module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 201,
"column": 2
} | {
"line": 205,
"column": 72
} | {
"line": 207,
"column": 0
} | [
{
"pp": "E : Type u_5\nF : Type u_8\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ E\nn : ℕ\nm : Fin n → E\nf : 𝓢(E, F)\n⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | induction n with
| zero => simp
| succ n IH =>
rw [iteratedLineDerivOp_succ_left]
exact (tsupport_lineDerivOp_subset (m 0) _).trans (IH <| Fin.tail m) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 31
} | {
"line": 163,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : DecidableEq ι\ni : ι\na : G i\nf : ↥(lp G 2)\nj : ι\nh : ¬j = i\n⊢ ⟪Pi.single i a j, ↑f j⟫ = 0",
"ppTerm": "?neg✝",
"a... | [] | · simp [Pi.single_eq_of_ne h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 21
} | {
"line": 235,
"column": 21
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nx y : AddCircle T\nhxy : ↑x.toCircle = ↑y.toCircle\n⊢ x = y",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"Complex.instNormedField",
"Subtype.coe_inj",
"SeminormedRing.toRi... | [
"T : ℝ\nhT : Fact (0 < T)\nx y : AddCircle T\nhxy : x.toCircle = y.toCircle\n⊢ x = y"
] | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 11
} | {
"line": 104,
"column": 2
} | [
{
"pp": "x y : ℝ\n⊢ cexp (2 * ↑π * ↑y * I) = cexp (2 * ↑π * I * 1 * ↑y / 1)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 9
} | {
"line": 183,
"column": 0
} | [
{
"pp": "case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 174,
"column": 6
} | {
"line": 175,
"column": 73
} | {
"line": 176,
"column": 4
} | [
{
"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 ⊆... | [] | rw [setIntegral_union disjoint_sdiff_inter (hs.inter u_open.measurableSet)
(h''i.mono_set sdiff_subset) (h''i.mono_set inter_subset_left)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 202,
"column": 6
} | {
"line": 202,
"column": 23
} | {
"line": 203,
"column": 6
} | [
{
"pp": "case hcg\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns : Set α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nhs : Measura... | [
"case hcg\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns : Set α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nhs : MeasurableSet s\nt ... | rw [← sub_self a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 11
} | {
"line": 381,
"column": 11
} | [
{
"pp": "E : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_4\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module ℝ V\ninst✝² : MeasurableSpace V\nW : Type u_5\ninst✝¹ : AddCommGroup W\ninst✝ : Module ℝ W\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nμ : Measure V\nf : V → E\nw : W\n⊢ ∫ (v : V), Complex.exp (↑(-... | [] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 443,
"column": 73
} | {
"line": 443,
"column": 80
} | {
"line": 443,
"column": 80
} | [
{
"pp": "V : Type u_1\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteDimensional ℝ V\nf : V → E\nw : V\n⊢ ∫ (v : V), Complex.exp (↑(-(2 * π * ⟪v, w⟫)) * Comple... | [] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 27
} | {
"line": 283,
"column": 28
} | [
{
"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 ... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 56
} | {
"line": 207,
"column": 0
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\nh : IntegrableOn (fun x ↦ ‖↑x ^ s‖) (Ioi t) volume\na : ℝ\nha : a ∈ Ioi t\n⊢ ‖↑a ^ s‖ = a ^ s.re",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"Real.instZero",
"congrArg"... | [] | rw [Complex.norm_cpow_eq_rpow_re_of_pos (ht.trans ha)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 221,
"column": 41
} | {
"line": 222,
"column": 47
} | {
"line": 222,
"column": 47
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\nhs : s.re ≤ 0\nx : ℝ\nhx : x ∈ Ioi t\n⊢ deriv (fun x ↦ ‖↑x ^ s‖) x = ?m.36 x",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.d... | [] | by
rw [deriv_norm_ofReal_cpow _ (ht.trans hx)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 11
} | {
"line": 228,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nX : ℝ\nhX : X ∈ Ici 0\n⊢ (fun X ↦ s * s.partialGamma X - ↑X ^ s * ↑(rexp (-X))) X = s * s.partialGamma X - ↑(rexp (-X)) * ↑X ^ s",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 440,
"column": 8
} | {
"line": 440,
"column": 27
} | {
"line": 440,
"column": 28
} | [
{
"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... | [
"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\nhφ : ∀ (x ... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 69
} | {
"line": 454,
"column": 2
} | [
{
"pp": "case hf\ns : ℝ\nhs : 0 < s\nthis : (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0\n⊢ ∀ x ∈ Ioi 0, 0 x ≤ (fun x ↦ rexp (-x) * x ^ (s - 1)) x",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_po... | [] | exact fun x hx => (mul_pos (exp_pos _) (rpow_pos_of_pos hx _)).le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 23
} | {
"line": 56,
"column": 24
} | [
{
"pp": "b : ℂ\nc T : ℝ\n⊢ ‖cexp (-b * (↑T + ↑c * I) ^ 2)‖ = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"Complex.im",
"Real.instSu... | [
"b : ℂ\nc T : ℝ\n⊢ rexp (-b * (↑T + ↑c * I) ^ 2).re = rexp (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2))"
] | Complex.norm_exp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 9
} | {
"line": 60,
"column": 0
} | [
{
