module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Matrix.Spectrum | {
"line": 212,
"column": 18
} | {
"line": 212,
"column": 38
} | {
"line": 212,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nx : ℝ\n| x ∈ spectrum ℝ A",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Matrix.smul",
"Rea... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nx : ℝ\n| x ∈ spectrum ℝ (((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues)))"
] | hA.spectral_theorem, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 26
} | {
"line": 219,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nh : hA.eigenvalues = 0\n⊢ A = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"Real",
"Algebra.to_s... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nh : hA.eigenvalues = 0\n⊢ ((conjStarAlgAut 𝕜 (Matrix n n 𝕜)) hA.eigenvectorUnitary) (diagonal (RCLike.ofReal ∘ hA.eigenvalues)) = 0"
] | hA.spectral_theorem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 96
} | {
"line": 56,
"column": 2
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ A.trace = 0 ↔ A = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.smul",
"MulOne.toOne",
"Real.instLE",
"Real",
"Alge... | [
"n : Type u_2\n𝕜 : Type u_3\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nhA : A.PosSemidef\n⊢ (∀ i ∈ Finset.univ, (RCLike.ofReal ∘ ⋯.eigenvalues) i = 0) ↔ A = 0"
] | conv_lhs => rw [hA.1.spectral_theorem, conjStarAlgAut_apply, trace_mul_cycle, coe_star_mul_self,
one_mul, trace_diagonal, Finset.sum_eq_zero_iff_of_nonneg (by simp [hA.eigenvalues_nonneg])] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 46
} | {
"line": 464,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F)\n⊢ ⇑({ toF... | rw [ContDiffMapSupportedIn.iteratedFDerivLM] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 14
} | {
"line": 60,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : DecidableEq n\ni : n\nx : E\nj k : n\nhij : i ≠ j\n⊢ gram 𝕜 (Pi.single i x) j k = single i i (⟪x, x⟫_𝕜) j k",
"ppTerm": "?inl",
"assigned": true... | [
"case inr\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : DecidableEq n\ni : n\nx : E\nk : n\n⊢ gram 𝕜 (Pi.single i x) i k = single i i (⟪x, x⟫_𝕜) i k"
] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 6
} | {
"line": 88,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : BorelSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopologyEither α E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : Fact (1 ≤ p)\n⊢ (toLpHom p μ).range = Lp.boundedContinuousFunct... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : BorelSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopologyEither α E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : Fact (1 ≤ p)\n⊢ Lp.boundedContinuousFunction E p μ = (toLpHom p μ).range"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 50
} | {
"line": 236,
"column": 51
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ns t : Set α\ninst✝ : CompleteSpace E\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : μ s ≠ ∞\nhμt : μ t ≠ ∞\na b : E\nthis : InnerProdu... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ns t : Set α\ninst✝ : CompleteSpace E\nhs : MeasurableSet s\nht : MeasurableSet t\nhμs : μ s ≠ ∞\nhμt : μ t ≠ ∞\na b : E\nthis : InnerProductSpace ℝ E ... | inner_indicatorConstLp_eq_inner_setIntegral, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 963,
"column": 6
} | {
"line": 963,
"column": 30
} | {
"line": 964,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\nn : ℕ∞\nK : Compacts E\nm : MeasurableSpace E\ninst✝¹⁰ : OpensMeasurableSpace E\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedSpac... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\nn : ℕ∞\nK : Compacts E\nm : MeasurableSpace E\ninst✝¹⁰ : OpensMeasurableSpace E\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedSpace 𝕜 F₁\nins... | filter_upwards [] with x | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 39
} | {
"line": 88,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : 1 ≤ p\nf :... | have hε₂ : 0 < ε / 2 := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 114,
"column": 6
} | {
"line": 119,
"column": 68
} | {
"line": 120,
"column": 6
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf : ↥(lp G 2)\n⊢ failed to pretty print expression (use 'set_option... | [
"case calc_1\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf : ↥(lp G 2)\n⊢ 0 < ENNReal.toReal 2",
"case calc_2\nι : Type u... | calc
‖f‖ ^ 2 = ‖f‖ ^ (2 : ℝ≥0∞).toReal := by norm_cast
_ = ∑' i, ‖f i‖ ^ (2 : ℝ≥0∞).toReal := lp.norm_rpow_eq_tsum ?_ f
_ = ∑' i, ‖f i‖ ^ (2 : ℕ) := by norm_cast
_ = ∑' i, re ⟪f i, f i⟫ := by simp
_ = re (∑' i, ⟪f i, f i⟫) := (RCLike.reCLM.map_tsum ?_).symm | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 233,
