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.Analysis.SpecificLimits.Basic | {
"line": 629,
"column": 2
} | {
"line": 635,
"column": 46
} | [
{
"pp": "ε : ℝ≥0\nhε : ε ≠ 0\nι : Type u_4\ninst✝ : Countable ι\n⊢ ∃ ε', (∀ (i : ι), 0 < ε' i) ∧ ∃ c, HasSum ε' c ∧ c < ε",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Real.instLE",
"Real",
... | cases nonempty_encodable ι
obtain ⟨a, a0, aε⟩ := exists_between (pos_iff_ne_zero.2 hε)
obtain ⟨ε', hε', c, hc, hcε⟩ := posSumOfEncodable a0 ι
exact
⟨fun i ↦ ⟨ε' i, (hε' i).le⟩, fun i ↦ NNReal.coe_lt_coe.1 <| hε' i,
⟨c, hasSum_le (fun i ↦ (hε' i).le) hasSum_zero hc⟩, NNReal.hasSum_coe.1 hc,
aε.tran... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.PiSystem | {
"line": 458,
"column": 31
} | {
"line": 465,
"column": 25
} | [
{
"pp": "α : Type u_3\nι : Type u_4\nπ : ι → Set (Set α)\nS : Set ι\ni : ι\nhis : i ∈ S\n⊢ π i ⊆ piiUnionInter π S",
"usedConstants": [
"Eq.mpr",
"piiUnionInter",
"congrArg",
"Set.Subset.trans",
"Set.univ",
"Membership.mem",
"Set.instUnion",
"Eq.mp",
"Se... | by
have h_ss : {i} ⊆ S := by
intro j hj
rw [mem_singleton_iff] at hj
rwa [hj]
refine Subset.trans ?_ (piiUnionInter_mono_right h_ss)
rw [piiUnionInter_singleton]
exact subset_union_left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 30
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns : ℕ → Set α\nhs : ∑' (i : ℕ), μ (s i) ≠ ∞\n⊢ μ (limsup s atTop) = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CompleteLattice.toConditionallyCompleteLattice",
"id",
"... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 30
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns : ℕ → Set α\nh : ∑' (i : ℕ), μ (s i) ≠ ∞\n⊢ μ (liminf s atTop) = 0",
"usedConstants": [
"Eq.mpr",
"Filter.liminf",
"congrArg",
"CompleteLattice.toConditionallyCompleteLattice... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 253,
"column": 17
} | {
"line": 253,
"column": 20
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhμst : μ (s ∆ t) ≠ ∞\nu v : Set α\nhμuv : μ (u ∆ v) ≠ ∞\n⊢ μ u = ∞ → μ v = ∞",
"usedConstants": [
"MeasureTheory.Measure",
"ENNReal",
"ENNReal.instTop",
"Top.top",
"Eq",
"DFunLike.coe",
"Measu... | hμu | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 78,
"column": 7
} | {
"line": 78,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν : Measure α\ns✝ t : Set α\na : α\ninst✝ : SFinite μ\ns : Set α\n⊢ μ.restrict s = sum fun n ↦ (sfiniteSeq μ n).restrict s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congr... | rw [← restrict_sum_of_countable, sum_sfiniteSeq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 78,
"column": 7
} | {
"line": 78,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν : Measure α\ns✝ t : Set α\na : α\ninst✝ : SFinite μ\ns : Set α\n⊢ μ.restrict s = sum fun n ↦ (sfiniteSeq μ n).restrict s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congr... | rw [← restrict_sum_of_countable, sum_sfiniteSeq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 78,
"column": 7
} | {
"line": 78,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ ν : Measure α\ns✝ t : Set α\na : α\ninst✝ : SFinite μ\ns : Set α\n⊢ μ.restrict s = sum fun n ↦ (sfiniteSeq μ n).restrict s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congr... | rw [← restrict_sum_of_countable, sum_sfiniteSeq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 542,
"column": 2
} | {
"line": 542,
"column": 23
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Preorder ι\ninst✝¹ : IsCodirectedOrder ι\ninst✝ : atBot.IsCountablyGenerated\ns : ι → Set α\nhs : Monotone s\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nhfin : ∃ i, μ (s i) ≠ ∞\n⊢ ⨅ i, μ (s i) ≤ μ (⋂ i, s i)",
"usedConstant... | have := hfin.nonempty | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 421,
"column": 10
} | {
"line": 421,
"column": 40
} | [
{
"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) ≠ ∞\nw : ℕ → Set α := fun n ↦ toMeasurable μ (t ∩ v n)\nhw : ∀ (n : ℕ), μ (w n) < ∞\nt' : Set α := ⋃ n, toMeasurable μ (t ∩ disjointed w n)\nht... | have : x ∈ t ∩ v n := ⟨hx, hn⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 732,
"column": 2
} | {
"line": 732,
"column": 28
} | [
{
"pp": "case h.e_f.h\nα : Type u_1\nm : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (x : α), μ (measurableAtom x) = ν (measurableAtom x)\ns : Set α\nhs : MeasurableSet s\nh1 : s = ⋃₀ (measurableAtom '' s)\nh_count : (measurableAtom '' s).Countable\nh_disj : (measurableAtom '' s).Pairwise Dis... | obtain ⟨x, hxs, hx⟩ := hs' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 73
} | [
{
"pp": "β : Type u_2\nb₁ : Set (Set β)\nhb₁c : b₁.Countable\nb₂ : Set (Set β)\nhb₂c : b₂.Countable\n⊢ CountablyGenerated β",
"usedConstants": [
"Lattice.toSemilatticeSup",
"MeasurableSpace.generateFrom_sup_generateFrom",
"CompleteLattice.toConditionallyCompleteLattice",
"Set.instUni... | exact @mk _ (_ ⊔ _) ⟨_, hb₁c.union hb₂c, generateFrom_sup_generateFrom⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CountablySeparated α\nm' : MeasurableSpace α\nleft✝¹ : CountablyGenerated α\nleft✝ : SeparatesPoints α\nm'le : m' ≤ inst✝¹\n⊢ Measurable (mapNatBool α)",
"usedConstants": [
"le_rfl",
"MeasurableSpace.instPartialOrder",
"PartialOrde... | exact (measurable_mapNatBool _).mono m'le le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1099,
"column": 2
} | {
