module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Visible | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 72
} | [
{
"pp": "V : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module ℝ V\ns : Set V\ny : V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul ℝ V\nhs : IsClosed[inst✝²] s\nhy : y ∈ s\nx : V\nt : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s\nht₁ : 1 ∈ t\nht : BddBelow t\nδ : ℝ := sInf... | rintro hδ₀; simp [hδ₀] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 50
} | [
{
"pp": "case e_r\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volum... | exact intervalIntegral.integral_of_le (f_nn ω) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 75
} | [
{
"pp": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : ↥(Lp E 2 μ)\nx : α\nh_two : 2 = ↑2\n⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)",
"usedConstants": [
... | rw [← Real.rpow_natCast _ 2, ← h_two, ←
ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 56
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nr : 𝕜\nx : α\nhx : ↑↑(r • f) x = (r • ↑↑f) x\n⊢ ⟪↑↑(r • f) x, ↑↑g x⟫ = (starRingEnd 𝕜) r • ⟪↑↑f x, ↑↑g x⟫",
"... | rw [smul_eq_mul, ← inner_smul_left, hx, Pi.smul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 75,
"column": 72
} | {
"line": 83,
"column": 43
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nN : ℕ\n⊢ ∃ k, ∀ n ≤ N, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k",
"usedConstants": [
"Real.instI... | by
choose k hk using hf_temperate.isBigO
use (Finset.range (N + 1)).sup k
intro n hn
refine (hk n).trans (isBigO_of_le _ fun x ↦ ?_)
rw [Real.norm_of_nonneg (by positivity), Real.norm_of_nonneg (by positivity)]
gcongr
· simp
· exact Finset.le_sup (by simpa using hn) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 56
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\... | simp [hgv hv, show x ∉ s from fun h => hv (hsv h)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 56
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\... | simp [hgv hv, show x ∉ s from fun h => hv (hsv h)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 56
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\... | simp [hgv hv, show x ∉ s from fun h => hv (hsv h)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 361,
"column": 4
} | {
"line": 361,
"column": 33
} | [
{
"pp": "H : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤... | have := le_sup_right.trans hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 527,
"column": 12
} | {
"line": 527,
"column": 47
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : CompleteSpace E\nv : ι → E\nhv : Orthonormal 𝕜 v\nhsp : (... | ← orthogonal_orthogonal_eq_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 15
} | [
{
"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\nf : 𝓓^{n}_{K}(E, F)\n⊢ ⇑(... | rw [fderivLM] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 54
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : ... | rw [(schwartz_withSeminorms 𝕜 E F).withSeminorms_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 664,
"column": 6
} | {
"line": 664,
"column": 78
} | [
{
"pp": "case hbc\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✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedAddCommGro... | exact norm_iteratedFDeriv_clm_apply_const (f.smooth _).contDiffAt le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 49,
"column": 2
} | {
"line": 51,
"column": 86
} | [
{
"pp": "c : ℝ\n⊢ ∫ (x : ℝ) in Iic c, rexp x = rexp c",
"usedConstants": [
"Real.instIsOrderedRing",
"InnerProductSpace.toNormedSpace",
"Real.partialOrder",
"Real",
"instIsCountablyGenerated_atBot",
"Real.instArchimedean",
"integrableOn_exp_Iic",
"MeasureTheor... | refine
tendsto_nhds_unique
(intervalIntegral_tendsto_integral_Iic _ (integrableOn_exp_Iic _) tendsto_id) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 897,
"column": 26
} | {
"line": 897,
"column": 56
} | [
{
"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✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | ← le_seminorm 𝕜 (k + 1) n f x, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 167,
"column": 8
} | {
"line": 167,
"column": 45
} | [
{
"pp": "case inr\ns : ℝ\nh : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume\nhs : -1 < s\nthis : IntegrableOn (fun x ↦ x ^ s) (Ioi 1) volume\n⊢ False",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Real.in... | integrableOn_Ioi_rpow_iff zero_lt_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.ExpDecay | {
