module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Logic.Hydra | {
"line": 80,
"column": 40
} | {
"line": 80,
"column": 57
} | {
"line": 80,
"column": 57
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\nx' x : α\nh✝ : r x' x\na : α\nh : a ∈ {x'}\n⊢ r a x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Multiset",
"id",
"Multiset.instSingleton",
"Multiset.instMembershi... | [
"α : Type u_1\nr : α → α → Prop\nx' x : α\nh✝ : r x' x\na : α\nh : a ∈ {x'}\n⊢ r x' x"
] | mem_singleton.1 h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Hydra | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 25
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\ns : Multiset α\n⊢ ¬∃ t a, (∀ a' ∈ t, r a' a) ∧ a ∈ 0 ∧ s = erase 0 a + t",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"False",
"HEq.refl",
"List.Mem.tail",
"False.elim",
"Classical.propDecidable... | [] | rintro ⟨_, _, _, ⟨⟩, _⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Logic.Hydra | {
"line": 125,
"column": 24
} | {
"line": 125,
"column": 47
} | {
"line": 125,
"column": 48
} | [
{
"pp": "case inl.refine_2\nα : Type u_1\nr : α → α → Prop\ns₁ s₂ t : Multiset α\na : α\nhr : ∀ a' ∈ t, r a' a\nhe : (s₁ + s₂ + t).erase a + {a} = s₁ + s₂ + t\nh : a ∈ s₁\n⊢ (s₁.erase a + t, s₂).1 + (s₁.erase a + t, s₂).2 = (s₁ + (s₂ + t)).erase a",
"ppTerm": "?inl.refine_2",
"assigned": true,
"used... | [
"case inl.refine_2\nα : Type u_1\nr : α → α → Prop\ns₁ s₂ t : Multiset α\na : α\nhr : ∀ a' ∈ t, r a' a\nhe : (s₁ + s₂ + t).erase a + {a} = s₁ + s₂ + t\nh : a ∈ s₁\n⊢ (s₁.erase a + t, s₂).1 + (s₁.erase a + t, s₂).2 = s₁.erase a + (s₂ + t)"
] | erase_add_left_pos _ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.ClosedCompactCylinders | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 65
} | {
"line": 45,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\n⊢ ∅ ∈ closedCompactCylinders X",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.topologicalSpace",
"Iff.of_eq",
"congrArg",
"_private.Mathlib.MeasureTheory.Construc... | [
"ι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\n⊢ ∃ i i_1, ∃ (_ : IsClosed i_1) (_ : IsCompact i_1), ∅ = cylinder i i_1"
] | simp_rw [closedCompactCylinders, mem_iUnion, mem_singleton_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.SetSemiring | {
"line": 351,
"column": 49
} | {
"line": 360,
"column": 40
} | {
"line": 362,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nhI : ↑I ⊆ C\n⊢ Disjoint I (hC.disjointOfDiffUnion hs hI)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"False",
"Finset.decidableDisjoint",
"congrArg",
"Finset... | [] | by
by_contra h
rw [Finset.not_disjoint_iff] at h
obtain ⟨u, huI, hu_disjointOfDiffUnion⟩ := h
have h_disj : u ≤ ⊥ :=
hC.disjoint_sUnion_disjointOfDiffUnion hs hI (subset_sUnion_of_mem huI)
(subset_sUnion_of_mem hu_disjointOfDiffUnion)
simp only [Set.bot_eq_empty, subset_empty_iff] at h_disj
refine h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.SetSemiring | {
"line": 508,
"column": 12
} | {
"line": 508,
"column": 14
} | {
"line": 508,
"column": 15
} | [
{
"pp": "case inr.inr\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u < u'\nhv : v' < v\na : Set α\n⊢ a ∈ ↑{Set.Ioc u u', Set.Ioc v' v} →\n ∀ ⦃y : Set α⦄, y ∈ ↑{Set.Ioc u u', Set.Ioc v' v} → a ≠ y → Function.onFun Disjoint id a y",
"ppTe... | [
"case inr.inr\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u < u'\nhv : v' < v\na : Set α\nha : a ∈ ↑{Set.Ioc u u', Set.Ioc v' v}\n⊢ ∀ ⦃y : Set α⦄, y ∈ ↑{Set.Ioc u u', Set.Ioc v' v} → a ≠ y → Function.onFun Disjoint id a y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 96,
"column": 11
} | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 37
} | [
{
"pp": "case pos\nα : Type u_1\n𝒜 : Set (Set α)\nι : Type u_2\nh𝒜 : IsSetAlgebra 𝒜\ns : ι → Set α\nS : Finset ι\nhs : ∀ i ∈ S, s i ∈ 𝒜\nh : S = ∅\n⊢ ⋂ i ∈ ∅, s i ∈ 𝒜",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
"Finset",
... | [
"case pos\nα : Type u_1\n𝒜 : Set (Set α)\nι : Type u_2\nh𝒜 : IsSetAlgebra 𝒜\ns : ι → Set α\nS : Finset ι\nhs : ∀ i ∈ S, s i ∈ 𝒜\nh : S = ∅\n⊢ ⋂ x ∈ ↑∅, s x ∈ 𝒜"
] | ← Finset.set_biInter_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.WithTop | {
"line": 121,
"column": 10
} | {
"line": 121,
"column": 20
} | {
"line": 122,
"column": 10
} | [
{
"pp": "case a.inl.coe.coe.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx... | [
"case a.inl.coe.coe.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 30
} | {
"line": 237,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nhC : IsSetSemiring C\nm : AddContent G C\ns : Set α\nJ : Finset (Set α)\nhJ : ↑J ⊆ C\nh'J : (↑J).PairwiseDisjoint id\nhs : s = ⋃₀ ↑J\nh : ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s = ⋃₀ ↑J\n⊢ ⋃₀ ↑h.choose = s",
"ppTerm": "?m.71",... | [] | exact h.choose_spec.2.2.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.WithTop | {
"line": 161,
"column": 10
} | {
"line": 161,
"column": 20
} | {
"line": 162,
"column": 10
} | [
{
"pp": "case a.inr.coe.coe.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx... | [
"case a.inr.coe.coe.refine_1\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Order.WithTop | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 57
} | {
"line": 283,
"column": 58
