module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 37
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhz : ∀ᶠ (x : α) in l, v x ≠ 0\n⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Asymptotics.isEquivalent_of_tendsto_one"
],
"usedFVars": [
... | [] | exact isEquivalent_of_tendsto_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 37
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhz : ∀ᶠ (x : α) in l, v x ≠ 0\n⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Asymptotics.isEquivalent_of_tendsto_one"
],
"usedFVars": [
... | [] | exact isEquivalent_of_tendsto_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 37
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhz : ∀ᶠ (x : α) in l, v x ≠ 0\n⊢ Tendsto (u / v) l (𝓝 1) → u ~[l] v",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Asymptotics.isEquivalent_of_tendsto_one"
],
"usedFVars": [
... | [] | exact isEquivalent_of_tendsto_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 256,
"column": 52
} | {
"line": 256,
"column": 62
} | {
"line": 257,
"column": 4
} | [
{
"pp": "case h₂\nα : Type u_1\nE : Type u_2\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\na b : α → 𝕜\nu v : α → E\nl : Filter α\nhab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖\nφ : α → 𝕜\nhabφ : a =ᶠ[l] φ * b\nthis : ((fun x ↦ a x • u x)... | [] | exact huvx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 648,
"column": 10
} | {
"line": 648,
"column": 12
} | {
"line": 648,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : IsOrderedModule ℝ E\ninst✝¹ : ClosedIciTopology E\nβ : Type u_6\ninst✝ : Preorder β\nf : α → β → E\ns : Set β\nhf_m... | [
"α : Type u_1\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : IsOrderedModule ℝ E\ninst✝¹ : ClosedIciTopology E\nβ : Type u_6\ninst✝ : Preorder β\nf : α → β → E\ns : Set β\nhf_mono : ∀ᵐ (x ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 348,
"column": 26
} | {
"line": 350,
"column": 81
} | {
"line": 352,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedField β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\nu v : α → β\nl : Filter α\ninst✝ : ClosedIicTopology β\nh : u ~[l] v\nhv : ∀ᶠ (t : α) in l, 0 < v t\n⊢ ∀ᶠ (x : α) in l, 0 < u x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants":... | [] | by
obtain ⟨φ, hφ, h_eq⟩ := h.exists_pos_eq_mul
exact (hφ.and (hv.and h_eq)).mono (fun x ⟨hφ, hv, h_eq⟩ ↦ h_eq ▸ mul_pos hφ hv) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 656,
"column": 10
} | {
"line": 656,
"column": 12
} | {
"line": 656,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : IsOrderedModule ℝ E\ninst✝¹ : ClosedIciTopology E\nβ : Type u_6\ninst✝ : Preorder β\nf : α → β → E\ns : Set β\nhf_a... | [
"α : Type u_1\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : IsOrderedModule ℝ E\ninst✝¹ : ClosedIciTopology E\nβ : Type u_6\ninst✝ : Preorder β\nf : α → β → E\ns : Set β\nhf_anti : ∀ᵐ (x ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 666,
"column": 10
} | {
"line": 666,
"column": 12
} | {
"line": 666,
"column": 13
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | [
"α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : α → β → E\ns... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 773,
"column": 35
} | {
"line": 773,
"column": 37
} | {
"line": 773,
"column": 38
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\nf' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x\na : α\n⊢... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\nf' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x\na : α\nha : Monotone... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 506,
"column": 2
} | {
"line": 507,
"column": 22
} | {
"line": 509,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ninst✝² : IsBoundedSMul 𝕜 ... | [] | rw [setToL1_eq_setToL1' hT h_smul, setToL1_eq_setToL1' hT h_smul]
exact map_smul _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 506,
"column": 2
} | {
"line": 507,
"column": 22
} | {
"line": 509,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_6\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜 F\ninst✝² : IsBoundedSMul 𝕜 ... | [] | rw [setToL1_eq_setToL1' hT h_smul, setToL1_eq_setToL1' hT h_smul]
exact map_smul _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 859,
"column": 47
} | {
"line": 859,
"column": 49
} | {
"line": 859,
"column": 50
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 738,
"column": 95
} | {
"line": 742,
"column": 28
} | {
"line": 744,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhf : Integrable f μ\nhg : Inte... | [] | by
by_cases hF : CompleteSpace F; swap
· simp [setToFun, hF]
rw [setToFun_eq hT (hf.add hg), setToFun_eq hT hf, setToFun_eq hT hg, Integrable.toL1_add,
(L1.setToL1 hT).map_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 895,
"column": 36
} | {
"line": 895,
"column": 38
} | {
"line": 895,
"column": 39
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 897,
"column": 37
} | {
"line": 897,
"column": 39
} | {
"line": 897,
"column": 40
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 78
} | {
"line": 175,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\ninst✝¹ : Nonempty α\ninst✝ : CompactSpace α\nw : ∀ (x : α), dist (f x) (g x) < C\nc : Continuous[inst✝³, _] fun x ↦ dist (f x) (g x)\nx : α\nle : IsMaxOn (fun x ↦ dist (f x) (g x)) univ x\n⊢ dist f g... | [] | exact lt_of_le_of_lt (dist_le_iff_of_nonempty.mpr fun y => le trivial) (w x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 102,
"column": 2
} | {
"line": 105,
"column": 62
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\n⊢ (E.indicator fun x ↦ 1) ≤ thickenedIndicatorAux δ E",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"_private.Mathlib.Topology.MetricSpace.ThickenedIndicator.0.indicato... | [] | intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_notMem, not_false_iff, zero_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 102,
