module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 217,
"column": 4
} | {
"line": 218,
"column": 23
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.image_mono",
... | 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.MeasureTheory.OuterMeasure.OfFunction | {
"line": 298,
"column": 88
} | {
"line": 301,
"column": 60
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • boundedBy m = boundedBy (c • m)",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"False",
"instHSMul",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"MeasureTheory.OuterMeasure.bounde... | 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": 92,
"column": 10
} | {
"line": 92,
"column": 19
} | [
{
"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))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddComm... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 28
} | [
{
"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))",
... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 522,
"column": 6
} | {
"line": 522,
"column": 54
} | [
{
"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... | simp only [ia, Finset.sum_insert, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 210,
"column": 4
} | {
"line": 215,
"column": 30
} | [
{
"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",
"usedConstants": ... | · 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": 1036,
"column": 11
} | {
"line": 1036,
"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... | ← upperSemicontinuousWithinAt_univ_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 1096,
"column": 11
} | {
"line": 1096,
"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... | ← upperSemicontinuousWithinAt_univ_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 28
} | [
{
"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... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 130,
"column": 12
} | {
"line": 130,
"column": 67
} | [
{
"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)... | inducedOuterMeasure_eq m0 mU (MeasurableSet.iUnion hf), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 567,
"column": 2
} | {
"line": 570,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nC : ℝ\nf : ℕ → α\nhu₂ : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ dist (f n) a ≤ C / 2 ^ n",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"NonUnitalCommR... | convert dist_le_tsum_of_dist_le_of_tendsto _ hu₂ (summable_geometric_two' C) ha n
simp only [add_comm n, pow_add, ← div_div]
symm
exact ((hasSum_geometric_two' C).div_const _).tsum_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 567,
"column": 2
} | {
"line": 570,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nC : ℝ\nf : ℕ → α\nhu₂ : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C / 2 / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ dist (f n) a ≤ C / 2 ^ n",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"NonUnitalCommR... | convert dist_le_tsum_of_dist_le_of_tendsto _ hu₂ (summable_geometric_two' C) ha n
simp only [add_comm n, pow_add, ← div_div]
symm
exact ((hasSum_geometric_two' C).div_const _).tsum_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 311,
"column": 47
} | {
"line": 314,
"column": 54
} | [
{
"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
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ s ⊆ toMeasurable μ s",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.propDecidable",
"Exists",
"Filter.EventuallyE... | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ MeasurableSet (toMeasurable μ s)",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.propDecidable",
"Exists",
"Filter... | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 348,
"column": 6
} | {
"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",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Classical.propDecidable",
"Exists",
"Filter.Event... | toMeasurable_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 28
} | [
{
"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)",
"usedConstants": [
"nonempty_encodable"
]
}
] | 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
} | [
{
"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)",
"usedConstants": [
"cond",
"Iff.mpr",
"Eq.mpr",
"Function.onFun",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Disjoint... | 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": 639,
"column": 4
} | {
"line": 641,
"column": 78
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Function.onFun",
"CompleteBooleanAlgebra.toCompleteDistrib... | 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": 639,
"column": 4
} | {
"line": 641,
"column": 78
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Function.onFun",
"CompleteBooleanAlgebra.toCompleteDistrib... | 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.Analysis.SpecificLimits.Basic | {
"line": 685,
"column": 22
} | {
"line": 685,
"column": 36
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.cast_succ",
"Real",
"DivInvMonoid.toInv",
"HMul.hMul",
"Ad... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 748,
"column": 2
} | {
"line": 750,
"column": 23
} | [
{
"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",
"usedConstants": [
... | · 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.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | [
