module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 83,
"column": 6
} | {
"line": 87,
"column": 12
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case hbc\nb : ℂ\nhb : 0 < b.re\nc✝ T✝ : ℝ\nhT✝ : 0 ≤ T✝\nT : ℝ\nhT : 0 ≤ T\nc y : ℝ\nhy : |y| ≤ |c|\n⊢ 2 * b.im * y ≤ 2 * |b.im| * |c|",
"ppTerm": "?hbc✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul"... | [] | (conv_lhs => rw [mul_assoc]); (conv_rhs => rw [mul_assoc])
gcongr _ * ?_
refine (le_abs_self _).trans ?_
rw [abs_mul]
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 83,
"column": 6
} | {
"line": 87,
"column": 12
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case hbc\nb : ℂ\nhb : 0 < b.re\nc✝ T✝ : ℝ\nhT✝ : 0 ≤ T✝\nT : ℝ\nhT : 0 ≤ T\nc y : ℝ\nhy : |y| ≤ |c|\n⊢ 2 * b.im * y ≤ 2 * |b.im| * |c|",
"ppTerm": "?hbc✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Semigroup.toMul"... | [] | (conv_lhs => rw [mul_assoc]); (conv_rhs => rw [mul_assoc])
gcongr _ * ?_
refine (le_abs_self _).trans ?_
rw [abs_mul]
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Pi | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 45
} | {
"line": 151,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝³ : Fintype ι\nX : ι → Type u_3\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\nf : X i → E\nhf : AEStronglyMeasurable f (μ i)\n⊢ ∫ (x : (i... | [] | rw [← (measurePreserving_eval μ i).map_eq, integral_map]
· exact Measurable.aemeasurable (by fun_prop)
· rwa [(measurePreserving_eval μ i).map_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Pi | {
"line": 147,
"column": 2
} | {
"line": 149,
"column": 45
} | {
"line": 151,
"column": 0
} | [
{
"pp": "ι : Type u_2\ninst✝³ : Fintype ι\nX : ι → Type u_3\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\nf : X i → E\nhf : AEStronglyMeasurable f (μ i)\n⊢ ∫ (x : (i... | [] | rw [← (measurePreserving_eval μ i).map_eq, integral_map]
· exact Measurable.aemeasurable (by fun_prop)
· rwa [(measurePreserving_eval μ i).map_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1148,
"column": 4
} | {
"line": 1148,
"column": 15
} | {
"line": 1148,
"column": 16
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1149,
"column": 27
} | {
"line": 1149,
"column": 38
} | {
"line": 1149,
"column": 39
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 64
} | {
"line": 87,
"column": 4
} | [
{
"pp": "V : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Inte... | [
"case e'_3\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nh'f : Integ... | convert! tendsto_integral_cexp_sq_smul this using 4 with c w | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 17
} | {
"line": 100,
"column": 18
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc T : ℝ\nhT : 0 ≤ T\nvert_norm_bound :\n ∀ {T : ℝ},\n 0 ≤ T →\n ∀ {c y : ℝ},\n |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))\ny : ℝ\nhy : y ∈ uIoc 0 c\n⊢ |y| ≤ |c|",
"ppTerm": "?m.477",
"assigned"... | [
"b : ℂ\nhb : 0 < b.re\nc T : ℝ\nhT : 0 ≤ T\nvert_norm_bound :\n ∀ {T : ℝ},\n 0 ≤ T →\n ∀ {c y : ℝ},\n |y| ≤ |c| → ‖cexp (-b * (↑T + ↑y * I) ^ 2)‖ ≤ rexp (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2))\ny : ℝ\nhy : y ∈ uIoc 0 c\n⊢ |y| ≤ |c|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1198,
"column": 6
} | {
"line": 1198,
"column": 72
} | {
"line": 1198,
"column": 73
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulC... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1249,
"column": 30
} | {
"line": 1250,
"column": 65
} | {
"line": 1250,
"column": 66
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : ProperSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedS... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : ProperSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 21
} | {
"line": 299,
"column": 22
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)\n⊢ ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f (-x)",
"ppTerm": "?m.133",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | [
"b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : ∫ (x : ℝ), cexp (-b * ↑x ^ 2) = (↑π / b) ^ (1 / 2)\n⊢ ∫ (x : ℝ) in Iic 0, cexp (-(b * ↑x ^ 2)) = ∫ (x : ℝ) in Ioi 0, cexp (-(b * ↑x ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 304,
"column": 39
} | {
"line": 304,
"column": 50
} | {
"line": 304,
"column": 51
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)\nh_eq : ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f x\n⊢ (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)",
"ppTerm": "?m.249",
"... | [
"b : ℂ\nhb : 0 < b.re\nf : ℝ → ℂ := fun x ↦ cexp (-b * ↑x ^ 2)\nfull_integral : (∫ (x : ℝ) in Ioi 0, cexp (-b * ↑x ^ 2)) * 2 = (↑π / b) ^ (1 / 2)\nh_eq : ∫ (x : ℝ) in Iic 0, f x = ∫ (x : ℝ) in Ioi 0, f x\n⊢ (∫ (x : ℝ) in Ioi 0, cexp (-(b * ↑x ^ 2))) * 2 = (↑π / b) ^ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1313,