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nα : Type u_2\nf : Filter α\nx : α → WithTop ι\nh : Nonempty ι\n⊢ (∀ i < ⊤, ∀ᶠ (x_1 : α) in f, x x_1 ∈ Ioi i) ↔ ∀ (i : ι), ∀ᶠ (a : α) in f, ↑i < x a",
"ppTerm": "?inr",
"assigned": true,
"us... | [
"case inr\nι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nα : Type u_2\nf : Filter α\nx : α → WithTop ι\nh : Nonempty ι\n⊢ (∀ i < ⊤, ∀ᶠ (x_1 : α) in f, x x_1 ∈ Ioi i) ↔ ∀ a ∈ range some, ∀ᶠ (a_2 : α) in f, a < x a_2"
] | ← Set.forall_mem_range (p := (∀ᶠ a in f, · < x a)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.WithTop | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 26
} | {
"line": 290,
"column": 26
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : Nonempty ι\n⊢ Tendsto some atTop (𝓝 ⊤)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"WithTop.instPreorde... | [
"case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : NoMaxOrder ι\nh : Nonempty ι\n⊢ ∀ (i : ι), ∀ᶠ (a : ι) in atTop, ↑i < ↑a"
] | tendsto_nhds_top_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 447,
"column": 10
} | {
"line": 448,
"column": 64
} | {
"line": 449,
"column": 6
} | [
{
"pp": "case inl.refine_2\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n ... | [] | rw [mem_dilatransvections_iff_finrank_quotient,
auxTransvection_mul_fixed hfv, hrank, hn0, zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 447,
"column": 10
} | {
"line": 448,
"column": 64
} | {
"line": 449,
"column": 6
} | [
{
"pp": "case inl.refine_2\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n ... | [] | rw [mem_dilatransvections_iff_finrank_quotient,
auxTransvection_mul_fixed hfv, hrank, hn0, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Transvection.Generation | {
"line": 447,
"column": 10
} | {
"line": 448,
"column": 64
} | {
"line": 449,
"column": 6
} | [
{
"pp": "case inl.refine_2\nK : Type u_1\ninst✝³ : DivisionRing K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nn✝ : ℕ\ne : V ≃ₗ[K] V\nhe : ∀ (a : K), ∃ x, e.fixedReduce x ≠ a • x\nn : ℕ\nhind :\n ∀ {e : V ≃ₗ[K] V},\n (∀ (a : K), ∃ x, ¬e.fixedReduce x = a • x) →\n ... | [] | rw [mem_dilatransvections_iff_finrank_quotient,
auxTransvection_mul_fixed hfv, hrank, hn0, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 38
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case neg\nΩ : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\... | [
"case neg\nΩ : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[ℝ]... | have : (ae μ).NeBot := ae_neBot.2 hμ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 77
} | {
"line": 290,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nhp_ne_zero : ¬p = 0\nhp_ne_top : ¬p = ∞\nM : ℝ\nhM' : 0 ≤ M\nhM :\n ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indica... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nhp_ne_zero : ¬p = 0\nhp_ne_top : ¬p = ∞\nM : ℝ\nhM' : 0 ≤ M\nhM :\n ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicator (fun x ↦... | rw [enorm_indicator_eq_indicator_enorm, enorm_indicator_eq_indicator_enorm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 141,
"column": 40
} | {
"line": 141,
"column": 42
} | {
"line": 141,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 141,
"column": 56
} | {
"line": 141,
"column": 58
} | {
"line": 141,
"column": 58
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 52
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₂ : ¬s ≤[i] 0\nn : ℕ\nhn : ¬s ≤[i \\ ⋃ k, ⋃ (_ : k < n + 1), s.restrictNonposSeq i k] 0\nh₁ : ¬s ≤[i \\ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k] 0\n⊢ 0 < s (s.someExistsOneDivLT (i \\ ⋃ k, ⋃ (_ : k ≤ n), s.restric... | [
"case succ\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₂ : ¬s ≤[i] 0\nn : ℕ\nhn : ¬s ≤[i \\ ⋃ k, ⋃ (_ : k < n + 1), s.restrictNonposSeq i k] 0\nh₁ : ¬s ≤[i \\ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSeq i k] 0\nleft✝¹ :\n s.someExistsOneDivLT (i \\ ⋃ k, ⋃ (_ : k ≤ n), s.restrictNonposSe... | rcases someExistsOneDivLT_spec h₁ with ⟨_, _, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 39
} | {
"line": 362,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε\nδ : ℝ\nh... | refine ⟨δ, hδpos, fun s hs hμs => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 206,
"column": 40
} | {
"line": 206,
"column": 42
} | {
"line": 206,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 206,
"column": 56
} | {
"line": 206,
"column": 58
} | {
"line": 206,
"column": 58
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 212,
"column": 40
} | {
"line": 212,
"column": 42
} | {
"line": 212,
"column": 43
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 212,
"column": 56
} | {
"line": 212,
"column": 58
} | {
"line": 212,
"column": 58
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 903,
"column": 94
} | {
"line": 910,
"column": 8
} | {
"line": 912,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝⁵ : MeasurableSpace β\nM : Type u_3\ninst✝⁴ : AddCommMonoid M\ninst✝³ : TopologicalSpace M\nR : Type u_4\ninst✝² : Semiring R\ninst✝¹ : DistribMulAction R M\ninst✝ : ContinuousConstSMul R M\nv : VectorMeasure α M\nf : α → β\nc : R\n⊢ (c • v).map f... | [] | by
by_cases hf : Measurable f
· ext i hi
simp [map_apply _ hf hi]
· simp only [map, dif_neg hf]
-- `smul_zero` does not work since we do not require `ContinuousAdd`