{
"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",
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"MeasureTheory.Measure.toOuterMeasure"
]
}
] | exact μ.empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | [
{
"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",
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"MeasureTheory.Measure.toOuterMeasure"
]
}
] | exact μ.empty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 17
} | [
{
"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",
"usedConstants": [
"MeasureTheory.OuterMeasure.empty",
"MeasureTheory.Measure.toOuterMeasure"
]
}
] | exact μ.empty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.BorelCantelli | {
"line": 80,
"column": 42
} | {
"line": 82,
"column": 55
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\np : ℕ → α → Prop\nhp : ∑' (i : ℕ), μ {x | p i x} ≠ ∞\n⊢ μ {x | ∃ᶠ (n : ℕ) in atTop, p n x} = 0",
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"iSup",
"Set.iInter",
... | by
simpa only [limsup_eq_iInf_iSup_of_nat, frequently_atTop, ← bex_def, setOf_forall,
setOf_exists] using measure_limsup_atTop_eq_zero hp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 379,
"column": 2
} | {
"line": 380,
"column": 60
} | [
{
"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)",
"usedConstants": [
"MeasureTheory.Measure",
"Set.Countable.toEncodable",
"congrArg",
"Membership.mem",
... | haveI := hc.toEncodable
simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 379,
"column": 2
} | {
"line": 380,
"column": 60
} | [
{
"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)",
"usedConstants": [
"MeasureTheory.Measure",
"Set.Countable.toEncodable",
"congrArg",
"Membership.mem",
... | haveI := hc.toEncodable
simp only [biUnion_eq_iUnion, measure_iUnion_toMeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 584,
"column": 2
} | {
"line": 587,
"column": 64
} | [
{
"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)))",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"iSup",... | 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": 584,
"column": 2
} | {
"line": 587,
"column": 64
} | [
{
"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)))",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"iSup",... | 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": 612,
"column": 2
} | {
"line": 612,
"column": 30
} | [
{
"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... | 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.Typeclasses.Finite | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 85
} | [
{
"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 ... | 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": 1030,
"column": 28
} | {
"line": 1030,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.toPreorder",
"_private.Mathlib.MeasureT... | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1030,
"column": 28
} | {
"line": 1030,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.toPreorder",
"_private.Mathlib.MeasureT... | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1030,
"column": 28
} | {
"line": 1030,
"column": 79
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\n⊢ ¬ν ≤ μ ↔ ∃ s, MeasurableSet s ∧ μ s < ν s",
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasurableSet",
"Iff.of_eq",
"congrArg",
"PartialOrder.toPreorder",
"_private.Mathlib.MeasureT... | simp only [le_iff, not_forall, not_le, exists_prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 629,
"column": 2
} | {
"line": 629,
"column": 62
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen s ∧ μ s < ∞}\nT : Set (Se... | obtain ⟨n, rfl⟩ : ∃ n : ℕ, f n = t := by simpa only using tT | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 55
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",
"congrArg",
"MeasureTheory.Measu... | 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
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",
"congrArg",
"MeasureTheory.Measu... | 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
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nhm : m ≤ m0\n⊢ Measure.trim 0 hm = 0",
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSpace.instLE",
"MeasureTheory.OuterMeasure.caratheodory",
"MeasureTheory.OuterMeasure.instZero",
"congrArg",
"MeasureTheory.Measu... | 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
} | [
{
"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",
"usedConstants": [
"MeasureTheory.Measure",
"PartialOrder.toPreorder",
"MeasureTheory.Measure.trim",
"Preorder.toLE",
"id",
"LE.le",
"ENNReal",
"ENN... | 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
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLatti... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 911,
"column": 45
} | {
"line": 912,
"column": 75
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.M... | by
rw [← hf.map_apply, hf.map_comap, restrict_apply' hf.measurableSet_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1046,
"column": 12
} | {
"line": 1046,
"column": 45
} | [
{
"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 =ᶠ[ae (μ.restrict s)] f",
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"piecewise_ae_eq_restrict",
"Zero.toOf... | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1046,
