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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.MetricSpace.Basic | {
"line": 305,
"column": 52
} | {
"line": 305,
"column": 69
} | {
"line": 308,
"column": 0
} | [
{
"pp": "X : Type u_2\nm : MetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"MetricSpace.ext",
"MetricSpace.replaceDist",
"Dist.ext",
"MetricSpace.toPseudoMetricSpace",
"PseudoMetricSpace.toDi... | [] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 15
} | {
"line": 201,
"column": 2
} | [
{
"pp": "β : Type u_2\ninst✝² : PseudoEMetricSpace β\nK : ℝ≥0\nα : Type u_4\ninst✝¹ : EMetricSpace α\ninst✝ : Nontrivial α\nf : α → β\nhf : AntilipschitzWith K f\n⊢ 0 < K",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder... | [
"β : Type u_2\ninst✝² : PseudoEMetricSpace β\nK : ℝ≥0\nα : Type u_4\ninst✝¹ : EMetricSpace α\ninst✝ : Nontrivial α\nf : α → β\nhf : AntilipschitzWith K f\nh₀ : K ≤ 0\n⊢ False"
] | by_contra! h₀ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 165,
"column": 2
} | {
"line": 167,
"column": 41
} | {
"line": 169,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝³ : FunLike F (Set α) ℝ≥0∞\ninst✝² : OuterMeasureClass F α\nμ : F\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\ns : Set α\nhs : μ s ≠ 0\n⊢ ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENN... | [] | contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 165,
"column": 2
} | {
"line": 167,
"column": 41
} | {
"line": 169,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝³ : FunLike F (Set α) ℝ≥0∞\ninst✝² : OuterMeasureClass F α\nμ : F\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\ns : Set α\nhs : μ s ≠ 0\n⊢ ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENN... | [] | contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 84,
"column": 77
} | {
"line": 87,
"column": 40
} | {
"line": 89,
"column": 0
} | [
{
"pp": "α : Type u_1\na : ℝ≥0∞\nf : α → ℝ≥0∞\nu : Filter α\nha : a ≠ ∞\nhf : ∀ (x : α), f x ≠ ∞\n⊢ Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 a.toNNReal) ↔ Tendsto f u (𝓝 a)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNN... | [] | by
refine ⟨fun h => ?_, fun h => (ENNReal.tendsto_toNNReal ha).comp h⟩
rw [← coe_comp_toNNReal_comp hf]
exact (tendsto_coe_toNNReal ha).comp h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 141,
"column": 46
} | {
"line": 141,
"column": 68
} | {
"line": 141,
"column": 69
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (n : ℕ), ∀ᶠ (a : α) in f, ↑n < m a\nx : ℝ≥0\nn : ℕ\nhn : x < ↑n\nx✝ : α\n⊢ ↑x < ↑n",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.ofNNReal",
"Preo... | [
"α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (n : ℕ), ∀ᶠ (a : α) in f, ↑n < m a\nx : ℝ≥0\nn : ℕ\nhn : x < ↑n\nx✝ : α\n⊢ ↑x < ↑↑n"
] | ← ENNReal.coe_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 441,
"column": 4
} | {
"line": 444,
"column": 80
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case neg\na : ℝ≥0∞\na_infty : ¬a = ∞\n⊢ Continuous fun x ↦ x - a",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Continuous",
"eq_false",
"congrArg",
"ContinuousOn.comp_continuous",
"ENNReal.continuousOn_sub",
"co... | [] | rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply continuousOn_sub.comp_continuous (by fun_prop)
intro x
simp only [a_infty, Ne, mem_setOf_eq, Prod.mk_inj, and_false, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 441,
"column": 4
} | {
"line": 444,
"column": 80
} | {
"line": 446,
"column": 0
} | [
{
"pp": "case neg\na : ℝ≥0∞\na_infty : ¬a = ∞\n⊢ Continuous fun x ↦ x - a",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Continuous",
"eq_false",
"congrArg",
"ContinuousOn.comp_continuous",
"ENNReal.continuousOn_sub",
"co... | [] | rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply continuousOn_sub.comp_continuous (by fun_prop)
intro x
simp only [a_infty, Ne, mem_setOf_eq, Prod.mk_inj, and_false, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Defs | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 22
} | {
"line": 294,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : sign a = -1\n⊢ a < 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"PartialOrder.toPreorder",
"SignType.instLinear... | [
"α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = -1\n⊢ a < 0"
] | rw [sign_apply] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Sign.Defs | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 22
} | {
"line": 313,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : sign a = 0\n⊢ a = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"Semilat... | [
"α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = 0\n⊢ a = 0"
] | rw [sign_apply] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 729,
"column": 74
} | {
"line": 730,
"column": 60
} | {
"line": 732,
"column": 0
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Icc a b) = b - a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_o... | [] | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 732,
"column": 74
} | {
"line": 733,
"column": 60
} | {
"line": 735,
"column": 0
