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.MeasureTheory.Integral.PeakFunction | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 21
} | {
"line": 319,
"column": 4
} | [
{
"pp": "case hab\nα : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\... | [] | exact ht x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 21
} | {
"line": 319,
"column": 4
} | [
{
"pp": "case hab\nα : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\... | [] | exact ht x hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 21
} | {
"line": 319,
"column": 4
} | [
{
"pp": "case hab\nα : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\... | [] | exact ht x hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 403,
"column": 2
} | {
"line": 424,
"column": 83
} | {
"line": 426,
"column": 0
} | [] | [] | ∑ i ∈ Finset.range (k + 1),
k.choose i * ‖iteratedFDeriv ℝ i (fun (y : V) ↦ T (fun _ ↦ L y)) v‖ *
‖iteratedFDeriv ℝ (k - i) f v‖
≤ ∑ i ∈ Finset.range (k + 1),
k.choose i * (n.descFactorial i * ‖L‖ ^ n * ‖v‖ ^ (n - i)) *
‖iteratedFDeriv ℝ (k - i) f v‖ := by
gcongr with i _hi
exact I₃ ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 339,
"column": 11
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | [
"α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCompact s\nhμ ... | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 548,
"column": 10
} | {
"line": 548,
"column": 97
} | {
"line": 548,
"column": 97
} | [
{
"pp": "E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝... | [
"E : Type u_1\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : NormedSpace ℝ V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : FiniteDi... | LinearIsometryEquiv.integrable_comp_iff (𝕜 := ℝ) (φ := fderiv ℝ (iteratedFDeriv ℝ n f)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 425,
"column": 63
} | {
"line": 426,
"column": 55
} | {
"line": 427,
"column": 6
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : FiniteDimensional ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure F\ninst✝ : μ.IsAddHaarMeasure\nφ : F → ℝ... | [] | by
gcongr; exact mul_nonneg (by positivity) (hφ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 185,
"column": 56
} | {
"line": 187,
"column": 56
} | {
"line": 189,
"column": 0
} | [
{
"pp": "a : ℂ\nha : a.re < -1\nc : ℝ\nhc : 0 < c\n⊢ IntegrableOn (fun t ↦ ‖↑t ^ a‖) (Ioi c) volume",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"instClosedIicTopology",
"Re... | [] | by
refine (integrableOn_Ioi_rpow_of_lt ha hc).congr_fun (fun x hx => ?_) measurableSet_Ioi
rw [Complex.norm_cpow_eq_rpow_re_of_pos (hc.trans hx)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 31
} | {
"line": 229,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nthis : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma\n⊢ Tendsto (s + 1).partialGamma atTop (𝓝 (s * s.GammaIntegral))",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"R... | [
"s : ℂ\nhs : 0 < s.re\nthis : (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) =ᶠ[atTop] (s + 1).partialGamma\n⊢ Tendsto (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) atTop (𝓝 (s * s.GammaIntegral))"
] | refine Tendsto.congr' this ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 301,
"column": 10
} | {
"line": 301,
"column": 26
} | {
"line": 301,
"column": 27
} | [
{
"pp": "case succ\ns : ℂ\nn : ℕ\nh1 : -s.re < ↑n\nk : ℕ\nhk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s\ni0 : -s.re < ↑⌊1 - s.re⌋₊\n⊢ 0 ≤ ↑k",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.toAddZeroC... | [
"case succ\ns : ℂ\nn : ℕ\nh1 : -s.re < ↑n\nk : ℕ\nhk : GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s\ni0 : -s.re < ↑⌊1 - s.re⌋₊\n⊢ ↑0 ≤ ↑k"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 302,
"column": 44
} | {
"line": 302,
"column": 61
} | {
"line": 302,
"column": 61
} | [
{
"pp": "s : ℂ\nn : ℕ\nh1 : -s.re < ↑n\nu : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s\n⊢ n = ⌊1 - s.re⌋₊ + (n - ⌊1 - s.re⌋₊)",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"congrArg"... | [
"s : ℂ\nn : ℕ\nh1 : -s.re < ↑n\nu : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s\n⊢ n = n",
"s : ℂ\nn : ℕ\nh1 : -s.re < ↑n\nu : ∀ (k : ℕ), GammaAux (⌊1 - s.re⌋₊ + k) s = Gamma s\n⊢ ⌊1 - s.re⌋₊ ≤ n"
] | Nat.add_sub_of_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 129,
"column": 27
} | {
"line": 133,
"column": 23
} | {
"line": 135,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : RCLike 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℂ E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : MeasurableSpace V\ni... | [] | by
intro f
ext x
rw [fourierInv_coe, fourier_coe, f.continuous.fourierInv_fourier_eq f.integrable
(𝓕 f).integrable] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 58,
"column": 85
} | {
"line": 59,
"column": 98
} | {
"line": 61,
"column": 0
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nn : d → ℤ\nx : UnitAddTorus d\nm : d → ℤ\n⊢ (mFourier (m + n)) x = (mFourier m) x * (mFourier n) x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Continuous",
"HMul.hMul",
... | [] | by
simp only [mFourier, Pi.add_apply, fourier_add, ContinuousMap.coe_mk, ← Finset.prod_mul_distrib] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 61,
"column": 50
} | {
"line": 64,
"column": 28
} | {
