module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Regular | {
"line": 233,
"column": 55
} | {
"line": 238,
"column": 72
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np q : Set α → Prop\nU : Set α\nε : ℝ≥0∞\nH : μ.InnerRegularWRT p q\nh0 : p ∅\nhU : q U\nhμU : μ U ≠ ∞\nhε : ε ≠ 0\n⊢ ∃ K ⊆ U, p K ∧ μ U < μ K + ε",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"ENNReal.instAd... | by
rcases eq_or_ne (μ U) 0 with h₀ | h₀
· refine ⟨∅, empty_subset _, h0, ?_⟩
rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
· rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 568,
"column": 2
} | {
"line": 569,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\nhs : MeasurableSet s\nr : ℝ≥0∞\nhr : r < μ s\nB : ℕ → Set α := spanningSets μ\nhBU : ⋃ n, s ∩ B n = s\n⊢ ∃ K ⊆ s, (fun s ↦ MeasurableSet s ∧ μ s ≠ ∞) K ∧ r < μ K",
"usedConstants": [
"Eq.mpr",
"Me... | have : μ s = ⨆ n, μ (s ∩ B n) := by
rw [← (monotone_const.inter (monotone_spanningSets μ)).measure_iUnion, hBU] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nβ : Type u_3\nγ : Type u_4\ninst✝ : MeasurableSingletonClass α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nx : α\nf : γ → β\nhf : MeasurableEmbedding f\ns : Set γ\nhs : MeasurableSet s\n⊢ MeasurableSet ((fun y ↦ (x, f y)) '' s)",
"usedConstants": [
... | convert! (MeasurableSet.singleton x).prod (hf.measurableSet_image.mpr hs) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 570,
"column": 6
} | {
"line": 570,
"column": 87
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ≥0\nhf : Measurable f\nμ : Measure α\ninst✝ : SigmaFinite μ\nsigma_finite_sets : ℕ → Set α := spanningSets μ\nnorm_sets : ℕ → Set α := fun n ↦ {x | f x ≤ ↑n}\nnorm_sets_spanning : ⋃ n, norm_sets n = univ\nsets : ℕ → Set α := fun n ↦ sigma_finite_sets n ∩ n... | refine Set.iUnion_inter_of_monotone (monotone_spanningSets μ) fun i j hij x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 60
} | [
{
"pp": "case h\nα : Type u_1\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ⇑φ ≤ fun a ↦ f a\nh : ¬∀ᵐ (a : α) ∂μ, φ a ≠ ∞\nh_meas : μ (⇑φ ⁻¹' {∞}) ≠ 0\nb : ℝ≥0∞\nhb : b < ∞\nn : ℕ\nhn : b < ↑n * μ (⇑φ ⁻¹' {∞})\n⊢ (∀ (x : α), (⇑φ ⁻¹' {∞}).indicator (fun x ↦ ↑(Function.const α (↑n) x)) ... | refine ⟨indicator_le fun x hx => le_trans ?_ (hφ _), hn⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 51
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh : ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < (SimpleFunc.map ofNNReal φ).lintegral μ + ε\nhb : ∀ (i : α →ₛ ℝ... | simp only [add_apply, sub_apply, add_tsub_eq_max] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_3\nf : ι → α → ℝ≥0∞\nu : Filter ι\ninst✝ : u.IsCountablyGenerated\nh_meas : ∀ (i : ι), AEMeasurable (f i) μ\nhu : ¬u.NeBot\n⊢ ∫⁻ (a : α), liminf (fun i ↦ f i a) u ∂μ ≤ liminf (fun i ↦ ∫⁻ (a : α), f i a ∂μ) u",
"usedConstants":... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_3\nf : ι → α → ℝ≥0∞\nu : Filter ι\ninst✝ : u.IsCountablyGenerated\nh_meas : ∀ (i : ι), AEMeasurable (f i) μ\nhu : ¬u.NeBot\n⊢ ∫⁻ (a : α), liminf (fun i ↦ f i a) u ∂μ ≤ liminf (fun i ↦ ∫⁻ (a : α), f i a ∂μ) u",
"usedConstants":... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 12
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_3\nf : ι → α → ℝ≥0∞\nu : Filter ι\ninst✝ : u.IsCountablyGenerated\nh_meas : ∀ (i : ι), AEMeasurable (f i) μ\nhu : ¬u.NeBot\n⊢ ∫⁻ (a : α), liminf (fun i ↦ f i a) u ∂μ ≤ liminf (fun i ↦ ∫⁻ (a : α), f i a ∂μ) u",
"usedConstants":... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 545,
"column": 68
} | {
"line": 545,
"column": 82
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\nr : ℝ≥0∞\nthis : ∀ x ∈ {x | f x = r}, f x = r\n⊢ r * μ (univ ∩ {x | f x = r}) = r * μ {x | f x = r}",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
"Co... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1028,
"column": 61
} | {
"line": 1031,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nι : Type u_5\nf : α →ₛ ℝ≥0∞\nμ : ι → Measure α\n⊢ f.lintegral (Measure.sum μ) = ∑' (i : ι), f.lintegral (μ i)",
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"MeasurableSet"... | by
