module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Jensen | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 47
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_4\nι : Type u_5\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst... | [] | · grind [hf.map_sum_eq_iff_of_pos h₀ h₁ hmem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 27
} | {
"line": 253,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁷ : SeminormedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : SeminormedRing 𝕜₂\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\nι : Type u_11\ninst✝ : None... | [
"case mp\n𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁷ : SeminormedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : SeminormedRing 𝕜₂\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\nι : Type u_11\ninst✝ : Nonempt... | constructor <;> intro h i | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 809,
"column": 2
} | {
"line": 810,
"column": 29
} | {
"line": 810,
"column": 30
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p LinearMap.id\n⊢ p.moduleFilterBasis.topology =... | [
"case refine_1\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p LinearMap.id\n⊢ Continuous[p.moduleFilterBa... | refine le_antisymm ?_ ?_ <;>
rw [← continuous_id_iff_le] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Probability.UniformOn | {
"line": 141,
"column": 25
} | {
"line": 145,
"column": 23
} | {
"line": 147,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\nht : s ⊆ t\n⊢ (uniformOn s) t = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
... | [] | by
haveI := isProbabilityMeasure_uniformOn hs hs'
refine eq_of_le_of_not_lt prob_le_one ?_
rw [not_lt, ← uniformOn_self hs hs']
exact measure_mono ht | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.UniformOn | {
"line": 153,
"column": 21
} | {
"line": 153,
"column": 53
} | {
"line": 154,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns t : Set Ω\nhsf : s.Finite\nh : #⋯.toFinset = #hsf.toFinset\nthis : s ∩ t = s\n⊢ s ⊆ t",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Eq.rec",
"LE.le",
"Set.... | [] | by exact this ▸ fun x hx => hx.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 267,
"column": 29
} | {
"line": 267,
"column": 64
} | {
"line": 268,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\nC : (i : ι) → Set (Set (α i))\nhC : ∀ (i : ι), generateFrom (C i) = inst✝ i\nh2C : ∀ (i : ι), IsPiSystem (C i)\nh3C : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)\nμν : Measure ((i : ... | [] | exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 23
} | {
"line": 86,
"column": 24
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\ns : Set α\nc : ε\nhμs : μ s ≠ 0\nh : eLpNormEssSup (s.indicator fun x ↦ c) μ < ‖c‖ₑ\nh' : μ {a | ‖c‖ₑ ≤ ‖s.indicator (fun x ↦ c) a‖ₑ} = 0\nx : α\nhx_mem : x ∈ s\n⊢ x ∈ {a | ‖c... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\ns : Set α\nc : ε\nhμs : μ s ≠ 0\nh : eLpNormEssSup (s.indicator fun x ↦ c) μ < ‖c‖ₑ\nh' : μ {a | ‖c‖ₑ ≤ ‖s.indicator (fun x ↦ c) a‖ₑ} = 0\nx : α\nhx_mem : x ∈ s\n⊢ ‖c‖ₑ ≤ ‖s.indicator (fu... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 35
} | {
"line": 82,
"column": 35
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c * ‖... | ← lintegral_const_mul' _ _ this | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 471,
"column": 26
} | {
"line": 471,
"column": 57
} | {
"line": 471,
"column": 57
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhq_pos : 0 < q\nh0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ eLpNorm' (fun x ↦ ‖f x‖ₑ ^ q) p.toReal μ = eLpNorm' f (p.toReal * (ENNReal.ofReal q).toReal) μ ^ q",
"ppTerm": "?neg✝",
"assi... | [
"case neg\nα : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhq_pos : 0 < q\nh0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ eLpNorm' (fun x ↦ ‖f x‖ₑ ^ q) p.toReal μ = eLpNorm' f (p.toReal * q) μ ^ q"
] | ENNReal.toReal_ofReal hq_pos.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 41,
"column": 58
} | {
"line": 60,
"column": 58
} | {
"line": 62,
"column": 0
} | [
{
"pp": "⊢ StrictConvexOn ℝ univ rexp",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toN... | [] | by
apply strictConvexOn_of_slope_strict_mono_adjacent convex_univ
rintro x y z - - hxy hyz
trans exp y
· have h1 : 0 < y - x := by linarith
have h2 : x - y < 0 := by linarith
rw [div_lt_iff₀ h1]
calc
exp y - exp x = exp y - exp y * exp (x - y) := by rw [← exp_add]; ring_nf
_ = exp y * (1... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 6
} | {
"line": 478,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\n⊢ eLpNorm (fun x ↦ ‖f x‖ ^ q) p μ = eLpNorm (fun x ↦ ‖f x‖ₑ ^ q) p μ",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"α : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\n⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ q) p μ = eLpNorm (fun x ↦ ‖f x‖ ^ q) p μ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 479,
"column": 27
} | {
"line": 479,
"column": 40
} | {
"line": 479,
"column": 40
} | [
{
"pp": "case e'_2.e'_5\nα : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\nx✝ : α\n⊢ ‖f x✝‖ₑ ^ q = ENNReal.ofReal (‖f x✝‖ ^ q)",
"ppTerm": "?e'_2.e'_5",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"case e'_2.e'_5\nα : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\nx✝ : α\n⊢ ENNReal.ofReal ‖f x✝‖ ^ q = ENNReal.ofReal (‖f x✝‖ ^ q)"
] | ← ofReal_norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 108,
"column": 20
} | {
"line": 108,
"column": 91
} | {
"line": 109,
"column": 2
} | [
{
"pp": "s : ℝ\nhs✝ : -1 ≤ s\np : ℝ\nhp : 1 < p\nhp' : 0 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs' : 1 + p * s = 1\n⊢ s = 0",