"line": 66,
"column": 0
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\n⊢ mFourier 0 = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Continuous",
"Finset.univ",
"Complex.commRing",
"Pi.topologicalSpace",
"... | [] | by
ext x
simp only [mFourier, Pi.zero_apply, fourier_zero, Finset.prod_const_one, ContinuousMap.coe_mk,
ContinuousMap.one_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 569,
"column": 77
} | {
"line": 572,
"column": 72
} | {
"line": 574,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢'(E, F)\nm : E\n⊢ 𝓕 (∂_{m} f) = (2 * ↑π * Complex.I) • (smulLeftCL... | [] | by
ext u
have : (inner ℝ · m).HasTemperateGrowth := by fun_prop
simp [SchwartzMap.lineDerivOp_fourier_eq, ← smulLeftCLM_ofReal ℂ this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 95
} | {
"line": 101,
"column": 95
} | [
{
"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... | [] | simpa using hf₁ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 95
} | {
"line": 101,
"column": 95
} | [
{
"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... | [] | simpa using hf₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 80
} | {
"line": 101,
"column": 95
} | {
"line": 101,
"column": 95
} | [
{
"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... | [] | simpa using hf₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 62
} | {
"line": 65,
"column": 2
} | [
{
"pp": "G : 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\n⊢ ψ₂ * ψ₁⁻¹ ≠ 1",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid.toD... | [] | rwa [Ne, mul_inv_eq_one, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 328,
"column": 47
} | {
"line": 328,
"column": 54
} | {
"line": 328,
"column": 54
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : InnerProductSpace ℂ F\ninst✝ : CompleteSpace F\ns : ℝ\nf : 𝓢'(E, F)\nhf : MemSobolev s 2... | [
"E : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : InnerProductSpace ℂ F\ninst✝ : CompleteSpace F\ns : ℝ\nf : 𝓢'(E, F)\nhf : MemSobolev s 2 f\nm x : E\... | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nn : ℕ\nhn : Squarefree n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ 1 - f p = 1 + ((↑μ).pmul f) p",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ArithmeticFunction.pmul",
"HM... | [
"R : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nn : ℕ\nhn : Squarefree n\np : ℕ\nhp : p ∈ n.primeFactors\n⊢ 1 - f p = 1 + ↑μ p * f p"
] | pmul_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 153,
"column": 13
} | {
"line": 153,
"column": 24
} | {
"line": 153,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nn : ℕ\nhn : Squarefree n\n⊢ ∑ d ∈ n.divisors, ((↑μ).pmul f) d = ∑ d ∈ n.divisors, ↑(μ d) * f d",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Int.cast",
"ArithmeticFunction.pmul",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nn : ℕ\nhn : Squarefree n\n⊢ ∑ d ∈ n.divisors, ↑μ d * f d = ∑ d ∈ n.divisors, ↑(μ d) * f d"
] | pmul_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 60
} | {
"line": 148,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atTop] fun x ↦ |x| ^ (-b)\nR S : ℝ\nclaim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)\nc : ℝ\nhc : c > 0\nd : ℝ\nhd : ∀ (b_1 : ℝ), d ≤ b_1 → ‖f b_1‖ ≤ c * ‖|b_1| ^ (-... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[atTop] fun x ↦ |x| ^ (-b)\nR S : ℝ\nclaim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)\nc : ℝ\nhc : c > 0\nd : ℝ\nhd : ∀ (b_1 : ℝ), d ≤ b_1 → ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖\nx : ℝ\n... | have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 89
} | {
"line": 121,
"column": 4
} | [
{
"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... | refine ⟨B₀.toNNReal + 1, add_pos_of_nonneg_of_pos B₀.toNNReal.coe_nonneg one_pos, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 64
} | {
"line": 312,
"column": 0
} | [
{
"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... | [] | simpa only [this] using sum_eq_iff_sum_smul_moebius_eq_on s hs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 19
} | {
"line": 190,
"column": 20
} | [
{
"pp": "case neg\nM : Type u_1\ninst✝ : CommMonoid M\nk i : ℕ\nhi : i.Coprime k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ (ζ ^ (↑i * i.gcdA k)) ^ l * (ζ ^ (↑k * i.gcdB k)) ^ l = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"zpow_natCast",
... | [
"case neg\nM : Type u_1\ninst✝ : CommMonoid M\nk i : ℕ\nhi : i.Coprime k\nh0 : ¬k = 0\nζ : Mˣ\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : (ζ ^ i) ^ l = 1\n⊢ (ζ ^ (↑i * i.gcdA k)) ^ ↑l * (ζ ^ (↑k * i.gcdB k)) ^ l = 1"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 8
} | {
"line": 204,
"column": 8
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nk : ℕ\nζ : M\nh : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni : ℕ\nhi : IsPrimitiveRoot (ζ ^ i) k\na : ℕ\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"HMul.hMul",
... | [
"M : Type u_1\ninst✝ : CommMonoid M\nk : ℕ\nζ : M\nh : IsPrimitiveRoot ζ k\nh0 : 0 < k\ni a : ℕ\nhi : IsPrimitiveRoot (ζ ^ (i.gcd k * a)) k\nha : i = i.gcd k * a\nb : ℕ\nhb : k = i.gcd k * b\n⊢ b = k"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 262,
"column": 33
} | {
"line": 266,
"column": 93
} | {
