module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 6
} | {
"line": 78,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\nι : Type u_4\ns : ι → Finset α\nhs : ∀ (t : Finset α), ∃ i, t ⊆ s i\n⊢ ⨆ s, ∑ a ∈ s, f a = ⨆ i, ∑ a ∈ s i, f a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"ENNReal.instAddCommMonoid",
"iSup",
"Finset",
"ConditionallyCompleteL... | [
"α : Type u_1\nf : α → ℝ≥0∞\nι : Type u_4\ns : ι → Finset α\nhs : ∀ (t : Finset α), ∃ i, t ⊆ s i\n⊢ ⨆ i, ∑ a ∈ s i, f a = ⨆ s, ∑ a ∈ s, f a"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.EReal.Inv | {
"line": 153,
"column": 72
} | {
"line": 157,
"column": 73
} | {
"line": 159,
"column": 0
} | [
{
"pp": "⊢ Covariant { x // 0 < x } EReal (fun x y ↦ ↑x * y) fun x1 x2 ↦ x1 ≤ x2",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"EReal.abs",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"Si... | [] | by
rintro ⟨x, x0⟩ a b h
simp only [le_iff_sign, EReal.sign_mul, sign_pos x0, one_mul, EReal.abs_mul] at h ⊢
exact h.imp_right <| Or.imp (And.imp_right <| And.imp_right (mul_le_mul_right · _)) <|
Or.imp_right <| And.imp_right <| And.imp_right (mul_le_mul_right · _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 342,
"column": 2
} | {
"line": 342,
"column": 60
} | {
"line": 343,
"column": 2
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\nhsum : ∑' (x : α), f x ≠ ∞\n⊢ HasSum (fun x ↦ (f x).toReal) (∑' (x : α), (f x).toReal)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.canLift",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"PseudoMetri... | [
"α : Type u_1\nf : α → ℝ≥0\nhsum : ∑' (x : α), (fun i ↦ ↑(f i)) x ≠ ∞\n⊢ HasSum (fun x ↦ ((fun i ↦ ↑(f i)) x).toReal) (∑' (x : α), ((fun i ↦ ↑(f i)) x).toReal)"
] | lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hsum | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 186,
"column": 68
} | {
"line": 188,
"column": 18
} | {
"line": 190,
"column": 0
} | [
{
"pp": "⊢ Tendsto toReal (𝓝[≠] ⊤) atTop",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Filter.tendsto_id",
"Filter.map",
"Compl.compl",
"EReal.instTopologicalSpace",
"nhdsWithin",
"EReal",
"Filter... | [] | by
rw [nhdsWithin_top, tendsto_map'_iff]
exact tendsto_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 61
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nP : Set α → Prop\nm_top : ∀ (s : Set α), ¬P s → m s = ∞\ns : Set α\nt : ℕ → Set α\nht_subset : s ⊆ iUnion t\nht : ∀ (i : ℕ), P (t i)\n⊢ ⨅ (_ : ∀ (i : ℕ), P (t i)), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) ≤ ∑' (n : ℕ), m (t n)",
"ppTerm"... | [
"case neg\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nP : Set α → Prop\nm_top : ∀ (s : Set α), ¬P s → m s = ∞\ns : Set α\nt : ℕ → Set α\nht_subset : s ⊆ iUnion t\nht : ¬∀ (i : ℕ), P (t i)\n⊢ ⨅ (_ : ∀ (i : ℕ), P (t i)), ⨅ (_ : s ⊆ ⋃ i, t i), ∑' (i : ℕ), m (t i) ≤ ∑' (n : ℕ), m (t n)"
] | · exact iInf_le_of_le ht (iInf_le_of_le ht_subset le_rfl) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 210,
"column": 8
} | {
"line": 211,
"column": 55
} | {
"line": 212,
"column": 8
} | [
{
"pp": "𝕜 : 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⊢ Tendsto (fun n ↦ |r| ^ n) atTop atTop",... | [
"𝕜 : 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|\nb : 𝕜\n⊢ ∃ a, b ≤ |r| ^ a"
] | refine (pow_right_strictMono₀ <| lt_of_le_of_ne (le_of_not_gt hr_le)
hr).monotone.tendsto_atTop_atTop (fun b ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 462,
"column": 6
} | {
"line": 462,
"column": 44
} | {
"line": 463,
"column": 6
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set 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 : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set 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 : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithinAt g s x\nhcont : Cont... | filter_upwards [hf z₁ z₁lt] with z h₁z | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 17
} | {
"line": 287,
"column": 18
} | [
{
"pp": "α : Type u_1\n⊢ boundedBy ⊤ = ⊤",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"MeasureTheory.OuterMeasure.boundedBy",
"eq_top_iff",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
... | [
"α : Type u_1\n⊢ ⊤ ≤ boundedBy ⊤"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 20
} | {
"line": 300,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ OuterMeasure.ofFunction (c • fun s ↦ ⨆ (_ : s.Nonempty), m s) ⋯ =\n OuterMeasure.ofFunction (fun s ↦ ⨆ (_ : s.Nonempty), (c • m) s) ⋯",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"instHSMul",
"instSMulOfMul"... | [
"α : Type u_1\nm : Set α → ℝ≥0∞\nc : ℝ≥0∞\nhc : c ≠ ∞\ns : Set α\n⊢ (c • fun s ↦ ⨆ (_ : s.Nonempty), m s) s = ⨆ (_ : s.Nonempty), (c • m) s"
] | congr 1 with s : 1 | Batteries.Tactic._aux_Batteries_Tactic_Congr___macroRules_Batteries_Tactic_congrConfigWith_1 | Batteries.Tactic.congrConfigWith |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 495,
"column": 2
} | {
"line": 498,
"column": 55
} | {
"line": 500,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ edist (f n) a ≤ 2 * C / 2 ^ n",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpac... | [] | simp only [div_eq_mul_inv, ENNReal.inv_pow] at *
rw [mul_assoc, mul_comm]
convert! edist_le_of_edist_le_geometric_of_tendsto 2⁻¹ C hu ha n using 1
rw [ENNReal.one_sub_inv_two, div_eq_mul_inv, inv_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 495,
"column": 2
} | {
"line": 498,
"column": 55
} | {