"column": 2
} | {
"line": 105,
"column": 62
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nE : Set α\n⊢ (E.indicator fun x ↦ 1) ≤ thickenedIndicatorAux δ E",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"_private.Mathlib.Topology.MetricSpace.ThickenedIndicator.0.indicato... | [] | intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_notMem, not_false_iff, zero_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LeftRightLim | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 88
} | {
"line": 345,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\n⊢ Tendsto f (𝓝[<] x) (𝓝[≤] leftLim f x)",
"ppTerm": "?m.... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\n⊢ ∀ᶠ (x_1 : α) in 𝓝[<] x, f x_1 ∈ Iic (leftLim f x)"
] | apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within f (hf.tendsto_leftLim x) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 37
} | {
"line": 177,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns : Set X\nμ : Measure X\nhs : MeasurableSet s\nhfi : ¬IntegrableOn f s μ\n⊢ ¬Integrable (s.indicator f) μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants":... | [] | rwa [integrable_indicator_iff hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1372,
"column": 32
} | {
"line": 1372,
"column": 34
} | {
"line": 1373,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\ninst✝¹ : CompleteSpace E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nin... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\ninst✝¹ : CompleteSpace E\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\ninst✝ : Counta... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 79
} | {
"line": 289,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set X\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nh_mono : Monotone s\nhne : atTop.NeBot... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set X\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nh_mono : Monotone s\nhne : atTop.NeBot\nthis✝ : Is... | rw [← withDensity_apply₀ _ (hSm.diff (hsm _)), ← hν, measure_sdiff hsub hsm'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1479,
"column": 35
} | {
"line": 1479,
"column": 37
} | {
"line": 1479,
"column": 38
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nX : Type u_7\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nhT : Dominated... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Content | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 85
} | {
"line": 325,
"column": 2
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\n⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \\ U) ≤ μ.outerMeasure ↑U'",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [... | [
"G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\nS : MeasurableSpace G\ninst✝ : BorelSpace G\nU : Set G\nhU : U ∈ {s | IsOpen s}\nU' : Opens G\nthis : Nonempty { L // ↑L ⊆ ↑U' ∩ U }\n⊢ (⨆ x, μ ↑x) + μ.outerMeasure (↑U' \\ U) ≤ μ.outerMeasure ↑U'"
] | have : Nonempty { L : Compacts G // (L : Set G) ⊆ U' ∩ U } := ⟨⟨⊥, empty_subset _⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 28
} | {
"line": 353,
"column": 2
} | [
{
"pp": "G : Type u_1\nα : Type u_3\nE : Type u_5\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nhs : IsFundamentalDomain G s μ\nht : IsF... | [
"case hf\nG : Type u_1\nα : Type u_3\nE : Type u_5\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ns t : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nhs : IsFundamentalDomain G s μ\nht : IsFund... | rw [hs.setLIntegral_eq ht] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 44
} | {
"line": 546,
"column": 2
} | [
{
"pp": "case neg\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 : R\nha : ¬IsBot a\nb : R\nhb : ... | [
"case neg\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 : R\nha : ¬IsBot a\nb : R\nhb : b < a\nu : ℕ... | replace u_lt_a n : u n < a := (u_lt_a n).2 | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 221,
"column": 2
} | {
"line": 224,
"column": 62
} | {
"line": 225,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index (K... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index (K₁.carrier ∪ ... | refine
le_trans
(add_le_add (this K₁.1 <| Subset.trans subset_union_left h1s)
(this K₂.1 <| Subset.trans subset_union_right h1s)) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 842,
"column": 2
} | {
"line": 851,
"column": 29
} | {
"line": 852,
"column": 2
} | [
{
"pp": "case refine_1\nG : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\nν : Measure α\ninst✝³ : SMulInvariantMeasure G α ν\ninst✝² : Countable G\ninst✝¹ : MeasurableConstSMul G α\ni : SigmaFinite ν\ni' : HasFundamentalDomain G α ν\nμ : Measure (Quotient α_mod_G)... | [
"case refine_2\nG : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\nν : Measure α\ninst✝³ : SMulInvariantMeasure G α ν\ninst✝² : Countable G\ninst✝¹ : MeasurableConstSMul G α\ni : SigmaFinite ν\ni' : HasFundamentalDomain G α ν\nμ : Measure (Quotient α_mod_G)\ninst✝ : Qu... | · obtain ⟨s, fund_dom_s⟩ := i'
have : π ⁻¹' π '' (A n) = _ := MulAction.quotient_preimage_image_eq_union_mul (A n) (G := G)
have measπAn : MeasurableSet (π '' A n) := by
rw [measurableSet_quotient, Quotient.mk''_eq_mk, this]
apply MeasurableSet.iUnion
exact fun g ↦ MeasurableSet.const_smul (hA... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 881,
"column": 4
} | {
"line": 881,
"column": 22
} | {
"line": 882,
"column": 4
} | [
{
"pp": "case pos.hf\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH : Integrable f μ\ng : X → ℝ := ⋯\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\nx : X\n⊢ g x ≤ ENNReal.toRe... | [