"line": 268,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nζ : M\nn a b : ℕ\nhn : 0 < n\nh : IsPrimitiveRoot ζ n\nhprod : n = a * b\n⊢ IsPrimitiveRoot (ζ ^ a) b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOn... | [] | by
subst n
simp only [iff_def, ← pow_mul, h.pow_eq_one, true_and]
intro l hl
exact Nat.dvd_of_mul_dvd_mul_left (Nat.pos_of_mul_pos_right hn) <| h.dvd_of_pow_eq_one _ hl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 342,
"column": 24
} | {
"line": 342,
"column": 79
} | {
"line": 342,
"column": 79
} | [
{
"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.... | [] | by simpa [mul_eq_mul_right_iff, or_iff_left hα] using e | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 492,
"column": 2
} | {
"line": 493,
"column": 64
} | {
"line": 495,
"column": 0
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\ni : ℕ\n⊢ h.zmodEquivZPowers ↑i = Additive.ofMul ⟨ζ ^ i, ⋯⟩",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Subtype.mk.... | [] | have : (i : ZMod k) = (i : ℤ) := by norm_cast
simp only [this, zmodEquivZPowers_apply_coe_int, zpow_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 492,
"column": 2
} | {
"line": 493,
"column": 64
} | {
"line": 495,
"column": 0
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\ni : ℕ\n⊢ h.zmodEquivZPowers ↑i = Additive.ofMul ⟨ζ ^ i, ⋯⟩",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Subtype.mk.... | [] | have : (i : ZMod k) = (i : ℤ) := by norm_cast
simp only [this, zmodEquivZPowers_apply_coe_int, zpow_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 561,
"column": 8
} | {
"line": 561,
"column": 23
} | {
"line": 561,
"column": 24
} | [
{
"pp": "case refine_2\nR : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\nn : ℤ\nhξ : ζ ^ n ∈ rootsOfUnity k R\nhk0 : 0 < ↑k\ni : ℤ := n % ↑k\ni₀ : ℕ\nhi₀ : ↑i₀ = i\n⊢ ζ ^ i₀ = ζ ^ n",
"ppTerm": "?refine_2",
"assigned": true,
"usedConst... | [
"case refine_2\nR : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\nn : ℤ\nhξ : ζ ^ n ∈ rootsOfUnity k R\nhk0 : 0 < ↑k\ni : ℤ := n % ↑k\ni₀ : ℕ\nhi₀ : ↑i₀ = i\n⊢ ζ ^ ↑i₀ = ζ ^ n"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 360,
"column": 84
} | {
"line": 362,
"column": 70
} | {
"line": 363,
"column": 2
} | [
{
"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": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSem... | [] | by
simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 625,
"column": 6
} | {
"line": 625,
"column": 74
} | {
"line": 626,
"column": 4
} | [
{
"pp": "case refine_1.refine_1\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_7\ninst✝¹ : CommSemiring S\ninst✝ : Algebra S R\nζ₂ : R\nk₁ k₂ : ℕ\nhζ₂ : IsPrimitiveRoot ζ₂ k₂\nhk₁ : k₁ ≠ 0\nhk₂ : k₂ ≠ 0\nζ : R\nhζ : IsPrimitiveRoot ζ (k₁.lcm k₂)\nthis : NeZero (k₁.lcm k₂)\nw✝ : ℕ\nleft✝ : w... | [] | exact Subalgebra.pow_mem _ (Algebra.self_mem_adjoin_singleton S _) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 74
} | {
"line": 629,
"column": 2
} | [
{
"pp": "case refine_1.refine_2\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_7\ninst✝¹ : CommSemiring S\ninst✝ : Algebra S R\nζ₁ : R\nk₁ k₂ : ℕ\nhζ₁ : IsPrimitiveRoot ζ₁ k₁\nhk₁ : k₁ ≠ 0\nhk₂ : k₂ ≠ 0\nζ : R\nhζ : IsPrimitiveRoot ζ (k₁.lcm k₂)\nthis : NeZero (k₁.lcm k₂)\nw✝ : ℕ\nleft✝ : w... | [] | exact Subalgebra.pow_mem _ (Algebra.self_mem_adjoin_singleton S _) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 669,
"column": 28
} | {
"line": 678,
"column": 91
} | {
"line": 680,
"column": 0
} | [
{
"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\n⊢ (nthRoots n a).Nodup",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"IsDomain",
"CommRing... | [] | by
obtain (rfl | hn) := n.eq_zero_or_pos; · simp
by_cases! h : ∃ α, α ^ n = a
· obtain ⟨α, hα⟩ := h
by_cases hα' : α = 0
· exact (ha (by rwa [hα', zero_pow hn.ne', eq_comm] at hα)).elim
rw [nthRoots_eq h hα, Multiset.nodup_map_iff_inj_on (Multiset.nodup_range n)]
exact h.injOn_pow_mul hα'
· suff... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 29
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case inr.inl\nn : ℕ\nhnzero : n > 0\nh : n ≠ 0\n⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic 0 ℚ)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Rat",
"Polynomial.cyclotomic",
"Rat.commSemiring",
"Rat.instDivisionRing",
"DivisionRing.toRing",
"... | [] | exact isCoprime_one_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 29
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case inr.inl\nn : ℕ\nhnzero : n > 0\nh : n ≠ 0\n⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic 0 ℚ)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Rat",
"Polynomial.cyclotomic",
"Rat.commSemiring",
"Rat.instDivisionRing",
"DivisionRing.toRing",
"... | [] | exact isCoprime_one_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 29
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case inr.inl\nn : ℕ\nhnzero : n > 0\nh : n ≠ 0\n⊢ IsCoprime (cyclotomic n ℚ) (cyclotomic 0 ℚ)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Rat",
"Polynomial.cyclotomic",
"Rat.commSemiring",
"Rat.instDivisionRing",
"DivisionRing.toRing",
"... | [] | exact isCoprime_one_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 81
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case refine_2\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.... | [] | intro hx
