": 2
} | {
"line": 547,
"column": 9
} | {
"line": 547,
"column": 10
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (c * x)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonA... | [
"b c : ℝ\n⊢ (fun x ↦ (log b)⁻¹ * log (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 24
} | {
"line": 550,
"column": 25
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (x * c)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommR... | [
"b c : ℝ\n⊢ (fun x ↦ logb b (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 61
} | {
"line": 42,
"column": 62
} | [
{
"pp": "case mpr.refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\ninst✝ : R1Space α\nh : μ.InnerRegularWRT IsCompact IsClosed\nA : Set α\nhA : IsClosed A\nr : ℝ≥0∞\nhr : r < μ A\nK : Set α\nhK1 : K ⊆ A\nhK2 : IsCompact K\nhK3 : r < μ K\n⊢ (IsCompact ∘ closure) (cl... | [
"case mpr.refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\ninst✝ : R1Space α\nh : μ.InnerRegularWRT IsCompact IsClosed\nA : Set α\nhA : IsClosed A\nr : ℝ≥0∞\nhr : r < μ A\nK : Set α\nhK1 : K ⊆ A\nhK2 : IsCompact K\nhK3 : r < μ K\n⊢ IsCompact (closure K)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 627,
"column": 12
} | {
"line": 627,
"column": 23
} | {
"line": 627,
"column": 24
} | [
{
"pp": "case zero\nP : ℝ → Prop\nx₀ r : ℝ\nhr : 1 < r\nhx₀ : 0 < x₀\nbase : ∀ x ∈ Ico x₀ (r * x₀), P x\nstep : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z\n⊢ ∀ x ∈ Ico x₀ (r ^ (0 + 1) * x₀), P x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [... | [
"case zero\nP : ℝ → Prop\nx₀ r : ℝ\nhr : 1 < r\nhx₀ : 0 < x₀\nbase : ∀ x ∈ Ico x₀ (r * x₀), P x\nstep : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z\n⊢ ∀ (x : ℝ), x₀ ≤ x → x < r * x₀ → P x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 160,
"column": 59
} | {
"line": 160,
"column": 70
} | {
"line": 160,
"column": 71
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), f x ∂μs i) L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 70
} | {
"line": 161,
"column": 71
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), f x ∂μs i) L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 182,
"column": 59
} | {
"line": 182,
"column": 70
} | {
"line": 182,
"column": 71
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i ↦ ∫ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 183,
"column": 59
} | {
"line": 183,
"column": 70
} | {
"line": 183,
"column": 71
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf (fun i ↦ ∫ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 34
} | {
"line": 92,
"column": 35
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ Polynomial.eval (↑n - 1) (descPochhammer ℝ n) = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"descPochhammer",
... | [
"n : ℕ\nhn : 0 < n\n⊢ Polynomial.eval (↑(n - 1)) (descPochhammer ℝ n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 54
} | {
"line": 117,
"column": 6
} | [
{
"pp": "case h_lim\nE : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nP : Measure E\ninst✝ : IsFiniteMeasure P\nε : ℝ\ng : E →ᵇ ℝ\nhε : 0 < ε\nx : E\n⊢ Tendsto (fun x_1 ↦ ((1 + (↑x_1)⁻¹ • -(ε • g * g)) ^ x_1) x) atTop (𝓝 (rexp (-(ε * g x * g x))))",