"line": 500,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nC : ℝ≥0∞\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C / 2 ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ edist (f n) a ≤ 2 * C / 2 ^ n",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpac... | [] | simp only [div_eq_mul_inv, ENNReal.inv_pow] at *
rw [mul_assoc, mul_comm]
convert! edist_le_of_edist_le_geometric_of_tendsto 2⁻¹ C hu ha n using 1
rw [ENNReal.one_sub_inv_two, div_eq_mul_inv, inv_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 6
} | {
"line": 570,
"column": 2
} | [
{
"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⊢ C / 2 ^ n = ∑' (m : ℕ), C / 2 / 2 ^ m / 2 ^ n",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Real",
"in... | [
"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⊢ ∑' (m : ℕ), C / 2 / 2 ^ m / 2 ^ n = C / 2 ^ n"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.PiSystem | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 26
} | {
"line": 87,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Set α\n⊢ IsPiSystem {S}",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.inter_self",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instInter",
"Inter.inter",
"Set.Nonempty",
... | [] | intro s h_s t h_t _
rw [Set.mem_singleton_iff.1 h_s, Set.mem_singleton_iff.1 h_t, Set.inter_self,
Set.mem_singleton_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.PiSystem | {
"line": 83,
"column": 2
} | {
"line": 85,
"column": 26
} | {
"line": 87,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Set α\n⊢ IsPiSystem {S}",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.inter_self",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instInter",
"Inter.inter",
"Set.Nonempty",
... | [] | intro s h_s t h_t _
rw [Set.mem_singleton_iff.1 h_s, Set.mem_singleton_iff.1 h_t, Set.inter_self,
Set.mem_singleton_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 736,
"column": 4
} | {
"line": 736,
"column": 46
} | {
"line": 737,
"column": 4
} | [
{
"pp": "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)\nx : R\nhx : 1 ≤ x\n⊢ ↑⌊a * x⌋₊ / x ≤ a",
"ppTerm": "?hfh",
... | [
"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)\nx : R\nhx : 1 ≤ x\n⊢ ↑⌊a * x⌋₊ ≤ a * x"
] | rw [div_le_iff₀ (zero_lt_one.trans_le hx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 1285,
"column": 6
} | {
"line": 1285,
"column": 30
} | {
"line": 1287,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\nh₁ : LowerSemicontinuousWithinAt f s x\nh₂ : UpperSemicontinuousWithinAt f s x\nv : Set γ\nhv : v ∈ 𝓝 (f x)\nHl : ∀ (l : γ), f x... | [] | exact mem_of_mem_nhds hv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 215,
"column": 93
} | {
"line": 215,
"column": 97
} | {
"line": 216,
"column": 4
} | [
{
"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)\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ ... | [
"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)\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 214,
"column": 60
} | {
"line": 216,
"column": 64
} | {
"line": 218,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ ν : Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ μ t = μ s ∧ ν t = ν s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"cond",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"Exists",
"... | [] | by
simpa only [Bool.forall_bool.trans and_comm] using!
exists_measurable_superset_forall_eq (fun b => cond b μ ν) s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.PiSystem | {
"line": 633,
"column": 15
} | {
"line": 633,
"column": 36
} | {
"line": 634,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nα : Type u_3\nd : DynkinSystem α\ns : Set α\nh : d.Has s\n⊢ d.Has (∅ ∩ s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.empty_inter",
"Set.instInter",
"Inter.inter",
"MeasurableSpace.DynkinSystem.Has",
... | [] | by simp [d.has_empty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 71
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\nthis : Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)\n⊢ Measurable f",
"ppTerm": "?m.50",
... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 71
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\nthis : Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)\n⊢ Measurable f",
"ppTerm": "?m.50",
... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 71
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nH : Measurable (g ∘ f)\nthis : Measurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f)\n⊢ Measurable f",
"ppTerm": "?m.50",
... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 34
} | {
"line": 146,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set (Set α)\nhs : s.Countable\nh : ∀ t ∈ s, NullMeasurableSet t μ\n⊢ NullMeasurableSet (⋃ i ∈ s, i) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MeasureTheory.NullMeasurableSpace.instMeasurableSpace",
"Meas... | [] | exact MeasurableSet.biUnion hs h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 34
} | {
"line": 63,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis✝ : Nonempty ↑s\nhsub : ¬s ⊆ Iio b\nthis : ¬(comap Subtype.val (𝓝[<] b) = atTop ↔ s ⊆ Iio b ∧ ∀ a < b, (s ∩ Ioo a b).Nonempty)\n⊢ Tendsto Subtype.val atTop (𝓝[<] b)",
... | [] | simp_all [tendsto_iff_comap] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 30
} | {
"line": 392,
"column": 2
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\nh : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)\nt : Set α\nht : MeasurableSet t\nD : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i\n⊢ (μ.restrict (⋃ i, s i)) t = (ν.restrict (⋃ i, s i)) t"... | [