"case pos\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH✝ : Integrable f μ\ng : X → ℝ := fun x ↦ max (f x) 0\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\nx : X\nH : x ∈ s\n⊢ g x... | by_cases H : x ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 337,
"column": 60
} | {
"line": 343,
"column": 6
} | {
"line": 345,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\n⊢ normSq (⟪x, y⟫ • x - ⟪x, x⟫ • y) = normSq x * (normSq x * normSq y - ‖⟪x, y⟫‖ ^ 2)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"A... | [] | by
rw [← @ofReal_inj 𝕜, ofReal_normSq_eq_inner_self]
simp only [inner_sub_sub_self, inner_smul_left, inner_smul_right, conj_ofReal, mul_sub, ←
ofReal_normSq_eq_inner_self x, ← ofReal_normSq_eq_inner_self y]
rw [← mul_assoc, mul_conj, RCLike.conj_mul, mul_left_comm, ← inner_conj_symm y, mul_conj]
push_cast
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 28
} | {
"line": 419,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫ + ‖y‖ ^ 2"
] | repeat' rw [← sq (M := ℝ)] | Lean.Elab.Tactic.evalRepeat' | Lean.Parser.Tactic.repeat' |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 28
} | {
"line": 444,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ‖x - y‖ ^ 2 = ‖x‖ ^ 2 - 2 * re ⟪x, y⟫ + ‖y‖ ^ 2"
] | repeat' rw [← sq (M := ℝ)] | Lean.Elab.Tactic.evalRepeat' | Lean.Parser.Tactic.repeat' |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 153,
"column": 94
} | {
"line": 154,
"column": 95
} | {
"line": 156,
"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\nl₁ l₂ : ι →₀ 𝕜\n⊢ ⟪(linearCombination 𝕜 v) l₁, (linearCombination 𝕜 v) l₂⟫ = l₁.sum fun i y ↦ (starRingEnd 𝕜) y * l₂ i",
"ppTerm": ... | [] | by
simp [l₁.linearCombination_apply, Finsupp.sum_inner, hv.inner_right_finsupp, inner_smul_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 75
} | {
"line": 237,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nh : IsComplete ↑K\nthis : InnerProductSpace ℝ E := rclikeToReal 𝕜 E\nK' : Submodule ℝ E := restrictScalars ℝ K\n⊢ ∀ (u : E), ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖",
"ppTer... | [] | exact exists_norm_eq_iInf_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 249,
"column": 6
} | {
"line": 251,
"column": 29
} | {
"line": 252,
"column": 6
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\n⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Norm.norm",
"norm_eq_iInf_iff_real_inner_le_zer... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nh : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0\n⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0"
] | have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
rwa [norm_eq_iInf_iff_real_inner_le_zero] at h
exacts [K.convex, hv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 16
} | {
"line": 253,
"column": 6
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nh : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0\n⊢ ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"InnerProductSpace... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nh : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0\nw : F\nhw : w ∈ K\n⊢ ⟪u - v, w⟫_ℝ = 0"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 235,
"column": 36
} | {
"line": 236,
"column": 78
} | {
"line": 237,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx : E\nh : 0 < ‖x‖\n⊢ ‖x‖ * ‖x‖ = ‖⟪x, x⟫‖",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
... | [] | by
rw [← sq, inner_self_eq_norm_sq_to_K, norm_pow, norm_ofReal, abs_norm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 273,
"column": 8
} | {
"line": 273,
"column": 18
} | {
"line": 274,
"column": 8
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0\n⊢ ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0",
"ppTerm": "?m.252",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submo... | [
"F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh : ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0\nw : F\nhw : w ∈ K\n⊢ ⟪u - v, w - v⟫_ℝ ≤ 0"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 14
} | {
"line": 294,
"column": 4
} | [
{
"pp": "case mp\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 := rclikeToReal 𝕜 E\nK' : Submodule ℝ E := restrictScalars ℝ K\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u... | [
"case mp\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 := rclikeToReal 𝕜 E\nK' : Submodule ℝ E := restrictScalars ℝ K\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 16
} | {
"line": 305,
"column": 6
} | [
{
"pp": "𝕜 : 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 := rclikeToReal 𝕜 E\nK' : Submodule ℝ E := restrictScalars ℝ K\nH : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0\n⊢ ∀ w ∈ K', ⟪u - v, w⟫_ℝ... | [
"𝕜 : 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 := rclikeToReal 𝕜 E\nK' : Submodule ℝ E := restrictScalars ℝ K\nH : ∀ w ∈ K, ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w ∈ K'\n⊢ ⟪u - v, w⟫_ℝ =... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 363,
"column": 56
} | {
"line": 371,
"column": 46
} | {
"line": 373,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : TopologicalSpace N\ninst✝⁴ : Module R N\ninst✝³ : ContinuousAdd N\ninst✝² : ContinuousConstSMul R N\ninst✝¹ : ContinuousAdd M\ni... | [] | by
ext x
simp only [mapEquiv_apply, toSubmodule_sup, Submodule.carrier_eq_coe, Submodule.map_coe,
LinearEquiv.coe_coe, ContinuousLinearEquiv.coe_toLinearEquiv, coe_toSubmodule,
Submodule.coe_closure, Set.mem_image]
have : f = f.toLinearEquiv.toLinearMap := by
exact LinearMap.ext (congrFun rfl)
rw [←... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 287,
"column": 53
} | {
"line": 287,
"column": 56
} | {
"line": 287,
"column": 57
} | [
{
"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, T v⟫ = 0",
"ppTerm": "?m.158",