ext i
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = (-r).toNNReal • j.negPart",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"MeasureTheory.JordanDecomposition.posPart",
"Eq.mpr",
... | [] | rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = (-r).toNNReal • j.negPart",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"MeasureTheory.JordanDecomposition.posPart",
"Eq.mpr",
... | [] | rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 71
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = (-r).toNNReal • j.negPart",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"MeasureTheory.JordanDecomposition.posPart",
"Eq.mpr",
... | [] | rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 35
} | {
"line": 234,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂\nhν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃\nk : Set α\nhk : MeasurableSet k\n⊢ ... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂\nhν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃\nk : Set α\nhk : MeasurableSet k\n⊢ (s.toMeasure... | toSignedMeasure_sub_apply hk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1077,
"column": 2
} | {
"line": 1077,
"column": 28
} | {
"line": 1079,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nhi₁ : MeasurableSet i\nhi₂ : v ≤[i] w\nhj₁ : MeasurableSe... | [] | · rintro (_ | _) <;> simpa | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 39
} | {
"line": 399,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nε : ℝ\nhε : 0 < ε\nhℒp : MemLp f p μ\nright✝ : eLpNorm f p μ < ∞\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᵐ[μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nε : ℝ\nhε : 0 < ε\nhℒp : MemLp f p μ\nright✝ : eLpNorm f p μ < ∞\nf' : α → β\nhf' : StronglyMeasurable f'\nheq : f =ᵐ[μ] f'\nδ : ℝ\nhδpos : 0 < δ\nhδ : ∀ (s : Set α),... | refine ⟨δ, hδpos, fun s hs hμs => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 463,
"column": 69
} | {
"line": 464,
"column": 39
} | {
"line": 466,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\n⊢ IsFiniteMeasure s.totalVariation",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"MeasureTheory.JordanDecomposition.posPart",
"MeasureTheory.SignedMeasure.totalVariation",
"MeasureTheory.isFiniteMeasu... | [] | by
unfold totalVariation; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1331,
"column": 20
} | {
"line": 1331,
"column": 34
} | {
"line": 1331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n : MeasurableSpace α\nv : VectorMeasure α ... | [] | rw [if_neg hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1331,
"column": 20
} | {
"line": 1331,
"column": 34
} | {
"line": 1331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n : MeasurableSpace α\nv : VectorMeasure α ... | [] | rw [if_neg hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1331,
"column": 20
} | {
"line": 1331,
"column": 34
} | {
"line": 1331,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n : MeasurableSpace α\nv : VectorMeasure α ... | [] | rw [if_neg hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 430,
"column": 4
} | {
"line": 430,
"column": 29
} | {
"line": 431,
"column": 4
} | [
{
"pp": "case left.hA\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : MeasurableSet i\nhj : MeasurableSet j\nhi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → 0 j ≤ s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → s j ≤ 0 j\nhj' :\n (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j →... | [
"case left.hB\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : MeasurableSet i\nhj : MeasurableSet j\nhi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → 0 j ≤ s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → s j ≤ 0 j\nhj' :\n (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → 0 j_1 ≤ s j... | · exact hj.compl.inter hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 75
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case e'_3.e'_6\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : SignedMeasure α\ninst✝ : s.HaveLebesgueDecomposition μ\n⊢ s.toJordanDecomposition.negPart =\n s.toJordanDecomposition.negPart.singularPart μ + μ.withDensity (s.toJordanDecomposition.negPart.rnDeriv μ)",
"ppTerm": "?e'_3.e'_6... | [] | exact s.toJordanDecomposition.negPart.haveLebesgueDecomposition_add μ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 438,
"column": 4
} | {
"line": 441,
"column": 45
} | {
"line": 442,
"column": 4
} | [
{
"pp": "case right.h\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : MeasurableSet i\nhj : MeasurableSet j\nhi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → 0 j ≤ s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → s j ≤ 0 j\nhj' :\n (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j →... | [
"case right.hA\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni j : Set α\nhi : MeasurableSet i\nhj : MeasurableSet j\nhi' : (∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → 0 j ≤ s j) ∧ ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ iᶜ → s j ≤ 0 j\nhj' :\n (∀ ⦃j_1 : Set α⦄, MeasurableSet j_1 → j_1 ⊆ j → 0 j_1 ≤ s ... | · exact
Set.disjoint_of_subset_left Set.inter_subset_left
(Set.disjoint_of_subset_right Set.inter_subset_right