"line": 34,
"column": 2
} | {
"line": 36,
"column": 90
} | [
{
"pp": "a b : ℝ\nh : 0 < b\n⊢ IntegrableOn (fun x ↦ rexp (-b * x)) (Ioi a) volume",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Filter.Tendsto.neg",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"NonU... | have : Tendsto (fun x => -exp (-b * x) / b) atTop (𝓝 (-0 / b)) := by
refine Tendsto.div_const (Tendsto.neg ?_) _
exact tendsto_exp_atBot.comp (tendsto_id.const_mul_atTop_of_neg (neg_neg_iff_pos.2 h)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 418,
"column": 4
} | {
"line": 421,
"column": 89
} | [
{
"pp": "case h.h₁.hbc.hbc\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\nK : ℕ∞ω\nC : ℝ\nhf : ContDiff ℝ K f\nn... | · norm_cast
calc n.descFactorial i ≤ n ^ i := Nat.descFactorial_le_pow _ _
_ ≤ (n + 1) ^ i := by gcongr; lia
_ ≤ (n + 1) ^ k := by gcongr; exacts [le_add_self, Finset.mem_range_succ_iff.mp hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 90
} | [
{
"pp": "case hf\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... | apply (smulRightL ℝ (fun (_ : Fin n) ↦ W) E).continuous₂.comp_aestronglyMeasurable₂ _ hf | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1282,
"column": 88
} | {
"line": 1284,
"column": 28
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : MeasurableSpace E\ninst✝³ : OpensMeasurableSpace E\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\np :... | by
refine congrArg (eLpNorm · p μ) (funext fun x ↦ ?_)
simp [(h_one_add x).ne'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 40
} | [
{
"pp": "s : ℂ\nhs : 0 < s.re\nX : ℝ\nhX : 0 ≤ X\n⊢ 0 < (s + 1).re",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.neg_mul",
... | · simp only [add_re, one_re]; linarith | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 182,
"column": 4
} | {
"line": 184,
"column": 8
} | [
{
"pp": "case right\ns : ℂ\nhs : 0 < s.re\nY : ℝ\nhY : 0 ≤ Y\nthis : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))\n⊢ ∀ᵐ (x : ℝ), x ∈ Ioc 0 Y → rexp (-x) * x ^ (s.re - 1) = ‖↑(rexp (-x))‖ * ‖↑x ^ (s - 1)‖",
"usedConstants": [
"MeasureTheory.ae",
"Norm.no... | filter_upwards with x hx
rw [Complex.norm_of_nonneg (exp_pos _).le, norm_cpow_eq_rpow_re_of_pos hx.1]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 182,
"column": 4
} | {
"line": 184,
"column": 8
} | [
{
"pp": "case right\ns : ℂ\nhs : 0 < s.re\nY : ℝ\nhY : 0 ≤ Y\nthis : (fun x ↦ ↑(rexp (-x)) * (s * ↑x ^ (s - 1))) = fun x ↦ s * (↑(rexp (-x)) * ↑x ^ (s - 1))\n⊢ ∀ᵐ (x : ℝ), x ∈ Ioc 0 Y → rexp (-x) * x ^ (s.re - 1) = ‖↑(rexp (-x))‖ * ‖↑x ^ (s - 1)‖",
"usedConstants": [
"MeasureTheory.ae",
"Norm.no... | filter_upwards with x hx
rw [Complex.norm_of_nonneg (exp_pos _).le, norm_cpow_eq_rpow_re_of_pos hx.1]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 25
} | [
{
"pp": "s : ℝ\nhs : s ≠ 0\n⊢ (Complex.Gamma ↑(s + 1)).re = s * (Complex.Gamma ↑s).re",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"Complex.ofReal_add",
"Complex.Gamma",
"id",
"Complex.ofReal",
"Real.instAdd",
"Complex.re",
... | Complex.ofReal_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 423,
"column": 13
} | {
"line": 423,
"column": 32
} | [
{
"pp": "n : ℕ\n⊢ (Complex.Gamma ↑(↑n + 1)).re = ↑n !",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Complex.ofReal_add",
"Complex.Gamma",
"id",
"Complex.ofReal",
"Nat.cast",
"Real.instAdd",
"Complex.re",
"Real.instOne",
"instHAdd"... | Complex.ofReal_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 23
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ (Function.support fun x ↦ rexp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"Set.Ioi",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Real.instSub",
"HSub.hSub",
"id",
"H... | rw [inter_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 37
} | [
{
"pp": "case e_f.h\nb : ℂ\nhb : 0 < b.re\np : ℝ × ℝ\n⊢ cexp (-b * ↑p.1 ^ 2) * cexp (-b * ↑p.2 ^ 2) = cexp (-b * (↑p.1 ^ 2 + ↑p.2 ^ 2))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Real",
"HMul.hMul",
"Complex.exp_add",
"congrArg",
"Complex.instMul",
... | rw [← Complex.exp_add, mul_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 82
} | [
{
"pp": "case h.e'_3\n⊢ ↑π ^ (1 / 2) = ↑√π",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",
