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.Typeclasses.SFinite | {
"line": 407,
"column": 2
} | {
"line": 462,
"column": 32
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ n, v n\nh'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞\n⊢ μ (toMeasurable μ t ∩ s) = μ (t ∩ s)",
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"Eq.mpr",
"MeasureT... | have A : ∃ t', t' ⊇ t ∧ MeasurableSet t' ∧ ∀ u, MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u) := by
let w n := toMeasurable μ (t ∩ v n)
have hw : ∀ n, μ (w n) < ∞ := by
intro n
simp_rw [w, measure_toMeasurable]
exact (h'v n).lt_top
set t' := ⋃ n, toMeasurable μ (t ∩ disjointed w n) with ht'
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 407,
"column": 2
} | {
"line": 462,
"column": 32
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\nv : ℕ → Set α\nhv : t ⊆ ⋃ n, v n\nh'v : ∀ (n : ℕ), μ (t ∩ v n) ≠ ∞\n⊢ μ (toMeasurable μ t ∩ s) = μ (t ∩ s)",
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"Eq.mpr",
"MeasureT... | have A : ∃ t', t' ⊇ t ∧ MeasurableSet t' ∧ ∀ u, MeasurableSet u → μ (t' ∩ u) = μ (t ∩ u) := by
let w n := toMeasurable μ (t ∩ v n)
have hw : ∀ n, μ (w n) < ∞ := by
intro n
simp_rw [w, measure_toMeasurable]
exact (h'v n).lt_top
set t' := ⋃ n, toMeasurable μ (t ∩ disjointed w n) with ht'
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 494,
"column": 2
} | {
"line": 495,
"column": 83
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ ⨆ i, (μ.restrict (spanningSets μ i)) s = μ s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"iSup",
"MeasureTheory.Measure.restrict",
"id",
"Condit... | rw [← measure_toMeasurable s,
← iSup_restrict_spanningSets_of_measurableSet (measurableSet_toMeasurable _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 55
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh : SeparatesPoints α\nthis : MeasurableSpace α := generateFrom S\nx y : α\nhxy : ∀ (s : Set α), MeasurableSet s → (x ∈ s ↔ y ∈ s)\n⊢ x = y",
"usedConstants": [
"MeasurableSet",
"Membership.mem",
"MeasurableSpace.generateFrom",
"Iff.mp",
... | exact separatesPoints_def <| fun _ hs ↦ (hxy _ hs).mp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 592,
"column": 39
} | {
"line": 592,
"column": 68
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable α\nhμ : ∀ (a : α), μ {a} < ∞\nh✝ : Nonempty α\nf : ℕ → α\nhf : Surjective f\n⊢ ∀ (i : ℕ), μ ((fun n ↦ {f n}) i) < ∞",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"Function.Surjective.forall",
... | by simpa [hf.forall] using hμ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 602,
"column": 2
} | {
"line": 603,
"column": 36
} | [
{
"pp": "case pos\nα : Type u_1\nμ : Measure α\nh this : SigmaFinite μ\ns : ℕ → Set α := spanningSets μ\nhs_univ : ⋃ i, s i = univ\nhs_meas : ∀ (i : ℕ), s i = ∅ ∨ s i = univ\nh_univ_empty : univ = ∅\n⊢ μ univ < ∞",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
... | · rw [h_univ_empty, measure_empty]
exact ENNReal.zero_ne_top.lt_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 383,
"column": 8
} | {
"line": 383,
"column": 14
} | [
{
"pp": "case refine_2\nα : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\nx✝ : ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S\nS : Finset (Set α)\nhS_subset : ↑S ⊆ memPartition t n\nhS_eq : s = ⋃₀ ↑S\n⊢ MeasurableSet s",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Finset",
... | hS_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 55
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\n⊢ generateFrom (memPartition t n) ≤ generateFrom (range t)",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.instLE",
"congrArg",
"Eq.rec",
"memPartition",
"id",
"LE.le",
"MeasurableSpace.generateFrom",
"Measura... | conv_rhs => rw [← generateFrom_iUnion_memPartition t] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Topology.GDelta.MetrizableSpace | {
"line": 52,
"column": 37
} | {
"line": 61,
"column": 78
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nY : Type u_2\ninst✝¹ : TopologicalSpace Y\ninst✝ : PseudoMetrizableSpace Y\nf : X → Y\n⊢ IsGδ {x | ContinuousAt f x}",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Set.instSProd",
"UniformSpace",
"Eq.mpr",
"Fi... | by
let := pseudoMetrizableSpaceUniformity Y
have := pseudoMetrizableSpaceUniformity_countably_generated Y
obtain ⟨U, _, hU⟩ := (@uniformity_hasBasis_open_symmetric Y _).exists_antitone_subbasis
simp only [Uniform.continuousAt_iff_prod, nhds_prod_eq]
simp only [(nhds_basis_opens _).prod_self.tendsto_iff hU.toH... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 415,
"column": 48
} | {
"line": 420,
"column": 51
} | [
{
"pp": "α : Type u_1\nG : Type u\ninst✝³ : DivInvMonoid G\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableMul₂ G\ninst✝ : MeasurableInv G\nn : ℤ\n⊢ Measurable fun x ↦ (x, n).1 ^ (x, n).2",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"DivInvMonoid.toInv",
"congrArg",
"measurab... | by
rcases n with n | n