"column": 31
} | {
"line": 233,
"column": 42
} | {
"line": 233,
"column": 42
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nx y : AddCircle T\nhxy : x ≠ y\n⊢ ↑x.toCircle ≠ (fourier 1) y",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"ContinuousMap",
"Complex.instNormedFiel... | [
"T : ℝ\nhT : Fact (0 < T)\nx y : AddCircle T\nhxy : x ≠ y\n⊢ ↑x.toCircle ≠ ↑y.toCircle"
] | fourier_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 214,
"column": 51
} | {
"line": 221,
"column": 12
} | {
"line": 223,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthog... | [] | by
rw [hV.linearIsometry_apply, ← tsum_ite_eq i (fun _ ↦ V i x)]
congr
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; simp
· simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 516,
"column": 53
} | {
"line": 516,
"column": 73
} | {
"line": 517,
"column": 4
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nn m : ℤ\n⊢ fourierCoeff (↑↑(fourierLp 2 n)) m = Pi.single n 1 m",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | [
"T : ℝ\nhT : Fact (0 < T)\nn m : ℤ\n⊢ ↑(fourierBasis.repr (fourierLp 2 n)) m = Pi.single n 1 m"
] | ← fourierBasis_repr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 567,
"column": 10
} | {
"line": 567,
"column": 49
} | {
"line": 567,
"column": 50
} | [
{
"pp": "a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\n| 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =\n 1 / (-2 * ↑π ... | [
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nx : ℝ\n| (fourier (-n)) ↑x * f x",
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\... | pattern (occs := 1 2 3) fourier _ _ * _ | Lean.Elab.Tactic.Conv.evalPattern | Lean.Parser.Tactic.Conv.pattern |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 159,
"column": 8
} | {
"line": 164,
"column": 69
} | {
"line": 165,
"column": 6
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆... | [] | refine setIntegral_mono_on ?_ ?_ (hs.diff u_open.measurableSet) fun x hx => ?_
· exact IntegrableOn.mono_set h''i.norm sdiff_subset
· exact IntegrableOn.mono_set (hmg.norm.const_mul _) sdiff_subset
rw [norm_smul]
gcongr
simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 159,
"column": 8
} | {
"line": 164,
"column": 69
} | {
"line": 165,
"column": 6
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆... | [] | refine setIntegral_mono_on ?_ ?_ (hs.diff u_open.measurableSet) fun x hx => ?_
· exact IntegrableOn.mono_set h''i.norm sdiff_subset
· exact IntegrableOn.mono_set (hmg.norm.const_mul _) sdiff_subset
rw [norm_smul]
gcongr
simpa only [Pi.zero_apply, dist_zero_left] using (hi x hx).l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 594,
"column": 8
} | {
"line": 594,
"column": 25
} | {
"line": 595,
"column": 8
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup V\ninst✝⁸ : InnerProductSpace ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : InnerProductSpace ℝ W\ninst✝³ : MeasurableSpace ... | [
"V : Type u_1\nW : Type u_2\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℂ E\ninst✝⁹ : NormedAddCommGroup V\ninst✝⁸ : InnerProductSpace ℝ V\ninst✝⁷ : MeasurableSpace V\ninst✝⁶ : BorelSpace V\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : InnerProductSpace ℝ W\ninst✝³ : MeasurableSpace W\ninst✝² : ... | rw [h, smul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 54
} | {
"line": 177,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : ProperSpace E\nf : 𝓢(E, F)\n⊢ Tendsto (⇑f) (cocompact E) (𝓝 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",... | [] | apply (isBigO_cocompact_rpow f (-1)).trans_tendsto
simp_rw [Real.rpow_neg_one]
exact tendsto_norm_cocompact_atTop.inv_tendsto_atTop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 54
} | {
"line": 177,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : ProperSpace E\nf : 𝓢(E, F)\n⊢ Tendsto (⇑f) (cocompact E) (𝓝 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",... | [] | apply (isBigO_cocompact_rpow f (-1)).trans_tendsto
simp_rw [Real.rpow_neg_one]
exact tendsto_norm_cocompact_atTop.inv_tendsto_atTop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 322,
"column": 6
} | {
"line": 322,
"column": 57
} | {
"line": 322,
"column": 57
} | [
{
"pp": "case h\nι : 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✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nk n : ℕ\nx : E\n⊢ ‖x‖ ^ k * ‖iteratedFD... | [] | grw [f.decay_neg_aux k n x, f.le_seminormAux k n x] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 336,
"column": 6
} | {
"line": 338,
"column": 86
} | {
"line": 338,
"column": 86
} | [
{
"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✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : 𝓢(E, F)\nk n : ℕ\n⊢ ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iterate... | [] | use f.seminormAux k n + g.seminormAux k n