simp only [lintegral, Measure.sum_apply, f.measurableSet_preimage, ← Finset.tsum_subtype, ←
ENNReal.tsum_mul_left]
apply ENNReal.tsum_comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1273,
"column": 8
} | {
"line": 1275,
"column": 53
} | [
{
"pp": "case h\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → ... | rw [Set.image_subset_iff]
convert! Set.subset_univ _
exact preimage_const_of_mem (mem_singleton _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1273,
"column": 8
} | {
"line": 1275,
"column": 53
} | [
{
"pp": "case h\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → ... | rw [Set.image_subset_iff]
convert! Set.subset_univ _
exact preimage_const_of_mem (mem_singleton _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1315,
"column": 8
} | {
"line": 1317,
"column": 53
} | [
{
"pp": "case h\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty γ\nP : (α →ₛ γ) → Prop\nconst : ∀ (c : γ), P (SimpleFunc.const α c)\npcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)\nc : γ := Classical.ofNonempty\nx : γ\ns : Finset γ\nhxs : ... | rw [Set.image_subset_iff]
convert! Set.subset_univ _
exact preimage_const_of_mem (mem_singleton _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1315,
"column": 8
} | {
"line": 1317,
"column": 53
} | [
{
"pp": "case h\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty γ\nP : (α →ₛ γ) → Prop\nconst : ∀ (c : γ), P (SimpleFunc.const α c)\npcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)\nc : γ := Classical.ofNonempty\nx : γ\ns : Finset γ\nhxs : ... | rw [Set.image_subset_iff]
convert! Set.subset_univ _
exact preimage_const_of_mem (mem_singleton _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 331,
"column": 6
} | {
"line": 332,
"column": 44
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\n⊢ {x | ‖x⁻¹ * a‖ < b 0} ∈ 𝓝 a",
"usedConstants"... | simp_rw [← dist_eq_norm_inv_mul]
exact Metric.ball_mem_nhds _ (b_pos _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 331,
"column": 6
} | {
"line": 332,
"column": 44
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\n⊢ {x | ‖x⁻¹ * a‖ < b 0} ∈ 𝓝 a",
"usedConstants"... | simp_rw [← dist_eq_norm_inv_mul]
exact Metric.ball_mem_nhds _ (b_pos _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 98,
"column": 66
} | {
"line": 98,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\n⊢ |a ⊓ b - c ⊓ d| = |a ⊓ b - c ⊓ b + (c ⊓ b - c ⊓ d)|",
"usedConstants": [
"Eq.mpr",
"sub_add_sub_cancel",
"abs",
"congrArg",
"AddCommGr... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 111,
"column": 66
} | {
"line": 111,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\n⊢ |a ⊔ b - c ⊔ d| = |a ⊔ b - c ⊔ b + (c ⊔ b - c ⊔ d)|",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"sub_add_sub_cancel",
"abs",
... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 78
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : NormedDivisionRing α\na b : α\n⊢ ‖1‖ * ‖1‖ = ‖1‖ * 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"HMul.hMul",
"congrArg",
"NormedDivisionRing.toNorm",
"Normed... | ← norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 273,
"column": 4
} | {
"line": 275,
"column": 29
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : DenselyNormedField α\n⊢ ∀ (a₁ a₂ : ↑(Set.range nnnorm)), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Preorder.toLT",
"NormedField.exists_lt_nnnorm_lt",
"Seminormed... | rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy
let ⟨z, h⟩ := exists_lt_nnnorm_lt α hxy
exact ⟨⟨‖z‖₊, z, rfl⟩, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 273,
"column": 4
} | {
"line": 275,
"column": 29
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_4\ninst✝ : DenselyNormedField α\n⊢ ∀ (a₁ a₂ : ↑(Set.range nnnorm)), a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Preorder.toLT",
"NormedField.exists_lt_nnnorm_lt",
"Seminormed... | rintro ⟨-, x, rfl⟩ ⟨-, y, rfl⟩ hxy
let ⟨z, h⟩ := exists_lt_nnnorm_lt α hxy
exact ⟨⟨‖z‖₊, z, rfl⟩, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Ring.Lemmas | {
"line": 167,
"column": 36
} | {