· simp_rw [Int.ofNat_eq_natCast, zpow_natCast]
exact measurable_id.pow_const _
· simp_rw [zpow_negSucc]
exact (measurable_id.pow_const (n + 1)).inv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 741,
"column": 6
} | {
"line": 742,
"column": 73
} | [
{
"pp": "α✝ : Type u_1\nM : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace M\ninst✝⁴ : MeasurableSpace α\ninst✝³ : SMul M α\ninst✝² : SMul Mᵐᵒᵖ α\ninst✝¹ : IsCentralScalar M α\ninst✝ : MeasurableSMul₂ M α\n⊢ Measurable fun x ↦ MulOpposite.op (unop x.1) • x.2",
"usedConstants": [
"Eq.mpr",
"ins... | simp_rw [op_smul_eq_smul]
exact (measurable_mul_unop.comp measurable_fst).smul measurable_snd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 741,
"column": 6
} | {
"line": 742,
"column": 73
} | [
{
"pp": "α✝ : Type u_1\nM : Type u_2\nα : Type u_3\ninst✝⁵ : MeasurableSpace M\ninst✝⁴ : MeasurableSpace α\ninst✝³ : SMul M α\ninst✝² : SMul Mᵐᵒᵖ α\ninst✝¹ : IsCentralScalar M α\ninst✝ : MeasurableSMul₂ M α\n⊢ Measurable fun x ↦ MulOpposite.op (unop x.1) • x.2",
"usedConstants": [
"Eq.mpr",
"ins... | simp_rw [op_smul_eq_smul]
exact (measurable_mul_unop.comp measurable_fst).smul measurable_snd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 31
} | [
{
"pp": "G : Type v\nX : Type w\ninst✝³ : PseudoMetricSpace X\ninst✝² : Group G\ninst✝¹ : MulAction G X\ninst✝ : IsIsometricSMul G X\nc : G\nx : X\nr : ℝ\n⊢ (fun x ↦ c • x) ⁻¹' ball x r = ball (c⁻¹ • x) r",
"usedConstants": [
"Metric.smul_ball",
"Eq.mpr",
"instHSMul",
"DivInvOneMonoi... | rw [preimage_smul, smul_ball] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 31
} | [
{
"pp": "G : Type v\nX : Type w\ninst✝³ : PseudoMetricSpace X\ninst✝² : Group G\ninst✝¹ : MulAction G X\ninst✝ : IsIsometricSMul G X\nc : G\nx : X\nr : ℝ\n⊢ (fun x ↦ c • x) ⁻¹' ball x r = ball (c⁻¹ • x) r",
"usedConstants": [
"Metric.smul_ball",
"Eq.mpr",
"instHSMul",
"DivInvOneMonoi... | rw [preimage_smul, smul_ball] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 31
} | [
{
"pp": "G : Type v\nX : Type w\ninst✝³ : PseudoMetricSpace X\ninst✝² : Group G\ninst✝¹ : MulAction G X\ninst✝ : IsIsometricSMul G X\nc : G\nx : X\nr : ℝ\n⊢ (fun x ↦ c • x) ⁻¹' ball x r = ball (c⁻¹ • x) r",
"usedConstants": [
"Metric.smul_ball",
"Eq.mpr",
"instHSMul",
"DivInvOneMonoi... | rw [preimage_smul, smul_ball] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 52
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\npa qa : Set α → Prop\nH : μ.InnerRegularWRT pa qa\nf : α → β\nhf : AEMeasurable f μ\npb qb : Set β → Prop\nhAB : ∀ (U : Set β), qb U → qa (f ⁻¹' U)\nhAB' : ∀ (K : Set α), pa K → pb (f '' K)\nhB₂ : ∀ (U : S... | rw [map_apply_of_aemeasurable hf (hB₂ _ hU)] at hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Action | {
"line": 133,
"column": 97
} | {
"line": 134,
"column": 81
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (s ∆ (c⁻¹ • t)) = μ ((c • s) ∆ t)",
"usedConstants": [
"Set.smul_set_symmDiff",
"Eq.mpr",
"MulOne.toOne",
"DivI... | by
rw [← measure_smul _ c, smul_set_symmDiff, smul_smul, mul_inv_cancel, one_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Group.Action | {
"line": 224,
"column": 46
} | {
"line": 224,
"column": 51
} | [
{
"pp": "M : Type uM\nα : Type uα\nβ : Type uβ\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : SMul M α\ninst✝² : SMul M β\ninst✝¹ : MeasurableConstSMul M β\nμ : Measure α\ninst✝ : SMulInvariantMeasure M α μ\nf : α → β\nhsmul : ∀ (m : M) (a : α), f (m • a) = m • f a\nhf : Measurable f\nm : M\n... | hsmul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace α\nν : Measure α\ninst✝¹ : μ.OuterRegular\ninst✝ : ν.OuterRegular\nhμν : ∀ (U : Set α), IsOpen[inst✝²] U → μ U = ν U\n⊢ μ = ν",
"usedConstants": [
"MeasureTheory.Measure.ext",
"Set"
]
}
] | ext s ms | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 72
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ MeasurableSpace.generateFrom {s | ∃ a, s = Ioi a ∨ s = Iio a} ≤ MeasurableSpace.generateFrom (range Iio)",
"usedConstants": [
"PartialOrder.toPreord... | letI : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 26
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh_mono : Monotone f\nc : ℝ≥0 → ℝ≥0∞ := ofNNReal\nF : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a\n⊢ lintegral μ F ≤ ⨆ n, ∫⁻ (a : α), f n a ∂μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
... | rw [lintegral_eq_nnreal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 290,
"column": 8
} | {
"line": 290,
"column": 36
} | [
{
"pp": "case e_f.h.hf\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n ↦ (eapprox f n) a) i ≤ (fun n ↦ (eapprox f n) a) j",
"usedConstants": [
"MeasureTheory.SimpleFunc.monotone_eapprox"
]
}
] | exact monotone_eapprox _ h a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 292,
"column": 8
} | {
"line": 292,
"column": 36
} | [
{
"pp": "case e_f.h.hg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\na : α\ni j : ℕ\nh : i ≤ j\n⊢ (fun n ↦ (eapprox g n) a) i ≤ (fun n ↦ (eapprox g n) a) j",