"α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\nh : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)\nt : Set α\nht : MeasurableSet t\nD : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i\n⊢ (μ.restrict (⋃ t, ⋃ i ∈ t, s i)) t = (ν.restrict (⋃ t, ⋃ i ∈ t, s ... | rw [iUnion_eq_iUnion_finset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 6
} | {
"line": 342,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : ∀ s ∈ C, MeasurableSet s\nm : ℝ≥0∞ := ⨆ D ∈ {D | D ⊆ C ∧ D.Countable}, μ (⋃₀ D)\nD : Set (Set α)\nD_mem : D ∈ {D | D ⊆ C ∧ D.Countable}\nhD : μ (⋃₀ D) = m\ns : Set α\nhs : s ∈ C\n⊢ s ∪ ⋃₀ D =ᵐ[μ] ⋃₀ D"... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : ∀ s ∈ C, MeasurableSet s\nm : ℝ≥0∞ := ⋯\nD : Set (Set α)\nD_mem : D ∈ {D | D ⊆ C ∧ D.Countable}\nhD : μ (⋃₀ D) = m\ns : Set α\nhs : s ∈ C\n⊢ ⋃₀ D =ᵐ[μ] s ∪ ⋃₀ D"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 37
} | {
"line": 85,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : CountablyGenerated α\ns : Set α\nhs : s ∈ countableGeneratingSet α\n⊢ MeasurableSet s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasurableSpace.measurableSet_generateFrom",
"MeasurableSpace.countableGeneratingSet... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 59
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ninst✝ : CountablyGenerated α\np : ℕ → Prop\nn : ℕ\n⊢ MeasurableSet (if p n then natGeneratingSequence α n else (natGeneratingSequence α n)ᶜ)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasurableSet.ite'",
"Compl.compl",
... | [] | exact MeasurableSet.ite' (fun _ ↦ measurableSet_natGeneratingSequence n)
(fun _ ↦ (measurableSet_natGeneratingSequence n).compl) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 905,
"column": 29
} | {
"line": 905,
"column": 70
} | {
"line": 907,
"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 α\ns t : Set β\nht : MeasurableSet t\n⊢ ((Measure.map f μ).restrict s) t = (Measure.map f (μ.restrict (f ⁻¹' s))) t",
"ppTerm": "?m.27",
"assigned": true,
"usedCons... | [] | simp [hf.map_apply, ht, hf.measurable ht] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 905,
"column": 29
} | {
"line": 905,
"column": 70
} | {
"line": 907,
"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 α\ns t : Set β\nht : MeasurableSet t\n⊢ ((Measure.map f μ).restrict s) t = (Measure.map f (μ.restrict (f ⁻¹' s))) t",
"ppTerm": "?m.27",
"assigned": true,
"usedCons... | [] | simp [hf.map_apply, ht, hf.measurable ht] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 905,
"column": 29
} | {
"line": 905,
"column": 70
} | {
"line": 907,
"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 α\ns t : Set β\nht : MeasurableSet t\n⊢ ((Measure.map f μ).restrict s) t = (Measure.map f (μ.restrict (f ⁻¹' s))) t",
"ppTerm": "?m.27",
"assigned": true,
"usedCons... | [] | simp [hf.map_apply, ht, hf.measurable ht] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 564,
"column": 6
} | {
"line": 564,
"column": 55
} | {
"line": 564,
"column": 55
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ ν₁ ν₂ : Measure α\ninst✝ : SigmaFinite μ\nh : μ + ν₁ = μ + ν₂\n⊢ ν₁ = ν₂",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.restrict",
"id",
... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ ν₁ ν₂ : Measure α\ninst✝ : SigmaFinite μ\nh : μ + ν₁ = μ + ν₂\n⊢ ∀ (i : ℕ), ν₁.restrict (spanningSets μ i) = ν₂.restrict (spanningSets μ i)"
] | ext_iff_of_iUnion_eq_univ (iUnion_spanningSets μ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 593,
"column": 73
} | {
"line": 593,
"column": 91
} | {
"line": 593,
"column": 91
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable α\nhμ : ∀ (a : α), μ {a} < ∞\nh✝ : Nonempty α\nf : ℕ → α\nhf : Surjective f\n⊢ ⋃ i, {f i} = univ",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Set.iUnion_singleton_e... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 593,
"column": 73
} | {
"line": 593,
"column": 91
} | {
"line": 593,
"column": 91
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable α\nhμ : ∀ (a : α), μ {a} < ∞\nh✝ : Nonempty α\nf : ℕ → α\nhf : Surjective f\n⊢ ⋃ i, {f i} = univ",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Set.iUnion_singleton_e... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 593,
"column": 73
} | {
"line": 593,
"column": 91
} | {
"line": 593,
"column": 91
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : Countable α\nhμ : ∀ (a : α), μ {a} < ∞\nh✝ : Nonempty α\nf : ℕ → α\nhf : Surjective f\n⊢ ⋃ i, {f i} = univ",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.univ",
"Set.iUnion_singleton_e... | [] | simp [hf.range_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 71
} | {
"line": 271,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nμ : Measure α\nH : AEMeasurable (g ∘ f) μ\nthis : AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) μ\n⊢ AEMeasurable f μ... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 71
} | {
"line": 271,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nμ : Measure α\nH : AEMeasurable (g ∘ f) μ\nthis : AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) μ\n⊢ AEMeasurable f μ... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 71
} | {
"line": 271,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : α → β\ng : β → γ\nhg : MeasurableEmbedding g\nμ : Measure α\nH : AEMeasurable (g ∘ f) μ\nthis : AEMeasurable ((rangeSplitting g ∘ rangeFactorization g) ∘ f) μ\n⊢ AEMeasurable f μ... | [] | rwa [(rightInverse_rangeSplitting hg.injective).comp_eq_id] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 808,
