module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 39
} | {
"line": 42,
"column": 6
} | [
{
"pp": "case hf\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal) (μ.restric... | [
"case hf\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal) μ"
] | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.RadonNikodym | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 39
} | {
"line": 42,
"column": 6
} | [
{
"pp": "case hg\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal) (μ.restric... | [
"case hg\nα : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\ninst✝ : SigmaFinite μ\nh : s.toJordanDecomposition.posPart ≪ μ ∧ s.toJordanDecomposition.negPart ≪ μ\ni : Set α\nhi : MeasurableSet i\n⊢ Integrable (fun x ↦ (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal) μ"
] | refine Integrable.integrableOn ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 88
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ ENNReal.ofReal (lpNorm f p μ) = eLpNorm f p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofRea... | [] | rw [← toReal_eLpNorm hf.aestronglyMeasurable, ENNReal.ofReal_toReal hf.eLpNorm_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [] | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 47,
"column": 2
} | {
"line": 53,
"column": 84
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ = (∫ (x : α), ‖f x‖ ^ p.toReal ∂μ) ^ p.toReal⁻¹",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants":... | [] | rw [← toReal_eLpNorm hf, eLpNorm_eq_lintegral_rpow_enorm_toReal hp₀ hp, ← ENNReal.toReal_rpow,
← integral_toReal]
· simp [← ENNReal.toReal_rpow]
· simp_rw [← ofReal_norm]
borelize E
fun_prop
· exact .of_forall fun x ↦ ENNReal.rpow_lt_top_of_nonneg (by positivity) (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 69
} | {
"line": 95,
"column": 70
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable (-f) μ\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable (-f) μ\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 20
} | {
"line": 141,
"column": 21
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nf : α → E\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nhc : c ≠ 0\nh : AEStronglyMeasurable (c • f) μ\n⊢ AEStr... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝³ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : NormSMulClass 𝕜 E\nc : 𝕜\nf : α → E\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nhc : c ≠ 0\nh : AEStronglyMeasurable (c • f) μ\n⊢ AEStronglyMeasura... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 38
} | {
"line": 146,
"column": 39
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nf : α → E\nμ : Measure α\n⊢ lpNorm (n • f) p μ = n • lpNorm f p μ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toA... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nf : α → E\nμ : Measure α\n⊢ lpNorm (n • f) p μ = ↑n * lpNorm f p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 33
} | {
"line": 152,
"column": 34
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nn : ℕ\nf : α → 𝕜\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nn : ℕ\nf : α → 𝕜\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (↑n * f) p μ = ↑n * lpNorm f p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 29
} | {
"line": 159,
"column": 30
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nf : α → 𝕜\nn : ℕ\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\nm : MeasurableSpace α\n𝕜 : Type u_3\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace ℝ 𝕜\nf : α → 𝕜\nn : ℕ\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (f * ↑n) p μ = lpNorm f p μ * ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 182,
"column": 47
} | {
"line": 182,
"column": 58
} | {
"line": 182,
"column": 59
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\nhfg : MemLp (f + g) p μ\n⊢ MemLp g p μ",
"ppTerm": "?m.185",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\nhfg : MemLp (f + g) p μ\n⊢ MemLp g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 24
} | {
"line": 187,
"column": 25
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 30
} | {
"line": 191,
"column": 31
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instL... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\n⊢ lpNorm (f + -g) p μ ≤ lpNorm f p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (f - g) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 30