intro x
grw [decay_add_le_aux k n f g x, f.le_seminormAux k n x, g.le_seminormAux k n x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 336,
"column": 6
} | {
"line": 338,
"column": 86
} | {
"line": 338,
"column": 86
} | [
{
"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✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : 𝓢(E, F)\nk n : ℕ\n⊢ ∃ C, ∀ (x : E), ‖x‖ ^ k * ‖iterate... | [] | use f.seminormAux k n + g.seminormAux k n
intro x
grw [decay_add_le_aux k n f g x, f.le_seminormAux k n x, g.le_seminormAux k n x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 510,
"column": 2
} | {
"line": 510,
"column": 31
} | {
"line": 511,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nμ : M... | [
"case pos\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nμ : Mea... | by_cases h'f : Integrable f μ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Integral.Asymptotics | {
"line": 170,
"column": 2
} | {
"line": 179,
"column": 52
} | {
"line": 181,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\nf : α → E\ng : α → F\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : SecondCountableTopology α\ninst✝⁸ : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : AddCommGroup α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsOrdered... | [] | have h_int := (hf.locallyIntegrableOn (Ici 0)).integrableOn_of_isBigO_atTop ho hg
rw [← integrableOn_univ, ← Iic_union_Ici_of_le le_rfl, integrableOn_union]
refine ⟨?_, h_int⟩
have h_map_neg : (μ.restrict (Ici 0)).map Neg.neg = μ.restrict (Iic 0) := by
conv => rhs; rw [← Measure.map_neg_eq_self μ, measurableE... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Asymptotics | {
"line": 170,
"column": 2
} | {
"line": 179,
"column": 52
} | {
"line": 181,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\nf : α → E\ng : α → F\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : SecondCountableTopology α\ninst✝⁸ : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : AddCommGroup α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsOrdered... | [] | have h_int := (hf.locallyIntegrableOn (Ici 0)).integrableOn_of_isBigO_atTop ho hg
rw [← integrableOn_univ, ← Iic_union_Ici_of_le le_rfl, integrableOn_union]
refine ⟨?_, h_int⟩
have h_map_neg : (μ.restrict (Ici 0)).map Neg.neg = μ.restrict (Iic 0) := by
conv => rhs; rw [← Measure.map_neg_eq_self μ, measurableE... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 585,
"column": 4
} | {
"line": 585,
"column": 14
} | {
"line": 586,
"column": 4
} | [
{
"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... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 219,
"column": 73
} | {
"line": 221,
"column": 61
} | {
"line": 222,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nthis : Tendsto (s + 1).partialGamma atTop (𝓝 (s * s.GammaIntegral))\n⊢ (s + 1).GammaIntegral = s * s.GammaIntegral",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | by
refine tendsto_nhds_unique ?_ this
apply tendsto_partialGamma; rw [add_re, one_re]; linarith | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 443,
"column": 6
} | {
"line": 443,
"column": 26
} | {
"line": 443,
"column": 26
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ 0 < Gamma s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instPow",
"Real",
"Set.Ioi",
"HMul.hMul",
"Real.instZero",
"Real.instRCLike",
"congrArg",
... | [
"s : ℝ\nhs : 0 < s\n⊢ 0 < ∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1)"
] | Gamma_eq_integral hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 6
} | {
"line": 142,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ × ℝ → E\n⊢ ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p) = ∫ (p : ℝ × ℝ), f p",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"DistribMulAction.toD... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ × ℝ → E\n⊢ ∫ (p : ℝ × ℝ), f p = ∫ (p : ℝ × ℝ) in polarCoord.target, p.1 • f (↑polarCoord.symm p)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 2
} | [
{
"pp": "f : ℝ × ℝ → ℝ≥0∞\n⊢ ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑polarCoord.symm p) = ∫⁻ (p : ℝ × ℝ), f p",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"instSMulOfMul",
"ENNReal.ofReal",
"CommSemiring.toSemiring... | [
"f : ℝ × ℝ → ℝ≥0∞\n⊢ ∫⁻ (p : ℝ × ℝ), f p = ∫⁻ (p : ℝ × ℝ) in polarCoord.target, ENNReal.ofReal p.1 • f (↑polarCoord.symm p)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 847,
"column": 21
} | {
"line": 847,
"column": 51
} | {