"column": 29
} | {
"line": 810,
"column": 53
} | {
"line": 812,
"column": 0
} | [
{
"pp": "M : Type u_2\nα : Type u_4\ninst✝² : CommMonoid M\ninst✝¹ : MeasurableSpace M\ninst✝ : MeasurableMul₂ M\nm : MeasurableSpace α\nμ : Measure α\nl : Multiset (α → M)\nhl : ∀ f ∈ l, AEMeasurable f μ\n⊢ AEMeasurable l.prod μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"AEMea... | [] | by
rcases l with ⟨l⟩
simpa using l.aemeasurable_prod (by simpa using hl) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 189,
"column": 18
} | {
"line": 189,
"column": 22
} | {
"line": 189,
"column": 22
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α✝\nα : Type u_6\ninst✝ : TopologicalSpace α\np : α → Prop\nhα : BorelSpace α\nthis✝ : MeasurableSpace α := borel α\n⊢ instMeasurableSpace = borel (Subtype p)",
"ppTerm": "?m.26",
"assigned": true,
... | [
"α✝ : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α✝\nα : Type u_6\ninst✝ : TopologicalSpace α\np : α → Prop\nhα : BorelSpace α\nthis✝ : MeasurableSpace α := ⋯\n⊢ borel (Subtype p) = instMeasurableSpace"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 65
} | {
"line": 641,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝² : BorelSpace α\nmβ : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\n⊢ Prod.instMeasurableSpace ≤ borel (α × β)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝² : BorelSpace α\nmβ : TopologicalSpace β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\n⊢ Prod.instMeasurableSpace ≤ borel (α × β)"
] | rw [‹BorelSpace α›.measurable_eq, ‹BorelSpace β›.measurable_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 153,
"column": 20
} | {
"line": 153,
"column": 62
} | {
"line": 153,
"column": 62
} | [
{
"pp": "δ : ℝ\nX : Type u\ninst✝ : PseudoMetricSpace X\nE : Set X\nx z : X\n⊢ edist x z < ENNReal.ofReal δ ↔ (edist x z).toReal < δ",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"Preorder.toLT",
... | [
"δ : ℝ\nX : Type u\ninst✝ : PseudoMetricSpace X\nE : Set X\nx z : X\n⊢ (edist x z).toReal < δ ↔ (edist x z).toReal < δ"
] | lt_ofReal_iff_toReal_lt (edist_ne_top _ _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 10
} | {
"line": 381,
"column": 6
} | [
{
"pp": "α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S}\nth... | [
"α : Type u_5\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := ⋯\nthis : MeasurableSpace α := ⋯\na : α\nt : Set α\nhts : t... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 397,
"column": 77
} | {
"line": 398,
"column": 41
} | {
"line": 400,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\ns : Set α\n⊢ cthickening δ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s) = cthickening δ s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"ENNReal.ofReal",
"congrArg",
"setOf",
"id",
... | [] | by
simp_rw [cthickening, infEDist_closure] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 702,
"column": 31
} | {
"line": 703,
"column": 61
} | {
"line": 705,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nh : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nhs : s.Nonempty\n⊢ x ∉ s ↔ 0 < infDist x s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT"... | [] | by
simp [h.mem_iff_infDist_zero hs, infDist_nonneg.lt_iff_ne'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 496,
"column": 4
} | {
"line": 496,
"column": 40
} | {
"line": 497,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_nn : 0 ≤ δ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty\nE : Set α\nε : ℝ\nhε : ε ∈ s\n⊢ cthickening δ E ⊆ thickening ε E",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Set... | [
"case refine_1\nα : Type u\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_nn : 0 ≤ δ\ns : Set ℝ\nhsδ : s ⊆ Ioi δ\nhs : ∀ (ε : ℝ), δ < ε → (s ∩ Ioc δ ε).Nonempty\nE : Set α\nε : ℝ\nhε : ε ∈ s\nε' : ℝ\nhε' : ε' ∈ Ioc δ ε\n⊢ cthickening δ E ⊆ thickening ε E"
] | obtain ⟨ε', -, hε'⟩ := hs ε (hsδ hε) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 542,
"column": 57
} | {
"line": 544,
"column": 49
} | {
"line": 546,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nE : Set α\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E = ⋂ δ, ⋂ (_ : 0 < δ), thickening δ E",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Eq.ge",
"Real.instZero",
"co... | [] | by
rw [← cthickening_zero]
exact cthickening_eq_iInter_thickening rfl.ge E | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 144,
"column": 2
} | {
"line": 146,
"column": 83
} | {
"line": 148,
"column": 0
} | [
{
"pp": "β : Type u_2\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\n⊢ ∃ c, ∀ (x : α), f x = c",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"MeasurableSet",
"CompleteLattice.toLattice",
"Set.univ",
"OrderBot.toBot",
"Partial... | [] | have hf_meas := @SimpleFunc.measurableSet_fiber α _ ⊥ f
simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas
exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc ↦ congr_fun hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 144,
"column": 2
} | {
"line": 146,
"column": 83
} | {
"line": 148,
"column": 0
} | [
{
"pp": "β : Type u_2\nα : Type u_5\nf : α →ₛ β\ninst✝ : Nonempty β\n⊢ ∃ c, ∀ (x : α), f x = c",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"MeasurableSet",
"CompleteLattice.toLattice",
"Set.univ",
"OrderBot.toBot",
"Partial... | [] | have hf_meas := @SimpleFunc.measurableSet_fiber α _ ⊥ f
