module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Content | {
"line": 177,
"column": 33
} | {
"line": 177,
"column": 51
} | {
"line": 177,
"column": 51
} | [
{
"pp": "case e'_3\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nU : ℕ → Opens G\nh3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ i, U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact ... | [
"case e'_3\nG : Type w\ninst✝¹ : TopologicalSpace G\nμ : Content G\ninst✝ : R1Space G\nU : ℕ → Opens G\nh3 : ∀ (t : Finset ℕ) (K : ℕ → Compacts G), μ (t.sup K) ≤ ∑ i ∈ t, μ (K i)\nK : Compacts G\nhK : ↑K ⊆ ↑(⨆ i, U i)\nt : Finset ℕ\nht : ↑K ⊆ ⋃ i ∈ t, ↑(U i)\nK' : ℕ → Set G\nh1K' : ∀ (i : ℕ), IsCompact (K' i)\nh2K'... | Finset.sup_eq_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 577,
"column": 4
} | {
"line": 577,
"column": 8
} | {
"line": 578,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b < a\n⊢ 0 = ofReal... | [
"case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b < a\n⊢ ofReal (↑f b - leftLim... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 584,
"column": 4
} | {
"line": 584,
"column": 8
} | {
"line": 585,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b ≤ a\n⊢ 0 = ofReal... | [
"case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b ≤ a\n⊢ ofReal (leftLim (↑f) b... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 8
} | {
"line": 601,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b ≤ a\n⊢ 0 = ofReal... | [
"case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : b ≤ a\n⊢ ofReal (leftLim (↑f) b... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 603,
"column": 4
} | {
"line": 604,
"column": 85
} | {
"line": 606,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\nA : Disjoint... | [] | simp [← Icc_union_Ioo_eq_Ico le_rfl hab, -singleton_union, f.mono.leftLim_le,
measure_union A measurableSet_Ioo, f.mono.le_leftLim hab, ← ENNReal.ofReal_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 860,
"column": 6
} | {
"line": 860,
"column": 56
} | {
"line": 860,
"column": 56
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nf_int : Integrable f μ\nf_nonneg : 0 ≤ᵐ[μ] f\ns : Set X\nhs : ∀ x ∈ s, 1 ≤ f x\n⊢ μ s ≤ ENNReal.ofReal (∫ (x : X), f x ∂μ)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTh... | [
"X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nf_int : Integrable f μ\nf_nonneg : 0 ≤ᵐ[μ] f\ns : Set X\nhs : ∀ x ∈ s, 1 ≤ f x\n⊢ μ s ≤ ∫⁻ (x : X), ENNReal.ofReal (f x) ∂μ"
] | ofReal_integral_eq_lintegral_ofReal f_int f_nonneg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 170,
"column": 64
} | {
"line": 170,
"column": 72
} | {
"line": 170,
"column": 73
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : (interior V).Nonempty\ns : Finset G\nh1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀\nh2s : s.card = index ↑K ↑K₀\nt : Finset G\nh1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ ... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK : Compacts G\nV : Set G\nhV : (interior V).Nonempty\ns : Finset G\nh1s : ↑K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' ↑K₀\nh2s : s.card = index ↑K ↑K₀\nt : Finset G\nh1t : ↑K₀ ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V... | intro g₂ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 183,
"column": 39
} | {
"line": 183,
"column": 43
} | {
"line": 184,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : (interior V).Nonempty\nh1t : ∅ ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}\ng : G\nhg : g ∈ interior ↑K\n⊢ ∅ = ⋃ g ∈ ∅, (fun h ↦ g * h) ⁻¹' V",
"ppTerm": "?m.142",
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : (interior V).Nonempty\nh1t : ∅ ∈ {t | ↑K ⊆ ⋃ g ∈ t, (fun h ↦ g * h) ⁻¹' V}\ng : G\nhg : g ∈ interior ↑K\n⊢ ⋃ g ∈ ∅, (fun h ↦ g * h) ⁻¹' V = ∅"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 751,
"column": 2
} | {
"line": 751,
"column": 25
} | {
"line": 752,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhfg : f.meas... | [] | cases le_total x y with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 6
} | {
"line": 219,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx : E\n⊢ re ⟪x, x⟫ = ‖↑(re ⟪x, x⟫)‖",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"AddMonoid.toAddSemigroup",
"Inner.i... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx : E\n⊢ ‖↑(re ⟪x, x⟫)‖ = re ⟪x, x⟫"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 26
} | {
"line": 397,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f K₂ - f K₁\nthis : Continuous eval\nU : Set G\nleft✝ : U ⊆ ↑⊤.toOpens\nh2U : IsOpen[inst✝¹] U\nh3U : 1 ∈ U\n⊢ pr... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\neval : (Compacts G → ℝ) → ℝ := ⋯\nthis : Continuous eval\nU : Set G\nleft✝ : U ⊆ ↑⊤.toOpens\nh2U : IsOpen[inst✝¹] U\nh3U : 1 ∈ U\n⊢ (interior U).Nonempty"
] | apply prehaar_mono _ h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 8
} | {
"line": 360,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhz : ∀ (i : ι), v i ≠ 0\nho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\ns : Finset ι\ng : ι → 𝕜\nhg : ∑ i ∈ s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ g i * ⟪v i, v i⟫ = ∑ i_1 ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhz : ∀ (i : ι), v i ≠ 0\nho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\ns : Finset ι\ng : ι → 𝕜\nhg : ∑ i ∈ s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ ∑ i_1 ∈ s, ⟪v i, g i_1 • v i_1⟫ = g i... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 18
} | {
"line": 430,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁\nh2U₂ :... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\nK₁ K₂ : Compacts G\nh : Disjoint K₁.carrier K₂.carrier\nh₂ : IsClosed[inst✝¹] K₂.carrier\nU₁ U₂ : Set G\nh1U₁ : IsOpen[inst✝¹] U₁\nh1U₂ : IsOpen[inst✝¹] U₂\nh2U₁ : K₁.carrier ⊆ U₁\nh2U₂ : K₂.carrier ... | let V := V₁ ∩ V₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 96,
