module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Continuous | {
"line": 84,
"column": 82
} | {
"line": 86,
"column": 63
} | {
"line": 88,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nr r' : ℝ\nhf : ConcaveOn ℝ (ball x₀ r) f\nhr : r' < r\nhf' : IsBounded (f '' ball x₀ r)\n⊢ ∃ K, LipschitzOnWith K f (ball x₀ r')",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AddGroup... | [] | by
replace hf' : IsBounded ((-f) '' ball x₀ r) := by convert! hf'.neg; ext; simp [neg_eq_iff_eq_neg]
simpa using hf.neg.exists_lipschitzOnWith_of_isBounded hr hf' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Continuous | {
"line": 132,
"column": 18
} | {
"line": 132,
"column": 45
} | {
"line": 132,
"column": 45
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhC' : C.Nonempty\nhf : ConvexOn ℝ C f\ntfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C\ntfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, Cont... | [] | simpa using! hC.mem_nhds hx | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Convex.Continuous | {
"line": 132,
"column": 18
} | {
"line": 132,
"column": 45
} | {
"line": 132,
"column": 45
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhC' : C.Nonempty\nhf : ConvexOn ℝ C f\ntfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C\ntfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, Cont... | [] | simpa using! hC.mem_nhds hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Continuous | {
"line": 132,
"column": 18
} | {
"line": 132,
"column": 45
} | {
"line": 132,
"column": 45
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhC' : C.Nonempty\nhf : ConvexOn ℝ C f\ntfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C\ntfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, Cont... | [] | simpa using! hC.mem_nhds hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.KreinMilman | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 45
} | {
"line": 69,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : T2Space E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhscomp : IsCompact s\nhsnemp : s.Nonempty\nS : Set (Set E) := {t | t.Nonempty ∧ IsClosed... | [
"E : Type u_1\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : T2Space E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhscomp : IsCompact s\nhsnemp : s.Nonempty\nS : Set (Set E) := {t | t.Nonempty ∧ IsClosed t ∧ IsExtre... | obtain ⟨⟨x, hxt⟩, htclos, hst⟩ := ht.prop | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Join | {
"line": 162,
"column": 8
} | {
"line": 162,
"column": 11
} | {
"line": 162,
"column": 12
} | [
{
"pp": "case refine_4\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nx₁ : E\nhx₁ : x₁ ∈ s\ny₁ : E\nhy₁ : y₁ ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab... | [
"case refine_4\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nx₁ : E\nhx₁ : x₁ ∈ s\ny₁ : E\nhy₁ : y₁ ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.KreinMilman | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 23
} | {
"line": 123,
"column": 23
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : T2Space E\ninst✝⁶ : IsTopologicalAddGroup E\ninst✝⁵ : ContinuousSMul ℝ E\ninst✝⁴ : LocallyConvexSpace ℝ E\ns : Set E\ninst✝³ : AddCommGroup F\ninst✝² : Module ℝ F\ninst✝¹ : TopologicalSpace ... | [
"E : Type u_1\nF : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : T2Space E\ninst✝⁶ : IsTopologicalAddGroup E\ninst✝⁵ : ContinuousSMul ℝ E\ninst✝⁴ : LocallyConvexSpace ℝ E\ns : Set E\ninst✝³ : AddCommGroup F\ninst✝² : Module ℝ F\ninst✝¹ : TopologicalSpace F\ninst✝ : T... | mem_extremePoints | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 288,
"column": 12
} | {
"line": 288,
"column": 58
} | {
"line": 290,
"column": 0
} | [
{
"pp": "R : Type u_1\nI : Type u_6\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\ninst✝ : IsStrictOrderedRing R\nw w' : StdSimplex R I\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\n⊢ (convexCombPair s t hs ht hst w w').weights = s • w.weights + t • w'.weights",
"ppTerm": "?m.57",
"assigned": true,... | [] | simp [convexCombPair, sum_add_index, add_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 288,
"column": 12
} | {
"line": 288,
"column": 58
} | {
"line": 290,
"column": 0
} | [
{
"pp": "R : Type u_1\nI : Type u_6\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\ninst✝ : IsStrictOrderedRing R\nw w' : StdSimplex R I\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\n⊢ (convexCombPair s t hs ht hst w w').weights = s • w.weights + t • w'.weights",
"ppTerm": "?m.57",
"assigned": true,... | [] | simp [convexCombPair, sum_add_index, add_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 288,
"column": 12
} | {
"line": 288,
"column": 58
} | {
"line": 290,
"column": 0
} | [
{
"pp": "R : Type u_1\nI : Type u_6\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\ninst✝ : IsStrictOrderedRing R\nw w' : StdSimplex R I\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nhst : s + t = 1\n⊢ (convexCombPair s t hs ht hst w w').weights = s • w.weights + t • w'.weights",
"ppTerm": "?m.57",
"assigned": true,... | [] | simp [convexCombPair, sum_add_index, add_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Radon | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 44
} | [
{
"pp": "case hn\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nn k : ℕ\nh_card : finrank 𝕜 E + 1 ≤ k\nhk :\n ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},\n (∀ i ∈ ... | [
"case hn\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nn k : ℕ\nh_card : finrank 𝕜 E + 1 ≤ k\nhk :\n ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},\n (∀ i ∈ s, Convex 𝕜... | simp only [coe_mem, card_erase_of_mem] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.SimplicialComplex.AffineIndependentUnion | {
"line": 54,
"column": 19
} | {
"line": 57,
"column": 25
} | {