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ico a b) = b - a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_o... | [] | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 735,
"column": 74
} | {
"line": 736,
"column": 60
} | {
"line": 738,
"column": 0
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ioc a b) = b - a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Set.Ioc",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"cov... | [] | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 738,
"column": 74
} | {
"line": 739,
"column": 60
} | {
"line": 741,
"column": 0
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ioo a b) = b - a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_o... | [] | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | [] | simp [hβ, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | [] | simp [hβ, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | [] | simp [hβ, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 450,
"column": 72
} | {
"line": 453,
"column": 32
} | {
"line": 455,
"column": 0
} | [
{
"pp": "α : Type u_1\nf g : α → ℝ≥0\nsf sg : ℝ≥0\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhf : HasSum f sf\nhg : HasSum g sg\n⊢ sf < sg",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Real.partialOrder",
"Real.ins... | [] | by
have A : ∀ a : α, (f a : ℝ) ≤ g a := fun a => NNReal.coe_le_coe.2 (h a)
have : (sf : ℝ) < sg := hasSum_lt A (NNReal.coe_lt_coe.2 hi) (hasSum_coe.2 hf) (hasSum_coe.2 hg)
exact NNReal.coe_lt_coe.1 this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 768,
"column": 11
} | {
"line": 768,
"column": 37
} | {
"line": 768,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\ns : Set α\n⊢ LowerHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ t) → f x ⊆ t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\ns : Set α\n⊢ (∀ x ∈ s, LowerHemicontinuousWithinAt f s x) ↔\n ∀ x ∈ s, ∀ (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ t) → f x ⊆ t"
] | lowerHemicontinuousOn_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.EReal.Inv | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 84
} | {
"line": 433,
"column": 2
} | [
{
"pp": "b : EReal\nh : b < 0\nh' : b ≠ ⊥\na a' : EReal\na_lt_a' : a < a'\n⊢ a' / b < a / b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"EReal.instDivInvMonoid",
"instHDiv",
"PartialOrder.toPreorder",
"EReal",
"le_of_lt",
"HDiv.hDiv",
"instZero... | [
"b : EReal\nh : b < 0\nh' : b ≠ ⊥\na a' : EReal\na_lt_a' : a < a'\n⊢ a' / b ≠ a / b"
] | apply lt_of_le_of_ne <| div_le_div_right_of_nonpos (le_of_lt h) (le_of_lt a_lt_a') | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 85,
"column": 2
} | {
"line": 93,
"column": 28
} | {
"line": 94,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\nf : α → β\ninst✝ : LinearOrder β\ns : Set α\nne_s : s.Nonempty\nhs : IsCompact s\nhf : LowerSemicontinuousOn f s\nx✝¹ : Nonempty α\nx✝ : Nonempty ↑s\nφ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)\nℱ : Filter α := ⨅ a, φ (f ↑a)\n⊢ ∃ a ∈ s, ∀ x ... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\nf : α → β\ninst✝ : LinearOrder β\ns : Set α\nne_s : s.Nonempty\nhs : IsCompact s\nhf : LowerSemicontinuousOn f s\nx✝¹ : Nonempty α\nx✝ : Nonempty ↑s\nφ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)\nℱ : Filter α := ⨅ a, φ (f ↑a)\nthis : ℱ.NeBot\n⊢ ∃ a ∈ s, ... | have : ℱ.NeBot := by
apply iInf_neBot_of_directed _ _
· change Directed GE.ge (fun x ↦ (φ ∘ (fun (a : s) ↦ f ↑a)) x)
exact Directed.mono_comp GE.ge (fun x y hxy ↦
principal_mono.mpr (inter_subset_inter_right _ (preimage_mono <| Iic_subset_Iic.mpr hxy)))
(Std.Total.directed _)
· intro x... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 206,
"column": 4
} | {
"line": 209,
"column": 41
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ContinuousAt (fun x ↦ ENNReal.ofReal x.toReal) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [] | refine tendsto_nhds_of_eventually_eq ?_
rw [h_bot, nhds_bot_basis.eventually_iff]
simpa [toReal_bot, ENNReal.ofReal_zero, ENNReal.ofReal_eq_zero, true_and] using
⟨0, fun _ hx ↦ toReal_nonpos hx.le⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 206,
"column": 4
} | {
"line": 209,
"column": 41
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ContinuousAt (fun x ↦ ENNReal.ofReal x.toReal) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [] | refine tendsto_nhds_of_eventually_eq ?_
rw [h_bot, nhds_bot_basis.eventually_iff]
simpa [toReal_bot, ENNReal.ofReal_zero, ENNReal.ofReal_eq_zero, true_and] using
⟨0, fun _ hx ↦ toReal_nonpos hx.le⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 305,
"column": 2
} | {
"line": 314,
"column": 46
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ limsup (fun x ↦ c * u x) f = c * limsup u f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LE.le.eq_or_lt",
"EReal.instDivInvMonoid",
"False",
... | [] | obtain rfl | h₃ := h₁.eq_or_lt
· simp
simp_rw [EReal.mul_comm (x := c)]
apply eq_of_le_of_ge
· rw [limsup_le_iff]
simpa [← EReal.lt_div_iff (by aesop) (by aesop)]
using fun _ ↦ eventually_lt_of_limsup_lt
· rw [le_limsup_iff]