"Complex.ofReal_cpow",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.inst... | rw [sqrt_eq_rpow, ofReal_cpow pi_pos.le, ofReal_div, ofReal_ofNat, ofReal_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 82
} | [
{
"pp": "case h.e'_3\n⊢ ↑π ^ (1 / 2) = ↑√π",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",
"Complex.ofReal_cpow",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.inst... | rw [sqrt_eq_rpow, ofReal_cpow pi_pos.le, ofReal_div, ofReal_ofNat, ofReal_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 349,
"column": 4
} | {
"line": 349,
"column": 82
} | [
{
"pp": "case h.e'_3\n⊢ ↑π ^ (1 / 2) = ↑√π",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"Real.instZero",
"congrArg",
"Real.instDivInvMonoid",
"Complex.ofReal_cpow",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.inst... | rw [sqrt_eq_rpow, ofReal_cpow pi_pos.le, ofReal_div, ofReal_ofNat, ofReal_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 863,
"column": 2
} | {
"line": 871,
"column": 41
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nN : ℕ∞\nn : ℕ\nhf : ContDiff ℝ (↑N) f\nh'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume\nhn : ↑n ≤ N\n⊢ 𝓕 (iteratedDeriv n f) = fun x ↦ (2 * ↑π * I * ↑x) ^ n • 𝓕 f x",
"usedConstants": [
"LinearI... | ext x : 1
have A : ∀ (n : ℕ), n ≤ N → Integrable (iteratedFDeriv ℝ n f) := by
intro n hn
rw [iteratedFDeriv_eq_equiv_comp]
exact (LinearIsometryEquiv.integrable_comp_iff _).2 (h'f n hn)
change 𝓕 (fun x ↦ iteratedDeriv n f x) x = _
simp_rw [iteratedDeriv, ← fourier_continuousMultilinearMap_apply (A n ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 863,
"column": 2
} | {
"line": 871,
"column": 41
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\nN : ℕ∞\nn : ℕ\nhf : ContDiff ℝ (↑N) f\nh'f : ∀ (n : ℕ), ↑n ≤ N → Integrable (iteratedDeriv n f) volume\nhn : ↑n ≤ N\n⊢ 𝓕 (iteratedDeriv n f) = fun x ↦ (2 * ↑π * I * ↑x) ^ n • 𝓕 f x",
"usedConstants": [
"LinearI... | ext x : 1
have A : ∀ (n : ℕ), n ≤ N → Integrable (iteratedFDeriv ℝ n f) := by
intro n hn
rw [iteratedFDeriv_eq_equiv_comp]
exact (LinearIsometryEquiv.integrable_comp_iff _).2 (h'f n hn)
change 𝓕 (fun x ↦ iteratedDeriv n f x) x = _
simp_rw [iteratedDeriv, ← fourier_continuousMultilinearMap_apply (A n ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 346,
"column": 82
} | {
"line": 355,
"column": 8
} | [
{
"pp": "b : ℂ\nV : Type u_1\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nhb : 0 < b.re\nx w : V\n⊢ 𝓕 (fun v ↦ cexp (-b * ↑‖v‖ ^ 2 + 2 * ↑π * I * ↑⟪x, v⟫)) w =\n (↑π / b) ^ (↑(Module.finrank ℝ V) / 2) * cexp... | by
simp only [neg_mul, fourier_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,
smul_eq_mul, ← Complex.exp_add, real_inner_comm w]
convert! integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v
· congr 1
simp [inner_sub_left]
ring
· have : b ≠ 0 := by contrapose! hb; rw [hb, zero... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 52,
"column": 2
} | {
"line": 84,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\nin... | refine mkCLM ((𝓕 : (V → E) → (V → E)) ·) ?_ ?_ ?_ ?_
· intro f g
simp [fourier_eq, integral_add ((fourierIntegral_convergent_iff _).mpr f.integrable)
((fourierIntegral_convergent_iff _).mpr g.integrable)]
· simp [fourier_eq, smul_comm, integral_smul]
· exact fun f ↦ contDiff_fourier (fun n _ ↦ integrab... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 52,
"column": 2
} | {
"line": 84,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\nin... | refine mkCLM ((𝓕 : (V → E) → (V → E)) ·) ?_ ?_ ?_ ?_
· intro f g
simp [fourier_eq, integral_add ((fourierIntegral_convergent_iff _).mpr f.integrable)
((fourierIntegral_convergent_iff _).mpr g.integrable)]
· simp [fourier_eq, smul_comm, integral_smul]
· exact fun f ↦ contDiff_fourier (fun n _ ↦ integrab... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 264,
"column": 2
} | {
"line": 264,
"column": 19
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : InnerProductSpace ℂ F\ninst✝ : CompleteSpace F\ns : ℝ\nhs : ↑(Module.finrank ℝ E) < 2 * s... | use this.toLp • u | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 42
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nx : UnitAddTorus d\n⊢ HasSum (fun i ↦ mFourierCoeff (⇑f) i • (mFourier i) x) (f x)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simpa only [map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
(hasSum_mFourier_series_of_summable h) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 42