"pp": "b : ℂ\nc T : ℝ\n⊢ rexp\n (-(b.re * ((T + (c * 0 - 0 * 1)) * (T + (c * 0 - 0 * 1)) - (0 + (c * 1 + 0 * 0)) * (0 + (c * 1 + 0 * 0))) -\n b.im * ((T + (c * 0 - 0 * 1)) * (0 + (c * 1 + 0 * 0)) + (0 + (c * 1 + 0 * 0)) * (T + (c * 0 - 0 * 1))))) =\n rexp (-(b.re * (T * T) - 2 * b.im * c * T ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 90
} | {
"line": 130,
"column": 2
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\nthis : Integrable (fun x ↦ rexp (-(b.re * x ^ 2))) volume\n⊢ HasFiniteIntegral (fun a ↦ rexp (c ^ 2 * (b.im ^ 2 / b.re + b.re)) * rexp (-(b.re * (a - b.im * c / b.re) ^ 2)))\n volume",
"ppTerm": "?m.179",
"assigned": true,
"usedConstants": [
"NormedComm... | [] | exact (Integrable.comp_sub_right this (b.im * c / b.re)).hasFiniteIntegral.const_mul _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 90
} | {
"line": 130,
"column": 2
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\nthis : Integrable (fun x ↦ rexp (-(b.re * x ^ 2))) volume\n⊢ HasFiniteIntegral (fun a ↦ rexp (c ^ 2 * (b.im ^ 2 / b.re + b.re)) * rexp (-(b.re * (a - b.im * c / b.re) ^ 2)))\n volume",
"ppTerm": "?m.179",
"assigned": true,
"usedConstants": [
"NormedComm... | [] | exact (Integrable.comp_sub_right this (b.im * c / b.re)).hasFiniteIntegral.const_mul _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 90
} | {
"line": 130,
"column": 2
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\nthis : Integrable (fun x ↦ rexp (-(b.re * x ^ 2))) volume\n⊢ HasFiniteIntegral (fun a ↦ rexp (c ^ 2 * (b.im ^ 2 / b.re + b.re)) * rexp (-(b.re * (a - b.im * c / b.re) ^ 2)))\n volume",
"ppTerm": "?m.179",
"assigned": true,
"usedConstants": [
"NormedComm... | [] | exact (Integrable.comp_sub_right this (b.im * c / b.re)).hasFiniteIntegral.const_mul _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.Inversion | {
"line": 118,
"column": 10
} | {
"line": 118,
"column": 29
} | {
"line": 118,
"column": 30
} | [
{
"pp": "case h'φ\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\n... | [
"case h'φ\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nv : V\nh'f :... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 13
} | {
"line": 123,
"column": 6
} | [
{
"pp": "case h'φ\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\n... | [
"case h'φ\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\nv : V\nh'f :... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 162,
"column": 39
} | {
"line": 162,
"column": 46
} | {
"line": 162,
"column": 47
} | [
{
"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 * ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 11
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case e_a\nb : ℂ\nhb : 0 < b.re\nc : ℂ\nthis✝ : b ≠ 0\nh : (-↑π * b).re < 0\nt : ℝ\nthis :\n ∀ (x : ℝ),\n ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =\n -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0\n⊢ 0 * 4 * b - -(2 ^ 2 * ↑π * (-(I * ↑t) + c) ^ 2) = -(↑π * 4 * (↑... | [
"case e_a\nb : ℂ\nhb : 0 < b.re\nc : ℂ\nthis✝ : b ≠ 0\nh : (-↑π * b).re < 0\nt : ℝ\nthis :\n ∀ (x : ℝ),\n ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =\n -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0\n⊢ -(↑π * I * ↑t * c * 8) + ↑π * I ^ 2 * ↑t ^ 2 * 4 + ↑π * c ^ 2 * 4 =\n -(... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Fourier.Inversion | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 25
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case e_f.e_a.e_a\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f ... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Distribution.FourierMultiplier | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 11
} | {
"line": 122,
"column": 4
} | [
{
"pp": "case e_f.e_a\nE : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : NormedSpace ℂ F\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\nι : Type := Fin (Mo... | [
"case e_f.e_a\nE : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : NormedSpace ℂ F\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\nι : Type := Fin (Module.finrank... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Distribution.FourierMultiplier | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 11
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case e_f.e_a\nE : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : NormedSpace ℂ F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : 𝓢'(E, F)\nι : Type := Fin (Module.finrank ℝ E)\nb : Or... | [
"case e_f.e_a\nE : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : NormedSpace ℂ F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : 𝓢'(E, F)\nι : Type := Fin (Module.finrank ℝ E)\nb : OrthonormalBas... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 123,
"column": 70
} | {
"line": 123,
"column": 85
} | {
"line": 123,
"column": 85
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nx y : UnitAddTorus d\ni : d\nhi : ¬x i = y i\n⊢ ¬↑(AddCircle.toCircle (x i)) = ↑(AddCircle.toCircle (y i))",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",... | [
"d : Type u_1\ninst✝ : Fintype d\nx y : UnitAddTorus d\ni : d\nhi : ¬x i = y i\n⊢ ¬AddCircle.toCircle (x i) = AddCircle.toCircle (y i)"
] | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 73,
"column": 2
} | {
"line": 74,
"column": 25
} | {
"line": 76,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nx y : W\n⊢ char he hL (x... | [] | ext