"ppTerm": "?m.201",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"False",
"Real.partialOrder",
... | [] | rwa [add_eq_left, mul_eq_zero, eq_false_intro hp'.ne', false_or] at hs' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.Real.ConjExponents | {
"line": 359,
"column": 89
} | {
"line": 361,
"column": 24
} | {
"line": 363,
"column": 0
} | [
{
"pp": "p q : ℝ≥0\n⊢ p.HolderConjugate q ↔ 1 < p ∧ p⁻¹ + q⁻¹ = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"congrArg",
"Real.instInv",
"NNReal.coe_lt_coe._simp_1",... | [] | by
rw [← holderConjugate_coe_iff, Real.holderConjugate_iff, ← coe_one]
exact_mod_cast Iff.rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.ConjExponents | {
"line": 493,
"column": 6
} | {
"line": 493,
"column": 26
} | {
"line": 493,
"column": 26
} | [
{
"pp": "p q : ℝ≥0∞\nh : p.toReal.HolderConjugate q.toReal\n⊢ p.HolderConjugate q",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"Real.HolderTriple",
"Eq.mp",
"Real.HolderConjugate.eq_1",
"Real.instOne",
"ENNReal.toReal",
... | [
"p q : ℝ≥0∞\nh : p.toReal.HolderTriple q.toReal 1\n⊢ p.HolderConjugate q"
] | Real.HolderConjugate | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.ConjExponents | {
"line": 557,
"column": 2
} | {
"line": 558,
"column": 87
} | {
"line": 560,
"column": 0
} | [
{
"pp": "case inr.inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\nhq : q ≠ ∞\n⊢ p * q = p + q",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"add_mul",
"Distrib.leftDistribClass",
"Eq.mpr",
"ENNReal.instAdd",
"False",
"HMul.hMul",
"eq_fal... | [] | simpa [add_comm p, mul_add, add_mul, hp, hq, ne_zero p q, ne_zero q p, ENNReal.mul_inv_cancel,
ENNReal.inv_mul_cancel_right] using congr(p * $((inv_add_inv_eq_one p q).symm) * q) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Slope | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 21
} | {
"line": 269,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\n⊢ StrictMonoOn f (s ∩ Set.Ici y)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Pr... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f x < f y\nu : 𝕜\nhu : u ∈ s ∩ Set.Ici y\nv : 𝕜\nhv : v ∈ s ∩ Set.Ici y\nhuv : u < v\n⊢ f u < f v"
] | intro u hu v hv huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 238,
"column": 20
} | {
"line": 238,
"column": 28
} | {
"line": 238,
"column": 29
} | [
{
"pp": "t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ (1 + t) / 2 * rexp x + (1 - t) / 2 * rexp (-x) = cosh x + t * sinh x",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"Rea... | [
"t : ℝ\nht : -1 ≤ t ∧ t ≤ 1\nx : ℝ\n⊢ (1 + t) / 2 * rexp x + (1 - t) / 2 * rexp (-x) = (rexp x + rexp (-x)) / 2 + t * sinh x"
] | cosh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Monovary | {
"line": 340,
"column": 97
} | {
"line": 341,
"column": 52
} | {
"line": 342,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : Semifield α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Semifield β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\nf : ι → α\ng : ι → β\nhf : StrongLT 0 f\nhg : StrongLT 0 g\n⊢ Monovary f⁻¹ g⁻¹ ↔ Monovary f g",
"ppTer... | [] | by
rw [monovary_inv_left₀ hf, antivary_inv_right₀ hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 24
} | {
"line": 236,
"column": 25
} | [
{
"pp": "case refine_2\nι : Type u\ns : Finset ι\nw z : ι → ℝ≥0∞\nhw' : ∑ i ∈ s, w i = 1\np : ℝ\nhp : 1 ≤ p\nhp_pos : 0 < p\nhp_nonneg : 0 ≤ p\nhp_not_neg : ¬p < 0\nh_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞\n⊢ (∑ i ∈ s, w i * z i) ^ p ≠ ∞ →\n ∑ i ∈ s, w i * z i ^ p ≠ ∞ → ((∑ i ∈ s, w i ... | [
"case refine_2\nι : Type u\ns : Finset ι\nw z : ι → ℝ≥0∞\nhw' : ∑ i ∈ s, w i = 1\np : ℝ\nhp : 1 ≤ p\nhp_pos : 0 < p\nhp_nonneg : 0 ≤ p\nhp_not_neg : ¬p < 0\nh_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞\nh_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞\n⊢ ∑ i ∈ s, w i * z i ^ p ≠ ∞ → ((∑ i ∈ s, w i *... | intro h_top_rpow_sum | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.Convex.Mul | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 37
} | {
"line": 183,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\n⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ -[n+1]",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"DivInvMonoid.toInv",
"instSMulOfMul",
"Div... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nn : ℕ\n⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x⁻¹ ^ (n + 1)"
] | simp_rw [zpow_negSucc, ← inv_pow] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 17
} | {
"line": 338,
"column": 2
} | [
{
"pp": "p : ℝ≥0∞\n⊢ p.LpAddConst < ∞",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Set.decidableMemIoo",
"Eq.mpr",
"Real",
"Preorder.toLT",
"instHDiv",
"congrArg",
"ENNReal.instPowReal",
"ENNReal.LpAddConst.eq_1",
"Real.instDivInvMon... | [
"p : ℝ≥0∞\n⊢ (if p ∈ Set.Ioo 0 1 then 2 ^ (1 / p.toReal - 1) else 1) < ∞"
] | rw [LpAddConst] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 64
} | {
"line": 59,
"column": 2
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nq : ℝ\nhq_pos : 0 < q\nh_nnnorm_le_eLpNorm_ess_sup : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ eLpNormEssSup f μ\n⊢ ∀ᵐ (a : α) ∂μ, ‖f a‖ₑ ^ q ≤ eLpNormEssSup f μ ^ q",
"ppTerm": "?m.69... | [] | exact h_nnnorm_le_eLpNorm_ess_sup.mono fun x hx => by gcongr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 20
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhpq : p ≤ q\nhfq_m : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm f q μ < ∞\nhp0 : p ≠ 0\n⊢ MemLp f p μ",
"ppTerm"... | [
