module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Hall.Finite | {
"line": 250,
"column": 8
} | {
"line": 250,
"column": 39
} | {
"line": 250,
"column": 39
} | [
{
"pp": "case mpr\nι : Type u_1\nα : Type u_2\ninst✝¹ : Finite ι\ninst✝ : DecidableEq α\nt : ι → Finset α\nf : ι → α\nhf₁ : Function.Injective f\nhf₂ : ∀ (x : ι), f x ∈ t x\ns : Finset ι\n⊢ #s ≤ #(s.biUnion t)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",... | [
"case mpr\nι : Type u_1\nα : Type u_2\ninst✝¹ : Finite ι\ninst✝ : DecidableEq α\nt : ι → Finset α\nf : ι → α\nhf₁ : Function.Injective f\nhf₂ : ∀ (x : ι), f x ∈ t x\ns : Finset ι\n⊢ #(image f s) ≤ #(s.biUnion t)"
] | ← card_image_of_injective s hf₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Approximation | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 38
} | {
"line": 191,
"column": 4
} | [
{
"pp": "case refine_3\n𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhφc ... | [] | simpa using! hb ▸ h ⟨x, trivial⟩ | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Convex.Approximation | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 38
} | {
"line": 191,
"column": 4
} | [
{
"pp": "case refine_3\n𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhφc ... | [] | simpa using! hb ▸ h ⟨x, trivial⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Approximation | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 38
} | {
"line": 191,
"column": 4
} | [
{
"pp": "case refine_3\n𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhφc ... | [] | simpa using! hb ▸ h ⟨x, trivial⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.Limits.Konig | {
"line": 128,
"column": 40
} | {
"line": 138,
"column": 51
} | {
"line": 140,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝⁴ : SmallCategory J\nF : J ⥤ TopCat\ninst✝³ : IsCofilteredOrEmpty J\ninst✝² : ∀ (j : J), Nonempty ↑(F.obj j)\ninst✝¹ : ∀ (j : J), CompactSpace ↑(F.obj j)\ninst✝ : ∀ (j : J), T2Space ↑(F.obj j)\n⊢ Nonempty ↑(limitCone F).pt",
"ppTerm": "?m.27",
"assigned": true,
"usedConstan... | [] | by
classical
obtain ⟨u, hu⟩ :=
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed (fun G => partialSections F _)
(partialSections.directed F) (fun G => partialSections.nonempty F _)
(fun G => IsClosed.isCompact (partialSections.closed F _)) fun G =>
partialSections.closed F _
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 119,
"column": 2
} | {
"line": 122,
"column": 40
} | {
"line": 127,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ Equiv.Perm.permMatrix R σ ∈ rowStochastic R n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLik... | [] | rw [mem_rowStochastic_iff_sum]
refine ⟨fun i j => ?g1, ?g2⟩
case g1 => aesop
case g2 => simp [Equiv.toPEquiv_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 119,
"column": 2
} | {
"line": 122,
"column": 40
} | {
"line": 127,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nσ : Equiv.Perm n\n⊢ Equiv.Perm.permMatrix R σ ∈ rowStochastic R n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PEquiv.instFunLik... | [] | rw [mem_rowStochastic_iff_sum]
refine ⟨fun i j => ?g1, ?g2⟩
case g1 => aesop
case g2 => simp [Equiv.toPEquiv_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Body | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 71
} | {
"line": 222,
"column": 4
} | [
{
"pp": "case refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε\nx : V\nh : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K\nC : ℝ\nhC_pos : C > 0\nhC_bdd : ∀ x ∈ ↑K,... | [
"case refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖↑(u n) - ↑0‖ < ε\nC : ℝ\nhC_pos : C > 0\nhC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C\nε : ℝ\nhε : 0 < ε\nn : ℕ\nhn : ∀ (n_1 : ℕ... | obtain ⟨y, hyK, rfl⟩ := Set.mem_smul_set.mp (Set.mem_iInter.mp h n) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Between | {
"line": 990,
"column": 2
} | {
"line": 991,
"column": 11
} | {
"line": 993,
"column": 0
} | [
{
"pp": "case neg\nR : 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\nw z : P\nt₁ : R\nht₁ : t₁ ∈ Set.Icc 0 1\nt₂ : R\nht₂ : t₂ ∈ Set.Icc 0 1\nh : ¬1 - t₂ * t₁ = 0\n⊢ ((t₁ - t₂ *... | [] | · rw [div_self h]
ring_nf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Between | {
"line": 971,
"column": 2
} | {
"line": 991,
"column": 11
} | {
"line": 993,
"column": 0
} | [
{
"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\nw x y z : P\nh₁ : Wbtw R w y z\nh₂ : Wbtw R w x y\n⊢ Wbtw R x y z",
"ppTerm": "?m.27",
"assigned": true,
"... | [] | rcases h₁ with ⟨t₁, ht₁, rfl⟩
rcases h₂ with ⟨t₂, ht₂, rfl⟩
refine
⟨(t₁ - t₂ * t₁) / (1 - t₂ * t₁),
⟨div_nonneg (sub_nonneg.2 (mul_le_of_le_one_left ht₁.1 ht₂.2))
(sub_nonneg.2 (mul_le_one₀ ht₂.2 ht₁.1 ht₁.2)), div_le_one_of_le₀
(sub_le_sub_right ht₁.2 _) (sub_nonneg.2 (mul_le_one₀ h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 971,
"column": 2
} | {
"line": 991,
"column": 11
} | {
"line": 993,
"column": 0
} | [
{
"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\nw x y z : P\nh₁ : Wbtw R w y z\nh₂ : Wbtw R w x y\n⊢ Wbtw R x y z",
"ppTerm": "?m.27",
"assigned": true,
"... | [] | rcases h₁ with ⟨t₁, ht₁, rfl⟩
rcases h₂ with ⟨t₂, ht₂, rfl⟩
refine
⟨(t₁ - t₂ * t₁) / (1 - t₂ * t₁),
⟨div_nonneg (sub_nonneg.2 (mul_le_of_le_one_left ht₁.1 ht₂.2))
(sub_nonneg.2 (mul_le_one₀ ht₂.2 ht₁.1 ht₁.2)), div_le_one_of_le₀
(sub_le_sub_right ht₁.2 _) (sub_nonneg.2 (mul_le_one₀ h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 1014,
"column": 6
} | {
"line": 1014,
"column": 29
} | {
"line": 1015,
"column": 6
} | [
{
"pp": "case refine_1.right\nR : 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\nw x y z : P\nh_ne : x ≠ y\nt₁ : R\nht₁ : t₁ ∈ Set.Icc 0 1\nhx : (lineMap w y) t₁ = x\nt₂ : R\nht₂... | [