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nx : UnitAddTorus d\n⊢ HasSum (fun i ↦ mFourierCoeff (⇑f) i • (mFourier i) x) (f x)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simpa only [map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
(hasSum_mFourier_series_of_summable h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 42
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : C(UnitAddTorus d, ℂ)\nh : Summable (mFourierCoeff ⇑f)\nx : UnitAddTorus d\n⊢ HasSum (fun i ↦ mFourierCoeff (⇑f) i • (mFourier i) x) (f x)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | simpa only [map_smul] using (ContinuousMap.evalCLM ℂ x).hasSum
(hasSum_mFourier_series_of_summable h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.Convolution | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 86
} | [
{
"pp": "case h.e'_6.h\nE : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : InnerProductSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : BorelSpace E\ninst✝⁸ : NormedAddCommGroup F₁\ninst✝⁷ : NormedSpace ℂ F₁\ninst✝⁶ : NormedAddCo... | rw [← fourier_convolution_apply B f g, fourier_convolution, pairing_apply_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 77,
"column": 2
} | {
"line": 79,
"column": 47
} | [
{
"pp": "G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Finite G\n⊢ LinearIndependent R DFunLike.coe",
"usedConstants": [
"Iff.mpr",
"Pi.Function.module",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"S... | cases nonempty_fintype G
exact linearIndependent_of_ne_zero_of_wInner_cWeight_eq_zero coe_ne_zero
fun ψ₁ ψ₂ ↦ wInner_cWeight_eq_zero_iff_ne.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 77,
"column": 2
} | {
"line": 79,
"column": 47
} | [
{
"pp": "G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\ninst✝ : Finite G\n⊢ LinearIndependent R DFunLike.coe",
"usedConstants": [
"Iff.mpr",
"Pi.Function.module",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"S... | cases nonempty_fintype G
exact linearIndependent_of_ne_zero_of_wInner_cWeight_eq_zero coe_ne_zero
fun ψ₁ ψ₂ ↦ wInner_cWeight_eq_zero_iff_ne.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 178,
"column": 2
} | {
"line": 179,
"column": 84
} | [
{
"pp": "G : Type u_1\ninst✝¹ : CommGroup G\ninst✝ : Finite G\nι : Type\ninst : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (Additive G ≃+ ⨁ (i : ι), ZMod (n i))\n⊢ ∃ ι x n, (∀ (i : ι), 1 < n i) ∧ Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))",
"usedConstants": [
"MulEquiv.t... | exact ⟨ι, inst, n, h₁, ⟨MulEquiv.toAdditive.symm <| h₂.some.trans <|
(DirectSum.addEquivProd _).trans (MulEquiv.piMultiplicative _).toAdditiveRight⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 25
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nk : ℕ\nζ : M\nhk : 0 < k\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl' : 0 < l\nhl : l < k\n⊢ ζ ^ l ≠ 1",
"usedConstants": [
"IsPrimitiveRoot.eq_orderOf",
"congrArg",
"Eq.mp",
"orderOf",
"CommMonoid.toMonoid",
"Nat",
"LT.lt",
... | rw [h.eq_orderOf] at hl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 33
} | [
{
"pp": "G : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\nl : ℕ\nhl : ζ⁻¹ ^ l = 1\n⊢ k ∣ l",
"usedConstants": [
"DivisionCommMonoid.toCommMonoid",
"IsPrimitiveRoot.dvd_of_pow_eq_one"
]
}
] | apply h.dvd_of_pow_eq_one l | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 65
} | [
{
"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\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"usedConstants": [
"Int.lt_of_not_ge",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"LT.lt.le",
... | have : 0 ≤ -i := (Int.neg_pos_of_neg <| Int.lt_of_not_ge h0).le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 243,
"column": 2
} | {
"line": 245,
"column": 26
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ map (Int.castRingHom R) (cyclotomic n ℤ) = cyclotomic n R",
"usedConstants": [
"Polynomial.map_one",
"of_eq_false",
"Polynomial.instOne",
"WithBot",
"dite_congr",
"instDecidableTrue",
"eq_false",
"Complex.commRin... | by_cases hzero : n = 0
· simp only [hzero, cyclotomic, dif_pos, Polynomial.map_one]
simp [cyclotomic, hzero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 243,
"column": 2
} | {
"line": 245,
"column": 26
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ map (Int.castRingHom R) (cyclotomic n ℤ) = cyclotomic n R",
"usedConstants": [
"Polynomial.map_one",
"of_eq_false",