"ppTerm": "?... | [
"case h_lim\nE : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nP : Measure E\ninst✝ : IsFiniteMeasure P\nε : ℝ\ng : E →ᵇ ℝ\nhε : 0 < ε\nx : E\n⊢ Tendsto (fun x_1 ↦ (1 + -(ε * (g x * g x)) / ↑x_1) ^ x_1) atTop (𝓝 (rexp (-(ε * (g x * g x)))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 68
} | {
"line": 103,
"column": 69
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\nf : ℝ → ℝ := (Set.Ici (↑n - 1)).piecewise (fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) 0\nh_jensen : f (∑ i ∈ t, w i • ↑(p i)) ≤ ∑ i ∈ t, w i • ... | [
"n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\nf : ℝ → ℝ := (Set.Ici (↑n - 1)).piecewise (fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) 0\nh_jensen : f (∑ i ∈ t, w i • ↑(p i)) ≤ ∑ i ∈ t, w i • f ↑(p i)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 113,
"column": 31
} | {
"line": 113,
"column": 67
} | {
"line": 113,
"column": 67
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\n⊢ Polynomial.eval (∑ i ∈ t, w i * ↑(p i)) (descPochhammer ℝ n) / ↑n.factorial ≤\n (∑ i ∈ t, w i * Polynomial.eval (↑(p i)) (descPochhammer ℝ n)) ... | [
"n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\n⊢ Polynomial.eval (∑ i ∈ t, w i * ↑(p i)) (descPochhammer ℝ n) / ↑n.factorial ≤\n (∑ x ∈ t, w x * ↑((p x).descFactorial n)) / ↑n.factorial"
] | descPochhammer_eval_eq_descFactorial | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 40
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case neg\nε : ℝ\nE : Type u_2\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nf : E →ᵇ ℝ\nA : Subalgebra ℝ (E →ᵇ ℝ)\nhA : (Subalgebr... | [
"case neg\nε : ℝ\nE : Type u_2\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nf : E →ᵇ ℝ\nA : Subalgebra ℝ (E →ᵇ ℝ)\nhA : (Subalgebra.map (toCon... | have hgA : g ∈ A := hg'A.choose_spec.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 30
} | {
"line": 105,
"column": 4
} | [
{
"pp": "L : PeriodPair\n⊢ L.ω₁ / 2 ∉ L.lattice",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\n⊢ L.ω₁ / 2 ∉ L.lattice"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 30
} | {
"line": 109,
"column": 4
} | [
{
"pp": "L : PeriodPair\n⊢ L.ω₂ / 2 ∉ L.lattice",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\n⊢ L.ω₂ / 2 ∉ L.lattice"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 7
} | {
"line": 134,
"column": 0
} | [
{
"pp": "L : PeriodPair\ns : Set ℂ\nhs : s ⊆ ↑L.lattice\n⊢ IsClosed s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe._simp_1",
"congrArg",
"Set.ofPred",
"HEq.refl",
... | [] | convert!
L.isClosed_lattice.isClosedMap_subtype_val _ (isClosed_discrete (α := L.lattice) ((↑) ⁻¹' s))
convert! Set.image_preimage_eq_inter_range.symm using 1
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 7
} | {
"line": 134,
"column": 0
} | [
{
"pp": "L : PeriodPair\ns : Set ℂ\nhs : s ⊆ ↑L.lattice\n⊢ IsClosed s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe._simp_1",
"congrArg",
"Set.ofPred",
"HEq.refl",
... | [] | convert!