"case refine_1.right.h\nR : 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\nw x y z : P\nh_ne : x ≠ y\nt₁ : R\nht₁ : t₁ ∈ Set.Icc 0 1\nhx : (lineMap w y) t₁ = x\nt₂ : R\nht₂ : t₂ ∈ Se... | apply div_le_one_of_le₀ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 52
} | {
"line": 125,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀ : x₀ ∈ K\nf ... | [] | simpa using hv 0 C.zero_mem | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Continuous | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 83
} | {
"line": 126,
"column": 83
} | [
{
"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! h.eventually (eventually_le_nhds (by simp)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Convex.Continuous | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 83
} | {
"line": 126,
"column": 83
} | [
{
"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! h.eventually (eventually_le_nhds (by simp)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Continuous | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 83
} | {
"line": 126,
"column": 83
} | [
{
"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! h.eventually (eventually_le_nhds (by simp)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 124,
"column": 25
} | {
"line": 124,
"column": 59
} | {
"line": 124,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\nx : P\n⊢ Subtype.val '' frontier (Subtype.val ⁻¹' {x}) = ∅",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\nx : P\n⊢ Subtype.val '' frontier univ = ∅"
] | preimage_coe_affineSpan_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Join | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case inr.refine_1\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 u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\n... | [] | linear_combination a₂ * hab₁ + hab₂ | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Analysis.Convex.Join | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case inr.refine_1\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 u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\n... | [] | linear_combination a₂ * hab₁ + hab₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Join | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case inr.refine_1\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 u : Set E\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ t\na₁ b₁ : 𝕜\nha₁ : 0 ≤ a₁\nhb₁ : 0 ≤ b₁\nhab₁ : a₁ + b₁ = 1\nz : E\nhz : z ∈ u\n... | [] | linear_combination a₂ * hab₁ + hab₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 78
} | {
"line": 204,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformS... | [] | simpa only [average_eq_integral] using hg.map_average_le hgc hsc hfs hfi hgi | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Integral | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 78
} | {
"line": 204,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformS... | [] | simpa only [average_eq_integral] using hg.map_average_le hgc hsc hfs hfi hgi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Integral | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 78
} | {
"line": 204,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ng : E → ℝ\ninst✝ : IsProbabilityMeasure μ\nhg : ConvexOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformS... | [] | simpa only [average_eq_integral] using hg.map_average_le hgc hsc hfs hfi hgi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 169,
"column": 7
} | {
"line": 169,
"column": 31
} | {
"line": 171,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_9\ninst✝ : IsStrictOrderedRing R\nx : StdSimplex R M\na✝ : M\n⊢ (single x).join.weights a✝ = x.weights a✝",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.smulZeroClass"... | [] | simp [join, ← mk_single] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Convex.ConvexSpace.Defs | {
"line": 208,
"column": 41
} | {
"line": 208,
"column": 68
} | {
"line": 208,
"column": 68
} | [
{
"pp": "X : Type u_2\nK : Type u_8\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : IsDomain K\nw : StdSimplex K X\ns : Set X\nhs : ∃ x ∈ s, w.weights x ≠ 0\ninst✝ : DecidablePred fun x ↦ x ∈ s\nthis : ((filter (fun x ↦ x ∈ s) w.weights).sum fun x k ↦ k) ≠ 0\n⊢ ((filter (... | [] | by convert inv_ne_zero this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Convex.Set | {
"line": 200,
"column": 42
} | {
"line": 200,
"column": 77
} | {
"line": 200,
"column": 77
} | [
{
"pp": "K : Type u_4\nX : Type u_5\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : ConvexSpace K X\ns : Set X\nhs : ∀ (a b : K) (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1), ∀ x ∈ s, ∀ y ∈ s, convexCombPair a b ha hb hab x y ∈ s\nt✝ : Finset X\nx : X\nt : Finset X\nhx : x ∉ t\... | [] | by simpa [hy] using congr(y ∈ $hsw) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 101,
"column": 16
} | {
"line": 101,
"column": 63
} | {
"line": 102,
"column": 4
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| t * dist x y",
"ppT... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| dist x y - s * dist x y"
] | rw [eq_sub_iff_add_eq'.mpr h, sub_mul, one_mul] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 101,
"column": 16
} | {
"line": 101,
"column": 63
} | {
"line": 102,
"column": 4
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| t * dist x y",
"ppT... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| dist x y - s * dist x y"
] | rw [eq_sub_iff_add_eq'.mpr h, sub_mul, one_mul] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 101,
"column": 16
} | {
"line": 101,
"column": 63
} | {
"line": 102,
"column": 4
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| t * dist x y",
"ppT... | [
"X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ns t : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : X\nH :\n ∀ {s t : ℝ} (hs : 0 ≤ s) (ht : 0 ≤ t) (h : s + t = 1) (x y : X),\n dist (convexCombPair s t hs ht h x y) x ≤ t * dist x y\n| dist x y - s * dist x y"