} | {
"line": 199,
"column": 31
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm g p μ + lpNorm (g - f) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm f p μ ≤ lpNorm (f + g) p μ + lpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g h : α → E\nhf : MemLp f p μ\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": []... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g h : α → E\nhf : MemLp f p μ\nhg : MemLp g p μ\nhp : 1 ≤ p\n⊢ lpNorm (f - h) p μ ≤ lpNorm (f - g) p μ + lpNorm (g - h) p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | {
"line": 215,
"column": 14
} | [
{
"pp": "case hb\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nι : Type u_3\ns : Finset ι\nf : ι → α → E\nhf : ∀ i ∈ s, MemLp (f i) p μ\nhp : 1 ≤ p\n⊢ ∑ i ∈ s, eLpNorm (f i) p μ ≠ ∞",
"ppTerm": "?hb",
"assigned": true,
"usedConstants": [
... | [
"case hb\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nι : Type u_3\ns : Finset ι\nf : ι → α → E\nhf : ∀ i ∈ s, MemLp (f i) p μ\nhp : 1 ≤ p\n⊢ ∀ x ∈ s, ¬eLpNorm (f x) p μ = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 22
} | {
"line": 240,
"column": 23
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.61",
"assigned": false,
"usedConstants": [... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 282,
"column": 2
} | {
"line": 284,
"column": 72
} | {
"line": 285,
"column": 2
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : SignedMeasure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ\nhadd : s = t + μ.withDensityᵥ f\nhtμ' : t ⟂ᵥ μ.toENNRealVectorMeasure\n⊢ t.toJordan... | [
"case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : SignedMeasure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ\nhadd : s = t + μ.withDensityᵥ f\nhtμ' : t ⟂ᵥ μ.toENNRealVectorMeasure\n⊢ t.toJordanDecompositio... | · have hfpos : Measurable fun x => ENNReal.ofReal (f x) := by fun_prop
refine eq_singularPart hfpos htμ.1 ?_
rw [toJordanDecomposition_eq_of_eq_add_withDensity hf hfi htμ' hadd] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 20
} | {
"line": 254,
"column": 21
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nf : α → E\nc : ℝ≥0\nhf : ¬AEStronglyMeasurable f μ\nhp₀ : p ≠ 0\nhc : c ≠ 0\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.113",
"assigned": fal... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nhp : p ≠ ∞\nf : α → E\nc : ℝ≥0\nhf : ¬AEStronglyMeasurable f μ\nhp₀ : p ≠ 0\nhc : c ≠ 0\nh : AEStronglyMeasurable f (c • μ)\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 95
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case pos.hf\nα : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable ((starRingEnd (α → K)) f) μ",
"ppTerm": "?pos.hf✝",
"assigned": true,
"usedConstants": [
"AEMeasurable.aestrongly... | [] | exact (continuous_star.measurable.comp_aemeasurable hf.aemeasurable).aestronglyMeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 264,
"column": 4
} | {
"line": 265,
"column": 11
} | {
"line": 265,
"column": 12
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ\n⊢ AEStronglyMeasurable f μ",
"ppTerm": "?m.106",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"α : Type u_1\nm : MeasurableSpace α\nK : Type u_3\ninst✝ : RCLike K\nf : α → K\np : ℝ≥0∞\nμ : Measure α\nhf : ¬AEStronglyMeasurable f μ\nh : AEStronglyMeasurable ((starRingEnd (α → K)) f) μ\n⊢ AEStronglyMeasurable f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 62
} | {
"line": 69,
"column": 63
} | [
{
"pp": "case refine_1\nΩ : Type u_1\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : MeasurableSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ... | [
"case refine_1\nΩ : Type u_1\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : MeasurableSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c\nf : Ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 63
} | {
"line": 70,
"column": 64
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : MeasurableSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ... | [