"column": 2
} | {
"line": 1313,
"column": 13
} | {
"line": 1313,
"column": 14
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nC : ℝ\nleft✝ : 0 < C\nhC : ∀ (x : E), ‖x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1338,
"column": 17
} | {
"line": 1338,
"column": 89
} | {
"line": 1339,
"column": 4
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\np : ℝ≥0∞\nμ : Measure E\nhp₁ : p ≠ 0\nhp₂ :... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\np : ℝ≥0∞\nμ : Measure E\nhp₁ : p ≠ 0\nhp₂ : p ≠ ∞\nhμ :... | MeasureTheory.MemLp.eLpNorm_eq_integral_rpow_norm hp₁ hp₂ (f.memLp p μ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1343,
"column": 2
} | {
"line": 1343,
"column": 13
} | {
"line": 1343,
"column": 14
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nhμ : μ.HasTemperateGrowth\n⊢... | [
"E : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : SecondCountableTopologyEither E F\nf : 𝓢(E, F)\nμ : Measure E\nhμ : μ.HasTemperateGrowth\n⊢ ‖f.toLp 1 μ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1353,
"column": 15
} | {
"line": 1353,
"column": 33
} | {
"line": 1353,
"column": 34
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : MeasurableSpace E\ninst✝² : OpensMeasurableSpace E\ninst✝¹ : SecondCountableTopologyEither E F\np : ℝ≥0∞\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst... | [
"E : Type u_5\nF : Type u_6\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : MeasurableSpace E\ninst✝² : OpensMeasurableSpace E\ninst✝¹ : SecondCountableTopologyEither E F\np : ℝ≥0∞\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝ : μ.IsOpen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 56
} | {
"line": 341,
"column": 57
} | [
{
"pp": "case e'_2\n⊢ Gamma (1 / 2) = ↑(Real.Gamma (1 / 2))",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instD... | [
"case e'_2\n⊢ Gamma 2⁻¹ = ↑(Real.Gamma 2⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1365,
"column": 2
} | {
"line": 1365,
"column": 13
} | {
"line": 1365,
"column": 14
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : SMulCommClass ℝ 𝕜 F\nin... | [
"𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : SMulCommClass ℝ 𝕜 F\ninst✝ : Second... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1387,
"column": 2
} | {
"line": 1387,
"column": 82
} | {
"line": 1388,
"column": 2
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : SecondCountableTopologyEither E F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\np... | [
"E : Type u_5\nF : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : OpensMeasurableSpace E\ninst✝³ : SecondCountableTopologyEither E F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\np : ℝ≥0∞\nhp ... | refine (mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall).2 fun ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 363,
"column": 14
} | {
"line": 363,
"column": 45
} | {
"line": 363,
"column": 46
} | [
{
"pp": "case succ\nk : ℕ\n⊢ Gamma (↑(k + 1) + 1 / 2) = ↑(2 * (k + 1) - 1)‼ * √π / 2 ^ (k + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"AddMonoid.toAddSemigro... | [
"case succ\nk : ℕ\n⊢ Gamma (↑k + 1 + 1 / 2) = ↑(2 * k + 1)‼ * √π / 2 ^ (k + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 152,
"column": 4
} | {
"line": 154,
"column": 55
} | {
"line": 154,
"column": 56
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\nI₁ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nHI₁ : I₁ = fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nI₂ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)\nI₄ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (↑T + ↑y * I) ^ 2... | [
"b : ℂ\nhb : 0 < b.re\nc : ℝ\nI₁ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nHI₁ : I₁ = fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * (↑x + ↑c * I) ^ 2)\nI₂ : ℝ → ℂ := fun T ↦ ∫ (x : ℝ) in -T..T, cexp (-b * ↑x ^ 2)\nI₄ : ℝ → ℂ := fun T ↦ ∫ (y : ℝ) in 0..c, cexp (-b * (↑T + ↑y * I) ^ 2)\nI₅ : ℝ → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 9
} | {
"line": 193,
"column": 10
} | [
{
"pp": "b : ℂ\nhb : b.re < 0\nc d : ℂ\nhb' : b ≠ 0\nH : ¬Integrable (fun x ↦ cexp (b * ↑x ^ 2 + c * ↑x + d)) volume\n⊢ False",
"ppTerm": "?m.73",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : ℂ\nhb : b.re < 0\nc d : ℂ\nhb' : b ≠ 0\nH : ¬Integrable (fun x ↦ cexp (b * ↑x ^ 2 + c * ↑x + d)) volume\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 197,
"column": 27
} | {
"line": 197,
"column": 38
} | {
"line": 197,
"column": 39
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc d : ℂ\n⊢ (-b).re < 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp_1",