"column": 12
} | {
"line": 98,
"column": 18
} | {
"line": 99,
"column": 12
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nne : K.Nonempty\nh₁ : IsComplete K\nh₂ : Convex ℝ K\nu : F\nδ : ℝ := ⨅ w, ‖u - ↑w‖\nthis : Nonempty ↑K := Set.Nonempty.to_subtype ne\nzero_le_δ : 0 ≤ δ\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖\n... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nne : K.Nonempty\nh₁ : IsComplete K\nh₂ : Convex ℝ K\nu : F\nδ : ℝ := ⨅ w, ‖u - ↑w‖\nthis : Nonempty ↑K := Set.Nonempty.to_subtype ne\nzero_le_δ : 0 ≤ δ\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖\nw : ℕ → ↑K\n... | have eq₂ : u + u - (wq + wp) = a + b := by
change u + u - (wq + wp) = u - wq + (u - wp)
abel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 475,
"column": 76
} | {
"line": 478,
"column": 54
} | {
"line": 480,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ‖x‖ = 0\n⊢ ⟪x, y⟫ = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"norm_eq_zero",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | [] | by
rw [← norm_eq_zero]
refine le_antisymm ?_ (by positivity)
exact norm_inner_le_norm _ _ |>.trans <| by simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1196,
"column": 8
} | {
"line": 1197,
"column": 28
} | {
"line": 1198,
"column": 2
} | [
{
"pp": "case e_a\nY : Type u_2\nE : Type u_3\nF : Type u_4\nX : Type u_5\nG : Type u_6\n𝕜 : Type u_7\ninst✝¹¹ : TopologicalSpace X\ninst✝¹⁰ : TopologicalSpace Y\ninst✝⁹ : MeasurableSpace Y\ninst✝⁸ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst... | [] | refine (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
simp [hfs q y hq hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1196,
"column": 8
} | {
"line": 1197,
"column": 28
} | {
"line": 1198,
"column": 2
} | [
{
"pp": "case e_a\nY : Type u_2\nE : Type u_3\nF : Type u_4\nX : Type u_5\nG : Type u_6\n𝕜 : Type u_7\ninst✝¹¹ : TopologicalSpace X\ninst✝¹⁰ : TopologicalSpace Y\ninst✝⁹ : MeasurableSpace Y\ninst✝⁸ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst... | [] | refine (setIntegral_eq_integral_of_forall_compl_eq_zero (fun y hy ↦ ?_)).symm
simp [hfs q y hq hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 73,
"column": 25
} | {
"line": 73,
"column": 38
} | {
"line": 73,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nh : Orthonormal 𝕜 v\ni : ι\n⊢ ‖v i‖ₑ = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNor... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nh : Orthonormal 𝕜 v\ni : ι\n⊢ ENNReal.ofReal ‖v i‖ = 1"
] | ← ofReal_norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 66
} | {
"line": 179,
"column": 4
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nh : Convex ℝ K\nu v : F\nhv : v ∈ K\nthis✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩\neq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nw : F\nhw : w ∈ K\nδ : ℝ := ⨅ w, ‖u - ↑w‖\np : ℝ := ⟪u - v, w - v⟫_ℝ\nq : ℝ := ‖w - v‖ ^ 2\nδ_le : ∀ (... | [] | exact le_of_sub_nonneg (nonneg_of_mul_nonneg_right this hθ₁) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 90,
"column": 21
} | {
"line": 90,
"column": 74
} | {
"line": 92,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhv : Orthonormal 𝕜 v\ni j : ι\nhij : i ≠ j\na : (fun _i ↦ 𝕜) i\nb : (fun _i ↦ 𝕜) j\n⊢ ⟪((fun i ↦ LinearIsometry.toSpanSingleton 𝕜 E ⋯) i) a, ((fun i ↦ LinearI... | [] | by simp [inner_smul_left, inner_smul_right, hv.2 hij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 10
} | {
"line": 296,
"column": 6
} | [
{
"pp": "case him\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := ⋯\nK' : Submodule ℝ E := ⋯\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w ∈ ... | [
"case him\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := ⋯\nK' : Submodule ℝ E := ⋯\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w ∈ K\n⊢ im 0 = ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 286,
"column": 57
} | {
"line": 286,
"column": 73
} | {
"line": 286,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\ny : E\nhy : T y = S (S v)\n⊢ ⟪y, 0⟫ = 0",
"ppTerm": "?m.159",
"assigned": true,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\ny : E\nhy : T y = S (S v)\n⊢ 0 = 0"
] | inner_zero_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 34
} | {
"line": 303,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\nhUV : IsCompl U V\nh : U ⟂ V\nx y : E\n⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, (U.projection V hUV) y + (V.projection U ⋯) y⟫ =\n ⟪(U.projection V ... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\nhUV : IsCompl U V\nh : ∀ u ∈ U, ∀ v ∈ V, ⟪u, v⟫ = 0\nx y : E\n⊢ ⟪(U.subtype ∘ₗ U.projectionOnto V hUV) x, (U.projection V hUV) y + (V.projection U ⋯) y⟫ =\n ⟪(U.pr... | rw [isOrtho_iff_inner_eq] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 8
} | {
"line": 163,
"column": 4
} | [
{
"pp": "case zero\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nφ : F ≃ₗᵢ[ℝ] F\nhn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ 0\nthis : (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker = ⊤\n⊢ φ = 1",
"ppTerm": "?zero",
"assigned": t... | [
"case zero\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nφ : F ≃ₗᵢ[ℝ] F\nhn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ 0\nthis : (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker = ⊤\n⊢ 1 = φ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 643,
"column": 6
} | {
"line": 643,
"column": 22
} | {
"line": 643,
"column": 22
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\nh : K.starProjection v = 0\nw : E\nhw : w ∈ K\n⊢ ⟪w, 0⟫ = 0",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\nh : K.starProjection v = 0\nw : E\nhw : w ∈ K\n⊢ 0 = 0"