simp [e.map_add_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 73,
"column": 2
} | {
"line": 74,
"column": 25
} | {
"line": 76,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nx y : W\n⊢ char he hL (x... | [] | ext
simp [e.map_add_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 9
} | {
"line": 99,
"column": 2
} | [
{
"pp": "case h\nV : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhe' : e ≠ 1\nhL : Continuous fun p ↦ (L p.1) p.2\nhL'... | [
"case h\nV : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhe' : e ≠ 1\nhL : Continuous fun p ↦ (L p.1) p.2\nhL' : ∀ (v : V)... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 129,
"column": 69
} | {
"line": 130,
"column": 48
} | {
"line": 132,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℂ F\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : SecondCountableTopology E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\nf : 𝓢(E, F)\ng : 𝓢(E, ℂ)\n⊢ ((t... | [] | by
simp [toTemperedDistributionCLM, comp_apply _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 216,
"column": 66
} | {
"line": 227,
"column": 27
} | {
"line": 229,
"column": 0
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : Is... | [] | by
rw [LinearMap.ker_eq_bot', ContinuousLinearMap.coe_coe]
intro f hf
rw [eq_zero_iff_ae_eq_zero]
apply ae_eq_zero_of_integral_contDiff_smul_eq_zero
· exact (Lp.memLp f).locallyIntegrable hp.elim
· intro g g_smooth g_cpt
have hg₁ : HasCompactSupport (Complex.ofRealCLM ∘ g) := g_cpt.comp_left rfl
hav... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 250,
"column": 4
} | {
"line": 262,
"column": 10
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case right\nE : 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\nhs : ↑(Module.finrank ... | [] | · rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top (by norm_num) (by norm_num)]
suffices h : ∫⁻ a : E, ENNReal.ofReal ‖(1 + ‖a‖ ^ 2) ^ (-s)‖ < ⊤ from by
norm_cast
simp_rw [ofReal_norm] at h
simp_rw [← enorm_pow]
convert h
rw [← Real.rpow_mul_natCast (by positivity)]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.Convolution | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 52
} | {
"line": 144,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_2\nE : Type u_3\nF : Type u_4\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹⁷ : RCLike 𝕜\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : InnerProductSpace ℝ E\ninst✝¹⁴ : FiniteDimensional ℝ E\ninst✝¹³ : MeasurableSpace E\ninst✝¹² : BorelSpace E\ninst✝¹¹ : NormedAddCommGroup ... | [] | simp [FourierTransform.fourier_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.Convolution | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 52
} | {
"line": 144,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_2\nE : Type u_3\nF : Type u_4\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹⁷ : RCLike 𝕜\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : InnerProductSpace ℝ E\ninst✝¹⁴ : FiniteDimensional ℝ E\ninst✝¹³ : MeasurableSpace E\ninst✝¹² : BorelSpace E\ninst✝¹¹ : NormedAddCommGroup ... | [] | simp [FourierTransform.fourier_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.Convolution | {
"line": 143,
"column": 17
} | {
"line": 143,
"column": 52
} | {
"line": 144,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nR : Type u_2\nE : Type u_3\nF : Type u_4\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹⁷ : RCLike 𝕜\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : InnerProductSpace ℝ E\ninst✝¹⁴ : FiniteDimensional ℝ E\ninst✝¹³ : MeasurableSpace E\ninst✝¹² : BorelSpace E\ninst✝¹¹ : NormedAddCommGroup ... | [] | simp [FourierTransform.fourier_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 13
} | {
"line": 275,
"column": 6
} | [
{
"pp": "case e'_3\nE : 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\nhs : ↑(Module.finrank ℝ... | [
"case e'_3\nE : 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\nhs : ↑(Module.finrank ℝ E) < 2 * s\... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 168,
"column": 32
} | {
"line": 168,
"column": 49
} | {
"line": 168,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ doubleDualEmb a = doubleDualEmb 0 ↔ a = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.commRing",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAddGroup"... | [
"α : Type u_1\ninst✝¹ : AddCommGroup α\na : α\ninst✝ : Finite α\n⊢ a = 0 ↔ a = 0"
] | doubleDualEmb_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 78,
"column": 32
} | {
"line": 78,
"column": 39
} | {
"line": 78,
"column": 39
} | [
{
"pp": "case a\nE : 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\nw : V\nhw : w ≠ 0\nhiw : ⟪i w, w⟫ = 1 / 2\nv : V\nH : ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 133,
"column": 43
} | {
"line": 133,
"column": 89
} | {
"line": 133,
"column": 89
} | [
{
"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... | div_mul_cancel₀ _ (norm_eq_zero.not.mpr hw_ne) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 380,
"column": 36
} | {
"line": 380,
"column": 48
} | {
"line": 380,
"column": 48
} | [
{
"pp": "case neg\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : i.gcd ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ ↑i') k",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Di... | [
"case neg\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : i.gcd ↑k = 1\nh0 : ¬0 ≤ i\ni' : ℕ\nhi' : ↑i' = -i\n⊢ IsPrimitiveRoot (ζ ^ i') k"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 11