simpa [← EReal.div_lt_iff (by aesop) (by aesop)]
using fun _ ↦ fr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 305,
"column": 2
} | {
"line": 314,
"column": 46
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ limsup (fun x ↦ c * u x) f = c * limsup u f",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LE.le.eq_or_lt",
"EReal.instDivInvMonoid",
"False",
... | [] | obtain rfl | h₃ := h₁.eq_or_lt
· simp
simp_rw [EReal.mul_comm (x := c)]
apply eq_of_le_of_ge
· rw [limsup_le_iff]
simpa [← EReal.lt_div_iff (by aesop) (by aesop)]
using fun _ ↦ eventually_lt_of_limsup_lt
· rw [le_limsup_iff]
simpa [← EReal.div_lt_iff (by aesop) (by aesop)]
using fun _ ↦ fr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 91,
"column": 10
} | {
"line": 91,
"column": 19
} | {
"line": 91,
"column": 20
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁵ : DivisionSemiring 𝕜\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\nx : 𝕜\n⊢ 𝓝 1 = 𝓝 (1 / (1 + x * 0))",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_4\ninst✝⁵ : DivisionSemiring 𝕜\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\nx : 𝕜\n⊢ 𝓝 1 = 𝓝 (1 / (1 + 0))"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 28
} | {
"line": 95,
"column": 4
} | [
{
"pp": "case convert_6\n𝕜 : Type u_4\ninst✝⁵ : DivisionSemiring 𝕜\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\nx : 𝕜\nthis : 𝓝 1 = 𝓝 (1 / (1 + x * 0))\n⊢ Tendsto (fun n ↦ x / ↑n) atTop (𝓝 (x * 0))",
... | [
"case convert_6\n𝕜 : Type u_4\ninst✝⁵ : DivisionSemiring 𝕜\ninst✝⁴ : TopologicalSpace 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\nx : 𝕜\nthis : 𝓝 1 = 𝓝 (1 / (1 + x * 0))\n⊢ Tendsto (fun n ↦ x * (↑n)⁻¹) atTop (𝓝 (x * 0))"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 63
} | {
"line": 511,
"column": 0
} | [
{
"pp": "case h.refine_2.refine_2.refine_2\nx : ℝ\n⊢ (⊤, ⊤).2 ∈ Ioi 1",
"ppTerm": "?h.refine_2.refine_2.refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Membe... | [] | · rw [Set.mem_Ioi, ← EReal.coe_one]; exact EReal.coe_lt_top 1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 209,
"column": 4
} | {
"line": 214,
"column": 30
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_4\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nr : 𝕜\nh : Tendsto (abs ∘ fun n ↦ r ^ n) atTop (𝓝 0)\nhr_le : ¬|r| < 1\nhr : ¬1 = |r|\n⊢ False",
"ppTerm": "?neg✝"... | [] | · apply @not_tendsto_nhds_of_tendsto_atTop 𝕜 ℕ _ _ _ _ atTop _ (fun n ↦ |r| ^ n) _ 0 _
· refine (pow_right_strictMono₀ <| lt_of_le_of_ne (le_of_not_gt hr_le)
hr).monotone.tendsto_atTop_atTop (fun b ↦ ?_)
obtain ⟨n, hn⟩ := (pow_unbounded_of_one_lt b (lt_of_le_of_ne (le_of_not_gt hr_le) hr))
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 558,
"column": 6
} | {
"line": 558,
"column": 54
} | {
"line": 559,
"column": 6
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na✝ : ι\ns✝ : Finset ι\nia... | [
"case insert\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na✝ : ι\ns✝ : Finset ι\nia : a✝ ∉ s✝\n... | simp only [ia, Finset.sum_insert, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 1072,
"column": 11
} | {
"line": 1072,
"column": 49
} | {
"line": 1072,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : UpperSemicontinuousAt f x\nhg : UpperSemicontinuousAt g x\nhcont : ContinuousAt (fun p... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhcont : ContinuousAt (fun p ↦ p.1 + p.2) (f x, g x)\nhf : UpperSemicontinuousWithinAt f univ x\nhg : Up... | ← upperSemicontinuousWithinAt_univ_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 1132,
"column": 11
} | {
"line": 1132,
"column": 49
} | {
"line": 1132,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, UpperSemicontinuousAt... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, UpperSemicontinuousWithinAt (f i)... | ← upperSemicontinuousWithinAt_univ_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 23
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ⊆ ⋃ n, (range f)ᶜ ∪ f '' t n",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | rw [← union_iUnion, ← inter_subset, ← image_preimage_eq_inter_range, ← image_iUnion]
exact image_mono ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 23
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ⊆ ⋃ n, (range f)ᶜ ∪ f '' t n",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | rw [← union_iUnion, ← inter_subset, ← image_preimage_eq_inter_range, ← image_iUnion]
exact image_mono ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | {
"line": 566,
"column": 0
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"ppTerm": "?top.top",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Insta... | [] | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | {
"line": 566,
"column": 0
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"ppTerm": "?top.top",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Insta... | [] | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | {
"line": 566,
"column": 0
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"ppTerm": "?top.top",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Insta... | [] | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 297,