"case refine_2\nΩ : Type u_1\ninst✝⁴ : TopologicalSpace Ω\ninst✝³ : MeasurableSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\ninst✝¹ : L.IsCountablyGenerated\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nfs : ι → Ω →ᵇ ℝ≥0\nc : ℝ≥0\nfs_le_const : ∀ᶠ (i : ι) in L, ∀ᵐ (ω : Ω) ∂μ, (fs i) ω ≤ c\nf : Ω... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\nx : Ω\n⊢ ‖↑((thickenedIndicator δ_pos F) x)‖ ≤ 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"... | [
"Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\nx : Ω\n⊢ (thickenedIndicatorAux δ F x).toNNReal ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 33
} | {
"line": 143,
"column": 33
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝... | [
"Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nF_closed : IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] F\nδs : ℕ → ℝ\nδs_pos : ∀ (n : ℕ), 0 < δs n\nδs_lim : Tendsto δs atTop (𝓝 0)\nfs : ℕ ... | lintegral_coe_eq_integral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 395,
"column": 2
} | {
"line": 399,
"column": 44
} | {
"line": 401,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : SignedMeasure α\nμ : Measure α\ninst✝² : s.HaveLebesgueDecomposition μ\ninst✝¹ : t.HaveLebesgueDecomposition μ\ninst✝ : (s + t).HaveLebesgueDecomposition μ\n⊢ μ.withDensityᵥ ((s + t).rnDeriv μ) = μ.withDensityᵥ (s.rnDeriv μ + t.rnDeriv μ)",
"ppTerm": "?m.6... | [] | rw [← add_right_inj ((s + t).singularPart μ), singularPart_add_withDensity_rnDeriv_eq,
withDensityᵥ_add (integrable_rnDeriv _ _) (integrable_rnDeriv _ _), singularPart_add,
add_assoc, add_comm (t.singularPart μ), add_assoc, add_comm _ (t.singularPart μ),
singularPart_add_withDensity_rnDeriv_eq, ← add_assoc,... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 38
} | {
"line": 95,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : ¬m ≤ m0\n⊢ ∫ (x : α)... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α), |f x| ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 52
} | {
"line": 97,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : ¬Sigm... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\nhm : m ≤ m0\nhsig : ¬SigmaFinite (μ.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 38
} | {
"line": 117,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 49
} | {
"line": 119,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 52
} | {
"line": 121,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf :... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 38
} | {
"line": 132,
"column": 39
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : ¬m ≤ m0\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inn... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 49
} | {
"line": 134,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 52
} | {
"line": 136,
"column": 53
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : Integrable f μ\nhsig : ¬SigmaFinite (μ.trim hm)\n⊢ ∫ (x : α) in s, μ[f | m] x ∂μ ≤ ∫ (x : α) in s, f x ∂μ",
"ppTerm": "?pos✝",
"assigned":... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf : α → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhm : m ≤ m0\nhfint : Integrable f μ\nhsig : ¬SigmaFinite (μ.trim hm)\n⊢ 0 ≤ ∫ (x : α) in s, f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 66
} | {
"line": 134,
"column": 67
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 9
} | {
"line": 209,
"column": 10
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : ProbabilityMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFV... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : ProbabilityMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 13
} | {
"line": 212,
"column": 14
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\ns : Set Ω\n⊢ μ s ≤ 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\ns : Set Ω\n⊢ μ s ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 66
} | {
"line": 138,
"column": 67
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\ns✝ t : Set Ω\nμ ν : FiniteMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\ns✝ t : Set Ω\nμ ν : FiniteMeasure Ω\nh : (fun μ s ↦ (↑μ s).toNNReal) μ = (fun μ s ↦ (↑μ s).toNNReal) ν\ns : Set Ω\nx✝ : MeasurableSet s\n⊢ ↑μ s = ↑ν s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 48
} | {
"line": 227,
"column": 49
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : ProbabilityMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 49
} | {