rw [pow_succ]
exact repr_mul_isIntegral hB (fun _ => hn _ le_rfl (fun _ => hx _) _) hx hmin | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 81
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case refine_2\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.... | [] | intro hx
rw [pow_succ]
exact repr_mul_isIntegral hB (fun _ => hn _ le_rfl (fun _ => hx _) _) hx hmin | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 134,
"column": 2
} | {
"line": 136,
"column": 31
} | {
"line": 137,
"column": 2
} | [
{
"pp": "R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : ¬p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ (p - 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | [
"R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : ¬p ∣ n\nthis : Algebra (ZMod p) R := ZMod.algebra R p\n⊢ cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1)"
] | suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ (p - 1) by
rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
this, Polynomial.map_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 79
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case succ.succ.zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\n⊢ (1 < x → 0 < eval x (cyclotomic (0 + 1 + 1) R)) ∧ (1 ≤ x → 0 ≤ eval x (cyclotomic (0 + 1 + 1) R))",
"ppTerm": "?succ.succ.zero",
"assigned": true,
"usedConstants": [
... | [
"case succ.succ.zero\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\n⊢ (1 < x → 0 < x + 1) ∧ (1 ≤ x → 0 ≤ x + 1)"
] | simp only [zero_add, reduceAdd, cyclotomic_two, eval_add, eval_X, eval_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 26
} | {
"line": 252,
"column": 4
} | [
{
"pp": "n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhno : Odd n\nζ x : K\nhζ : IsPrimitiveRoot ζ n\nhx : IsOfFinOrder x\n⊢ 2 * n = orderOf (-1) * orderOf ζ",
"ppTerm": "?m.121",
"assigned": true,
"usedConstants": [
"Is... | [
"n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhno : Odd n\nζ x : K\nhζ : IsPrimitiveRoot ζ n\nhx : IsOfFinOrder x\n⊢ (orderOf (-1)).Coprime (orderOf ζ)"
] | · simp [hζ.eq_orderOf] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 179,
"column": 4
} | {
"line": 181,
"column": 36
} | {
"line": 182,
"column": 4
} | [
{
"pp": "case refine_1\nS T : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension (S ∪ T) A B\n⊢ {b | ∃ n ∈ S ∪ T, n ≠ 0 ∧ b ^ n = 1} ⊆ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} ∪ {b | ∃ n ∈ T, n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?refine_1",
"assign... | [
"case refine_2\nS T : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension (S ∪ T) A B\n⊢ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} ∪ {b | ∃ n ∈ T, n ≠ 0 ∧ b ^ n = 1} ⊆ {b | ∃ n ∈ S ∪ T, n ≠ 0 ∧ b ^ n = 1}"
] | · rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 53
} | {
"line": 295,
"column": 2
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nC : Type u_1\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B ≃ₐ[A] C\n⊢ IsCyclotomicExtension S A C",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [
"S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nC : Type u_1\ninst✝¹ : CommRing C\ninst✝ : Algebra A C\nh : IsCyclotomicExtension S A B\nf : B ≃ₐ[A] C\nthis : Algebra B C := (↑f).toAlgebra\n⊢ IsCyclotomicExtension S A C"
] | let : Algebra B C := f.toAlgHom.toRingHom.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 50
} | {
"line": 344,
"column": 2
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}\n⊢ f x = g x... | [] | induction hx using Algebra.adjoin_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 407,
"column": 16
} | {
"line": 407,
"column": 59
} | {
"line": 408,
"column": 6
} | [
{
"pp": "case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p... | [
"case refine_2\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + ... | rw [IntermediateField.adjoin_simple_le_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 28
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case e'_1\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (... | [] | rw [Nat.sub_add_comm hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 422,
"column": 4
} | {
"line": 422,
"column": 28
} | {
"line": 423,
"column": 2
} | [
{
"pp": "case e'_4\np : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (... | [] | rw [Nat.sub_add_comm hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.Units | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 19
} | {
"line": 106,
"column": 20
} | [
{
"pp": "N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d * (↑a₀ * p + ↑N₀ * q) = ↑d\n⊢ IsCoprime ↑a₀ ↑N₀",
"ppTerm": "?m.305",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocComm... | [
"N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d * (p * ↑a₀ + ↑N₀ * q) = ↑d\n⊢ IsCoprime ↑a₀ ↑N₀"
] | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 84
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case succ\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nhp : 1 < n✝ + 1\nh1 : (X ^ (n✝ + 1) - X).roots = univ.val\nh2 : (X ^ (n✝ + 1) - X).natDegree = n✝ + 1\n⊢ (X ^ (n✝ + 1) - X).Splits",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Finset.card_univ",
"Eq.mpr",
... | [] | rw [splits_iff_card_roots, h1, ← Finset.card_def, Finset.card_univ, h2, ZMod.card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 60