] | rw [eq_sub_iff_add_eq'.mpr h, sub_mul, one_mul] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.Convex.Radon | {
"line": 60,
"column": 20
} | {
"line": 60,
"column": 76
} | {
"line": 60,
"column": 76
} | [
{
"pp": "ι : Type u_1\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\nf : ι → E\ns : Finset ι\nw : ι → 𝕜\nh_wsum : s.sum w = 0\nh_vsum : ∑ e ∈ s, w e • f e = 0\nnonzero_w_index : ι\nh1 : nonzero_w_index ∈ s... | [] | by simp only [I, mem_filter, h1', h2'.le, and_self, h2'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 186,
"column": 2
} | {
"line": 188,
"column": 77
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns : Finset E\nhs : s ∈ K.faces\nhs' : ¬(s ∈ K.faces ∧ ∀ ⦃t : Finset E⦄, t ∈ K.faces → s ⊆ t → s = t)\n⊢ ∃ t ∈ K.faces, s ⊂ t",
"ppTerm": ... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns : Finset E\nhs : s ∈ K.faces\n⊢ (∃ t ∈ K.faces, s ⊂ t) → s ∉ K.facets"
] | · push Not at hs'
obtain ⟨t, ht⟩ := hs' hs
exact ⟨t, ht.1, ⟨ht.2.1, fun hts => ht.2.2 (Subset.antisymm ht.2.1 hts)⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Radon | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 18
} | {
"line": 254,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : Set (Set E)\nh_convex : ∀ X ∈ F, Convex 𝕜 X\nh_compact : ∀ X ∈ F... | [
"case a\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : Set (Set E)\nh_convex : ∀ X ∈ F, Convex 𝕜 X\nh_compact : ∀ X ∈ F, Is... | apply h_inter G' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 34
} | {
"line": 349,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G ≠ ⊥ ↔ ∃ a b, G.Adj a b",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Classical.not_not._simp_1",
"congrArg",
"SimpleGraph.Adj",
"Exists",
"Ne",
"Bot.bot",
"_private.Mathlib.Combinatorics.SimpleGraph.Bas... | [] | simp [eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 34
} | {
"line": 349,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G ≠ ⊥ ↔ ∃ a b, G.Adj a b",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Classical.not_not._simp_1",
"congrArg",
"SimpleGraph.Adj",
"Exists",
"Ne",
"Bot.bot",
"_private.Mathlib.Combinatorics.SimpleGraph.Bas... | [] | simp [eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 34
} | {
"line": 349,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G ≠ ⊥ ↔ ∃ a b, G.Adj a b",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Classical.not_not._simp_1",
"congrArg",
"SimpleGraph.Adj",
"Exists",
"Ne",
"Bot.bot",
"_private.Mathlib.Combinatorics.SimpleGraph.Bas... | [] | simp [eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 26
} | {
"line": 832,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\n⊢ w ∈ G.neighborSet v ∪ Gᶜ.neighborSet v ↔ w ∈ {v}ᶜ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"SimpleGraph.Adj",
"SimpleGraph.ne_of_adj",
"Ne"
],
"usedFVars": [
"V",
"G"
],
"usedGoals": [
... | [
"V : Type u\nG : SimpleGraph V\nv w : V\nh : ∀ {a b : V}, G.Adj a b → a ≠ b\n⊢ w ∈ G.neighborSet v ∪ Gᶜ.neighborSet v ↔ w ∈ {v}ᶜ"
] | have h := @ne_of_adj _ G | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Visible | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 73
} | {
"line": 127,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module 𝕜 V\ns : Set V\nx y : V\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : Co... | [
"𝕜 : Type u_1\nV : Type u_2\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module 𝕜 V\ns : Set V\nx y : V\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : OrderTopology 𝕜\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul... | suffices h : ∀ᶠ (_δ : 𝕜) in 𝓝[>] 0, False by obtain ⟨_, ⟨⟩⟩ := h.exists | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Convex.Side | {
"line": 888,
"column": 2
} | {
"line": 889,
"column": 47
} | {
"line": 890,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : Fin (n + 1)\... | let w₃ : Fin (n + 1) → R :=
w₂ - w₂ i • (w₁ i)⁻¹ • (w₁ - Pi.single j 1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.Visible | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 55
} | {
"line": 172,
"column": 55
} | [
{
"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... | [
"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 t\nhδ₁ : δ ... | rintro hδ₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Side | {
"line": 988,
"column": 2
} | {
"line": 994,
"column": 83
} | {
"line": 996,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j,... | [] | · by_cases h0 : w₁ i = 0
· exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0)
· by_cases h0' : w₂ i = 0
· exact wSameSide_of_right_mem _
((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).2 h0')
simp only [h0, h0', or_self, or_false] at h... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 15
} | {
"line": 343,
"column": 2
} | [
{
"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 <... | [] | exact ⟨⟨s⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 51
} | {
"line": 500,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\n⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\n⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}"
] | rw [lintegral_eq_lintegral_meas_le μ f_nn f_mble] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 56
} | {
"line": 535,
"column": 4
} | [
{
"pp": "case e'_3.refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nkey : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}\nlhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞\nrhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ... | [
"case e'_3.refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nkey : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}\nlhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞\nrhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f ... | · exact Eventually.of_forall (fun x ↦ by positivity) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 524,