"case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhpq : p ≤ q\nhfq_m : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm f q μ < ∞\nhp0 : p ≠ 0\n⊢ eLpNorm f p μ < ∞"
] | refine ⟨hfq_m, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 272,
"column": 31
} | {
"line": 272,
"column": 72
} | {
"line": 273,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nhp1 : 1 ≤ p\n⊢ ∫⁻ (a : α), 2 ^ (p - 1) * f a ^ p ∂μ + ∫⁻ (a : α), 2 ^ (p - 1) * g a ^ p ∂μ < ∞",
"ppTerm": "?m.141",
"a... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nhp1 : 1 ≤ p\n⊢ 2 ^ (p - 1) * ∫⁻ (a : α), f a ^ p ∂μ + ∫⁻ (a : α), 2 ^ (p - 1) * g a ^ p ∂μ < ∞",
"case hf\nα : Type u_1\ninst✝ : Measurab... | lintegral_const_mul'' _ (hf.pow_const p), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"instHDiv... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"instHDiv... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p / ENNReal.ofReal p + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"instHDiv... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"inst... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"inst... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q / ENNReal.ofReal q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"DivInvMonoid.toInv",
"inst... | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 30
} | {
"line": 557,
"column": 4
} | [
{
"pp": "case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\na b : ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nh : a = ∞ ∨ b = ∞\n⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹"
] | rw [div_eq_mul_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 303,
"column": 8
} | {
"line": 303,
"column": 12
} | {
"line": 304,
"column": 8
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | [
"α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := ⋯\nq2 : ℝ := ⋯\nhp2... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.MeanInequalities | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 57
} | {
"line": 619,
"column": 2
} | [
{
"pp": "case inl\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : ∑ i ∈ s, f i ^ p = 0\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Mean... | [
"case inr\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : 0 < ∑ i ∈ s, f i ^ p\n⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)"
] | · exact inner_le_Lp_mul_Lp_of_norm_eq_zero s f g hpq hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 336,
"column": 6
} | {
"line": 339,
"column": 11
} | {
"line": 340,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nh_add_m : AEMeasurable (fun a ↦ (f + g) a ^ (p - 1)) μ\n⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ =\n ∫⁻ (a : α), f a * (f + g) a ^ (p - 1)... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nh_add_m : AEMeasurable (fun a ↦ (f + g) a ^ (p - 1)) μ\nh_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p -... | have h_add_apply :
(∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 39
} | {
"line": 240,
"column": 4
} | [
{
"pp": "case inr.inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\... | [
"case inr.inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F →... | have : r = p := by simpa using hpqr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 781,
"column": 4
} | {
"line": 781,
"column": 61
} | {
"line": 782,
"column": 4
} | [
{
"pp": "case left\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) ∈ (fun g ↦ ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1}",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
... | [
"case h\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ (fun i ↦ f i ^ p / f i / (∑ i ∈ s, f i ^ p) ^ (1 / q)) ∈ {g | ∑ i ∈ s, g i ^ q ≤ 1} ∧\n ((fun g ↦ ∑ i ∈ s, f i * g i) fun i ↦ f i ^ p / f i / (∑ i ∈ s, f i ^ p) ^ (1 / q)) = (∑ i ∈ s, f i ^ p) ^ (1 / p)"
] | use fun i => f i ^ p / f i / (∑ i ∈ s, f i ^ p) ^ (1 / q) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 349,
"column": 61
} | {
"line": 350,
"column": 44
} | {
"line": 352,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\n𝕜 : Type u_3\nx✝ : MeasurableSpace α\ninst✝ : NormedCommRing 𝕜\nμ : Measure α\nf : ι → α → 𝕜\np : ι → ℝ≥0∞\ns : Finset ι\nhf : ∀ i ∈ s, MemLp (f i) (p i) μ\n⊢ MemLp (fun ω ↦ ∏ i ∈ s, f i ω) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ",
"ppTerm": "?m.32",
"assigned": true,
"usedCon... | [] | by
simpa [Finset.prod_fn] using MemLp.prod hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 8
} | {
"line": 396,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nh1 : ¬p = 1\n⊢ 1 ≠ p",
"ppTerm": "?m.177",
"assigned": true,
"u... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nh1 : ¬p = 1\n⊢ p ≠ 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 36
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SeminormedAddGroup E\nf : ℕ → α → E\np : ℝ\nhp_pos : 0 < p\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))\nh_pow_liminf : liminf (fun n ↦ eLpNorm'... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SeminormedAddGroup E\nf : ℕ → α → E\np : ℝ\nhp_pos : 0 < p\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nh_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))\nh_pow_liminf : liminf (fun n ↦ eLpNorm' (f n) p μ) ... | inv_mul_cancel₀ hp_pos.ne.symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 95,
"column": 2
} | {
"line": 97,
"column": 20
} | {
"line": 98,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : SeminormedAddGroup E\nι : Type u_3\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nC : ℝ≥0∞\nbound : ∀ᶠ (n : ι) in u, eLpNorm (f n) p μ ≤ C\nhf : ∀ (n : ι), AEStronglyMeasurab... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : SeminormedAddGroup E\nι : Type u_3\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nC : ℝ≥0∞\nbound : ∀ᶠ (n : ι) in u, eLpNorm (f n) p μ ≤ C\nhf : ∀ (n : ι), AEStronglyMeasurable (f n) μ\n... | have : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f (v n) x) atTop (𝓝 (g x)) := by