"Real.instZero",
"instIsLeftCancelAddOfAddLeftR... | [
"b : ℂ\nhb : 0 < b.re\nc d : ℂ\n⊢ 0 < b.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 66
} | {
"line": 212,
"column": 67
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℂ\nthis : b ≠ 0\n⊢ (-↑π * b).re < 0",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
... | [
"b : ℂ\nhb : 0 < b.re\nc : ℂ\nthis : b ≠ 0\n⊢ 0 < π * b.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 49
} | {
"line": 233,
"column": 50
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\n⊢ (𝓕 fun x ↦ cexp (-↑π * b * ↑x ^ 2)) = fun t ↦ 1 / b ^ (1 / 2) * cexp (-↑π / b * ↑t ^ 2)",
"ppTerm": "?m.94",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : ℂ\nhb : 0 < b.re\n⊢ (𝓕 fun x ↦ cexp (-↑π * b * ↑x ^ 2)) = fun t ↦ 1 / b ^ (1 / 2) * cexp (-↑π / b * ↑t ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 289,
"column": 29
} | {
"line": 289,
"column": 40
} | {
"line": 289,
"column": 41
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nb : ι → ℂ\nhb : ∀ (i : ι), 0 < (b i).re\nc : ι → ℂ\ni : ι\n⊢ (-b i).re < 0",
"ppTerm": "?m.198",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Left.neg_neg_iff._simp... | [
"ι : Type u_2\ninst✝ : Fintype ι\nb : ι → ℂ\nhb : ∀ (i : ι), 0 < (b i).re\nc : ι → ℂ\ni : ι\n⊢ 0 < (b i).re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 13
} | {
"line": 332,
"column": 14
} | [
{
"pp": "b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\n⊢ ∫ (v : V), cexp (-b * ↑‖v‖ ^ 2) = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2)",
"ppTerm": "?m.66",
"assigned": true,
... | [
"b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\n⊢ ∫ (v : V), cexp (-(b * ↑‖v‖ ^ 2)) = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 13
} | {
"line": 364,
"column": 14
} | [
{
"pp": "b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\nw : V\n⊢ 𝓕 (fun v ↦ cexp (-b * ↑‖v‖ ^ 2)) w = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-↑π ^ 2 * ↑‖w‖ ^ 2 / b)",
... | [
"b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\nw : V\n⊢ 𝓕 (fun v ↦ cexp (-(b * ↑‖v‖ ^ 2))) w = (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp (-(↑π ^ 2 * ↑‖w‖ ^ 2) / b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | {
"line": 176,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\nin... | [
"𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 13
} | {
"line": 289,
"column": 14
} | [
{
"pp": "E : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℂ E\nV : Type u_3\ninst✝¹⁰ : NormedAddCommGroup V\ninst✝⁹ : InnerProductSpace ℝ V\ninst✝⁸ : FiniteDimensional ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℂ F... | [
"E : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℂ E\nV : Type u_3\ninst✝¹⁰ : NormedAddCommGroup V\ninst✝⁹ : InnerProductSpace ℝ V\ninst✝⁸ : FiniteDimensional ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℂ F\nG : Type u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 13
} | {
"line": 306,
"column": 14
} | [
{
"pp": "V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖(𝓕 f).toBoundedContinuousFunction x‖ ≤ ‖f.to... | [
"V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖(𝓕 f) x‖ ≤ ‖f.toLp 1 volume‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 311,
"column": 8
} | {
"line": 311,
"column": 19
} | {
"line": 311,
"column": 20
} | [
{
"pp": "V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ 0 (⇑(𝓕 f)) x‖ ≤ ‖... | [
"V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖(𝓕 f) x‖ ≤ ‖f.toLp 1 volume‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 79
} | {
"line": 328,
"column": 4
} | [
{
"pp": "V : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : FiniteDimensional ℝ V\ninst✝⁴ : MeasurableSpace V\ninst✝³ : BorelSpace V\nH : Type u_4\ninst✝² : NormedAddCommGroup H\ninst✝¹ : InnerProductSpace ℂ H\ninst✝ : CompleteSpace H\nf : 𝓢(V, H)\n⊢ Complex.ofRealLI (∫ (ξ : ... | [
"V : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : FiniteDimensional ℝ V\ninst✝⁴ : MeasurableSpace V\ninst✝³ : BorelSpace V\nH : Type u_4\ninst✝² : NormedAddCommGroup H\ninst✝¹ : InnerProductSpace ℂ H\ninst✝ : CompleteSpace H\nf : 𝓢(V, H)\n⊢ ∫ (x : V), ↑‖(𝓕 f) x‖ ^ 2 = ∫ (x : V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.Support | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | {
"line": 120,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nF : Type u_6\nV : Type u_10\ninst✝³ : FunLike F α β\ninst✝² : TopologicalSpace α\ninst✝¹ : Zero β\ninst✝ : Zero V\nf : F → V\nx : α\n⊢ x ∈ dsupport f ↔ ∀ (i : Set α), x ∈ i ∧ IsOpen[inst✝²] i → ¬IsVanishingOn f i",
"ppTerm": "?m.44",
"assigned": true,
"usedConsta... | [