] | inner_zero_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 928,
"column": 4
} | {
"line": 932,
"column": 91
} | {
"line": 934,
"column": 0
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p ... | [] | · have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0
rw [prod_nnnorm_eq_add hpt, prod_nnnorm_eq_add hpt, one_div, NNReal.rpow_inv_eq_iff hp0.ne',
NNReal.mul_rpow, ← NNReal.rpow_mul, inv_mul_cancel₀ hp0.ne', NNReal.rpow_one, mul_add,
← NNReal... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 766,
"column": 29
} | {
"line": 766,
"column": 52
} | {
"line": 768,
"column": 0
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp 1 β\n⊢ dist x y = ∑ x_1, dist (x.ofLp x_1) (y.ofLp x_1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"PiLp.dist_eq_of_L1"
],
"usedFVars": [
"ι",
... | [] | exact dist_eq_of_L1 _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 77
} | {
"line": 143,
"column": 0
} | [
{
"pp": "a r : ℝ\n⊢ volume (closedBall a r) = ofReal (2 * r)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"sub_add",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"cong... | [] | rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 77
} | {
"line": 143,
"column": 0
} | [
{
"pp": "a r : ℝ\n⊢ volume (closedBall a r) = ofReal (2 * r)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"sub_add",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"cong... | [] | rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 77
} | {
"line": 143,
"column": 0
} | [
{
"pp": "a r : ℝ\n⊢ volume (closedBall a r) = ofReal (2 * r)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"sub_add",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"cong... | [] | rw [closedBall_eq_Icc, volume_Icc, ← sub_add, add_sub_cancel_left, two_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 1024,
"column": 6
} | {
"line": 1024,
"column": 18
} | {
"line": 1025,
"column": 4
} | [
{
"pp": "case top.refine_1.inr\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝ : DecidableEq ι\ni : ι\nb : β i\nthis : Nonempty ι\nhp : Fact (1 ≤ ∞)\nj : ι\nhij : i ≠ j\n⊢ ‖(single ∞ i b).ofLp j‖₊ ≤ ‖b‖₊",
"ppTerm": "?top.refine_1.inr",
"assign... | [] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 1029,
"column": 62
} | {
"line": 1029,
"column": 86
} | {
"line": 1030,
"column": 6
} | [
{
"pp": "case coe\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝ : DecidableEq ι\ni : ι\nb : β i\nthis : Nonempty ι\np : ℝ≥0\nhp : Fact (1 ≤ ↑p)\nhp0 : ↑p ≠ 0\n⊢ (∑ i_1, ‖(single (↑p) i b).ofLp i_1‖₊ ^ ↑p) ^ (1 / ↑p) = ‖b‖₊",
"ppTerm": "?coe",
... | [
"case coe\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝ : DecidableEq ι\ni : ι\nb : β i\nthis : Nonempty ι\np : ℝ≥0\nhp : Fact (1 ≤ ↑p)\nhp0 : ↑p ≠ 0\n⊢ (‖(single (↑p) i b).ofLp i‖₊ ^ ↑p) ^ (1 / ↑p) = ‖b‖₊",
"case coe\nι : Type u_2\nβ : ι → Type u_4\ni... | Fintype.sum_eq_single i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 27
} | {
"line": 114,
"column": 6
} | [
{
"pp": "case h₂\nι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b\nF : ℝ → Fin 1 → ℝ := ⋯\n⊢ F '' Icc 0 1 ⊆ Icc 0 1",
"ppTerm": "?h₂",
"assigned": true,
"usedConstants": [
"Real",
"Pi.preorder",
"Real... | [
"case h₂\nι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b\nF : ℝ → Fin 1 → ℝ := fun t _i ↦ t\ny : ℝ\nhy : y ∈ Icc 0 1\n⊢ F y ∈ Icc 0 1"
] | rintro x ⟨y, hy, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 149,
"column": 48
} | {
"line": 149,
"column": 73
} | {
"line": 149,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nu : ℕ → E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjo... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nu : ℕ → E\nsb : Bornology.IsBounded s\nhu : Bornology.IsBounded (range u)\nhs : Pairwise (Disjoint on fun n... | iUnion_singleton_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 33
} | {
"line": 382,
"column": 33
} | [
{
"pp": "n : ℕ\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nx : ℝ\nhx : x ∈ Ioc (I.lower i) (I.upper i)\ny : Fin n → ℝ\nhy : y ∈ ↑(I.face i)\n⊢ x ∈ Ioc (I.lower i) (I.upper i) ∧\n ∀ (i_1 : Fin n), i.insertNth x y (i.succAbove i_1) ∈ Ioc (I.lower (i.succAbove i_1)) (I.upper (i.succAbove i_1))",
"ppTerm": "?m.... | [
"n : ℕ\nI : Box (Fin (n + 1))\ni : Fin (n + 1)\nx : ℝ\nhx : x ∈ Ioc (I.lower i) (I.upper i)\ny : Fin n → ℝ\nhy : y ∈ ↑(I.face i)\n⊢ x ∈ Ioc (I.lower i) (I.upper i) ∧ ∀ (i_1 : Fin n), y i_1 ∈ Ioc (I.lower (i.succAbove i_1)) (I.upper (i.succAbove i_1))"
] | Fin.insertNth_apply_succAbove | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 466,
"column": 23
} | {
"line": 466,
"column": 58
} | {
"line": 466,
"column": 58
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\ni : ι\n⊢ nndist I.lower I.upper / nndist (I.lower i) (I.upper i) * nndist (I.lower i) (I.upper i) ≤\n (Finset.univ.sup fun i ↦ nndist I.lower I.upper / nndist (I.lower i) (I.upper i)) * nndist (I.lower i) (I.upper i)",
"ppTerm": "?m.82",
"assigned"... | [
"ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\ni : ι\n⊢ nndist I.lower I.upper / nndist (I.lower i) (I.upper i) * nndist (I.lower i) (I.upper i) ≤\n nndist I.lower I.upper / nndist (I.lower i) (I.upper i) * nndist (I.lower i) (I.upper i)"
] | ← Finset.le_sup (Finset.mem_univ i) | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 60,
"column": 2
} | {
"line": 63,
"column": 28
} | {
"line": 65,
"column": 0
} | [
{
"pp": "case pos\nι : Type u_1\nI : Box ι\ns : Set ι\ny : ι → ℝ\ni : ι\nhs : i ∈ s\n⊢ y i ∈ Ioc ((I.lower i + I.upper i) / 2) (I.upper i) ↔\n y i ∈ Ioc (I.lower i) (I.upper i) ∧ (I.lower i + I.upper i) / 2 < y i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Se... | [] | exacts [⟨fun H ↦ ⟨⟨(left_lt_add_div_two.2 (I.lower_lt_upper i)).trans H.1, H.2⟩, H.1⟩,