"usedConstants": [
"MeasureTheory.SimpleFunc.monotone_eapprox"
]
}
] | exact monotone_eapprox _ h a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 297,
"column": 8
} | {
"line": 297,
"column": 76
} | [
{
"pp": "case e_s.h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\nn : ℕ\n⊢ ∫⁻ (a : α), (⇑(eapprox f n) + ⇑(eapprox g n)) a ∂μ = (eapprox f n).lintegral μ + (eapprox g n).lintegral μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.instAdd",
... | rw [← SimpleFunc.add_lintegral, ← SimpleFunc.lintegral_eq_lintegral] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Map | {
"line": 122,
"column": 59
} | {
"line": 122,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ng : α → β\nhg : MeasurePreserving g μ ν\nf : β → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (a : α), f (g a) ∂μ = ∫⁻ (b : β), f b ∂Measure.map g μ",
"usedConstants": [
"Eq.mpr",
"congrArg... | lintegral_map hf hg.measurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 134,
"column": 6
} | {
"line": 139,
"column": 13
} | [
{
"pp": "case pos\nι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝¹ : AddCommGroup G\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\nf : α → G\nt : Finset ι\na✝ : t ∈ s.powerset\nht : t.Nonempty\n⊢ (-1) ^ #t • ∑ a ∈ t.inf' ht S, f a =\n ∑ x ∈ s.biUnion S, ((-1) ^ #t * ((∏ i ∈ s \\ t, 1) * ∏ i ∈ t, (↑... | · obtain ⟨i, hi⟩ := ht
simp only [prod_const_one, prod_indicator_apply]
simp only [smul_sum, Set.indicator, Set.mem_iInter, mem_coe, Pi.one_apply, mul_ite, mul_one,
mul_zero, ite_smul, zero_smul, sum_ite, not_forall, sum_const_zero, add_zero]
congr
aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1046,
"column": 18
} | {
"line": 1046,
"column": 70
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nb : α\nhb : ¬f b = 0\n⊢ f b * μ (⇑(f.restrict s) ⁻¹' {f b}) = f b * μ (⇑f ⁻¹' {f b} ∩ s)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | rw [restrict_preimage_singleton _ hs hb, inter_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1046,
"column": 18
} | {
"line": 1046,
"column": 70
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nb : α\nhb : ¬f b = 0\n⊢ f b * μ (⇑(f.restrict s) ⁻¹' {f b}) = f b * μ (⇑f ⁻¹' {f b} ∩ s)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | rw [restrict_preimage_singleton _ hs hb, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1046,
"column": 18
} | {
"line": 1046,
"column": 70
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\nb : α\nhb : ¬f b = 0\n⊢ f b * μ (⇑(f.restrict s) ⁻¹' {f b}) = f b * μ (⇑f ⁻¹' {f b} ∩ s)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | rw [restrict_preimage_singleton _ hs hb, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Int | {
"line": 38,
"column": 49
} | {
"line": 38,
"column": 100
} | [
{
"pp": "n : ℤ\n⊢ ↑↑n.natAbs = ↑↑n.natAbs",
"usedConstants": [
"Int.cast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"AddMonoidWithOne.toNatCast",
"NNReal",
"Real.instRing",
... | by simp only [Int.cast_natCast, NNReal.coe_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1268,
"column": 4
} | {
"line": 1276,
"column": 28
} | [
{
"pp": "case insert\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 ⇑... | have Pg : motive g := by
apply ih
simp only [g, SimpleFunc.coe_piecewise, range_piecewise]
rw [image_compl_preimage, union_diff_distrib, diff_diff_comm, h, Finset.coe_insert,
insert_diff_self_of_notMem, diff_eq_empty.mpr, Set.empty_union]
· rw [Set.image_subset_iff]
convert! Set.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 75
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type w\nX : Type x\nY : Type y\ninst✝⁷ : UniformSpace X\ninst✝⁶ : UniformSpace Y\ninst✝⁵ : SMul M X\ninst✝⁴ : SMul N X\ninst✝³ : SMul M N\ninst✝² : UniformContinuousConstSMul M X\ninst✝¹ : UniformContinuousConstSMul N X\ninst✝ : IsScalarTower M N X\nm : M\nn : N\nx : Complet... | exact congr_arg (fun f => Completion.map f x) (funext (smul_assoc _ _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 76
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type w\nX : Type x\nY : Type y\ninst✝⁶ : UniformSpace X\ninst✝⁵ : UniformSpace Y\ninst✝⁴ : SMul M X\ninst✝³ : SMul N X\ninst✝² : SMulCommClass M N X\ninst✝¹ : UniformContinuousConstSMul M X\ninst✝ : UniformContinuousConstSMul N X\nm : M\nn : N\nx : Completion X\nhmn : m • n ... | · exact congr_arg (fun f => Completion.map f x) (funext (smul_comm _ _)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 98,
"column": 56
} | {
"line": 99,
"column": 58
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a * b‖ ≤ ‖a‖ + ‖b‖",
"usedConstants": [
"dist_triangle",
"Norm.norm",
"Real.instLE",
"Real",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass... | by