"column": 88
} | {
"line": 300,
"column": 60
} | {
"line": 302,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • boundedBy m = boundedBy (c • m)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"False",
"instHSMul",
"Lattice.toSemilatticeSup",
"instSMu... | [] | by
simp only [boundedBy, smul_ofFunction hc]
congr 1 with s : 1
rcases s.eq_empty_or_nonempty with (rfl | hs) <;> simp [*] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 685,
"column": 22
} | {
"line": 685,
"column": 36
} | {
"line": 685,
"column": 37
} | [
{
"pp": "k : ℕ\nhn : 0 < k.succ\n⊢ (∏ i ∈ Finset.range k.succ, ↑(i + 1)) * (∏ _k ∈ Finset.range k.succ, ↑k.succ)⁻¹ ≤ (↑k.succ)⁻¹",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.cast_succ",
"Real",
"Div... | [
"k : ℕ\nhn : 0 < k.succ\n⊢ (∏ i ∈ Finset.range k.succ, ↑(i + 1)) * (∏ _k ∈ Finset.range k.succ, (↑k + 1))⁻¹ ≤ (↑k + 1)⁻¹"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 748,
"column": 2
} | {
"line": 750,
"column": 23
} | {
"line": 751,
"column": 2
} | [
{
"pp": "case hgf\nR : Type u_4\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x ↦ a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, a ≤ ↑⌈a * b⌉₊ / b",
"ppTerm": "?hgf",
... | [
"case hfh\nR : Type u_4\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x ↦ a + x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, ↑⌈a * b⌉₊ / b ≤ a + b⁻¹"
] | · refine eventually_atTop.2 ⟨1, fun x hx ↦ ?_⟩
rw [le_div_iff₀ (zero_lt_one.trans_le hx)]
exact Nat.le_ceil _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 28
} | {
"line": 140,
"column": 2
} | [
{
"pp": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nβ : Type u_2\ninst✝ : Countable β\nf... | [
"case intro\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i : ℕ), m (f i) ⋯\nβ : Type u_2\ninst✝ : Countable β\nf... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 129,
"column": 12
} | {
"line": 129,
"column": 67
} | {
"line": 129,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ ⋯ = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) → m (⋃ i, f i)... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nm : (s : Set α) → MeasurableSet s → ℝ≥0∞\nm0 : m ∅ ⋯ = 0\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (h : ∀ (i : ℕ), MeasurableSet (f i)),\n Pairwise (Disjoint on f) → m (⋃ i, f i) ⋯ = ∑' (i :... | inducedOuterMeasure_eq m0 mU (MeasurableSet.iUnion hf), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 311,
"column": 47
} | {
"line": 314,
"column": 54
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_6\nβ : Type u_7\ninst✝ : MeasurableSpace β\nμ : Measure β\nC : β → Set α → Prop\ns : Set (Set α)\nm : MeasurableSpace α\nh_eq : m = generateFrom s\nh_inter : IsPiSystem s\nh_empty : ∀ᵐ (x : β) ∂μ, C x ∅\nh_basic : ∀ᵐ (x : β) ∂μ, ∀ t ∈ s, C x t\nh_compl : ∀ᵐ (x : β) ∂μ, ∀ (t : Set α), Measura... | [] | by
filter_upwards [h_empty, h_basic, h_compl, h_union] with x hx_empty hx_basic hx_compl hx_union
using MeasurableSpace.induction_on_inter (C := fun t _ ↦ C x t)
h_eq h_inter hx_empty hx_basic hx_compl hx_union | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 22
} | {
"line": 333,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ s ⊆ toMeasurable μ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.propDecidable... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ s ⊆\n if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose\n else\n if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose\n else ⋯.choose"
] | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 22
} | {
"line": 342,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ MeasurableSet (toMeasurable μ s)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.p... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ MeasurableSet\n (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose\n else\n if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose\n else ⋯.choose)"
] | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 22
} | {
"line": 348,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ μ (toMeasurable μ s) = μ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.propDec... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ μ\n (if h : ∃ t ⊇ s, MeasurableSet t ∧ t =ᵐ[μ] s then h.choose\n else\n if h' : ∃ t ⊇ s, MeasurableSet t ∧ ∀ (u : Set α), MeasurableSet u → μ (t ∩ u) = μ (s ∩ u) then h'.choose\n else ⋯.choose) =\n μ s"
] | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 28
} | {
"line": 544,
"column": 2
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\nβ : Type u_4\ninst✝ : Countable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\nh : ∀ (i : β), d.Has (f i)\n⊢ d.Has (⋃ i, f i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"nonempty_encodable",
"Nonempty.intro",
"MeasurableSp... | [
"case intro\nα : Type u_3\nd : DynkinSystem α\nβ : Type u_4\ninst✝ : Countable β\nf : β → Set α\nhd : Pairwise (Disjoint on f)\nh : ∀ (i : β), d.Has (f i)\nval✝ : Encodable β\n⊢ d.Has (⋃ i, f i)"
] | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.PiSystem | {
"line": 550,
"column": 23