filter_upwards [h_tendsto] with x hx
exact hx.comp hv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 47
} | [
{
"pp": "α✝ : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE✝ : Type u_4\nF : Type u_5\nm✝ : MeasurableSpace α✝\np✝ : ℝ≥0∞\nμ✝ : Measure α✝\ninst✝² : NormedAddCommGroup E✝\ninst✝¹ : NormedAddCommGroup F\nα : Type ?u.18\nE : Type u_6\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf : ... | [
"α✝ : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE✝ : Type u_4\nF : Type u_5\nm✝ : MeasurableSpace α✝\np✝ : ℝ≥0∞\nμ✝ : Measure α✝\ninst✝² : NormedAddCommGroup E✝\ninst✝¹ : NormedAddCommGroup F\nα : Type ?u.18\nE : Type u_6\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf : α →ₘ[μ] E\nh... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 24
} | {
"line": 176,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : EDist E\nl : Filter ι\nf f' : ι → α → E\ng g' : α → E\nh_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i\nh_right : g =ᵐ[μ] g'\nh_tendsto : TendstoInMeasure μ f l g\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist ... | [
"α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : EDist E\nl : Filter ι\nf f' : ι → α → E\ng g' : α → E\nh_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i\nh_right : g =ᵐ[μ] g'\nh_tendsto : TendstoInMeasure μ f l g\nε : ℝ≥0∞\nhε : 0 < ε\n⊢ (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) =ᶠ... | suffices
(fun i ↦ μ { x | ε ≤ edist (f' i x) (g' x) }) =ᶠ[l] fun i ↦ μ { x | ε ≤ edist (f i x) (g x) } by
rw [tendsto_congr' this]
exact h_tendsto ε hε | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.MeanInequalities | {
"line": 1063,
"column": 2
} | {
"line": 1068,
"column": 46
} | {
"line": 1070,
"column": 0
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np : ℝ\nhp : 1 ≤ p\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ p\n⊢ (Summable fun i ↦ (f i + g i) ^ p) ∧\n (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ ... | [] | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 1063,
"column": 2
} | {
"line": 1068,
"column": 46
} | {
"line": 1070,
"column": 0
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np : ℝ\nhp : 1 ≤ p\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ p\n⊢ (Summable fun i ↦ (f i + g i) ^ p) ∧\n (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ ... | [] | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast0 at *
exact NNReal.Lp_add_le_tsum hp hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 221,
"column": 13
} | {
"line": 221,
"column": 26
} | {
"line": 221,
"column": 26
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nhn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i\na : α\n⊢ ‖∑ i ∈ Finse... | [
"case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nhn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i\na : α\n⊢ ENNReal.ofReal ‖∑ i ∈ Fi... | ← ofReal_norm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpOrder | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 27
} | {
"line": 57,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup E\ninst✝¹ : PartialOrder E\ninst✝ : IsOrderedAddMonoid E\nf g₁ g₂ : ↥(Lp E p μ)\nhg₁₂ : g₁ ≤ g₂\n⊢ f + g₁ ≤ f + g₂",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Measure... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup E\ninst✝¹ : PartialOrder E\ninst✝ : IsOrderedAddMonoid E\nf g₁ g₂ : ↥(Lp E p μ)\nhg₁₂ : ↑↑g₁ ≤ᵐ[μ] ↑↑g₂\n⊢ ↑↑(f + g₁) ≤ᵐ[μ] ↑↑(f + g₂)"
] | rw [← coeFn_le] at hg₁₂ ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 23
} | {
"line": 321,
"column": 24
} | [
{
"pp": "case h\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k ↦ ... | [
"case h\nα : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 21
} | {
"line": 317,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhp : 1 ≤ p\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - ... | [
"α : Type u_1\nm✝ : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhp : 1 ≤ p\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m : ℕ), N ≤ n → N ≤ m → eLpNorm (f n - f m) p μ < B N\... | intro N n m hn hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 132,
"column": 80
} | {
"line": 138,
"column": 9
} | {
"line": 140,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\nn : ℕ\n⊢ ∀ᶠ (t : R) in 𝓝 0, (↑x + t)⁻¹ʳ = (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * (↑x + t)⁻¹ʳ",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"add_mul",
"Units.val",
... | [] | by
have hzero : Tendsto (-(↑x⁻¹ : R) * ·) (𝓝 0) (𝓝 0) :=
(mulLeft_continuous _).tendsto' _ _ <| mul_zero _
filter_upwards [inverse_add x, hzero.eventually (inverse_one_sub_nth_order n)] with t ht ht'
rw [neg_mul, sub_neg_eq_add] at ht'
conv_lhs => rw [ht, ht', add_mul, ← neg_mul, mul_assoc]
rw [ht] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Multilinear.Topology | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 77
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case h₂.h₁\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nF✝ : Type u_4\ninst✝¹⁴ : NormedField 𝕜\ninst✝¹³ : (i : ι) → TopologicalSpace (E i)\ninst✝¹² : (i : ι) → AddCommGroup (E i)\ninst✝¹¹ : (i : ι) → Module 𝕜 (E i)\ninst✝¹⁰ : AddCommGroup F✝\ninst✝⁹ : Module 𝕜 F✝\ninst✝⁸ : UniformSpace F✝\ninst✝⁷... | [