"line": 208,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α), ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 186,
"column": 2
} | {
"line": 187,
"column": 9
} | {
"line": 187,
"column": 10
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : FiniteMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\nμ : FiniteMeasure Ω\nf : ι → Set Ω\n⊢ Tendsto (fun i ↦ μ (accumulate f i)) atTop (𝓝 (μ (⋃ i, f i)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 13
} | {
"line": 209,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ",
"ppTerm": "?neg✝",
"assigned": false,
"use... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α), ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α), ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 209,
"column": 55
} | {
"line": 209,
"column": 66
} | {
"line": 209,
"column": 67
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖ ^ 1) μ",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Norm.... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 48
} | {
"line": 227,
"column": 49
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : FiniteMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ ν : FiniteMeasure Ω\nh : ∀ (s : Set Ω), MeasurableSet s → μ s = ν s\ns : Set Ω\ns_mble : MeasurableSet s\n⊢ ↑μ s = ↑ν s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 43
} | {
"line": 216,
"column": 44
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : ¬m ≤ m0\n⊢ ∫ (x : α) i... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : ¬m ≤ m0\n⊢ 0 ≤ ∫ (x : α) in s, ‖f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 57
} | {
"line": 276,
"column": 58
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
... | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ ↑μ (⋃ n, f n) ≤ ∑' (a : ℕ), ↑μ (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 57
} | {
"line": 218,
"column": 58
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaF... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaFinite (μ.tri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 98
} | {
"line": 219,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : ¬SigmaF... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : SigmaFinite (μ.trim... | · simpa [condExp_of_not_sigmaFinite hm hsig, hp'] using integral_nonneg (fun x => by positivity) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 57
} | {
"line": 364,
"column": 58
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ μ (⋃ n, f n) ≤ ∑' (n : ℕ), μ (f n)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"_private.Mathlib.MeasureTheory.Measure.Finit... | [
"Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : FiniteMeasure Ω\nf : ℕ → Set Ω\nhf : Summable fun n ↦ μ (f n)\n⊢ ↑μ (⋃ n, f n) ≤ ∑' (a : ℕ), ↑μ (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 40
} | {
"line": 466,
"column": 41
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nle_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)\nω : Ω\nle' : f ω ≤ g ω + nndist f g\n⊢ ↑(f ω) ≤ ↑(g ω) + ↑(nndist f g)",
"ppTerm": "?m.128",
"assign... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\nle_dist : ∀ (ω : Ω), dist (f ω) (g ω) ≤ ↑(nndist f g)\nω : Ω\nle' : f ω ≤ g ω + nndist f g\n⊢ ↑(f ω) ≤ ↑(g ω + nndist f g)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 49
} | {
"line": 232,
"column": 50
} | [
{
"pp": "case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : ¬Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ",
"ppTer... | [
"case pos\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : ¬Integrable f μ\n⊢ 0 ≤ ∫ (x : α) in s, ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 36
} | {
"line": 472,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\n⊢ LipschitzWith μ.mass fun f ↦ μ.testAgainstNN f",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeasure.mass",
"Eq.mpr",
... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\n⊢ ∀ (x y : Ω →ᵇ ℝ≥0), dist (μ.testAgainstNN x) (μ.testAgainstNN y) ≤ ↑μ.mass * dist x y"
] | rw [lipschitzWith_iff_dist_le_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 13
} | {
"line": 233,
"column": 14
} | [
{
"pp": "case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ",
"ppTerm... | [
"case neg\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ ∫ (x : α) in s, ‖μ[f | m] x‖ ∂μ ≤ ∫ (x : α) in s, ‖f x‖ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 233,