} | {
"line": 552,
"column": 82
} | {
"line": 554,
"column": 0
} | [
{
"pp": "α : Type u_3\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : d.Has s₁\nh₂ : d.Has s₂\nh : Disjoint s₁ s₂\n⊢ d.Has (s₁ ∪ s₂)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"cond",
"Iff.mpr",
"Eq.mpr",
"Function.onFun",
"CompleteBooleanAlgebra.toCompleteDistri... | [] | by
rw [union_eq_iUnion]
exact d.has_iUnion (pairwise_disjoint_on_bool.2 h) (Bool.forall_bool.2 ⟨h₂, h₁⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.PiSystem | {
"line": 641,
"column": 4
} | {
"line": 643,
"column": 78
} | {
"line": 645,
"column": 0
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nα : Type u_3\nd : DynkinSystem α\ns : Set α\nh : d.Has s\nf : ℕ → Set α\nhd : Pairwise (Disjoint on f)\nhf : ∀ (i : ℕ), d.Has (f i ∩ s)\n⊢ d.Has ((⋃ i, f i) ∩ s)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.onFun",
... | [] | rw [iUnion_inter]
refine d.has_iUnion_nat ?_ hf
exact hd.mono fun i j => Disjoint.mono inter_subset_left inter_subset_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.PiSystem | {
"line": 641,
"column": 4
} | {
"line": 643,
"column": 78
} | {
"line": 645,
"column": 0
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nα : Type u_3\nd : DynkinSystem α\ns : Set α\nh : d.Has s\nf : ℕ → Set α\nhd : Pairwise (Disjoint on f)\nhf : ∀ (i : ℕ), d.Has (f i ∩ s)\n⊢ d.Has ((⋃ i, f i) ∩ s)",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.onFun",
... | [] | rw [iUnion_inter]
refine d.has_iUnion_nat ?_ hf
exact hd.mono fun i j => Disjoint.mono inter_subset_left inter_subset_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | {
"line": 245,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable ι\nf : ι → Set α\nhn : Pairwise (Disjoint on f)\nh : ∀ (i : ι), MeasurableSet (f i)\n⊢ μ ∅ = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"Meas... | [] | exact μ.empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | {
"line": 245,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable ι\nf : ι → Set α\nhn : Pairwise (Disjoint on f)\nh : ∀ (i : ι), MeasurableSet (f i)\n⊢ μ ∅ = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"Meas... | [] | exact μ.empty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | {
"line": 245,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable ι\nf : ι → Set α\nhn : Pairwise (Disjoint on f)\nh : ∀ (i : ι), MeasurableSet (f i)\n⊢ μ ∅ = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"Meas... | [] | exact μ.empty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 361,
"column": 64
} | {
"line": 365,
"column": 100
} | {
"line": 367,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh₁ : s ≤ᵐ[μ] t\nh₂ : μ t ≤ μ s\nhsm : NullMeasurableSet s μ\nht : μ t ≠ ∞\n⊢ s =ᵐ[μ] t",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.ae",
"Iff.m... | [] | by
refine eventuallyLE_antisymm_iff.mpr ⟨h₁, ae_le_set.mpr ?_⟩
replace h₂ : μ t = μ s := h₂.antisymm (measure_mono_ae h₁)
replace ht : μ s ≠ ∞ := h₂ ▸ ht
rw [measure_sdiff' t hsm ht, measure_congr (union_ae_eq_left_iff_ae_subset.mpr h₁), h₂, tsub_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 414,
"column": 2
} | {
"line": 415,
"column": 60
} | {
"line": 417,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nI : Set β\nhc : I.Countable\ns : β → Set α\n⊢ μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Set.Countable.toEncodable",
... | [] | haveI := hc.toEncodable
simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 414,
"column": 2
} | {
"line": 415,
"column": 60
} | {
"line": 417,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nI : Set β\nhc : I.Countable\ns : β → Set α\n⊢ μ (⋃ b ∈ I, toMeasurable μ (s b)) = μ (⋃ b ∈ I, s b)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Set.Countable.toEncodable",
... | [] | haveI := hc.toEncodable
simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 85
} | {
"line": 424,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure\nh : ∀ (b : β), m {b}ᶜ ≠ 0\ninhabited_h : Inhabited β\nthis ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝³ : TopologicalSpace β\ninst✝² : T1Space β\ninst✝¹ : SecondCountableTopology β\ninst✝ : Nonempty β\nf : α → β\nm : OuterMeasure β := (OuterMeasure.map f) μ.toOuterMeasure\nh : ∀ (b : β), m {b}ᶜ ≠ 0\ninhabited_h : Inhabited β\nthis : m univ ≠ 0... | rcases exists_mem_forall_mem_nhdsWithin_pos_measure (h b) with ⟨a, hab : a ≠ b, ha⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 650,
"column": 2
} | {
"line": 653,
"column": 64
} | {
"line": 655,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhm : Monotone s\n⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (μ (⋃ n, s n)))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureThe... | [] | refine .of_neBot_imp fun h ↦ ?_
have := (atTop_neBot_iff.1 h).2
rw [hm.measure_iUnion]
exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 650,
"column": 2
} | {
"line": 653,
"column": 64
} | {
"line": 655,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhm : Monotone s\n⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (μ (⋃ n, s n)))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureThe... | [] | refine .of_neBot_imp fun h ↦ ?_