"case h₂.h₂\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nF✝ : Type u_4\ninst✝¹⁴ : NormedField 𝕜\ninst✝¹³ : (i : ι) → TopologicalSpace (E i)\ninst✝¹² : (i : ι) → AddCommGroup (E i)\ninst✝¹¹ : (i : ι) → Module 𝕜 (E i)\ninst✝¹⁰ : AddCommGroup F✝\ninst✝⁹ : Module 𝕜 F✝\ninst✝⁸ : UniformSpace F✝\ninst✝⁷ : IsUniform... | · exact isClosed_iInter fun m ↦ isClosed_iInter fun i ↦
isClosed_iInter fun x ↦ isClosed_iInter fun y ↦ isClosed_eq H (H.add H) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Maps.Strict.Basic | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 87
} | {
"line": 127,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nf_quot : IsQuotientMap f\nΦ : ↑(range (g ∘ f)) ≃ₜ ↑(range g) := Homeomorph.setCongr ⋯\nkey : rangeFactorization g ∘ f = ⇑Φ ∘ rangeFactorization (g ∘ f)\n... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nf_quot : IsQuotientMap f\nΦ : ↑(range (g ∘ f)) ≃ₜ ↑(range g) := Homeomorph.setCongr ⋯\nkey : rangeFactorization g ∘ f = ⇑Φ ∘ rangeFactorization (g ∘ f)\n⊢ IsQuotient... | simp_rw [isStrictMap_iff_isQuotientMap_rangeFactorization, ← f_quot.of_comp_iff, key] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Real | {
"line": 143,
"column": 4
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns₁ s₂ : Set α\nh : μ (s₁ ∪ s₂) < ∞\n⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"False",
"Real.instLE",
"Real",
... | [] | have A : μ s₁ ≠ ∞ := measure_ne_top_of_subset subset_union_left h.ne
have B : μ s₂ ≠ ∞ := measure_ne_top_of_subset subset_union_right h.ne
simp only [Measure.real, ← ENNReal.toReal_add A B]
exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Real | {
"line": 143,
"column": 4
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns₁ s₂ : Set α\nh : μ (s₁ ∪ s₂) < ∞\n⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"False",
"Real.instLE",
"Real",
... | [] | have A : μ s₁ ≠ ∞ := measure_ne_top_of_subset subset_union_left h.ne
have B : μ s₂ ≠ ∞ := measure_ne_top_of_subset subset_union_right h.ne
simp only [Measure.real, ← ENNReal.toReal_add A B]
exact ENNReal.toReal_mono (by simp [A, B]) (measure_union_le _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 23
} | {
"line": 425,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α := ⋯\nhu :... | [
"α : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α := {x | x ∈ s ∧ f x ≠ ... | rw [union_sdiff_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 601,
"column": 45
} | {
"line": 609,
"column": 47
} | {
"line": 611,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_8\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nf : α → 𝕜\nhf : Integrable f μ\ng : α → β\ng_aestronglyMeasurable : AEStronglyMeasurable g μ\ness_sup_g :... | [] | by
rw [← memLp_one_iff_integrable] at *
refine ⟨hf.1.smul g_aestronglyMeasurable, ?_⟩
have hg' : eLpNorm g ∞ μ ≠ ∞ := by rwa [eLpNorm_exponent_top]
calc
eLpNorm (fun x : α => f x • g x) 1 μ ≤ _ := by
simpa using! MeasureTheory.eLpNorm_smul_le_mul_eLpNorm g_aestronglyMeasurable hf.1
(p := 1) (q... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 733,
"column": 2
} | {
"line": 734,
"column": 38
} | {
"line": 735,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : TopologicalSpace β\ninst✝ : BorelSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\na✝ : Nontrivial α\ninhabited... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : TopologicalSpace β\ninst✝ : BorelSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\na✝ : Nontrivial α\ninhabited_h : Inhabit... | obtain ⟨u, u_open, hu⟩ : ∃ u : Set α, IsOpen u ∧ f ⁻¹' t ∩ s = u ∩ s :=
_root_.continuousOn_iff'.1 hf t ht | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 754,
"column": 2
} | {
"line": 754,
"column": 54
} | {
"line": 755,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetrizableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nh's : IsSep... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetrizableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : MeasurableSet s\nh's : IsSeparable s\nth... | rw [aestronglyMeasurable_iff_aemeasurable_separable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 782,
"column": 2
} | {
"line": 782,
"column": 54
} | {
"line": 783,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : Mea... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : MeasurableSpace... | rw [aestronglyMeasurable_iff_aemeasurable_separable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 39
} | {
"line": 143,
"column": 2
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁴ : MeasurableSpace X\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ns : Set X\ninst✝ : SecondCountableTopology X\nhf : LocallyIntegrableOn f s μ\nu : ↑s → Set X\nu_open : ∀ (x : ↑s), IsOpen[inst✝³] (u x)... | [] | exact ⟨T, hT_count, by rwa [hT_un]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 52
} | {
"line": 223,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_6\ninst✝² : MeasurableSpace X\ninst✝¹ : TopologicalSpace X\ninst✝ : NormedAddCommGroup E\nμ : Measure X\ns : Set X\nf : X → E\n⊢ (∀ x ∈ s, IntegrableAtFilter (-f) (𝓝[s] x) μ) ↔ ∀ x ∈ s, IntegrableAtFilter f (𝓝[s] x) μ",
"ppTerm": "?m.21",
"assigned": true,
"usedCo... | [] | simp_rw [MeasureTheory.integrableAtFilter_neg_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1172,
"column": 10
} | {
"line": 1172,
"column": 50
} | {
"line": 1172,
"column": 51
} | [
{
"pp": "case h₂\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹² : Fintype ι'\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁸ : (i : ι) → S... | [
"case h₂\n𝕜 : Type u\nι : Type v\nι' : Type v'\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nG : Type wG\nG' : Type wG'\ninst✝¹² : Fintype ι'\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁸ : (i : ι) → SeminormedAdd... | prod_subtype _ (fun _ ↦ s.mem_toFinset), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 271,