simpa [dist_eq_norm_inv_mul] using dist_triangle a⁻¹ 1 b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Algebra | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : PseudoMetricSpace α\ninst✝⁵ : PseudoMetricSpace β\ninst✝⁴ : Zero α\ninst✝³ : Zero β\ninst✝² : SMul α β\ninst✝¹ : IsBoundedSMul α β\nX : Type u_3\nι : Type u_4\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nG : ι → X → β\nf : X → α\ng : X → β\nl : Filter ι\nh... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 179,
"column": 54
} | {
"line": 179,
"column": 76
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx✝ : α\n⊢ x✝ ∈ {x | 0 ≤ x} ↔ x✝ ∈ negPart ⁻¹' {0}",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | simp [negPart_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 54,
"column": 23
} | {
"line": 54,
"column": 45
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedGroup E\na b : E\n⊢ dist b (b * a) = dist 1 a",
"usedConstants": [
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedGroup.toGroup",
... | ← dist_mul_left b 1 a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 33
} | [
{
"pp": "α : Type u_5\nm : MeasurableSpace α\ninst✝¹ : CountablyGenerated α\ninst✝ : MeasurableSpace.SeparatesPoints α\ns : Set (ℕ → Bool)\nf : α ≃ᵐ ↑s\n⊢ ∃ x, SecondCountableTopology α ∧ T4Space α ∧ BorelSpace α",
"usedConstants": [
"MeasurableEquiv.instEquivLike",
"Pi.topologicalSpace",
... | letI := induced f inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NormedDivisionRing α\nX : Type u_4\nι : Type u_5\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nf : X → α\nl : Filter ι\nhF : TendstoLocallyUniformlyOn F f l s\nhf : ∀ x ∈ s, Disjoint (map f (𝓝[s] x)) (𝓝 0)\n⊢ TendstoLocallyUniformlyOn F⁻¹ f⁻¹ l s",
"usedConstants":... | tendstoLocallyUniformlyOn_iff_forall_tendsto | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 322,
"column": 2
} | {
"line": 323,
"column": 52
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E ⊆ thickening δ E",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Metric.cthickening_zero",
"id",
... | rw [← cthickening_zero]
exact cthickening_subset_thickening' δ_pos δ_pos E | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 322,
"column": 2
} | {
"line": 323,
"column": 52
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E ⊆ thickening δ E",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Metric.cthickening_zero",
"id",
... | rw [← cthickening_zero]
exact cthickening_subset_thickening' δ_pos δ_pos E | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 384,
"column": 2
} | {
"line": 390,
"column": 14
} | [
{
"pp": "ε : ℝ\nα : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nhε : 0 ≤ ε\n⊢ diam (thickening ε s) ≤ diam s + 2 * ε",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Metric.diam_mono",
"LE.le.eq_or_lt",
"Real.partialOrder",
"Real.instLE",
"Real",
... | by_cases hs : IsBounded s
· exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans
(diam_cthickening_le _ hε)
obtain rfl | hε := hε.eq_or_lt
· simp [thickening_of_nonpos, diam_nonneg]
· rw [diam_eq_zero_of_unbounded (mt (IsBounded.subset · <| self_subset_thickening hε _) hs)]
posit... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 384,
"column": 2
} | {
"line": 390,
"column": 14
} | [
{
"pp": "ε : ℝ\nα : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nhε : 0 ≤ ε\n⊢ diam (thickening ε s) ≤ diam s + 2 * ε",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Metric.diam_mono",
"LE.le.eq_or_lt",
"Real.partialOrder",
"Real.instLE",
"Real",
... | by_cases hs : IsBounded s
· exact (diam_mono (thickening_subset_cthickening _ _) hs.cthickening).trans
(diam_cthickening_le _ hε)
obtain rfl | hε := hε.eq_or_lt
· simp [thickening_of_nonpos, diam_nonneg]
· rw [diam_eq_zero_of_unbounded (mt (IsBounded.subset · <| self_subset_thickening hε _) hs)]
posit... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 663,
"column": 2
} | {
"line": 665,
"column": 74
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\ns : Set α\n⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"le_refl",
"Real",
"ENNReal.instAddCommMonoid",
"ENNReal.ofReal",
... | intro x
simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
exact fun hx => infEDist_le_infEDist_cthickening_add.trans (by grw [hx]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 663,
"column": 2
} | {
"line": 665,
"column": 74
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\ns : Set α\n⊢ cthickening ε (cthickening δ s) ⊆ cthickening (ε + δ) s",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"le_refl",
"Real",
"ENNReal.instAddCommMonoid",
"ENNReal.ofReal",
... | intro x
simp_rw [mem_cthickening_iff, ENNReal.ofReal_add hε hδ]
exact fun hx => infEDist_le_infEDist_cthickening_add.trans (by grw [hx]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 21
} | [
{
"pp": "case right\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\ng : α → ℝ≥0∞\nhmg : Measurable g\nhgf : g ≤ f\nhifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ\ns : Set α\nhms : MeasurableSet s\n⊢ ∫⁻ (a : α), g a ∂μ - ∫⁻ (x : α) in sᶜ, f x ∂μ ≤ ∫⁻ (x : α), g... | gcongr; apply hgf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 21