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case e'_3\nq : ℚ\nh : Even q.num\nn : ℤ\nhn : q.num = 2 • n\n⊢ cexp (↑π * I * (2 * ↑n / ↑q.den)) = cexp (2 * ↑π * I * (↑n / ↑q.den))",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.mpr",
"NonAssocSem... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 11
} | {
"line": 79,
"column": 2
} | [
{
"pp": "case e'_3\nq : ℚ\nh : Odd q.num\n⊢ cexp (↑π * I * ↑q) = cexp (2 * ↑π * I * (↑q / 2))",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.d... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 125,
"column": 73
} | {
"line": 128,
"column": 53
} | {
"line": 130,
"column": 0
} | [
{
"pp": "k : ℕ\n⊢ (primitiveRoots k ℂ).card = φ k",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"instHDiv",
"Real.pi",
"HMul.hMul",
"Complex.commRing",
"congrArg",
"Finset",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instDivInvMonoid",
"C... | [] | by
by_cases h : k = 0
· simp [h]
exact (isPrimitiveRoot_exp k h).card_primitiveRoots | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 53
} | {
"line": 197,
"column": 4
} | [
{
"pp": "K : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nx : ℕ\nhx : x ∈ k.properDivisors\n... | [
"K : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nx : ℕ\nhx : x ∈ k.properDivisors\nxsmall : x <... | obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hx).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 732,
"column": 25
} | {
"line": 733,
"column": 81
} | {
"line": 735,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R... | [] | simp [← pow_mul, mul_comm b,
pow_eq_pow_of_modEq h (isPrimitiveRoot_of_mem_primitiveRoots x.2).pow_eq_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 11
} | {
"line": 221,
"column": 12
} | [
{
"pp": "K : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P ↦ map (Int.castRingHom K) P = cyclotomic' (↑n) K) ... | [
"K : Type u_2\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nζ : K\nn : ℕ+\nh : IsPrimitiveRoot ζ ↑n\nP : ℤ[X]\nhP : map (Int.castRingHom K) P = cyclotomic' (↑n) K ∧ P.degree = (cyclotomic' (↑n) K).degree ∧ P.Monic\nQ : ℤ[X]\nhQ : (fun P ↦ map (Int.castRingHom K) P = cyclotomic' (↑n) K) Q\n⊢ map (In... | hP.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 327,
"column": 2
} | {
"line": 328,
"column": 27
} | {
"line": 330,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\n⊢ (cyclotomic n R).natDegree ≤ n.totient",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"le_refl",
"Nat.instMulZeroClass",
"Polynomial.natDegree_of_subsingleton",
"LinearOrderedCommMon... | [] | nontriviality R
rw [natDegree_cyclotomic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 327,
"column": 2
} | {
"line": 328,
"column": 27
} | {
"line": 330,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\n⊢ (cyclotomic n R).natDegree ≤ n.totient",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"le_refl",
"Nat.instMulZeroClass",
"Polynomial.natDegree_of_subsingleton",
"LinearOrderedCommMon... | [] | nontriviality R
rw [natDegree_cyclotomic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 24
} | {
"line": 396,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nhn : n > 0\n⊢ d ∈ n.divisors.erase 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.mem_divisors._simp_1",
"False",
"Dvd.dvd",
"eq_false",
"congrArg",
"and_self",
... | [] | simp [hd, hdn, hn.ne'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 878,
"column": 67
} | {
"line": 878,
"column": 91
} | {
"line": 878,
"column": 91
} | [
{
"pp": "G : Type u_7\nG' : Type u_8\ninst✝³ : Group G\ninst✝² : IsCyclic G\ninst✝¹ : Finite G\ninst✝ : CommGroup G'\na : G\nha : a ≠ 1\ninst : Fintype G := Fintype.ofFinite G\nζ : G'\nhζ : IsPrimitiveRoot ζ (Nat.card G)\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ Nat.card G ∣ Nat.card G",
"ppTerm": "... | [
"G : Type u_7\nG' : Type u_8\ninst✝³ : Group G\ninst✝² : IsCyclic G\ninst✝¹ : Finite G\ninst✝ : CommGroup G'\na : G\nha : a ≠ 1\ninst : Fintype G := Fintype.ofFinite G\nζ : G'\nhζ : IsPrimitiveRoot ζ (Nat.card G)\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ Fintype.card G ∣ Fintype.card G"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 38
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case neg\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\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : ¬n = 0\nhpos : 0 < n\nP : ℤ[X] := minpoly ℤ μ\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhdiff : ¬P = Q\nPmonic : P.Monic\nQmonic : Q... | [
"case neg\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\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : ¬n = 0\nhpos : 0 < n\nP : ℤ[X] := minpoly ℤ μ\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhdiff : ¬P = Q\nPmonic : P.Monic\nQmonic : Q.Monic\nPirr... | rw [coe_mapRingHom, Polynomial.map_mul, Polynomial.map_sub, Polynomial.map_one,
Polynomial.map_pow, map_X] at prod | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 44
} | {
"line": 83,
"column": 2
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : (cyclotomic n K).IsRoot μ\nhμn : μ ^ n = 1\nhnμ : ¬IsPrimitiveRoot μ n\nho : 0 < orderOf μ\ni : ℕ\nhio : i ∈ (orderOf μ).divisors\nhiμ : (cyclotomic i K).IsRoot μ\n⊢ False",
"ppTerm": "?m.111",