"α : Type u_2\nβ : Type u_3\nF : Type u_6\nV : Type u_10\ninst✝³ : FunLike F α β\ninst✝² : TopologicalSpace α\ninst✝¹ : Zero β\ninst✝ : Zero V\nf : F → V\nx : α\n⊢ x ∈ dsupport f ↔ ∀ (i : Set α), x ∈ i → IsOpen[inst✝²] i → ¬IsVanishingOn f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 91
} | {
"line": 221,
"column": 4
} | [
{
"pp": "d : Type u_1\ninst✝¹ : Fintype d\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\n⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"d : Type u_1\ninst✝¹ : Fintype d\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\n⊢ (span ℂ (range (mFourierLp p))).topologicalClosure = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 55
} | {
"line": 287,
"column": 56
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ mFourierCoeff (↑↑f) i • mFourierLp 2 i) f",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.to... | [
"d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ mFourierCoeff (↑↑f) i • mFourierBasis i) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 36
} | {
"line": 303,
"column": 37
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ ‖mFourierCoeff (↑↑f) i‖ ^ 2) (∫ (t : UnitAddTorus d), ‖↑↑f t‖ ^ 2)",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\n⊢ HasSum (fun i ↦ ‖mFourierCoeff (↑↑f) i‖ ^ 2) (∫ (t : UnitAddTorus d), ‖↑↑f t‖ ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 54
} | {
"line": 324,
"column": 55
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nsum_L2 : HasSum (fun i ↦ mFourierCoeff (⇑f) i • mFourierLp 2 i) ((ContinuousMap.toLp 2 volume ℂ) f)\n⊢ Summable fun a ↦ ‖mFourierCoeff (⇑f) a • mFourier a‖",
"ppTerm": "?m.83",
"assigned": true,
"use... | [
"d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nsum_L2 : HasSum (fun i ↦ mFourierCoeff (⇑f) i • mFourierLp 2 i) ((ContinuousMap.toLp 2 volume ℂ) f)\n⊢ Summable fun a ↦ ‖mFourierCoeff (⇑f) a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 55
} | {
"line": 176,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℂ F\ninst✝² : CompleteSpace F\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nf : ↥(Lp F p μ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℂ F\ninst✝² : CompleteSpace F\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nf : ↥(Lp F p μ)\ng : 𝓢(E,... | filter_upwards [g.coeFn_toLp (1 - p⁻¹)⁻¹ μ] with x hg | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Fourier.Convolution | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 24
} | {
"line": 48,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\... | [
"𝕜 : Type u_1\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedAddCommGroup F₂\ninst✝⁷ : NormedAddCommGroup F₃\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 222,
"column": 4
} | {
"line": 227,
"column": 27
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\nin... | [] | intro g g_smooth g_cpt
have hg₁ : HasCompactSupport (Complex.ofRealCLM ∘ g) := g_cpt.comp_left rfl
have hg₂ : ContDiff ℝ ∞ (Complex.ofRealCLM ∘ g) := by fun_prop
calc
_ = toTemperedDistributionCLM F μ p f (hg₁.toSchwartzMap hg₂) := by simp
_ = _ := by simp [hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 222,
"column": 4
} | {
"line": 227,
"column": 27
} | {
"line": 229,
"column": 0
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\nin... | [] | intro g g_smooth g_cpt
have hg₁ : HasCompactSupport (Complex.ofRealCLM ∘ g) := g_cpt.comp_left rfl
have hg₂ : ContDiff ℝ ∞ (Complex.ofRealCLM ∘ g) := by fun_prop
calc
_ = toTemperedDistributionCLM F μ p f (hg₁.toSchwartzMap hg₂) := by simp
_ = _ := by simp [hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Inner | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 85
} | {
"line": 151,
"column": 86
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\n𝕜 : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : RCLike 𝕜\nf : κ → ι → 𝕜\nhf : ∀ (k : κ), f k ≠ 0\nhinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0\nh✝ : Nonempty ι\n⊢ Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0",
"ppTerm": "?m.81",
"assigned": false,
"usedConstan... | [
"ι : Type u_1\nκ : Type u_2\n𝕜 : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : RCLike 𝕜\nf : κ → ι → 𝕜\nhf : ∀ (k : κ), f k ≠ 0\nhinner : Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫ₙ_[𝕜] = 0\nh✝ : Nonempty ι\n⊢ Pairwise fun k₁ k₂ ↦ ⟪f k₁, f k₂⟫_[𝕜] = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 30
} | {
"line": 66,
"column": 31
} | [
{
"pp": "case neg\nG : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\nψ₁ ψ₂ : AddChar G R\nh : ¬ψ₁ = ψ₂\nthis : ψ₂ * ψ₁⁻¹ ≠ 1\n⊢ 𝔼 i, ψ₂ i * (ψ₁ i)⁻¹ = 0",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"case neg\nG : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\nψ₁ ψ₂ : AddChar G R\nh : ¬ψ₁ = ψ₂\nthis : ψ₂ * ψ₁⁻¹ ≠ 1\n⊢ 𝔼 i, ψ₂ i * (ψ₁ i)⁻¹ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 55