"line": 59,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\nG : SimpleGraph ι\n⊢ ∀ (v : ι), {v} ∈ {s | ∃ v, s = {v}} ∪ Sym2.toFinset '' G.edgeSet",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"_private.Mathlib.Analysis.Convex.SimplicialComplex.A... | [] | by
simp only [Set.mem_union, Set.mem_setOf_eq, Set.mem_image]
intro v
exact Or.inl ⟨v, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.SpecificFunctions.Pow | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 48
} | {
"line": 58,
"column": 0
} | [
{
"pp": "case inr.inr\np : ℝ\nhp₀✝ : 0 ≤ p\nhp₁✝ : p ≤ 1\nhp₀ : 0 < p\nhp₁ : p < 1\n⊢ ConcaveOn ℝ≥0 univ fun x ↦ x ^ p",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real",
"Semiring.toModule",
"Set.univ",
"NNReal",
"NNReal.instPartialOrder",
"HPow.... | [] | exact (strictConcaveOn_rpow hp₀ hp₁).concaveOn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.SpecificFunctions.Pow | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 48
} | {
"line": 90,
"column": 0
} | [
{
"pp": "case inr.inr\np : ℝ\nhp₀✝ : 0 ≤ p\nhp₁✝ : p ≤ 1\nhp₀ : 0 < p\nhp₁ : p < 1\n⊢ ConcaveOn ℝ (Ici 0) fun x ↦ x ^ p",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.strictConcaveOn_rpow",
"Real.partialOrder",
"Real",
"Semiring.toMod... | [] | exact (strictConcaveOn_rpow hp₀ hp₁).concaveOn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Side | {
"line": 282,
"column": 6
} | {
"line": 282,
"column": 15
} | {
"line": 282,
"column": 15
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : 0 ≤ t\n⊢ SameRay R ((t • (x -ᵥ p₁) +ᵥ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : 0 ≤ t\n⊢ SameRay R (t • (x -ᵥ p₁)) (x -ᵥ p₁)"
] | vadd_vsub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 440,
"column": 87
} | {
"line": 440,
"column": 96
} | {
"line": 440,
"column": 96
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y p₂ : P\nh : p₂ ∈ s\n⊢ ((x ∉ s ∧ y ∉ s) ∧ ∃ p₂_1 ∈ s, SameRay R (y -ᵥ p₂) (x -ᵥ p₂_1)) ↔\n ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y p₂ : P\nh : p₂ ∈ s\n⊢ (x ∉ s ∧ y ∉ s ∧ ∃ p₂_1 ∈ s, SameRay R (y -ᵥ p₂) (x -ᵥ p₂_1)) ↔\n x ∉ s ∧ y ∉... | and_assoc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 129,
"column": 33
} | {
"line": 129,
"column": 38
} | {
"line": 129,
"column": 39
} | [
{
"pp": "case inr\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nr : ℝ\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne : dist x z ≠ 0\na b : ℝ\nleft... | [
"case inr\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nr : ℝ\nx y z : PE\nhxy : dist x y = r * dist x z\nhyz : dist y z = (1 - r) * dist x z\nhne : dist x z ≠ 0\na b : ℝ\nleft✝¹ : 0 ≤ a\n... | ← H', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Visible | {
"line": 82,
"column": 8
} | {
"line": 82,
"column": 27
} | {
"line": 82,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nhx : x ∉ (c... | [
"𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nhx : x ∉ (convexHull 𝕜... | ← one_smul 𝕜 (a i), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 161,
"column": 6
} | {
"line": 164,
"column": 42
} | {
"line": 164,
"column": 42
} | [
{
"pp": "α : 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 volume 0 t\nint... | [
"α : 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 volume 0 t\nintegrand_eq : ... | show
(Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ {a : α | s ≤ f a} =
(Ioi 0).indicator (fun _x : ℝ => 1 * μ {a : α | s ≤ f a}) s
by by_cases h : 0 < s <;> simp [h] | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 292,
"column": 6
} | {
"line": 293,
"column": 67
} | {
"line": 294,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\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\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 <... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\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\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 < ∫ (t : ℝ) i... | obtain ⟨s, hs, uns⟩ : ∃ s, g =ᶠ[ae (Measure.restrict volume (Ioc 0 s))] 0 ∧ u n < s :=
exists_lt_of_lt_csSup (Set.nonempty_of_mem zero_mem) (uM n) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 331,
"column": 4
} | {
"line": 333,
"column": 47
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case h\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\n⊢ ∀ (x : H), ‖‖x‖ ^ 2‖ ≤ 1 * (1 + ‖x‖) ^ 2",
"ppTerm": "?h✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing... | [] | intro x
rw [norm_pow, norm_norm, one_mul, add_pow_two]
exact le_add_of_nonneg_left (by positivity) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 331,
"column": 4
} | {
"line": 333,
"column": 47
} | {
"line": 335,
"column": 0
} | [
{
"pp": "case h\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\n⊢ ∀ (x : H), ‖‖x‖ ^ 2‖ ≤ 1 * (1 + ‖x‖) ^ 2",
"ppTerm": "?h✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing... | [] | intro x
rw [norm_pow, norm_norm, one_mul, add_pow_two]
exact le_add_of_nonneg_left (by positivity) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 135,
"column": 43
} | {
"line": 135,
"column": 46
} | {
"line": 135,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nM : ℝ := ⨆ x, |T.rayleighQuotient x|\nnonneg : 0 ≤ M\nhM : ∀ (x : E), |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2\nx y : E\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ M * (2 * (‖... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nM : ℝ := ⨆ x, |T.rayleighQuotient x|\nnonneg : 0 ≤ M\nhM : ∀ (x : E), |re ⟪T x, x⟫_𝕜| ≤ M * ‖x‖ ^ 2\nx y : E\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ M * (2 * (1 ^ 2 + ‖y‖ ^... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 162,
"column": 52
} | {
"line": 169,
"column": 7
} | {
"line": 171,
"column": 0
} | [
{
"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✝ : Nontrivial E\nhT' : (algebraMap ℝ 𝕜) ‖T‖ ∈ resolventSet 𝕜 T\n⊢ ∃ ε > 0, ∀ (x : E), T.rayleighQuotient x ≤ ‖T‖ - ε",
"ppTerm": "?m.50",
"assigned": true,
... | [] | by
by_cases hT0 : T = 0
· simp [hT0, spectrum.mem_resolventSet_iff] at hT'
obtain ⟨ε, hε0, hε⟩ := T.rayleighQuotient_le_of_mem_resolventSet ‖T‖ (by positivity) hT'