"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⊢ ⟪y, 0⟫ = 0"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 12
} | {
"line": 168,
"column": 2
} | [
{
"pp": "𝕜 : 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\n⊢ v - K.starProjection v ∈ Kᗮ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toN... | [
"𝕜 : 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 w : E\nhw : w ∈ K\n⊢ ⟪w, v - K.starProjection v⟫ = 0"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 16
} | {
"line": 180,
"column": 6
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\... | [
"F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\nhn : finran... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 16
} | {
"line": 192,
"column": 6
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\... | [
"F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\nhn : finran... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 16
} | {
"line": 197,
"column": 6
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\... | [
"F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nn : ℕ\nIH :\n ∀ (φ : F ≃ₗᵢ[ℝ] F),\n finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ n →\n ∃ l, l.length ≤ n ∧ φ = (List.map (fun v ↦ (ℝ ∙ v)ᗮ.reflection) l).prod\nφ : F ≃ₗᵢ[ℝ] F\nhn : finran... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 374,
"column": 16
} | {
"line": 374,
"column": 18
} | {
"line": 375,
"column": 8
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\na : E\n⊢ a ∈ v → ⟪a, e⟫_𝕜 = ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\na : E\nha : a ∈ v\n⊢ ⟪a, e⟫_𝕜 = 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 45
} | {
"line": 528,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E →L[𝕜] F\nG : Submodule 𝕜 F\nh : IsCompl (↑f).ran... | [] | exact isClosed_univ.prod isClosed_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 677,
"column": 5
} | {
"line": 678,
"column": 51
} | {
"line": 678,
"column": 51
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x y ≤ edist x.ofLp y.ofLp",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ENNReal.div_top",
"WithLp",
"NonAssocSemiring.toAddC... | [] | by simpa only [ENNReal.div_top, ENNReal.toReal_zero, NNReal.rpow_zero, ENNReal.coe_one,
one_mul] using antilipschitzWith_ofLp ∞ β x y | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 629,
"column": 2
} | {
"line": 629,
"column": 76
} | {
"line": 630,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | [
"case eq_top\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Norme... | refine ⟨fun h ↦ ⟨?eq_top, ?anti⟩, fun ⟨hd, c, hf⟩ ↦ ⟨hf.injective, ?surj⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 231,
"column": 64
} | {
"line": 234,
"column": 40
} | {
"line": 236,
"column": 0
} | [
{
"pp": "p : ℝ → Prop\na : ℝ\nh : ∀ᶠ (x : ℝ) in 𝓝 a, p x\n⊢ 0 < volume {x | p x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Real.instIsOrderedRing",
"Eq.mpr",
"sub_pos._simp_1",
"le_refl",
"Re... | [] | by
rcases h.exists_Ioo_subset with ⟨l, u, hx, hs⟩
grw [← hs]
simpa [-mem_Ioo] using hx.1.trans hx.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 94,
"column": 32
} | {
"line": 94,
"column": 42
} | {
"line": 94,
"column": 43
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nthis✝ : Icc 0 1 = ⇑K '' Icc 0 1\nx : E\nthis : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑... | [
"ι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nthis✝ : Icc 0 1 = ⇑K '' Icc 0 1\nx : E\nthis : ∀ (z : ι' → ℝ), ∑ x, z (e.symm x) • v x = ∑ i, z i • v ... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 127,
"column": 2
} | {
"line": 136,
"column": 14
} | {
"line": 138,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\n⊢ parallelepiped v = ∑ i, segment ℝ 0 (v i)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.partialOrder",
"Real",
"_pri... | [] | ext
simp only [mem_parallelepiped_iff, Set.mem_finsetSum, Finset.mem_univ, forall_true_left,
segment_eq_image, smul_zero, zero_add, ← Set.pi_univ_Icc, Set.mem_univ_pi]
constructor
· rintro ⟨t, ht, rfl⟩
exact ⟨t • v, fun {i} => ⟨t i, ht _, by simp⟩, rfl⟩
rintro ⟨g, hg, rfl⟩
choose t ht hg using @hg
r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 127,
"column": 2
} | {
"line": 136,
"column": 14
} | {
"line": 138,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝² : Fintype ι\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\n⊢ parallelepiped v = ∑ i, segment ℝ 0 (v i)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.partialOrder",
"Real",
"_pri... | [] | ext
simp only [mem_parallelepiped_iff, Set.mem_finsetSum, Finset.mem_univ, forall_true_left,
segment_eq_image, smul_zero, zero_add, ← Set.pi_univ_Icc, Set.mem_univ_pi]
constructor
· rintro ⟨t, ht, rfl⟩
exact ⟨t • v, fun {i} => ⟨t i, ht _, by simp⟩, rfl⟩
rintro ⟨g, hg, rfl⟩
choose t ht hg using @hg
r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 170,
"column": 53
} | {
"line": 170,
"column": 70
} | {
"line": 170,
"column": 71
} | [
{
"pp": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁶ : NormedField K\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace K E\nb : Basis ι K E\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ (b.repr m) i - (b.repr ↑(floor b m)) i = Int.fract... | [
"E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁶ : NormedField K\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace K E\nb : Basis ι K E\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ (b.repr m) i - ↑⌊(b.repr m) i⌋ = Int.fract ((b.repr m) i)"
] | repr_floor_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 234,
"column": 8
} | {
"line": 234,
"column": 25
} | {
"line": 234,
"column": 26
} | [
{