"line": 1099,
"column": 90
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nm : Set (Measure α)\ns : Set α\nhs : MeasurableSet s\nt : Set α\nμ : Measure α\nhμ : μ ∈ m\nu : Set α\nhtu : t ⊆ u\n_hu : MeasurableSet u\nhm : ∀ {s t : Set α}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t\n⊢ (⨅ μ ∈ toOuterMeasure '' m, μ (t ∩ s)) + ⨅... | exact add_le_add (hm <| inter_subset_inter_left _ htu) (hm <| diff_subset_diff_left htu) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 354,
"column": 2
} | {
"line": 386,
"column": 53
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\n⊢ MeasurableSet s ↔ ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Codisjoint",
"Set.diff_subset",
"Lattice.toSemilatticeSup",
"Finset.coe_singleton",
"Finset.coe_empty",
... | refine ⟨fun h ↦ ?_, fun ⟨S, hS_subset, hS_eq⟩ ↦ ?_⟩
· induction s, h using generateFrom_induction with
| hC u hu _ => exact ⟨{u}, by simp [hu], by simp⟩
| empty => exact ⟨∅, by simp, by simp⟩
| compl u _ hu =>
obtain ⟨S, hS_subset, rfl⟩ := hu
classical
refine ⟨(memPartition t n).toFinset... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 354,
"column": 2
} | {
"line": 386,
"column": 53
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\n⊢ MeasurableSet s ↔ ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Codisjoint",
"Set.diff_subset",
"Lattice.toSemilatticeSup",
"Finset.coe_singleton",
"Finset.coe_empty",
... | refine ⟨fun h ↦ ?_, fun ⟨S, hS_subset, hS_eq⟩ ↦ ?_⟩
· induction s, h using generateFrom_induction with
| hC u hu _ => exact ⟨{u}, by simp [hu], by simp⟩
| empty => exact ⟨∅, by simp, by simp⟩
| compl u _ hu =>
obtain ⟨S, hS_subset, rfl⟩ := hu
classical
refine ⟨(memPartition t n).toFinset... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1152,
"column": 6
} | {
"line": 1156,
"column": 22
} | [
{
"pp": "case refine_1.refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\nx✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}\nt' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅\nht' : t' = fun n ↦ if n = 0 the... | by_cases hxt : x ∈ t
· refine mem_iUnion.2 ⟨0, ?_⟩
simp [hx, hxt]
· refine mem_iUnion.2 ⟨1, ?_⟩
simp [hx, hxt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1152,
"column": 6
} | {
"line": 1156,
"column": 22
} | [
{
"pp": "case refine_1.refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\nx✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}\nt' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅\nht' : t' = fun n ↦ if n = 0 the... | by_cases hxt : x ∈ t
· refine mem_iUnion.2 ⟨0, ?_⟩
simp [hx, hxt]
· refine mem_iUnion.2 ⟨1, ?_⟩
simp [hx, hxt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 246,
"column": 2
} | {
"line": 248,
"column": 65
} | [
{
"pp": "case inr\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nμ : Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\n⊢ AEMeasurable f μ",
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"MeasureTheory.Measure",
"AEMeasurable"... | · haveI := ae_neBot.2 hμ
rcases h.exists with ⟨x, hx⟩
exact ⟨const α (f x), measurable_const, EventuallyEq.symm hx⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 404,
"column": 2
} | {
"line": 410,
"column": 70
} | [
{
"pp": "case refine_2\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\na✝ : Nontrivial β\ninhabited_h : Inhabited β\nS : Set (Set β)\nhSc : S.Countable\nhc : MeasurableSpace.CountablyGenerated β\nh : NullMeasurable f μ\nT : Set β → Set α\nhTf : ∀ s ∈ S, T s ⊆ f ⁻¹' s\nhTm : ∀ s ∈ ... | · rw [restrict_piecewise_compl, restrict_eq]
refine measurable_generateFrom fun s hs ↦ .of_subtype_image ?_
rw [preimage_comp, Subtype.image_preimage_coe]
convert! (hTm s hs).diff hvm using 1
rw [inter_comm]
refine Set.ext fun x ↦ and_congr_left fun hxv ↦ ⟨fun hx ↦ ?_, fun hx ↦ hTf s hs hx⟩
exac... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 172,
"column": 48
} | {
"line": 177,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nn : ℕ\n⊢ Measurable fun x ↦ (x, n).1 ^ (x, n).2",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.recAux",
"Measurable.mul",
"HMul.hMul",
"Monoid.toMulOneClass"... | by
induction n with
| zero => simp only [pow_zero, ← Pi.one_def, measurable_one]
| succ n ih =>
simp only [pow_succ]
exact ih.mul measurable_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.ProjIcc | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b x : α\nh : a < b\n⊢ projIcc a b ⋯ x = ⟨b, ⋯⟩ ↔ b ≤ x",
"usedConstants": [
"Iff.mpr",
"Set.right_mem_Icc",
"False",
"Subtype.mk.congr_simp",
"Lattice.toSemilatticeSup",
"eq_false",
"congrArg",
"PartialOrder.toPr... | simp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_ge] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Set.ProjIcc | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b x : α\nh : a < b\n⊢ projIcc a b ⋯ x = ⟨b, ⋯⟩ ↔ b ≤ x",
"usedConstants": [
"Iff.mpr",
"Set.right_mem_Icc",
"False",
"Subtype.mk.congr_simp",
"Lattice.toSemilatticeSup",
"eq_false",
"congrArg",
"PartialOrder.toPr... | simp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_ge] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.ProjIcc | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 71
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b x : α\nh : a < b\n⊢ projIcc a b ⋯ x = ⟨b, ⋯⟩ ↔ b ≤ x",
"usedConstants": [
"Iff.mpr",
"Set.right_mem_Icc",
"False",
"Subtype.mk.congr_simp",