(IsCompl.disjoint isCompl_compl)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 176,
"column": 4
} | {
"line": 181,
"column": 72
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : MemLp g p μ\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [] | rw [← toReal_eLpNorm (hf.add hg).aestronglyMeasurable,
← toReal_eLpNorm hf.aestronglyMeasurable, ← toReal_eLpNorm hg.aestronglyMeasurable,
← ENNReal.toReal_add hf.eLpNorm_ne_top hg.eLpNorm_ne_top]
gcongr
exacts [ENNReal.add_ne_top.2 ⟨hf.eLpNorm_ne_top, hg.eLpNorm_ne_top⟩,
eLpNorm_add_le hf.aes... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 176,
"column": 4
} | {
"line": 181,
"column": 72
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : MemLp g p μ\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [] | rw [← toReal_eLpNorm (hf.add hg).aestronglyMeasurable,
← toReal_eLpNorm hf.aestronglyMeasurable, ← toReal_eLpNorm hg.aestronglyMeasurable,
← ENNReal.toReal_add hf.eLpNorm_ne_top hg.eLpNorm_ne_top]
gcongr
exacts [ENNReal.add_ne_top.2 ⟨hf.eLpNorm_ne_top, hg.eLpNorm_ne_top⟩,
eLpNorm_add_le hf.aes... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 47,
"column": 2
} | {
"line": 48,
"column": 95
} | {
"line": 49,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\nf : α → ℝ\nhf : Integrable f μ\n⊢ ∀ (s : Set α),\n MeasurableSet s → μ s < ∞ → IntegrableOn (SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm)) s μ",
"ppTerm": "?refine_... | [
"case refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhμm : SigmaFinite (μ.trim hm)\nf : α → ℝ\nhf : Integrable f μ\n⊢ ∀ (s : Set α),\n MeasurableSet s →\n μ s < ∞ →\n ∫ (x : α) in s, SignedMeasure.rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm) x ∂μ = ∫ (x : α) in s, f... | · exact fun _ _ _ => (integrable_of_integrable_trim hm
(SignedMeasure.integrable_rnDeriv ((μ.withDensityᵥ f).trim hm) (μ.trim hm))).integrableOn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 374,
"column": 2
} | {
"line": 378,
"column": 44
} | {
"line": 380,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : s.HaveLebesgueDecomposition μ\n⊢ (-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
... | [] | refine
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg ?_
rw [withDensityᵥ_neg, ← add_right_inj ((-s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_neg, ← neg_add,
singularPart_add_withDensity_rnDeriv_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 374,
"column": 2
} | {
"line": 378,
"column": 44
} | {
"line": 380,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : s.HaveLebesgueDecomposition μ\n⊢ (-s).rnDeriv μ =ᵐ[μ] -s.rnDeriv μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
... | [] | refine
Integrable.ae_eq_of_withDensityᵥ_eq (integrable_rnDeriv _ _) (integrable_rnDeriv _ _).neg ?_
rw [withDensityᵥ_neg, ← add_right_inj ((-s).singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, singularPart_neg, ← neg_add,
singularPart_add_withDensity_rnDeriv_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 78,
"column": 35
} | {
"line": 78,
"column": 37
} | {
"line": 78,
"column": 38
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : PartialOrder E\ninst✝² : ClosedIciTopology E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nc : E\nhc : 0 ≤ c\nhfc : ... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : PartialOrder E\ninst✝² : ClosedIciTopology E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nc : E\nhc : 0 ≤ c\nhfc : ∀ᵐ (x : α) ∂... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 90,
"column": 50
} | {
"line": 90,
"column": 52
} | {
"line": 90,
"column": 53
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhfint : Integrable f μ\nh1 : μ[f | ... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhfint : Integrable f μ\nh1 : μ[f | m] ≤ᵐ[μ] μ[f... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 94
} | {
"line": 98,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : ¬Sigm... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : SigmaFinite (μ.tr... | · simpa [condExp_of_not_sigmaFinite hm hsig] using integral_nonneg (fun x => abs_nonneg f x) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 94
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | · simpa [condExp_of_not_sigmaFinite hm hsig] using integral_nonneg (fun x => abs_nonneg f x) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 27
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case h_univ\nΩ : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν ... | [] | exact key isClosed_univ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 27
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case h_univ\nΩ : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν ... | [] | exact key isClosed_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 27
} | {
"line": 265,
"column": 2
} | [
{
"pp": "case h_univ\nΩ : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : HasOuterApproxClosed Ω\ninst✝¹ : BorelSpace Ω\nμ ν : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh : ∀ (f : Ω →ᵇ ℝ≥0), ∫⁻ (x : Ω), ↑(f x) ∂μ = ∫⁻ (x : Ω), ↑(f x) ∂ν\nkey : ∀ {F : Set Ω}, IsClosed[inst✝³] F → μ F = ν ... | [] | exact key isClosed_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 44
} | {