"column": 70
} | {
"line": 280,
"column": 52
} | {
"line": 282,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\nhp_pos : p ≠ 0\nhp_ne_top : p ≠ ∞\nf : α →ₛ E\nhf : MemLp (⇑f) p μ\ny : E\nhy_ne : y ≠ 0\n⊢ μ (⇑f ⁻¹' {y}) < ∞",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Meas... | [] | by
have h_fin : (f.map fun x ↦ ‖x‖ₑ ^ p.toReal).FinMeasSupp μ := by
refine FinMeasSupp.of_lintegral_ne_top ?_
rw [← (f.map fun x ↦ ‖x‖ₑ ^ p.toReal).lintegral_eq_lintegral μ]
exact (lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_pos hp_ne_top hf.eLpNorm_lt_top).ne
have hf_fin : f.FinMeasSupp μ := by
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 412,
"column": 6
} | {
"line": 412,
"column": 15
} | {
"line": 413,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nk : 𝕜\nf : ↥(Lp E p μ)\ns : ... | [
"case h\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nE : Type u_4\nF : Type u_5\n𝕜 : Type u_6\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nk : 𝕜\nf : ↥(Lp E p μ)\ns : fail... | use k • s | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 212,
"column": 32
} | {
"line": 212,
"column": 57
} | {
"line": 212,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nT : Set α → β\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : μ s < ∞\n⊢ ‖(-T) s‖ ≤ C * μ.real s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [] | simpa using hT.2 s hs hμs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 212,
"column": 32
} | {
"line": 212,
"column": 57
} | {
"line": 212,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nT : Set α → β\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : μ s < ∞\n⊢ ‖(-T) s‖ ≤ C * μ.real s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [] | simpa using hT.2 s hs hμs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 212,
"column": 32
} | {
"line": 212,
"column": 57
} | {
"line": 212,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : SeminormedAddCommGroup β\nT : Set α → β\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\ns : Set α\nhs : MeasurableSet s\nhμs : μ s < ∞\n⊢ ‖(-T) s‖ ≤ C * μ.real s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [] | simpa using hT.2 s hs hμs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 6
} | {
"line": 321,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nm : MeasurableSpace α\nT : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ setToSimpleFunc T f = ∑ x ∈ f.range with x ≠ 0... | [
"α : Type u_1\nF : Type u_3\nF' : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup F'\ninst✝¹ : NormedSpace ℝ F'\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nm : MeasurableSpace α\nT : Set α → F →L[ℝ] F'\nf : α →ₛ F\n⊢ ∑ x ∈ f.range with x ≠ 0, (T (⇑f ⁻¹' {x})) x = setToSimple... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 732,
"column": 4
} | {
"line": 732,
"column": 50
} | {
"line": 733,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := ⋯\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero_mem : 0 ∈ (Set.range... | [
"α : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := ⋯\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0})... | apply IsSeparable.mono _ Set.inter_subset_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 772,
"column": 4
} | {
"line": 772,
"column": 8
} | {
"line": 773,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero... | [
"case refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : PartialOrder G\nhp : Fact (1 ≤ p)\nhp_ne_top : p ≠ ∞\ng : { g // 0 ≤ g }\nthis✝¹ : MeasurableSpace G := ⋯\nthis✝ : BorelSpace G\nhg_memLp : MemLp (↑↑↑g) p μ\nzero_mem : 0 ∈ (Set.ra... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 397,
"column": 30
} | {
"line": 397,
"column": 47
} | {
"line": 397,
"column": 48
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ ... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nh_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0\nh_add : FinMeasAdditive μ T\nf g : α →... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 310,
"column": 48
} | {
"line": 312,
"column": 58
} | {
"line": 314,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedSpace ℝ E\nf : α →ₛ E\nhf : Integrable (⇑f) μ\n⊢ ‖integral μ f‖ ≤ integral μ (map norm f)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MulOne... | [] | by
refine (norm_setToSimpleFunc_le_integral_norm _ (fun s _ _ => ?_) hf).trans (one_mul _).le
exact (norm_weightedSMul_le s).trans (one_mul _).symm.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 327,
"column": 70
} | {
"line": 329,
"column": 61
} | {
"line": 331,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nf : α → G\n⊢ ‖∫ (a : α), f a ∂μ‖ ≤ (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Semin... | [] | by
simp only [integral_eq_setToFun]
exact (norm_setToFun_le_toReal _ (by simp)).trans (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 23
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nhuv : u ~[l] v\n⊢ (fun x ↦ -u x) ~[l] fun x ↦ -v x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_sub",
"neg_add_eq_sub",
"congr... | [] | rw [IsEquivalent]
convert! huv.isLittleO.neg_left.neg_right
simp [neg_add_eq_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 169,
"column": 2
} | {
"line": 171,
"column": 23
} | {
"line": 173,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nhuv : u ~[l] v\n⊢ (fun x ↦ -u x) ~[l] fun x ↦ -v x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_sub",
"neg_add_eq_sub",
"congr... | [] | rw [IsEquivalent]
convert! huv.isLittleO.neg_left.neg_right