} | {
"line": 86,
"column": 4
} | [
{
"pp": "G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\n⊢ card (AddChar G R) ≤ card G",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Fintype G\n⊢ card (AddChar G R) ≤ card G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 35
} | {
"line": 98,
"column": 8
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [
"E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 35
} | {
"line": 101,
"column": 46
} | {
"line": 101,
"column": 47
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [
"E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 91
} | {
"line": 101,
"column": 92
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [
"E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E\ninst✝⁵ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 13
} | {
"line": 534,
"column": 14
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ (𝓕 ((toTemperedD... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ ∫ (x : E), (𝓕 g) x • f x = ∫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.Convolution | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | [] | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Fourier.Convolution | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | [] | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.Convolution | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 62
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case hf\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCommGrou... | [] | exact f.integrable.integrable_convolution B g.integrable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 76
} | {
"line": 68,
"column": 4
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx y : ℤ\nh : zmod n ↑x = zmod n ↑y\nhn : ↑n ≠ 0\n⊢ ↑x = ↑y",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"id",
"Int",
"AddGroupWithOne.toIntCast",
"Nat.cast",
"ZMod",
... | [
"n : ℕ\ninst✝ : NeZero n\nx y : ℤ\nh : zmod n ↑x = zmod n ↑y\nhn : ↑n ≠ 0\n⊢ x ≡ y [ZMOD ↑n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 86,
"column": 50
} | {
"line": 86,
"column": 74
} | {
"line": 86,
"column": 75
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : DecidableEq ι\nn : ι → ℕ\ninst✝ : ∀ (i : ι), NeZero (n i)\nf g : (i : ι) → ZMod (n i)\nh : (fun i ↦ zmod (n i) (f i)) = fun i ↦ zmod (n i) (g i)\n⊢ f = g",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"ZMod",
"_privat... | [
"ι : Type u_2\ninst✝¹ : DecidableEq ι\nn : ι → ℕ\ninst✝ : ∀ (i : ι), NeZero (n i)\nf g : (i : ι) → ZMod (n i)\nh : (fun i ↦ zmod (n i) (f i)) = fun i ↦ zmod (n i) (g i)\n⊢ ∀ (x : ι), f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ (∀ (ψ : AddChar α ℂ), ψ a = 1) ↔ a = 0",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ (∀ (ψ : AddChar α ℂ), ψ a = 1) ↔ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 13
} | {
"line": 190,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\na : α\n⊢ ∑ ψ, ψ a = if a = 0 then ↑(Fintype.card α) else 0",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\na : α\n⊢ ∑ ψ, ψ a = if a = 0 then ↑(Fintype.card α) else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\na : α\n⊢ 𝔼 ψ, ψ a = if a = 0 then 1 else 0",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\na : α\n⊢ 𝔼 ψ, ψ a = if a = 0 then 1 else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 13
} | {
"line": 240,
"column": 14
} | [
{
"pp": "R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : Module.Finite ℤ R\nA B : Submodule R K\nn : ℕ\nhn : n ≠ 0\nhfg : A.FG\nh : ∀ ⦃x : K⦄, x ∈ map (LinearMap.mulLeft R ↑n) A → x ∈ B\nthis✝ : A.toAddSubgroup.FG\nthis : (AddSubgroup.map (nsmulAddMonoidHom n)... | [
"R : Type u_1\nK : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : Module.Finite ℤ R\nA B : Submodule R K\nn : ℕ\nhn : n ≠ 0\nhfg : A.FG\nh : ∀ ⦃x : K⦄, x ∈ map (LinearMap.mulLeft R ↑n) A → x ∈ B\nthis✝ : A.toAddSubgroup.FG\nthis : (AddSubgroup.map (nsmulAddMonoidHom n) A.toAddSubg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Arithmeti... | [] | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Arithmeti... | [] | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 12
} | {
"line": 134,
"column": 0
} | [
{
"pp": "n m : ℕ\nhn : n ≠ 0\nhm : m ≠ 0\nhnm : n.Coprime m\n⊢ μ (n * m) = μ n * μ m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"ite_zero_mul_ite_zero",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"MulZeroClass.toMul",
... | [] | simp only [moebius, coe_mk, squarefree_mul hnm, ite_zero_mul_ite_zero, cardFactors_mul hn hm,
pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 231,
"column": 6
} | {
"line": 232,
"column": 18
} | {
"line": 233,
"column": 6