"Polynomial.instOne",
"WithBot",
"dite_congr",
"instDecidableTrue",
"eq_false",
"Complex.commRin... | by_cases hzero : n = 0
· simp only [hzero, cyclotomic, dif_pos, Polynomial.map_one]
simp [cyclotomic, hzero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 70
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRi... | simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 70
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRi... | simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 70
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRi... | simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 497,
"column": 4
} | {
"line": 505,
"column": 54
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\na✝ : Nontrivial R\nhP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1\n⊢ P = cyclotomic n R",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"WithBot.instPreorder",
... | have prod_monic : (∏ i ∈ Nat.properDivisors n, cyclotomic i R).Monic := by
apply monic_prod_of_monic
intro i _
exact cyclotomic.monic i R
rw [@cyclotomic_eq_X_pow_sub_one_div R _ _ hpos, (div_modByMonic_unique P 0 prod_monic _).1]
refine ⟨by rwa [zero_add, mul_comm], ?_⟩
rw [degree_zero, b... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 497,
"column": 4
} | {
"line": 505,
"column": 54
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nhpos : 0 < n\nP : R[X]\na✝ : Nontrivial R\nhP : P * ∏ i ∈ n.properDivisors, cyclotomic i R = X ^ n - 1\n⊢ P = cyclotomic n R",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"WithBot.instPreorder",
... | have prod_monic : (∏ i ∈ Nat.properDivisors n, cyclotomic i R).Monic := by
apply monic_prod_of_monic
intro i _
exact cyclotomic.monic i R
rw [@cyclotomic_eq_X_pow_sub_one_div R _ _ hpos, (div_modByMonic_unique P 0 prod_monic _).1]
refine ⟨by rwa [zero_add, mul_comm], ?_⟩
rw [degree_zero, b... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 807,
"column": 41
} | {
"line": 807,
"column": 84
} | [
{
"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✝⁷ : CommMonoid M\ninst✝⁶ : CommMonoid N\ninst✝⁵ : DivisionCommMonoid G\nk l : ℕ\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\nμ : S\nn : ℕ\nhμ : IsPrimitiveRoot μ n\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\ninst✝ : ... | simpa only [rootsOfUnity.coe_pow] using hxy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 807,
"column": 41
} | {
"line": 807,
"column": 84
} | [
{
"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✝⁷ : CommMonoid M\ninst✝⁶ : CommMonoid N\ninst✝⁵ : DivisionCommMonoid G\nk l : ℕ\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\nμ : S\nn : ℕ\nhμ : IsPrimitiveRoot μ n\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\ninst✝ : ... | simpa only [rootsOfUnity.coe_pow] using hxy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 807,
"column": 41
} | {
"line": 807,
"column": 84
} | [
{
"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✝⁷ : CommMonoid M\ninst✝⁶ : CommMonoid N\ninst✝⁵ : DivisionCommMonoid G\nk l : ℕ\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\nμ : S\nn : ℕ\nhμ : IsPrimitiveRoot μ n\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\ninst✝ : ... | simpa only [rootsOfUnity.coe_pow] using hxy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℚ\n⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j",
"usedConstants": [
"Nat.Prime",
"Fact.mk"
]
}
] | haveI : Fact p.Prime := ⟨hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 21
} | [
{
"pp": "S : Type u_2\ninst✝⁶ : CommRing S\nR : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\nhmin : minpoly S B.gen = Polynomial.map (algebraMap R S) (... | by_cases hQ : Q = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 49
} | [
{
"pp": "S : Type u_2\ninst✝⁶ : CommRing S\nR : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\nx y : A\nhx : ∀ (i : Fin B.dim), IsIntegral R ((B.basis.re... | simp only [Algebra.smul_mul_assoc, Algebra.mul_smul_comm, map_smulₛₗ, RingHom.id_apply,
Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 31
} | [
{
"pp": "case inr.refine_1.refine_2\np n : ℕ\nhp : Nat.Prime p\nhdiv : ¬p ∣ n\nR : Type u_1\ninst✝ : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p))) (n * p)\n⊢ minpoly ℚ (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p))) ... | refine minpoly.dvd ℚ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 31
} | [
{