"line": 167,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NonUnitalSeminormedRing α\nx e : α × α\n⊢ ‖e.1 * e.2 - x.1 * x.2‖ ≤ ‖e.1 * e.2 - e.1 * x.2 + (e.1 * x.2 - x.1 * x.2)‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"sub_add_sub_cancel",
"HMul.... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 85
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nF : Type u_3\ninst✝³ : PseudoMetricSpace X\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : EquivLike F X Y\ninst✝ : DilationEquivClass F X Y\ne : F\n⊢ map (⇑e) (cobounded X) = cobounded Y",
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg... | rw [← Dilation.comap_cobounded e, map_comap_of_surjective (EquivLike.surjective e)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 85
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nF : Type u_3\ninst✝³ : PseudoMetricSpace X\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : EquivLike F X Y\ninst✝ : DilationEquivClass F X Y\ne : F\n⊢ map (⇑e) (cobounded X) = cobounded Y",
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg... | rw [← Dilation.comap_cobounded e, map_comap_of_surjective (EquivLike.surjective e)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 85
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nF : Type u_3\ninst✝³ : PseudoMetricSpace X\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : EquivLike F X Y\ninst✝ : DilationEquivClass F X Y\ne : F\n⊢ map (⇑e) (cobounded X) = cobounded Y",
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg... | rw [← Dilation.comap_cobounded e, map_comap_of_surjective (EquivLike.surjective e)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.MulAction | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 35
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\n⊢ p ∈ (fun x ↦ s • x) '' ball x ε ↔ p ∈ ball (s • x) (‖s‖ * ε)",
"usedConstants": [
"Norm.norm",
... | simp_rw [Set.mem_image, mem_ball] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 68
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : PseudoMetrizableSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : BorelSpace β\nι : Type u_3\nμ : Measure α\nf : ι → α → β\ng : α → β\nu : Filter ι\nhu : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (... | exact @aeSeq.fun_prop_of_mem_aeSeqSet _ α β _ _ _ _ _ h'f x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 218,
"column": 8
} | {
"line": 220,
"column": 46
} | [
{
"pp": "case neg\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | simp only [SimpleFunc.piecewise_apply, H, ite_false]
apply hg'
simpa [H] using (mem_union _ _ _).1 hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 218,
"column": 8
} | {
"line": 220,
"column": 46
} | [
{
"pp": "case neg\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Conti... | simp only [SimpleFunc.piecewise_apply, H, ite_false]
apply hg'
simpa [H] using (mem_union _ _ _).1 hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 91
} | [
{
"pp": "E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\ny : E\nhy : y ≠ 0\n⊢ dist (x + r • ‖y‖⁻¹ • y) (x - r • ‖y‖⁻¹ • y) ≤ diam (sphere x r)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Seminorme... | apply dist_le_diam_of_mem isBounded_sphere <;> simp [norm_smul, hy, abs_of_nonneg hr] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 91
} | [
{
"pp": "E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\ny : E\nhy : y ≠ 0\n⊢ dist (x + r • ‖y‖⁻¹ • y) (x - r • ‖y‖⁻¹ • y) ≤ diam (sphere x r)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Seminorme... | apply dist_le_diam_of_mem isBounded_sphere <;> simp [norm_smul, hy, abs_of_nonneg hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 91
} | [
{
"pp": "E : Type u_6\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : 0 ≤ r\ny : E\nhy : y ≠ 0\n⊢ dist (x + r • ‖y‖⁻¹ • y) (x - r • ‖y‖⁻¹ • y) ≤ diam (sphere x r)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Seminorme... | apply dist_le_diam_of_mem isBounded_sphere <;> simp [norm_smul, hy, abs_of_nonneg hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 134,
"column": 7
} | {
"line": 136,
"column": 62
} | [] | a * r' ≤ 2 * r' := by gcongr