have := (atTop_neBot_iff.1 h).2
rw [hm.measure_iUnion]
exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 30
} | {
"line": 679,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhs : ∀ (i : ι), NullMeasurableSet (s i) μ\nhm : Antitone s\nhf : ∃ i, μ (s i) ≠ ∞\nh : atTop.NeBot\nthis : IsDirectedOrder ι\n⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (μ (⋂ n, s... | [
"α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhs : ∀ (i : ι), NullMeasurableSet (s i) μ\nhm : Antitone s\nhf : ∃ i, μ (s i) ≠ ∞\nh : atTop.NeBot\nthis : IsDirectedOrder ι\n⊢ Tendsto (⇑μ ∘ s) atTop (𝓝 (⨅ i, μ (s i)))"
] | rw [hm.measure_iInter hs hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 55
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",... | [] | simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 55
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",... | [] | simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 55
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",... | [] | simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 24
} | {
"line": 58,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\n⊢ μ s ≤ (μ.trim hm) s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"MeasureTheory.Measure.trim",
"id",
"LE.le",
"ENNReal.instLE",
"ENNReal... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\n⊢ μ s ≤ (μ.toMeasure ⋯) s"
] | simp_rw [Measure.trim] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Set.MemPartition | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 37
} | {
"line": 152,
"column": 37
} | [
{
"pp": "α : Type u_1\nf : ℕ → Set α\nn : ℕ\na : α\ns : Set α\nhs : s ∈ memPartition f n\nh : a ∈ s\nh_ne : ¬memPartitionSet f n a = s\nh_disj : Disjoint s (memPartitionSet f n a)\n⊢ ¬Disjoint s (memPartitionSet f n a)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u_1\nf : ℕ → Set α\nn : ℕ\na : α\ns : Set α\nhs : s ∈ memPartition f n\nh : a ∈ s\nh_ne : ¬memPartitionSet f n a = s\nh_disj : Disjoint s (memPartitionSet f n a)\n⊢ (s ∩ memPartitionSet f n a).Nonempty"
] | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 910,
"column": 45
} | {
"line": 911,
"column": 75
} | {
"line": 913,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\nm1 : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\nμ : Measure β\ns : Set β\n⊢ (Measure.comap f μ) (f ⁻¹' s) = μ (s ∩ range f)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measu... | [] | by
rw [← hf.map_apply, hf.map_comap, restrict_apply' hf.measurableSet_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1105,
"column": 28
} | {
"line": 1105,
"column": 79
} | {
"line": 1107,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.... | [] | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1105,
"column": 28
} | {
"line": 1105,
"column": 79
} | {
"line": 1107,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.... | [] | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1105,
"column": 28
} | {
"line": 1105,
"column": 79
} | {
"line": 1107,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.... | [] | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1045,
"column": 12
} | {
"line": 1045,
"column": 45
} | {
"line": 1047,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\nhs : MeasurableSet s\n⊢ s.indicator f =ᵐ[μ.restrict s] f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"pie... | [] | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1045,
"column": 12
} | {
"line": 1045,
"column": 45
} | {
"line": 1047,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\nhs : MeasurableSet s\n⊢ s.indicator f =ᵐ[μ.restrict s] f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"pie... | [] | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1045,
"column": 12
} | {
"line": 1045,
"column": 45
} | {
"line": 1047,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\nhs : MeasurableSet s\n⊢ s.indicator f =ᵐ[μ.restrict s] f",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"pie... | [] | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1056,
"column": 2
} | {
"line": 1056,
"column": 41
} | {
"line": 1057,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → f x = 0 x\nx : α\nhx : x ∈ sᶜ → f x = 0 x\nhxs : x ∈ s\n⊢ s.indicator f x = f x",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case neg\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → β\ninst✝ : Zero β\nhs : MeasurableSet s\nhf : ∀ᵐ (x : α) ∂μ, x ∈ sᶜ → f x = 0 x\nx : α\nhx : x ∈ sᶜ → f x = 0 x\nhxs : x ∉ s\n⊢ s.indicator f x = f x"
] | · simp only [hxs, Set.indicator_of_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 26
} | {
"line": 96,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : α → β\ninst✝ : Countable ι\nμ : ι → Measure α\nh : ∀ (i : ι), AEMeasurable f (μ i)\na✝ : Nontrivial β\ninhabited_h : Inhabited β\ns : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ... | [] | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 536,
"column": 2
} | {
"line": 538,
"column": 95
} | {