"Lattice.toSemilatticeSup",
"eq_false",
"congrArg",
"PartialOrder.toPr... | simp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_ge] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.ProjIcc | {
"line": 244,
"column": 4
} | {
"line": 247,
"column": 55
} | [
{
"pp": "case h.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nf : α → β\nha : ∀ x < a, f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"... | rcases le_or_gt x b with hxb | hbx
· lift x to Icc a b using ⟨hax, hxb⟩
rw [IccExtend_val, comp_apply]
· simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.ProjIcc | {
"line": 244,
"column": 4
} | {
"line": 247,
"column": 55
} | [
{
"pp": "case h.inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\na b : α\nh : a ≤ b\nf : α → β\nha : ∀ x < a, f x = f a\nhb : ∀ (x : α), b < x → f x = f b\nx : α\nhax : a ≤ x\n⊢ IccExtend h (f ∘ Subtype.val) x = f x",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"... | rcases le_or_gt x b with hxb | hbx
· lift x to Icc a b using ⟨hax, hxb⟩
rw [IccExtend_val, comp_apply]
· simp [IccExtend_of_right_le _ _ hbx.le, hb x hbx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 47,
"column": 2
} | {
"line": 48,
"column": 11
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ LipschitzWith K f ↔ ∀ (x y : α), dist (f x) (f y) ≤ ↑K * dist x y",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real.ins... | simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 47,
"column": 2
} | {
"line": 48,
"column": 11
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ LipschitzWith K f ↔ ∀ (x y : α), dist (f x) (f y) ≤ ↑K * dist x y",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real.ins... | simp only [LipschitzWith, edist_nndist, dist_nndist]
norm_cast | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 38
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns : Set α\nh : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s\n⊢ x ∈ s ↔ infEDist x s = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Metric.mem_closure_iff_infEDist_zero",
... | ← mem_closure_iff_infEDist_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 60
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nhs : IsCompact s\nhne : s.Nonempty\nx : α\nA : Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] fun y ↦ edist x y\n⊢ ∃ y ∈ s, infEDist x s = edist x y",
"usedConstants": [
"instClosedIicTopology",
"PseudoEMetricSpac... | obtain ⟨y, ys, hy⟩ := hs.exists_isMinOn hne A.continuousOn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 268,
"column": 2
} | {
"line": 269,
"column": 43
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\n⊢ hausdorffEDist s s = 0",
"usedConstants": [
"Eq.mpr",
"Metric.hausdorffEDist_def",
"_private.Mathlib.Topology.MetricSpace.HausdorffDistance.0.Metric.hausdorffEDist_self._simp_1_1",
"congrArg",
"iSup",
"Member... | simp only [hausdorffEDist_def, sup_idem, ENNReal.iSup_eq_zero]
exact fun x hx => infEDist_zero_of_mem hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 268,
"column": 2
} | {
"line": 269,
"column": 43
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\n⊢ hausdorffEDist s s = 0",
"usedConstants": [
"Eq.mpr",
"Metric.hausdorffEDist_def",
"_private.Mathlib.Topology.MetricSpace.HausdorffDistance.0.Metric.hausdorffEDist_self._simp_1_1",
"congrArg",
"iSup",
"Member... | simp only [hausdorffEDist_def, sup_idem, ENNReal.iSup_eq_zero]
exact fun x hx => infEDist_zero_of_mem hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 405,
"column": 28
} | {
"line": 410,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\nmβ : MeasurableSpace β\ninst✝¹ : TopologicalSpace β\nμ : Measure β\ninst✝ : μ.OuterRegular\nf : α → β\nf_cont : Continuous[inst✝², inst✝¹] f\nf_me : MeasurableEmbedding f\nA : Set α\nhA : MeasurableSet A\nr : ℝ≥0∞\nhr ... | by
rw [f_me.comap_apply] at hr
obtain ⟨U, hUA, Uopen, hμU⟩ := OuterRegular.outerRegular (f_me.measurableSet_image' hA) r hr
refine ⟨f ⁻¹' U, by rwa [Superset, ← image_subset_iff], Uopen.preimage f_cont, ?_⟩
rw [f_me.comap_apply]
exact (measure_mono (image_preimage_subset _ _)).trans_lt hμU | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 435,
"column": 2
} | {
"line": 466,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : ℕ → Set α\nh : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular\nh' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)\nh'' : univ ⊆ ⋃ n, s n\n⊢ μ.OuterRegular",
"usedConstants": [
"ENNReal.inst... | refine ⟨fun A hA r hr => ?_⟩
have HA : μ A < ∞ := lt_of_lt_of_le hr le_top
have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
-- Note that `A = ⋃ n, A ∩ disjointed s n`. We replace `A` with this sequence.
obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
∃ A' : ℕ → Set α,
(∀ n, MeasurableSet (A' n)) ∧... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 435,
"column": 2
} | {
"line": 466,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : ℕ → Set α\nh : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular\nh' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)\nh'' : univ ⊆ ⋃ n, s n\n⊢ μ.OuterRegular",
"usedConstants": [
"ENNReal.inst... | refine ⟨fun A hA r hr => ?_⟩
have HA : μ A < ∞ := lt_of_lt_of_le hr le_top
have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
-- Note that `A = ⋃ n, A ∩ disjointed s n`. We replace `A` with this sequence.
obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
∃ A' : ℕ → Set α,
(∀ n, MeasurableSet (A' n)) ∧... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 549,
"column": 33
} | {
"line": 549,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\nh : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞\nA s : Set α\ns_meas : MeasurableSet s\nhs : μ (s ∩ A) ≠ ∞\nr : ℝ≥0∞\nhr : r < (μ.restrict A) s\nK : Set α\nK_subs : K ⊆ toMeasurable μ (s ∩ A) ∩ s\npK : p K\nrK : r < μ K... | rwa [restrict_toMeasurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 549,
"column": 33
} | {
"line": 549,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\nh : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞\nA s : Set α\ns_meas : MeasurableSet s\nhs : μ (s ∩ A) ≠ ∞\nr : ℝ≥0∞\nhr : r < (μ.restrict A) s\nK : Set α\nK_subs : K ⊆ toMeasurable μ (s ∩ A) ∩ s\npK : p K\nrK : r < μ K... | rwa [restrict_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 549,
"column": 33
} | {
"line": 549,
"column": 60
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\nh : μ.InnerRegularWRT p fun s ↦ MeasurableSet s ∧ μ s ≠ ∞\nA s : Set α\ns_meas : MeasurableSet s\nhs : μ (s ∩ A) ≠ ∞\nr : ℝ≥0∞\nhr : r < (μ.restrict A) s\nK : Set α\nK_subs : K ⊆ toMeasurable μ (s ∩ A) ∩ s\npK : p K\nrK : r < μ K... | rwa [restrict_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 102,
"column": 8
} | {
"line": 105,
"column": 86
} | [
{
"pp": "case refine_1\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : z < x\n⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"s... | have ⟨q, h₁, h₂⟩ := exists_rat_btwn (max_lt lt (sub_lt_self x ε_pos))
rw [max_lt_iff] at h₁
refine ⟨q, ⟨conn.out hz hx ⟨h₁.1.le, h₂.le⟩, q, rfl⟩, ?_⟩
simpa only [abs_sub_comm, abs_of_pos (sub_pos.mpr h₂), sub_lt_comm] using h₁.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 102,
"column": 8
} | {
"line": 105,
"column": 86
} | [
{
"pp": "case refine_1\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : z < x\n⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"s... | have ⟨q, h₁, h₂⟩ := exists_rat_btwn (max_lt lt (sub_lt_self x ε_pos))
rw [max_lt_iff] at h₁
refine ⟨q, ⟨conn.out hz hx ⟨h₁.1.le, h₂.le⟩, q, rfl⟩, ?_⟩
simpa only [abs_sub_comm, abs_of_pos (sub_pos.mpr h₂), sub_lt_comm] using h₁.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 31
} | [
{
"pp": "⊢ borel ℝ = generateFrom (⋃ a, {Ioi ↑a})",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"congrArg",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toUniformSpace",
"SemilatticeInf.toPartialOrder",
"DistribLatti... | borel_eq_generateFrom_Ioi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 95
} | [
{
"pp": "⊢ IsPiSystem (⋃ a, {Ici ↑a})",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Set.image_univ",
"Real",
"Set.Ici",
"congrArg",
"HEq.refl",
"Set.univ",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"_private.Mathlib.MeasureTheo... | convert! isPiSystem_image_Ici (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 95
} | [
{
"pp": "⊢ IsPiSystem (⋃ a, {Ici ↑a})",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Set.image_univ",
"Real",
"Set.Ici",
"congrArg",
"HEq.refl",
"Set.univ",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"_private.Mathlib.MeasureTheo... | convert! isPiSystem_image_Ici (((↑) : ℚ → ℝ) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 39
} | [
{
"pp": "X : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : SigmaCompactSpace X\ninst✝ : MeasurableSpace X\nμ : Measure X\nF : Set X\nhF : IsClosed[inst✝²] F\nr : ℝ≥0∞\nhr : r < μ F\nB : ℕ → Set X := compactCovering X\nhBc : ∀ (n : ℕ), IsCompact (F ∩ B n)\nhBU : ⋃ n, F ∩ B n = F\n⊢ μ F = ⨆ n, μ (F ∩ B n)",
... | rw [← Monotone.measure_iUnion, hBU] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\n⊢ borel α = MeasurableSpace.generateFrom (range Iic)",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"PartialOrder.toPreorder",
"Semila... | borel_eq_generateFrom_Ioi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 154,
"column": 4
} | {
"line": 157,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α\ninst✝³ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\nmδ : MeasurableSpace δ\ninst✝¹ : Preorder α\na b x : α\nμ : Measure α\ninst✝ : R1Space α\n⊢ ∀ ⦃s : Set α⦄, s ∈ cocompact α → ∃ t ∈ coc... | intro _ hs
obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
exact ⟨(closure t)ᶜ, ht.closure.compl_mem_cocompact, isClosed_closure.measurableSet.compl,
(compl_subset_compl.2 subset_closure).trans hts⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 154,
"column": 4
} | {
"line": 157,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α\ninst✝³ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\nmδ : MeasurableSpace δ\ninst✝¹ : Preorder α\na b x : α\nμ : Measure α\ninst✝ : R1Space α\n⊢ ∀ ⦃s : Set α⦄, s ∈ cocompact α → ∃ t ∈ coc... | intro _ hs
obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
exact ⟨(closure t)ᶜ, ht.closure.compl_mem_cocompact, isClosed_closure.measurableSet.compl,
(compl_subset_compl.2 subset_closure).trans hts⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 561,
"column": 4
} | {
"line": 561,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u_5\ninst✝⁴ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : BorelSpace α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)\ns ... | rcases hsd.exists_ge' hst x with ⟨u, hus, hxu⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 223,