_ ≤ _ := by simpa only [← Metric.diam_sphere_eq x hr'.le]
using diam_mono (sphere_subset_ball hr'') isBounded_ball | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 78,
"column": 44
} | {
"line": 78,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nhf : ∀ a ∈ s, a ∈ t → f a ≤ 1\nhf' : ∀ a ∈ s, a ∉ t → f a = 0\n⊢ ∀ a ∈ tᶜ, s.indicator f a = 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Set.indicator",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 78,
"column": 44
} | {
"line": 78,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nhf : ∀ a ∈ s, a ∈ t → f a ≤ 1\nhf' : ∀ a ∈ s, a ∉ t → f a = 0\n⊢ ∀ a ∈ tᶜ, s.indicator f a = 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Set.indicator",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Markov | {
"line": 78,
"column": 44
} | {
"line": 78,
"column": 52
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : MeasurableSet s\nf : α → ℝ≥0∞\nhf : ∀ a ∈ s, a ∈ t → f a ≤ 1\nhf' : ∀ a ∈ s, a ∉ t → f a = 0\n⊢ ∀ a ∈ tᶜ, s.indicator f a = 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Set.indicator",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhfi : ∫⁻ (x : α), f x ∂μ = ∞\n⊢ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), g x - f x ∂μ + ∫⁻ (x : α), f x ∂μ",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"add_top",
"congrArg"... | rw [hfi, add_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 174,
"column": 57
} | {
"line": 174,
"column": 65
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nhf₀ : ¬∫⁻ (a : α), f a ∂μ = 0\ng : α → ℝ≥0∞\nhgf : g ≤ f\nhg_meas : Measurable g\nhgsupp : μ (support g) < ∞\nhgε : ∫⁻ (a : α), f a ∂μ - ε < ∫⁻ (a : α), g a ∂μ\nx✝ : α\n⊢ x✝ ∈ (suppo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 680,
"column": 2
} | {
"line": 690,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nι : Type u_5\ninst✝² : PseudoMetrizableSpace β\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → β\ng : α → β\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nlim : ∀ᵐ (x : α) ∂μ, Te... | borelize β
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨?_, ?_⟩
· exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).aemeasurable) lim
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ := fun n =>
(aestronglyMeasurable_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 680,
"column": 2
} | {
"line": 690,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nι : Type u_5\ninst✝² : PseudoMetrizableSpace β\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → β\ng : α → β\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nlim : ∀ᵐ (x : α) ∂μ, Te... | borelize β
refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨?_, ?_⟩
· exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).aemeasurable) lim
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ := fun n =>
(aestronglyMeasurable_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 91
} | [
{
"pp": "case inr.refine_1\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : Nonempty β\ninst✝ : T2Space β\nhα : Nonempty α\nhf : StronglyMeasurable f\nfs : ℕ → α →ₛ β := hf.approx\n⊢ f = fun x ↦ f hα.some",
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLatt... | have h_fs_tendsto : ∀ x, Tendsto (fun n => fs n x) atTop (𝓝 (f x)) := hf.tendsto_approx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Count | {
"line": 34,
"column": 75
} | {
"line": 34,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty α\n⊢ count ≠ 0",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.dirac",
"MeasureTheory.Measure.instZero",
"forall_const._simp_1",
"True",
"of_eq_true",... | simp [count] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Count | {
"line": 34,
"column": 75
} | {
"line": 34,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty α\n⊢ count ≠ 0",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.dirac",
"MeasureTheory.Measure.instZero",
"forall_const._simp_1",
"True",
"of_eq_true",... | simp [count] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Count | {
"line": 34,
"column": 75
} | {
"line": 34,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty α\n⊢ count ≠ 0",