} | {
"line": 181,
"column": 0
} | [
{
"pp": "p : ℕ\nh_prime : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Algebra (ZMod p) K\ninst✝ : Finite K\nthis : Fintype K\n⊢ (Polynomial.map (algebraMap (ZMod p) K) (X ^ Fintype.card K - X)).Splits",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"HSub.hSub",
... | [] | exact (FiniteField.isSplittingField_sub K (ZMod p)).splits | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 176,
"column": 10
} | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 13
} | [
{
"pp": "C : Type v\ninst✝ : CommMonoid C\nn : ℕ\nψ : AddChar (ZMod n) C\nhψ : ∀ (a : ZMod n), ψ a = 1 → a = 0\na : ZMod n\n⊢ a ≠ 0 → ψ.mulShift a ≠ 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"CommSemiring.toSemiring",
"Ne",
"ZMod",
"Co... | [
"C : Type v\ninst✝ : CommMonoid C\nn : ℕ\nψ : AddChar (ZMod n) C\nhψ : ∀ (a : ZMod n), ψ a = 1 → a = 0\na : ZMod n\nha : a ≠ 0\n⊢ ψ.mulShift a ≠ 1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 187,
"column": 10
} | {
"line": 187,
"column": 12
} | {
"line": 188,
"column": 2
} | [
{
"pp": "C : Type v\ninst✝¹ : CommMonoid C\nn : ℕ\ninst✝ : NeZero n\nζ : C\nh : IsPrimitiveRoot ζ n\na : ZMod n\n⊢ (zmodChar n ⋯) a = 1 → a = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Dvd.dvd",
"ZMod.commRing",
"Monoid.toMulOneClass",
... | [
"C : Type v\ninst✝¹ : CommMonoid C\nn : ℕ\ninst✝ : NeZero n\nζ : C\nh : IsPrimitiveRoot ζ n\na : ZMod n\nha : (zmodChar n ⋯) a = 1\n⊢ a = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 77,
"column": 10
} | {
"line": 77,
"column": 31
} | {
"line": 77,
"column": 31
} | [
{
"pp": "case refine_2.refine_2\nM : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nζ : M\nh : IsPrimitiveRoot ζ n\nζ' : ↥(rootsOfUnity n M) := ⟨⋯.unit, ?refine_2.refine_1⟩\n⊢ orderOf ⋯.unit ∣ Monoid.exponent ↥(rootsOfUnity n M)",
"ppTerm": "?refine_2.refine_2"... | [
"case refine_2.refine_2\nM : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nζ : M\nh : IsPrimitiveRoot ζ n\nζ' : ↥(rootsOfUnity n M) := ⟨⋯.unit, ?refine_2.refine_1⟩\n⊢ orderOf ⟨⋯.unit, ?refine_2.refine_2.ha⟩ ∣ Monoid.exponent ↥(rootsOfUnity n M)",
"case refine_2.refi... | ← Subgroup.orderOf_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 83
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case h\nn : ℕ\ninst✝² : NeZero n\nM : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nhm✝ : HasEnoughRootsOfUnity M n\ne : ↥(rootsOfUnity n M) ≃* ↥(rootsOfUnity n N)\nm : M\nhm : IsPrimitiveRoot m n\n⊢ IsPrimitiveRoot hm.toRootsOfUnity n",
"ppTerm": "?h",
"assigned": true,
... | [] | rwa [← IsPrimitiveRoot.coe_submonoidClass_iff, ← IsPrimitiveRoot.coe_units_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 14
} | {
"line": 108,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\na : R\n⊢ ¬IsUnit a → (if IsUnit a then 1 else 0) = 0",
"ppTerm": "?m.178",
"assigned": true,
"usedConstants": [
"IsUnit",
"CommMonoid.toMonoid",
"Not"
],
"usedFVars": [
"R",
... | [
"R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\na : R\nha : ¬IsUnit a\n⊢ (if IsUnit a then 1 else 0) = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 175,
"column": 12
} | {
"line": 175,
"column": 14
} | {
"line": 176,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\na : R\n⊢ ¬IsUnit a → (if hx : IsUnit a then ↑(f hx.unit) else 0) = 0",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"IsUnit",
"CommMonoid.toMonoid",
"Not"
],
... | [
"R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\na : R\nha : ¬IsUnit a\n⊢ (if hx : IsUnit a then ↑(f hx.unit) else 0) = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 827,
"column": 95
} | {
"line": 827,
"column": 97
} | {
"line": 827,
"column": 97
} | [
{
"pp": "case refine_4\nn : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ ... | [
"case refine_4\nn : ℕ\ninst✝¹⁰ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : IsDomain A\ninst✝ : NeZero ↑n\... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 89
} | {
"line": 155,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : n ∣ Fintype.card F - 1\nζ : R\nhζ : IsPrimitiveRoot ζ n\nhn₀ : 0 < n\ne : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card Fˣ) R) := equiv_rootsOfUnity F R\nζ' : Rˣ := ⋯.unit\n⊢ ∃ χ, orderOf χ = n",
"ppTerm"... | [
"F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nh : n ∣ Fintype.card F - 1\nζ : R\nhζ : IsPrimitiveRoot ζ n\nhn₀ : 0 < n\ne : MulChar F R ≃* ↥(rootsOfUnity (Fintype.card Fˣ) R) := equiv_rootsOfUnity F R\nζ' : Rˣ := ⋯.unit\nh' : ζ' ^ Fintype.card Fˣ = 1\n⊢ ∃ χ, orderOf χ... | have h' : ζ' ^ (Fintype.card Fˣ : ℕ) = 1 :=
Units.ext_iff.mpr <| (hζ.pow_eq_one_iff_dvd _).mpr <| Fintype.card_units (α := F) ▸ h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 899,
"column": 4
} | {
"line": 899,
"column": 84
} | {
"line": 900,
"column": 2
} | [
{
"pp": "case a\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ A[ζ] ≤ adjoin A {b | n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": ... | [] | · exact adjoin_mono <| Set.singleton_subset_iff.mpr ⟨NeZero.ne n, hζ.pow_eq_one⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 386,
"column": 11
} | {
"line": 386,
"column": 45
} | {