"column": 6
} | {
"line": 524,
"column": 11
} | {
"line": 525,
"column": 6
} | [
{
"pp": "case coe.inr\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\nhlk : ↑l ≤ ↑k * ↑p\n⊢ ∃ k... | [
"case h\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⋯\nhlk : ↑l ≤ ↑k * ↑p\n⊢ eLpNorm (fun x ↦ (1 + ‖x‖) ^ (... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 526,
"column": 8
} | {
"line": 526,
"column": 48
} | {
"line": 527,
"column": 8
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\nhlk : ↑l ≤ ↑k * ↑p\nthis : HasFiniteInt... | [
"E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\nhlk : ↑l ≤ ↑k * ↑p\nthis : ∫⁻ (a : E), ‖(1 + ‖a‖) ^... | rw [hasFiniteIntegral_iff_enorm] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 692,
"column": 42
} | {
"line": 697,
"column": 26
} | {
"line": 699,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nΩ : Opens E\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nn : ℕ∞\nm : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : Measure E\nH : LocallyIntegrableOn (fun x ↦ 1) (↑Ω) μ\nf : 𝓓^{n}(Ω, F)\n⊢ Integr... | [] | by
rw [← integrableOn_iff_integrable_of_support_subset (subset_tsupport f)]
replace H := H.integrableOn_compact_subset f.tsupport_subset f.hasCompactSupport
suffices IntegrableOn ((1 : ℝ) • f) (tsupport f) μ by simpa
rw [IntegrableOn, ← memLp_one_iff_integrable] at H ⊢
exact f.memLp_top.smul H | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 516,
"column": 4
} | {
"line": 516,
"column": 36
} | {
"line": 517,
"column": 4
} | [
{
"pp": "case coe\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nx✝ : ↑p = 0 ∨ ↑p ≠ 0\n⊢ ∃ k, eLpNorm (fun x ↦ (1 + ‖x‖) ^ (-↑k)) (↑p) μ < ∞",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.a... | [] | cases eq_or_ne (p : ℝ≥0∞) 0 with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Analysis.Normed.Operator.Compact.Basic | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 61
} | {
"line": 430,
"column": 2
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NormedField 𝕜₂\nσ₁₂ : 𝕜₁ →+* 𝕜₂\nM₁ : Type u_3\nM₂ : Type u_4\ninst✝⁸ : SeminormedAddCommGroup M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : NormedSpace 𝕜₁ M₁\ninst✝⁵ : Module 𝕜₂ M₂\ninst✝⁴ : UniformSpace M₂\ninst✝³ : IsUnifo... | [] | exact this.closure.isCompact_of_isClosed isClosed_closure | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 203,
"column": 8
} | {
"line": 203,
"column": 49
} | {
"line": 204,
"column": 4
} | [
{
"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.smul_mem _ μ (hf_mem hmn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 203,
"column": 8
} | {
"line": 203,
"column": 49
} | {
"line": 204,
"column": 4
} | [
{
"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.smul_mem _ μ (hf_mem hmn) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 203,
"column": 8
} | {
"line": 203,
"column": 49
} | {
"line": 204,
"column": 4
} | [
{
"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.smul_mem _ μ (hf_mem hmn) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 149,
"column": 2
} | {
"line": 152,
"column": 56
} | {
"line": 154,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\ni : n\n⊢ hA.eigenvalues i = RCLike.re (star (hA.eigenvectorBasis i).ofLp ⬝ᵥ A *ᵥ (hA.eigenvectorBasis i).ofLp)",
"ppTerm": "?m.45",
"assigned": true,
"usedConst... | [] | rw [dotProduct_comm]
simp only [mulVec_eigenvectorBasis, smul_dotProduct, ← EuclideanSpace.inner_eq_star_dotProduct,
inner_self_eq_norm_sq_to_K, RCLike.smul_re, hA.eigenvectorBasis.orthonormal.1 i,
mul_one, algebraMap.coe_one, one_pow, RCLike.one_re] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 149,
"column": 2
} | {
"line": 152,
"column": 56
} | {
"line": 154,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\ni : n\n⊢ hA.eigenvalues i = RCLike.re (star (hA.eigenvectorBasis i).ofLp ⬝ᵥ A *ᵥ (hA.eigenvectorBasis i).ofLp)",
"ppTerm": "?m.45",
"assigned": true,
"usedConst... | [] | rw [dotProduct_comm]
simp only [mulVec_eigenvectorBasis, smul_dotProduct, ← EuclideanSpace.inner_eq_star_dotProduct,
inner_self_eq_norm_sq_to_K, RCLike.smul_re, hA.eigenvectorBasis.orthonormal.1 i,
mul_one, algebraMap.coe_one, one_pow, RCLike.one_re] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 406,
"column": 56
} | {
"line": 407,
"column": 61
} | {
"line": 409,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : 𝕜\nv : E\nT : E →ₗ[𝕜] E\nh : T v = μ • v\n⊢ ⟪v, T v⟫ = μ * ↑‖v‖ ^ 2",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Norm.norm",
"InnerProductSpace.toN... | [] | by
simp only [h, inner_smul_right, inner_self_eq_norm_sq_to_K] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.Order | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 70
} | {
"line": 89,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nx✝³ x✝² : Matrix n n 𝕜\nx✝¹ : x✝³ ≤ x✝²\nx✝ : Matrix n n 𝕜\n⊢ x✝³ + x✝ ≤ x✝² + x✝",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matrix.le_iff",
"AddGroupWithOne.toAddGroup",
"congrArg",
"M... | [] | by rwa [le_iff, add_sub_add_right_eq_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 58
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case hf\nα : 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 : ↥(Lp E 2 μ)\n⊢ 0 ≤ᵐ[μ] fun x ↦ ‖↑↑f x‖ ^ 2",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"MeasureT... | [
"case hfm\nα : 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 : ↥(Lp E 2 μ)\n⊢ AEStronglyMeasurable (fun x ↦ ‖↑↑f x‖ ^ 2) μ",
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : Measu... | · exact Filter.Eventually.of_forall fun x => sq_nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions | {
"line": 155,
"column": 25
} | {
"line": 155,
"column": 52
} | {