"line": 540,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nt : ℕ → Set α\nu : Set α\nhu : u ∈ range t\n⊢ MeasurableSet u",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"_private.Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated.0.MeasurableSpace.generateFrom_iUnion_memP... | [] | · simp only [mem_range] at hu
obtain ⟨n, rfl⟩ := hu
exact generateFrom_mono (subset_iUnion _ _) _ (measurableSet_generateFrom_memPartition t n) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1342,
"column": 25
} | {
"line": 1342,
"column": 76
} | {
"line": 1344,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm0 : MeasurableSpace α\nμ : ι → Measure α\ni : ι\ns : Set α\nhs : MeasurableSet s\n⊢ (μ i) s ≤ (sum μ) s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
... | [] | simpa only [sum_apply μ hs] using ENNReal.le_tsum i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1342,
"column": 25
} | {
"line": 1342,
"column": 76
} | {
"line": 1344,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm0 : MeasurableSpace α\nμ : ι → Measure α\ni : ι\ns : Set α\nhs : MeasurableSet s\n⊢ (μ i) s ≤ (sum μ) s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
... | [] | simpa only [sum_apply μ hs] using ENNReal.le_tsum i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1342,
"column": 25
} | {
"line": 1342,
"column": 76
} | {
"line": 1344,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm0 : MeasurableSpace α\nμ : ι → Measure α\ni : ι\ns : Set α\nhs : MeasurableSet s\n⊢ (μ i) s ≤ (sum μ) s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
... | [] | simpa only [sum_apply μ hs] using ENNReal.le_tsum i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 318,
"column": 4
} | {
"line": 319,
"column": 35
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : atTop.IsCountablyGenerated\nx : α\ng : α → β\ng_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nI... | [] | rw [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]
exact aemeasurable_zero_measure | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 318,
"column": 4
} | {
"line": 319,
"column": 35
} | {
"line": 321,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : atTop.IsCountablyGenerated\nx : α\ng : α → β\ng_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\nI... | [] | rw [Ioc_eq_empty (not_lt.mpr h), Measure.restrict_empty]
exact aemeasurable_zero_measure | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 312,
"column": 20
} | {
"line": 314,
"column": 95
} | {
"line": 316,
"column": 0
} | [
{
"pp": "γ : Type u_3\nδ : Type u_5\ninst✝³ : TopologicalSpace γ\ninst✝² : MeasurableSpace γ\ninst✝¹ : BorelSpace γ\ninst✝ : MeasurableSpace δ\nf : δ → γ\nhf : ∀ (s : Set γ), IsClosed[inst✝³] s → MeasurableSet (f ⁻¹' s)\n⊢ Measurable f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | by
apply measurable_of_isOpen; intro s hs
rw [← MeasurableSet.compl_iff, ← preimage_compl]; apply hf; rw [isClosed_compl_iff]; exact hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 55
} | {
"line": 233,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nhs : IsCompact s\nhne : s.Nonempty\nx : α\n⊢ ∃ y ∈ s, infEDist x s = edist x y",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Continuous",
"continuous_const",
... | [
"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"
] | have A : Continuous fun y => edist x y := by fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 234,
"column": 59
} | {
"line": 234,
"column": 79
} | {
"line": 234,
"column": 79
} | [
{
"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\ny : α\nys : y ∈ s\nhy : IsMinOn (fun y ↦ edist x y) s y\n⊢ edist x y ≤ infEDist x s",
"ppTerm": "?m.84",
"as... | [] | by rwa [le_infEDist] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 40
} | {
"line": 434,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns₁ s₂ t₁ t₂ : Set α\n⊢ hausdorffEDist (s₁ ∪ s₂) (t₁ ∪ t₂) ≤ max (hausdorffEDist s₁ t₁) (hausdorffEDist s₂ t₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"iSup",
"CompletelyDi... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns₁ s₂ t₁ t₂ : Set α\n⊢ hausdorffEDist (⋃ b, bif b then s₁ else s₂) (⋃ b, bif b then t₁ else t₂) ≤\n ⨆ b, bif b then hausdorffEDist s₁ t₁ else hausdorffEDist s₂ t₂"
] | simp_rw [union_eq_iUnion, sup_eq_iSup] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 55
} | {
"line": 627,
"column": 56
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nr : ℝ\nhs : s.Nonempty\n⊢ (∀ y ∈ s, ENNReal.ofReal r ≤ edist x y) ↔ ∀ ⦃y : α⦄, y ∈ s → r ≤ dist x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nr : ℝ\nhs : s.Nonempty\n⊢ (∀ y ∈ s, r ≤ (edist x y).toReal) ↔ ∀ ⦃y : α⦄, y ∈ s → r ≤ dist x y"
] | ENNReal.ofReal_le_iff_le_toReal (edist_ne_top _ _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.AEMeasurableSequence | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 49
} | {
"line": 70,
"column": 50
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : ι → α → β\nμ : Measure α\np : α → (ι → β) → Prop\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ aeSeqSet hf p ⊆ {x | p x fun n ↦ f n x}",
"ppTerm": "?m.54",
"assigned": ... | [