"pp": "ι : Type u_2\nK : Type u_3\ninst✝⁵ : NormedField K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsStrictOrderedRing K\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : Unique ι\nk : K\nx✝ : ι\n⊢ ((Basis.singleton ι K).repr ↑(floor (Basis.singleton ι K) k)) x✝ = ((Basis.singleton ι K).repr ↑⌊k⌋) x✝",
"ppTerm"... | [
"ι : Type u_2\nK : Type u_3\ninst✝⁵ : NormedField K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsStrictOrderedRing K\ninst✝² : FloorRing K\ninst✝¹ : Fintype ι\ninst✝ : Unique ι\nk : K\nx✝ : ι\n⊢ ↑⌊((Basis.singleton ι K).repr k) x✝⌋ = ((Basis.singleton ι K).repr ↑⌊k⌋) x✝"
] | repr_floor_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 37
} | {
"line": 191,
"column": 2
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ : Prepartition I\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ #({J ∈ π.boxes | x ∈ Box.Icc J}) ≤ Fintype.card (Set ι)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Real",
"Finset.univ",
"Set.ofPred",
"Finset",
... | [
"ι : Type u_1\nI : Box ι\nπ : Prepartition I\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ InjOn (fun J ↦ {i | J.lower i = x i}) ↑({J ∈ π.boxes | x ∈ Box.Icc J})"
] | refine Finset.card_le_card_of_injOn (fun J : Box ι => { i | J.lower i = x i })
(fun _ _ => Finset.mem_univ _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 45
} | {
"line": 335,
"column": 46
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nhJ : J ∈ π.biUnion πi\n⊢ J ∈ πi (π.biUnionIndex πi J)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Exists.choose_spec",
"outParam",
"BoxIntegral.Prepartition",
... | [
"case e'_2\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nhJ : J ∈ π.biUnion πi\n⊢ π.biUnionIndex πi J = ⋯.choose",
"case e'_3\nι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nhJ : J ∈ π.biUnion πi\ne_2✝ : Prepartition (π.biUnionIndex πi J) = P... | convert! (π.mem_biUnion.1 hJ).choose_spec.2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 444,
"column": 87
} | {
"line": 445,
"column": 74
} | {
"line": 447,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI J J₁ : Box ι\nπ : Prepartition I\n⊢ J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, ↑J₁ = ↑J ∩ ↑J'",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.BoxIntegral.Partition.Basic.0.BoxIntegral.Prepartition.mem_restrict'._simp_1_1",
"Real",
"W... | [] | by
simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 751,
"column": 10
} | {
"line": 751,
"column": 25
} | {
"line": 751,
"column": 25
} | [
{
"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 : ... | [
"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 : Set E\nht : ... | ENNReal.add_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction | {
"line": 123,
"column": 2
} | {
"line": 131,
"column": 8
} | {
"line": 133,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nr : (ι → ℝ) → ↑(Ioi 0)\nz : ι → ℝ\nx✝ : z ∈ Box.Icc I\n⊢ ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J →\n Box.Icc J ⊆ U →\n (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m... | [] | · refine ⟨Box.Icc I ∩ closedBall z (r z),
inter_mem_nhdsWithin _ (closedBall_mem_nhds _ (r z).coe_prop), ?_⟩
intro J _ n Hmem HIcc Hsub
rw [Set.subset_inter_iff] at HIcc
refine ⟨single _ _ le_rfl _ Hmem, isPartition_single _, isHenstock_single _,
(isSubordinate_single _ _).2 HIcc.2, ?_, distorti... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 597,
"column": 4
} | {
"line": 598,
"column": 31
} | {
"line": 599,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ni... | [] | rwa [add_smul, neg_smul, SetLike.mem_coe, ← fract_eq_fract, Int.cast_smul_eq_zsmul ℚ,
Int.cast_smul_eq_zsmul ℚ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 776,
"column": 8
} | {
"line": 776,
"column": 49
} | {
"line": 777,
"column": 6
} | [
{
"pp": "case h0\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\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBa... | [] | exact (measure_closedBall_pos μ _ hr).ne' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 776,
"column": 8
} | {
"line": 776,
"column": 49
} | {
"line": 777,
"column": 6
} | [
{
"pp": "case h0\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\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBa... | [] | exact (measure_closedBall_pos μ _ hr).ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 776,
"column": 8
} | {
"line": 776,
"column": 49
} | {
"line": 777,
"column": 6
} | [
{
"pp": "case h0\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\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBa... | [] | exact (measure_closedBall_pos μ _ hr).ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 816,
"column": 2
} | {
"line": 816,
"column": 44
} | {
"line": 818,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype ι\nx : EuclideanSpace 𝕜 ι\ni : ι\n⊢ ⟪(basisFun ι 𝕜) i, x⟫ = x.ofLp i",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"fact_one_le_two_ennreal",
"Inner.inner",
"congrArg",
"Nat.instAtLeastTwo... | [] | simp [← OrthonormalBasis.repr_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 816,
"column": 2
} | {
"line": 816,
"column": 44
} | {
"line": 818,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype ι\nx : EuclideanSpace 𝕜 ι\ni : ι\n⊢ ⟪(basisFun ι 𝕜) i, x⟫ = x.ofLp i",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"fact_one_le_two_ennreal",
"Inner.inner",
"congrArg",
"Nat.instAtLeastTwo... | [] | simp [← OrthonormalBasis.repr_apply_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 816,
"column": 2
} | {
"line": 816,
"column": 44
} | {
"line": 818,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype ι\nx : EuclideanSpace 𝕜 ι\ni : ι\n⊢ ⟪(basisFun ι 𝕜) i, x⟫ = x.ofLp i",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"fact_one_le_two_ennreal",
"Inner.inner",
"congrArg",
"Nat.instAtLeastTwo... | [] | simp [← OrthonormalBasis.repr_apply_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 63