"column": 12
} | {
"line": 1046,
"column": 45
} | [
{
"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 =ᶠ[ae (μ.restrict s)] f",
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"piecewise_ae_eq_restrict",
"Zero.toOf... | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1046,
"column": 12
} | {
"line": 1046,
"column": 45
} | [
{
"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 =ᶠ[ae (μ.restrict s)] f",
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"piecewise_ae_eq_restrict",
"Zero.toOf... | exact piecewise_ae_eq_restrict hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 1057,
"column": 2
} | {
"line": 1057,
"column": 41
} | [
{
"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",
"usedConstants": [
"congrArg",
"Set... | · simp only [hxs, Set.indicator_of_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1258,
"column": 25
} | {
"line": 1258,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
"PartialOrder.toPreorder",
"Preor... | 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": 1258,
"column": 25
} | {
"line": 1258,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
"PartialOrder.toPreorder",
"Preor... | simpa only [sum_apply μ hs] using ENNReal.le_tsum i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1258,
"column": 25
} | {
"line": 1258,
"column": 76
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
"PartialOrder.toPreorder",
"Preor... | simpa only [sum_apply μ hs] using ENNReal.le_tsum i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 26
} | [
{
"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": 414,
"column": 2
} | {
"line": 416,
"column": 95
} | [
{
"pp": "case refine_2\nα : Type u_1\nt : ℕ → Set α\nu : Set α\nhu : u ∈ range t\n⊢ MeasurableSet u",
"usedConstants": [
"MeasurableSet",
"_private.Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated.0.MeasurableSpace.generateFrom_iUnion_memPartition._simp_1_2",
"Membership.mem",
... | · 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.AEMeasurable | {
"line": 318,
"column": 4
} | {
"line": 319,
"column": 35
} | [
{
"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
} | [
{
"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.Measure.AEMeasurable | {
"line": 390,
"column": 2
} | {
"line": 410,
"column": 70
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhc : MeasurableSpace.CountablyGenerated β\nh : NullMeasurable f μ\n⊢ AEMeasurable f μ",
"usedConstants": [
"MeasureTheory.ae",
"Nontrivial",
"Iff.mpr",
"Set.ext",
"... | classical
nontriviality β; inhabit β
rcases hc.1 with ⟨S, hSc, rfl⟩
choose! T hTf hTm hTeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_subset_ae_eq
choose! U hUf hUm hUeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_superset_ae_eq
set v := ⋃ s ∈ S, U s \ T s
have hvm : MeasurableSet v := ... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 390,
"column": 2
} | {
"line": 410,
"column": 70
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhc : MeasurableSpace.CountablyGenerated β\nh : NullMeasurable f μ\n⊢ AEMeasurable f μ",
"usedConstants": [
"MeasureTheory.ae",
"Nontrivial",
"Iff.mpr",
"Set.ext",
"... | classical
nontriviality β; inhabit β
rcases hc.1 with ⟨S, hSc, rfl⟩
choose! T hTf hTm hTeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_subset_ae_eq
choose! U hUf hUm hUeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_superset_ae_eq
set v := ⋃ s ∈ S, U s \ T s
have hvm : MeasurableSet v := ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 390,
"column": 2
} | {
"line": 410,
"column": 70
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhc : MeasurableSpace.CountablyGenerated β\nh : NullMeasurable f μ\n⊢ AEMeasurable f μ",
"usedConstants": [
"MeasureTheory.ae",
"Nontrivial",
"Iff.mpr",
"Set.ext",
"... | classical
nontriviality β; inhabit β
rcases hc.1 with ⟨S, hSc, rfl⟩
choose! T hTf hTm hTeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_subset_ae_eq
choose! U hUf hUm hUeq using fun s hs ↦ (h <| .basic s hs).exists_measurable_superset_ae_eq
set v := ⋃ s ∈ S, U s \ T s
have hvm : MeasurableSet v := ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 312,
"column": 20
} | {
"line": 314,
"column": 95
} | [
{
"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 s → MeasurableSet (f ⁻¹' s)\n⊢ Measurable f",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
... | 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
} | [
{
"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",
"usedConstants": [
"Continuous",
"continuous_const",
"continuous_id'",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.toUniformSpace... | 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
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nhs : IsCompact s\nhne : s.Nonempty\nx : α\nA : Continuous 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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder... | by rwa [le_infEDist] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 40
} | [
{
"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)",
"usedConstants": [
"cond",
"Eq.mpr",
"congrArg",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"Parti... | 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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreo... | 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