"column": 61
} | {
"line": 233,
"column": 72
} | {
"line": 233,
"column": 73
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖ ^ 1) μ",
"ppTerm": "?m.88",
"assigned": true,
... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\ns : Set α\nhs : MeasurableSet s\nf : α → E\nhfint : Integrable f μ\n⊢ Integrable (fun x ↦ ‖f x‖) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 29
} | {
"line": 480,
"column": 30
} | [
{
"pp": "case left\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₂ ≤ μ.testAgainstNN f₁ + μ.mass * nndist f₂ f₁\n⊢ ↑(μ.testAgainstNN f₂) ≤ ↑(μ.testAgainstNN f₁) + ↑μ.mass * dist f₁ f₂",
"p... | [
"case left\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₂ ≤ μ.testAgainstNN f₁ + μ.mass * nndist f₂ f₁\n⊢ ↑(μ.testAgainstNN f₂) ≤ ↑(μ.testAgainstNN f₁) + ↑μ.mass * dist f₁ f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 15
} | {
"line": 484,
"column": 16
} | [
{
"pp": "case right\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₁ ≤ μ.testAgainstNN f₂ + μ.mass * nndist f₁ f₂\n⊢ ↑(μ.testAgainstNN f₁) ≤ ↑(μ.testAgainstNN f₂) + ↑μ.mass * dist f₁ f₂",
"... | [
"case right\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\nkey : μ.testAgainstNN f₁ ≤ μ.testAgainstNN f₂ + μ.mass * nndist f₁ f₂\n⊢ ↑(μ.testAgainstNN f₁) ≤ ↑(μ.testAgainstNN f₂) + ↑μ.mass * dist f₁ f₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 791,
"column": 8
} | {
"line": 791,
"column": 19
} | {
"line": 791,
"column": 20
} | [
{
"pp": "case pos.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i... | [
"case pos.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 793,
"column": 8
} | {
"line": 793,
"column": 19
} | {
"line": 793,
"column": 20
} | [
{
"pp": "case neg.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i... | [
"case neg.h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indica... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 35
} | {
"line": 123,
"column": 36
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E",
"pp... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 35
} | {
"line": 126,
"column": 36
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\ni : ι\n⊢ (μs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 21
} | {
"line": 132,
"column": 22
} | [
{
"pp": "case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ =... | [
"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : limsup (fun i ↦ (μs i) E) L ≤ μ E\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 35
} | {
"line": 147,
"column": 36
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E",
"pp... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\n⊢ μ Eᶜ = 1 - μ E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 132,
"column": 2
} | {
"line": 142,
"column": 33
} | {
"line": 144,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝⁴ : TopologicalSpace 𝓧\ninst✝³ : TopologicalSpace 𝓨\ninst✝² : MeasurableSpace 𝓨\ninst✝¹ : OpensMeasurableSpace 𝓨\ninst✝ : T2Space 𝓨\nhS : IsTightMeasureSet S\nf : 𝓧 → 𝓨\nhf : Continuous[inst✝⁴, inst✝³] f\n⊢ IsTigh... | [] | rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le] at hS ⊢
simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
intro ε hε
obtain ⟨K, hK_compact, hKS⟩ := hS ε hε
refine ⟨f '' K, hK_compact.image hf, fun μ hμS ↦ ?_⟩
by_cases hf_meas : AEMeasurable f μ
swap; · simp [Measure.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 132,
"column": 2
} | {
"line": 142,
"column": 33
} | {
"line": 144,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝⁴ : TopologicalSpace 𝓧\ninst✝³ : TopologicalSpace 𝓨\ninst✝² : MeasurableSpace 𝓨\ninst✝¹ : OpensMeasurableSpace 𝓨\ninst✝ : T2Space 𝓨\nhS : IsTightMeasureSet S\nf : 𝓧 → 𝓨\nhf : Continuous[inst✝⁴, inst✝³] f\n⊢ IsTigh... | [] | rw [isTightMeasureSet_iff_exists_isCompact_measure_compl_le] at hS ⊢
simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂]
intro ε hε
obtain ⟨K, hK_compact, hKS⟩ := hS ε hε
refine ⟨f '' K, hK_compact.image hf, fun μ hμS ↦ ?_⟩
by_cases hf_meas : AEMeasurable f μ
swap; · simp [Measure.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 35
} | {
"line": 150,
"column": 36
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\n... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\ni : ι\n⊢ (μs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 761,