"usedConstants": [
"False",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.dirac",
"MeasureTheory.Measure.instZero",
"forall_const._simp_1",
"True",
"of_eq_true",... | simp [count] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UnitInterval | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝³ : AddCommGroup α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderedAddMonoid α\na b : α\nh : a ≤ b\nδ : α\ninst✝ : Archimedean α\nhδ : 0 < δ\nm : ℕ\nhm : b - a ≤ m • δ\n⊢ ∃ m, ∀ n ≥ m, ↑(addNSMul h δ n) = b",
"usedConstants": [
"PartialOrder.toPreorder",
"Membership.mem",
... | refine ⟨m, fun n hn ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.UnitInterval | {
"line": 398,
"column": 4
} | {
"line": 403,
"column": 13
} | [
{
"pp": "case neg\na b : ℝ\nx y z : ↑(Icc a b)\ns t : ↑unitInterval\nhs : ¬↑s = 1\nht : ¬↑t = 1\n⊢ (1 - ↑s) * ↑x + ↑s * ((1 - ↑t) * ↑y + ↑t * ↑z) =\n (1 - ↑s * ↑t) * ((1 - ↑s * (1 - ↑t) / (1 - ↑s * ↑t)) * ↑x + ↑s * (1 - ↑t) / (1 - ↑s * ↑t) * ↑y) + ↑s * ↑t * ↑z",
"usedConstants": [
"Mathlib.Tactic.R... | · have : (1 - s * t : ℝ) ≠ 0 := by
intro h
have : 1 ≤ (t : ℝ) := by nlinarith [s.2.2, t.2.1]
grind
field_simp
ring_nf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UnitInterval | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"instReflLe",
"congrArg",
"Real.in... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.UnitInterval | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"instReflLe",
"congrArg",
"Real.in... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UnitInterval | {
"line": 431,
"column": 4
} | {
"line": 431,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"instReflLe",
"congrArg",
"Real.in... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UnitInterval | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.UnitInterval | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UnitInterval | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 12
} | [
{
"pp": "case pos\na b : ℝ\nx y z : ↑(Icc a b)\nhxy : x ≤ y\nhyz : y ≤ z\nh : ↑z - ↑x = 0\n⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Count | {
"line": 178,
"column": 2
} | {
"line": 181,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Function.Injective f\nh2f : Measurable f\n⊢ map f count ≤ count",
"usedConstants": [
"MeasurableSet.preimage",
"Eq.mpr",
"Set.encard",
"MeasureTheory.Measure",
"MeasureTh... | refine le_intro fun s hs _ ↦ ?_
rw [map_apply h2f hs, count_apply (hs.preimage h2f), count_apply hs, ← hf.encard_image]
have := image_preimage_subset f s
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Count | {
"line": 178,
"column": 2
} | {
"line": 181,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : Function.Injective f\nh2f : Measurable f\n⊢ map f count ≤ count",
"usedConstants": [
"MeasurableSet.preimage",
"Eq.mpr",
"Set.encard",
"MeasureTheory.Measure",
"MeasureTh... | refine le_intro fun s hs _ ↦ ?_
rw [map_apply h2f hs, count_apply (hs.preimage h2f), count_apply hs, ← hf.encard_image]
have := image_preimage_subset f s
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ∀ s ∈ range singleton, μ.restrict s = ν.restrict s",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"instSMul... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ∀ s ∈ range singleton, μ.restrict s = ν.restrict s",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"instSMul... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 67
} | {
"line": 112,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ∀ s ∈ range singleton, μ.restrict s = ν.restrict s",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"instSMul... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 198,
"column": 53
} | {
"line": 198,
"column": 67
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nthis : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))\nf : ℕ × ℕ → Set α :=\n fun p ↦\n (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFiniteSetGE ν p.1\ne ... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Perfect | {
"line": 171,
"column": 42
} | {