} | [
{
"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}",
"usedConstants": [
"Eq.mpr",
... | ← compl_compl { x | p x fun n => f n x }, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 932,
"column": 2
} | {
"line": 936,
"column": 49
} | [
{
"pp": "case xz\nα : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis : ε / 2 ≠ 0\nK : Se... | · calc
μ (U \ s) ≤ μ (U \ K) := by gcongr
_ < ε / 2 := by
apply measure_diff_lt_of_lt_add hKcl.nullMeasurableSet hKU _ hμU
exact ne_top_of_le_ne_top hμs (by gcongr) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 1068,
"column": 92
} | {
"line": 1070,
"column": 43
} | [
{
"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",
"usedConstants": [
"MeasureTheory.Measure.WeaklyRegular.mk",
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.Me... | by
haveI := OuterRegular.smul μ hx
exact ⟨WeaklyRegular.innerRegular.smul x⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 863,
"column": 4
} | {
"line": 863,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 863,
"column": 4
} | {
"line": 863,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 863,
"column": 4
} | {
"line": 863,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 872,
"column": 4
} | {
"line": 872,
"column": 26
} | [
{
"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 ∅",
"usedConstants": [
"Set.instEmptyCollection",
"measurable_const",
"Conditiona... | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 879,
"column": 6
} | {
"line": 882,
"column": 33
} | [] | { 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.Constructions.BorelSpace.Order | {
"line": 981,
"column": 2
} | {
"line": 981,
"column": 53
} | [
{
"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 :... | 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
} | [
{
"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",
"usedConstants": [
"Int.instCommMonoid",
"E... | 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 |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 93,
"column": 6
} | {
"line": 96,
"column": 26
} | [
{
"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",
"usedConstants": [
"Int.instCommMonoid",
"E... | 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.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 29
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\nf : α → G\n⊢ ∑ a ∈ s.inf fun i ↦ (S i)ᶜ, f a = ∑ a, f a - ∑ a ∈ s.biUnion S, f a",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | ← Finset.compl_sup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 28
} | [
{
"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",
"usedConstants": [
"nonempty_encodable"
]
}... | 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
} | [
{
"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
} | [
{
"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
} | [
{
"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.Function.SimpleFunc | {
"line": 1296,
"column": 34
} | {
"line": 1323,
"column": 20
} | [
{
"pp": "α : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty γ\nP : (α →ₛ γ) → Prop\nconst : ∀ (c : γ), P (SimpleFunc.const α c)\npcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)\nf : α →ₛ γ\n⊢ P f",
"usedConstants": [
"MeasureTheory.Si... | by
let c : γ := Classical.ofNonempty
classical
generalize h : f.range \ {c} = s
rw [← Finset.coe_inj, Finset.coe_sdiff, Finset.coe_singleton, SimpleFunc.coe_range] at h
induction s using Finset.induction generalizing f with
| empty =>
rw [Finset.coe_empty, diff_eq_empty, range_subset_singleton] at h
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sequences | {
"line": 332,
"column": 2
} | {
"line": 333,
"column": 36
} | [
{
"pp": "X : Type u_1\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atT... | obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V :=
huφ.cauchySeq.mem_entourage V_in | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.Completion | {
"line": 539,
"column": 8
} | {
"line": 539,
"column": 37
} | [
{
"pp": "case refine_2\nα✝ : Type u_1\ninst✝³ : UniformSpace α✝\nβ : Type u_2\ninst✝² : UniformSpace β\nγ : Type u_3\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\na✝ : Completion α\na : α\n⊢ Completion.extension (lift' coe') (Completion.map SeparationQuotient.mk ↑a) = ↑a",
"usedConstants": [... | map_coe uniformContinuous_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Completion | {
"line": 530,
"column": 56
} | {
"line": 540,
"column": 41
} | [
{
"pp": "α✝ : Type u_1\ninst✝³ : UniformSpace α✝\nβ : Type u_2\ninst✝² : UniformSpace β\nγ : Type u_3\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\n⊢ Completion (SeparationQuotient α) ≃ Completion α",
"usedConstants": [
"Iff.mpr",
"UniformSpace.Completion.map",
"Eq.mpr",
... | by
refine ⟨Completion.extension (lift' ((↑) : α → Completion α)),
Completion.map SeparationQuotient.mk, fun a ↦ ?_, fun a ↦ ?_⟩
· refine induction_on a (isClosed_eq (continuous_map.comp continuous_extension) continuous_id) ?_
refine SeparationQuotient.surjective_mk.forall.2 fun a ↦ ?_
rw [extension_coe ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 331,
"column": 15
} | {
"line": 331,
"column": 37
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\n⊢ {x | ‖x⁻¹ * a‖ < b 0} ∈ 𝓝 a",
"usedConstants": [
"Filter.instMembership",