L.isClosed_lattice.isClosedMap_subtype_val _ (isClosed_discrete (α := L.lattice) ((↑) ⁻¹' s))
convert! Set.image_preimage_eq_inter_range.symm using 1
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 180,
"column": 61
} | {
"line": 180,
"column": 72
} | {
"line": 180,
"column": 73
} | [
{
"pp": "L : PeriodPair\nf : ↥L.lattice → ℂ → ℂ\nu : ℝ → ↥L.lattice → ℝ\nhu : ∀ r > 0, Summable (u r)\nhf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l\nx : ℂ\nr : ℝ\nhr : 0 < r\nhr' : 𝓝 x ≤ 𝓟 (Metric.ball 0 r)\nR : ℝ\nhR : ∀ (b : ℝ), R ≤ b → ∀ (x : ℂ), ‖x... | [
"L : PeriodPair\nf : ↥L.lattice → ℂ → ℂ\nu : ℝ → ↥L.lattice → ℝ\nhu : ∀ r > 0, Summable (u r)\nhf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l\nx : ℂ\nr : ℝ\nhr : 0 < r\nhr' : 𝓝 x ≤ 𝓟 (Metric.ball 0 r)\nR : ℝ\nhR : ∀ (b : ℝ), R ≤ b → ∀ (x : ℂ), ‖x‖ < r → ∀ (l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 176,
"column": 35
} | {
"line": 176,
"column": 68
} | {
"line": 176,
"column": 69
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy : dist 0 ((1 - ε / 4) • x) < ε / 4\n⊢ 1 - ε / 4 < ε / 4",
"ppTerm": "?m.179",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy : dist 0 ((1 - ε / 4) • x) < ε / 4\n⊢ 1 - ε / 4 < ε / 4"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 17
} | {
"line": 180,
"column": 18
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\n⊢ ‖y‖ ≤ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖",
"ppTerm": "?m.261",
"assig... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\n⊢ ‖y‖ ≤ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 195,
"column": 4
} | {
"line": 196,
"column": 11
} | {
"line": 196,
"column": 12
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\nhy₁ : ‖y‖ < 1\nu : E := ‖y‖⁻¹ • y\nhu₁ : ‖u‖ = 1\nhyx : dist y x < ε / 2\nH : u - y = (1 ... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\nhy₁ : ‖y‖ < 1\nu : E := ‖y‖⁻¹ • y\nhu₁ : ‖u‖ = 1\nhyx : dist y x < ε / 2\nH : u - y = (1 - ‖y‖) • u\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 89,
"column": 14
} | {
"line": 89,
"column": 25
} | {
"line": 89,
"column": 26
} | [
{
"pp": "case succ\nn a : ℕ\nH : ∃ c, a < (c + 1) ^ (n + 1)\nk : ℕ\nhc : k + 1 = Nat.find H\n⊢ (k + 1) ^ (n + 1) ≤ a",
"ppTerm": "?succ",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case succ\nn a : ℕ\nH : ∃ c, a < (c + 1) ^ (n + 1)\nk : ℕ\nhc : k + 1 = Nat.find H\n⊢ (k + 1) ^ (n + 1) ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 223,
"column": 6
} | {
"line": 224,
"column": 13
} | {
"line": 224,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\nthis✝ : Nontrivial E\nthis : ∀ {ε : ℝ}, 0 < ε → ε ≤ 2 → ↑(toSphereBa... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\nthis✝ : Nontrivial E\nthis : ∀ {ε : ℝ}, 0 < ε → ε ≤ 2 → ↑(toSphereBallBound (dim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 229,
"column": 2
} | {
"line": 231,
"column": 43
} | {
"line": 232,
"column": 2
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\n⊢ HasSumLocallyUniformly (fun l z ↦ if ↑l = l₀ then 0 else 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2) ℘[L - l₀]",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | [
"L : PeriodPair\nl₀ : ℂ\nr : ℝ\nhr : r > 0\n⊢ ∀ (b : ℝ),\n 2 * r ≤ b →\n ∀ (x : ℂ),\n ‖x‖ < r →\n ∀ (l : ↥L.lattice), ‖↑l‖ = b → ‖if ↑l = l₀ then 0 else 1 / (x - ↑l) ^ 2 - 1 / ↑l ^ 2‖ ≤ 10 * r * ‖l‖ ^ (-3)"
] | refine L.hasSumLocallyUniformly_aux (u := (10 * · * ‖·‖ ^ (-3 : ℝ))) _
(fun _ _ ↦ (ZLattice.summable_norm_rpow _ _ (by simp; norm_num)).mul_left _) fun r hr ↦
Filter.eventually_atTop.mpr ⟨2 * r, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 21
} | {
"line": 190,
"column": 22
} | [
{
"pp": "a b : Multiset ℂ\nh✝ : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k\nn : ℕ\nhn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n\nj : ℂ\nhj : j ∈ b\nm : ℕ\nh' : j + ↑n = -↑m\nthis : -j.re < ↑n\nh : j = -↑m - ↑n\n⊢ ↑m < 0",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