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\nf' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }\ng' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }\nn : ℕ\n⊢ (μ • g') (n + 1) = f' (n + 1) ↔ n + 1 > 0 → ∑ x ... | [
"case succ\nR : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\nf' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }\ng' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }\nn : ℕ\n⊢ (∑ x ∈ (n + 1).divisorsAntidiagonal, μ x.1 • if x.2 = 0 then... | simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
ite_false] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 113,
"column": 4
} | {
"line": 115,
"column": 12
} | {
"line": 115,
"column": 13
} | [
{
"pp": "case e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f (... | [
"case e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f (x + ↑n) = (∑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 66
} | {
"line": 117,
"column": 67
} | [
{
"pp": "E : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhf1 :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopolog... | [
"E : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhf1 :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 43
} | {
"line": 308,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nP : ℕ → Prop\nn : ℕ\nh : n > 0 → n ∈ s → P n\nhn : n ∈ s\nhs₀ : n ≤ 0\n⊢ 0 ∈ s",
"ppTerm": "?m.86",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nP : ℕ → Prop\nn : ℕ\nh : n > 0 → n ∈ s → P n\nhn : n ∈ s\nhs₀ : n ≤ 0\n⊢ 0 ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 25
} | {
"line": 309,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nhs₀ : 0 ∉ s\nthis : ∀ (P : ℕ → Prop), (∀ n ∈ s, P n) ↔ ∀ n > 0, n ∈ s → P n\n⊢ (∀ n ∈ s, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n ∈ s, ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n",
"ppTerm": "?m.6... | [
"R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nhs₀ : 0 ∉ s\nthis : ∀ (P : ℕ → Prop), (∀ n ∈ s, P n) ↔ ∀ n > 0, n ∈ s → P n\n⊢ (∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 328,
"column": 29
} | {
"line": 328,
"column": 54
} | {
"line": 328,
"column": 55
} | [
{
"pp": "M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : Nontrivial M₀\nl : ℕ\nhl : 0 ^ l = 1\n⊢ 0 ∣ l",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Nat.instSemigroupWithZero",
"SemigroupWithZero.toMulZeroClass",
"id",
"ins... | [
"M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : Nontrivial M₀\nl : ℕ\nhl : 0 ^ l = 1\n⊢ l = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 342,
"column": 27
} | {
"line": 342,
"column": 77
} | {
"line": 342,
"column": 78
} | [
{
"pp": "M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\nn : ℕ\nζ : M₀\nhζ : IsPrimitiveRoot ζ n\nα : M₀\nhα : α ≠ 0\ni : ℕ\nhi : i ∈ ↑(range n)\nj : ℕ\nhj : j ∈ ↑(range n)\ne : (fun x ↦ ζ ^ x * α) i = (fun x ↦ ζ ^ x * α) j\n⊢ (fun x ↦ ζ ^ x) i = (fun x ↦ ζ ^ x) j",
"ppTerm": "?m.... | [
"M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\nn : ℕ\nζ : M₀\nhζ : IsPrimitiveRoot ζ n\nα : M₀\nhα : α ≠ 0\ni : ℕ\nhi : i ∈ ↑(range n)\nj : ℕ\nhj : j ∈ ↑(range n)\ne : (fun x ↦ ζ ^ x * α) i = (fun x ↦ ζ ^ x * α) j\n⊢ ζ ^ i = ζ ^ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 465,
"column": 22
} | {
"line": 465,
"column": 61
} | {
"line": 465,
"column": 62
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh✝ h : IsPrimitiveRoot ζ k\n⊢ ∀ (x y : ℤ),\n Additive.ofMul ⟨(fun x ↦ ζ ^ x) (x + y), ⋯⟩ =\n Addi... | [] | by intro i j; simp only [zpow_add]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 22
} | {
"line": 112,
"column": 23
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx : ℂˣ\nhn0 : ↑n ≠ 0\nh : ↑x ^ n = 1\n⊢ ∃ i < n, cexp (2 * ↑π * I / ↑n) ^ i = ↑x",
"ppTerm": "?m.89",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\ninst✝ : NeZero n\nx : ℂˣ\nhn0 : ↑n ≠ 0\nh : ↑x ^ n = 1\n⊢ ∃ i < n, cexp (2 * ↑π * I / ↑n) ^ i = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 553,
"column": 56
} | {
"line": 562,
"column": 24
} | {
"line": 564,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ ξ : Rˣ\nh : IsPrimitiveRoot ζ k\nhξ : ξ ∈ rootsOfUnity k R\n⊢ ∃ i < k, ζ ^ i = ξ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"MulOne.toOne",
"In... | [] | by
obtain ⟨n, rfl⟩ : ∃ n : ℤ, ζ ^ n = ξ := by rwa [← h.zpowers_eq] at hξ
have hk0 : (0 : ℤ) < k := mod_cast NeZero.pos k
let i := n % k
have hi0 : 0 ≤ i := Int.emod_nonneg _ (ne_of_gt hk0)
lift i to ℕ using hi0 with i₀ hi₀
refine ⟨i₀, ?_, ?_⟩
· zify; rw [hi₀]; exact Int.emod_lt_of_pos _ hk0
· rw [← zpow... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 65
} | {
"line": 117,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 65
} | {
"line": 117,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 61