fun H ↦ ⟨H.2, H.1.2⟩⟩,
⟨fun H ↦ ⟨⟨H.1, H.2.trans (add_div_two_lt_right.2 (I.lower_lt_upper i)).le⟩, H.2⟩,
fun H ↦ ⟨H.1.1, H.2⟩⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 487,
"column": 47
} | {
"line": 498,
"column": 50
} | {
"line": 500,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\n⊢ μ (closedBall 0 1) = μ (ball 0 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [] | by
apply le_antisymm _ (measure_mono ball_subset_closedBall)
have A : Tendsto
(fun r : ℝ => ENNReal.ofReal (r ^ finrank ℝ E) * μ (closedBall (0 : E) 1)) (𝓝[<] 1)
(𝓝 (ENNReal.ofReal ((1 : ℝ) ^ finrank ℝ E) * μ (closedBall (0 : E) 1))) := by
refine ENNReal.Tendsto.mul ?_ (by simp) tendsto_const_nh... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 415,
"column": 64
} | {
"line": 415,
"column": 81
} | {
"line": 416,
"column": 4
} | [
{
"pp": "E : Type u_1\nι : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝³ : Fintype ι\ninst✝² : MeasurableSpace E\nμ : Measure E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nthis : FiniteDimensional ℝ E\nx : E\nhx : x ∈ {x | ∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.rep... | [
"E : Type u_1\nι : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝³ : Fintype ι\ninst✝² : MeasurableSpace E\nμ : Measure E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nthis : FiniteDimensional ℝ E\nx : E\nhx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ x ∉ fu... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 353,
"column": 2
} | {
"line": 356,
"column": 9
} | {
"line": 358,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\n⊢ Orthonormal 𝕜 fun i ↦ single i 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Euc... | [] | simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,
PiLp.single_apply]
intros
trivial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 353,
"column": 2
} | {
"line": 356,
"column": 9
} | {
"line": 358,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\n⊢ Orthonormal 𝕜 fun i ↦ single i 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Euc... | [] | simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,
PiLp.single_apply]
intros
trivial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 670,
"column": 2
} | {
"line": 676,
"column": 33
} | {
"line": 677,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt u : Set E\nh'u... | have A : Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t')) / μ ({x} + r • u')) (𝓝[>] 0) (𝓝 0) := by
apply tendsto_addHaar_inter_smul_zero_of_density_zero_aux1 μ s x h t' u'
· simp only [u', h'u, (pow_pos Rpos _).ne', abs_nonpos_iff, addHaar_smul, not_false_iff,
ENNReal.ofReal_eq_zero, inv_eq_zero, inv_pow,... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 708,
"column": 4
} | {
"line": 712,
"column": 19
} | {
"line": 713,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (�... | [] | calc
μ (s ∩ ({x} + r • t)) ≤ μ ({x} + r • t) := measure_mono inter_subset_right
_ = 0 := by
simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,
mul_zero] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 54
} | {
"line": 378,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc : ℝ≥0\nl : IntegrationParams\nπ : TaggedPrepartition I\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhπ : l.MemBaseSet I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : π₁.iUnion = ↑I \\ π.iUnion\nhc : π₁.distortion ≤ c\nπ₂ : TaggedPrepartiti... | [
"case refine_1\nι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nc : ℝ≥0\nl : IntegrationParams\nπ : TaggedPrepartition I\nr : (ι → ℝ) → ↑(Set.Ioi 0)\nhπ : l.MemBaseSet I c r π\np : Box ι → Prop\nhD : l.bDistortion = true\nπ₁ : Prepartition I\nhπ₁U : π₁.iUnion = ↑I \\ π.iUnion\nhc : π₁.distortion ≤ c\nπ₂ : TaggedPrepart... | refine ⟨π₁.disjUnion π₂.toPrepartition this, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Oscillation | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 50
} | {
"line": 147,
"column": 6
} | [
{
"pp": "case neg\nE : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nS_open : ∀ ... | [
"case neg\nE : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nS_open : ∀ r > 0, IsOpe... | rw [not_nonempty_iff_eq_empty] at T_nonempty | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 85
} | {
"line": 71,
"column": 85
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA : μ (s ∩ Box.Icc I)... | [
"ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nhl : l.bRiemann = false\ns : Set (ι → ℝ)\nhs : MeasurableSet s\nI : Box ι\ny : E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nε : ℝ≥0\nε0 : 0 < ε\nA : μ (s ∩ Box.Icc I) ≠ ∞\nB : μ ... | refine ⟨fun _ => r, fun c => l.rCond_of_bRiemann_eq_false hl, fun c π hπ hπp => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 77,
"column": 2
} | {
"line": 90,
"column": 45
} | {
"line": 92,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nh : IsBounded s\n⊢ ∃ B, hasIntegralVertices B ∧ s ⊆ ↑B",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
... | [] | have := Fintype.ofFinite ι
obtain ⟨R, hR₁, hR₂⟩ := IsBounded.subset_ball_lt h 0 0
let C : ℕ := ⌈R⌉₊
have hC := Nat.ceil_pos.mpr hR₁
let I : Box ι := Box.mk (fun _ ↦ -C) (fun _ ↦ C)
(fun _ ↦ by simp [C, neg_lt_self_iff, Nat.cast_pos, hC])
refine ⟨I, ⟨fun _ ↦ - C, fun _ ↦ C, fun i ↦ (Int.cast_neg_natCast C)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 77,
"column": 2
} | {
"line": 90,
"column": 45
} | {
"line": 92,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Finite ι\ns : Set (ι → ℝ)\nh : IsBounded s\n⊢ ∃ B, hasIntegralVertices B ∧ s ⊆ ↑B",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
... | [] | have := Fintype.ofFinite ι
obtain ⟨R, hR₁, hR₂⟩ := IsBounded.subset_ball_lt h 0 0
let C : ℕ := ⌈R⌉₊
have hC := Nat.ceil_pos.mpr hR₁