... | [
"a b : Multiset ℂ\nh✝ : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k\nn : ℕ\nhn : (b.toFinset.sup fun x ↦ ⌈-x.re⌉₊) + 1 ≤ n\nj : ℂ\nhj : j ∈ b\nm : ℕ\nh' : j + ↑n = -↑m\nthis : -j.re < ↑n\nh : j = -↑m - ↑n\n⊢ ↑m < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 15
} | {
"line": 200,
"column": 16
} | [
{
"pp": "a : Multiset ℂ\nthis : ∀ i ∈ a, Tendsto (fun n ↦ i / ↑n + 1) atTop (𝓝 ((fun x ↦ 1) i))\n⊢ Tendsto (fun n ↦ (Multiset.map (fun x ↦ x / ↑n + 1) a).prod) atTop (𝓝 1)",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Multiset ℂ\nthis : ∀ i ∈ a, Tendsto (fun n ↦ i / ↑n + 1) atTop (𝓝 ((fun x ↦ 1) i))\n⊢ Tendsto (fun n ↦ (Multiset.map (fun x ↦ x / ↑n + 1) a).prod) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | {
"line": 202,
"column": 14
} | [
{
"pp": "a : Multiset ℂ\ni : ℂ\nhi : i ∈ a\n⊢ Tendsto (fun n ↦ i / ↑n + 1) atTop (𝓝 ((fun x ↦ 1) i))",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"Complex.instNormedField",
"PseudoMetricSpace.toUniformSpace",
... | [
"a : Multiset ℂ\ni : ℂ\nhi : i ∈ a\n⊢ Tendsto (fun n ↦ i / ↑n + 1) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 398,
"column": 8
} | {
"line": 398,
"column": 19
} | {
"line": 398,
"column": 20
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀",
"ppTerm": "?m.286",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 19
} | {
"line": 410,
"column": 20
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhxs : x ∈ s\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀",
"ppTerm": "?m.463",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhxs : x ∈ s\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 40
} | {
"line": 427,
"column": 41
} | [
{
"pp": "L : PeriodPair\n⊢ ℘'[L - 0] 0 = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\n⊢ ℘'[L - 0] 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 450,
"column": 17
} | {
"line": 450,
"column": 28
} | {
"line": 450,
"column": 29
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nh₁ : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\nh₂ : x + ↑l ∈ ↑L.lattice\nh₃ : x + ↑l ≠ l₀\n⊢ x ∈ ↑L.lattice",
"ppTerm": "?m.232",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [
"L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nh₁ : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\nh₂ : x + ↑l ∈ ↑L.lattice\nh₃ : x + ↑l ≠ l₀\n⊢ x ∈ L.lattice"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | {
"line": 70,
"column": 0
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real.partialOrder",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"... | [] | norm_num [sigmoid] | Mathlib.Tactic._aux_Mathlib_Tactic_NormNum_Core___elabRules_Mathlib_Tactic_normNum_1 | Mathlib.Tactic.normNum |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | {
"line": 70,
"column": 0
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real.partialOrder",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"... | [] | norm_num [sigmoid] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | {
"line": 70,
"column": 0
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real.partialOrder",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"... | [] | norm_num [sigmoid] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 456,
"column": 6
} | {
"line": 456,
"column": 17
} | {
"line": 456,
"column": 18
} | [
{
"pp": "case a\nL : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\n⊢ x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp... | [
"case a\nL : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\n⊢ x ∈ L.lattice → x = l₀ - ↑l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 459,
"column": 27
} | {
"line": 459,
"column": 38
} | {
"line": 459,
"column": 39