"line": 848,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nN : ℕ∞\nn : ℕ\nhf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume\nhn : ↑n ≤ N\nx : ℝ\nA : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume\nB : AEStronglyMeasurable f volume\n⊢ ((iteratedFDeri... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nN : ℕ∞\nn : ℕ\nhf : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun x ↦ x ^ n • f x) volume\nhn : ↑n ≤ N\nx : ℝ\nA : ∀ (n : ℕ), ↑n ≤ N → Integrable (fun v ↦ ‖v‖ ^ n * ‖f v‖) volume\nB : AEStronglyMeasurable f volume\n⊢ ((𝓕 (fun v ↦ fourierPowSMu... | iteratedFDeriv_fourier A B hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 95
} | {
"line": 278,
"column": 0
} | [
{
"pp": "case e'_2\nι : Type u_1\ninst✝² : Fintype ι\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : (ι → ℝ × ℝ) → E\nx : ι → ℝ × ℝ\nhx : x ∈ Set.univ.pi fun x ↦ polarCoord.target\n⊢ ((∏ i, (x i).1) • f fun i ↦ ↑polarCoord.symm (x i)) =\n |(fderivPiPolarCoordSymm x).det| • f fun i ... | [] | simp_rw [det_fderivPiPolarCoordSymm, Finset.abs_prod, abs_fst_of_mem_pi_polarCoord_target hx] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1049,
"column": 48
} | {
"line": 1050,
"column": 33
} | {
"line": 1052,
"column": 0
} | [
{
"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✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : No... | [] | by
grw [le_seminorm 𝕜 k n f x] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1148,
"column": 4
} | {
"line": 1148,
"column": 90
} | {
"line": 1149,
"column": 2
} | [
{
"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... | [] | simpa using one_add_le_sup_seminorm_apply (m := m) (k := n) (n := 0) le_rfl le_rfl f x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1361,
"column": 2
} | {
"line": 1361,
"column": 41
} | {
"line": 1362,
"column": 2
} | [
{
"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... | refine ⟨k, C, C.coe_nonneg, fun f ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 162,
"column": 15
} | {
"line": 162,
"column": 95
} | {
"line": 162,
"column": 95
} | [
{
"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 → ... | (fun a b => by rw [sq]; ring_nf : ∀ a b : ℂ, (a - b * I) ^ 2 = (-a + b * I) ^ 2) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 32
} | {
"line": 98,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\nn✝ : ℕ\na b : α\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite α\nψ : AddChar α Circle\n⊢ (fun ψ ↦ { toFun := fun a ↦ ⟨ψ a, ⋯⟩, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ })\n ((fun ψ ↦ toMonoidHomEquiv.symm (coeHom.comp ψ.toMonoidHom)) ψ) =\n ψ",
"ppTerm"... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 32
} | {
"line": 98,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\nn✝ : ℕ\na b : α\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite α\nψ : AddChar α Circle\n⊢ (fun ψ ↦ { toFun := fun a ↦ ⟨ψ a, ⋯⟩, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ })\n ((fun ψ ↦ toMonoidHomEquiv.symm (coeHom.comp ψ.toMonoidHom)) ψ) =\n ψ",
"ppTerm"... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 98,
"column": 20
} | {
"line": 98,
"column": 33
} | {
"line": 99,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\nn✝ : ℕ\na b : α\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite α\nψ : AddChar α ℂ\n⊢ (fun ψ ↦ toMonoidHomEquiv.symm (coeHom.comp ψ.toMonoidHom))\n ((fun ψ ↦ { toFun := fun a ↦ ⟨ψ a, ⋯⟩, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ }) ψ) =\n ψ",
"ppTerm": "?m... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 98,
"column": 20
} | {
"line": 98,
"column": 33
} | {
"line": 99,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\nn✝ : ℕ\na b : α\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite α\nψ : AddChar α ℂ\n⊢ (fun ψ ↦ toMonoidHomEquiv.symm (coeHom.comp ψ.toMonoidHom))\n ((fun ψ ↦ { toFun := fun a ↦ ⟨ψ a, ⋯⟩, map_zero_eq_one' := ⋯, map_add_eq_mul' := ⋯ }) ψ) =\n ψ",
"ppTerm": "?m... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.PNat.Prime | {
"line": 200,
"column": 36
} | {
"line": 200,
"column": 40
} | {
"line": 201,
"column": 2
} | [
{
"pp": "m n k : ℕ+\nh : k.Coprime m\n⊢ (k * n).gcd m = m.gcd n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"PNat.gcd",
"HMul.hMul",
"instMulPNat",
"Eq.symm",
"instHMul",
"PNat"
],
"usedFVars": [
"m",
"n",
"k"
],
"us... | [
"m n k : ℕ+\nh : k.Coprime m\n⊢ m.gcd n = (k * n).gcd m"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.PNat.Prime | {
"line": 200,
"column": 36
} | {
"line": 200,
"column": 40
} | {
"line": 201,
"column": 2
} | [
{
"pp": "m n k : ℕ+\nh : k.Coprime m\n⊢ n.gcd m = (k * n).gcd m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"PNat.gcd",
"HMul.hMul",
"instMulPNat",
"Eq.symm",
"instHMul",
"PNat"
],
"usedFVars": [
"k",
"n",
"m"
],
"us... | [
"m n k : ℕ+\nh : k.Coprime m\n⊢ (k * n).gcd m = n.gcd m"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 85
} | {
"line": 236,
"column": 0
} | [