refine ⟨ε, hε0, fun x ↦ ?_⟩
grw [hε]
field_simp
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 402,
"column": 43
} | {
"line": 405,
"column": 12
} | {
"line": 406,
"column": 4
} | [
{
"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 ≤... | [] | by
gcongr 1
apply le_trans _ hk₂
gcongr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 420,
"column": 12
} | {
"line": 420,
"column": 52
} | {
"line": 422,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nΩ : Opens E\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace 𝕜 F\nn : ℕ∞\ninst✝¹ : Algebra ℝ 𝕜\ninst✝ : IsScalarTower ℝ 𝕜 F\nf g : 𝓓... | [] | by simp [toBoundedContinuousFunctionCLM] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Compact.Basic | {
"line": 108,
"column": 84
} | {
"line": 112,
"column": 73
} | {
"line": 114,
"column": 0
} | [
{
"pp": "M₁ : Type u_2\nM₂ : Type u_3\ninst✝³ : TopologicalSpace M₁\ninst✝² : AddCommMonoid M₁\ninst✝¹ : TopologicalSpace M₂\ninst✝ : T2Space M₂\nf : M₁ → M₂\n⊢ IsCompactOperator f ↔ ∃ V ∈ 𝓝 0, IsCompact (closure[inst✝¹] (f '' V))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Fil... | [] | by
rw [isCompactOperator_iff_exists_mem_nhds_image_subset_compact]
exact
⟨fun ⟨V, hV, K, hK, hKV⟩ => ⟨V, hV, hK.closure_of_subset hKV⟩,
fun ⟨V, hV, hVc⟩ => ⟨V, hV, closure (f '' V), hVc, subset_closure⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 73,
"column": 30
} | {
"line": 73,
"column": 33
} | {
"line": 73,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nhK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖\nC : 𝕜\nhC : 1 < ‖C‖\nε : ... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nhK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖\nC : 𝕜\nhC : 1 < ‖C‖\nε : ℝ\nhε : ε > ... | hx, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 230,
"column": 31
} | {
"line": 230,
"column": 34
} | {
"line": 230,
"column": 35
} | [
{
"pp": "case h\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\nx : Eigenvalues T := ⟨μ, hμ⟩\nhx : x = ⟨μ, hμ⟩\ni : ... | [
"case h\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\nx : Eigenvalues T := ⟨μ, hμ⟩\nhx : x = ⟨μ, hμ⟩\ni : Fin n\nhi : ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 106,
"column": 28
} | {
"line": 106,
"column": 42
} | {
"line": 106,
"column": 43
} | [
{
"pp": "m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosSemidef\nx✝² x✝¹ : n → 𝕜\nx✝ : 𝕜\n⊢ M *ᵥ x✝¹ ⬝ᵥ star (x✝ • x✝²) = (starRingEnd 𝕜) x✝ * M *ᵥ x✝¹ ⬝ᵥ star x✝²",
"ppTerm": "?m.50",
"assigned": true,
"usedCon... | [
"m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosSemidef\nx✝² x✝¹ : n → 𝕜\nx✝ : 𝕜\n⊢ M *ᵥ x✝¹ ⬝ᵥ star (x✝ • x✝²) = (starRingEnd 𝕜) x✝ • (M *ᵥ x✝¹ ⬝ᵥ star x✝²)"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 612,
"column": 80
} | {
"line": 618,
"column": 35
} | {
"line": 620,
"column": 0
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nn : ℕ∞\nK : Compacts E\nX : Type u_5\ninst✝ : TopologicalSpace X\nφ : X → 𝓓^{n}_{K}(E, F)\n⊢ Continuous[inst✝, _] φ ↔ ∀ (i : ℕ), ↑i ≤ n → Continuous[inst✝, _] (... | [] | by
rw [continuous_iff_comp]
congrm (∀ i, ?_)
by_cases hin : i ≤ n <;> simp only [hin, true_imp_iff, false_imp_iff, iff_true]
refine continuous_zero.congr fun x ↦ ?_
ext t : 1
simp [hin, structureMapCLM_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 34
} | {
"line": 61,
"column": 2
} | [
{
"pp": "case inr\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : DecidableEq n\ni : n\nx : E\nk : n\n⊢ gram 𝕜 (Pi.single i x) i k = single i i (⟪x, x⟫_𝕜) i k",
"ppTerm": "?inr",
"assigned": true,
"usedCons... | [
"case inr.inl\nE : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : DecidableEq n\ni : n\nx : E\nk : n\nhik : i ≠ k\n⊢ gram 𝕜 (Pi.single i x) i k = single i i (⟪x, x⟫_𝕜) i k",
"case inr.inr\nE : Type u_1\nn : Type u_2\n𝕜 : Ty... | obtain hik | rfl := ne_or_eq i k | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 56,
"column": 59
} | {
"line": 62,
"column": 6
} | {
"line": 64,
"column": 0
} | [
{
"pp": "E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : DecidableEq n\ni : n\nx : E\n⊢ gram 𝕜 (Pi.single i x) = single i i ⟪x, x⟫_𝕜",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm... | [] | by
ext j k
obtain hij | rfl := ne_or_eq i j
· simp [hij]
obtain hik | rfl := ne_or_eq i k
· simp [hik]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 120,
"column": 70
} | {
"line": 121,
"column": 42
} | {
"line": 123,
"column": 0
} | [
{
"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... | [] | by
rw [abs, ha.star_eq, sqrt_mul_self a ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 185,
"column": 28
} | {
"line": 185,
"column": 42
} | {
"line": 185,
"column": 43
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nr : 𝕜\n⊢ ∫ (a : α), ⟪↑↑(r • f) a, ↑↑g a⟫ ∂μ = (starRingEnd 𝕜) r * ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ ∂μ",
"ppTerm": "?m.53"... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nr : 𝕜\n⊢ ∫ (a : α), ⟪↑↑(r • f) a, ↑↑g a⟫ ∂μ = (starRingEnd 𝕜) r • ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ ∂μ"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 188,
"column": 38
} | {
"line": 188,
"column": 41
} | {
"line": 188,
"column": 42
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : 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⟫ = ⟪r • ↑↑f x, ↑↑g x⟫",
"ppTerm": "?m.113",
... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : 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⟫ = ⟪r • ↑↑f x, ↑↑g x⟫"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 24
} | {
"line": 110,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : Fact (1 ≤ ... | apply eLpNorm_congr_ae | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 178,
"column": 35
} | {
"line": 182,
"column": 12
} | {
"line": 184,
"column": 0
} | [
{
"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 : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthogo... | [] | by