} | {
"line": 355,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : Finite ι\nπ : Prepartition I\nh : π.IsPartition\n⊢ π.compl = ⊥",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition",
"congrArg",
"BoxIntegral.Box.toSet",
"BoxIntegral.Prepa... | [] | rw [← iUnion_eq_empty, iUnion_compl, h.iUnion_eq, sdiff_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 63
} | {
"line": 355,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : Finite ι\nπ : Prepartition I\nh : π.IsPartition\n⊢ π.compl = ⊥",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition",
"congrArg",
"BoxIntegral.Box.toSet",
"BoxIntegral.Prepa... | [] | rw [← iUnion_eq_empty, iUnion_compl, h.iUnion_eq, sdiff_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 63
} | {
"line": 355,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ninst✝ : Finite ι\nπ : Prepartition I\nh : π.IsPartition\n⊢ π.compl = ⊥",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition",
"congrArg",
"BoxIntegral.Box.toSet",
"BoxIntegral.Prepa... | [] | rw [← iUnion_eq_empty, iUnion_compl, h.iUnion_eq, sdiff_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 745,
"column": 6
} | {
"line": 745,
"column": 31
} | {
"line": 745,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : NormedField K\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nL : Submodule ℤ E\ne : F ≃ₗ[K] E\nx✝ : ↥(ZLattice.comap K L ↑e)\nx : F\nhx : x ∈ ZLattice.comap K L ↑e\n⊢ ((↑ℤ ↑e.symm).rest... | [] | by simp [Subtype.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1027,
"column": 50
} | {
"line": 1028,
"column": 55
} | {
"line": 1030,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\nA : ι → Submodule 𝕜 E\ninst✝¹ : DecidableEq ι\nh : IsInternal A\nα : ι → Type u_7\ninst✝ : (i : ι) → Fintype (α i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(A i... | [] | by
simp [DirectSum.IsInternal.collectedOrthonormalBasis] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Oscillation | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 49
} | {
"line": 124,
"column": 4
} | [
{
"pp": "E : 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, IsO... | [
"E : 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, IsOpen[PseudoEM... | have : oscillationWithin f D x < ε := hK x hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 353,
"column": 2
} | {
"line": 357,
"column": 39
} | {
"line": 359,
"column": 0
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite ι\nx : ι → ℝ\nhx : x ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))\n⊢ tag n (index n x) = x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.inst... | [] | rw [mem_smul_span_iff] at hx
ext i
obtain ⟨a, ha⟩ : ∃ a : ℤ, a = n * x i := hx i
rwa [tag_apply, index_apply, Int.cast_sub, Int.cast_one, sub_add_cancel, ← ha, Int.ceil_intCast,
div_eq_iff (NeZero.ne _), mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 353,
"column": 2
} | {
"line": 357,
"column": 39
} | {
"line": 359,
"column": 0
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Finite ι\nx : ι → ℝ\nhx : x ∈ (↑n)⁻¹ • span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))\n⊢ tag n (index n x) = x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.inst... | [] | rw [mem_smul_span_iff] at hx
ext i
obtain ⟨a, ha⟩ : ∃ a : ℤ, a = n * x i := hx i
rwa [tag_apply, index_apply, Int.cast_sub, Int.cast_one, sub_add_cancel, ← ha, Int.ceil_intCast,
div_eq_iff (NeZero.ne _), mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 197,
"column": 12
} | {
"line": 197,
"column": 14
} | {
"line": 198,
"column": 4
} | [
{
"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\nn : ι\nh₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)\nh : gramSchmidt 𝕜 f n = 0\na ... | [
"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\nn : ι\nh₀ : LinearIndependent 𝕜 (f ∘ Subtype.val)\nh : gramSchmidt 𝕜 f n = 0\na : ι\nha : a ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 327,
"column": 4
} | {
"line": 328,
"column": 27
} | {
"line": 330,
"column": 0
} | [
{
"pp": "case insert\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nα : Type u_1\nf : α → (ι → ℝ) → E\ng : α → F\na : α\ns : Fins... | [] | simp only [Finset.sum_insert ha]; rw [Finset.forall_mem_insert] at h
exact h.1.add (ihs h.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 327,
"column": 4
} | {
"line": 328,
"column": 27
} | {
"line": 330,
"column": 0
} | [
{
"pp": "case insert\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nα : Type u_1\nf : α → (ι → ℝ) → E\ng : α → F\na : α\ns : Fins... | [] | simp only [Finset.sum_insert ha]; rw [Finset.forall_mem_insert] at h
exact h.1.add (ihs h.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SumOverResidueClass | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 25
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case neg\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nk l : ZMod m\nhs : Summable ({n | ↑n = k}.indicator f)\nhf₀ : ∃ n, f n < 0\n⊢ Summable ({n | ↑n = l}.indicator f)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"ZMod.commRin... | [
"case neg\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nk l : ZMod m\nhs : Summable ({n | ↑n = k}.indicator f)\nn : ℕ\nhn : f n < 0\n⊢ Summable ({n | ↑n = l}.indicator f)"
] | obtain ⟨n, hn⟩ := hf₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SumOverResidueClass | {
"line": 94,
"column": 2
} | {
"line": 99,
"column": 30
} | {
"line": 101,
"column": 0
} | [
{
"pp": "m : ℕ\ninst✝ : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nk : ZMod m\n⊢ Summable ({n | ↑n = k}.indicator f) ↔ Summable f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Pi.addCommMonoid",
... | [] | refine ⟨fun H ↦ ?_, fun H ↦ Summable.indicator H _⟩
rw [Finset.sum_indicator_mod m f]
convert!