"column": 2
} | {
"line": 227,
"column": 70
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (a : α) ∂μ, f a ≤ g a\nt : Set α\nhts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t\nht : MeasurableSet t\nht0 : μ t = 0\nthis : ∀ᵐ (x : α) ∂μ, x ∉ t\ns : α →ₛ ℝ≥0∞\nhfs : ⇑s ≤ fun a ↦ f a\n⊢ ⇑(s.restrict tᶜ) ≤ fun a ↦ g a",... | · intro a
by_cases h : a ∈ t <;>
simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true,
indicator_of_notMem, zero_le, not_false_eq_true, indicator_of_mem]
exact le_trans (hfs a) (by_contradiction fun hnfg => h (hts hnfg)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 1083,
"column": 77
} | {
"line": 1119,
"column": 25
} | [
{
"pp": "α : Type u_5\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\n⊢ μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) + ∑' (n : ℤ), μ (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"usedConstants": [
"ENNReal.instCanonicallyOrdere... | by
have A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0) := by
rw [← measure_union]
· rw [← inter_union_distrib_left, ← preimage_union, singleton_union, Ioi_insert,
← _root_.bot_eq_zero, Ici_bot, preimage_univ, inter_univ]
· exact disjoint_singleton_left.mpr self_notMem_Ioi
|>.preimage f ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1084,
"column": 6
} | {
"line": 1086,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf✝ g✝ : α →ₛ ℝ≥0∞\nh : f✝ ≤ g✝\nf g : α →ₛ ℝ≥0∞\n⊢ (map Prod.fst (f.pair g)).lintegral μ ≤ (map (fun p ↦ max p.1 p.2) (f.pair g)).lintegral μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"Eq.mpr",
"MeasureTheory.Mea... | simp only [map_lintegral]
gcongr
exact le_sup_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1084,
"column": 6
} | {
"line": 1086,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf✝ g✝ : α →ₛ ℝ≥0∞\nh : f✝ ≤ g✝\nf g : α →ₛ ℝ≥0∞\n⊢ (map Prod.fst (f.pair g)).lintegral μ ≤ (map (fun p ↦ max p.1 p.2) (f.pair g)).lintegral μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"Eq.mpr",
"MeasureTheory.Mea... | simp only [map_lintegral]
gcongr
exact le_sup_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 319,
"column": 2
} | {
"line": 320,
"column": 77
} | [
{
"pp": "α : Type uα\ninst✝² : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\npkg' : AbstractCompletion.{vα', uα} α\nγ : Type uγ\ninst✝¹ : TopologicalSpace γ\ninst✝ : T3Space γ\nf : α → γ\ncont_f : Continuous[inst✝².toTopologicalSpace, inst✝¹] f\nx✝¹ : UniformSpace pkg'.space := pkg'.uniformStruct\nx✝ : U... | apply (IsDenseInducing.extend_unique (AbstractCompletion.isDenseInducing _) this
(Continuous.comp _ (uniformContinuous_compare pkg' pkg).continuous)).symm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 57
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\n⊢ (𝓝 1).HasBasis (fun ε ↦ 0 < ε) fun ε ↦ {y | ‖y‖ < ε}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"Monoid.toMulOneCl... | convert! NormedGroup.nhds_basis_norm_lt (1 : E) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 44
} | [
{
"pp": "case neg.inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : ¬Bornology.IsBounded s\nhε : 0 ≤ 0\n⊢ diam (thickening 0 s) ≤ diam s + 2 * 0",
"usedConstants": [
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"instReflLe",
... | simp [thickening_of_nonpos, diam_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 44
} | [
{
"pp": "case neg.inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : ¬Bornology.IsBounded s\nhε : 0 ≤ 0\n⊢ diam (thickening 0 s) ≤ diam s + 2 * 0",
"usedConstants": [
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"instReflLe",
... | simp [thickening_of_nonpos, diam_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 44
} | [
{
"pp": "case neg.inl\nα : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : ¬Bornology.IsBounded s\nhε : 0 ≤ 0\n⊢ diam (thickening 0 s) ≤ diam s + 2 * 0",
"usedConstants": [
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"instReflLe",
... | simp [thickening_of_nonpos, diam_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 109,
"column": 6
} | {
"line": 109,
"column": 62
} | [
{
"pp": "case neg.inl\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : OpensMeasurableSpace α\ne : ℕ → α\nx : α\nN : ℕ\nihN : ∀ {k : ℕ}, k ≤ N → edist ((nearestPt e N) x) x ≤ edist (e k) x\nl : ℕ\nhlN : l ≤ N\nhxl : edist (e l) x ≤ edist (e (N + 1)) x\nhk : N + 1 ≤ N + 1\n⊢ edis... | exacts [(ihN hlN).trans hxl, ihN (Nat.lt_succ_iff.1 hk)] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 354,
"column": 22
} | {
"line": 355,
"column": 83
} | [
{
"pp": "𝕜✝ : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\nα : Type u_5\ninst✝⁵ : NormedField 𝕜✝\ninst✝⁴ : SeminormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜✝ 𝕜'\n𝕜 : Type ?u.75379\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : CharZero 𝕜\ninst✝ : NormedAlgebra ℝ 𝕜\nq : ℚ\nx : 𝕜\n⊢ ‖q • x‖ ≤ ‖q‖ * ‖x‖",
... | by
rw [← smul_one_smul ℝ q x, Rat.smul_one_eq_cast, norm_smul, Rat.norm_cast_real] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 39
} | [
{
"pp": "α : 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\n⊢ Tendsto (fun n ↦ (hf.approxBounded c n) x) atTop (𝓝 (f x))",
"usedConstants": [
"MeasureTheory.StronglyMeasurable.... | have h_tendsto := hf.tendsto_approx x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 24
} | [
{
"pp": "α : 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_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ ... | let F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