"line": 155,
"column": 52
} | [
{
"pp": "case e'_3\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SecondCountableTopologyEither α E\ninst✝⁵ : CompactSpace α\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p... | [
"case e'_3\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁹ : TopologicalSpace α\ninst✝⁸ : BorelSpace α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SecondCountableTopologyEither α E\ninst✝⁵ : CompactSpace α\ninst✝⁴ : IsFiniteMeasure μ\n𝕜 : Type u_3\ninst✝³ : Fact (1 ≤ p)\ninst✝² : ... | ContinuousLinearMap.coe_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 206,
"column": 2
} | {
"line": 210,
"column": 7
} | {
"line": 212,
"column": 0
} | [
{
"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\ns : Set α\nf : ↥(Lp E 2 μ)\nhs : MeasurableSet s\nc : E\nhμs : μ s ≠ ∞\n⊢ ⟪indicatorConstLp 2 hs hμs c, f⟫ = ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ"... | [] | rw [inner_def, ← integral_indicator hs]
refine integral_congr_ae ((@indicatorConstLp_coeFn _ _ _ 2 μ _ s hs hμs c).mono fun x hx ↦ ?_)
have : ⟪indicatorConstLp 2 hs hμs c x, f x⟫ = s.indicator (fun x ↦ ⟪c, f x⟫) x := by
by_cases hxs : x ∈ s <;> simp [hx, hxs]
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 206,
"column": 2
} | {
"line": 210,
"column": 7
} | {
"line": 212,
"column": 0
} | [
{
"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\ns : Set α\nf : ↥(Lp E 2 μ)\nhs : MeasurableSet s\nc : E\nhμs : μ s ≠ ∞\n⊢ ⟪indicatorConstLp 2 hs hμs c, f⟫ = ∫ (x : α) in s, ⟪c, ↑↑f x⟫ ∂μ"... | [] | rw [inner_def, ← integral_indicator hs]
refine integral_congr_ae ((@indicatorConstLp_coeFn _ _ _ 2 μ _ s hs hμs c).mono fun x hx ↦ ?_)
have : ⟪indicatorConstLp 2 hs hμs c x, f x⟫ = s.indicator (fun x ↦ ⟪c, f x⟫) x := by
by_cases hxs : x ∈ s <;> simp [hx, hxs]
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 74
} | {
"line": 105,
"column": 4
} | [
{
"pp": "case neg\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst... | [] | · simpa [hx'] using this.and (f_cont.continuousAt.eventually_ne hx') | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 35
} | {
"line": 98,
"column": 36
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : AddCircle T → E\n⊢ ∫ (t : AddCircle T), f t ∂haarAddCircle = T⁻¹ • ∫ (t : AddCircle T), f t",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCircle.volume_e... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : AddCircle T → E\n⊢ ∫ (t : AddCircle T), f t ∂haarAddCircle = T⁻¹ • ∫ (t : AddCircle T), f t ∂ENNReal.ofReal T • haarAddCircle"
] | volume_eq_smul_haarAddCircle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 33
} | {
"line": 308,
"column": 34
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nn : ℤ\na : ℝ\nthis : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x\n⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle = (1 / T) • ∫ (b : AddCircle ... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nn : ℤ\na : ℝ\nthis : ∀ (x : ℝ), (fourier (-n)) ↑x • f ↑x = (fun z ↦ (fourier (-n)) z • f z) ↑x\n⊢ ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle =\n (1 / T) • ∫ (b : AddCircle T), (fo... | volume_eq_smul_haarAddCircle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 56
} | {
"line": 357,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nK : Submodule 𝕜 E\nhK : CompleteSpace ↥K\nthis : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)\n⊢ ⊤ ≤ (⨆ i, bif i then K else Kᗮ).topologicalClosure",
"pp... | [
"𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nK : Submodule 𝕜 E\nhK : CompleteSpace ↥K\nthis : ∀ (b : Bool), CompleteSpace ↥(bif b then K else Kᗮ)\n⊢ ⊤ ≤ ⨆ i, bif i then K else Kᗮ"
] | refine le_trans ?_ (Submodule.le_topologicalClosure _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 568,
"column": 2
} | {
"line": 571,
"column": 97
} | {
"line": 572,
"column": 2
} | [
{
"pp": "a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\n⊢ 1 / ↑(b - a) * ∫ (x : ℝ) in a..b, f x * (fourier (-n)) ↑x =\n 1 / (-2 * ↑π ... | [
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\n⊢ 1 / ↑(b - a) *\n (f b * (↑(b - a) / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑b) -\n ... | rw [integral_mul_deriv_eq_deriv_mul_of_hasDeriv_right hf
(fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.continuousAt |>.continuousWithinAt) hff'
(fun x _ ↦ has_antideriv_at_fourier_neg hT hn x |>.hasDerivWithinAt) hf'
(((map_continuous (fourier (-n))).comp (AddCircle.continuous_mk' _)).intervalIntegrabl... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 58
} | {
"line": 114,
"column": 8
} | [
{
"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\nint_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume\n⊢ ∫ (x : ℝ) in Iic 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x",
"ppTerm": "?m.363",
"assigned": true,
"usedConstants": [
... | [
"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\nint_Iic : IntegrableOn (fun x ↦ f |x|) (Iic 0) volume\n⊢ ∫ (x : ℝ) in Iic 0, f |x| = ∫ (x : ℝ) in Iic 0, f (-x)"
] | rw [← neg_zero, ← integral_comp_neg_Iic, neg_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 64
} | {
"line": 145,
"column": 4
} | [
{
"pp": "s t : ℝ\nht : 0 < t\nh : -1 ≤ s\nH : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume\nH' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume\n⊢ ∀ᵐ (a : ℝ) ∂volume.restrict (Ioi (max 1 t)), ‖a⁻¹‖ ≤ a ^ s",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
... | [
"s t : ℝ\nht : 0 < t\nh : -1 ≤ s\nH : IntegrableOn (fun x ↦ x ^ s) (Ioi t) volume\nH' : IntegrableOn (fun x ↦ x ^ s) (Ioi (max 1 t)) volume\nx : ℝ\nhx : x ∈ Ioi (max 1 t)\n⊢ ‖x⁻¹‖ ≤ x ^ s"
] | filter_upwards [ae_restrict_mem measurableSet_Ioi] with x hx | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 197,
"column": 90
} | {
"line": 198,
"column": 41
} | {
"line": 200,
"column": 0
} | [
{