let I : Box ι := Box.mk (fun _ ↦ -C) (fun _ ↦ C)
(fun _ ↦ by simp [C, neg_lt_self_iff, Nat.cast_pos, hC])
refine ⟨I, ⟨fun _ ↦ - C, fun _ ↦ C, fun i ↦ (Int.cast_neg_natCast C)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 178,
"column": 8
} | {
"line": 178,
"column": 39
} | {
"line": 178,
"column": 40
} | [
{
"pp": "case e_a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nhf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0\ni j : ι\nhj : j ∈ Finset.... | [
"case e_a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nhf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0\ni j : ι\nhj : j ∈ Finset.Iio i\n⊢ ↑((... | Submodule.starProjection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 178,
"column": 40
} | {
"line": 178,
"column": 61
} | {
"line": 178,
"column": 61
} | [
{
"pp": "case e_a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nhf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0\ni j : ι\nhj : j ∈ Finset.... | [
"case e_a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nhf : Pairwise fun x1 x2 ↦ ⟪f x1, f x2⟫_𝕜 = 0\ni j : ι\nhj : j ∈ Finset.Iio i\n⊢ (𝕜... | Submodule.coe_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 121,
"column": 4
} | {
"line": 123,
"column": 25
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case succ\nι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc... | [] | · refine (measure_mono_null ?_ hf).le.trans_lt ?_
· exact fun x hxN hxf => n.succ_ne_zero ((Eq.symm hxN).trans <| N0.2 hxf)
· simp [(δ0 _).ne'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 223,
"column": 48
} | {
"line": 223,
"column": 65
} | {
"line": 223,
"column": 66
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nν : ι → ℤ\n⊢ ν ∈ {ν | ↑(box n ν) ⊆ ↑B} ↔ box n ν ≤ B",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"BoxIntegral.Box.toSet",
"setOf",
"Membershi... | [
"ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nν : ι → ℤ\n⊢ ↑(box n ν) ⊆ ↑B ↔ box n ν ≤ B"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 264,
"column": 2
} | {
"line": 271,
"column": 40
} | {
"line": 273,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f)",
"ppTe... | [] | unfold Orthonormal
constructor
· simp only [gramSchmidtNormed_unit_length, h₀, imp_true_iff]
· intro i j hij
simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, RCLike.conj_inv,
RCLike.conj_ofReal, mul_eq_zero, inv_eq_zero, RCLike.ofReal_eq_zero, norm_eq_zero]
repeat' right
exact gr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 264,
"column": 2
} | {
"line": 271,
"column": 40
} | {
"line": 273,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ Orthonormal 𝕜 (gramSchmidtNormed 𝕜 f)",
"ppTe... | [] | unfold Orthonormal
constructor
· simp only [gramSchmidtNormed_unit_length, h₀, imp_true_iff]
· intro i j hij
simp only [gramSchmidtNormed, inner_smul_left, inner_smul_right, RCLike.conj_inv,
RCLike.conj_ofReal, mul_eq_zero, inv_eq_zero, RCLike.ofReal_eq_zero, norm_eq_zero]
repeat' right
exact gr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 298,
"column": 81
} | {
"line": 299,
"column": 66
} | {
"line": 301,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\n⊢ span 𝕜 (Set.range (gramSchmidtNormed 𝕜 f)) = span 𝕜 (Set.range (gramSchmidt �... | [] | by
simpa only [image_univ.symm] using span_gramSchmidtNormed f univ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Orientation | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 74
} | {
"line": 302,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : LinearOrder R\ninst✝⁵ : IsStrictOrderedRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : Nonempty ι\ne : Basis ι R M\nx : Orientation R M ι\ni : ι\nh : ¬e.orientatio... | [] | · by_cases hi : i = Classical.arbitrary ι <;> simp [unitsSMul_apply, hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 32
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case zero.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (-positiveOrientation).volumeForm = -positiveOrientation.volumeForm",
"ppTerm": "?zero.refine_1",
"assigned": true,
"usedConstants": [
"AlternatingMap.instAdd... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 32
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case zero.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (-positiveOrientation).volumeForm = -positiveOrientation.volumeForm",
"ppTerm": "?zero.refine_1",
"assigned": true,
"usedConstants": [
"AlternatingMap.instAdd... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 32
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case zero.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (-positiveOrientation).volumeForm = -positiveOrientation.volumeForm",
"ppTerm": "?zero.refine_1",
"assigned": true,
"usedConstants": [
"AlternatingMap.instAdd... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 32
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case zero.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (- -positiveOrientation).volumeForm = -(-positiveOrientation).volumeForm",
"ppTerm": "?zero.refine_2",
"assigned": true,
"usedConstants": [
"AlternatingMap.in... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 32
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case zero.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (- -positiveOrientation).volumeForm = -(-positiveOrientation).volumeForm",
"ppTerm": "?zero.refine_2",
"assigned": true,
"usedConstants": [
"AlternatingMap.in... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 32
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case zero.refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\n⊢ (- -positiveOrientation).volumeForm = -(-positiveOrientation).volumeForm",
"ppTerm": "?zero.refine_2",
"assigned": true,
"usedConstants": [