then h.choose else ∞ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 261,
"column": 34
} | {
"line": 270,
"column": 13
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ [μ ⟂ₘ ν, Disjoint μ ν, Disjoint (ae μ) (ae ν)].TFAE",
"usedConstants": [
"MeasureTheory.ae",
"List.IsChain.cons_cons",
"MeasureTheory.Measure",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder",
... | by
tfae_have 1 → 2
| h => h.disjoint
tfae_have 2 → 1
| h => mutuallySingular_of_disjoint h
tfae_have 1 → 3
| h => h.disjoint_ae
tfae_have 3 → 2
| h => disjoint_of_disjoint_ae h
tfae_finish | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 715,
"column": 70
} | {
"line": 726,
"column": 82
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\nmb : MeasurableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nμ : Measure α\nf : α → β\nhf : AEStronglyMeasurable f μ\ns : Set β\nhs : MeasurableSet s\nh_nonempty : s.Nonempty\nh_mem : ∀ᵐ (x : α) ∂μ, f x ∈... | by
obtain ⟨f', hf', hff'⟩ := hf
classical
refine ⟨(f' ⁻¹' s).piecewise f' (fun _ ↦ h_nonempty.some), ?meas, ?subset, ?ae_eq⟩
case meas => exact hf'.piecewise (hf'.measurable hs) stronglyMeasurable_const
case subset =>
rw [← Set.range_subset_iff]
simpa [Set.range_piecewise] using fun _ _ ↦ h_nonempty.s... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 222,
"column": 2
} | {
"line": 223,
"column": 24
} | [
{
"pp": "α : 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 : ℕ), F a ≤... | let F' : α → ℝ≥0∞ := fun a ↦ if h : ∃ l, Tendsto (fun i ↦ f i a) atTop (𝓝 l)
then h.choose else ∞ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 959,
"column": 36
} | {
"line": 959,
"column": 49
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : SecondCountableTopology G\nf : α → G\n_m0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ (∃ g, FinStronglyMeasurable g μ ∧ f =ᶠ[ae μ] g) ↔ AEMeasurable f μ",
"use... | AEMeasurable, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 144,
"column": 4
} | {
"line": 146,
"column": 32
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → α\nhf : Function.Injective f\ns : Set β\ns_mble : MeasurableSet s\nfs_mble : MeasurableSet (f '' s)\nhs : ¬s.Finite\n⊢ count (f '' s) = count s",
"usedConstants": [
"Eq.mpr",
"MeasureThe... | rw [count_apply_infinite hs]
rw [← finite_image_iff hf.injOn] at hs
rw [count_apply_infinite hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Count | {
"line": 144,
"column": 4
} | {
"line": 146,
"column": 32
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : β → α\nhf : Function.Injective f\ns : Set β\ns_mble : MeasurableSet s\nfs_mble : MeasurableSet (f '' s)\nhs : ¬s.Finite\n⊢ count (f '' s) = count s",
"usedConstants": [
"Eq.mpr",
"MeasureThe... | rw [count_apply_infinite hs]
rw [← finite_image_iff hf.injOn] at hs
rw [count_apply_infinite hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ : Measure α\nmeasurableAtoms : Set (Set α) := measurableAtom '' univ\nh_nonempty : ∀ (s : ↑measurableAtoms), (↑s).Nonempty\npoints : ↑measurableAtoms → α := fun s ↦ ⋯.some\nx y : α\ns : Set α\nhyx : ¬y = points ⟨measurableAtom x, ⋯⟩\nhyx'... | rw [← hz, ← hsy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Perfect | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 26
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : T1Space α\nh : Perfect (closure[inst✝¹] C)\nx : α\nxC : x ∈ C\nH : AccPt x (𝓟 (closure[inst✝¹] C))\n⊢ AccPt x (𝓟 C)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"nhds",
"Eq.mp",
... | accPt_iff_frequently | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 53
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nhs : MeasurableSet s\nhs_subset : s ⊆ (μ.sigmaFiniteSetWRT' ν)ᶜ\nhνs : ν s ≠ 0\nhsσ : SigmaFinite (μ.restrict s)\n⊢ Disjoint (μ.sigmaFiniteSetWRT' ν) s",
"usedConstants": [
"ChainCompletePartialOrder.... | · exact disjoint_compl_right.mono_right hs_subset | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Perfect | {
"line": 232,
"column": 2
} | {
"line": 233,
"column": 59
} | [
{
"pp": "case refine_1\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 ... | · refine hclosed.sdiff (isOpen_biUnion fun _ ↦ ?_)
exact fun ⟨Ub, _⟩ ↦ IsTopologicalBasis.isOpen bbasis Ub | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 138,
"column": 63
} | {
"line": 138,
"column": 83
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\ns : Set α\nhs : s.Countable\n⊢ ∫⁻ (a : α) in s, f a ∂μ = ∫⁻ (a : α) in ⋃ x ∈ s, {x}, f a ∂μ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Set.biUnion_of... | biUnion_of_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.MetricSpace.Lipschitz | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 56
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\ns : Set α\nf : α → β\nhf : LipschitzOnWith K f s\nu : ℕ → α\nhu : CauchySeq u\nh'u : range u ⊆ s\nb : ℕ → ℝ\nb_nonneg : ∀ (n : ℕ), 0 ≤ b n\nhb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m)... | exact mul_le_mul_of_nonneg_left (hb n m N hn hm) K.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.CantorScheme | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 24
} | [
{
"pp": "case a.a.h.succ\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\nx y : ↑(inducedMap A).fst\nhxy : (inducedMap A).snd x = (inducedMap A).snd y\nn : ℕ\nih : res (↑x) n = res (↑y) n\nhA : ¬↑x n = ↑y n\n⊢ ¬_root_.Disjoint (A (↑x n :: res (↑x) n)) (A (↑y n :: res (↑x) n))",