} | {
"line": 608,
"column": 62
} | [
{
"pp": "case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ (Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)).card",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConsta... | [
"case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 192,
"column": 2
} | {
"line": 198,
"column": 38
} | {
"line": 199,
"column": 2
} | [
{
"pp": "case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\n⊢ cyclotomic' k K ∈ li... | [
"case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nBint : B ∈ lifts (Int.castRingHom ... | have Bint : B ∈ lifts (Int.castRingHom K) := by
refine Subsemiring.prod_mem (lifts (Int.castRingHom K)) ?_
intro x hx
have xsmall := (Nat.mem_properDivisors.1 hx).2
obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hx).1
rw [mul_comm] at hd
exact ihk x xsmall (h.pow hpos hd) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 74
} | {
"line": 206,
"column": 75
} | [
{
"pp": "case right\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ⋯\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.castRingHom K) B₁ = B\nleft✝ : B₁.... | [
"case right\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.castRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 723,
"column": 6
} | {
"line": 723,
"column": 44
} | {
"line": 723,
"column": 45
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R... | [
"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R)\nhr : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 728,
"column": 6
} | {
"line": 728,
"column": 44
} | {
"line": 728,
"column": 45
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R... | [
"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R)\nhr : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 345,
"column": 2
} | {
"line": 346,
"column": 47
} | {
"line": 346,
"column": 48
} | [
{
"pp": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\ninteger : ∏ i ∈ n.divisors, cyclotomic i ℤ = X ^ n - 1\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\ninteger : ∏ i ∈ n.divisors, cyclotomic i ℤ = X ^ n - 1\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 351,
"column": 4
} | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 30
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nthis : cyclotomic n ℤ ∣ X ^ n - 1\n⊢ cyclotomic n R ∣ X ^ n - 1",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nR : Type u_1\ninst✝ : Ring R\nthis : cyclotomic n ℤ ∣ X ^ n - 1\n⊢ cyclotomic n R ∣ X ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 29
} | {
"line": 362,
"column": 30
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []... | [
"n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 69
} | {
"line": 366,
"column": 0
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Int.instAddCommMonoid",
"Finset.prod_erase_mul",
"Polynomial.in... | [] | rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
cyclotomic_one, geom_sum_mul, prod_cyclotomic_eq_X_pow_sub_one h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 99
} | {
"line": 371,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : ℕ\nhp : Fact (Nat.Prime p)\nthis : cyclotomic p ℤ = ∑ i ∈ range p, X ^ i\n⊢ cyclotomic p R = ∑ i ∈ range p, X ^ i",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Ring R\np : ℕ\nhp : Fact (Nat.Prime p)\nthis : cyclotomic p ℤ = ∑ i ∈ range p, X ^ i\n⊢ cyclotomic p R = ∑ i ∈ range p, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 99
} | {
"line": 389,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nthis : cyclotomic d ℤ ∣ ∑ i ∈ range n, X ^ i\n⊢ cyclotomic d R ∣ ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nthis : cyclotomic d ℤ ∣ ∑ i ∈ range n, X ^ i\n⊢ cyclotomic d R ∣ ∑ i ∈ range n, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 878,
"column": 4
} | {
"line": 878,
"column": 92
} | {
"line": 880,
"column": 2
} | [
{
"pp": "G : Type u_7\nG' : Type u_8\ninst✝³ : Group G\ninst✝² : IsCyclic G\ninst✝¹ : Finite G\ninst✝ : CommGroup G'\na : G\nha : a ≠ 1\ninst : Fintype G := Fintype.ofFinite G\nζ : G'\nhζ : IsPrimitiveRoot ζ (Nat.card G)\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ orderOf ζ ∣ orderOf g",
"ppTerm": "?m... | [] | rw [← hζ.eq_orderOf, orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 88,
"column": 39
} | {
"line": 88,
"column": 83
} | {
"line": 88,
"column": 84
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : X - C μ ∣ cyclotomic n K\nhμn : orderOf μ ∣ n\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : X - C μ ∣ cyclotomic i K\nhio : i ∣ orderOf μ\nkey : i < n\nkey' : i ∣ n\n⊢ {i, n} ⊆ n.divisors",