rw [hV.summable_iff_norm_sq_summable]
convert! (lp.memℓp f).summable _
· norm_cast
· norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 339,
"column": 77
} | {
"line": 339,
"column": 91
} | {
"line": 340,
"column": 4
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ ∫ (t : AddCircle T), (c * (fourier (-n)) t) • f t ∂haarAddCircle =\n c • ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle",
"ppTerm": "?m.33",
"assig... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ ∫ (t : AddCircle T), (c • (fourier (-n)) t) • f t ∂haarAddCircle =\n c • ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 491,
"column": 2
} | {
"line": 492,
"column": 48
} | {
"line": 494,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)",
"ppTerm": "?m.34",
... | [] | simpa only [b.repr_apply_apply, inner_orthogonalProjectionOnto_eq_of_mem_left] using
b.hasSum_repr (U.orthogonalProjectionOnto x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 491,
"column": 2
} | {
"line": 492,
"column": 48
} | {
"line": 494,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)",
"ppTerm": "?m.34",
... | [] | simpa only [b.repr_apply_apply, inner_orthogonalProjectionOnto_eq_of_mem_left] using
b.hasSum_repr (U.orthogonalProjectionOnto x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 491,
"column": 2
} | {
"line": 492,
"column": 48
} | {
"line": 494,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)",
"ppTerm": "?m.34",
... | [] | simpa only [b.repr_apply_apply, inner_orthogonalProjectionOnto_eq_of_mem_left] using
b.hasSum_repr (U.orthogonalProjectionOnto x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 116,
"column": 2
} | {
"line": 117,
"column": 62
} | {
"line": 119,
"column": 0
} | [
{
"pp": "case e_f\n𝕜 : Type u_1\ninst✝⁹ : CommRing 𝕜\nV : Type u_2\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module 𝕜 V\ninst✝⁶ : MeasurableSpace V\nW : Type u_3\ninst✝⁵ : AddCommGroup W\ninst✝⁴ : Module 𝕜 W\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\ninst✝¹ : MeasurableAdd V\ne : AddCh... | [] | rw [← smul_assoc, smul_eq_mul, ← Circle.coe_mul, ← e.map_add_eq_mul, ← LinearMap.neg_apply,
← sub_eq_add_neg, ← LinearMap.sub_apply, map_sub, neg_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 79
} | {
"line": 220,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝... | [
"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : SecondCou... | let F' : W → V → W →L[ℝ] E := fun w' v ↦ 𝐞 (-L v w') • fourierSMulRight L f v | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 56
} | {
"line": 101,
"column": 56
} | [
{
"pp": "f : ℝ → ℝ\neq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x\nhf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume\n⊢ IntegrableOn (fun x ↦ f |x|) (Iic 0) volume",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"... | [
"f : ℝ → ℝ\neq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x\nhf : IntegrableOn (fun x ↦ f |x|) (Ioi 0) volume\n⊢ IntegrableOn (fun x ↦ f |x|) (Iic 0) (Measure.map Neg.neg volume)"
] | ← Measure.map_neg_eq_self (volume : Measure ℝ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 24
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\ns : ℝ\nh : IntegrableOn (fun x ↦ x ^ s) (Ioi 0) volume\nhs : -1 < s\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real.partialOrder",
"Real",
"Set.Ioi",
"MeasureTheory.M... | [] | · have : IntegrableOn (fun x ↦ x ^ s) (Ioi (1 : ℝ)) := h.mono (Ioi_subset_Ioi zero_le_one) le_rfl
rw [integrableOn_Ioi_rpow_iff zero_lt_one] at this
exact hs.not_gt this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 443,
"column": 2
} | {
"line": 445,
"column": 72
} | {
"line": 447,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nμ : M... | [] | refine (hf.const_mul ((2 * π * ‖L‖) ^ n)).mono' (h'f.fourierPowSMulRight L n) ?_
filter_upwards with v
exact (norm_fourierPowSMulRight_le L f v n).trans (le_of_eq (by ring)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 443,
"column": 2
} | {
"line": 445,
"column": 72
} | {
"line": 447,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\nμ : M... | [] | refine (hf.const_mul ((2 * π * ‖L‖) ^ n)).mono' (h'f.fourierPowSMulRight L n) ?_
filter_upwards with v
exact (norm_fourierPowSMulRight_le L f v n).trans (le_of_eq (by ring)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 97,
"column": 2
} | {
"line": 103,
"column": 8
} | {
"line": 105,
"column": 0
} | [
{
"pp": "case right\ns : ℂ\nhs : 0 < s.re\n⊢ HasFiniteIntegral (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) (volume.restrict (Ioi 0))",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCo... | [] | · rw [← hasFiniteIntegral_norm_iff]
refine HasFiniteIntegral.congr (Real.GammaIntegral_convergent hs).2 ?_
apply (ae_restrict_iff' measurableSet_Ioi).mpr
filter_upwards with x hx
rw [norm_mul, Complex.norm_of_nonneg <| le_of_lt <| exp_pos <| -x,
norm_cpow_eq_rpow_re_of_pos hx _]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 73
} | {
"line": 696,
"column": 2
} | [
{
"pp": "case right\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✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ :... | [
"case right\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✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAlgeb... | simp_rw [← ContinuousLinearMap.bilinearRestrictScalars_apply_apply ℝ B] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 317,
"column": 4
} | {
"line": 319,
"column": 34
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case e'_3\nb : ℝ\nhb : 0 < b\n⊢ ↑(√(π / b) / 2) = (↑π / ↑b) ^ (1 / 2) / 2",
"ppTerm": "?e'_3✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instPow",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"inst... | [] | rw [sqrt_eq_rpow, ← ofReal_div, ofReal_div, ofReal_cpow]
· simp
· exact (div_pos pi_pos hb).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 317,
"column": 4
} | {
"line": 319,
"column": 34
} | {
"line": 322,
"column": 0
} | [
{