simp_rw [MeasurableSpace.measurableSet_bot_iff] at hf_meas
exact (exists_eq_const_of_preimage_singleton hf_meas).imp fun c hc ↦ congr_fun hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 949,
"column": 21
} | {
"line": 949,
"column": 46
} | {
"line": 949,
"column": 47
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\n⊢ (hausdorffEDist {x} {y}).toReal = dist x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"congrArg",
"Set.instSingletonSet",
"... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\n⊢ (edist x y).toReal = dist x y"
] | hausdorffEDist_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 653,
"column": 2
} | {
"line": 655,
"column": 33
} | {
"line": 657,
"column": 0
} | [
{
"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 : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)\n⊢ Measurable f",
"ppTerm": "?m... | [] | convert! measurable_generateFrom (α := δ) _
· exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)
· rintro _ ⟨x, rfl⟩; exact hf x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 653,
"column": 2
} | {
"line": 655,
"column": 33
} | {
"line": 657,
"column": 0
} | [
{
"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 : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nf : δ → α\nhf : ∀ (x : α), MeasurableSet (f ⁻¹' Ioi x)\n⊢ Measurable f",
"ppTerm": "?m... | [] | convert! measurable_generateFrom (α := δ) _
· exact BorelSpace.measurable_eq.trans (borel_eq_generateFrom_Ioi _)
· rintro _ ⟨x, rfl⟩; exact hf x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 671,
"column": 2
} | {
"line": 671,
"column": 37
} | {
"line": 672,
"column": 2
} | [
{
"pp": "β : Type u_2\nδ : Type u_4\ninst✝⁶ : TopologicalSpace β\nmβ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\nmδ : MeasurableSpace δ\ninst✝⁴ : CompleteLinearOrder β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\nι : Type u_5\ninst✝¹ : TopologicalSpace ι\ninst✝ : SeparableSpace ι\nf : ι → δ → ... | [
"β : Type u_2\nδ : Type u_4\ninst✝⁶ : TopologicalSpace β\nmβ : MeasurableSpace β\ninst✝⁵ : BorelSpace β\nmδ : MeasurableSpace δ\ninst✝⁴ : CompleteLinearOrder β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\nι : Type u_5\ninst✝¹ : TopologicalSpace ι\ninst✝ : SeparableSpace ι\nf : ι → δ → β\nmf : ∀ (t... | refine ⟨fun ⟨i, hi⟩ ↦ ?_, by grind⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 41
} | {
"line": 54,
"column": 4
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set α\nhs : s ∈ C\nt : Set β\nht : t ∈ D\n⊢ MeasurableSet s",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"MeasurableSpace.measurableSet_generateFrom"... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 41
} | {
"line": 54,
"column": 4
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set α\nhs : s ∈ C\nt : Set β\nht : t ∈ D\n⊢ MeasurableSet s",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"MeasurableSpace.measurableSet_generateFrom"... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 41
} | {
"line": 54,
"column": 4
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set α\nhs : s ∈ C\nt : Set β\nht : t ∈ D\n⊢ MeasurableSet s",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"MeasurableSpace.measurableSet_generateFrom"... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 10
} | {
"line": 833,
"column": 2
} | [
{
"pp": "case e_f\nα : Type u_1\nβ : Type u_2\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable... | [
"case e_f\nα : Type u_1\nβ : Type u_2\ninst✝⁷ : MeasurableSpace α\ninst✝⁶ : SemilatticeSup β\ninst✝⁵ : OrderBot β\ninst✝⁴ : Zero β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderClosedTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\ni : ℕ → β\nf : α → β\nn : ℕ\na : α\nhf : Measurable f\nk : ℕ\n⊢... | funext k | _aux_Init_NotationExtra___macroRules_tacticFunext____1 | tacticFunext___ |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 728,
"column": 8
} | {
"line": 728,
"column": 22
} | {
"line": 729,
"column": 8
} | [
{
"pp": "case pos\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι),... | [
"case neg\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurable ... | · simp [hb, B] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 734,
"column": 4
} | {
"line": 735,
"column": 76
} | {
"line": 736,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι),... | [
"case inr\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurable ... | suffices ∀ b, IsLUB { a | ∃ i, f' i b = a } (g b) from
Measurable.isLUB (fun i ↦ Measurable.piecewise hs (hf i) g'_meas) this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 761,
"column": 17
} | {
"line": 761,
"column": 34
} | {
"line": 761,
"column": 35
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\nμ : Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι... | [
"α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\nμ : Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), AEMeasura... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 22
} | {
"line": 107,
"column": 22
} | [
{
"pp": "case refine_2\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : x < z\nq : ℚ\nh₁ : x < ↑q\nh₂ : ↑q < min z (x + ε)\n⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε",
"ppTerm": "?refine_2",
"assigned": true,
"usedConsta... | [
"case refine_2\ns : Set ℝ\nconn : s.OrdConnected\nnt : s.Nontrivial\nx : ℝ\nhx : x ∈ s\nε : ℝ\nε_pos : ε > 0\nz : ℝ\nhz : z ∈ s\nne : z ≠ x\nlt : x < z\nq : ℚ\nh₁ : x < ↑q\nh₂ : ↑q < z ∧ ↑q < x + ε\n⊢ ∃ y ∈ s ∩ range Rat.cast, |y - x| < ε"
] | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1175,
"column": 4
} | {
"line": 1175,
"column": 56
} | {
"line": 1176,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : Zero β\nμ : Measure α\nf : α →ₛ β\nh : f.FinMeasSupp μ\ny : β\nhy : y ≠ 0\n⊢ ⇑f ⁻¹' {y} ⊆ {x | (fun x ↦ f x = 0 x) x}ᶜ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc",
"Members... | [] | exact fun x hx (H : f x = 0) => hy <| H ▸ Eq.symm hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 968,
"column": 4
} | {
"line": 972,
"column": 40
} | {
"line": 974,
"column": 0
} | [
{
"pp": "case neg\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\ns : Set ι\nf : ι → δ → α\nhs : s.Countable\nh... | [] | have : ∀ b, ⨆ i ∈ s, f i b = (⨆ (i : s), f i b) ⊔ sSup ∅ :=
fun b ↦ cbiSup_eq_of_not_forall (f := fun i ↦ f i b) H
simp only [this]
apply Measurable.sup _ measurable_const
exact .iSup (fun (i : s) ↦ hf i i.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 968,
"column": 4
} | {
"line": 972,
"column": 40
} | {
"line": 974,
"column": 0
} | [
{
"pp": "case neg\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\ns : Set ι\nf : ι → δ → α\nhs : s.Countable\nh... | [] | have : ∀ b, ⨆ i ∈ s, f i b = (⨆ (i : s), f i b) ⊔ sSup ∅ :=
fun b ↦ cbiSup_eq_of_not_forall (f := fun i ↦ f i b) H
simp only [this]
apply Measurable.sup _ measurable_const
exact .iSup (fun (i : s) ↦ hf i i.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 30
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : PseudoMetrizableSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : BorelSpace β\nι : Type u_3\nμ : Measure α\nf : ι → α → β\ng : α → β\nu : Filter ι\nhu : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 30
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : PseudoMetrizableSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : BorelSpace β\nι : Type u_3\nμ : Measure α\nf : ι → α → β\ng : α → β\nu : Filter ι\nhu : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 30
} | {
"line": 75,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : PseudoMetrizableSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : BorelSpace β\nι : Type u_3\nμ : Measure α\nf : ι → α → β\ng : α → β\nu : Filter ι\nhu : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 73,
"column": 64
} | {
"line": 73,
"column": 89
} | {
"line": 73,
"column": 89
} | [
{
"pp": "⊢ generateFrom (range fun a ↦ Ioi ↑a) = generateFrom (⋃ a, {Iic ↑a})",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"congrArg",
"Real.instRatCast",
"Rat",
"Set.iUnion_singleton_eq_range",
"Set.instSingle... | [
"⊢ generateFrom (range fun a ↦ Ioi ↑a) = generateFrom (range fun a ↦ Iic ↑a)"
] | iUnion_singleton_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 36
} | {
"line": 81,
"column": 37
} | [
{
"pp": "⊢ borel ℝ = generateFrom (⋃ a, {Ici ↑a})",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ici",
"Real.borel_eq_generateFrom_Iio_rat",
"congrArg",
"Real.instRatCast",
"Rat",
"PseudoMetricSpace.toUniformSpace",
... | [
"⊢ generateFrom (⋃ a, {Iio ↑a}) = generateFrom (⋃ a, {Ici ↑a})"
] | borel_eq_generateFrom_Iio_rat, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 81,
"column": 64
} | {
"line": 81,
"column": 89
} | {
"line": 81,
"column": 89
} | [
{
"pp": "⊢ generateFrom (range fun a ↦ Iio ↑a) = generateFrom (⋃ a, {Ici ↑a})",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ici",
"congrArg",
"Real.instRatCast",
"Rat",
"Set.iUnion_singleton_eq_range",
"Set.instSingle... | [
"⊢ generateFrom (range fun a ↦ Iio ↑a) = generateFrom (range fun a ↦ Ici ↑a)"
] | iUnion_singleton_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 30
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nι : Type u_5\nf : ι → α → ℝ≥0∞\ng : α → ℝ≥0∞\nμ : Measure α\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nhlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))\nv : ℕ → ι\nhv : Tendsto v atTop ... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 30
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nι : Type u_5\nf : ι → α → ℝ≥0∞\ng : α → ℝ≥0∞\nμ : Measure α\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nhlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))\nv : ℕ → ι\nhv : Tendsto v atTop ... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 30
} | {
"line": 310,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nι : Type u_5\nf : ι → α → ℝ≥0∞\ng : α → ℝ≥0∞\nμ : Measure α\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nhlim : ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) u (𝓝 (g a))\nv : ℕ → ι\nhv : Tendsto v atTop ... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 51
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nc : ℝ≥0∞\n⊢ ∫⁻ (x : α) in s, c ∂μ = c * μ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | [] | rw [lintegral_const, Measure.restrict_apply_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 51
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nc : ℝ≥0∞\n⊢ ∫⁻ (x : α) in s, c ∂μ = c * μ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | [] | rw [lintegral_const, Measure.restrict_apply_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 51
} | {
"line": 124,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nc : ℝ≥0∞\n⊢ ∫⁻ (x : α) in s, c ∂μ = c * μ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg",
... | [] | rw [lintegral_const, Measure.restrict_apply_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 20
} | {