} | [
{
"pp": "case right\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : ∫⁻ (a : α), f a ∂μ ≠ ∞\ng : α → ℝ≥0∞\nhmg : Measurable g\nhgf : g ≤ f\nhifg : ∫⁻ (a : α), f a ∂μ = ∫⁻ (a : α), g a ∂μ\ns : Set α\nhms : MeasurableSet s\n⊢ ∫⁻ (a : α), g a ∂μ - ∫⁻ (x : α) in sᶜ, f x ∂μ ≤ ∫⁻ (x : α), g... | gcongr; apply hgf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 152,
"column": 2
} | {
"line": 157,
"column": 15
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetrizableSpace α\ninst✝ : OpensMeasurableSpace α\ns : Set α\nh1 : IsSeparable s\nh2 : μ sᶜ = 0\na✝ : Nontrivial α\na : α\nh : StronglyMeasurable Subtype.val\n⊢ StronglyMeasurable ((closure[inst✝²] s).piece... | have : (closure s).piecewise id (fun _ ↦ a) =
((↑) : closure s → α).extend ((↑) : closure s → α) (fun _ ↦ a) := by
ext x
by_cases hx : x ∈ closure s
· simp [Function.extend_val_apply, hx]
· simp [hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 176,
"column": 26
} | {
"line": 179,
"column": 97
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nβ : Type u_2\nx✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nhμν : μ ⟂ₘ ν\n⊢ map f μ ⟂ₘ map f ν",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Compl.compl",
... | by
refine ⟨f '' hμν.nullSet, hf.measurableSet_image' hμν.measurableSet_nullSet, ?_, ?_⟩
· rw [hf.map_apply, hf.injective.preimage_image, hμν.measure_nullSet]
· rw [hf.map_apply, Set.preimage_compl, hf.injective.preimage_image, hμν.measure_compl_nullSet] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 31
} | [
{
"pp": "case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.ap... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 31
} | [
{
"pp": "case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.ap... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 31
} | [
{
"pp": "case hbc\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : NormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nhf : StronglyMeasurable f\nc : ℝ\nx : α\nhfx : ‖f x‖ ≤ c\nh_tendsto : Tendsto (fun n ↦ (hf.approx n) x) atTop (𝓝 0)\nhfx0 : f x = 0\nh_tendsto_norm : Tendsto (fun n ↦ ‖(hf.ap... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 26
} | [
{
"pp": "case h.inl\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) 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\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i... | exact pure_le_nhds _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UnitInterval | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 42
} | [
{
"pp": "case mp.left\na t : ℝ\nha : 0 < a\nh₁ : 0 ≤ a * t\nh₂ : a * t ≤ 1\n⊢ 0 ≤ t",
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.semiring",
"nonneg_of_mul_nonneg_right",
"IsStrictOrderedRing.toPosMulStrictMono",
"Real.linearOrder",
"Real.instIsStrictOrder... | exact nonneg_of_mul_nonneg_right h₁ ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UnitInterval | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 42
} | [
{
"pp": "case mp.left\na t : ℝ\nha : 0 < a\nh₁ : 0 ≤ a * t\nh₂ : a * t ≤ 1\n⊢ 0 ≤ t",
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.semiring",
"nonneg_of_mul_nonneg_right",
"IsStrictOrderedRing.toPosMulStrictMono",
"Real.linearOrder",
"Real.instIsStrictOrder... | exact nonneg_of_mul_nonneg_right h₁ ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UnitInterval | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 42
} | [
{
"pp": "case mp.left\na t : ℝ\nha : 0 < a\nh₁ : 0 ≤ a * t\nh₂ : a * t ≤ 1\n⊢ 0 ≤ t",
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.semiring",
"nonneg_of_mul_nonneg_right",
"IsStrictOrderedRing.toPosMulStrictMono",
"Real.linearOrder",
"Real.instIsStrictOrder... | exact nonneg_of_mul_nonneg_right h₁ ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UnitInterval | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nδ : α\n⊢ ↑(addNSMul h δ 0) = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"congrArg",
"SMulWithZero.toSMulZeroClass",
"AddCommGroup.toAddCo... | rw [addNSMul, zero_smul, add_zero, projIcc_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UnitInterval | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nδ : α\n⊢ ↑(addNSMul h δ 0) = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"congrArg",
"SMulWithZero.toSMulZeroClass",
"AddCommGroup.toAddCo... | rw [addNSMul, zero_smul, add_zero, projIcc_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UnitInterval | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nδ : α\n⊢ ↑(addNSMul h δ 0) = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"congrArg",
"SMulWithZero.toSMulZeroClass",
"AddCommGroup.toAddCo... | rw [addNSMul, zero_smul, add_zero, projIcc_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 603,
"column": 72
} | {
"line": 605,
"column": 59
} | [
{
"pp": "α : Type u_1\nM : Type u_5\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : Multiset (α → M)\nhl : ∀ f ∈ l, StronglyMeasurable f\n⊢ StronglyMeasurable l.prod",
"usedConstants": [