summable_sum (s := Finset.univ) fun a _ ↦
summable_indicator_mod_iff_summable_indicator_mod hf a H
simp only [Finset.sum_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SumOverResidueClass | {
"line": 94,
"column": 2
} | {
"line": 99,
"column": 30
} | {
"line": 101,
"column": 0
} | [
{
"pp": "m : ℕ\ninst✝ : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nk : ZMod m\n⊢ Summable ({n | ↑n = k}.indicator f) ↔ Summable f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Pi.addCommMonoid",
... | [] | refine ⟨fun H ↦ ?_, fun H ↦ Summable.indicator H _⟩
rw [Finset.sum_indicator_mod m f]
convert!
summable_sum (s := Finset.univ) fun a _ ↦
summable_indicator_mod_iff_summable_indicator_mod hf a H
simp only [Finset.sum_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 29,
"column": 39
} | {
"line": 32,
"column": 64
} | {
"line": 34,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : NormedField 𝕜\ninst✝ : Norm E\na : 𝕜\nf : 𝕜 → E\nh : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (𝓝[≠] a) (norm ∘ f)\n⊢ f =o[𝓝[≠] a] fun x ↦ (x - a)⁻¹",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"tendsto_norm_sub_self_... | [] | by
refine (h.isBigO_const (one_ne_zero' ℝ)).trans_isLittleO (isLittleO_const_left.2 <| Or.inr ?_)
simp only [Function.comp_def, norm_inv]
exact (tendsto_norm_sub_self_nhdsNE a).inv_tendsto_nhdsGT_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 88
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case h\nα : Type u_1\nu v : α → ℝ\nl : Filter α\nhv : 0 ≤ v\nh : u ~[l] v\nr : ℝ\nφ : α → ℝ\nhφ : Tendsto φ l (𝓝 1)\nhuφv : u =ᶠ[l] φ * v\nhφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)\n⊢ u ^ r =ᶠ[l] fun x ↦ φ x ^ r * v x ^ r",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Nor... | [
"α : Type u_1\nu v : α → ℝ\nl : Filter α\nhv : 0 ≤ v\nh : u ~[l] v\nr : ℝ\nφ : α → ℝ\nhφ : Tendsto φ l (𝓝 1)\nhuφv : u =ᶠ[l] φ * v\nhφr : Tendsto ((fun x ↦ x ^ r) ∘ φ) l (𝓝 1)\nx : α\nhφ_pos : 0 < φ x\nhuv' : u x = (φ * v) x\n⊢ (u ^ r) x = φ x ^ r * v x ^ r"
] | filter_upwards [Tendsto.eventually_const_lt (zero_lt_one) hφ, huφv] with x hφ_pos huv' | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.PSeries | {
"line": 162,
"column": 7
} | {
"line": 162,
"column": 26
} | {
"line": 162,
"column": 26
} | [
{
"pp": "f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ f 0 + ∑ x ∈ range n, 2 ^ x * f (2 ^ x) ≤ f 0 + ∑' (k : ℕ), 2 ^ k * f (2 ^ k)",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.instAdd",
"le_refl... | [
"f : ℕ → ℝ≥0∞\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\nn : ℕ\n⊢ f 0 + ∑' (x : ℕ), 2 ^ x * f (2 ^ x) ≤ f 0 + ∑' (k : ℕ), 2 ^ k * f (2 ^ k)"
] | ENNReal.sum_le_tsum | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.PSeries | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 36
} | {
"line": 226,
"column": 2
} | [
{
"pp": "C : ℕ\nu : ℕ → ℕ\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nf : ℕ → ℝ≥0\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (fun i ↦ ↑(f i)) n ≤ (fun i ↦ ↑(f i)) m\n⊢ (Summable fun k ↦ (↑(u (k + 1)) - ↑(u k)) * (fun i ↦ ↑(f i)) (u k)) ↔ Summable fun i ↦... | [
"C : ℕ\nu : ℕ → ℕ\nh_pos : ∀ (n : ℕ), 0 < u n\nhu_strict : StrictMono u\nhC_nonzero : C ≠ 0\nh_succ_diff : SuccDiffBounded C u\nf : ℕ → ℝ≥0\nhf : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\n⊢ (Summable fun k ↦ (↑(u (k + 1)) - ↑(u k)) * ↑(f (u k))) ↔ Summable fun i ↦ ↑(f i)"
] | simp only [NNReal.coe_le_coe] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 249,
"column": 6
} | {
"line": 249,
"column": 31
} | {
"line": 250,
"column": 6
} | [
{
"pp": "case mpr.ht\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nt : Set D\nhcompact : IsCompact t\nh : tᶜ ⊆ {x | ‖f x‖ ≤ c}\n⊢ ∃ C, ∀ x ∈ f '' tᶜ, ‖x‖ ≤ C",
"ppTe... | [
"case mpr.ht\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nD : Type u_2\ninst✝ : TopologicalSpace D\nf : D → E\nhf : Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nc : ℝ\nt : Set D\nhcompact : IsCompact t\nh : tᶜ ⊆ {x | ‖f x‖ ≤ c}\nx : E\nhx : x ∈ f '' tᶜ\n⊢ ‖x‖ ≤ c"
] | refine ⟨c, fun x hx ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.PSeries | {
"line": 286,
"column": 86
} | {
"line": 286,
"column": 99
} | {
"line": 287,
"column": 8
} | [
{
"pp": "case inl\np : ℝ\nhp : 0 ≤ p\n⊢ (Summable fun k ↦ 2 ^ ↑k * (2 ^ (↑k * p))⁻¹) ↔ 1 < p",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real.instPow",
"Real",
"NonUnitalCommRing.toNonU... | [
"case inl\np : ℝ\nhp : 0 ≤ p\n⊢ (Summable fun k ↦ 2 ^ ↑k * (2 ^ (p * ↑k))⁻¹) ↔ 1 < p"
] | mul_comm _ p, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.SMulAntidiagonal | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 51