"line": 175,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : T25Space α\nhC : Perfect C\ny : α\nyC : y ∈ C\n⊢ ∃ x ∈ C, x ≠ y",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Set.univ",
"Perfect.acc",
"accPt_iff_nhds",
"Membership.mem",
"Exists",
... | by
have := hC.acc _ yC
rw [accPt_iff_nhds] at this
rcases this univ univ_mem with ⟨x, xC, hxy⟩
exact ⟨x, xC.2, hxy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Perfect | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 37
} | [
{
"pp": "case hs\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed[inst✝¹] C\nb : Set (Set α)\nbct : b.Countable\nleft✝ : ∅ ∉ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := ⋯\nV : Set α := ⋯\nD : Set α := ⋯\n⊢ v.Countable",
"usedConstants": [
... | · exact bct.mono (sep_subset _ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1108,
"column": 10
} | {
"line": 1108,
"column": 33
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ... | ← Set.union_iUnion tᶜ T | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 93
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\na : α → ℝ≥0∞\na_mble : Measurable a\nc : ℝ≥0∞\ntsum_le_c : ∫⁻ (a_1 : α), a a_1 ∂count ≤ c\nε : ℝ≥0∞\nε_ne_zero : ε ≠ 0\nε_ne_top : ε ≠ ∞\n⊢ count {i | ε ≤ a i} ≤ c / ε",
"usedConstants": [
"MeasureTheory.Measure",
... | apply (MeasureTheory.meas_ge_le_lintegral_div a_mble.aemeasurable ε_ne_zero ε_ne_top).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nμ : Measure α\ninst✝ : SFinite μ\nh : IsFiniteMeasure μ\ng : ℕ → α → ℝ≥0∞\nhgm : ∀ (n : ℕ), Measurable (g n)\nhgf : ∀ (n : ℕ), g n ≤ f\nhgle : ∀ (n : ℕ), g n ≤ ↑n\nhgint : ∀ (n : ℕ), ∫⁻ (a : α), min (f a) ↑n ∂μ = ∫⁻ (a : α), g n a ... | rcases ψ.range.bddAbove with ⟨C, hC⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 85
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nC : ℝ≥0∞\nf : Set α → ℝ≥0∞\nhf : ∀ (s : Set α), MeasurableSet s → μ s ≠ ∞ → f s ≤ C\nh_F_lim : ∀ (S : ℕ → Set α), (∀ (n : ℕ), MeasurableSet (S n)) → Monotone S → f (⋃ n, S n) ≤ ⨆ n, f (S n)\nS : ℕ → Set... | exact ((le_trim hm).trans_lt (@measure_spanningSets_lt_top _ m (μ.trim hm) _ n)).ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.FunctionSeries | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 10
} | [
{
"pp": "β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\ns : Set β\nε : ℝ\nεpos : 0 < ε\nt : Finset ι\nht : ∀ (b : Finset ι), t ⊆ b → ∑' (a : { x // x ∉ b }), u ↑a < ε\nN : Set ι\nhN : N ∈ cofinite\nHN : ∀ y ∈ N, ∀ x ∈ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : x = y\nh✝¹ : x = z\nh✝ : y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : x = y\nh✝¹ : x = z\nh✝ : ¬y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : x = y\nh✝¹ : ¬x = z\nh✝ : y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : x = y\nh✝¹ : ¬x = z\nh✝ : ¬y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : ¬x = y\nh✝¹ : x = z\nh✝ : y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : ¬x = y\nh✝¹ : x = z\nh✝ : ¬y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : ¬x = y\nh✝¹ : ¬x = z\nh✝ : y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"False",
"Real.instLE",
"Real",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 271,
"column": 65
} | {
"line": 271,
"column": 73
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y z : X\nh✝² : ¬x = y\nh✝¹ : ¬x = z\nh✝ : ¬y = z\n⊢ (if x = z then 0 else 1) ≤ (if x = y then 0 else 1) + if y = z then 0 else 1",
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 12
} | [
{
"pp": "case h.e'_3\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nm : MetricSpace X :=\n { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,\n uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 12
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nm : MetricSpace X :=\n { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,\n uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.MetricSpace.Polish | {
"line": 66,
"column": 21
} | {
"line": 69,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : IsCompletelyMetrizableSpace α\n⊢ PolishSpace α",
"usedConstants": [
"TopologicalSpace.UpgradedIsCompletelyMetrizableSpace.toMetricSpace",