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ L.lattice → x + ↑l = l₀\nH : x + ↑l ∈ L.lattice\n⊢ x ∈ L.lattice",
"ppTerm": "?m.310",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ L.lattice → x + ↑l = l₀\nH : x + ↑l ∈ L.lattice\n⊢ x ∈ L.lattice"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 17
} | {
"line": 303,
"column": 18
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\n⊢... | [
"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\n⊢ ∫ (x : ↑{0}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 14
} | [
{
"pp": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis✝ : sin ∘ arcsin =ᶠ[𝓝[≥] (-1)] id\nthis : HasDerivWithinAt id (cos (arcsin (-1)) * derivWithin arcsin (Ici (-1)) (-1)) (Ici (-1)) (-1)\n⊢ False",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis✝ : sin ∘ arcsin =ᶠ[𝓝[≥] (-1)] id\nthis : HasDerivWithinAt id (cos (arcsin (-1)) * derivWithin arcsin (Ici (-1)) (-1)) (Ici (-1)) (-1)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 58
} | {
"line": 89,
"column": 59
} | [
{
"pp": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\nthis : DifferentiableWithinAt ℝ (fun i ↦ -(arcsin ∘ Neg.neg) i) (Ici (-x)) (-x)\n⊢ x ≠ 1",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"Real",
"id",
"Ne",
"Real.instOne",
"One.... | [
"x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\nthis : DifferentiableWithinAt ℝ (fun i ↦ -(arcsin ∘ Neg.neg) i) (Ici (-x)) (-x)\n⊢ ¬x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff | {
"line": 64,
"column": 42
} | {
"line": 70,
"column": 8
} | {
"line": 72,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f [[a, b]]\n⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.to... | [] | by
rcases le_or_gt a b with hab | hab
· simp only [uIcc_of_le hab] at h
exact integral_deriv_of_contDiffOn_Icc h hab
· simp only [uIcc_of_ge hab.le] at h
rw [integral_symm, integral_deriv_of_contDiffOn_Icc h hab.le]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 27
} | {
"line": 137,
"column": 28
} | [
{
"pp": "n : ℕ\nr : ℝ := (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr : r = (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr1 : r < 1\nthis : HasSum (fun j ↦ r ^ (j + 1) / 3) (1 / (12 * ↑(n + 1) * (↑(n + 1) + 1)))\nj : ℕ\n⊢ 1 / (2 * ↑(j + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (j + 1) ≤ r ^ (j + 1) / 3",
"ppTerm": "?m.359",
"a... | [
"n : ℕ\nr : ℝ := (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr : r = (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr1 : r < 1\nthis : HasSum (fun j ↦ r ^ (j + 1) / 3) (1 / (12 * ↑(n + 1) * (↑(n + 1) + 1)))\nj : ℕ\n⊢ 3 ≤ 2 * (↑j + 1) + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 48
} | {
"line": 153,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ 12⁻¹",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real",
"instOfNatNat",
"Real.log",
"Stirling.stirlingSeq",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"OfNat.of... | [
"n : ℕ\nf : ℕ → ℝ := fun k ↦ log (stirlingSeq (k + 1))\n⊢ log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ 12⁻¹"
] | let f (k : ℕ) : ℝ := log (stirlingSeq (k + 1)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 672,
"column": 10
} | {
"line": 672,
"column": 71
} | {
"line": 672,
"column": 72
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ 1 ∈ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ ∀ x_1 ∈ L.lattice, ¬x_1 = l₀ → ‖z - x‖ < ‖x_1 - x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 188,
"column": 68
} | {
"line": 188,
"column": 93
} | {
"line": 188,
"column": 94
} | [
{
"pp": "x : ℝ\nx_pos : 0 < x\nhx : ∀ (n : ℕ), x ≤ stirlingSeq (n + 1)\n⊢ x ∈ lowerBounds (Set.range (stirlingSeq ∘ succ))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"lowerBounds",
"congrArg",
"Set.ofPred",
"Fu... | [