"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : ι → α → β\nμ : Measure α\np : α → (ι → β) → Prop\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nx : α\nhx : x ∈ aeSeqSet hf p\n⊢ aeSeqSet hf p ⊆ {x | p x fun n ↦ f n x}ᶜᶜ"
] | ← compl_compl { x | p x fun n => f n x }, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 1090,
"column": 92
} | {
"line": 1092,
"column": 43
} | {
"line": 1094,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\ninst✝ : μ.WeaklyRegular\nx : ℝ≥0∞\nhx : x ≠ ∞\n⊢ (x • μ).WeaklyRegular",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.WeaklyRegular.mk",
"instHSMul",
"Me... | [] | by
haveI := OuterRegular.smul μ hx
exact ⟨WeaklyRegular.innerRegular.smul x⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 877,
"column": 6
} | {
"line": 880,
"column": 33
} | {
"line": 881,
"column": 2
} | [] | [] | { a : α | ennrealRatEmbed b ≤ f a }.indicator (fun _ => ennrealRatEmbed b) a ≤
ennrealRatEmbed b :=
indicator_le_self _ _ a
_ < ⊤ := ENNReal.coe_lt_top | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 28
} | {
"line": 156,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), Measurable (f b)\nh_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f\n⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ",
"ppTerm": "?m.30",
"assigned": true,
"usedConst... | [
"case intro\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), Measurable (f b)\nh_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f\nval✝ : Encodable β\n⊢ ∫⁻ (a : α), ⨆ b, f b a ∂μ = ⨆ b, ∫⁻ (a : α), f b a ∂μ"
] | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 49
} | {
"line": 170,
"column": 6
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), Measurable (f b)\nh_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f\nval✝ : Encodable β\nh✝ : Nonempty β\ninhabited_h : Inhabited β\nthis : ∀ (a : α), ⨆ b, f b a = ⨆ n, f ... | [] | exact le_iSup (fun b => ∫⁻ a, f b a ∂μ) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 49
} | {
"line": 170,
"column": 6
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), Measurable (f b)\nh_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f\nval✝ : Encodable β\nh✝ : Nonempty β\ninhabited_h : Inhabited β\nthis : ∀ (a : α), ⨆ b, f b a = ⨆ n, f ... | [] | exact le_iSup (fun b => ∫⁻ a, f b a ∂μ) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 49
} | {
"line": 170,
"column": 6
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nhf : ∀ (b : β), Measurable (f b)\nh_directed : Directed (fun x1 x2 ↦ x1 ≤ x2) f\nval✝ : Encodable β\nh✝ : Nonempty β\ninhabited_h : Inhabited β\nthis : ∀ (a : α), ⨆ b, f b a = ⨆ n, f ... | [] | exact le_iSup (fun b => ∫⁻ a, f b a ∂μ) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 891,
"column": 4
} | {
"line": 891,
"column": 26
} | {
"line": 893,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sSup ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 891,
"column": 4
} | {
"line": 891,
"column": 26
} | {
"line": 893,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sSup ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 891,
"column": 4
} | {
"line": 891,
"column": 26
} | {
"line": 893,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sSup ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 900,
"column": 4
} | {
"line": 900,
"column": 26
} | {
"line": 902,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sInf ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 900,
"column": 4
} | {
"line": 900,
"column": 26
} | {
"line": 902,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sInf ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 900,
"column": 4
} | {
"line": 900,
"column": 26
} | {
"line": 902,
"column": 0
} | [
{
"pp": "case neg\nδ : Type u_4\nmδ : MeasurableSpace δ\nα : Type u_5\nmα : MeasurableSpace α\ninst✝ : ConditionallyCompleteLattice α\np : Prop\nf : δ → α\nhf : Measurable f\nh : ¬p\n⊢ Measurable fun b ↦ sInf ∅",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Set.instEmptyCollection"... | [] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 1009,
"column": 2
} | {
"line": 1009,
"column": 53
} | {
"line": 1010,
"column": 2
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nι' : Type u_6\nf : ι → δ → α\nv : Filter ι\nhf : ∀ (i :... | [
"case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nι' : Type u_6\nf : ι → δ → α\nv : Filter ι\nhf : ∀ (i : ι... | rcases isEmpty_or_nonempty (Subtype p) with hp | hp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 93,
"column": 6
} | {
"line": 96,
"column": 26
} | {
"line": 97,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝ : AddCommGroup G\ns : Finset ι\nS : ι → Set α\nf : α → G\na : α\nha : a ∈ ⋃ i ∈ s, S i\n⊢ ∑ t ∈ s.powerset, (-1) ^ #t • (⋂ i ∈ t, S i).indicator f a = (∏ i ∈ s, (1 - (S i).indicator 1 a)) • f a",
"ppTerm": "?m.375",
"assigned": true,
"usedCons... | [] | simp only [Int.reduceNeg, prod_sub, prod_const_one, mul_one, sum_smul]
congr! 1 with t
simp only [prod_const_one, prod_indicator_apply]
simp [Set.indicator] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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