"pp": "case e'_3\nb : ℝ\nhb : 0 < b\n⊢ ↑(√(π / b) / 2) = (↑π / ↑b) ^ (1 / 2) / 2",
"ppTerm": "?e'_3✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instPow",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"inst... | [] | rw [sqrt_eq_rpow, ← ofReal_div, ofReal_div, ofReal_cpow]
· simp
· exact (div_pos pi_pos hb).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 35
} | {
"line": 101,
"column": 50
} | {
"line": 101,
"column": 50
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [] | simpa using hf₂ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 35
} | {
"line": 101,
"column": 50
} | {
"line": 101,
"column": 50
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [] | simpa using hf₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.Convolution | {
"line": 101,
"column": 35
} | {
"line": 101,
"column": 50
} | {
"line": 101,
"column": 50
} | [
{
"pp": "E : Type u_3\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup F₁\ninst✝¹¹ : NormedAddCommGroup F₂\ninst✝¹⁰ : NormedAddCommGroup F₃\ninst✝⁹ : InnerProductSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : BorelSpace E... | [] | simpa using hf₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 269,
"column": 4
} | {
"line": 275,
"column": 10
} | {
"line": 276,
"column": 2
} | [
{
"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... | [] | · congr
ext x
rw [Pi.mul_apply]
norm_cast
rw [← Real.rpow_add (by positivity)]
ring_nf
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 79,
"column": 45
} | {
"line": 79,
"column": 48
} | {
"line": 79,
"column": 49
} | [
{
"pp": "case add\nι : Type\ninst✝ : DecidableEq ι\np n : ι → ℕ\nx y : ⨁ (i : { i // n i ≠ 0 }), ZMod (p ↑i ^ n ↑i)\nhx :\n (DirectSum.toAddMonoid fun i ↦ if h : n i = 0 then 0 else DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩)\n ((directSumNeZeroMulHom p n) x) =\n x\nhy :\n (DirectSum.toAddMonoid... | [
"case add\nι : Type\ninst✝ : DecidableEq ι\np n : ι → ℕ\nx y : ⨁ (i : { i // n i ≠ 0 }), ZMod (p ↑i ^ n ↑i)\nhx :\n (DirectSum.toAddMonoid fun i ↦ if h : n i = 0 then 0 else DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩)\n ((directSumNeZeroMulHom p n) x) =\n x\nhy :\n (DirectSum.toAddMonoid fun i ↦ if ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 88,
"column": 45
} | {
"line": 88,
"column": 48
} | {
"line": 88,
"column": 49
} | [
{
"pp": "case add\nι : Type\ninst✝ : DecidableEq ι\np n : ι → ℕ\nx y : ⨁ (i : ι), ZMod (p i ^ n i)\nhx :\n (directSumNeZeroMulHom p n)\n ((DirectSum.toAddMonoid fun i ↦ if h : n i = 0 then 0 else DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩) x) =\n x\nhy :\n (directSumNeZeroMulHom p n)\n ((Di... | [
"case add\nι : Type\ninst✝ : DecidableEq ι\np n : ι → ℕ\nx y : ⨁ (i : ι), ZMod (p i ^ n i)\nhx :\n (directSumNeZeroMulHom p n)\n ((DirectSum.toAddMonoid fun i ↦ if h : n i = 0 then 0 else DirectSum.of (fun j ↦ ZMod (p ↑j ^ n ↑j)) ⟨i, h⟩) x) =\n x\nhy :\n (directSumNeZeroMulHom p n)\n ((DirectSum.toAd... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 224,
"column": 80
} | {
"line": 226,
"column": 30
} | {
"line": 228,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝³ : CommRing R\ninst✝² : Module.Finite ℤ R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nA : Submodule R M\nhfg : A.FG\n⊢ A.toAddSubgroup.FG",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"AddCommGroup.intI... | [] | by
rw [← AddSubgroup.toIntSubmodule_toAddSubgroup A.toAddSubgroup, ← fg_iff_addSubgroup_fg]
exact FG.restrictScalars hfg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 180,
"column": 82
} | {
"line": 181,
"column": 64
} | {
"line": 183,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\n⊢ ↑ζ * ↑μ = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"ArithmeticFunction.intCoe_one",
"HMul.hMul",
"ArithmeticFunction.instMul",
"AddGroupWithOne.toAddGroup",
... | [] | by
rw [← coe_coe, ← intCoe_mul, coe_zeta_mul_moebius, intCoe_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 66,
"column": 2
} | {
"line": 81,
"column": 26
} | {
"line": 83,
"column": 0
} | [
{
"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\nw : V\nhw : w ≠ 0\n⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f (v + i w) = -... | [] | have hiw : ⟪i w, w⟫ = 1 / 2 := by
rw [inner_smul_left, inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id,
RCLike.conj_to_real, ← div_div, div_mul_cancel₀]
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
have :
(fun v : V => 𝐞 (-⟪v, w⟫) • f (v + i w)) =
fun v : V => (fun x : V => -(𝐞 (-⟪x, w⟫) ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 66,
"column": 2
} | {
"line": 81,
"column": 26
} | {
"line": 83,
"column": 0
} | [
{
"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\nw : V\nhw : w ≠ 0\n⊢ ∫ (v : V), 𝐞 (-⟪v, w⟫) • f (v + i w) = -... | [] | have hiw : ⟪i w, w⟫ = 1 / 2 := by
rw [inner_smul_left, inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id,
RCLike.conj_to_real, ← div_div, div_mul_cancel₀]
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
have :
(fun v : V => 𝐞 (-⟪v, w⟫) • f (v + i w)) =
fun v : V => (fun x : V => -(𝐞 (-⟪x, w⟫) ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 639,
"column": 2
} | {
"line": 641,
"column": 91
} | {
"line": 643,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\na : R\nh : ¬∃ α, α ^ n = a\n⊢ (nthRoots n a).card = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"IsD... | [] | · obtain (rfl | hn) := n.eq_zero_or_pos; · simp
push Not at h
simpa only [Multiset.card_eq_zero, Multiset.eq_zero_iff_forall_notMem, mem_nthRoots hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 656,
"column": 2
} | {
"line": 660,
"column": 7
} | {
"line": 662,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nn : ℕ\ninst✝ : NeZero n\n⊢ Nat.card ↥(rootsOfUnity n R) = n ↔ ∃ ζ, IsPrimitiveRoot ζ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Units.val",
"rootsOfUnity.isCyclic",
"congrArg",
"SubgroupClass.t... | [] | refine ⟨fun h ↦ ?_, fun ⟨ζ, hζ⟩ ↦ hζ.card_rootsOfUnity⟩