"line": 386,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoidWithZero R\nn : ℕ\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nχ : DirichletCharacter R n\na : ℤ\nha : IsCoprime a ↑χ.conductor\nthis✝ : (changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter = χ.primitiveCharacter\nthis : ((changeLevel ⋯) ((changeLevel... | [
"R : Type u_1\ninst✝² : CommMonoidWithZero R\nn : ℕ\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nχ : DirichletCharacter R n\na : ℤ\nha : IsCoprime a ↑χ.conductor\nthis✝ : (changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter = χ.primitiveCharacter\nthis : ((changeLevel hm) χ).primitiveCharacter ↑a... | changeLevel_eq_cast_of_dvd' _ _ ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 388,
"column": 29
} | {
"line": 388,
"column": 60
} | {
"line": 389,
"column": 6
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommMonoidWithZero R\nn : ℕ\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nχ : DirichletCharacter R n\na : ℤ\nha : IsCoprime a ↑χ.conductor\n⊢ (changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter = (changeLevel ⋯) χ.primitiveCharacter",
"ppTerm": "?... | [
"case pos\nR : Type u_1\ninst✝² : CommMonoidWithZero R\nn : ℕ\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nχ : DirichletCharacter R n\na : ℤ\nha : IsCoprime a ↑χ.conductor\n⊢ (changeLevel hm) χ = (changeLevel ⋯) χ.primitiveCharacter"
] | changeLevel_primitiveCharacter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 441,
"column": 4
} | {
"line": 441,
"column": 35
} | {
"line": 441,
"column": 36
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ ψ : DirichletCharacter R n\nhn : NeZero n\nh : χ.conductor.lcm ψ.conductor ∣ n\n⊢ χ * ψ = (changeLevel ⋯) χ.primitiveCharacter * (changeLevel ⋯) ψ.primitiveCharacter",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nR : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ ψ : DirichletCharacter R n\nhn : NeZero n\nh : χ.conductor.lcm ψ.conductor ∣ n\n⊢ χ * ψ = χ * (changeLevel ⋯) ψ.primitiveCharacter"
] | changeLevel_primitiveCharacter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 28,
"column": 14
} | {
"line": 28,
"column": 16
} | {
"line": 28,
"column": 16
} | [
{
"pp": "N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\n⊢ e.mulShift ↑(↑(↑u).val - 1) = ... | [
"N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\n⊢ e.mulShift ↑(↑d * a) = 1"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Grade | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 17
} | {
"line": 315,
"column": 18
} | [
{
"pp": "case succ\n𝕆 : Type u_1\nℙ : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝⁴ : Preorder 𝕆\ninst✝³ : Preorder ℙ\ninst✝² : Preorder α\ninst✝¹ : Preorder β\nn✝ : ℕ\ninst✝ : GradeMinOrder (Fin (n✝ + 1)) α\na : Fin (n✝ + 1)\nh : IsMin a\n⊢ IsMin ↑a",
"ppTerm": "?succ",
"assigned": true,
"usedCons... | [
"case succ\n𝕆 : Type u_1\nℙ : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝⁴ : Preorder 𝕆\ninst✝³ : Preorder ℙ\ninst✝² : Preorder α\ninst✝¹ : Preorder β\nn✝ : ℕ\ninst✝ : GradeMinOrder (Fin (n✝ + 1)) α\na : Fin (n✝ + 1)\nh : IsMin a\n⊢ IsMin ↑⊥"
] | h.eq_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.GaussSum | {
"line": 270,
"column": 40
} | {
"line": 270,
"column": 62
} | {
"line": 270,
"column": 63
} | [
{
"pp": "case succ\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\nn : ℕ\nih : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ\n⊢ χ (↑p ^ n) ^ p * ga... | [
"case succ\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\nn : ℕ\nih : gaussSum χ ψ ^ p ^ n = χ (↑p ^ n) * gaussSum χ ψ\n⊢ χ (↑p ^ n) ^ p * (χ ↑p * gaussS... | hχ.gaussSum_frob _ hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 98,
"column": 4
} | {
"line": 100,
"column": 46
} | {
"line": 102,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\n⊢ auxDFT ((fun Φ k ↦ (↑N)⁻¹ • auxDFT Φ (-k)) Φ) = Φ",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"MulOn... | [] | ext1 j
simp only [← Pi.smul_def, auxDFT_smul, auxDFT_neg, auxDFT_auxDFT, neg_neg, ← mul_smul,
inv_mul_cancel₀ (NeZero.ne _), one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.ZMod | {
"line": 98,
"column": 4
} | {
"line": 100,
"column": 46
} | {
"line": 102,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\n⊢ auxDFT ((fun Φ k ↦ (↑N)⁻¹ • auxDFT Φ (-k)) Φ) = Φ",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"MulOn... | [] | ext1 j
simp only [← Pi.smul_def, auxDFT_smul, auxDFT_neg, auxDFT_auxDFT, neg_neg, ← mul_smul,
inv_mul_cancel₀ (NeZero.ne _), one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 40
} | {
"line": 349,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\n⊢ 2 ^ (Fintype.card F / 2) = ↑(χ₈ ↑(Fintype.card F))",
"ppTerm": "?m.65",
"assign... | [
"F : Type u_1\ninst✝¹ : Fintype F\ninst✝ : Field F\nhF : ringChar F ≠ 2\nhp2 : ∀ (n : ℕ), 2 ^ n ≠ 0\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nFF : Type u_1 := CyclotomicField 8 F\nhchar : ringChar F = ringChar FF\n⊢ 2 ^ (Fintype.card F / 2) = ↑(χ₈ ↑(Fintype.card F))"
] | have hchar := Algebra.ringChar_eq F FF | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Finset.Interval | {
"line": 81,
"column": 20
} | {
"line": 81,
"column": 30
} | {
"line": 81,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ⊆ u ∧ u ⊂ t ↔ u ∈ image (fun x ↦ s ∪ x) (t \\ s).ssubsets",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Finset.instUnion",