"x : ℝ\nx_pos : 0 < x\nhx : ∀ (n : ℕ), x ≤ stirlingSeq (n + 1)\n⊢ ∀ (a : ℕ), x ≤ stirlingSeq (a + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 675,
"column": 6
} | {
"line": 675,
"column": 17
} | {
"line": 675,
"column": 18
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x l",
... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ∀ l ∈ L.lattice, ¬l = l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 676,
"column": 4
} | {
"line": 676,
"column": 64
} | {
"line": 676,
"column": 65
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\nthis : ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x ... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\nthis : ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x l\n⊢ ‖z - x‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 689,
"column": 6
} | {
"line": 689,
"column": 28
} | {
"line": 689,
"column": 29
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nthis : |κ⁻¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Niven | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "q : ℚ\n⊢ IsIntegral ℤ (cexp (-(↑q * ↑π) * I))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.instMul",
"id",
"NonUnitalNonAssocR... | [
"q : ℚ\n⊢ IsIntegral ℤ (cexp (-(↑q * ↑π * I)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 698,
"column": 33
} | {
"line": 698,
"column": 58
} | {
"line": 698,
"column": 59
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 699,
"column": 6
} | {
"line": 699,
"column": 42
} | {
"line": 699,
"column": 43
} | [
{
"pp": "L : PeriodPair\nl₀ z : ℂ\nκ : ℝ\nhκ : 1 < κ\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summable fun i ↦ (↑i + 2) * κ ^ (-↑i)\np : ℕ × ↥L.lattice\nhp :\n ¬↑((Equiv.prodCongrLeft fun x ↦\n ... | [
"L : PeriodPair\nl₀ z : ℂ\nκ : ℝ\nhκ : 1 < κ\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summable fun i ↦ (↑i + 2) * κ ^ (-↑i)\np : ℕ × ↥L.lattice\nhp :\n ¬↑((Equiv.prodCongrLeft fun x ↦\n { toFun :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Niven | {
"line": 168,
"column": 46
} | {
"line": 168,
"column": 57
} | {
"line": 168,
"column": 58
} | [
{
"pp": "r : ℚ\nθ : ℝ\nh : ↑r * π = θ\nhcos : ∃ q, cos θ = ↑q\nh_bnd : θ ∈ Set.Icc (0 * π) (1 * π)\n⊢ θ ∈ Set.Icc 0 π",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
... | [
"r : ℚ\nθ : ℝ\nh : ↑r * π = θ\nhcos : ∃ q, cos θ = ↑q\nh_bnd : θ ∈ Set.Icc (0 * π) (1 * π)\n⊢ 0 ≤ θ ∧ θ ≤ π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Niven | {
"line": 178,
"column": 68
} | {
"line": 178,
"column": 87
} | {
"line": 178,
"column": 88
} | [
{
"pp": "r q : ℚ\nhq : cos (↑r * π) = ↑q\n⊢ cos (↑r * π - ↑⌊r⌋ * π) = (-1) ^ ⌊r⌋ * ↑q",
"ppTerm": "?m.164",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"... | [
"r q : ℚ\nhq : cos (↑r * π) = ↑q\n⊢ (-1) ^ ⌊r⌋ * cos (↑r * π) = (-1) ^ ⌊r⌋ * ↑q"
] | cos_sub_int_mul_pi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Finset.Box | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 62
} | {
"line": 34,
"column": 63
} | [
{
"pp": "case ha\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ -↑n ≤ -↑m",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
... | [
"case ha\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ ↑m ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Finset.Box | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 62
} | {
"line": 34,
"column": 63
} | [
{
"pp": "case hb\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ ↑m ≤ ↑n",
"ppTerm": "?hb",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hb\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ ↑m ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 722,
"column": 4
} | {
"line": 722,
"column": 26
} | {
"line": 722,
"column": 27