{
"pp": "R : 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\nx : ℕ × ℕ\nhx : x ∈ (n + 1).divisorsAntidiagonal\n⊢ (μ x.1 • ... | [] | rw [if_neg (pos_of_mem_divisors (snd_mem_divisors_of_mem_antidiagonal hx)).ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 179,
"column": 35
} | {
"line": 179,
"column": 43
} | {
"line": 179,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[cocompact ℝ] fun x ↦ |x| ^ (-b)\nK : Compacts ℝ\nr : ℝ\nhr : ↑K ⊆ Icc (0 - r) r\n⊢ (fun x ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x ↦ |x| ^ (-b)",
"ppTerm": "?m.96... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : C(ℝ, E)\nb : ℝ\nhb : 0 < b\nhf : ⇑f =O[cocompact ℝ] fun x ↦ |x| ^ (-b)\nK : Compacts ℝ\nr : ℝ\nhr : ↑K ⊆ Icc (-r) r\n⊢ (fun x ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x ↦ |x| ^ (-b)"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 448,
"column": 39
} | {
"line": 449,
"column": 31
} | {
"line": 451,
"column": 0
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ (∑ i ∈ range k, X ^ i).IsRoot ζ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.eval",
"congrArg",
"CommSemiring.toSemiring",
"Polynomia... | [] | by
simp [geom_sum_eq_zero hζ hk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 472,
"column": 6
} | {
"line": 477,
"column": 41
} | {
"line": 478,
"column": 6
} | [
{
"pp": "case left\nM : 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⊢ Function.Injective\n ⇑(((Int.castAddHom (ZMod k)).liftOfRight... | [
"case right\nM : 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⊢ Function.Surjective\n ⇑(((Int.castAddHom (ZMod k)).liftOfRightInverse ZM... | · rw [injective_iff_map_eq_zero]
intro i hi
rw [Subtype.ext_iff] at hi
have := (h.zpow_eq_one_iff_dvd _).mp hi
rw [← (CharP.intCast_eq_zero_iff (ZMod k) k _).mpr this, eq_comm]
exact ZMod.intCast_rightInverse i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nhmonic : (X ^ n - C 1).Monic\n⊢ X ^ n - 1 = (Multiset.map (fun ζ ↦ X - C ζ) (X ^ n - C 1).roots).prod",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Polynomial.C",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nhmonic : (X ^ n - C 1).Monic\n⊢ (Multiset.map (fun ζ ↦ X - C ζ) (X ^ n - C 1).roots).prod = X ^ n - 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 6
} | {
"line": 599,
"column": 6
} | [
{
"pp": "case neg\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 = Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"HMu... | [
"case neg\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⊢ Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n) = nthRoots n a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 700,
"column": 2
} | {
"line": 700,
"column": 6
} | {
"line": 701,
"column": 2
} | [
{
"pp": "case neg\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nk : ℕ\nh : IsPrimitiveRoot ζ k\nh0 : ¬k = 0\nthis : NeZero k\n⊢ #(primitiveRoots k R) = φ k",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Nat.totient",
"Nat",
"Finset.card",
"primiti... | [
"case neg\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nk : ℕ\nh : IsPrimitiveRoot ζ k\nh0 : ¬k = 0\nthis : NeZero k\n⊢ φ k = #(primitiveRoots k R)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 28
} | {
"line": 83,
"column": 29
} | [
{
"pp": "case inr\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhdiv : ¬p ∣ n\nhpos : n > 0\nthis : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed\n⊢ eval₂ (algebraMap ℤ K) μ ((expand ℤ p) (minpoly ℤ (μ ^ p))) = 0",
"ppTerm... | [
"case inr\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhdiv : ¬p ∣ n\nhpos : n > 0\nthis : IsIntegrallyClosed ℤ := GCDMonoid.toIsIntegrallyClosed\n⊢ eval₂ (algebraMap ℤ K) μ (eval₂ C (X ^ p) (minpoly ℤ (μ ^ p))) = 0"
] | coe_expand, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 6
} | {
"line": 288,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nhspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ\n⊢ cyclotomic 1 R = X - 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Polynomial.instOne",
"HSub.hSub",
"Polynomial.cyclotomic",
"instOfNatNat",
"Polynomial",... | [
"R : Type u_1\ninst✝ : Ring R\nhspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ\n⊢ X - 1 = cyclotomic 1 R"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 147,
"column": 2
} | {
"line": 152,
"column": 62
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"HMul.hMul",
"Algebra.algebraMap",
"Ring... | [] | letI : Algebra (ZMod p) R := ZMod.algebra _ _
suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p by
rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
this, Polynomial.map_pow]
rw [← ZMod.expand_card, ← map_cyclotomic_int n, ← map_expand,
cyclotom... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 147,
"column": 2