"line": 291,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm : MeasurableSpace α\nf : α → β\nm' m₀ : MeasurableSpace α\nμ : Measure α\ninst✝ : Zero β\nhm : m ≤ m₀\ns : Set α\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)\nhf : AEStronglyMeasurable f μ... | [
"case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm : MeasurableSpace α\nf : α → β\nm' m₀ : MeasurableSpace α\nμ : Measure α\ninst✝ : Zero β\nhm : m ≤ m₀\ns : Set α\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α), MeasurableSet (s ∩ t) → MeasurableSet (s ∩ t)\nhf : AEStronglyMeasurable f μ\nhf_zero : ... | · simp [hxs, hx] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 743,
"column": 4
} | {
"line": 743,
"column": 83
} | {
"line": 745,
"column": 0
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\nmb : MeasurableSpace β\ninst✝ : BorelSpace β\nm : MeasurableSpace α\nμ : Measure α\nf : α → β\ns : Set β\nhs : MeasurableSet s\nh_nonempty : s.Nonempty\nh_mem : ∀ᵐ (x : α) ∂μ, f x ∈ s\nf' : α → β\nhf' : StronglyM... | [] | exact Eq.symm <| (f' ⁻¹' s).piecewise_eq_of_mem f' _ (by simpa [hx'] using! hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 22
} | {
"line": 224,
"column": 4
} | [
{
"pp": "case neg.inl\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nc : ℝ\nhf : StronglyMeasurable f\nhc : 0 ≤ c\nn : ℕ\nx : α\nh0 : ¬‖(hf.approx n) x‖ = 0\nh : ‖(hf.approx n) x‖ ≤ c\n⊢ ‖min 1 (c / ‖(hf.approx n) x‖)‖ * ‖(hf.approx n) ... | [
"case neg.inl\nα : Type u_1\nβ : Type u_5\nf : α → β\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : NormedSpace ℝ β\nm : MeasurableSpace α\nc : ℝ\nhf : StronglyMeasurable f\nhc : 0 ≤ c\nn : ℕ\nx : α\nh0 : ¬‖(hf.approx n) x‖ = 0\nh : ‖(hf.approx n) x‖ ≤ c\n⊢ ‖1‖ * ‖(hf.approx n) x‖ ≤ c",
"α : Type u_1\nβ : Type u_5\n... | rw [min_eq_left _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 412,
"column": 29
} | {
"line": 412,
"column": 51
} | {
"line": 412,
"column": 51
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\ns : Set α\n⊢ ∫⁻ (a : α), f a ∂c • μ.restrict s = c • ∫⁻ (a : α) in s, f a ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\ns : Set α\n⊢ c • ∫⁻ (a : α) in s, f a ∂μ = c • ∫⁻ (a : α) in s, f a ∂μ"
] | lintegral_smul_measure | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 82
} | {
"line": 594,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nt : Set β\ns : β → Set α\nht : t.Countable\nhm : ∀ i ∈ t, NullMeasurableSet (s i) μ\nhd : t.Pairwise (AEDisjoint μ on s)\nf : α → ℝ≥0∞\nthis : Encodable ↑t\n⊢ ∫⁻ (a : α) in ⋃ i ∈ t, s i, f a ∂μ = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ",... | [] | rw [biUnion_eq_iUnion, lintegral_iUnion₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 280,
"column": 6
} | {
"line": 280,
"column": 30
} | {
"line": 281,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.rest... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 125,
"column": 6
} | {
"line": 127,
"column": 27
} | {
"line": 128,
"column": 4
} | [
{
"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\nval✝ : Encodable β\nh✝ : Nonempty β\ninhabited_h : Inh... | [] | rw [lintegral_iInf ?_ h_directed.sequence_anti]
· exact hf_int _
· exact fun n => hf _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 125,
"column": 6
} | {
"line": 127,
"column": 27
} | {
"line": 128,
"column": 4
} | [
{
"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\nval✝ : Encodable β\nh✝ : Nonempty β\ninhabited_h : Inh... | [] | rw [lintegral_iInf ?_ h_directed.sequence_anti]
· exact hf_int _
· exact fun n => hf _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 28
} | {
"line": 430,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nf : α → β\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : GroupWithZero β\ninst✝¹ : ContinuousInv₀ β\ninst✝ : MetrizableSpace β\nhf : StronglyMeasurable f\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nx : α\nthis : Measurable fun x ↦ ... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 28
} | {
"line": 456,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nf g : α → β\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : GroupWithZero β\ninst✝² : ContinuousMul β\ninst✝¹ : ContinuousInv₀ β\ninst✝ : MetrizableSpace β\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nthis✝¹ : MeasurableSpace β := borel β\n... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Norm | {
"line": 23,
"column": 11
} | {
"line": 23,
"column": 24
} | {
"line": 23,
"column": 24
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂μ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
"_private.Math... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\n⊢ ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ ≤ ∫⁻ (x : α), ENNReal.ofReal ‖f x‖ ∂μ"
] | ← ofReal_norm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 21
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_b... | [] | · exact h_tendsto | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Perfect | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 32
} | {
"line": 231,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : TopologicalSpace α\nC : Set α\ninst✝ : SecondCountableTopology α\nhclosed : IsClosed[inst✝¹] C\nb : Set (Set α)\nbct : b.Countable\nleft✝ : ∅ ∉ b\nbbasis : IsTopologicalBasis b\nv : Set (Set α) := ⋯\nV : Set α := ⋯\nD : Set α := ⋯\n⊢ ∀ a ∈ v, (a ∩ C).Countable",
"ppTe... | [] | · exact sep_subset_setOf _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.InfiniteSum | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 44
} | {