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2 =\n ∑' (i : ℕ), ((↑i + 1) * (↑l - x) ^ (-↑(i + 2)) - Nat.casesOn i (↑l ^ (-2)) 0) * (z - x) ^ i",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case neg\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ ((z - ↑l) ^ 2)⁻¹ - (↑l ^ 2)⁻¹ =\n ∑' (i : ℕ), (z - x) ^ i * ((↑i + 1) * (↑l - x) ^ (-2 + -↑i) - Nat.rec (↑l ^ 2)⁻¹ (fun n n_ih ↦ 0) i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 723,
"column": 10
} | {
"line": 723,
"column": 35
} | {
"line": 723,
"column": 36
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ ↑l ≠ x",
"ppTerm": "?m.167",
"assigned": true,
"usedConstants": [
"Submodule",
"Membership.mem",
"id",
"Ne",
"Int",
"Complex.addCommGroup",
... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ ¬↑l = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 724,
"column": 10
} | {
"line": 724,
"column": 39
} | {
"line": 724,
"column": 40
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ z - x ∈ Metric.eball 0 ‖↑l - x‖ₑ",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedC... | [
"L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ ‖z - x‖ < ‖↑l - x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 22
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case e'_4\nn : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\n⊢ 2 ^ (n - 1) = sumNodes n (fun i ↦ leadingCoeffC n i) (T ℝ ↑n)",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Polynomial.Chebyshev.T",
"con... | [] | · rw [sumNodes_T_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 715,
"column": 87
} | {
"line": 729,
"column": 56
} | {
"line": 731,
"column": 0
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ ℘[L - l₀] z = ∑' (i : ℕ), (L.weierstrassPExceptSeries l₀ x).coeff i * (z - x) ^ i",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Int.instAddCommGroup",
"AddGroup... | [] | by
trans ∑' (l : L.lattice) (i : ℕ), if l.1 = l₀ then 0 else
((i + 1) * (l.1 - x) ^ (- ↑(i + 2) : ℤ) - i.casesOn (l.1 ^ (-2 : ℤ)) 0) * (z - x) ^ i
· delta weierstrassPExcept
congr 1 with l
split_ifs with h
· simp
simpa [mul_comm] using ((Complex.one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnB... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 750,
"column": 6
} | {
"line": 751,
"column": 13
} | {
"line": 751,
"column": 14
} | [
{
"pp": "case r_le\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ‖x + ↑↑r - x‖ < ‖↑l - x‖",
"ppTerm": "?r_le",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Submodule",
"... | [
"case r_le\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ↑r < ‖↑l - x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 22
} | {
"line": 220,
"column": 0
} | [
{
"pp": "case e'_3\nn : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\n⊢ 2 ^ (n - 1) = sumNodes n (fun i ↦ leadingCoeffC n i) (T ℝ ↑n)",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Polynomial.Chebyshev.T",
"con... | [] | · rw [sumNodes_T_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 754,
"column": 31
} | {
"line": 754,
"column": 42
} | {
"line": 754,
"column": 43
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : z ∈ Metric.eball 0 ↑r\n⊢ ‖z‖ < ↑r",
"ppTerm": "?m.270",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : z ∈ Metric.eball 0 ↑r\n⊢ ‖z‖ < ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 758,
"column": 6
} | {
"line": 758,
"column": 67
} | {
"line": 759,
"column": 8
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : ‖z‖ < ↑r\nthis :\n (∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z‖ < ‖↑l - x‖) →\n HasSum (fun i ↦ (L.weierstrassPExceptSeries l₀ x).coeff i * z ^ i) (℘[L - l₀] (x + z))\nl : ↥L.lattice\nhl : ↑l ≠... | [
"L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : ‖z‖ < ↑r\nthis :\n (∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z‖ < ‖↑l - x‖) →\n HasSum (fun i ↦ (L.weierstrassPExceptSeries l₀ x).coeff i * z ^ i) (℘[L - l₀] (x + z))\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ↑r < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 769,