"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\nk n : ℕ\nf : 𝓢(E, F)\nx : E\n⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (-⇑f) x‖ = ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (⇑f) x‖",
"ppTerm": "?m.52",
"assigned": true,
... | [] | by
rw [iteratedFDeriv_neg_apply, norm_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 338,
"column": 11
} | {
"line": 338,
"column": 38
} | {
"line": 338,
"column": 39
} | [
{
"pp": "case h\nι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : 𝓢(E, F)\nk n : ℕ\nx : E\n⊢ ‖x‖ ^ k * ‖iterated... | [
"case h\nι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf g : 𝓢(E, F)\nk n : ℕ\nx : E\n⊢ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n (... | decay_add_le_aux k n f g x, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 38
} | {
"line": 636,
"column": 6
} | [
{
"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✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace ℝ E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace ℝ F\ninst✝¹⁰ : NormedField 𝕜\ninst✝⁹ : NormedField 𝕜'\ninst... | [
"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✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace ℝ E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace ℝ F\ninst✝¹⁰ : NormedField 𝕜\ninst✝⁹ : NormedField 𝕜'\ninst✝⁸ : NormedA... | rcases hbound with ⟨s, C, hC, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 79
} | {
"line": 158,
"column": 0
} | [
{
"pp": "s : ℂ\nX : ℝ\nhs : 0 < s.re\nhX : 0 ≤ X\n⊢ IntervalIntegrable (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) volume 0 X",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"HMul.h... | [] | rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hX]
exact IntegrableOn.mono_set (GammaIntegral_convergent hs) Ioc_subset_Ioi_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 79
} | {
"line": 158,
"column": 0
} | [
{
"pp": "s : ℂ\nX : ℝ\nhs : 0 < s.re\nhX : 0 ≤ X\n⊢ IntervalIntegrable (fun x ↦ ↑(rexp (-x)) * ↑x ^ (s - 1)) volume 0 X",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"HMul.h... | [] | rw [intervalIntegrable_iff_integrableOn_Ioc_of_le hX]
exact IntegrableOn.mono_set (GammaIntegral_convergent hs) Ioc_subset_Ioi_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 35
} | {
"line": 59,
"column": 6
} | [
{
"pp": "case right.inl\nx y : ℝ\nhxy : (x, y) ∈ {q | 0 < q.1}\n⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0",
"ppTerm": "?right.inl",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Equiv.instEquivLike",
"Real.instZer... | [
"case right.inr\nx y : ℝ\nhxy : (x, y) ∈ {q | q.2 ≠ 0}\n⊢ 0 ≤ (Complex.equivRealProd.symm (x, y)).re ∨ (Complex.equivRealProd.symm (x, y)).im ≠ 0"
] | · exact Or.inl (le_of_lt hxy) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 510,
"column": 8
} | {
"line": 510,
"column": 21
} | {
"line": 511,
"column": 8
} | [
{
"pp": "case hs\nn✝ : ℕ\nn_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0\ns : ℝ\nhs' : -↑(n✝ + 1) < s\nm : ℕ\nhs : s ≠ -↑(1 + m)\n⊢ s + 1 ≠ -↑m",
"ppTerm": "?hs",
"assigned": true,
"usedConstants": [
"Real",
"Mathlib.Tactic.Contrapose.contrapose₄",
"instOfNatNat",
... | [
"case hs\nn✝ : ℕ\nn_ih : ∀ {s : ℝ}, (∀ (m : ℕ), s ≠ -↑m) → -↑n✝ < s → Gamma s ≠ 0\ns : ℝ\nhs' : -↑(n✝ + 1) < s\nm : ℕ\nhs : s + 1 = -↑m\n⊢ s = -↑(1 + m)"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 58
} | {
"line": 122,
"column": 0
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\nc : ℝ\n⊢ Tendsto (HMul.hMul b.re) atTop atTop",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"HMul.hMul",
"Filter.tendsto_id",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"DistribL... | [] | exact (tendsto_const_mul_atTop_of_pos hb).mpr tendsto_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 56
} | {
"line": 306,
"column": 2
} | [
{
"pp": "V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ‖(𝓕 f).toBoundedContinuousFunction‖ ≤ ‖f.toLp 1 volume‖"... | [
"V : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : InnerProductSpace ℝ V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : 𝓢(V, F)\n⊢ ∀ (x : V), ‖(𝓕 f).toBoundedContinuousFunction x‖ ≤ ‖f.toLp 1 volume‖... | rw [BoundedContinuousFunction.norm_le (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 53
} | {
"line": 82,
"column": 4
} | [
{
"pp": "d : Type u_1\ninst✝¹ : Fintype d\ninst✝ : DecidableEq d\nz : d → AddCircle 1\ni : d\nthis :\n (fourier (Pi.single i 1 i)) (z i) * ∏ i_1 ∈ {i}ᶜ, (fourier (Pi.single i 1 i_1)) (z i_1) =\n ∏ i_1, (fourier (Pi.single i 1 i_1)) (z i_1)\nj : d\nhj : j ∈ {i}ᶜ\n⊢ (fourier (Pi.single i 1 j)) (z j) = 1",
... | [
"d : Type u_1\ninst✝¹ : Fintype d\ninst✝ : DecidableEq d\nz : d → AddCircle 1\ni : d\nthis :\n (fourier (Pi.single i 1 i)) (z i) * ∏ i_1 ∈ {i}ᶜ, (fourier (Pi.single i 1 i_1)) (z i_1) =\n ∏ i_1, (fourier (Pi.single i 1 i_1)) (z i_1)\nj : d\nhj : ¬j = i\n⊢ (fourier (Pi.single i 1 j)) (z j) = 1"
] | rw [Finset.mem_compl, Finset.mem_singleton] at hj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 38
} | {
"line": 130,
"column": 2
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nx : V →ᵇ ℂ\nhx : x ∈ (ch... | [
"V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nx : V →ᵇ ℂ\nhx : ∃ x_1, (charAlgHom ... | simp only [AlgHom.mem_range] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 217,
"column": 29
} | {
"line": 217,
"column": 56
} | {
"line": 217,
"column": 56
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : Is... | [
"E : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\ninst✝⁴ : CompleteSpace F\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : IsLocallyFinit... | ContinuousLinearMap.coe_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 20
} | {
"line": 85,
"column": 4
} | [
{