} | {
"line": 152,
"column": 62
} | {
"line": 154,
"column": 0
} | [
{
"pp": "R : Type u_1\np n : ℕ\nhp : Fact (Nat.Prime p)\ninst✝¹ : Ring R\ninst✝ : CharP R p\nhn : p ∣ n\n⊢ cyclotomic (n * p) R = cyclotomic n R ^ p",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"HMul.hMul",
"Algebra.algebraMap",
"Ring... | [] | letI : Algebra (ZMod p) R := ZMod.algebra _ _
suffices cyclotomic (n * p) (ZMod p) = cyclotomic n (ZMod p) ^ p by
rw [← map_cyclotomic _ (algebraMap (ZMod p) R), ← map_cyclotomic _ (algebraMap (ZMod p) R),
this, Polynomial.map_pow]
rw [← ZMod.expand_card, ← map_cyclotomic_int n, ← map_expand,
cyclotom... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 22
} | {
"line": 102,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 22
} | {
"line": 102,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 22
} | {
"line": 102,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 240,
"column": 8
} | {
"line": 240,
"column": 12
} | {
"line": 241,
"column": 8
} | [
{
"pp": "case e'_4.e'_5\nn : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nthis : ¬SameRay ℝ (↑q) (-ζ)\n⊢ q = ‖q‖",
"ppTerm": "?e'_4.e'_5",
"assigned": true,... | [
"case e'_4.e'_5\nn : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := ⋯\nhζ : IsPrimitiveRoot ζ n\nthis : ¬SameRay ℝ (↑q) (-ζ)\n⊢ ‖q‖ = q"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 323,
"column": 40
} | {
"line": 326,
"column": 36
} | {
"line": 327,
"column": 2
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n\nn : ℕ\nhn1 : n ∈ S\nhn2 : n ≠ 0\n⊢ ∃ r, IsPrimitiveRoot r n",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLik... | [] | by
obtain ⟨r, hr1, hr2⟩ := h n hn1 hn2
exact ⟨⟨r, subset_adjoin ⟨n, hn1, hn2, hr1⟩⟩, Subtype.val_injective hr1,
fun l hl ↦ hr2 l congr($hl.1)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 331,
"column": 6
} | {
"line": 332,
"column": 65
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n\nx✝ x : B\nhx : x ∈ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}\n⊢ ⟨x, ⋯⟩ ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?mem",
"assigned": tr... | [] | obtain ⟨n, hn1, hn2, hx⟩ := hx
exact subset_adjoin ⟨n, hn1, hn2, Subtype.val_injective hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 331,
"column": 6
} | {
"line": 332,
"column": 65
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n\nx✝ x : B\nhx : x ∈ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}\n⊢ ⟨x, ⋯⟩ ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?mem",
"assigned": tr... | [] | obtain ⟨n, hn1, hn2, hx⟩ := hx
exact subset_adjoin ⟨n, hn1, hn2, Subtype.val_injective hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 178,
"column": 2
} | {
"line": 180,
"column": 60
} | {
"line": 182,
"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\n⊢ (Polynomial.map (algebraMap (ZMod p) K) (X ^ Nat.card K - X)).Splits",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
... | [] | haveI : Fintype K := Fintype.ofFinite K
rw [Nat.card_eq_fintype_card]
exact (FiniteField.isSplittingField_sub K (ZMod p)).splits | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 178,
"column": 2
} | {
"line": 180,
"column": 60
} | {
"line": 182,
"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\n⊢ (Polynomial.map (algebraMap (ZMod p) K) (X ^ Nat.card K - X)).Splits",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.ofFinite",
... | [] | haveI : Fintype K := Fintype.ofFinite K
rw [Nat.card_eq_fintype_card]
exact (FiniteField.isSplittingField_sub K (ZMod p)).splits | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 35
} | {
"line": 513,
"column": 4
} | [
{
"pp": "case pos\np : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhk : k ≠ 0\nht... | [
"case pos\np : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhk : k ≠ 0\nhtwo : p ^ (k ... | simp only [add_eq_right] at hks | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 49,
"column": 17
} | {
"line": 49,
"column": 23
} | {
"line": 50,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 4),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) *\n match y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 49,
"column": 17
} | {
"line": 49,
"column": 23
} | {
"line": 50,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 4),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) *\n match y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 49,
"column": 17
} | {
"line": 49,
"column": 23
} | {
"line": 50,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 4),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) *\n match y with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 50,
"column": 21
} | {
"line": 50,
"column": 27
} | {
"line": 52,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 4),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n 0",