"usedConstants": [
"Eq.... | not_disjoint_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.CantorScheme | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 7
} | [
{
"pp": "β : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝ : PseudoMetricSpace α\nε : ℝ\nε_pos : 0 < ε\nx : ℕ → β\nhA : ∀ ε > 0, ∃ N, ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε\nn : ℕ\nhn : ∀ n_1 ≥ n, Metric.ediam (A (res x n_1)) ≤ ENNReal.ofReal (ε / 2)\n⊢ ∃ n, ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z <... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.MetricSpace.CantorScheme | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 9
} | [
{
"pp": "β : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝¹ : PseudoMetricSpace α\ninst✝ : CompleteSpace α\nhdiam : VanishingDiam A\nhanti : ClosureAntitone A\nhnonempty : ∀ (l : List β), (A l).Nonempty\nx : ℕ → β\nu : ℕ → α\nhu : ∀ (n : ℕ), u n ∈ A (res x n)\numem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)\n... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.MetricSpace.Polish | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 21
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nu : ι → UniformSpace α\nhcomp : ∀ (i : ι), CompleteSpace α\nhcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated\nht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i ↦ (u i).toTopologicalSpace) i ≤ t₀\nhut : ∀ (i : ι), (u i).toTopologicalSpace = (fun i ↦ (u i).toTopolo... | iInf_uniformity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 45
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : MetricSpace α\nC : Set α\nε : ℝ≥0∞\nhC : Perfect C\nε_pos : 0 < ε\nx : α\nxC : x ∈ C\nthis✝ : x ∈ eball x (ε / 2)\nthis : Perfect (closure (eball x (ε / 2) ∩ C)) ∧ (closure (eball x (ε / 2) ∩ C)).Nonempty\n⊢ closure (eball x (ε / 2) ∩ C) ⊆ C",
"usedConstants": [... | IsClosed.closure_subset_iff hC.closed | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Polish | {
"line": 238,
"column": 2
} | {
"line": 241,
"column": 25
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MetricSpace α\ns : Opens α\ninst✝ : CompleteSpace α\nu : ℕ → s.CompleteCopy\nhu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (fun x ↦ (1 / 2) ^ x) N\nA : CauchySeq fun n ↦ ↑(u n)\nx : α\nxlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)\nxs : x ∉ s\nC : ... | have I' : 1 / C ≤ infDist x sᶜ :=
have : Tendsto (fun n => infDist (u n).1 sᶜ) atTop (𝓝 (infDist x sᶜ)) :=
((continuous_infDist_pt (sᶜ : Set α)).tendsto x).comp xlim
ge_of_tendsto' this I | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 109,
"column": 76
} | {
"line": 111,
"column": 38
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nm : MeasurableSpace X\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\np : X → Prop\nhp : ∀ᵐ (x : X) ∂μ, p x\n⊢ Dense {x | p x}",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.interior_eq_empty_of_null",
"congrArg",
"Compl.compl",
... | by
rw [dense_iff_closure_eq, closure_eq_compl_interior_compl, compl_univ_iff]
exact μ.interior_eq_empty_of_null hp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 10
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nm : Measure (Measure α)\nf : α → ℝ≥0∞\nhf : AEMeasurable f m.join\nhfm : Measurable f\n⊢ ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range, x * ∫⁻ (μ : Measure α), μ (⇑(SimpleFunc.eapprox f n) ⁻¹' {x}) ∂m =\n ∫⁻ (μ : Measure α), ⨆ n, ∑ x ∈ (SimpleFunc.eapprox f n).range... | clear hf | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 500,
"column": 2
} | {
"line": 500,
"column": 20
} | [
{
"pp": "E : 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\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A ≤ firstDiff x y\n⊢ (s ∩ cylinder x (Nat.... | refine ⟨y, hy, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.Complement | {
"line": 202,
"column": 65
} | {
"line": 202,
"column": 93
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.t... | simpa using hx' (y, ⟨_, hy⟩) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Complement | {
"line": 202,
"column": 65
} | {
"line": 202,
"column": 93
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.t... | simpa using hx' (y, ⟨_, hy⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Complement | {
"line": 202,
"column": 65
} | {
"line": 202,
"column": 93
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S × ↑T\nhx : ↑x.1 * ↑x.2 = g\nhx' : ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = x\ny : ↑S\nhy : (fun s ↦ (↑s)⁻¹ * g ∈ T) y\n⊢ ?m.180 = ?m.181",
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.t... | simpa using hx' (y, ⟨_, hy⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Complement | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 55
} | [
{
"pp": "G : 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''\n⊢ ∃ S, S * ↑(map H.subtype H'') = ↑H ∧ Nat.card ↑S * Nat.card ↥(map H.subtype H'') = Nat.card ↥H",
"usedConstants": [
"Membership.mem",
"Subtype",
... | obtain ⟨S, cmem, -⟩ := H''.exists_isComplement_left 1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 659,
"column": 2
} | {
"line": 668,
"column": 42
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure ((α × β) × γ)\ninst✝ : IsFiniteMeasure μ\n⊢ μ = ν ↔\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n MeasurableSet s → MeasurableSet t → MeasurableSet u → μ ((s ×ˢ t) ×ˢ u) ... | rw [← MeasurableEquiv.prodAssoc.map_measurableEquiv_injective.eq_iff, ext_prod₃_iff]