"column": 2
} | {
"line": 765,
"column": 34
} | {
"line": 767,
"column": 0
} | [
{
"pp": "case refine_2\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nγ : Type u_2\n𝕜 : Type u_3\ninst✝ : RCLike 𝕜\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμ : FiniteMeasure Ω\nh : ∀ (f : Ω →ᵇ 𝕜), Tendsto (fun i ↦ ∫ (ω : Ω), f ω ∂↑(μs i)) F (𝓝 (∫ (ω ... | [] | · specialize h ((RCLike.ofRealAm (K := 𝕜)).compLeftContinuousBounded ℝ
RCLike.lipschitzWith_ofReal f)
simp only [AlgHom.compLeftContinuousBounded_apply_apply, RCLike.ofRealAm_coe,
integral_ofReal] at h
exact tendsto_ofReal_iff'.mp h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 21
} | {
"line": 156,
"column": 22
} | [
{
"pp": "case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ =... | [
"case inr\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nE : Set Ω\nE_mble : MeasurableSet E\nh : μ E ≤ liminf (fun i ↦ (μs i) E) L\nhne : L.NeBot\nmeas_Ec : μ Eᶜ = 1 - μ E\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 27
} | {
"line": 157,
"column": 28
} | [
{
"pp": "case h₁\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, ... | [
"case h₁\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 829,
"column": 2
} | {
"line": 859,
"column": 96
} | {
"line": 861,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nhfu : UniformIntegrable f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indic... | [] | obtain ⟨-, hfu, M, hM⟩ := hfu
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ { x | C ≤ ‖f i x‖₊ } ≤ ENNReal.ofReal δ := by
by_contra! hcon
choose ℐ hℐ using hcon
lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ eLpNorm (f (ℐ C)) p μ := by
int... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 829,
"column": 2
} | {
"line": 859,
"column": 96
} | {
"line": 861,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nhfu : UniformIntegrable f p μ\nε : ℝ\nhε : 0 < ε\n⊢ ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indic... | [] | obtain ⟨-, hfu, M, hM⟩ := hfu
obtain ⟨δ, hδpos, hδ⟩ := hfu hε
obtain ⟨C, hC⟩ : ∃ C : ℝ≥0, ∀ i, μ { x | C ≤ ‖f i x‖₊ } ≤ ENNReal.ofReal δ := by
by_contra! hcon
choose ℐ hℐ using hcon
lift δ to ℝ≥0 using hδpos.le
have : ∀ C : ℝ≥0, C • (δ : ℝ≥0∞) ^ (1 / p.toReal) ≤ eLpNorm (f (ℐ C)) p μ := by
int... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 27
} | {
"line": 160,
"column": 28
} | [
{
"pp": "case h₂\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, ... | [
"case h₂\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\ninst✝¹ : TopologicalSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\ninst✝ : TopologicalSpace 𝓨\nμ : Set (Measure (𝓧 × 𝓨))\nhμ₁ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ∧ ∀ μ_1 ∈ Measure.fst '' μ, μ_1 Kᶜ ≤ ε\nhμ₂ : ∀ (ε : ENNReal), 0 < ε → ∃ K, IsCompact K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 21
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (κ a) (a✝¹ i)\na : α... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (κ a) (a✝¹ i)\na : α\nha : κ a =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 21
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (η a) (a✝¹ i)\na : α... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝² : Finset ι\na✝¹ : ι → Set Ω\na✝ : ∀ i ∈ a✝², a✝¹ i ∈ π i\nh' : ∀ᵐ (a : α) ∂μ, (η a) (⋂ i ∈ a✝², a✝¹ i) = ∏ i ∈ a✝², (η a) (a✝¹ i)\na : α\nha : κ a =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | {
"line": 147,
"column": 21
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ * (κ a) a✝²\na : α\nha : κ a = η a\nh'a : (κ a) (a✝³ ∩ a... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (κ a) (a✝³ ∩ a✝²) = (κ a) a✝³ * (κ a) a✝²\na : α\nha : κ a = η a\nh'a : (κ a) (a✝³ ∩ a✝²) = (κ a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 20
} | {
"line": 147,
"column": 21
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (η a) (a✝³ ∩ a✝²) = (η a) a✝³ * (η a) a✝²\na : α\nha : κ a = η a\nh'a : (η a) (a✝³ ∩ a... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ η : Kernel α Ω\nμ : Measure α\ns1 s2 : Set (Set Ω)\nh : ⇑κ =ᵐ[μ] ⇑η\na✝³ a✝² : Set Ω\na✝¹ : a✝³ ∈ s1\na✝ : a✝² ∈ s2\nh' : ∀ᵐ (a : α) ∂μ, (η a) (a✝³ ∩ a✝²) = (η a) a✝³ * (η a) a✝²\na : α\nha : κ a = η a\nh'a : (η a) (a✝³ ∩ a✝²) = (η a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 905,