"assigned": true... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : (cyclotomic n K).IsRoot μ\nhμn : μ ^ n = 1\nhnμ : ¬IsPrimitiveRoot μ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : (cyclotomic i K).IsRoot μ\nhio : i ∣ orderOf μ\n⊢ False"
] | replace hio := Nat.dvd_of_mem_divisors hio | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 555,
"column": 64
} | {
"line": 555,
"column": 79
} | {
"line": 555,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1\nhn : 1 < n\nhprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1\n⊢ (cyclotomic n R * ∏ x ∈ n.properDivisors, cyclotomic x R).coeff 0 = -(cyclotomic n R).coeff 0",
"ppTerm": "?m.261",
"assigned... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1\nhn : 1 < n\nhprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1\n⊢ (cyclotomic n R).coeff 0 * (∏ x ∈ n.properDivisors, cyclotomic x R).coeff 0 = -(cyclotomic n R).coeff 0"
] | mul_coeff_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 232,
"column": 2
} | {
"line": 235,
"column": 35
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℚ\nthis : Fact (Nat.Prime p)\nP : ℚ[X] := ∑ i, C (α i) * X ^ ↑i\nhP : ∀ (i : Fin p), α i = P.coeff ↑i\nhP' : P.degree ≤ ↑(p - 1)\nx✝ : cyclotomic p ℚ ∣ P\nc : ℚ[X]\nh... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℚ\nthis : Fact (Nat.Prime p)\nP : ℚ[X] := ∑ i, C (α i) * X ^ ↑i\nhP : ∀ (i : Fin p), α i = P.coeff ↑i\nhP' : P.degree ≤ ↑(p - 1)\nH : ∀ (i j : Fin p), α i = α j\ni : ℕ\nh : i < p... | · rw [hc, degree_mul, degree_cyclotomic, Nat.totient_prime hp] at hP'
have : c.degree ≤ 0 := (WithBot.add_le_add_iff_left (x := ↑(p - 1)) (by simp)).mp (by simpa)
obtain ⟨c, rfl⟩ := natDegree_eq_zero.mp (natDegree_eq_zero_iff_degree_le_zero.mpr this)
simp [hP, hc, cyclotomic_prime] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 115,
"column": 85
} | {
"line": 121,
"column": 86
} | {
"line": 123,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\n⊢ (1 < x → 0 < eval x (cyclotomic n R)) ∧ (1 ≤ x → 0 ≤ eval x (cyclotomic n R))",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_add... | [] | by
rcases n with (_ | _ | _ | n)
· simp only [cyclotomic_zero, eval_one, zero_lt_one, implies_true, zero_le_one, and_self]
· simp
· simp only [zero_add, reduceAdd, cyclotomic_two, eval_add, eval_X, eval_one]
constructor <;> intro <;> linarith
· constructor <;> intro <;> [skip; apply le_of_lt] <;> apply cy... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 93,
"column": 19
} | {
"line": 93,
"column": 60
} | {
"line": 93,
"column": 60
} | [
{
"pp": "p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : n ≠ 0\nthis✝ : Fintype (GaloisField p n)\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly ≠ 0\nnat_degree_eq : g_poly.natDegree = p ^ n\nthis : g_poly.rootSet (GaloisField p n) = Set.univ\nkey : Fintype.card ↑Set.univ = p ^ n\n⊢ Module.fin... | [
"p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : n ≠ 0\nthis✝ : Fintype (GaloisField p n)\ng_poly : (ZMod p)[X] := X ^ p ^ n - X\nhp : 1 < p\naux : g_poly ≠ 0\nnat_degree_eq : g_poly.natDegree = p ^ n\nthis : g_poly.rootSet (GaloisField p n) = Set.univ\nkey : Fintype.card (GaloisField p n) = p ^ n\n⊢ Module.finrank... | ← Fintype.ofEquiv_card (Equiv.Set.univ _) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 224,
"column": 2
} | {
"line": 225,
"column": 88
} | {
"line": 226,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\n⊢ l ∣ 2 * n",
"ppTerm": "?m.45",
"assigned": true,
"usedConstant... | [
"n : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nkey : φ (l.lcm n) ≤ Module.finrank ℚ K\n⊢ l ∣ 2 * n"
] | have key := IsPrimitiveRoot.lcm_totient_le_finrank hζ hroot
(cyclotomic.irreducible_rat <| Nat.lcm_pos (Nat.pos_of_ne_zero hl.1) (NeZero.pos n)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 56
} | {
"line": 200,
"column": 4
} | [
{
"pp": "S T : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension T A B\nhS : S ⊆ T\nb : ↥(adjoin A {b | ∃ a ∈ S, a ≠ 0 ∧ b ^ a = 1})\n⊢ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} = ⊤",
"ppTerm": "?m.167",
"assigned": true,
"usedCon... | [
"S T : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension T A B\nhS : S ⊆ T\nb : ↥(adjoin A {b | ∃ a ∈ S, a ≠ 0 ∧ b ^ a = 1})\n⊢ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} = adjoin A {a | ∃ a_1 ∈ S, a_1 ≠ 0 ∧ ↑a ^ a_1 = 1}"
] | rw [← adjoin_adjoin_coe_preimage, preimage_setOf_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 40
} | {
"line": 219,
"column": 0
} | [
{
"pp": "case neg.refine_2\nn : ℕ\ninst✝³ : NeZero n\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhB : IsCyclotomicExtension S A B\nr : B\nhr : IsPrimitiveRoot r n\nhn : ¬n = 0\n⊢ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1} ≤ adjoin A {b | ∃ n_1 ∈ S ∪ {n}, n_1... | [] | exact Algebra.adjoin_mono (by aesop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 254,
"column": 6
} | {
"line": 255,
"column": 66
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case refine_2.inl\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nx : B\nm : ℕ\nhxpow : m ≠ 0 ∧ x ^ m = 1\ninst✝ : NeZero m\nh : ∃ s ∈ S, s ≠ 0 ∧ m ∣ s\nH : IsCyclotomicExtension (S ∪ {m}) A B\n⊢ ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1",
"ppTerm": "?refine_2.i... | [] | obtain ⟨y, ⟨hy, hy', ⟨z, rfl⟩⟩⟩ := h
exact ⟨_, ⟨hy, hy', by simp only [pow_mul, hxpow, one_pow]⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 254,
"column": 6
} | {
"line": 255,