"ppTerm": "?m.331",
"assigned": true,
"usedConstants": [
"Units.val",
"instDecidableNot",
"of_decide_eq_true",
"ZMod.commR... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 50,
"column": 21
} | {
"line": 50,
"column": 27
} | {
"line": 52,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 4),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n 0",
"ppTerm": "?m.331",
"assigned": true,
"usedConstants": [
"Units.val",
"instDecidableNot",
"of_decide_eq_true",
"ZMod.commR... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 50,
"column": 21
} | {
"line": 50,
"column": 27
} | {
"line": 52,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 4),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 1 => 1\n | 3 => -1) =\n 0",
"ppTerm": "?m.331",
"assigned": true,
"usedConstants": [
"Units.val",
"instDecidableNot",
"of_decide_eq_true",
"ZMod.commR... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 8
} | {
"line": 57,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 4), χ₄ a = 0 ∨ χ₄ a = 1 ∨ χ₄ a = -1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"ZMod.χ₄",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"AddGroupWithOne.toAddMonoid... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 10
} | {
"line": 69,
"column": 2
} | [
{
"pp": "n : ℤ\n⊢ ∀ (m : ℤ), 0 ≤ m → m < 4 → χ₄ ↑m = if m % 2 = 0 then 0 else if m = 1 then 1 else -1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.decidableLELT",
"ZMod.χ₄",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toCommMo... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 84,
"column": 91
} | {
"line": 84,
"column": 97
} | {
"line": 85,
"column": 2
} | [
{
"pp": "n : ℕ\nhn : n % 2 = 1\n⊢ ∀ m < 4, m % 2 = 1 → (if m = 1 then 1 else -1) = (-1) ^ (m / 2)",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"instHDiv",
"of_decide_eq_true",
"Int.instDecidableEq",
"id",
"HDiv.hDiv",
"Nat.instMod",
"instHMod",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 125,
"column": 17
} | {
"line": 125,
"column": 23
} | {
"line": 126,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 125,
"column": 17
} | {
"line": 125,
"column": 23
} | {
"line": 126,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 125,
"column": 17
} | {
"line": 125,
"column": 23
} | {
"line": 126,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 126,
"column": 21
} | {
"line": 126,
"column": 27
} | {
"line": 128,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 126,
"column": 21
} | {
"line": 126,
"column": 27
} | {
"line": 128,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 126,
"column": 21
} | {
"line": 126,
"column": 27
} | {
"line": 128,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 7 => 1\n | 3 => -1\n | 5 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 8
} | {
"line": 133,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8), χ₈ a = 0 ∨ χ₈ a = 1 ∨ χ₈ a = -1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"ZMod.χ₈",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"AddGroupWithOne.toAddMonoid... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 10
} | {
"line": 147,
"column": 2
} | [
{
"pp": "n : ℤ\n⊢ ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈ ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.decidableLELT",
"ZMod.χ₈",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 164,
"column": 17
} | {
"line": 164,
"column": 23
} | {
"line": 165,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => ... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 164,
"column": 17
} | {
"line": 164,
"column": 23
} | {
"line": 165,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 164,
"column": 17
} | {
"line": 164,
"column": 23
} | {
"line": 165,
"column": 2
} | [
{
"pp": "⊢ ∀ (x y : ZMod 8),\n (match x * y with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n (match x with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => ... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 27
} | {
"line": 167,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 27
} | {
"line": 167,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 165,
"column": 21
} | {
"line": 165,
"column": 27
} | {
"line": 167,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8),\n ¬IsUnit a →\n (match a with\n | 0 => 0\n | 2 => 0\n | 4 => 0\n | 6 => 0\n | 1 => 1\n | 3 => 1\n | 5 => -1\n | 7 => -1) =\n 0",
"ppTerm": "?m.613",
"assigned": true,
"usedConstants": [
"Units.val... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 8
} | {
"line": 172,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8), χ₈' a = 0 ∨ χ₈' a = 1 ∨ χ₈' a = -1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toSemiring",