"line": 321,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nι : Type u_3\nμ : FiniteMeasure Ω\nT : Finset ι\ns : ι → Set Ω\nhd : (↑T).Pairwise (Disjoint on s)\nhm : ∀ (i : ι), MeasurableSet (s i)\nt : Set Ω\nht : MeasurableSet t\n⊢ ((↑μ).restrict (⋃ i ∈ T, s i)) t = ∑ c ∈ T, ((↑μ).restrict (s c)) t",
"ppTerm": "?m.46... | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nι : Type u_3\nμ : FiniteMeasure Ω\nT : Finset ι\ns : ι → Set Ω\nhd : (↑T).Pairwise (Disjoint on s)\nhm : ∀ (i : ι), MeasurableSet (s i)\nt : Set Ω\nht : MeasurableSet t\n⊢ (Measure.sum fun i ↦ (↑μ).restrict (s ↑i)) t = ∑ c ∈ T, ((↑μ).restrict (s c)) t"
] | rw [Measure.restrict_biUnion_finset hd hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 702,
"column": 31
} | {
"line": 702,
"column": 33
} | {
"line": 703,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nfn : ι → α → β\nhUI : UnifIntegrable fn p μ\nhfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ\nε : ℝ\nhε :... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nfn : ι → α → β\nhUI : UnifIntegrable fn p μ\nhfn : ∀ (i : ι), AEStronglyMeasurable (fn i) μ\nε : ℝ\nhε : 0 < ε\nδ : ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 462,
"column": 9
} | {
"line": 464,
"column": 72
} | {
"line": 465,
"column": 2
} | [] | [] | f ω
_ ≤ g ω + nndist (f ω) (g ω) := NNReal.le_add_nndist (f ω) (g ω)
_ ≤ g ω + nndist f g := (add_le_add_iff_left (g ω)).mpr (le_dist ω) | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 748,
"column": 4
} | {
"line": 748,
"column": 36
} | {
"line": 749,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), eLpNorm (f... | [
"case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), eLpNorm (f i) p μ < ∞\... | refine ⟨C.toNNReal, fun i => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 753,
"column": 6
} | {
"line": 753,
"column": 30
} | {
"line": 754,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), eLpNorm (f... | [
"case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), eLpNorm (f i) p μ < ∞\... | obtain ⟨i, -, rfl⟩ := hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 751,
"column": 4
} | {
"line": 754,
"column": 16
} | {
"line": 755,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : Finite ι\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nval✝ : Fintype ι\nhι : Nonempty ι\nh✝ : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhf : ∀ (i : ι), eLpNorm (f... | [] | · refine ne_of_lt ((Finset.max'_lt_iff _ _).2 fun y hy => ?_)
rw [Finset.mem_image] at hy
obtain ⟨i, -, rfl⟩ := hy
exact hf i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 260,
"column": 12
} | {
"line": 260,
"column": 69
} | {
"line": 261,
"column": 10
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\np : ℝ≥0∞\nhp : 1 ≤ p\nhf : MemLp f p μ\nh✝ : NeZero μ\nhp' : 0 < p\nhm : m ≤ m0\nhsig : SigmaFinite (μ.trim hm)\nhpt : p ≠ ∞\n⊢ lpN... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\np : ℝ≥0∞\nhp : 1 ≤ p\nhf : MemLp f p μ\nh✝ : NeZero μ\nhp' : 0 < p\nhm : m ≤ m0\nhsig : SigmaFinite (μ.trim hm)\nhpt : p ≠ ∞\n⊢ lpNorm μ[f | m]... | lpNorm_eq_integral_norm_rpow_toReal hp'.ne.symm hpt hf.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 739,
"column": 2
} | {
"line": 743,
"column": 15
} | {
"line": 744,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\nh : ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))\nf : Ω →ᵇ ℝ\nf_pos : Ω →ᵇ... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\nh : ∀ (f : Ω →ᵇ ℝ≥0), Tendsto (fun i ↦ ∫⁻ (x : Ω), ↑(f x) ∂↑(μs i)) F (𝓝 (∫⁻ (x : Ω), ↑(f x) ∂↑μ))\nf : Ω →ᵇ ℝ\nf_pos : Ω →ᵇ ℝ≥0 := f.nn... | have aux :
∀ g : Ω →ᵇ ℝ≥0,
(ENNReal.toReal ∘ fun i : γ ↦ ∫⁻ x : Ω, ↑(g x) ∂(μs i : Measure Ω)) =
fun i : γ ↦ (∫⁻ x : Ω, ↑(g x) ∂(μs i : Measure Ω)).toReal :=
fun _ ↦ rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 138,
"column": 34
} | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 37
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (κ a) (a✝¹ i)\na : α... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (κ a) (a✝¹ i)\na : α\nha : κ a =... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 138,
"column": 34
} | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 37
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (η a) (a✝¹ i)\na : α... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (η a) (a✝¹ i)\na : α\nha : κ a =... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 146,
"column": 34
} | {
"line": 146,
"column": 36
} | {
"line": 146,
"column": 37
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ * (κ a) a✝²\na : α\n⊢ κ a = η a → (κ a) (a✝³ ∩ a✝²) = (κ... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ * (κ a) a✝²\na : α\nha : κ a = η a\n⊢ (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 146,
"column": 34
} | {
"line": 146,
"column": 36
} | {
"line": 146,
"column": 37
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (η a) (a✝³ ∩ a✝²) = (η a) a✝³ * (η a) a✝²\na : α\n⊢ κ a = η a → (η a) (a✝³ ∩ a✝²) = (η... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (η a) (a✝³ ∩ a✝²) = (η a) a✝³ * (η a) a✝²\na : α\nha : κ a = η a\n⊢ (η a) (a✝³ ∩ a✝²) = (η a) a✝³ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 178,
"column": 77
} | {
"line": 178,
"column": 79
} | {
"line": 179,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : iIndepSets π κ μ\na : α\n⊢ (κ a) (⋂ i ∈ ∅, univ) = ∏ i ∈ ∅, (κ a) univ → IsProbabilityMeasure (κ a)",