"column": 66
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case refine_2.inl\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\nx : B\nm : ℕ\nhxpow : m ≠ 0 ∧ x ^ m = 1\ninst✝ : NeZero m\nh : ∃ s ∈ S, s ≠ 0 ∧ m ∣ s\nH : IsCyclotomicExtension (S ∪ {m}) A B\n⊢ ∃ n ∈ S, n ≠ 0 ∧ x ^ n = 1",
"ppTerm": "?refine_2.i... | [] | obtain ⟨y, ⟨hy, hy', ⟨z, rfl⟩⟩⟩ := h
exact ⟨_, ⟨hy, hy', by simp only [pow_mul, hxpow, one_pow]⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 30
} | {
"line": 179,
"column": 30
} | [
{
"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 ^ Nat.card K - X)).Splits",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algeb... | [
"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"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 186,
"column": 11
} | {
"line": 186,
"column": 35
} | {
"line": 186,
"column": 35
} | [
{
"pp": "p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Algebra (ZMod p) K\nh : Nat.card K = p ^ n\nthis✝ : Finite K\nthis : Fintype K\n⊢ IsSplittingField (ZMod p) K (X ^ Nat.card K - X)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Algebra (ZMod p) K\nh : Nat.card K = p ^ n\nthis✝ : Finite K\nthis : Fintype K\n⊢ IsSplittingField (ZMod p) K (X ^ Fintype.card K - X)"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 37
} | {
"line": 232,
"column": 38
} | [
{
"pp": "K : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\nx : K'\nthis✝¹ : Finite K\nthis✝ : Fintype K\nthis : Fintype K'\n⊢ ∏ x_1, (⇑(frobeniusAlgEquivOfAlgebraic K K'))^[↑x_1] x = x ^ ((Fintype.card K' - 1) / (Fintype.card K - 1))",
"ppTerm": "?m.... | [
"K : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Algebra K K'\ninst✝ : Finite K'\nx : K'\nthis✝¹ : Finite K\nthis✝ : Fintype K\nthis : Fintype K'\n⊢ ∏ x_1, (fun x ↦ x ^ Fintype.card K)^[↑x_1] x = x ^ ((Fintype.card K' - 1) / (Fintype.card K - 1))"
] | coe_frobeniusAlgEquivOfAlgebraic, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 482,
"column": 2
} | {
"line": 484,
"column": 8
} | {
"line": 485,
"column": 2
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\n⊢ (Algebra.norm K) (ζ ^ 2 ^ k - 1... | [
"K : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (2 ^ (k + 1)) K)\nthis : IsPrimitiveRoot (ζ ^ 2 ^ k) 2\nH : -1 - 1 = (algebraMap K L) (-2)\n⊢ (Algebr... | have H : (-1 : L) - (1 : L) = algebraMap K L (-2) := by
simp only [map_neg, map_ofNat]
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 9
} | {
"line": 60,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℤ\n⊢ cexp (2 * ↑π / 1 * (↑j / ↑N) * I) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 635,
"column": 23
} | {
"line": 640,
"column": 58
} | {
"line": 640,
"column": 59
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type w\nL : Type z\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {n} K L\n⊢ adjoin K ((cyclotomic n K).rootSet L) = ⊤",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.mem_singleton",
"Eq.mpr",... | [] | by
rw [← ((iff_adjoin_eq_top {n} K L).1 inferInstance).2]
letI := Classical.decEq L
obtain ⟨ζ : L, hζ⟩ :=
IsCyclotomicExtension.exists_isPrimitiveRoot K L (mem_singleton n) (NeZero.ne _)
exact adjoin_roots_cyclotomic_eq_adjoin_nth_roots hζ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 721,
"column": 38
} | {
"line": 721,
"column": 76
} | {
"line": 722,
"column": 4
} | [
{
"pp": "n : ℕ\ninst✝⁹ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra A K\ninst✝¹ : IsDomain A\ninst✝ : IsFractionRing A K\n⊢ FaithfulSMul A (Cyclo... | [
"n : ℕ\ninst✝⁹ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : Algebra A K\ninst✝¹ : IsDomain A\ninst✝ : IsFractionRing A K\n⊢ Function.Injective ⇑(algebraMap A... | faithfulSMul_iff_algebraMap_injective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 77,
"column": 19
} | {
"line": 77,
"column": 93
} | {
"line": 77,
"column": 93
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ₀ χ₁ : MulChar R R'\nh : (fun χ ↦ (↑χ.toMonoidHom).toFun) χ₀ = (fun χ ↦ (↑χ.toMonoidHom).toFun) χ₁\n⊢ χ₀ = χ₁",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"MonoidHom"... | [] | by cases χ₀; cases χ₁; congr; apply MonoidHom.ext (fun _ => congr_fun h _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 187,
"column": 23
} | {
"line": 187,
"column": 36
} | {
"line": 187,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\nx : Rˣ\n⊢ ↑(χ.toUnitHom x) = χ ↑x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instFunLike",
"MonoidHom",
"congrA... | [
"R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ : MulChar R R'\nx : Rˣ\n⊢ χ ↑x = χ ↑x"
] | coe_toUnitHom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 373,
"column": 6
} | {
"line": 374,
"column": 43
} | {
"line": 375,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nχ ψ : MulChar R R'\nx✝ : Rˣ\n⊢ ↑((equivToUnitHom.toFun (χ * ψ)) x✝) = ↑((equivToUnitHom.toFun χ * equivToUnitHom.toFun ψ) x✝)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CommMonoidWithZer... | [] | simp only [Equiv.toFun_as_coe, coe_equivToUnitHom, coeToFun_mul, Pi.mul_apply,
MonoidHom.mul_apply, Units.val_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 101,
"column": 48
} | {
"line": 101,
"column": 72
} | {
"line": 101,
"column": 72
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : Fintype M\ninst✝¹ : DecidableEq M\nR : Type u_2\ninst✝ : CommMonoidWithZero R\nζ : Rˣ\nhζ : ζ ∈ rootsOfUnity (Fintype.card Mˣ) R\ng : Mˣ\nhg : ∀ (x : Mˣ), x ∈ Subgroup.zpowers g\nthis : orderOf ζ ∣ Fintype.card Mˣ\n⊢ Fintype.card Mˣ = Nat.card Mˣ",