"pp": "f : C(ℝ, ℂ)\nhf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nm : ℤ\ne : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }\nneK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖... | [] | simp_rw [eadd] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 20
} | {
"line": 91,
"column": 6
} | [
{
"pp": "f : C(ℝ, ℂ)\nhf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nm : ℤ\ne : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }\nneK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖... | [
"f : C(ℝ, ℂ)\nhf : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nm : ℤ\ne : C(ℝ, ℂ) := (fourier (-m)).comp { toFun := QuotientAddGroup.mk, continuous_toFun := ⋯ }\nneK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖ContinuousMap.restrict (↑K) (e * g)‖ = ‖ContinuousMa... | simp_rw [eadd] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 454,
"column": 40
} | {
"line": 454,
"column": 57
} | {
"line": 454,
"column": 58
} | [
{
"pp": "R : Type u_4\nk : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ∑ i ∈ range k.pred, ζ ^ i + ζ ^ k.pred = 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"AddGroupWithOne.toAd... | [
"R : Type u_4\nk : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nhζ : IsPrimitiveRoot ζ k\nhk : 1 < k\n⊢ ∑ x ∈ range (k.pred + 1), ζ ^ x = 0"
] | ← sum_range_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 476,
"column": 12
} | {
"line": 476,
"column": 64
} | {
"line": 476,
"column": 65
} | [
{
"pp": "case left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh✝ h : IsPrimitiveRoot ζ k\ni : ZMod k\nhi :\n ↑((((Int.castAddHom (ZMod k)).liftOfRightInver... | [
"case left\nM : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh✝ h : IsPrimitiveRoot ζ k\ni : ZMod k\nhi :\n ↑((((Int.castAddHom (ZMod k)).liftOfRightInverse ZMod.cast... | ← (CharP.intCast_eq_zero_iff (ZMod k) k _).mpr this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 31
} | {
"line": 173,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nζ : K\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nprod_monic : (∏ i ∈ n.properDivisors, cyclotomic' i K).Monic\n⊢ (∏ i ∈ n.properDivisors, cyclotomic' i K) * cyclotomic' n K =\n cyclotomic' n K * ∏ i ∈ n.properDivisors, cyclotomic' i K ∧\n ... | [
"K : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nζ : K\nn : ℕ\nhpos : 0 < n\nh : IsPrimitiveRoot ζ n\nprod_monic : (∏ i ∈ n.properDivisors, cyclotomic' i K).Monic\n⊢ ⊥ < (∏ i ∈ n.properDivisors, cyclotomic' i K).degree"
] | refine ⟨by rw [mul_comm], ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 622,
"column": 4
} | {
"line": 622,
"column": 43
} | {
"line": 623,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_7\ninst✝¹ : CommSemiring S\ninst✝ : Algebra S R\nζ₁ ζ₂ : R\nk₁ k₂ : ℕ\nhζ₁ : IsPrimitiveRoot ζ₁ k₁\nhζ₂ : IsPrimitiveRoot ζ₂ k₂\nhk₁ : k₁ ≠ 0\nhk₂ : k₂ ≠ 0\nζ : R\nhζ : IsPrimitiveRoot ζ (k₁.lcm k₂)\nthis : NeZero (k₁.lcm... | [
"case refine_1.refine_1\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nS : Type u_7\ninst✝¹ : CommSemiring S\ninst✝ : Algebra S R\nζ₁ ζ₂ : R\nk₁ k₂ : ℕ\nhζ₁ : IsPrimitiveRoot ζ₁ k₁\nhζ₂ : IsPrimitiveRoot ζ₂ k₂\nhk₁ : k₁ ≠ 0\nhk₂ : k₂ ≠ 0\nζ : R\nhζ : IsPrimitiveRoot ζ (k₁.lcm k₂)\nthis : NeZero (k₁.lcm k₂... | refine Set.pair_subset_iff.mpr ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 202,
"column": 2
} | {
"line": 206,
"column": 86
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (... | [
"case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.castRing... | have huniq : 0 + B * cyclotomic' k K = X ^ k - 1 ∧ (0 : K[X]).degree < B.degree := by
constructor
· rw [zero_add, mul_comm, ← prod_cyclotomic'_eq_X_pow_sub_one hpos h, ←
Nat.cons_self_properDivisors hpos.ne', Finset.prod_cons]
· simpa only [degree_zero, bot_lt_iff_ne_bot, Ne, degree_eq_bot] using Bm... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 285,
"column": 2
} | {
"line": 286,
"column": 78
} | {
"line": 287,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\n⊢ cyclotomic 1 R = X - 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Polynomial.map_one",
"Polynomial.instOne",
"Complex.commRing",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
"Polynomial.cyclotom... | [
"R : Type u_1\ninst✝ : Ring R\nhspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ\n⊢ cyclotomic 1 R = X - 1"
] | have hspec : map (Int.castRingHom ℂ) (X - 1) = cyclotomic' 1 ℂ := by
simp only [cyclotomic'_one, map_X, Polynomial.map_one, Polynomial.map_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 794,
"column": 38
} | {
"line": 794,
"column": 83
} | {
"line": 794,
"column": 83
} | [
{
"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✝ : ... | [] | by simpa only [rootsOfUnity.coe_pow] using! h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 89
} | {
"line": 129,
"column": 6
} | [
{
"pp": "case refine_1\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : ¬n = 0\nhpos : 0 < n\nP : ℤ[X] := minpoly ℤ μ\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhdiff : ¬P = Q\nPmonic : P.Monic\nQmoni... | [
"case refine_1\nn : ℕ\nK : Type u_1\ninst✝² : CommRing K\nμ : K\nh : IsPrimitiveRoot μ n\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nhprime : Fact (Nat.Prime p)\nhdiv : ¬p ∣ n\nhn : ¬n = 0\nhpos : 0 < n\nP : ℤ[X] := minpoly ℤ μ\nQ : ℤ[X] := minpoly ℤ (μ ^ p)\nhdiff : ¬P = Q\nPmonic : P.Monic\nQmonic : Q.Monic\... | have aux := IsPrimitive.Int.irreducible_iff_irreducible_map_cast Pmonic.isPrimitive | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 42
} | {
"line": 85,
"column": 2
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : (cyclotomic n K).IsRoot μ\nhμn : μ ^ n = 1\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : (cyclotomic i K).IsRoot μ\nhio : i ∣ orderOf μ\n⊢ False",
"ppTerm": "?m.129",
"assigned": true,