simp [neg_add_eq_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 372,
"column": 2
} | {
"line": 375,
"column": 51
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_1\nu v t w : α → ℝ\nl : Filter α\nhu : 0 ≤ v\nhw : 0 ≤ w\nhtu : u ~[l] v\nhvw : t ~[l] w\n⊢ u + t ~[l] v + w",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"abs",
"c... | [] | simp only [IsEquivalent, add_sub_add_comm]
change (fun x ↦ (u - v) x + (t - w) x) =o[l] (fun x ↦ v x + w x)
conv => enter [3, x]; rw [← abs_eq_self.mpr (hu x), ← abs_eq_self.mpr (hw x)]
simpa [← Real.norm_eq_abs] using .add_add htu hvw | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 372,
"column": 2
} | {
"line": 375,
"column": 51
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_1\nu v t w : α → ℝ\nl : Filter α\nhu : 0 ≤ v\nhw : 0 ≤ w\nhtu : u ~[l] v\nhvw : t ~[l] w\n⊢ u + t ~[l] v + w",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"abs",
"c... | [] | simp only [IsEquivalent, add_sub_add_comm]
change (fun x ↦ (u - v) x + (t - w) x) =o[l] (fun x ↦ v x + w x)
conv => enter [3, x]; rw [← abs_eq_self.mpr (hu x), ← abs_eq_self.mpr (hw x)]
simpa [← Real.norm_eq_abs] using .add_add htu hvw | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 132,
"column": 58
} | {
"line": 132,
"column": 77
} | {
"line": 132,
"column": 77
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nx y z : P\n⊢ ‖z -ᵥ y‖ = dist z y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"AddC... | [
"V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nx y z : P\n⊢ ‖z -ᵥ y‖ = ‖z -ᵥ y‖"
] | dist_eq_norm_vsub V | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 163,
"column": 61
} | {
"line": 163,
"column": 80
} | {
"line": 163,
"column": 80
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ - (p₂ -ᵥ p₄)‖ ≤ ‖p₁ -ᵥ p₃‖ + dist p₂ p₄",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"... | [
"V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ p₄ : P\n⊢ ‖p₁ -ᵥ p₃ - (p₂ -ᵥ p₄)‖ ≤ ‖p₁ -ᵥ p₃‖ + ‖p₂ -ᵥ p₄‖"
] | dist_eq_norm_vsub V | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 17
} | {
"line": 706,
"column": 17
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhfi : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nthis : f =ᵐ[μ] fun x ↦ ‖f x‖\n⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ = ∫⁻ (x : α), ENNReal.ofReal (f x) ∂μ",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAd... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhfi : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nthis : f =ᵐ[μ] fun x ↦ ‖f x‖\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.Analysis.Normed.Module.RieszLemma | {
"line": 51,
"column": 2
} | {
"line": 78,
"column": 55
} | {
"line": 80,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nr : ℝ\nhr : r < 1\n⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖",
"ppTerm": "?m.40",... | [] | classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1 :=... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 51,
"column": 2
} | {
"line": 78,
"column": 55
} | {
"line": 80,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nr : ℝ\nhr : r < 1\n⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖",
"ppTerm": "?m.40",... | [] | classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1 :=... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 51,
"column": 2
} | {
"line": 78,
"column": 55
} | {
"line": 80,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nr : ℝ\nhr : r < 1\n⊢ ∃ x₀ ∉ F, ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖",
"ppTerm": "?m.40",... | [] | classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1 :=... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Matrix | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 67
} | {
"line": 281,
"column": 68
} | [
{
"pp": "X : Type u_1\nl : Type u_3\nm : Type u_4\nn : Type u_5\np : Type u_6\nR : Type u_8\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace R\nA : X → Matrix n l R\nB : X → Matrix n m R\nC : X → Matrix p l R\nD : X → Matrix p m R\nhA : Continuous A\nhB : Continuous B\nhC : Continuous C\nhD : Continuous D... | [
"case inl.inl\nX : Type u_1\nl : Type u_3\nm : Type u_4\nn : Type u_5\np : Type u_6\nR : Type u_8\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace R\nA : X → Matrix n l R\nB : X → Matrix n m R\nC : X → Matrix p l R\nD : X → Matrix p m R\nhA : Continuous A\nhB : Continuous B\nhC : Continuous C\nhD : Continuous... | rintro (i | i) (j | j) <;> refine Continuous.matrix_elem ?_ i j | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 21
} | {
"line": 146,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\nhC : ∀ ⦃x : β⦄, x ∈ range ⇑f ∪ range ⇑g → ∀ ⦃y : β⦄, y ∈ range ⇑f ∪ range ⇑g → dist x y ≤ C\n⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [
"case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\nhC : ∀ ⦃x : β⦄, x ∈ range ⇑f ∪ range ⇑g → ∀ ⦃y : β⦄, y ∈ range ⇑f ∪ range ⇑g → dist x y ≤ C\nx : α\n⊢ f x ∈ range ⇑f",
"case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ :... | refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right] | Batteries.Tactic._aux_Batteries_Tactic_SeqFocus___macroRules_Batteries_Tactic_seq_focus_1 | Batteries.Tactic.seq_focus |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1182,
"column": 6
} | {
"line": 1183,
"column": 26
} | {
"line": 1184,
"column": 6
} | [
{
"pp": "case hp_ne_zero\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((... | [
"case hp_ne_top\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((fun x ↦ ‖f x‖... | · rw [Ne, ENNReal.ofReal_eq_zero, not_le]
exact hpq.symm.pos | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 444,
"column": 4