"congrArg",
"Fin... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\nh : s ⊆ t\nu : Finset α\n⊢ s ⊆ u ∧ u ⊂ t ↔ ∃ a ∈ (t \\ s).ssubsets, s ∪ a = u"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Hofer | {
"line": 60,
"column": 6
} | {
"line": 71,
"column": 60
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\n... | [
"case hi\nX : Type u_1\ninst✝¹ : MetricSpace X\ninst✝ : CompleteSpace X\nx : X\nε : ℝ\nε_pos : 0 < ε\nϕ : X → ℝ\ncont : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ϕ\nnonneg : ∀ (y : X), 0 ≤ ϕ y\nreformulation : ∀ (x' : X) (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x'\nthis : Nonem... | have A : d (u (n + 1)) x ≤ 2 * ε := by
rw [dist_comm]
let r := range (n + 1) -- range (n+1) = {0, ..., n}
calc
d (u 0) (u (n + 1)) ≤ ∑ i ∈ r, d (u i) (u <| i + 1) := dist_le_range_sum_dist u (n + 1)
_ ≤ ∑ i ∈ r, ε / 2 ^ i :=
(sum_le_sum fun i i_in => (IH i <| Nat.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 184,
"column": 16
} | {
"line": 184,
"column": 35
} | {
"line": 185,
"column": 4
} | [
{
"pp": "case h₂.hp\nι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i... | [] | exact fun _ _ ↦ hp₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 184,
"column": 16
} | {
"line": 184,
"column": 35
} | {
"line": 185,
"column": 4
} | [
{
"pp": "case h₂.hp\nι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i... | [] | exact fun _ _ ↦ hp₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 184,
"column": 16
} | {
"line": 184,
"column": 35
} | {
"line": 185,
"column": 4
} | [
{
"pp": "case h₂.hp\nι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i... | [] | exact fun _ _ ↦ hp₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 419,
"column": 8
} | {
"line": 419,
"column": 42
} | {
"line": 420,
"column": 8
} | [
{
"pp": "F : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nu : E → F\nhu : ContDiff ℝ 1 u\nh2u... | [
"case hf\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nu : E → F\nhu : ContDiff ℝ 1 u\nh2u : ... | rw [lintegral_mul_const, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 445,
"column": 2
} | {
"line": 451,
"column": 61
} | {
"line": 453,
"column": 0
} | [
{
"pp": "F : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nu : E → F\nhu : ContDiff ℝ 1 u\nh2u... | [] | have h0p : 0 < (p : ℝ) := hp.coe.symm.pos
rw [eLpNorm_one_eq_lintegral_enorm,
← ENNReal.rpow_le_rpow_iff h0p, ENNReal.mul_rpow_of_nonneg _ _ h0p.le,
← ENNReal.coe_rpow_of_nonneg _ h0p.le, eLpNormLESNormFDerivOneConst, ← NNReal.rpow_mul,
eLpNorm_nnreal_pow_eq_lintegral hp.symm.pos.ne',
inv_mul_cancel₀ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 445,
"column": 2
} | {
"line": 451,
"column": 61
} | {
"line": 453,
"column": 0
} | [
{
"pp": "F : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nu : E → F\nhu : ContDiff ℝ 1 u\nh2u... | [] | have h0p : 0 < (p : ℝ) := hp.coe.symm.pos
rw [eLpNorm_one_eq_lintegral_enorm,
← ENNReal.rpow_le_rpow_iff h0p, ENNReal.mul_rpow_of_nonneg _ _ h0p.le,
← ENNReal.coe_rpow_of_nonneg _ h0p.le, eLpNormLESNormFDerivOneConst, ← NNReal.rpow_mul,
eLpNorm_nnreal_pow_eq_lintegral hp.symm.pos.ne',
inv_mul_cancel₀ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 126,
"column": 43
} | {
"line": 126,
"column": 81
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ ... | [] | simp [hn, hε.le, mul_div_cancel_left₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 126,
"column": 43
} | {
"line": 126,
"column": 81
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ∈ ... | [] | simp [hn, hε.le, mul_div_cancel_left₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Basis.MulOpposite | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 89
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : Type u_1\nH : Type u_2\ninst✝² : DivisionRing R\ninst✝¹ : AddCommGroup H\ninst✝ : Module R H\nb : Basis (↑(Basis.ofVectorSpaceIndex R H)) R H := Basis.ofVectorSpace R H\n⊢ Module.finrank R Hᵐᵒᵖ = Module.finrank R H",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | rw [Module.finrank_eq_nat_card_basis b, Module.finrank_eq_nat_card_basis b.mulOpposite] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 185,
"column": 15
} | {
"line": 185,
"column": 17
} | {
"line": 186,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\na b : ℕ\nhl : b ≤ a\n⊢ a ∈ ↑T.s... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\na b : ℕ\nhl : b ≤ a\nha : a ∈ ↑T.singularVa... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 297,
"column": 2
} | {
"line": 298,
"column": 92
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case e_f\nV : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubsp... | [
"case e_f\nV : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nhs ... | have hu : MeasurableSet u :=
(ht.inter (closed_of_finiteDimensional _).measurableSet).preimage measurable_subtype_coe | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 286,
"column": 2
} | {
"line": 288,
"column": 6
} | {
"line": 290,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nm : Set X → ℝ≥0∞\ns : Set X\n⊢ Tendsto (fun n ↦ (pre m (↑n)⁻¹) s) atTop (𝓝 ((mkMetric' m) s))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"Set.Ioi",
"Preorder.toLT",