"AlternatingMap.in... | [] | simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 224,
"column": 2
} | {
"line": 226,
"column": 32
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case zero\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\n_i : Fact (finrank ℝ E = 0)\no : Orientation ℝ E (Fin 0)\n⊢ (-o).volumeForm = -o.volumeForm",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"AlternatingMap.instAddCommGroup",
"Altern... | [
"case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\n⊢ (-o).volumeForm = -o.volumeForm"
] | · refine o.eq_or_eq_neg_of_isEmpty.elim ?_ ?_ <;> rintro rfl
· simp [volumeForm_zero_neg]
· simp [volumeForm_zero_neg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 8
} | {
"line": 231,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\ne : OrthonormalBasis (Fin n.succ) ℝ E := Orientation.finOrthonormalBasis ⋯ ⋯ o\nh₁ : e.toBasis.orientation = o\n⊢ e.toBasis.orientation ≠ -o",
"ppTe... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\ne : OrthonormalBasis (Fin n.succ) ℝ E := ⋯\nh₁ : e.toBasis.orientation = o\n⊢ -o ≠ e.toBasis.orientation"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 266,
"column": 9
} | {
"line": 266,
"column": 14
} | {
"line": 266,
"column": 14
} | [
{
"pp": "case refine_2\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nl : IntegrationParams\nhl : l.bRiemann = false\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : ... | [
"case refine_2\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nl : IntegrationParams\nhl : l.bRiemann = false\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E... | ← hcε | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 8
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case e'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nx : Fin (n + 1) → E\nthis : FiniteDimensional ℝ E\n⊢ o = (map (Fin (n + 1)) φ.toLine... | [
"case e'_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nφ : E ≃ₗᵢ[ℝ] E\nhφ : 0 < LinearMap.det ↑φ.toLinearEquiv\nn : ℕ\n_i : Fact (finrank ℝ E = n + 1)\no : Orientation ℝ E (Fin (n + 1))\nx : Fin (n + 1) → E\nthis : FiniteDimensional ℝ E\n⊢ (map (Fin (n + 1)) φ.toLinearEquiv) o = o"
... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 55
} | {
"line": 161,
"column": 4
} | [
{
"pp": "ι : Type u_2\ninst✝⁴ : Fintype ι\nL₁ L₂ : Submodule ℤ (ι → ℝ)\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ≤ L₂\nb₁ : Basis ι ℤ ↥L₁ := ⋯\nb₂ : Basis ι ℤ ↥L₂ := ⋯\n⊢ |(Basis.ofZLatticeBasis ℝ L₂ b₂).det ⇑(Basis.ofZLatticeBasis ℝ... | [
"ι : Type u_2\ninst✝⁴ : Fintype ι\nL₁ L₂ : Submodule ℤ (ι → ℝ)\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ≤ L₂\nb₁ : Basis ι ℤ ↥L₁ := IsZLattice.basis L₁\nb₂ : Basis ι ℤ ↥L₂ := IsZLattice.basis L₂\n⊢ |((Basis.ofZLatticeBasis ℝ L₂ b₂).toMa... | rw [Basis.det_apply, Basis.det_apply, Int.cast_det] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 250,
"column": 54
} | {
"line": 250,
"column": 71
} | {
"line": 250,
"column": 72
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nι : Type u_2\ninst✝ : Fintype ι\nX : Set E\nhX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X\nF : E → ℝ\nhF₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x\nc : ℝ\nhc : 0 < c\nx : E\n⊢ c⁻¹ • x ∈ {x | x ∈ X ∧ F x ≤ 1} ↔ x... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nι : Type u_2\ninst✝ : Fintype ι\nX : Set E\nhX : ∀ ⦃x : E⦄ ⦃r : ℝ⦄, x ∈ X → 0 < r → r • x ∈ X\nF : E → ℝ\nhF₁ : ∀ (x : E) ⦃r : ℝ⦄, 0 ≤ r → F (r • x) = r ^ card ι * F x\nc : ℝ\nhc : 0 < c\nx : E\n⊢ c⁻¹ • x ∈ X ∧ F (c⁻¹ • x) ≤ 1 ↔ x ∈ X ∧ F x ≤ c... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 8
} | {
"line": 139,
"column": 4
} | [
{
"pp": "case refine_1\nM : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni : ι\nx : R[M]\nhx : 0 ∈ gradeBy R (⇑f) i\n⊢ (decomposeAux f) ↑⟨0, hx⟩ = (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨0, hx⟩",
"ppTerm... | [
"case refine_1\nM : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni : ι\nx : R[M]\nhx : 0 ∈ gradeBy R (⇑f) i\n⊢ (DirectSum.of (fun i ↦ ↥(gradeBy R (⇑f) i)) i) ⟨0, hx⟩ = (decomposeAux f) ↑⟨0, hx⟩"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 55
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset f... | [
"case neg\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc (... | replace hd : 1 ≤ d := by rwa [Nat.one_le_iff_ne_zero] | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 182,
"column": 4
} | {
"line": 183,
"column": 34
} | {
"line": 184,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nh✝ : Nontrivial ↥L\nI : Type u_1 := Free.ChooseBasisIndex ℤ ↥L\nthis : Fintype I\nb : Basis I ℤ ↥L := Free.chooseBasis ℤ ↥L\nd : ℕ := Fintype.card I\nhd : d ≠ 0\nε ... | simp only [EmbeddingLike.apply_eq_iff_eq, implies_true, Set.injOn_of_eq_iff_eq,
Finset.sum_image, ge_iff_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 359,
"column": 2
} | {
"line": 361,
"column": 55
} | {
"line": 362,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nL : Submodule ℤ E\ninst✝¹ : DiscreteTopology ↥L\ninst✝ : IsZLattice ℝ L\ns : Set E\nhs₁ : Bornology.IsBounded s\nhs₂ : MeasurableSet... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nL : Submodule ℤ E\ninst✝¹ : DiscreteTopology ↥L\ninst✝ : IsZLattice ℝ L\ns : Set E\nhs₁ : Bornology.IsBounded s\nhs₂ : MeasurableSet s\nhs₃ : volume (front... | · simp only [measureReal_def]
rw [volume_image_eq_volume_div_covolume' L b hs₂.nullMeasurableSet, ENNReal.toReal_div,
ENNReal.toReal_ofReal (covolume_pos L volume).le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.List.GetD | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 66
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nl : List α\nd : α\nn : ℕ\nf : α → β\n⊢ (map f l).getD n (f d) = f (l.getD n d)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"Option.getD",