"column": 2
} | {
"line": 769,
"column": 30
} | {
"line": 770,
"column": 2
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : ℝ\nh₁ : 0 < r\nh₂ : Metric.closedBall l r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants... | [
"L : PeriodPair\nl : ℂ\nr : NNReal\nh₁ : 0 < ↑r\nh₂ : Metric.closedBall l ↑r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l"
] | lift r to NNReal using h₁.le | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 770,
"column": 2
} | {
"line": 770,
"column": 40
} | {
"line": 771,
"column": 4
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : NNReal\nh₁ : 0 < ↑r\nh₂ : Metric.closedBall l ↑r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l",
"ppTerm": "?m.89",
"assigned": false,
"usedC... | [
"L : PeriodPair\nl : ℂ\nr : NNReal\nh₁ : 0 < ↑r\nh₂ : Metric.closedBall l ↑r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 73
} | {
"line": 115,
"column": 74
} | [
{
"pp": "z : ℍ\nc d : ℝ\nhd : 1 ≤ d ^ 2\nH1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2)\n⊢ r z ≤ ‖↑c * ↑z + ↑d‖",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"HMul.hMul",
... | [
"z : ℍ\nc d : ℝ\nhd : 1 ≤ d ^ 2\nH1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2)\n⊢ z.im ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2) ∨ √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 15
} | {
"line": 787,
"column": 16
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘[L - l] l / ↑n ! = if n = 0 then ℘[L - l] l else (↑n + 1) * L.sumInvPow l (n + 2)",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘[L - l] l / ↑n ! = if n = 0 then ℘[L - l] l else (↑n + 1) * L.sumInvPow l (n + 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 118,
"column": 49
} | {
"line": 129,
"column": 14
} | {
"line": 131,
"column": 0
} | [
{
"pp": "x : Fin 2 → ℤ\nhx : x ≠ 0\n⊢ 1 ≤ (↑(x 0) / ‖x‖) ^ 2 ∨ 1 ≤ (↑(x 1) / ‖x‖) ^ 2",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"max_choice",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"GroupWithZero.toM... | [] | by
refine (max_choice (x 0).natAbs (x 1).natAbs).imp (fun H0 ↦ ?_) (fun H1 ↦ ?_)
· have : x 0 ≠ 0 := by
rwa [← norm_ne_zero_iff, norm_eq_max_natAbs, H0, Nat.cast_ne_zero, Int.natAbs_ne_zero] at hx
simp only [norm_eq_max_natAbs, H0, Nat.cast_natAbs, Int.cast_abs, div_pow, sq_abs, ne_eq,
OfNat.ofNat_n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 55
} | {
"line": 84,
"column": 56
} | [
{
"pp": "case inr\nn : ℤ\nx : ℝ\nhx : 1 ≤ |x|\nthis : ∀ (n : ℤ) {x : ℝ}, 1 ≤ |x| → 0 ≤ x → 1 ≤ |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 ≤ |eval x (T ℝ n)|",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nn : ℤ\nx : ℝ\nhx : 1 ≤ |x|\nthis : ∀ (n : ℤ) {x : ℝ}, 1 ≤ |x| → 0 ≤ x → 1 ≤ |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 ≤ |eval x (T ℝ n)|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 55
} | {
"line": 90,
"column": 56
} | [
{
"pp": "case inr\nn : ℤ\nhn : n ≠ 0\nx : ℝ\nhx : 1 < |x|\nthis : ∀ {n : ℤ}, n ≠ 0 → ∀ {x : ℝ}, 1 < |x| → 0 ≤ x → 1 < |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 < |eval x (T ℝ n)|",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nn : ℤ\nhn : n ≠ 0\nx : ℝ\nhx : 1 < |x|\nthis : ∀ {n : ℤ}, n ≠ 0 → ∀ {x : ℝ}, 1 < |x| → 0 ≤ x → 1 < |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 < |eval x (T ℝ n)|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 95,
"column": 32
} | {
"line": 95,
"column": 43
} | {
"line": 95,
"column": 44
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\nx : ℝ\n⊢ |eval x (T ℝ n)| ≤ 1 → |x| ≤ 1",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℤ\nhn : n ≠ 0\nx : ℝ\n⊢ |eval x (T ℝ n)| ≤ 1 → |x| ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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