"Multiset.prod",
"Quot.ind",
"Multiset.mem_coe... | by
rcases l with ⟨l⟩
simpa using l.stronglyMeasurable_prod (by simpa using hl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 743,
"column": 4
} | {
"line": 750,
"column": 59
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : PseudoMetrizableSpace γ\ng : β → γ\nf : α → β\nhg : IsEmbedding g\nthis : PseudoMetricSpace γ := pseudoMetrizableSpacePseud... | let G : β → range g := rangeFactorization g
have hG : IsClosedEmbedding G :=
{ hg.codRestrict _ _ with
isClosed_range := by
rw [rangeFactorization_surjective.range_eq]
exact isClosed_univ }
have : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
exact hG.measurableE... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 743,
"column": 4
} | {
"line": 750,
"column": 59
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : PseudoMetrizableSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : PseudoMetrizableSpace γ\ng : β → γ\nf : α → β\nhg : IsEmbedding g\nthis : PseudoMetricSpace γ := pseudoMetrizableSpacePseud... | let G : β → range g := rangeFactorization g
have hG : IsClosedEmbedding G :=
{ hg.codRestrict _ _ with
isClosed_range := by
rw [rangeFactorization_surjective.range_eq]
exact isClosed_univ }
have : Measurable (G ∘ f) := Measurable.subtype_mk H.measurable
exact hG.measurableE... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 123,
"column": 40
} | {
"line": 123,
"column": 68
} | [
{
"pp": "case a.h.a\nα : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : Countable α\nμ1 μ2 : Measure α\ninst✝¹ : SigmaFinite μ1\ninst✝ : SigmaFinite μ2\nx : α\n⊢ μ1 {x} = μ2 {x} ↔ (μ1 {x}).toReal = (μ2 {x}).toReal",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"congr... | ENNReal.toReal_eq_toReal_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 99
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nm : MeasurableSpace α\nf g : α → E\ninst✝³ : TopologicalSpace E\ninst✝² : Preorder E\ninst✝¹ : OrderClosedTopology E\ninst✝ : PseudoMetrizableSpace E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {a | f a < g a}",
"usedConstants": [
"Eq.mpr... | simpa only [lt_iff_le_not_ge] using (hf.measurableSet_le hg).inter (hg.measurableSet_le hf).compl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 99
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nm : MeasurableSpace α\nf g : α → E\ninst✝³ : TopologicalSpace E\ninst✝² : Preorder E\ninst✝¹ : OrderClosedTopology E\ninst✝ : PseudoMetrizableSpace E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {a | f a < g a}",
"usedConstants": [
"Eq.mpr... | simpa only [lt_iff_le_not_ge] using (hf.measurableSet_le hg).inter (hg.measurableSet_le hf).compl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 943,
"column": 2
} | {
"line": 943,
"column": 99
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nm : MeasurableSpace α\nf g : α → E\ninst✝³ : TopologicalSpace E\ninst✝² : Preorder E\ninst✝¹ : OrderClosedTopology E\ninst✝ : PseudoMetrizableSpace E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {a | f a < g a}",
"usedConstants": [
"Eq.mpr... | simpa only [lt_iff_le_not_ge] using (hf.measurableSet_le hg).inter (hg.measurableSet_le hf).compl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 79
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\n⊢ Tendsto (fun i ↦ (⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 1 / ↑i) atTop\n (𝓝 ((⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s) - 0))",
"usedConstants": [
... | refine ENNReal.Tendsto.sub tendsto_const_nhds ?_ (Or.inr ENNReal.zero_ne_top) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Perfect | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 37
} | [
{
"pp": "case refine_3\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) := {U | U ∈ b ∧ (U ∩ C).Countable}\nV : Set α := ⋃ U ∈ v, U\nD : Set α ... | rw [inter_comm, inter_union_diff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Perfect | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 37
} | [
{
"pp": "case refine_3\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) := {U | U ∈ b ∧ (U ∩ C).Countable}\nV : Set α := ⋃ U ∈ v, U\nD : Set α ... | rw [inter_comm, inter_union_diff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Perfect | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 37
} | [
{
"pp": "case refine_3\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) := {U | U ∈ b ∧ (U ∩ C).Countable}\nV : Set α := ⋃ U ∈ v, U\nD : Set α ... | rw [inter_comm, inter_union_diff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1225,
"column": 2
} | {
"line": 1229,
"column": 68
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : BorelSpace β... | obtain ⟨t_sf, ht_sf⟩ :
∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
refine ⟨h_str_meas.approx, fun j x => ?_⟩
exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 26
} | [
{
"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 ... | rw [lintegral_eq_nnreal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1257,