"PseudoMetricSpace.toUniformSpace",
"EMetricSpace.t... | by
letI := upgradeIsCompletelyMetrizable α
haveI := UniformSpace.secondCountable_of_separable α
constructor | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 71
} | [
{
"pp": "case mp\nX : Type u\nY : Type v\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\ns : Set ((X ⊕ Y) × (X ⊕ Y))\nhsX : s ∈ Filter.map (fun p ↦ (Sum.inl p.1, Sum.inl p.2)) (𝓤 X)\nhsY : s ∈ Filter.map (fun p ↦ (Sum.inr p.1, Sum.inr p.2)) (𝓤 Y)\nεX : ℝ\nεX0 : εX > 0\nhX : ∀ ⦃a b : X⦄, dist a b < εX → (a, b)... | refine ⟨min (min εX εY) 1, lt_min (lt_min εX0 εY0) zero_lt_one, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 20
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nh : x ≠ y\n⊢ x (firstDiff x y) ≠ y (firstDiff x y)",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"PiNat.firstDiff",
"PiNat.firstDiff_def",
"congrArg",
"Classical.propDecidable",
"PiNat.definition._proof_1._@.Mathl... | firstDiff_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Polish | {
"line": 284,
"column": 26
} | {
"line": 284,
"column": 40
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed[inst✝¹] s\nthis✝ : PolishSpace ↑s\nt : Set α := sᶜ\nthis : PolishSpace ↑t\nf : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s\nhle : coinduced (⇑f) instTopologicalSpaceSum ≤ inst✝¹\n⊢ IsOpen[instTopologicalSp... | isOpen_sum_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Separation.CountableSeparatingOn | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 59
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : T0Space ↑s\ninst✝ : SecondCountableTopology ↑s\nx : ↑s\nx✝¹ : x ∈ univ\ny : ↑s\nx✝ : y ∈ univ\nh : ∀ s_1 ∈ countableBasis ↑s, x ∈ s_1 ↔ y ∈ s_1\n⊢ x = y",
"usedConstants": [
"Iff.mpr",
"TopologicalSpace.countableBasis",
... | exact ((isBasis_countableBasis _).inseparable_iff.2 h).eq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 74
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} → Perf... | apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 533,
"column": 6
} | {
"line": 533,
"column": 25
} | [
{
"pp": "case neg\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A - 1 < Nat.find A\ny : (n :... | mem_cylinder_iff_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 46
} | [
{
"pp": "case refine_3\nι : Type u_2\ninst✝¹ : Countable ι\nα : Type u_3\ninst✝ : MeasurableSpace α\ns t : ι → Set α\nu : ι → ι → Set α\nhsu : ∀ (m n : ι), s m ⊆ u m n\nhtu : ∀ (m n : ι), Disjoint (t n) (u m n)\nhu : ∀ (m n : ι), MeasurableSet (u m n)\nm : ι\n⊢ MeasurableSet (⋂ n, u m n)",
"usedConstants": ... | exact MeasurableSet.iInter fun n => hu m n | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 94
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm :\n ∀ {α : Type ?u.1622.10} {β : Type ?u.1622.9} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs :... | suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator (m hs)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1130,
"column": 4
} | {
"line": 1130,
"column": 12
} | [
{
"pp": "case inl\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ = ∅\n⊢ ∃ n, C ∈ comap (distDense... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1130,
"column": 4
} | {
"line": 1130,
"column": 12
} | [
{
"pp": "case inl\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ = ∅\n⊢ ∃ n, C ∈ comap (distDense... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1130,
"column": 4
} | {
"line": 1130,
"column": 12
} | [
{
"pp": "case inl\nX : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ = ∅\n⊢ ∃ n, C ∈ comap (distDense... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1140,
"column": 90
} | {
"line": 1140,
"column": 98
} | [
{
"pp": "X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn :... | simp [ε] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Complement | {
"line": 259,
"column": 2
} | {
"line": 262,
"column": 74
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\n⊢ IsComplement (range f) ↑H",
"usedConstants": [
"Eq.mpr",
"congrArg",
"QuotientGroup.mk",
"Subtype.casesOn",
"Membership.mem",
"Eq.rec",
"Set.Elem",
"id",
... | rw [isComplement_subgroup_right_iff_bijective]