"ppT... | [
"M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : Fintype M\ninst✝¹ : DecidableEq M\nR : Type u_2\ninst✝ : CommMonoidWithZero R\nζ : Rˣ\nhζ : ζ ∈ rootsOfUnity (Fintype.card Mˣ) R\ng : Mˣ\nhg : ∀ (x : Mˣ), x ∈ Subgroup.zpowers g\nthis : orderOf ζ ∣ Fintype.card Mˣ\n⊢ Fintype.card Mˣ = Fintype.card Mˣ"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 115,
"column": 64
} | {
"line": 116,
"column": 85
} | {
"line": 116,
"column": 85
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : Fintype M\ninst✝¹ : DecidableEq M\nR : Type u_2\ninst✝ : CommMonoidWithZero R\ninst_cyc : IsCyclic Mˣ\nχ : MulChar M R\n⊢ χ.toUnitHom (Classical.choose ⋯) ∈ rootsOfUnity (Fintype.card Mˣ) R",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants":... | [] | by
simp only [toUnitHom_eq, mem_rootsOfUnity, ← map_pow, pow_card_eq_one, map_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 121,
"column": 17
} | {
"line": 123,
"column": 68
} | {
"line": 124,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : Fintype M\ninst✝¹ : DecidableEq M\nR : Type u_2\ninst✝ : CommMonoidWithZero R\ninst_cyc : IsCyclic Mˣ\nζ : ↥(rootsOfUnity (Fintype.card Mˣ) R)\n⊢ (fun χ ↦ ⟨χ.toUnitHom (Classical.choose ⋯), ⋯⟩) ((fun ζ ↦ ofRootOfUnity ⋯ ⋯) ζ) = ζ",
"ppTerm": "?m.55",
... | [] | by
ext
simp only [toUnitHom_eq, coe_equivToUnitHom, ofRootOfUnity_spec] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 260,
"column": 2
} | {
"line": 263,
"column": 71
} | {
"line": 264,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\ninst✝ : NeZero n\n⊢ conductor 1 = 1",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"DirichletCharacter.conductor",
"DirichletCharacter.FactorsThrough",
"ZMod.commRing",
"MulChar.hasOne",
... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\ninst✝ : NeZero n\n⊢ FactorsThrough 1 1"
] | suffices FactorsThrough (1 : DirichletCharacter R n) 1 by
have h : conductor (1 : DirichletCharacter R n) ≤ 1 :=
Nat.sInf_le <| (mem_conductorSet_iff _).mpr this
exact Nat.le_antisymm h (Nat.pos_of_ne_zero <| conductor_ne_zero _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 561,
"column": 6
} | {
"line": 561,
"column": 20
} | {
"line": 561,
"column": 21
} | [
{
"pp": "S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nψ : DirichletCharacter S m\nx : ZMod m\nhψ : ψ (-1) = -1\n⊢ ψ (-x) = -ψ x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing... | [
"S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nψ : DirichletCharacter S m\nx : ZMod m\nhψ : ψ (-1) = -1\n⊢ ψ (-1 * x) = -ψ x"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 20
} | {
"line": 566,
"column": 21
} | [
{
"pp": "S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nψ : DirichletCharacter S m\nx : ZMod m\nhψ : ψ (-1) = 1\n⊢ ψ (-x) = ψ x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",... | [
"S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nψ : DirichletCharacter S m\nx : ZMod m\nhψ : ψ (-1) = 1\n⊢ ψ (-1 * x) = ψ x"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 571,
"column": 17
} | {
"line": 571,
"column": 31
} | {
"line": 571,
"column": 32
} | [
{
"pp": "S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nχ : DirichletCharacter S m\nhχ : χ.Even\nx✝ : ZMod m\n⊢ χ (-x✝) = χ x✝",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nχ : DirichletCharacter S m\nhχ : χ.Even\nx✝ : ZMod m\n⊢ χ (-1 * x✝) = χ x✝"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 575,
"column": 17
} | {
"line": 575,
"column": 31
} | {
"line": 575,
"column": 32
} | [
{
"pp": "S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nχ : DirichletCharacter S m\nhχ : χ.Odd\nx✝ : ZMod m\n⊢ χ (-x✝) = -χ x✝",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"S : Type u_2\ninst✝ : CommRing S\nm : ℕ\nχ : DirichletCharacter S m\nhχ : χ.Odd\nx✝ : ZMod m\n⊢ χ (-1 * x✝) = -χ x✝"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 43,
"column": 10
} | {
"line": 43,
"column": 17
} | {
"line": 44,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nx✝ : E\np : ℝ\nhp : 1 < p\nhx : x✝ = 0\nh2p : 0 < p - 1\nx : E\n⊢ ‖x‖ ^ p = ‖x‖ ^ (1 + (p - 1))",
"ppTerm": "?m.206",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zer... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 11
} | {
"line": 55,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nx : E\np : ℝ\nhp : 1 < p\nhx : ¬x = 0\n⊢ HasStrictFDerivAt (fun x ↦ ‖x‖ ^ p) ((p * ‖x‖ ^ (p - 2)) • (innerSL ℝ) x) x ↔\n HasStrictFDerivAt (fun x ↦ ‖x‖ ^ (↑2 * (p / 2))) ((p / 2 * ‖x‖ ^ (↑2 * (p / 2 - 1)) * ↑2) • (innerSL ℝ)... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 9
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case hbc\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : F → E\nhf : Differentiable ℝ f\nx : F\np : ℝ\nhp : 1 < p\n⊢ ‖f x‖ ^ (p - 2 + 1) * ‖fderiv ℝ f x‖ = ‖f x‖ ^ (p - 1) * ‖fderiv ℝ f x‖",
"ppT... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 9
} | {
"line": 97,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nx : E\np : ℝ\nhp : 1 < p\n⊢ p * ‖x‖ ^ (p - 2 + 1) = p * ‖x‖ ^ (p - 1)",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Norm.norm",
"Mathlib.Tacti... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Data.Finset.Grade | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 71