"ppTerm": "?m.47",
"assigned": true,
"use... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : iIndepSets π κ μ\na : α\nha : (κ a) (⋂ i ∈ ∅, univ) = ∏ i ∈ ∅, (κ a) univ\n⊢ IsProbabilityMeasure (κ a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 183,
"column": 71
} | {
"line": 183,
"column": 73
} | {
"line": 183,
"column": 74
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\ns : ι → Set Ω\ninst✝ : Fintype ι\nh : iIndepSets π κ μ\nhs : ∀ (i : ι), s i ∈ π i\na : α\n⊢ (κ a) (⋂ i ∈ Finset.univ, s i) = ∏ i, (κ a) (s i) → (κ a) (⋂ i, s i... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\ns : ι → Set Ω\ninst✝ : Fintype ι\nh : iIndepSets π κ μ\nhs : ∀ (i : ι), s i ∈ π i\na : α\nha : (κ a) (⋂ i ∈ Finset.univ, s i) = ∏ i, (κ a) (s i)\n⊢ (κ a) (⋂ i, s i) = ∏ i,... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 197,
"column": 78
} | {
"line": 197,
"column": 80
} | {
"line": 198,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\nm : ι → MeasurableSpace Ω\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set Ω\ninst✝ : Fintype ι\nh : iIndep m κ μ\nhs : ∀ (i : ι), MeasurableSet (s i)\na : α\n⊢ (κ a) (⋂ i ∈ Finset.univ, s i) = ∏ i, (κ a) (s i) → (κ ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\nm : ι → MeasurableSpace Ω\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set Ω\ninst✝ : Fintype ι\nh : iIndep m κ μ\nhs : ∀ (i : ι), MeasurableSet (s i)\na : α\nha : (κ a) (⋂ i ∈ Finset.univ, s i) = ∏ i, (κ a) (s i)\n⊢ (κ a) (⋂ i,... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 221,
"column": 48
} | {
"line": 221,
"column": 50
} | {
"line": 222,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nι' : Type u_5\ng : ι' → ι\nhg : Function.Injective g\nh : iIndepSets π κ μ\ns : Finset ι'\nf : ι' → Set Ω\nhf : ∀ i ∈ s, f i ∈ (π ∘ g) i\nf' : ι → Set Ω := Fun... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nι' : Type u_5\ng : ι' → ι\nhg : Function.Injective g\nh : iIndepSets π κ μ\ns : Finset ι'\nf : ι' → Set Ω\nhf : ∀ i ∈ s, f i ∈ (π ∘ g) i\nf' : ι → Set Ω := Function.extend... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 1010,
"column": 6
} | {
"line": 1010,
"column": 71
} | {
"line": 1011,
"column": 6
} | [
{
"pp": "case hf\nΩ' : Type u_2\ninst✝⁶ : MeasurableSpace Ω'\ninst✝⁵ : TopologicalSpace Ω'\ninst✝⁴ : BorelSpace Ω'\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : BorelSpace Ω\ninst✝ : NormalSpace Ω'\nf : Ω → Ω'\nhf : IsClosedEmbedding f\nM : Set (FiniteMeasure Ω') := {μ | μ (ra... | [
"case hs\nΩ' : Type u_2\ninst✝⁶ : MeasurableSpace Ω'\ninst✝⁵ : TopologicalSpace Ω'\ninst✝⁴ : BorelSpace Ω'\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : BorelSpace Ω\ninst✝ : NormalSpace Ω'\nf : Ω → Ω'\nhf : IsClosedEmbedding f\nM : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}... | · exact fun t ht ↦ hf.measurableEmbedding.measurableSet_image' ht | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 269,
"column": 42
} | {
"line": 269,
"column": 44
} | {
"line": 270,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns₁ s₂ : Set (Set Ω)\nh : IndepSets s₁ s₂ κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\na : α\n⊢ (κ a) (t2 ∩ t1) = (κ a) t2 * (κ a) t1 → (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2",
"ppTerm": ... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns₁ s₂ : Set (Set Ω)\nh : IndepSets s₁ s₂ κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\na : α\nha : (κ a) (t2 ∩ t1) = (κ a) t2 * (κ a) t1\n⊢ (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 1020,
"column": 6
} | {
"line": 1020,
"column": 71
} | {
"line": 1021,
"column": 6
} | [
{
"pp": "case hf\nΩ' : Type u_2\ninst✝⁶ : MeasurableSpace Ω'\ninst✝⁵ : TopologicalSpace Ω'\ninst✝⁴ : BorelSpace Ω'\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : BorelSpace Ω\ninst✝ : NormalSpace Ω'\nf : Ω → Ω'\nhf : IsClosedEmbedding f\nM : Set (FiniteMeasure Ω') := {μ | μ (ra... | [
"case hs\nΩ' : Type u_2\ninst✝⁶ : MeasurableSpace Ω'\ninst✝⁵ : TopologicalSpace Ω'\ninst✝⁴ : BorelSpace Ω'\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : BorelSpace Ω\ninst✝ : NormalSpace Ω'\nf : Ω → Ω'\nhf : IsClosedEmbedding f\nM : Set (FiniteMeasure Ω') := {μ | μ (range f)ᶜ = 0}... | · exact fun t ht ↦ hf.measurableEmbedding.measurableSet_image' ht | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 175,
"column": 43
} | {
"line": 175,
"column": 45
} | {
"line": 175,
"column": 46
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nhf :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nhf :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 397,
"column": 30
} | {
"line": 397,
"column": 32
} | {
"line": 398,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → Set Ω\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh : ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)\nS : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ S, f i ∈ (fun i ↦ {s i}) i\na : α\n⊢ (κ a) ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → Set Ω\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nh : ∀ (S : Finset ι), ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ S, s i) = ∏ i ∈ S, (κ a) (s i)\nS : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ S, f i ∈ (fun i ↦ {s i}) i\na : α\nha : (κ a) (⋂ i ∈ S,... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 17