"ZMod.fintype",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 10
} | {
"line": 178,
"column": 2
} | [
{
"pp": "n : ℤ\n⊢ ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈' ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 3 then 1 else -1",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.decidableLELT",
"of_decide_eq_true",
"ZMod.commRing",
"CommSemiring.toCommMonoidWith... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 8
} | {
"line": 189,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8), χ₈' a = χ₄ a.cast * χ₈ a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.χ₈",
"HMul.hMul",
"ZMod.cast",
"ZMod.χ₄",
"of_decide_eq_true",
"ZMod.commRing",
"ZMod.fintype",
"CommSemiring.toCommMonoidWithZero",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 8
} | {
"line": 189,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8), χ₈' a = χ₄ a.cast * χ₈ a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.χ₈",
"HMul.hMul",
"ZMod.cast",
"ZMod.χ₄",
"of_decide_eq_true",
"ZMod.commRing",
"ZMod.fintype",
"CommSemiring.toCommMonoidWithZero",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 8
} | {
"line": 189,
"column": 0
} | [
{
"pp": "⊢ ∀ (a : ZMod 8), χ₈' a = χ₄ a.cast * χ₈ a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.χ₈",
"HMul.hMul",
"ZMod.cast",
"ZMod.χ₄",
"of_decide_eq_true",
"ZMod.commRing",
"ZMod.fintype",
"CommSemiring.toCommMonoidWithZero",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 341,
"column": 4
} | {
"line": 343,
"column": 60
} | {
"line": 344,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\n⊢ ∀ (a b : MulChar R R'), a * b = b * a",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Units.val",
"HMul.hMul",
"CommMonoid.toCommSemigro... | [] | intro χ₁ χ₂
ext a
simp only [mul_comm, Pi.mul_apply, MulChar.coeToFun_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 341,
"column": 4
} | {
"line": 343,
"column": 60
} | {
"line": 344,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\n⊢ ∀ (a b : MulChar R R'), a * b = b * a",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Units.val",
"HMul.hMul",
"CommMonoid.toCommSemigro... | [] | intro χ₁ χ₂
ext a
simp only [mul_comm, Pi.mul_apply, MulChar.coeToFun_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 55
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\na : Rˣ\nn : ℕ\nih : (χ ^ n) ↑a = χ ↑a ^ n\n⊢ (χ ^ (n + 1)) ↑a = χ ↑a ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"... | [] | rw [pow_succ, pow_succ, mul_apply, ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 55
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\na : Rˣ\nn : ℕ\nih : (χ ^ n) ↑a = χ ↑a ^ n\n⊢ (χ ^ (n + 1)) ↑a = χ ↑a ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"... | [] | rw [pow_succ, pow_succ, mul_apply, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 351,
"column": 17
} | {
"line": 351,
"column": 55
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\na : Rˣ\nn : ℕ\nih : (χ ^ n) ↑a = χ ↑a ^ n\n⊢ (χ ^ (n + 1)) ↑a = χ ↑a ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"... | [] | rw [pow_succ, pow_succ, mul_apply, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 32
} | {
"line": 616,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : DecidableEq R\na : R\nx✝ : a ∈ Finset.univ\nh : IsUnit a\n⊢ 1 a = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MulChar.one_apply_coe",
"CommSemiring.to... | [] | exact one_apply_coe h.unit | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 32
} | {
"line": 616,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : DecidableEq R\na : R\nx✝ : a ∈ Finset.univ\nh : IsUnit a\n⊢ 1 a = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MulChar.one_apply_coe",
"CommSemiring.to... | [] | exact one_apply_coe h.unit | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 615,
"column": 6
} | {
"line": 615,
"column": 32
} | {
"line": 616,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : DecidableEq R\na : R\nx✝ : a ∈ Finset.univ\nh : IsUnit a\n⊢ 1 a = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MulChar.one_apply_coe",
"CommSemiring.to... | [] | exact one_apply_coe h.unit | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 932,
"column": 2
} | {
"line": 932,
"column": 78
} | {
"line": 933,
"column": 2
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝¹ : NeZero n₁\ninst✝ : NeZero n₂\nζ₁ : ↥C₁\nhζ₁ : IsPrimitiveRoot ζ₁ n₁\nζ₂ ... | [
"A : Type u\nB : Type v\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝¹ : NeZero n₁\ninst✝ : NeZero n₂\nζ₁ : ↥C₁\nζ₂ : ↥C₂\nhζ₂ : IsPrimitiveRoot ζ₂ n₂\nhζ₁ :... | replace hζ₁ := hζ₁.map_of_injective (FaithfulSMul.algebraMap_injective C₁ B) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
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