"Norm.norm",... | ← dist_eq_norm_inv_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 211,
"column": 29
} | {
"line": 211,
"column": 73
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖v‖ = ‖u⁻¹ * (u * v)‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"inv_mul_cancel",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass"... | by rw [← mul_assoc, inv_mul_cancel, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"congrArg",
"sub_zero",
... | have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0
simpa only [sup_idem, sub_zero] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"congrArg",
"sub_zero",
... | have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0
simpa only [sup_idem, sub_zero] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1007,
"column": 50
} | {
"line": 1009,
"column": 32
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na b : E\n⊢ 0 < ‖a / b‖ ↔ a ≠ b",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"norm_nonneg'",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder... | by
rw [(norm_nonneg' _).lt_iff_ne, ne_comm]
exact norm_div_eq_zero_iff.not | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 246,
"column": 35
} | {
"line": 246,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nF : Type u_4\ninst✝⁴ : PseudoEMetricSpace α\ninst✝³ : PseudoEMetricSpace β\ninst✝² : PseudoEMetricSpace γ\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf✝ : F\nf : α → β\nhf : Isometry f\n⊢ ∀ (x y : α), edist (f x) (f y) = ↑1 * edist x y",
"usedCons... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 246,
"column": 35
} | {
"line": 246,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nF : Type u_4\ninst✝⁴ : PseudoEMetricSpace α\ninst✝³ : PseudoEMetricSpace β\ninst✝² : PseudoEMetricSpace γ\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf✝ : F\nf : α → β\nhf : Isometry f\n⊢ ∀ (x y : α), edist (f x) (f y) = ↑1 * edist x y",
"usedCons... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Dilation | {
"line": 246,
"column": 35
} | {
"line": 246,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nF : Type u_4\ninst✝⁴ : PseudoEMetricSpace α\ninst✝³ : PseudoEMetricSpace β\ninst✝² : PseudoEMetricSpace γ\ninst✝¹ : FunLike F α β\ninst✝ : DilationClass F α β\nf✝ : F\nf : α → β\nhf : Isometry f\n⊢ ∀ (x y : α), edist (f x) (f y) = ↑1 * edist x y",
"usedCons... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 468,
"column": 14
} | {
"line": 468,
"column": 34
} | [
{
"pp": "case hbc.hbc\nα : Type u_2\ninst✝ : SeminormedRing α\na b : αˣ\n⊢ ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ ≤ ‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖",
"usedConstants": [
"Units.val",
"SeminormedAddGroup.toAddGroup",
"HMul.hMul",
"Ring.toNonAssocRing",
"norm_su... | exact norm_sub_le .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NormedDivisionRing α\nX : Type u_4\nι : Type u_5\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nf : X → α\nl : Filter ι\nhF : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (l ×ˢ 𝓝[s] x) (𝓤 α)\nhf : ∀ x ∈ s, Disjoint (map f (𝓝[s] x)) (𝓝 0)\nx : X\nhx : x ∈ s\nU : Set X... | simp [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NormedDivisionRing α\nX : Type u_4\nι : Type u_5\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nf : X → α\nl : Filter ι\nhF : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (l ×ˢ 𝓝[s] x) (𝓤 α)\nhf : ∀ x ∈ s, Disjoint (map f (𝓝[s] x)) (𝓝 0)\nx : X\nhx : x ∈ s\nU : Set X... | simp [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NormedDivisionRing α\nX : Type u_4\nι : Type u_5\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nf : X → α\nl : Filter ι\nhF : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (l ×ˢ 𝓝[s] x) (𝓤 α)\nhf : ∀ x ∈ s, Disjoint (map f (𝓝[s] x)) (𝓝 0)\nx : X\nhx : x ∈ s\nU : Set X... | simp [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ∃ R > 0, μ (thickening R s) ≠ ∞\nh's : IsClosed s\n⊢ Tendsto (fun r ↦ μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ s))",
"usedConstants": [
"Eq.mpr",
"Real",
... | convert tendsto_measure_thickening hs
exact h's.closure_eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ∃ R > 0, μ (thickening R s) ≠ ∞\nh's : IsClosed s\n⊢ Tendsto (fun r ↦ μ (thickening r s)) (𝓝[>] 0) (𝓝 (μ s))",
"usedConstants": [
"Eq.mpr",
"Real",
... | convert tendsto_measure_thickening hs
exact h's.closure_eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 94
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : PseudoMetrizableSpace β\ninst✝⁴ : MeasurableSpace β\ninst✝³ : BorelSpace β\nι : Type u_3\ninst✝² : Countable ι\ninst✝¹ : Nonempty ι\nμ : Measure α\nf : ι → α → β\nL : Filter ι\ninst✝ : L.IsCountablyG... | have h_ae_eq : ∀ᵐ x ∂μ, ∀ n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 67,
"column": 43
} | {
"line": 67,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℤ\nx : E\n⊢ ‖n • x‖ = ‖(n • 1) • x‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"AddCommGroup.intIsScalarTower",
"Real"... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nμ : Measure α\nhμ : μ ≠ 0\nhf : ∀ (b : β), Measurable (f b)\nhf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞\nh_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f\n⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
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