"pp": "case inr.refine_1.refine_3\np n : ℕ\nhp : Nat.Prime p\nhdiv : ¬p ∣ n\nR : Type u_1\ninst✝ : CommRing R\nhnpos : n > 0\nthis : NeZero n\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n\n⊢ minpoly ℚ (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) ∣\n (expand ℚ p) (minpoly ℚ (Comp... | refine minpoly.dvd ℚ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 87,
"column": 2
} | {
"line": 102,
"column": 13
} | [
{
"pp": "case a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\n⊢ IntermediateField.fixedField L.fixingSubgroup ≤ L",
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Eq.mpr",
"IsGalois.to_no... | · intro x hx
rw [IntermediateField.mem_fixedField_iff] at hx
have mem : x ∈ (adjoin L {x}).1 := subset_adjoin _ _ rfl
have : IntermediateField.fixedField (⊤ : Subgroup ((adjoin L {x}) ≃ₐ[L] (adjoin L {x}))) = ⊥ :=
(IsGalois.tfae.out 0 1).mp (by infer_instance)
have : ⟨x, mem⟩ ∈ (⊥ : IntermediateFi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 37
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : CommRing K\np q : ℕ\ninst✝ : CharP K p\nh : p ∣ q\n⊢ 1 * (X ^ q - X) + (X ^ q - X - 1) * derivative (X ^ q - X) = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"outParam",
"congrArg",
"CommSemiring.toSemirin... | ← CharP.cast_eq_zero_iff K[X] p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 50
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\n⊢ ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖",
"usedConstants": [
"Iff.mpr",
"Norm.norm"... | refine ⟨ζ, (mem_primitiveRoots hn).mpr hζ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 29
} | [
{
"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\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField p n)) = p ^ n\nnat_degree_eq : g_poly.natDegree = p ^ n\n⊢ g_poly.rootSet (GaloisField p n) = Se... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 24
} | [
{
"pp": "n : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : CommRing L\ninst✝² : IsDomain L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {n} K L\nζ : L\nhζ : IsPrimitiveRoot ζ n\n⊢ (IsPrimitiveRoot.powerBasis K hζ).gen ∈ K[ζ - 1]",
"usedConstants": [
"Subalgebra.instS... | rw [powerBasis_gen, adjoin_singleton_eq_range_aeval, AlgHom.mem_range]
exact ⟨X + 1, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 24
} | [
{
"pp": "n : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : CommRing L\ninst✝² : IsDomain L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {n} K L\nζ : L\nhζ : IsPrimitiveRoot ζ n\n⊢ (IsPrimitiveRoot.powerBasis K hζ).gen ∈ K[ζ - 1]",
"usedConstants": [
"Subalgebra.instS... | rw [powerBasis_gen, adjoin_singleton_eq_range_aeval, AlgHom.mem_range]
exact ⟨X + 1, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 50
} | [
{
"pp": "n : 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\n⊢ ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1",
"usedConstants": [
"Iff.mpr",
"Norm.norm"... | refine ⟨ζ, (mem_primitiveRoots hn).mpr hζ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 150,
"column": 68
} | {
"line": 150,
"column": 93
} | [
{
"pp": "p✝ : ℕ\ninst✝ : Fact (Nat.Prime p✝)\nn✝ p : ℕ\nh_prime : Fact (Nat.Prime p)\nn : ℕ\nh : X ^ p ^ 1 = X ^ Fintype.card (ZMod p)\n⊢ IsSplittingField (ZMod p) (ZMod p) (X ^ p ^ 1 - X)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroCl... | by rw [h]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 24
} | [
{
"pp": "case neg\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⊢ IsCyclotomicExtension (S ∪ {n}) A B",
"usedConstants": [
"Eq.mpr",
"Lat... | rw [iff_adjoin_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 194,
"column": 2
} | {
"line": 195,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n = 1\n⊢ χ = 1",
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"DirichletCharacter.level_one",
"instOfNatNat",
"ZMod",
"Nat",
"DirichletCharacter",
"Eq.ndrec",... | subst hn
exact level_one _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 194,
"column": 2
} | {
"line": 195,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nhn : n = 1\n⊢ χ = 1",
"usedConstants": [
"ZMod.commRing",
"MulChar.hasOne",
"DirichletCharacter.level_one",
"instOfNatNat",
"ZMod",
"Nat",
"DirichletCharacter",
"Eq.ndrec",... | subst hn
exact level_one _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\n⊢ conductor 1 ∣ n",
"usedConstants": [
"DirichletCharacter.conductor",
"Eq.mpr",
"Dvd.dvd",
"DirichletCharacter.isPrimitive_def",