"line": 131,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nε : Type u_5\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddCommMonoid ε\nf : ι → ε\ng : ι → ℝ≥0∞\na : ℝ≥0∞\nhg : HasSum g a\nh : ∀ (i : ι), ‖f i‖ₑ ≤ g i\nhf : Summable f\n⊢ ‖∑' (i : ι), f i‖ₑ ≤ a",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [
"case neg\nι : Type u_1\nε : Type u_5\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddCommMonoid ε\nf : ι → ε\ng : ι → ℝ≥0∞\na : ℝ≥0∞\nhg : HasSum g a\nh : ∀ (i : ι), ‖f i‖ₑ ≤ g i\nhf : ¬Summable f\n⊢ ‖∑' (i : ι), f i‖ₑ ≤ a"
] | · exact hf.hasSum.enorm_le_of_bounded hg h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 483,
"column": 53
} | {
"line": 486,
"column": 19
} | {
"line": 488,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : AEMeasurable f (μ.trim hm)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (x : α) in s, f x ∂μ.trim hm = ∫⁻ (x : α) in s, f x ∂μ",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Me... | [] | by
rw [← lintegral_trim_ae hm]
all_goals rw [← restrict_trim hm _ hs]
exact hf.restrict | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 33
} | {
"line": 218,
"column": 6
} | [
{
"pp": "case inl\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i :... | [
"case inl\nα : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f... | refine ⟨0, h.mono_right ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1139,
"column": 4
} | {
"line": 1139,
"column": 28
} | {
"line": 1140,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 41
} | {
"line": 217,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyMetrizableSpace X\n⊢ TopologicalSpace.MetrizableSpace X",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"TopologicalSpace.upgradeIsCompletelyMetrizable"
],
"usedFVars": [
"X",
"in... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyMetrizableSpace X\nthis : UpgradedIsCompletelyMetrizableSpace X := upgradeIsCompletelyMetrizable X\n⊢ TopologicalSpace.MetrizableSpace X"
] | letI := upgradeIsCompletelyMetrizable X | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 21
} | {
"line": 249,
"column": 4
} | [
{
"pp": "case mp\nX : Type u\nY : Type v\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\ns : Set ((X ⊕ Y) × (X ⊕ Y))\n⊢ s ∈ 𝓤 (X ⊕ Y) → ∃ ε > 0, ∀ (a b : X ⊕ Y), Sum.dist a b < ε → (a, b) ∈ s",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real",
... | [
"case mp\nX : Type u\nY : Type v\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\ns : Set ((X ⊕ Y) × (X ⊕ Y))\nhsX : s ∈ Filter.map (fun p ↦ (Sum.inl p.1, Sum.inl p.2)) (𝓤 X)\nhsY : s ∈ Filter.map (fun p ↦ (Sum.inr p.1, Sum.inr p.2)) (𝓤 Y)\n⊢ ∃ ε > 0, ∀ (a b : X ⊕ Y), Sum.dist a b < ε → (a, b) ∈ s"
] | rintro ⟨hsX, hsY⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 16
} | {
"line": 97,
"column": 16
} | [
{
"pp": "E : ℕ → Type u_1\nx y z : (n : ℕ) → E n\nh : x ≠ z\nH : firstDiff x z < min (firstDiff x y) (firstDiff y z)\n⊢ False",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"PiNat.firstDiff",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Semilattic... | [
"E : ℕ → Type u_1\nx y z : (n : ℕ) → E n\nh : x ≠ z\nH : firstDiff x z < firstDiff x y ∧ firstDiff x z < firstDiff y z\n⊢ False"
] | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 16
} | {
"line": 214,
"column": 6
} | [
{
"pp": "case mp.succ\nα : Type u_2\nx y : ℕ → α\nn : ℕ\nih : res x n = res y n → ∀ ⦃m : ℕ⦄, m < n → x m = y m\nh : res x (n + 1) = res y (n + 1)\n⊢ ∀ ⦃m : ℕ⦄, m < n + 1 → x m = y m",
"ppTerm": "?mp.succ",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"instHAdd",
"HAdd.hAdd"... | [
"case mp.succ\nα : Type u_2\nx y : ℕ → α\nn : ℕ\nih : res x n = res y n → ∀ ⦃m : ℕ⦄, m < n → x m = y m\nh : res x (n + 1) = res y (n + 1)\nm : ℕ\nhm : m < n + 1\n⊢ x m = y m"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 85
} | {
"line": 319,
"column": 2
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : dist x y < (1 / 2) ^ n\ni : ℕ\nhi : i ≤ n\nhne : x ≠ y\n⊢ n < firstDiff x y",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",... | [] | simpa [dist_eq_of_ne hne, inv_lt_inv₀, pow_lt_pow_iff_right₀, one_lt_two] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 652,
"column": 4
} | {
"line": 654,
"column": 92
} | {
"line": 656,
"column": 0
} | [] | [] | range (toInductiveLimit I i) ⊆ closure (toInductiveLimit I i '' (hsX i)) :=
(toInductiveLimit_isometry I i |>.continuous).range_subset_closure_image_dense (hdX i)
_ ⊆ closure s := closure_mono <| subset_iUnion (fun j ↦ toInductiveLimit I j '' hsX j) i | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Topology.MetricSpace.CantorScheme | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 29
} | {
"line": 170,
"column": 4
} | [
{
"pp": "β : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝¹ : PseudoMetricSpace α\ninst✝ : CompleteSpace α\nhdiam : VanishingDiam A\nhanti : ClosureAntitone A\nhnonempty : ∀ (l : List β), (A l).Nonempty\nx : ℕ → β\nu : ℕ → α\nhu : ∀ (n : ℕ), u n ∈ A (res x n)\numem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)\n... | [
"β : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝¹ : PseudoMetricSpace α\ninst✝ : CompleteSpace α\nhdiam : VanishingDiam A\nhanti : ClosureAntitone A\nhnonempty : ∀ (l : List β), (A l).Nonempty\nx : ℕ → β\nu : ℕ → α\nhu : ∀ (n : ℕ), u n ∈ A (res x n)\numem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)\n⊢ ∀ ε > 0, ∃... | rw [Metric.cauchySeq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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