"List.instGetElem?NatLtLength",
"List.getD",
"List.getEl... | [] | simp only [getD_eq_getElem?_getD, getElem?_map, Option.getD_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.GetD | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 66
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nl : List α\nd : α\nn : ℕ\nf : α → β\n⊢ (map f l).getD n (f d) = f (l.getD n d)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"Option.getD",
"List.instGetElem?NatLtLength",
"List.getD",
"List.getEl... | [] | simp only [getD_eq_getElem?_getD, getElem?_map, Option.getD_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.GetD | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 66
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nl : List α\nd : α\nn : ℕ\nf : α → β\n⊢ (map f l).getD n (f d) = f (l.getD n d)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"Option.getD",
"List.instGetElem?NatLtLength",
"List.getD",
"List.getEl... | [] | simp only [getD_eq_getElem?_getD, getElem?_map, Option.getD_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 36
} | {
"line": 71,
"column": 4
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nυ : Type u_3\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial τ R\ng : MvPolynomial τ R →ₐ[R] MvPolynomial υ R\nx : υ → R\ni : σ\n⊢ (eval₂Hom ((eval₂Hom (algebraMap R R) x).comp (algebraMap R (MvPolynomial υ R))) fun i ↦\n (eval₂Hom (alge... | [
"σ : Type u_1\nτ : Type u_2\nυ : Type u_3\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial τ R\ng : MvPolynomial τ R →ₐ[R] MvPolynomial υ R\nx : υ → R\ni : σ\n⊢ (eval₂Hom (algebraMap R R) x).comp (algebraMap R (MvPolynomial υ R)) = algebraMap R R"
] | refine eval₂Hom_congr ?_ rfl rfl | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 209,
"column": 11
} | {
"line": 209,
"column": 23
} | {
"line": 209,
"column": 24
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni j : σ →₀ ℕ\n⊢ (1 = 0 ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Finsupp.instLE",
"False",
... | [
"σ : Type u_1\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\ni j : σ →₀ ℕ\n⊢ (False ∨ i ≤ j) ∧ 1 ∣ 1 ↔ i ≤ j"
] | one_ne_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 242,
"column": 35
} | {
"line": 242,
"column": 57
} | {
"line": 242,
"column": 57
} | [
{
"pp": "case e'_3.e'_5.e'_1\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nm : Multiset α\n⊢ m.card = (Finsupp.toMultiset (toFinsupp m)).card",
"ppTerm": "?e'_3.e'_5.e'_1",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"Fins... | [
"case e'_3.e'_5.e'_1\nα : Type u_1\ninst✝ : DecidableEq α\na : α\nm : Multiset α\n⊢ m.card = m.card"
] | m.toFinsupp_toMultiset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 60,
"column": 2
} | {
"line": 72,
"column": 70
} | {
"line": 74,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\n⊢ IsUnit P ↔ IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"NonUnitalNo... | [] | refine ⟨fun H ↦ ⟨H.map constantCoeff, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· intro n hn
obtain ⟨i, hi⟩ : ∃ i, n i ≠ 0 := by simpa [Finsupp.ext_iff] using hn
let e := (optionEquivLeft _ _).symm.trans (renameEquiv R (Equiv.optionSubtypeNe i))
have H := (Polynomial.coeff_isUnit_isNilpotent_of_isUnit (H.map e.symm)).2 (n ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 60,
"column": 2
} | {
"line": 72,
"column": 70
} | {
"line": 74,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\n⊢ IsUnit P ↔ IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"NonUnitalNo... | [] | refine ⟨fun H ↦ ⟨H.map constantCoeff, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· intro n hn
obtain ⟨i, hi⟩ : ∃ i, n i ≠ 0 := by simpa [Finsupp.ext_iff] using hn
let e := (optionEquivLeft _ _).symm.trans (renameEquiv R (Equiv.optionSubtypeNe i))
have H := (Polynomial.coeff_isUnit_isNilpotent_of_isUnit (H.map e.symm)).2 (n ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Expand | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 30
} | {
"line": 213,
"column": 2
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nf : MvPolynomial σ R\nhf : IsUnit (coeff 0 f) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i ((expand p) f))\n⊢ IsUnit (coeff 0 f) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i f)",
"ppTerm": "?m.39",
"assigned": true,
"use... | [
"R : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\np : ℕ\nhp : p ≠ 0\nf : MvPolynomial σ R\nhf : IsUnit (coeff 0 f) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i ((expand p) f))\ni : σ →₀ ℕ\nhi : i ≠ 0\n⊢ IsNilpotent (coeff i f)"
] | refine ⟨hf.1, fun i hi ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 380,
"column": 6
} | {
"line": 382,
"column": 42
} | {
"line": 384,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, 2 ^ f i * (f i)! ≤\n ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, (2 * f i)!",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
... | [] | gcongr
rw [← doubleFactorial_two_mul]
exact doubleFactorial_le_factorial _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 380,
"column": 6
} | {
"line": 382,
"column": 42
} | {
"line": 384,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, 2 ^ f i * (f i)! ≤\n ((∑ i ∈ s, f i) ^ ∑ i ∈ s, f i) * (∑ i ∈ s, f i)! * ∏ i ∈ s, (2 * f i)!",
"ppTerm": "?m.223",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
... | [] | gcongr
rw [← doubleFactorial_two_mul]
exact doubleFactorial_le_factorial _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 39
} | {
"line": 113,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : DecidableEq ι\nx : Π₀ (i : ι), α i\ni : ι\nhs : Acc (DFinsupp.Lex r s) (single i (x i))\nhu : Acc (DFinsupp.Lex r s) (erase i x)\n⊢ ({i}, single i (x i), erase i x).2.1.piecewise ({i... | [] | convert! piecewise_single_erase x i | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Data.Sym.Card | {