obtain ⟨⟨ζ, hζ'⟩, hζ⟩ := (rootsOfUnity.isCyclic R n).exists_ofOrder_eq_natCard
rw [h, ← IsPrimitiveRoot.iff_orderOf, ← coe_submonoidClass_iff,
← IsPrimitiveRoot.coe_units_iff] at hζ
use ζ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 656,
"column": 2
} | {
"line": 660,
"column": 7
} | {
"line": 662,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nn : ℕ\ninst✝ : NeZero n\n⊢ Nat.card ↥(rootsOfUnity n R) = n ↔ ∃ ζ, IsPrimitiveRoot ζ n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Units.val",
"rootsOfUnity.isCyclic",
"congrArg",
"SubgroupClass.t... | [] | refine ⟨fun h ↦ ?_, fun ⟨ζ, hζ⟩ ↦ hζ.card_rootsOfUnity⟩
obtain ⟨⟨ζ, hζ'⟩, hζ⟩ := (rootsOfUnity.isCyclic R n).exists_ofOrder_eq_natCard
rw [h, ← IsPrimitiveRoot.iff_orderOf, ← coe_submonoidClass_iff,
← IsPrimitiveRoot.coe_units_iff] at hζ
use ζ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 340,
"column": 2
} | {
"line": 346,
"column": 91
} | {
"line": 348,
"column": 0
} | [
{
"pp": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.map_one",
"Eq.mpr",
"Polynomial.instOne",
"instHDiv",
"Real.pi",
"HMul.hMul",
... | [] | have integer : ∏ i ∈ Nat.divisors n, cyclotomic i ℤ = X ^ n - 1 := by
apply map_injective (Int.castRingHom ℂ) Int.cast_injective
simp only [Polynomial.map_prod, int_cyclotomic_spec, Polynomial.map_pow, map_X,
Polynomial.map_one, Polynomial.map_sub]
exact prod_cyclotomic'_eq_X_pow_sub_one hpos (Complex... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 340,
"column": 2
} | {
"line": 346,
"column": 91
} | {
"line": 348,
"column": 0
} | [
{
"pp": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Polynomial.map_one",
"Eq.mpr",
"Polynomial.instOne",
"instHDiv",
"Real.pi",
"HMul.hMul",
... | [] | have integer : ∏ i ∈ Nat.divisors n, cyclotomic i ℤ = X ^ n - 1 := by
apply map_injective (Int.castRingHom ℂ) Int.cast_injective
simp only [Polynomial.map_prod, int_cyclotomic_spec, Polynomial.map_pow, map_X,
Polynomial.map_one, Polynomial.map_sub]
exact prod_cyclotomic'_eq_X_pow_sub_one hpos (Complex... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 53
} | {
"line": 153,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn m : ℕ\nhzero : n ≠ 0\nthis : NeZero n\nhnm : cyclotomic n ℂ = cyclotomic m ℂ\nhroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m\nhmzero : NeZero m\nhprim : n = orderOf (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))\... | [] | rwa [← IsPrimitiveRoot.eq_orderOf hroot] at hprim | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 168,
"column": 53
} | {
"line": 168,
"column": 66
} | {
"line": 168,
"column": 67
} | [
{
"pp": "K : Type u_2\ninst✝⁴ : Field K\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nμ : R\nn : ℕ\ninst✝¹ : Algebra K R\nhμ : IsPrimitiveRoot μ n\nh : Irreducible (cyclotomic n K)\ninst✝ : NeZero ↑n\nthis : NeZero ↑n\n⊢ eval μ (cyclotomic n R) = 0",
"ppTerm": "?m.59",
"assigned": true,
"... | [
"K : Type u_2\ninst✝⁴ : Field K\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nμ : R\nn : ℕ\ninst✝¹ : Algebra K R\nhμ : IsPrimitiveRoot μ n\nh : Irreducible (cyclotomic n K)\ninst✝ : NeZero ↑n\nthis : NeZero ↑n\n⊢ (cyclotomic n R).IsRoot μ"
] | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 59,
"column": 74
} | {
"line": 59,
"column": 87
} | {
"line": 60,
"column": 8
} | [
{
"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⊢ eval (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p)) ^ p) (... | [
"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⊢ (cyclotomic n ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p))... | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 66,
"column": 58
} | {
"line": 66,
"column": 71
} | {
"line": 66,
"column": 72
} | [
{
"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⊢ eval (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n) ^ p)\n (map (algebraMap ℚ ℂ) (minpoly... | [
"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⊢ (map (algebraMap ℚ ℂ) (minpoly ℚ (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)))).IsRoot\n (Complex.ex... | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 100,
"column": 72
} | {
"line": 100,
"column": 85
} | {
"line": 101,
"column": 6
} | [
{
"pp": "case inr.refine_1\np n : ℕ\nhp : Nat.Prime p\nhdiv : p ∣ n\nR : Type u_1\ninst✝ : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p))) (n * p)\n⊢ eval (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p)) ^ p) (cyclotomic... | [
"case inr.refine_1\np n : ℕ\nhp : Nat.Prime p\nhdiv : p ∣ n\nR : Type u_1\ninst✝ : CommRing R\nhzero : n > 0\nthis : NeZero n\nhpos : 0 < n * p\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p))) (n * p)\n⊢ (cyclotomic n ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑(n * p)) ^ p)"
] | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 179,
"column": 9
} | {
"line": 179,
"column": 22
} | {
"line": 179,
"column": 23
} | [
{
"pp": "case inr.refine_1\nm k p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\nμ : R\ninst✝ : NeZero ↑m\nhk : k > 0\nh : eval μ (cyclotomic m R) = 0\n⊢ IsPrimitiveRoot μ m",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
... | [
"case inr.refine_1\nm k p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nhp : Fact (Nat.Prime p)\nhchar : CharP R p\nμ : R\ninst✝ : NeZero ↑m\nhk : k > 0\nh : (cyclotomic m R).IsRoot μ\n⊢ IsPrimitiveRoot μ m"
] | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 99,
"column": 29
} | {
"line": 99,
"column": 42
} | {
"line": 99,
"column": 43
} | [
{
"pp": "n : ℕ\ninst✝⁶ : NeZero n\nA : Type w\nB : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : IsCyclotomicExtension {n} A B\ninst✝¹ : IsDomain B\ninst✝ : NeZero ↑n\n⊢ eval (zeta n A B) (Polynomial.map (algebraMap A B) (cyclotomic n A)) = 0",
"ppTerm": "?m.37",
"assi... | [
"n : ℕ\ninst✝⁶ : NeZero n\nA : Type w\nB : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : IsCyclotomicExtension {n} A B\ninst✝¹ : IsDomain B\ninst✝ : NeZero ↑n\n⊢ (Polynomial.map (algebraMap A B) (cyclotomic n A)).IsRoot (zeta n A B)"
] | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 162,
"column": 72
} | {
"line": 162,
"column": 85
} | {
"line": 163,
"column": 10
} | [
{
"pp": "case mk\np n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclo... | [
"case mk\np n : ℕ\ninst✝¹² : NeZero n\nA : Type w\nB : Type z\nK : Type u\nL : Type v\nC✝ : Type w\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsCyclotomicExtension {n} A B\ninst✝⁷ : Field K\ninst✝⁶ : CommRing L\ninst✝⁵ : IsDomain L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtensi... | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 115,
"column": 4
} | {
"line": 116,
"column": 29
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\... | [
"case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | simp only [coeff_X_pow, coeff_X_zero, sub_zero, _root_.map_eq_zero, ite_eq_right_iff,
one_ne_zero, coeff_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 12
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\... | [
"case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | intro hn | Lean.Elab.Tactic.evalIntro | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 12
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\... | [
"case succ.refine_2\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | intro hn | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 350,
"column": 59
} | {
"line": 350,
"column": 75
} | {
"line": 350,
"column": 75
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝ : IsCyclotomicExtension {n} K L\nh : 2 < n\nhirr : Irreducible (cyclotomic n K)\nthis✝¹ : NeZero ↑n\nE : Type v := AlgebraicClosure L\nz : E\nhz : (cyclotom... | [
"n : ℕ\ninst✝⁴ : NeZero n\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝ : IsCyclotomicExtension {n} K L\nh : 2 < n\nhirr : Irreducible (cyclotomic n K)\nthis✝¹ : NeZero ↑n\nE : Type v := AlgebraicClosure L\nz : E\nhz : (cyclotomic n E).IsRo... | ← univ_eq_attach | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 122,
"column": 8
} | {
"line": 122,
"column": 11
} | {
"line": 122,
"column": 12
} | [
{
"pp": "case succ.refine_4\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\... | [
"case succ.refine_4\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 86
} | {
"line": 270,
"column": 4
} | [
{
"pp": "case neg.refine_1\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhS : ¬∃ s ∈ S, s ≠ 0\nH : IsCyclotomicExtension ∅ A B\n⊢ adjoin A {b | ∃ n ∈ {1}, n ≠ 0 ∧ b ^ n = 1} = ⊤",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedConstants": [
... | [] | simpa [adjoin_singleton_one] using subsingleton_iff_bot_eq_top.mpr inferInstance | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 125,
"column": 17
} | {
"line": 125,
"column": 20
} | {
"line": 125,
"column": 21
} | [
{
"pp": "case succ.refine_5\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑((X ^ (n✝ + 1) ^ n - X).rootSet (GaloisField (n✝ + 1) n))... | [
"case succ.refine_5\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑((X ^ (n✝ + 1) ^ n - X).rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 129,
"column": 8
} | {
"line": 129,
"column": 11
} | {
"line": 129,
"column": 12
} | [
{
"pp": "case succ.refine_6\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\... | [
"case succ.refine_6\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ^ n\nnat_degree_... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ZMod.Units | {
"line": 88,
"column": 47
} | {
"line": 88,
"column": 75
} | {
"line": 88,
"column": 75
} | [
{
"pp": "⊢ 0 = 1 * ↑0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | rw [Nat.cast_zero, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.Units | {
"line": 88,
"column": 47
} | {
"line": 88,
"column": 75
} | {
"line": 88,
"column": 75
} | [
{
"pp": "⊢ 0 = 1 * ↑0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | rw [Nat.cast_zero, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ZMod.Units | {
"line": 88,
"column": 47
} | {
"line": 88,
"column": 75
} | {
"line": 88,
"column": 75
} | [
{
"pp": "⊢ 0 = 1 * ↑0",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ZMod.commRing",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | rw [Nat.cast_zero, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ZMod.Units | {
"line": 132,
"column": 45
} | {
"line": 133,
"column": 49
} | {
"line": 135,
"column": 0
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nm : ℤ\nh : IsCoprime m ↑n\n⊢ ↑m * ↑(↑m)⁻¹.val = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"ZMod.instInv",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"Ad... | [] | by
rw [natCast_zmod_val, coe_int_mul_inv_eq_one h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 233,
"column": 31
} | {
"line": 233,
"column": 66
} | {
"line": 233,
"column": 67
} | [
{
"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 ^ ∑ i ∈ range (Module.finrank K K'), Fintype.card K ^ i = x ^ ((Fintype.card K' - 1) / (Fintype.card K - 1))",
"ppTerm": "... | [
"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 ^ ((Fintype.card K ^ Module.finrank K K' - 1) / (Fintype.card K - 1)) =\n x ^ ((Fintype.card K' - 1) / (Fintype.card K - 1))"
] | Nat.geomSum_eq Fintype.one_lt_card, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 151,
"column": 17
} | {
"line": 151,
"column": 80
} | {
"line": 152,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx✝¹ x✝ : ZMod n\n⊢ (ZMod.rootsOfUnityAddChar n) x✝¹ = (ZMod.rootsOfUnityAddChar n) x✝ → x✝¹ = x✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"MulEquiv.instEquivLike",
"ZMod.rootsOfUnityAddChar",
"ZMod.com... | [] | simp [ZMod.rootsOfUnityAddChar, ZMod.injective_toCircle.eq_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 151,
"column": 17
} | {
"line": 151,
"column": 80
} | {
"line": 152,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx✝¹ x✝ : ZMod n\n⊢ (ZMod.rootsOfUnityAddChar n) x✝¹ = (ZMod.rootsOfUnityAddChar n) x✝ → x✝¹ = x✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"MulEquiv.instEquivLike",