} | {
"line": 484,
"column": 18
} | [
{
"pp": "case inr.basic\nα : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m... | [] | | basic u hu => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 486,
"column": 32
} | {
"line": 486,
"column": 34
} | {
"line": 487,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet t1\nh : IsM... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 195,
"column": 32
} | {
"line": 195,
"column": 34
} | {
"line": 195,
"column": 35
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nh : ∀ (i : ι), ∀ᵐ (a : α) ∂μ, f i =ᵐ[κ a] g i\nh' : iIndepFun g κ μ\ni : ι\na : α\n⊢ f i =ᵐ[κ a] g i ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nh : ∀ (i : ι), ∀ᵐ (a : α) ∂μ, f i =ᵐ[κ a] g i\nh' : iIndepFun g κ μ\ni : ι\na : α\nha : f i =ᵐ[κ a] g i\n⊢ g i =ᵐ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 204,
"column": 31
} | {
"line": 204,
"column": 33
} | {
"line": 205,
"column": 2
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nγ : ι → Type u_9\nmβ : (i : ι) → MeasurableSpace (β i)\nmγ : (i : ι) → MeasurableSpace (γ i)\nf : (i : ι) → Ω → β i\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nγ : ι → Type u_9\nmβ : (i : ι) → MeasurableSpace (β i)\nmγ : (i : ι) → MeasurableSpace (γ i)\nf : (i : ι) → Ω → β i\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 493,
"column": 32
} | {
"line": 493,
"column": 34
} | {
"line": 494,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet t1\nh : IsM... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 516,
"column": 32
} | {
"line": 516,
"column": 34
} | {
"line": 517,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : Indep... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 κ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 526,
"column": 30
} | {
"line": 526,
"column": 32
} | {
"line": 527,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : Indep... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 κ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 571,
"column": 42
} | {
"line": 571,
"column": 44
} | {
"line": 572,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s x\nht1_eq ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s x\nht1_eq : t1 = ⋂ x ∈... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Basic | {
"line": 1083,
"column": 6
} | {
"line": 1083,
"column": 54
} | {
"line": 1084,
"column": 4
} | [
{
"pp": "ι : Type u_6\nΩ : Type u_7\nα : Type u_8\nβ : Type u_9\nmΩ : MeasurableSpace Ω\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure Ω\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun (fun i ω ↦ (X i... | [] | exact fun _ _ i _ _ ↦ .inr <| measure_ne_top _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 706,
"column": 8
} | {
"line": 706,
"column": 22
} | {
"line": 706,
"column": 23
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\na : ι\nS : Finset ι\nhp_ind : iIndepSets π κ μ\nhaS : a ∉ S\nt1 t2 : Set Ω\ns : Finset ι\nhs_mem : s ⊆ S\nft1 : ι → Set Ω\nhft1_mem : ∀ x ∈ s, ft1 x ∈ π x\nht1... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\na : ι\nS : Finset ι\nhp_ind : iIndepSets π κ μ\nhaS : a ∉ S\nt1 t2 : Set Ω\ns : Finset ι\nhs_mem : s ⊆ S\nft1 : ι → Set Ω\nhft1_mem : ∀ x ∈ s, ft1 x ∈ π x\nht1_eq : t1 = ⋂... | h_t1_inter_t2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Integration | {
"line": 171,
"column": 53
} | {
"line": 173,
"column": 53
} | {
"line": 174,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : ContinuousENorm E\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : OpensMeasurableSpace E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : ContinuousENorm F\ninst✝³ : MeasurableSpace F\ninst✝... | [] | by
rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.1.enorm hY.1.enorm
(hXY.comp measurable_enorm measurable_enorm)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.Covariance | {
"line": 153,
"column": 53
} | {
"line": 154,
"column": 56
} | {
"line": 156,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\nc : ℝ\n⊢ cov[X, fun ω ↦ Y ω / c; μ] = cov[X, Y; μ] / c",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"instHDiv",
"HMul.hMul",
"Divisio... | [] | by
simp_rw [← inv_mul_eq_div, covariance_const_mul_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.Covariance | {
"line": 160,
"column": 30
} | {
"line": 160,
"column": 50
} | {
"line": 160,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ cov[-1 • X, Y; μ] = -cov[X, Y; μ]",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
"ProbabilityTheo... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ -1 * cov[X, Y; μ] = -cov[X, Y; μ]"
] | covariance_smul_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 318,
"column": 2
} | {
"line": 323,
"column": 57
} | {
"line": 325,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nhX : AEMeasurable X (Measure.map Y μ)\nhY : AEMeasurable Y μ\n⊢ Var[X; Measure.map Y μ] = Var[X ∘ Y; μ]",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Prob... | [] | rw [variance_eq_integral hX, integral_map hY, variance_eq_integral (hX.comp_aemeasurable hY),
integral_map hY]
· congr
· exact hX.aestronglyMeasurable
· refine AEStronglyMeasurable.pow ?_ _