"line": 99,
"column": 4
} | {
"line": 103,
"column": 24
} | {
"line": 105,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ Fintype.card (Sym (Fin (n + 2)) (k + 1)) = (n + 2).multichoose (k + 1)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"instFintypeSum",
"congrArg",
"Equiv.trans",
"Sym.inst... | [] | rw [multichoose_succ_succ, ← card_sym_fin_eq_multichoose (n + 1) (k + 1),
← card_sym_fin_eq_multichoose (n + 2) k, add_comm (Fintype.card _), ← card_sum]
refine Fintype.card_congr (Equiv.symm ?_)
apply (Sym.e1.symm.sumCongr Sym.e2.symm).trans
apply Equiv.sumCompl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sym.Card | {
"line": 99,
"column": 4
} | {
"line": 103,
"column": 24
} | {
"line": 105,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ Fintype.card (Sym (Fin (n + 2)) (k + 1)) = (n + 2).multichoose (k + 1)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Fintype.card_congr",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"instFintypeSum",
"congrArg",
"Equiv.trans",
"Sym.inst... | [] | rw [multichoose_succ_succ, ← card_sym_fin_eq_multichoose (n + 1) (k + 1),
← card_sym_fin_eq_multichoose (n + 2) k, add_comm (Fintype.card _), ← card_sum]
refine Fintype.card_congr (Equiv.symm ?_)
apply (Sym.e1.symm.sumCongr Sym.e2.symm).trans
apply Equiv.sumCompl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.SchwartzZippel | {
"line": 96,
"column": 95
} | {
"line": 99,
"column": 50
} | {
"line": 100,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : DecidableEq R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\nhp : p ≠ 0\nS : Fin (n + 1) → Finset R\np' : Polynomial (MvPolynomial (Fin n) R) := (finSuccEquiv R n) p\nhp' : p' = (finSuccEquiv R n) p\nk : ℕ := p'.natDegree\nhk : k = p'.natDegree... | [] | by
rw [← add_div, ← Nat.cast_add, ← card_union_add_card_inter, filter_union_right,
← filter_and]
simp [← and_or_left, em, and_and_and_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Squarefree | {
"line": 163,
"column": 40
} | {
"line": 163,
"column": 46
} | {
"line": 163,
"column": 46
} | [
{
"pp": "n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ 0 < 2",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Squarefree | {
"line": 163,
"column": 40
} | {
"line": 163,
"column": 46
} | {
"line": 163,
"column": 46
} | [
{
"pp": "n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ 0 < 2",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 163,
"column": 40
} | {
"line": 163,
"column": 46
} | {
"line": 163,
"column": 46
} | [
{
"pp": "n k : ℕ\nn0 : 0 < n\ni : ℕ\ne : k = 2 * i + 3\nih : ∀ (m : ℕ), Prime m → m ∣ n → k ≤ m\nh : ¬n < k * k\nk2 : 2 ≤ k\n⊢ 0 < 2",
"ppTerm": "?m.109",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 923,
"column": 2
} | {
"line": 923,
"column": 27
} | {
"line": 925,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\n⊢ max (m.toSyn (m.degree f)) (m.toSyn (m.degree (m.leadingTerm f))) ≤ m.toSyn (m.degree f)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"MonomialOrder.linearOrderSyn",
"Nat.in... | [] | simp [degree_leadingTerm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Squarefree | {
"line": 195,
"column": 19
} | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 25
} | [
{
"pp": "d2 : 2 ∣ 0\nd4 : ¬2 ∣ 0 / 2\n⊢ 2 ∣ 0 / 2",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"instHDiv",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"HDiv.hDiv",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Squarefree | {
"line": 195,
"column": 19
} | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 25
} | [
{
"pp": "d2 : 2 ∣ 0\nd4 : ¬2 ∣ 0 / 2\n⊢ 2 ∣ 0 / 2",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"instHDiv",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"HDiv.hDiv",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 195,
"column": 19
} | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 25
} | [
{
"pp": "d2 : 2 ∣ 0\nd4 : ¬2 ∣ 0 / 2\n⊢ 2 ∣ 0 / 2",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"instHDiv",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"HDiv.hDiv",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Squarefree | {
"line": 197,
"column": 69
} | {
"line": 197,
"column": 75
} | {
"line": 197,
"column": 75
} | [
{
"pp": "n : ℕ\nd2 : 2 ∣ n\nd4 : ¬2 ∣ n / 2\nn0 : n > 0\n⊢ 0 < 2",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Squarefree | {
"line": 197,
"column": 69
} | {
"line": 197,
"column": 75
} | {
"line": 197,
"column": 75
} | [
{
"pp": "n : ℕ\nd2 : 2 ∣ n\nd4 : ¬2 ∣ n / 2\nn0 : n > 0\n⊢ 0 < 2",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 197,
"column": 69
} | {
"line": 197,
"column": 75
} | {
"line": 197,
"column": 75
} | [
{
"pp": "n : ℕ\nd2 : 2 ∣ n\nd4 : ¬2 ∣ n / 2\nn0 : n > 0\n⊢ 0 < 2",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
"Nat.decLt",
"Eq.refl",
"instLTNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Squarefree | {
"line": 202,
"column": 19
} | {
"line": 202,
"column": 25
} | {
"line": 202,
"column": 25
} | [
{
"pp": "d2 : ¬2 ∣ 0\n⊢ 2 ∣ 0",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Nat.Squarefree | {
"line": 202,
"column": 19
} | {
"line": 202,
"column": 25
} | {
"line": 202,
"column": 25
} | [
{
"pp": "d2 : ¬2 ∣ 0\n⊢ 2 ∣ 0",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 202,
"column": 19
} | {
"line": 202,
"column": 25
} | {
"line": 202,
"column": 25
} | [
{
"pp": "d2 : ¬2 ∣ 0\n⊢ 2 ∣ 0",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decidable_dvd",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instDvd",
"Nat",
"Bool",
"Eq.refl",
"OfNat.ofNat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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