} | {
"line": 445,
"column": 56
} | {
"line": 446,
"column": 4
} | [
{
"pp": "case refine_1.inl\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\nδ : Type u_2\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : α\n⊢ dist ((extend f g₁ h₁) (f x)) ((extend f g₂ h₂) (f x)) ≤\n max (dist g₁ g₂)... | [
"case refine_1.inr\nα : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\nδ : Type u_2\ninst✝¹ : TopologicalSpace δ\ninst✝ : DiscreteTopology δ\nf : α ↪ δ\ng₁ g₂ : α →ᵇ β\nh₁ h₂ : δ →ᵇ β\nx : δ\nhx : ¬∃ y, f y = x\n⊢ dist ((extend f g₁ h₁) x) ((extend f g₂ h₂) x) ≤\n max (dist g₁ g₂)... | · simp only [extend_apply]
exact (dist_coe_le_dist x).trans (le_max_left _ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 28
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nδseq_lim : Tendsto δseq atTop (𝓝 0)\nE : Set α\nx : α\nx_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\n⊢ Tendsto (fun i ↦ 1) atTop (𝓝 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedCon... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 148,
"column": 8
} | {
"line": 148,
"column": 58
} | {
"line": 149,
"column": 8
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns t : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\nhst : Disjoint s t\nx : (i : δ) → X i\ne : ((i : ↥s) → X ↑i) × ((i : ↥t) → ... | [
"δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns t : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\nhst : Disjoint s t\nx : (i : δ) → X i\ne : ((i : ↥s) → X ↑i) × ((i : ↥t) → X ↑i) ≃ᵐ ((i... | simp_rw [lmarginal, updateFinset_updateFinset hst] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1234,
"column": 2
} | {
"line": 1244,
"column": 6
} | {
"line": 1246,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhC : 0 ≤ C\n⊢ ‖setToFun μ T hT f... | [] | by_cases hF : CompleteSpace F; swap
· simp only [setToFun, hF, ↓reduceDIte, norm_zero, NNReal.coe_mk, ofReal_norm]
positivity
by_cases hf : Integrable f μ; swap
· simp only [setToFun_undef _ hf, norm_zero, NNReal.coe_mk, ofReal_norm]
positivity
apply (norm_setToFun_le hT hf hC).trans
gcongr
· simp
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1234,
"column": 2
} | {
"line": 1244,
"column": 6
} | {
"line": 1246,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhC : 0 ≤ C\n⊢ ‖setToFun μ T hT f... | [] | by_cases hF : CompleteSpace F; swap
· simp only [setToFun, hF, ↓reduceDIte, norm_zero, NNReal.coe_mk, ofReal_norm]
positivity
by_cases hf : Integrable f μ; swap
· simp only [setToFun_undef _ hf, norm_zero, NNReal.coe_mk, ofReal_norm]
positivity
apply (norm_setToFun_le hT hf hC).trans
gcongr
· simp
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1314,
"column": 2
} | {
"line": 1316,
"column": 35
} | {
"line": 1318,
"column": 0
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\... | [] | · filter_upwards [h_lim]
refine fun a h_lin => @Tendsto.comp _ _ _ (fun n => x (n + k)) (fun n => fs n a) _ _ _ h_lin ?_
rwa [tendsto_add_atTop_iff_nat] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1397,
"column": 2
} | {
"line": 1398,
"column": 84
} | {
"line": 1400,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝ : SFinite μ\nf : β → α → E\nhf : StronglyMeasurable (Function.uncurry f)\n⊢ MeasurableSet {x | Integrable (f x) μ}",
"ppTerm": "?m.61",
"assigned": true,
... | [] | simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
exact measurableSet_lt (Measurable.lintegral_prod_right hf.enorm) measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1397,
"column": 2
} | {
"line": 1398,
"column": 84
} | {
"line": 1400,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝ : SFinite μ\nf : β → α → E\nhf : StronglyMeasurable (Function.uncurry f)\n⊢ MeasurableSet {x | Integrable (f x) μ}",
"ppTerm": "?m.61",
"assigned": true,
... | [] | simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
exact measurableSet_lt (Measurable.lintegral_prod_right hf.enorm) measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 422,
"column": 6
} | {
"line": 422,
"column": 60
} | {
"line": 423,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 :=... | [] | exact (f.right_continuous q').mono Ioi_subset_Ici_self | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 389,
"column": 75
} | {
"line": 389,
"column": 79
} | {
"line": 389,
"column": 79
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0\n⊢ (fun x ↦ 0) =ᵐ[μ.restrict t] f",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Mea... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0\n⊢ f =ᵐ[μ.restrict t] fun x ↦ 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 6
} | {
"line": 483,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns : Set X\nμ : Measure X\nh : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0\n⊢ ∫ (x : X) in s, f x ∂μ = ∫ (x : X), f x ∂μ",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns : Set X\nμ : Measure X\nh : ∀ᵐ (x : X) ∂μ, x ∉ s → f x = 0\n⊢ ∫ (x : X), f x ∂μ = ∫ (x : X) in s, f x ∂μ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 6
} | {
"line": 502,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure X\ninst✝ : PartialOrder E\nf : X → E\nhf : AEStronglyMeasurable f μ\nh_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0}\nB : NullMeasurableSet {x | f x = 0} μ\n⊢ ∫ (x : X) in {x... | [
"X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure X\ninst✝ : PartialOrder E\nf : X → E\nhf : AEStronglyMeasurable f μ\nh_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0}\nB : NullMeasurableSet {x | f x = 0} μ\n⊢ ∫ (x : X) in {x | f x < 0} ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
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