} | {
"line": 96,
"column": 71
} | [
{
"pp": "α : Type u_1\ns t : Finset α\n⊢ (∃ a, ∃ (ha : a ∉ s), cons a s ha = t) → s ⋖ t",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Finset.cons",
"Finset",
"CovBy",
"PartialOrder.toPreorder",
"Membership.mem",
"Exists",
... | [] | by rintro ⟨a, ha, rfl⟩; exact covBy_cons _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.GaussSum | {
"line": 256,
"column": 45
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : 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'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | by
rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 66
} | {
"line": 50,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ u₁ = u₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instGeneralizedBooleanAlgeb... | [] | simpa only [disjUnion_inj_left, Subtype.ext_iff] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 66
} | {
"line": 50,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ u₁ = u₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instGeneralizedBooleanAlgeb... | [] | simpa only [disjUnion_inj_left, Subtype.ext_iff] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Interval | {
"line": 50,
"column": 10
} | {
"line": 50,
"column": 66
} | {
"line": 50,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq α\ns t : Finset α\nu₁ u₂ : ↥(t \\ s).powerset\nh : (fun u ↦ (↑u).disjUnion s ⋯) u₁ = (fun u ↦ (↑u).disjUnion s ⋯) u₂\n⊢ u₁ = u₂",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.instGeneralizedBooleanAlgeb... | [] | simpa only [disjUnion_inj_left, Subtype.ext_iff] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 161,
"column": 14
} | {
"line": 161,
"column": 21
} | {
"line": 162,
"column": 4
} | [
{
"pp": "ι : 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 : ι) → A i\... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Topology.Algebra.Module.LinearPMap | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 61
} | {
"line": 181,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : AddCommGroup F\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace F\nf : E →ₗ.[R] F\nhf : f.toFun.ker = ⊥\n⊢ IsClosed[instTopologicalSpaceProd] ↑(Submodule.map ... | [] | exact (ContinuousLinearEquiv.prodComm R E F).isClosed_image | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 422,
"column": 12
} | {
"line": 422,
"column": 47
} | {
"line": 422,
"column": 48
} | [
{
"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... | [
"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 : HasCompac... | ENNReal.mul_rpow_of_nonneg _ _ h0p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 82,
"column": 2
} | {
"line": 88,
"column": 14
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : NormedSpace 𝕜 E\nf : α → α\ng : α → E\nx : α\nh : Bornology.IsBounded (range fun x_1 ↦ g (f^[x_1] x))\n⊢ Tendsto (fun n ↦ birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) atTop (𝓝 0)",
... | [] | rcases Metric.isBounded_range_iff.1 h with ⟨C, hC⟩
have : Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
refine squeeze_zero_norm (fun n ↦ ?_) this
rw [← dist_eq_norm, dist_birkhoffAverage_apply_birkhoffAverage]
gcongr
exact hC n 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 82,
"column": 2
} | {
"line": 88,
"column": 14
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : NormedSpace 𝕜 E\nf : α → α\ng : α → E\nx : α\nh : Bornology.IsBounded (range fun x_1 ↦ g (f^[x_1] x))\n⊢ Tendsto (fun n ↦ birkhoffAverage 𝕜 f g n (f x) - birkhoffAverage 𝕜 f g n x) atTop (𝓝 0)",
... | [] | rcases Metric.isBounded_range_iff.1 h with ⟨C, hC⟩
have : Tendsto (fun n : ℕ ↦ C / n) atTop (𝓝 0) :=
tendsto_const_nhds.div_atTop tendsto_natCast_atTop_atTop
refine squeeze_zero_norm (fun n ↦ ?_) this
rw [← dist_eq_norm, dist_birkhoffAverage_apply_birkhoffAverage]
gcongr
exact hC n 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 430,
"column": 6
} | {
"line": 430,
"column": 41
} | {
"line": 430,
"column": 42
} | [
{
"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... | [
"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 : HasCompac... | ENNReal.mul_rpow_of_nonneg _ _ h0p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 52
} | {
"line": 72,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [] | simpa [H] using (hf.iterate n).dist_le_mul x 0 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 520,
"column": 4
} | {
"line": 520,
"column": 83
} | {
"line": 521,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [
"E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ℝ 1 u\nh2u :... | rw [← NNReal.coe_inj, ← inv_inj, hp', NNReal.coe_mul, h0γ, hn.coe.conjugate_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 178,
"column": 77
} | {
"line": 178,
"column": 94
} | {
"line": 179,
"column": 4
} | [
{
"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\nhS : ∀ m ∈ T.singularValues.sup... | [
"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\nhS : ∀ m ∈ T.singularValues.support, m < fi... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 42
} | {
"line": 539,
"column": 2
} | [
{
"pp": "case neg\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu :... | [
"case neg\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ℝ ... | let v : E → ℝ := fun x ↦ ‖u x‖ ^ (γ : ℝ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
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