"ZMod.rootsOfUnityAddChar",
"ZMod.com... | [] | simp [ZMod.rootsOfUnityAddChar, ZMod.injective_toCircle.eq_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 151,
"column": 17
} | {
"line": 151,
"column": 80
} | {
"line": 152,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\nx✝¹ x✝ : ZMod n\n⊢ (ZMod.rootsOfUnityAddChar n) x✝¹ = (ZMod.rootsOfUnityAddChar n) x✝ → x✝¹ = x✝",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"MulEquiv.instEquivLike",
"ZMod.rootsOfUnityAddChar",
"ZMod.com... | [] | simp [ZMod.rootsOfUnityAddChar, ZMod.injective_toCircle.eq_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Int.Range | {
"line": 37,
"column": 45
} | {
"line": 37,
"column": 82
} | {
"line": 39,
"column": 0
} | [
{
"pp": "P : ℤ → Prop\ninst✝ : DecidablePred P\nm n : ℤ\n⊢ (∀ (r : ℤ), r ∈ m.range n → P r) ↔ ∀ (r : ℤ), m ≤ r → r < n → P r",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"congrArg",
"_private.Mathlib.Data.Int.Range.0.Int.decidableLELT._simp_2",
"Membership.mem",
... | [] | by simp only [mem_range_iff, and_imp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 682,
"column": 28
} | {
"line": 682,
"column": 41
} | {
"line": 682,
"column": 42
} | [
{
"pp": "n : ℕ\ninst✝⁷ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : NeZero ↑n\nthis✝¹ : NeZero ↑n\nthis✝ : DecidableEq (CyclotomicField n K) := Classical.... | [
"n : ℕ\ninst✝⁷ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : NeZero ↑n\nthis✝¹ : NeZero ↑n\nthis✝ : DecidableEq (CyclotomicField n K) := Classical.decEq (Cyclo... | ← IsRoot.def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 153,
"column": 77
} | {
"line": 153,
"column": 80
} | {
"line": 153,
"column": 81
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd : ℕ\ninst✝ : NeZero n\nhd : d ∣ n\nx✝ : χ.FactorsThrough d\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap hd).ker\nw✝ : d ∣ n\nχ₀ : DirichletCharacter R d\nhχ₀ : χ = (changeLevel w✝) χ₀\n⊢ (toUnitHom χ₀) ((ZMod.un... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd : ℕ\ninst✝ : NeZero n\nhd : d ∣ n\nx✝ : χ.FactorsThrough d\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap hd).ker\nw✝ : d ∣ n\nχ₀ : DirichletCharacter R d\nhχ₀ : χ = (changeLevel w✝) χ₀\n⊢ (toUnitHom χ₀) 1 = 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 157,
"column": 36
} | {
"line": 157,
"column": 49
} | {
"line": 157,
"column": 50
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd : ℕ\ninst✝ : NeZero n\nhd : d ∣ n\nh : (ZMod.unitsMap hd).ker ≤ (toUnitHom χ).ker\nE : (ZMod d)ˣ →* Rˣ := ((ZMod.unitsMap hd).liftOfSurjective ⋯) ⟨toUnitHom χ, h⟩\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom... | [
"case refine_2\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd : ℕ\ninst✝ : NeZero n\nhd : d ∣ n\nh : (ZMod.unitsMap hd).ker ≤ (toUnitHom χ).ker\nE : (ZMod d)ˣ →* Rˣ := ((ZMod.unitsMap hd).liftOfSurjective ⋯) ⟨toUnitHom χ, h⟩\nhE : E.comp (ZMod.unitsMap hd) = toUnitHom χ\n⊢ equivT... | toUnitHom_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 319,
"column": 26
} | {
"line": 319,
"column": 53
} | {
"line": 319,
"column": 54
} | [
{
"pp": "R' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\nR : Type u_3\ninst✝ : CommMonoidWithZero R\nχ : MulChar R R'\na : R\na✝ : Nontrivial R\nha : ¬IsUnit a\n⊢ 0 = χ a⁻¹ʳ",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"cong... | [
"R' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\nR : Type u_3\ninst✝ : CommMonoidWithZero R\nχ : MulChar R R'\na : R\na✝ : Nontrivial R\nha : ¬IsUnit a\n⊢ 0 = χ 0"
] | Ring.inverse_non_unit a ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 167,
"column": 56
} | {
"line": 167,
"column": 59
} | {
"line": 167,
"column": 60
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd m : ℕ\ninst✝ : NeZero n\nhχ : χ.FactorsThrough d\nhd : d ∣ m\nhm : m ∣ n\nx : (ZMod n)ˣ\nhx : (ZMod.unitsMap hm) x = 1\n⊢ (ZMod.unitsMap hd) ((ZMod.unitsMap hm) x) = 1",
"ppTerm": "?m.72",
"assigned": true,
"u... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nd m : ℕ\ninst✝ : NeZero n\nhχ : χ.FactorsThrough d\nhd : d ∣ m\nhm : m ∣ n\nx : (ZMod n)ˣ\nhx : (ZMod.unitsMap hm) x = 1\n⊢ (ZMod.unitsMap hd) 1 = 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 91
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Finite Rˣ\nχ : MulChar R ℂ\n⊢ star χ = χ⁻¹",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
"DivisionCommMonoid.toDivisionMonoid",
"D... | [] | cases nonempty_fintype Rˣ
ext1 a
simp only [inv_apply_eq_inv']
exact (inv_eq_conj <| norm_eq_one_of_mem_rootsOfUnity <| χ.apply_mem_rootsOfUnity a).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 91
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Finite Rˣ\nχ : MulChar R ℂ\n⊢ star χ = χ⁻¹",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
"DivisionCommMonoid.toDivisionMonoid",
"D... | [] | cases nonempty_fintype Rˣ
ext1 a
simp only [inv_apply_eq_inv']
exact (inv_eq_conj <| norm_eq_one_of_mem_rootsOfUnity <| χ.apply_mem_rootsOfUnity a).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Lemmas | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 53
} | {
"line": 211,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Finite F\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nχ : MulChar F R\nn : ℕ\ninst✝ : NeZero n\nhχ : χ ^ n = 1\nμ : R\nhμ : IsPrimitiveRoot μ n\na : F\nha : a ≠ 0\nζ : Rˣ\nhζ₁ : ζ ∈ rootsOfUnity n R\nhζ₂ : ↑ζ = χ a\nhζ' : ↑ζ ^ n = 1\n⊢ ∃ k < n, χ a =... | [
"F : Type u_1\ninst✝⁴ : Field F\ninst✝³ : Finite F\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nχ : MulChar F R\nn : ℕ\ninst✝ : NeZero n\nhχ : χ ^ n = 1\nμ : R\nhμ : IsPrimitiveRoot μ n\na : F\nha : a ≠ 0\nζ : Rˣ\nhζ₁ : ζ ∈ rootsOfUnity n R\nhζ₂ : ↑ζ = χ a\nhζ' : ↑ζ ^ n = 1\nk : ℕ\nhk₁ : k < n\nhk₂ : μ ... | obtain ⟨k, hk₁, hk₂⟩ := hμ.eq_pow_of_pow_eq_one hζ' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
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