"column": 4
} | {
"line": 905,
"column": 15
} | {
"line": 905,
"column": 16
} | [
{
"pp": "case refine_2.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nhf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nε : ... | [
"case refine_2.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nhf₃ : ∃ C, ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nε : ℝ\nhε : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 23
} | {
"line": 179,
"column": 24
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : iIndepSets π κ μ\na : α\nha : (κ a) (⋂ i ∈ ∅, univ) = ∏ i ∈ ∅, (κ a) univ\n⊢ (κ a) univ = 1",
"ppTerm": "?m.54",
"assigned": false,
"usedConsta... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nh : iIndepSets π κ μ\na : α\nha : (κ a) (⋂ i ∈ ∅, univ) = ∏ i ∈ ∅, (κ a) univ\n⊢ (κ a) univ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 357,
"column": 70
} | {
"line": 357,
"column": 81
} | {
"line": 357,
"column": 82
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : HasOuterApproxClosed Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : μ (frontier E) = 0\n⊢ ↑μ (fr... | [
"Ω : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : HasOuterApproxClosed Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : μ (frontier E) = 0\n⊢ ↑μ (frontier ?m.34... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 28
} | {
"line": 410,
"column": 29
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : SFinite μ\ns : Set Ω\na b : ℝ\nhab : a < b\nmbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Function.onFun Disjoint fun r ↦ frontier (Me... | [
"Ω : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : SFinite μ\ns : Set Ω\na b : ℝ\nhab : a < b\nmbles : ∀ (r : ℝ), MeasurableSet (frontier (Metric.thickening r s))\ndisjs : Pairwise (Function.onFun Disjoint fun r ↦ frontier (Metric.thicken... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 16
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nμ : Measure α\ninst✝¹ : Subsingleton ι\nm : ι → Set (Set Ω)\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\ns : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ s, f i ∈ m i\n⊢ s = ∅ ∨ ∃ i, s = {i}",
"ppTerm": "?m.37",
"a... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nμ : Measure α\ninst✝¹ : Subsingleton ι\nm : ι → Set (Set Ω)\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\ns : Finset ι\nf : ι → Set Ω\nhf : ∀ i ∈ s, f i ∈ m i\n⊢ s = ∅ ∨ ∃ i, s = {i}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 914,
"column": 4
} | {
"line": 914,
"column": 15
} | {
"line": 914,
"column": 16
} | [
{
"pp": "case refine_3.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nC : ℝ≥0\nhC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nn... | [
"case refine_3.inr\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhp : 1 ≤ p\nf : ℕ → α → E\nhf₁ : ∀ (i : ℕ), AEStronglyMeasurable (f i) μ\nhf₂ : UnifIntegrable f p μ\nC : ℝ≥0\nhC : ∀ (i : ℕ), eLpNorm (f i) p μ ≤ ↑C\nn : ℕ\nhn : n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 941,
"column": 2
} | {
"line": 941,
"column": 13
} | {
"line": 941,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : TendstoInMeasure μ f u g\n⊢ MemLp g p μ",
"ppTer... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : TendstoInMeasure μ f u g\n⊢ MemLp g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 85
} | {
"line": 222,
"column": 86
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nι' : Type u_5\ng : ι' → ι\nhg : Function.Injective g\nh : iIndepSets π κ μ\ns : Finset ι'\nf : ι' → Set Ω\nhf : ∀ i ∈ s, f i ∈ (π ∘ g) i\nf' : ι → Set Ω := Fun... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nπ : ι → Set (Set Ω)\nι' : Type u_5\ng : ι' → ι\nhg : Function.Injective g\nh : iIndepSets π κ μ\ns : Finset ι'\nf : ι' → Set Ω\nhf : ∀ i ∈ s, f i ∈ (π ∘ g) i\nf' : ι → Set Ω := Function.extend... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 970,
"column": 2
} | {
"line": 970,
"column": 13
} | {
"line": 970,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))\... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup β\np : ℝ≥0∞\nκ : Type u_4\nu : Filter κ\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : κ → α → β\ng : α → β\nhUI : UniformIntegrable f p μ\nhtends : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) u (𝓝 (g x))\n⊢ MemLp g p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 270,