"ZMod.commRing",
"MulChar.hasOne",
"congrArg",
"id",
"instOfNatN... | rw [(isPrimitive_def _).mp isPrimitive_one_level_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 34
} | [
{
"pp": "case h\nR : Type u_1\nR' : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing R'\ninst✝ : StarRing R'\nχ χ' : MulChar R R'\na✝ : Rˣ\n⊢ (χ * χ').starComp ↑a✝ = (χ'.starComp * χ.starComp) ↑a✝",
"usedConstants": [
"Units.val",
"RingHom.instRingHomClass",
"HMul.hMul",
"CommMonoid.... | simp only [starComp_apply, starRingEnd, coeToFun_mul, Pi.mul_apply, map_mul, RingHom.coe_coe,
starRingAut_apply, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Grade | {
"line": 151,
"column": 27
} | {
"line": 151,
"column": 45
} | [
{
"pp": "α : Type u_1\ns : Finset α\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ IsAtom sᶜ ↔ ∃ a, s = {a}ᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Finset",
"PartialOrder.toPreorder",
"Exists",
"BooleanAlgebra.toCompl",
"SemilatticeInf.toPar... | Finset.isAtom_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 22
} | [
{
"pp": "case neg\nn' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nn : ℤ\nh : ¬n = 0\n⊢ 0 ^ n = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"Matrix",
"DivInvMonoid.toZPow",
"id",
"Int",
... | · rw [zero_zpow _ h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 183,
"column": 14
} | {
"line": 183,
"column": 25
} | [
{
"pp": "case bc.hq\nι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i... | · exact hk' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 200,
"column": 14
} | {
"line": 200,
"column": 25
} | [
{
"pp": "case hz\nι : Type u_1\nA : ι → Type u_2\ninst✝² : (i : ι) → MeasurableSpace (A i)\nμ : (i : ι) → Measure (A i)\ninst✝¹ : DecidableEq ι\np : ℝ\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nhp₀ : 0 ≤ p\ns : Finset ι\nhp : ↑(#s) * p ≤ 1\ni : ι\nhi : i ∉ s\nf : ((i : ι) → A i) → ℝ≥0∞\nhf : Measurable f\nx : (i : ... | · exact hk' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 64,
"column": 13
} | {
"line": 67,
"column": 76
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : StarOrderedRing R\ninst✝ : DecidableEq n\nd : n → R\nh : 0 ≤ d\nx : n →₀ R\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * diagonal d i j * xj",
"usedConstants": [
"Finsupp.instFunLike",
... | by
-- TODO: positivity
refine Finsupp.sum_nonneg fun i _ ↦ Finsupp.sum_nonneg fun j _ ↦ ?_
simp +contextual [diagonal, apply_ite, star_left_conjugate_nonneg (h _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 397,
"column": 13
} | {
"line": 397,
"column": 15
} | [
{
"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... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 50
} | [
{
"pp": "case h.e'_7.h.e'_3\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMaxOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\n⊢ ⨆ x, T.rayl... | have hx₀'' : x₀ ∈ sphere (0 : E) ‖x₀‖ := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 50
} | [
{
"pp": "case h.e'_7.h.e'_3\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsSelfAdjoint T\nx₀ : E\nhx₀ : x₀ ≠ 0\nhextr : IsMinOn T.reApplyInnerSelf (sphere 0 ‖x₀‖) x₀\nhx₀' : 0 < ‖x₀‖\n⊢ ⨅ x, T.rayl... | have hx₀'' : x₀ ∈ sphere (0 : E) ‖x₀‖ := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 204,
"column": 8
} | {
"line": 204,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := ⋯\nK : NNReal\nhK : AntilipschitzWith K ⇑S\nh... | exact Submodule.sub_mem _ (hf_mem' hmn.le) (hf_mem' le_rfl) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 204,
"column": 8
} | {
"line": 204,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := ⋯\nK : NNReal\nhK : AntilipschitzWith K ⇑S\nh... | exact Submodule.sub_mem _ (hf_mem' hmn.le) (hf_mem' le_rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 204,
"column": 8
} | {
"line": 204,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := ⋯\nK : NNReal\nhK : AntilipschitzWith K ⇑S\nh... | exact Submodule.sub_mem _ (hf_mem' hmn.le) (hf_mem' le_rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 687,
"column": 8
} | {
"line": 687,
"column": 58
} | [
{
"pp": "case h.e'_4.h.e'_6\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : Finite... | · simp [ENNReal.coe_toNNReal hs.measure_lt_top.ne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 421,
"column": 11
} | {
"line": 421,
"column": 22
} | [
{