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_1\nP : Type u_2\ninst✝³ : LinearOrder G\ninst✝² : LinearOrder P\ninst✝¹ : SMul G P\ninst✝ : IsOrderedCancelSMul G P\ns : Set G\nt : Set P\nhs : s.IsWF\nht : t.IsWF\nhns : s.Nonempty\nhnt : t.Nonempty\nb : P\nhat : b ∈ t\nhas : hs.min hns ∈ s\nhst : hs.min hns • b = hs.min hns • ht.m... | [] | exact ⟨rfl, IsCancelSMul.left_cancel _ _ _ hst⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 39,
"column": 63
} | {
"line": 39,
"column": 79
} | {
"line": 42,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max (lift.{u, v} #R) (lift.{v, u} #σ)) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 39,
"column": 63
} | {
"line": 39,
"column": 79
} | {
"line": 42,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max (lift.{u, v} #R) (lift.{v, u} #σ)) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 39,
"column": 63
} | {
"line": 39,
"column": 79
} | {
"line": 42,
"column": 0
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max (lift.{u, v} #R) (lift.{v, u} #σ)) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "σ R : Type u\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max #R #σ) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Cardinal.lift",
"Cardinal.... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "σ R : Type u\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max #R #σ) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Cardinal.lift",
"Cardinal.... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Cardinal | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "σ R : Type u\ninst✝² : CommSemiring R\ninst✝¹ : Nonempty σ\ninst✝ : Nontrivial R\n⊢ #(MvPolynomial σ R) = max (max #R #σ) ℵ₀",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Cardinal.lift",
"Cardinal.... | [] | simp [sup_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 26
} | {
"line": 67,
"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⊢ (g.comp f) (X i) = (aeval fun i ↦ g (X i)) (f (X i))",
"ppTerm": "?m.87",
"assigned": true,
"usedCons... | [
"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⊢ g (f (X i)) = (aeval fun i ↦ g (X i)) (f (X i))"
] | rw [AlgHom.comp_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 221,
"column": 4
} | {
"line": 224,
"column": 94
} | {
"line": 225,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\na : α\nf : α →₀ ℕ\nh : a ∈ f.support\n⊢ Nat.multinomial f.support ⇑f =\n (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",... | [
"case neg\nα : Type u_1\na : α\nf : α →₀ ℕ\nh : a ∉ f.support\n⊢ Nat.multinomial f.support ⇑f =\n (f.sum fun x ↦ id).choose (f a) * Nat.multinomial (f.update a 0).support ⇑(f.update a 0)"
] | · rw [← Finset.insert_erase h, Nat.multinomial_insert (Finset.notMem_erase a _),
Finset.add_sum_erase _ f h, support_update_zero]
congr 1
exact Nat.multinomial_congr fun _ h ↦ (Function.update_of_ne (mem_erase.1 h).1 0 f).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 62
} | {
"line": 42,
"column": 4
} | [
{
"pp": "case refine_3\nσ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\nP✝ : MvPolynomial σ R\ninst✝¹ : Finite σ\nthis : Fintype σ\nα : Type u_1\ninst✝ : Fintype α\nH : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)\nP : MvPolynomial (Option α) R\n⊢ IsNilpotent P ↔ ∀ (i : Opt... | [
"case refine_3\nσ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\nP✝ : MvPolynomial σ R\ninst✝¹ : Finite σ\nthis : Fintype σ\nα : Type u_1\ninst✝ : Fintype α\nH : ∀ (x : MvPolynomial α R), IsNilpotent x ↔ ∀ (i : α →₀ ℕ), IsNilpotent (coeff i x)\nP : Polynomial (MvPolynomial α R)\n⊢ IsNilpotent ((optionEquivLeft R α)... | obtain ⟨P, rfl⟩ := (optionEquivLeft _ _).symm.surjective P | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 337,
"column": 2
} | {
"line": 338,
"column": 25
} | {
"line": 339,
"column": 2
} | [
{
"pp": "case inl\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\nhR : Subsingleton R\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case inr\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\nhR : Nontrivial R\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r"
] | · simp only [Subsingleton.elim _ p, dvd_refl, and_self, and_true, exists_const, true_iff]
refine ⟨n, le_refl n⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 29
} | {
"line": 53,
"column": 30
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nP : MvPolynomial (Fin n) R\nf : Fin n ↪ σ\nH✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)\ni : σ →₀ ℕ\nH : i ∉ Set.range (Finsupp.embDomain f)\n⊢ IsNilpotent (coeff i ((rename ⇑f) P))",
"ppTerm": "?neg✝",
"assigned": true,
"... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nP : MvPolynomial (Fin n) R\nf : Fin n ↪ σ\nH✝ : ∀ (i : Fin n →₀ ℕ), IsNilpotent (coeff i P)\ni : σ →₀ ℕ\nH : i ∉ Set.range (Finsupp.embDomain f)\n⊢ IsNilpotent 0",
"case neg.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nn : ℕ\nP : MvPolynomial... | rw [coeff_rename_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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