"column": 2
} | {
"line": 1261,
"column": 68
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\nι : Type u_7\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu... | obtain ⟨t_sf, ht_sf⟩ :
∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id
refine ⟨h_str_meas.approx, fun j x => ?_⟩
exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 54
} | [
{
"pp": "case compl\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : α → Measure β\ninst✝ : ∀ (a : α), IsFiniteMeasure (μ a)\nS : Set (Set β)\nhgen : mβ = MeasurableSpace.generateFrom S\nhpi : IsPiSystem S\nh_basic : ∀ s ∈ S, Measurable fun a ↦ (μ a) s\nh_univ : Measurable fun a ... | simp only [measure_compl hsm (measure_ne_top _ _)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 29
} | [
{
"pp": "case refine_1\nα : 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 ... | exact map_mem ⟨_, hdom⟩ 0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 530,
"column": 2
} | {
"line": 531,
"column": 75
} | [
{
"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\n⊢ (s ∩ c... | obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) := by
simpa only [not_disjoint_iff, mem_cylinder_comm] using Nat.find_min A B | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 566,
"column": 6
} | {
"line": 566,
"column": 37
} | [
{
"pp": "case inr.inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nx y : (n : ℕ) → E n\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nH : longestPrefix x s < firstDiff x y\nxs : x ∉ s\nys : y ∉ s\nL : longestPrefix x ... | exact (A'y.not_disjoint Z).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 389,
"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)\n⊢ MeasurableSet (⋃ m, ⋂ n, u m n)",
"usedConstants": [
... | refine MeasurableSet.iUnion fun m => ?_
exact MeasurableSet.iInter fun n => hu m n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 389,
"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)\n⊢ MeasurableSet (⋃ m, ⋂ n, u m n)",
"usedConstants": [
... | refine MeasurableSet.iUnion fun m => ?_
exact MeasurableSet.iInter fun n => hu m n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 627,
"column": 11
} | {
"line": 627,
"column": 16
} | [
{
"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\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | ← fy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 668,
"column": 10
} | {
"line": 669,
"column": 27
} | [
{
"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\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | have : cylinder Ax.some (firstDiff x y) = cylinder Ay.some (firstDiff x y) := by
rw [I1, I2, I3] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 670,
"column": 20
} | {
"line": 670,
"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\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | ← fy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 1141,
"column": 4
} | {
"line": 1141,
"column": 80
} | [
{
"pp": "case inr.refine_2\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).Nonempty\nthis✝ : Non... | simpa using notMem_of_notMem_closure (mt infDist_le_dist_of_mem this.not_ge) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Complement | {
"line": 303,
"column": 2
} | {
"line": 305,
"column": 76
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nH' H : Subgroup G\nh : H' ≤ H\nH'' : Subgroup ↥H := comap H.subtype H'\nthis : H' = map H.subtype H''\nS : Set ↥H\ncmem : IsComplement S ↑H''\n⊢ Nat.card ↑(⇑H.subtype '' S) * Nat.card ↥(map H.subtype H'') = Nat.card ↥H",
"usedConstants": [
"Eq.mpr... | · rw [← cmem.card_mul_card]
refine congr_arg₂ (· * ·) ?_ ?_ <;>
exact Nat.card_congr (Equiv.Set.image _ _ <| subtype_injective H).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 688,
"column": 51
} | {
"line": 688,
"column": 70
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).pr... | preimage_swap_prod, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Complement | {
"line": 528,
"column": 70
} | {
"line": 530,
"column": 99
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : Quotient (QuotientGroup.rightRel H) → G\nhf : ∀ (q : Quotient (QuotientGroup.rightRel H)), Quotient.mk'' (f q) = q\nq : Quotient (QuotientGroup.rightRel H)\n⊢ ↑(⋯.rightQuotientEquiv q) = f q",
"usedConstants": [
"Subtype.coe_mk",
"Iff.m... | by
refine (Subtype.ext_iff.mp ?_).trans (Subtype.coe_mk (f q) ⟨q, rfl⟩)
exact (rightQuotientEquiv (isComplement_range_right hf)).apply_eq_iff_eq_symm_apply.2 (hf q).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Group.Prod | {
"line": 185,
"column": 67
} | {
"line": 195,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SFinite ν\ninst✝³ : SFinite μ\ninst✝² : MeasurableInv G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : ν.IsMulLeftInvariant\nf : G → G → ℝ≥0∞\nhf : AEMeasurable (uncurry f) (μ.prod ν)\n⊢ ∫⁻ (x : G)... | by
have h : Measurable fun z : G × G => (z.2 * z.1, z.1⁻¹) :=