"column": 7
} | {
"line": 270,
"column": 22
} | {
"line": 270,
"column": 23
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns₁ s₂ : Set (Set Ω)\nh : IndepSets s₁ s₂ κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\na : α\nha : (κ a) (t2 ∩ t1) = (κ a) t2 * (κ a) t1\n⊢ (κ a) (t1 ∩ t2) = (κ a) t1 * (κ a) t2",
"ppTer... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns₁ s₂ : Set (Set Ω)\nh : IndepSets s₁ s₂ κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ s₂\nht2 : t2 ∈ s₁\na : α\nha : (κ a) (t2 ∩ t1) = (κ a) t2 * (κ a) t1\n⊢ (κ a) (t2 ∩ t1) = (κ a) t1 * (κ a) t2"
] | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 17
} | {
"line": 170,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nπ : ι → Set (Set Ω)\nx✝ : MeasurableSpace Ω\nμ : Measure Ω\nh : iIndepSets π μ\n⊢ μ univ = 1",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nι : Type u_2\nπ : ι → Set (Set Ω)\nx✝ : MeasurableSpace Ω\nμ : Measure Ω\nh : iIndepSets π μ\n⊢ μ univ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 13
} | {
"line": 586,
"column": 14
} | [
{
"pp": "Ω : Type u_1\n_mΩ : MeasurableSpace Ω\ns t : Set Ω\nμ : Measure Ω\nh : IndepSet s t μ\n⊢ μ (s ∩ t) = μ s * μ t",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\n_mΩ : MeasurableSpace Ω\ns t : Set Ω\nμ : Measure Ω\nh : IndepSet s t μ\n⊢ μ (s ∩ t) = μ s * μ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Basic | {
"line": 617,
"column": 2
} | {
"line": 617,
"column": 13
} | {
"line": 617,
"column": 14
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : ι → Set Ω\nh : iIndepSet f μ\ns : Finset ι\n⊢ μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nf : ι → Set Ω\nh : iIndepSet f μ\ns : Finset ι\n⊢ μ (⋂ i ∈ s, f i) = ∏ i ∈ s, μ (f i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 494,
"column": 38
} | {
"line": 494,
"column": 77
} | {
"line": 494,
"column": 77
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm₂ m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh2 : m₂ ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m₂ = generateFrom p2\nhyp : IndepSets p1 p2 κ μ\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht1m : MeasurableSet t1\nh : IsM... | measure_iUnion hfd fun i ↦ h2 _ (hfm i) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 791,
"column": 6
} | {
"line": 791,
"column": 17
} | {
"line": 791,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nβ : ι → Type u_10\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh : iIndepFun f μ\n⊢ μ univ = 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nβ : ι → Type u_10\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nh : iIndepFun f μ\n⊢ μ univ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 522,
"column": 6
} | {
"line": 522,
"column": 21
} | {
"line": 522,
"column": 22
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : Indep... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 κ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 527,
"column": 8
} | {
"line": 527,
"column": 23
} | {
"line": 527,
"column": 24
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : Indep... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 κ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 858,
"column": 2
} | {
"line": 858,
"column": 18
} | {
"line": 858,
"column": 19
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : Fintype ι\nβ : ι → Type u_11\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂... | [
"Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : Fintype ι\nβ : ι → Type u_11\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nh :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → μ (⋂ i ∈ S, f i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 651,
"column": 67
} | {
"line": 651,
"column": 78
} | {
"line": 651,
"column": 79
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nh : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (↑(μs i)).real F) L ≤ (↑μ).rea... | [
"Ω : Type u_1\nι : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nh : ∀ (F : Set Ω), IsClosed[inst✝²] F → limsup (fun i ↦ (↑(μs i)).real F) L ≤ (↑μ).real F\n⊢ ∀ (F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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