"ENNReal.inv_pos._simp... | [] | refine (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, ?_⟩)
refine tendsto_principal.2 (Eventually.of_forall fun n => ?_)
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 286,
"column": 2
} | {
"line": 288,
"column": 6
} | {
"line": 290,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nm : Set X → ℝ≥0∞\ns : Set X\n⊢ Tendsto (fun n ↦ (pre m (↑n)⁻¹) s) atTop (𝓝 ((mkMetric' m) s))",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"False",
"Set.Ioi",
"Preorder.toLT",
"ENNReal.inv_pos._simp... | [] | refine (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, ?_⟩)
refine tendsto_principal.2 (Eventually.of_forall fun n => ?_)
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 327,
"column": 8
} | {
"line": 327,
"column": 36
} | {
"line": 327,
"column": 36
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\nt : Set P\n... | AffineSubspace.direction_mk' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 352,
"column": 71
} | {
"line": 352,
"column": 99
} | {
"line": 352,
"column": 99
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [] | AffineSubspace.direction_mk' | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 680,
"column": 2
} | {
"line": 680,
"column": 42
} | {
"line": 681,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\n⊢ μH[d] (f '' s) ≤ ↑C ^ d * μH[↑r ... | [
"case inl\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\ns : Set X\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f s\n⊢ μH[d] (f '' s) ≤ ↑0 ^ d * μH[↑r * d]... | rcases eq_zero_or_pos C with (rfl | hC0) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.InnerProductSpace.Trace | {
"line": 53,
"column": 44
} | {
"line": 53,
"column": 84
} | {
"line": 53,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nx y : E\n⊢ (trace 𝕜 𝕜) ↑((innerSL 𝕜) y ∘SL ContinuousLinearMap.toSpanSingleton 𝕜 x) = ⟪y, x⟫_𝕜",
"ppTerm": "?m.102",
"assigned": true,
"usedC... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nx y : E\n⊢ (trace 𝕜 𝕜) ↑(ContinuousLinearMap.toSpanSingleton 𝕜 (((innerSL 𝕜) y) x)) = ⟪y, x⟫_𝕜"
] | ContinuousLinearMap.comp_toSpanSingleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 973,
"column": 30
} | {
"line": 982,
"column": 48
} | {
"line": 983,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nY : Type u_3\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : EMetricSpace Y\ninst✝⁸ : MeasurableSpace X\ninst✝⁷ : BorelSpace X\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝... | [] | by
set e : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := ContinuousLinearEquiv.ofFinrankEq (by simp)
suffices μH[finrank ℝ E] (e '' K) < ⊤ by
rw [← e.symm_image_image K]
apply lt_of_le_of_lt <| e.symm.lipschitz.hausdorffMeasure_image_le (by simp) (e '' K)
rw [ENNReal.rpow_natCast]
exact ENNReal.mul_l... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 1096,
"column": 94
} | {
"line": 1099,
"column": 31
} | {
"line": 1101,
"column": 0
} | [
{
"pp": "E : Type u_5\nP : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace P\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor E P\ninst✝ : BorelSpace P\nx y : P\n⊢ μH[1] (affineSegment ℝ x y) = edist x y",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [] | by
rw [affineSegment, hausdorffMeasure_lineMap_image, hausdorffMeasure_real, Real.volume_Icc,
sub_zero, ENNReal.ofReal_one, ← Algebra.algebraMap_eq_smul_one]
exact (edist_nndist _ _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 43
} | {
"line": 226,
"column": 44
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nthis : ⟪o.rightAngleRotationAux₁ y, o.rightAngleRotationAux₁ x⟫ = ⟪y, x⟫\n⊢ ⟪y, o.rightAngleRotationAux₁ (o.rightAngleRotationAux₁ x)⟫ = ⟪y, -x⟫",
"ppTe... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nthis : ⟪o.rightAngleRotationAux₁ y, o.rightAngleRotationAux₁ x⟫ = ⟪y, x⟫\n⊢ -(o.areaForm y) (o.rightAngleRotationAux₁ x) = ⟪y, -x⟫"
] | o.inner_rightAngleRotationAux₁_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Matrix.LDL | {
"line": 116,
"column": 2
} | {
"line": 118,
"column": 47
} | {
"line": 119,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : WellFoundedLT n\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ lower hS * diag hS * (lower hS)ᴴ = S",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"L... | [
"𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nn : Type u_2\ninst✝³ : LinearOrder n\ninst✝² : WellFoundedLT n\ninst✝¹ : LocallyFiniteOrderBot n\nS : Matrix n n 𝕜\ninst✝ : Fintype n\nhS : S.PosDef\n⊢ diag hS = lowerInv hS * S * (lowerInv hS)ᴴ"
] | rw [LDL.lower, conjTranspose_nonsing_inv, Matrix.mul_assoc,
Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),
Matrix.mul_inv_eq_iff_eq_mul_of_invertible] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 425,
"column": 2
} | {
"line": 427,
"column": 41
} | {
"line": 429,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ (o.kahler (o.rightAngleRotation x)) y = -Complex.I * (o.kahler x) y",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"instInner... | [] | simp only [o.areaForm_rightAngleRotation_left, o.inner_rightAngleRotation_left,
o.kahler_apply_apply, Complex.ofReal_neg, Complex.real_smul]
linear_combination ω x y * Complex.I_sq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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