"ppTerm": "?m.1... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : X - C μ ∣ cyclotomic n K\nhμn : orderOf μ ∣ n\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : X - C μ ∣ cyclotomic i K\nhio : i ∣ orderOf μ\nkey : i < n\nkey' : i ∣ n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 49
} | {
"line": 169,
"column": 50
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nm n✝ : ℕ\nh : IsPrimitiveRoot μ n✝\nhn : Nat.Coprime 0 n✝\n⊢ μ = μ ^ 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOne... | [
"case refine_1\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nm n✝ : ℕ\nh : IsPrimitiveRoot μ n✝\nhn : Nat.Coprime 0 n✝\n⊢ μ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 38
} | {
"line": 149,
"column": 38
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn m : ℕ\nhzero : n ≠ 0\nthis : NeZero n\nhnm : cyclotomic n ℂ = cyclotomic m ℂ\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n\nhroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))\nhmzero... | [
"case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn m : ℕ\nhzero : n ≠ 0\nthis : NeZero n\nhnm : cyclotomic n ℂ = cyclotomic m ℂ\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n\nhroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m\nhmzero : NeZero m\n⊢ n =... | isRoot_cyclotomic_iff (R := ℂ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 57
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ (aeval μ) (cyclotomic n ℤ) = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Ring.toNonAssocRing",
"congrArg",
"... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ eval μ (cyclotomic n K) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 70
} | {
"line": 523,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔\n (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n +... | [
"R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔\n (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p,... | rw [← (eq_cyclotomic_iff (pow_pos hp.pos (n_n + 1 + 1)) _).mpr ?_] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 40
} | {
"line": 175,
"column": 41
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ (cyclotomic n ℤ).natDegree ≤ (minpoly ℤ μ).natDegree",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.natDegree_cyclotomic"... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ n.totient ≤ (minpoly ℤ μ).natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | {
"line": 242,
"column": 14
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℤ\n⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℤ\n⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 28
} | {
"line": 651,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nζ : R\nn : ℕ\nx y : R\ninst✝ : IsDomain R\nhodd : Odd n\nh : IsPrimitiveRoot ζ n\n⊢ x ^ n + y ^ n = ∏ ζ ∈ nthRootsFinset n 1, (x + ζ * y)",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\nζ : R\nn : ℕ\nx y : R\ninst✝ : IsDomain R\nhodd : Odd n\nh : IsPrimitiveRoot ζ n\n⊢ x ^ n + y ^ n = ∏ ζ ∈ nthRootsFinset n 1, (x + ζ * y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 27
} | {
"line": 163,
"column": 2
} | [
{
"pp": "case neg\nS : Type u_2\ninst✝⁶ : CommRing S\nR : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\nx : A\nhmin : minpoly S B.gen = Polynomial.map (... | [] | · exact isIntegral_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 36
} | [
{
"pp": "n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nthis : 0 < (fun x ↦ ↑x) (eval (↑(-1)) (cyclotomic n ℤ))\n⊢ 0 < eval (↑(-1)) (cyclotomic n ℤ)",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"used... | [
"n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nthis : 0 < (fun x ↦ ↑x) (eval (↑(-1)) (cyclotomic n ℤ))\n⊢ 0 < eval (↑(-1)) (cyclotomic n ℤ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 64,
"column": 69
} | {
"line": 64,
"column": 85
} | {
"line": 64,
"column": 86
} | [
{
"pp": "n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nh0 : eval 0 (cyclotomic n ℝ) = 1\nhx : eval (-1) (cyclotomic n ℝ) ≤ 0\nthis : Set.Icc (eval (-1) (cyclotomic n ℝ)) (eval 0 (cyclotomic n ℝ)) ⊆ Set.range fun x ↦ eval x (cyclot... | [
"n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nh0 : eval 0 (cyclotomic n ℝ) = 1\nhx : eval (-1) (cyclotomic n ℝ) ≤ 0\nthis : Set.Icc (eval (-1) (cyclotomic n ℝ)) (eval 0 (cyclotomic n ℝ)) ⊆ Set.range fun x ↦ eval x (cyclotomic n ℝ)\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | {
"line": 112,
"column": 16
} | [
{
"pp": "case inl\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 ≤ n\n⊢ Irreducible (cyclotomic (p ^ 0) R)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Po... | [
"case inl\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 ≤ n\n⊢ Irreducible (X - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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