"pp": "case h.mpr\nm : Type um\nn : Type un\nR : Type uR\ninst✝⁴ : Fintype n\ninst✝³ : Fintype m\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nA : Matrix m n R\nx : n → R\nh : A *ᵥ x = 0\n⊢ Aᵀ *ᵥ 0 = 0",
"usedConstants": [
"Eq.mpr",
"Pi.addCommMonoid",
"CommRi... | mulVec_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 97
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nμ : 𝕜\nhμ : ¬HasEigenvalue T μ\n⊢ {i | ↑(hT.unsortedEigenvalues hn i) = μ}.card =... | rw [Module.End.hasEigenvalue_iff.not_left.mp hμ, finrank_bot, Finset.card_filter_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\n⊢ HasEigenvector T (↑(hT.eigenvalues hn i)) ((hT.eigenvectorBasis hn) i)",
"u... | rw [eigenvalues_def, eigenvectorBasis_def, OrthonormalBasis.reindex_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 28
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\na✝ b✝ : Fin n\nh : a✝ ≤ b✝\n⊢ Fin.revPerm b✝ ≤ Fin.revPerm a✝",
"usedCons... | exact Fin.rev_le_rev.mpr h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 344,
"column": 64
} | {
"line": 344,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nw : EuclideanSpace 𝕜 (Fin n)\n⊢ ∑ x, w.ofLp x • T ((hT.eigenvectorBasis h... | apply_eigenvectorBasis | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Matrix.Order | {
"line": 144,
"column": 9
} | {
"line": 144,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : n → 𝕜\nB : Matrix n n 𝕜\nhA : (star B * B).PosSemidef\nh : B *ᵥ x = 0\n⊢ Bᴴ *ᵥ 0 = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"NonUnitalNonAssocSemiri... | mulVec_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Matrix.Order | {
"line": 188,
"column": 42
} | {
"line": 188,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → 𝕜\ny : Matrix n n 𝕜\nhx : (star y * y).PosSemidef\nh : IsUnit (star y * y)\nhv : y *ᵥ v = 0\n⊢ yᴴ *ᵥ 0 = 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing"... | mulVec_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 153,
"column": 4
} | {
"line": 156,
"column": 9
} | [
{
"pp": "case h1\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nv : n → E\nf : E →L[𝕜] F\nc : n →₀ 𝕜\n⊢ ∑ x ∈ c.support, ∑ y ∈ c.support, star (c x) * g... | rw [Finset.sum_comm]
simp [← inner_self_eq_norm_sq_to_K, inner_sum, sum_inner, inner_smul_left, inner_smul_right,
Finset.mul_sum, Finset.smul_sum, RCLike.real_smul_eq_coe_mul]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 153,
"column": 4
} | {
"line": 156,
"column": 9
} | [
{
"pp": "case h1\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nv : n → E\nf : E →L[𝕜] F\nc : n →₀ 𝕜\n⊢ ∑ x ∈ c.support, ∑ y ∈ c.support, star (c x) * g... | rw [Finset.sum_comm]
simp [← inner_self_eq_norm_sq_to_K, inner_sum, sum_inner, inner_smul_left, inner_smul_right,
Finset.mul_sum, Finset.smul_sum, RCLike.real_smul_eq_coe_mul]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 153,
"column": 4
} | {
"line": 156,
"column": 9
} | [
{
"pp": "case h2\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nv : n → E\nf : E →L[𝕜] F\nc : n →₀ 𝕜\n⊢ ‖f‖ ^ 2 • ↑‖∑ i ∈ c.support, c i • v i‖ ^ 2 =\n ... | rw [Finset.sum_comm]
simp [← inner_self_eq_norm_sq_to_K, inner_sum, sum_inner, inner_smul_left, inner_smul_right,
Finset.mul_sum, Finset.smul_sum, RCLike.real_smul_eq_coe_mul]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 153,
"column": 4
} | {
"line": 156,
"column": 9
} | [
{
"pp": "case h2\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nv : n → E\nf : E →L[𝕜] F\nc : n →₀ 𝕜\n⊢ ‖f‖ ^ 2 • ↑‖∑ i ∈ c.support, c i • v i‖ ^ 2 =\n ... | rw [Finset.sum_comm]
simp [← inner_self_eq_norm_sq_to_K, inner_sum, sum_inner, inner_smul_left, inner_smul_right,
Finset.mul_sum, Finset.smul_sum, RCLike.real_smul_eq_coe_mul]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 34
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simp [← sq, Real.sqrt_sq_eq_abs] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Trace | {
"line": 33,
"column": 58
} | {
"line": 33,
"column": 83
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype ι\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nb : OrthonormalBasis ι 𝕜 E\ni : ι\n⊢ (b.toBasis.repr (T (b i))) i = ⟪b i, T (b i)⟫_𝕜",
"usedConstants": [
"LinearIsometry... | b.coe_toBasis_repr_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nx✝ : T.IsSymmetric\nh : ∀ (x : F), 0 ≤ re ⟪T (f.symm x), f.symm x⟫\nx : E\n⊢ 0 ≤... | simpa using h (f x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
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