"usedConstan... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : (cyclotomic n K).IsRoot μ\nhμn : orderOf μ ∣ n\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : (cyclotomic i K).IsRoot μ\nhio : i ∣ orderOf μ\n⊢ False"
] | rw [← orderOf_dvd_iff_pow_eq_one] at hμn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 212,
"column": 4
} | {
"line": 216,
"column": 39
} | {
"line": 218,
"column": 0
} | [] | [] | n.totient = (primitiveRoots n K).card := h.card_primitiveRoots.symm
_ ≤ P_K.roots.toFinset.card := Finset.card_le_card (is_roots_of_minpoly h)
_ ≤ Multiset.card P_K.roots := Multiset.toFinset_card_le _
_ ≤ P_K.natDegree := card_roots' _
_ ≤ P.natDegree := natDegree_map_le | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 54
} | {
"line": 177,
"column": 2
} | [
{
"pp": "K : Type u_1\nS : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra K S\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : Algebra R S\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K S\nB B' : PowerBasis K S\nP : R[X]\nh : (aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B... | [
"K : Type u_1\nS : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra K S\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : Algebra R S\ninst✝¹ : Algebra R K\ninst✝ : IsScalarTower R K S\nB B' : PowerBasis K S\nP : R[X]\nh : (aeval B.gen) P = B'.gen\nhB : IsIntegral R B.gen\nhmin : minpoly K B.gen = Polyn... | refine repr_pow_isIntegral hB (fun i => ?_) hmin _ _ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 63
} | {
"line": 64,
"column": 6
} | [
{
"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\n⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)",
"ppTerm": "?inr.refine_1.refine_3",
"assigned": true,
"usedConstants": [
"instHDiv",
"Real.p... | [
"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⊢ cyclotomic n ℚ ∣ (expand ℚ p) (cyclotomic n ℚ)"
] | have hprim := Complex.isPrimitiveRoot_exp _ hnpos.ne.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 13
} | {
"line": 238,
"column": 4
} | [
{
"pp": "case inl\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = n\nineq : φ n * φ r ≤ φ (GCDMonoid... | [
"case inl\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = n\nineq : φ n * φ r ≤ φ (GCDMonoid.lcm l n)\nr... | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 167,
"column": 4
} | {
"line": 168,
"column": 57
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case refine_2\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Subsingleton B\nthis : Subsingleton (Subalgebra A B)\nhS : S ⊆ {0, 1}\ns : ℕ\nhs : s ∈ S\nhs' : s ≠ 0\n⊢ ∃ r, IsPrimitiveRoot r s",
"ppTerm": "?refine_2",
"assigned": true,
... | [] | · have : s = 1 := (subset_pair_iff.mp hS s hs).resolve_left hs'
exact ⟨0, this ▸ IsPrimitiveRoot.of_subsingleton 0⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 12
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case pos\nn : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : CommRing L\nζ : L\ninst✝³ : Field K\ninst✝² : Algebra K L\nhζ : IsPrimitiveRoot ζ n\ninst✝¹ : IsDomain L\ninst✝ : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic n K)\nhn : n = 2\n⊢ (Algebra.norm K) ζ = -1",
"ppTerm"... | [
"case pos\nK : Type u\nL : Type v\ninst✝⁵ : CommRing L\nζ : L\ninst✝⁴ : Field K\ninst✝³ : Algebra K L\ninst✝² : IsDomain L\ninst✝¹ : NeZero 2\nhζ : IsPrimitiveRoot ζ 2\ninst✝ : IsCyclotomicExtension {2} K L\nhirr : Irreducible (cyclotomic 2 K)\n⊢ (Algebra.norm K) ζ = -1"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 87
} | {
"line": 352,
"column": 4
} | [
{
"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... | refine Fintype.prod_equiv (hζ.embeddingsEquivPrimitiveRoots E hirr) _ _ fun σ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 56
} | {
"line": 366,
"column": 2
} | [
{
"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\nhn : IsPrimePow n\ninst✝ : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic n K)\nh : n ≠ 2\nthis : succ 1 < n\nhprime : Fact (Nat.Prime n.minFac) :=... | [
"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\nhn : IsPrimePow n\ninst✝ : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic n K)\nh : n ≠ 2\nthis : succ 1 < n\nhprime : Fact (Nat.Prime n.minFac) := { out := mi... | nth_rw 1 [← IsPrimePow.minFac_pow_factorization_eq hn] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 80
} | {
"line": 374,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n} A B\n⊢ Module.Finite A B",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"La... | [
"n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n} A B\n⊢ (Subalgebra.toSubmodule (adjoin A {b | ∃ n_1 ∈ {n}, n_1 ≠ 0 ∧ b ^ n_1 = 1})).FG"
] | rw [Module.finite_def, ← top_toSubmodule, ← ((iff_adjoin_eq_top _ _ _).1 h).2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ZMod.Units | {
"line": 105,
"column": 18
} | {
"line": 105,
"column": 31
} | {
"line": 105,
"column": 32
} | [
{
"pp": "N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d = ↑(d * a₀) * p + ↑(d * N₀) * q\n⊢ IsCoprime ↑a₀ ↑N₀",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"NonAssocSemiring... | [
"N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d = ↑d * ↑a₀ * p + ↑(d * N₀) * q\n⊢ IsCoprime ↑a₀ ↑N₀"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ZMod.Units | {
"line": 105,
"column": 32
} | {
"line": 105,
"column": 45
} | {
"line": 105,
"column": 46
} | [
{
"pp": "N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d = ↑d * ↑a₀ * p + ↑(d * N₀) * q\n⊢ IsCoprime ↑a₀ ↑N₀",
"ppTerm": "?m.261",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.... | [
"N : ℕ\na : ZMod N\nhN : N ≠ 0\nthis : NeZero N\nd : ℕ := a.val.gcd N\nhd : d ≠ 0\na₀ : ℕ\nha₀ : a.val = d * a₀\nN₀ : ℕ\nhN₀ : N = d * N₀\np q : ℤ\nhpq : ↑d = ↑d * ↑a₀ * p + ↑d * ↑N₀ * q\n⊢ IsCoprime ↑a₀ ↑N₀"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
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