refine ⟨?_, fun q ↦ ⟨⟨f q, q, rfl⟩, hf q⟩⟩
rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h
exact Subtype.ext <| congr_arg f <| ((hf q₁).symm.trans h).trans (hf q₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Complement | {
"line": 259,
"column": 2
} | {
"line": 262,
"column": 74
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\n⊢ IsComplement (range f) ↑H",
"usedConstants": [
"Eq.mpr",
"congrArg",
"QuotientGroup.mk",
"Subtype.casesOn",
"Membership.mem",
"Eq.rec",
"Set.Elem",
"id",
... | rw [isComplement_subgroup_right_iff_bijective]
refine ⟨?_, fun q ↦ ⟨⟨f q, q, rfl⟩, hf q⟩⟩
rintro ⟨-, q₁, rfl⟩ ⟨-, q₂, rfl⟩ h
exact Subtype.ext <| congr_arg f <| ((hf q₁).symm.trans h).trans (hf q₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1140,
"column": 90
} | {
"line": 1140,
"column": 98
} | [
{
"pp": "X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn :... | simp [ε] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1140,
"column": 90
} | {
"line": 1140,
"column": 98
} | [
{
"pp": "X : Type u_3\ninst✝¹ : MetricSpace X\ninst✝ : SeparableSpace X\nx : X\nC : Set X\nhxC : C ∈ 𝓝 x\nε : ℝ := min (infDist x (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ)) 1\nhC : (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] Cᶜ).Nonempty\nthis : Nonempty X\nn : ℕ\nhn :... | simp [ε] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Prod | {
"line": 171,
"column": 48
} | {
"line": 171,
"column": 63
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ns : Set G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\n⊢ EventuallyConst s (Filter.map Inv.inv (ae μ)) ↔ EventuallyConst s (ae μ)",
"usedConstants": [
"Measure... | Filter.map_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 17
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : StandardBorelSpace X\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : T0Space Y\ninst✝² : MeasurableSpace Y\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Measurable f\nhsurj : Surjective f\nd : Measurable f\n⊢... | letI := borel Y | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.MeasureTheory.Group.Prod | {
"line": 281,
"column": 28
} | {
"line": 281,
"column": 41
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\ns : Set G\ninst✝⁴ : MeasurableInv G\nμ' ν' : Measure G\ninst✝³ : SigmaFinite μ'\ninst✝² : SigmaFinite ν'\ninst✝¹ : μ'.IsMulLeftInvariant\ninst✝ : ν'.IsMulLeftInvariant\nh2s : ν' s ≠ 0\nh3s : ν' s ≠ ∞\nhν : ν' = 0\n⊢ ... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 716,
"column": 6
} | {
"line": 716,
"column": 48
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | exact ball_mem_nhds _ (half_pos (u_pos n)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LConvolution | {
"line": 145,
"column": 2
} | {
"line": 148,
"column": 17
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝⁴ : CommGroup G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : μ.IsInvInvariant\nf g : G → ℝ≥0∞\n⊢ f ⋆ₘₗ[μ] g = g ⋆ₘₗ[μ] f",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"... | ext x
simp only [mlconvolution_def]
rw [← lintegral_mul_left_eq_self _ x, ← lintegral_inv_eq_self]
simp [mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LConvolution | {
"line": 145,
"column": 2
} | {
"line": 148,
"column": 17
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝⁴ : CommGroup G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : μ.IsInvInvariant\nf g : G → ℝ≥0∞\n⊢ f ⋆ₘₗ[μ] g = g ⋆ₘₗ[μ] f",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"... | ext x
simp only [mlconvolution_def]
rw [← lintegral_mul_left_eq_self _ x, ← lintegral_inv_eq_self]
simp [mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 741,
"column": 12
} | {
"line": 741,
"column": 55
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : CommSemigroup G\nμ : Measure G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ Measure.map (fun x ↦ x * g) μ = μ",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"Measure... | by simp_rw [mul_comm, map_mul_left_eq_self] | [anonymous] | Lean.Parser.Term.byTactic |
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