(measurable_snd.mul measurable_fst).prodMk measurable_fst.inv
have h2f : AEMeasurable (uncurry fun x y => f (y * x) x⁻¹) (μ.prod ν) :=
hf.comp_quasiMeasurePreserving (measurePreserving_mul_prod_inv μ ν).quasiMeasurePreserving
simp_rw [lintegral... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 615,
"column": 2
} | {
"line": 616,
"column": 50
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : PolishSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T0Space Y\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Continuous[inst✝⁶, inst✝²] f\nhsurj : Surjective f\n⊢ MeasurableSpace.m... | borelize Y
exact hf.measurable.map_measurableSpace_eq hsurj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 615,
"column": 2
} | {
"line": 616,
"column": 50
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : PolishSpace X\ninst✝⁴ : MeasurableSpace X\ninst✝³ : BorelSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T0Space Y\ninst✝ : SecondCountableTopology Y\nf : X → Y\nhf : Continuous[inst✝⁶, inst✝²] f\nhsurj : Surjective f\n⊢ MeasurableSpace.m... | borelize Y
exact hf.measurable.map_measurableSpace_eq hsurj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 47
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nμ' : Measure α\ninst✝ : SFinite μ'\ni : ℕ × ℕ\ns : Set α\nt : Set β\nx✝¹ : MeasurableSet s\nx✝ : MeasurableSet t\n⊢ ((sfiniteSeq μ i.1).p... | simp_rw [add_apply, prod_prod, right_distrib] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 22
} | [
{
"pp": "case map_eq.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ninst✝¹ : SFinite μc\nf : α → β\ng : α → γ → δ\nhgm : Measurable (uncur... | simp [← hf.map_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 22
} | [
{
"pp": "case map_eq.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ninst✝¹ : SFinite μc\nf : α → β\ng : α → γ → δ\nhgm : Measurable (uncur... | simp [← hf.map_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 22
} | [
{
"pp": "case map_eq.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ninst✝¹ : SFinite μc\nf : α → β\ng : α → γ → δ\nhgm : Measurable (uncur... | simp [← hf.map_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 450,
"column": 6
} | {
"line": 451,
"column": 53
} | [
{
"pp": "case h.mpr\nG : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul G\nH : Type u_3\ninst✝² : Group H\nmH : MeasurableSpace H\ninst✝¹ : MeasurableMul H\nμ : Measure H\ninst✝ : μ.IsMulLeftInvariant\nf : G →* H\nhf : MeasurableEmbedding ⇑f\ng : G\ns : Set G\nhs : MeasurableSet ... | · intro ⟨y, yins, hy⟩
exact ⟨g⁻¹ * y, by simp [yins], by simp [hy]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1172,
"column": 6
} | {
"line": 1172,
"column": 19
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : IsProbabilityMeasure μ\ns : Set β\nhs : MeasurableSet s\n⊢ (μ.prod ν).snd s = ν s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",... | snd_apply hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 92
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\n⊢ (μ + ν).withDensity f = μ.withDensity f + ν.withDensity f",
"usedConstants": [
"MeasureTheory.lintegral_add_measure",
"ENNReal.instAdd",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
... | ext1 s hs
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 115,
"column": 2
} | {
"line": 116,
"column": 92
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\n⊢ (μ + ν).withDensity f = μ.withDensity f + ν.withDensity f",
"usedConstants": [
"MeasureTheory.lintegral_add_measure",
"ENNReal.instAdd",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
... | ext1 s hs
simp only [withDensity_apply f hs, restrict_add, lintegral_add_measure, Measure.add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 395,
"column": 73
} | {
"line": 395,
"column": 93
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh_mf : Measurable f\ng : ℕ → α → ℝ≥0∞\nh_mea_g : ∀ (n : ℕ), Measurable (g n)\nh_mono_g : Monotone g\nh_ind : ∀ (n : ℕ), ∫⁻ (a : α), g n a ∂μ.withDensity f = ∫⁻ (a : α), (f * g n) a ∂μ\nm n : ℕ\nhmn : m ≤ n\nx : α\n⊢ f x * g m x ≤ f x * ... | grw [h_mono_g hmn x] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.Algebra.QuadraticDiscriminant | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\na b c : K\nha : a ≠ 0\nh : ∀ (x : K), 0 < a * (x * x) + b * x + c\nthis : ∀ (x : K), 0 ≤ a * (x * x) + b * x + c\n⊢ discrim a b c < 0",
"usedConstants": [
"discrim_le_zero",
"Field.toDivisionRing",
... | refine lt_of_le_of_ne (discrim_le_zero this) fun h' ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Norm | {
"line": 122,
"column": 64
} | {
"line": 122,
"column": 86
} | [
{
"pp": "n : ℕ\n⊢ ↑‖↑n‖₊ = ↑↑n",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",
"Complex.instNormedAddCommGroup",
"Real.instZero",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"Complex.norm_natCast",
... | by simp [norm_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
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