exact AEMeasurable.aestronglyMeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Variance | {
"line": 318,
"column": 2
} | {
"line": 323,
"column": 57
} | {
"line": 325,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nhX : AEMeasurable X (Measure.map Y μ)\nhY : AEMeasurable Y μ\n⊢ Var[X; Measure.map Y μ] = Var[X ∘ Y; μ]",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Prob... | [] | rw [variance_eq_integral hX, integral_map hY, variance_eq_integral (hX.comp_aemeasurable hY),
integral_map hY]
· congr
· exact hX.aestronglyMeasurable
· refine AEStronglyMeasurable.pow ?_ _
exact AEMeasurable.aestronglyMeasurable (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Covariance | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 16
} | {
"line": 294,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ns : Finset ι\ninst✝ : IsFiniteMeasure μ\nι' : Type u_3\nY : ι' → Ω → ℝ\nt : Finset ι'\nhX : ∀ i ∈ s, MemLp (X i) 2 μ\nhY : ∀ i ∈ t, MemLp (Y i) 2 μ\n⊢ cov[fun ω ↦ ∑ i ∈ s, X i ω, fun ω ↦ ∑ j ∈ t, Y j ω; μ] = ∑ i ∈ s, ∑ j ... | [] | convert! covariance_sum_sum' hX hY
all_goals simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Covariance | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 16
} | {
"line": 294,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ns : Finset ι\ninst✝ : IsFiniteMeasure μ\nι' : Type u_3\nY : ι' → Ω → ℝ\nt : Finset ι'\nhX : ∀ i ∈ s, MemLp (X i) 2 μ\nhY : ∀ i ∈ t, MemLp (Y i) 2 μ\n⊢ cov[fun ω ↦ ∑ i ∈ s, X i ω, fun ω ↦ ∑ j ∈ t, Y j ω; μ] = ∑ i ∈ s, ∑ j ... | [] | convert! covariance_sum_sum' hX hY
all_goals simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 355,
"column": 51
} | {
"line": 355,
"column": 71
} | {
"line": 355,
"column": 71
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : ContinuousENorm E\ninst✝⁸ : MeasurableSpace E\ninst✝⁷ : OpensMeasurableSpace E\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : ContinuousENorm F\ninst✝⁴ : MeasurableSpace F\ninst✝³ : OpensMeas... | [] | by simp [enorm_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.IdentDistrib | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 22
} | {
"line": 233,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\ninst✝¹ : NormedAddCommGroup γ\ninst✝ : BorelSpace γ\nh : IdentDistrib f g μ ν\nhf : MemLp f 1 μ\n⊢ MemLp g 1 ν",
"ppTerm"... | [] | exact h.memLp_snd hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 427,
"column": 33
} | {
"line": 427,
"column": 35
} | {
"line": 428,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\nh... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 434,
"column": 33
} | {
"line": 434,
"column": 35
} | {
"line": 435,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\nh... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 476,
"column": 4
} | {
"line": 479,
"column": 21
} | {
"line": 480,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k : ι\... | [] | filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
filter_upwards [hi, hj] with ω hωi hωj
rw [← hωi, ← hωj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 476,
"column": 4
} | {
"line": 479,
"column": 21
} | {
"line": 480,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k : ι\... | [] | filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
filter_upwards [hi, hj] with ω hωi hωj
rw [← hωi, ← hωj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 503,
"column": 4
} | {
"line": 506,
"column": 21
} | {
"line": 507,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ... | [] | filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
filter_upwards [hi, hj] with ω hωi hωj
rw [← hωi, ← hωj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 503,
"column": 4
} | {
"line": 506,
"column": 21
} | {
"line": 507,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ... | [] | filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk,
Measure.ae_ae_of_ae_comp (hf_meas j).ae_eq_mk] with a hi hj
filter_upwards [hi, hj] with ω hωi hωj
rw [← hωi, ← hωj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 653,
"column": 33
} | {
"line": 653,
"column": 35
} | {
"line": 654,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns : Finset ι\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 741,
"column": 6
} | {
"line": 741,
"column": 54
} | {
"line": 742,
"column": 4
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\nβ : Type u_4\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\nX : ι → Ω → α\nY : ι → Ω → β\nf : ι → Set Ω\nt : ι → Set β\ns : Finset ι\ninst✝ : Finite ι\nhY : ∀ (i : ι), Measurable (Y i)\nhindep : iIndepFun... | [] | exact fun _ _ i _ _ ↦ .inr <| measure_ne_top _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.SpecialFunctions.Sinc | {
"line": 37,
"column": 25
} | {
"line": 40,
"column": 25
} | {
"line": 42,
"column": 0
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasure μ\n⊢ Integrable sinc μ",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.abs_sinc_le_one",
"Eq.mpr",
"Real.instLE",
"Real",
"MeasureTheory.Measure",
"Real.lattice",
"abs",
... | [] | by
refine Integrable.mono' (g := fun _ ↦ 1) (by fun_prop) (by fun_prop) <| ae_of_all _ fun x ↦ ?_
rw [Real.norm_eq_abs]
exact abs_sinc_le_one x | [anonymous] | Lean.Parser.Term.byTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.