have h_eq (ν : Measure ((α × β) × γ)) {s : Set α} {t : Set β} {u : Set γ}
(hs : MeasurableSet s) (ht : MeasurableSet t) (hu : MeasurableSet u) :
ν.map MeasurableEquiv.prodAssoc (s ×ˢ (t ×ˢ u)) = ν ((s ×ˢ t) ×ˢ u) := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 659,
"column": 2
} | {
"line": 668,
"column": 42
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure ((α × β) × γ)\ninst✝ : IsFiniteMeasure μ\n⊢ μ = ν ↔\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n MeasurableSet s → MeasurableSet t → MeasurableSet u → μ ((s ×ˢ t) ×ˢ u) ... | rw [← MeasurableEquiv.prodAssoc.map_measurableEquiv_injective.eq_iff, ext_prod₃_iff]
have h_eq (ν : Measure ((α × β) × γ)) {s : Set α} {t : Set β} {u : Set γ}
(hs : MeasurableSet s) (ht : MeasurableSet t) (hu : MeasurableSet u) :
ν.map MeasurableEquiv.prodAssoc (s ×ˢ (t ×ˢ u)) = ν ((s ×ˢ t) ×ˢ u) := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 786,
"column": 11
} | {
"line": 786,
"column": 58
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ninst✝ : SFinite μ\ny : β\ni : ℕ\ns : Set α\nt : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ (map (fun x ↦ (x, y)) (sfiniteSeq μ i)) (s ×ˢ t) = (sfiniteSeq μ i) s * (dirac y) t",
"... | map_apply measurable_prodMk_right (hs.prod ht), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.Measure | {
"line": 59,
"column": 37
} | {
"line": 59,
"column": 57
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ : Measure G\ninst✝ : μ.IsMulLeftInvariant\nc : ℝ≥0∞\ng : G\n⊢ c • Measure.map (fun x ↦ g * x) μ = c • μ",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
... | map_mul_left_eq_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1176,
"column": 2
} | {
"line": 1176,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nX : α → β\nY : α → γ\nμ : Measure α\nhX : AEMeasurable X μ\n⊢ (map (fun a ↦ (X a, Y a)) μ).snd = map Y μ",
"usedConstants": [
"MeasureTheory.Measure",
"AEMeasurab... | by_cases hY : AEMeasurable Y μ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Group.Measure | {
"line": 741,
"column": 34
} | {
"line": 741,
"column": 54
} | [
{
"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 ↦ g * x) μ = μ",
"usedConstants": [
"MeasureTheory.Measure",
"HMul.hMul",
"MeasureTheory.map_mul_l... | map_mul_left_eq_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Real.Sqrt | {
"line": 210,
"column": 47
} | {
"line": 210,
"column": 66
} | [
{
"pp": "x y : ℝ\nh : x ≤ y\n⊢ NNReal.sqrt x.toNNReal ≤ NNReal.sqrt y.toNNReal",
"usedConstants": [
"Eq.mpr",
"congrArg",
"NNReal.sqrt_le_sqrt",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"OrderIso",
"NNReal",
"LE.le",
"NNReal.sqrt",
"... | NNReal.sqrt_le_sqrt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 597,
"column": 94
} | {
"line": 599,
"column": 5
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\n⊢ NormedAddGroupHom.comp 0 f = 0",
"usedConstants": [
"NormedAddGroupHom.ext",
"NormedAddGroupHom",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 603,
"column": 45
} | {
"line": 605,
"column": 5
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝³ : SeminormedAddCommGroup V₁\ninst✝² : SeminormedAddCommGroup V₂\ninst✝¹ : SeminormedAddCommGroup V₃\nV₄ : Type u_5\ninst✝ : SeminormedAddCommGroup V₄\nh : NormedAddGroupHom V₃ V₄\ng : NormedAddGroupHom V₂ V₃\nf : NormedAddGroupHom V₁ V₂\n⊢ (h.comp g).... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 652,
"column": 90
} | {
"line": 654,
"column": 5
} | [
{
"pp": "V₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type u_5\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nh : g.comp f = 0\n⊢ (incl g.ker).comp (lift f g h) = f",
"usedConstants": [
"Normed... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 790,
"column": 35
} | {
"line": 792,
"column": 5
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nV₁ : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : SeminormedAddCommGroup W\ninst✝ : SeminormedAddCommGroup V₁\nf g : NormedAddGroupHom V W\nφ : NormedAddGroupHom V₁ V\nh : f.comp φ = g.comp φ\n⊢ (ι f g).comp (lift φ h) = φ",
"usedConstants": [
"NormedAddGr... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 821,
"column": 93
} | {
"line": 823,
"column": 5
} | [
{
"pp": "V₁ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\nW₁ : Type u_6\ninst✝ : SeminormedAddCommGroup W₁\nf₁ g₁ : NormedAddGroupHom V₁ W₁\n⊢ map (id V₁) (id W₁) ⋯ ⋯ = id ↥(f₁.equalizer g₁)",
"usedConstants": [
"NormedAddGroupHom.ext",
"NormedAddGroupHom",
"AddSubgroup.seminormedAddComm... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 832,
"column": 73
} | {
"line": 834,
"column": 5
} | [
{
"pp": "V₁ : Type u_3\nV₂ : Type u_4\nV₃ : Type u_5\ninst✝⁵ : SeminormedAddCommGroup V₁\ninst✝⁴ : SeminormedAddCommGroup V₂\ninst✝³ : SeminormedAddCommGroup V₃\nW₁ : Type u_6\nW₂ : Type u_7\nW₃ : Type u_8\ninst✝² : SeminormedAddCommGroup W₁\ninst✝¹ : SeminormedAddCommGroup W₂\ninst✝ : SeminormedAddCommGroup W₃... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Basic | {
"line": 313,
"column": 92
} | {
"line": 316,
"column": 78
} | [
{
"pp": "f : ℝ →+* ℂ\nh : Continuous ⇑f\n⊢ f = ofRealHom",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"NormedSpace.toIsBoundedSMul",
... | by
convert!
congr_arg AlgHom.toRingHom <|
Subsingleton.elim (AlgHom.mk' f <| map_real_smul f h) (Algebra.ofId ℝ ℂ) | [anonymous] | Lean.Parser.Term.byTactic |
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