module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 25
} | {
"line": 350,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\n⊢ ∑ x ∈ f.ran... | [
"α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\n⊢ ∀ i ∈ f.range, μ.real (... | apply Finset.sum_le_sum | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 359,
"column": 41
} | {
"line": 359,
"column": 52
} | {
"line": 359,
"column": 53
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\nx : α\nhx : ¬... | [
"α : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nν : Measure α\nf : α →ₛ F\nhf : 0 ≤ᵐ[ν] ⇑f\nhμν : μ ≤ ν\nhfν : Integrable (⇑f) ν\nx : α\nhx : ¬0 ≤ f x\nthi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 462,
"column": 4
} | {
"line": 462,
"column": 25
} | {
"line": 462,
"column": 26
} | [
{
"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 →₁ₛ[μ] E)\n⊢ ‖{ toFun := integral, map_add' := ⋯, map_smul' := ⋯ } f‖ ≤ 1 * ‖f‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"NormedCommRing.t... | [
"α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedSpace ℝ E\nf : ↥(α →₁ₛ[μ] E)\n⊢ ‖integral f‖ ≤ ‖↑f‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 11
} | {
"line": 474,
"column": 2
} | [
{
"pp": "case e'_3\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ↥(α →₁ₛ[μ] ℝ)\neq : ∀ (a : α), (toSimpleFunc f).posPart a = max ((toSimpleFunc f) a) 0\na✝¹ : α\na✝ : (toSimpleFunc (posPart f)) a✝¹ = ↑↑↑(posPart f) a✝¹\nh₂ : ↑↑(Lp.posPart ↑f) a✝¹ = max (↑↑↑f a✝¹) 0\nh₃ : (toSimpleFunc f) a✝¹ = ↑↑↑f a... | [] | rw [h₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 37
} | {
"line": 297,
"column": 38
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nL : Type u_6\ninst✝ : RCLike L\nr : L\nf : α → L\n⊢ ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nL : Type u_6\ninst✝ : RCLike L\nr : L\nf : α → L\n⊢ ∫ (a : α), f a / r ∂μ = (∫ (a : α), f a ∂μ) / r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 15
} | {
"line": 139,
"column": 16
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nhuv : u ~[l] v\nhu : u =o[l] fun _x ↦ 1\n⊢ v =o[l] fun _x ↦ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SeminormedAddGroup.toNorm",
... | [
"case pos\nα : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nhuv : u ~[l] v\nhu : u =o[l] fun _x ↦ 1\n⊢ Tendsto v l (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 13
} | {
"line": 203,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhuv : Tendsto (u / v) l (𝓝 1)\nh : ∃ᶠ (t : α) in l, (u / v) t = 0\n⊢ False",
"ppTerm": "?m.98",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝ : NormedField β\nu v : α → β\nl : Filter α\nhuv : Tendsto (u / v) l (𝓝 1)\nh : ∃ᶠ (t : α) in l, (u / v) t = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 35
} | {
"line": 295,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u v w : α → β\nl : Filter α\nhtu : t ~[l] u\nhvw : v ~[l] w\n⊢ t / v ~[l] u / w",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"inst... | [
"α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u v w : α → β\nl : Filter α\nhtu : t ~[l] u\nhvw : v ~[l] w\n⊢ t * v⁻¹ ~[l] u * w⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 299,
"column": 12
} | {
"line": 299,
"column": 23
} | {
"line": 299,
"column": 24
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\n⊢ t ^ 0 ~[l] u ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Monoid.toMulOneClass"... | [
"case zero\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\n⊢ 1 ~[l] 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 300,
"column": 17
} | {
"line": 300,
"column": 39
} | {
"line": 300,
"column": 40
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nn✝ : ℕ\nih : t ^ n✝ ~[l] u ^ n✝\n⊢ t ^ (n✝ + 1) ~[l] u ^ (n✝ + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"case succ\nα : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nn✝ : ℕ\nih : t ^ n✝ ~[l] u ^ n✝\n⊢ t ^ n✝ * t ~[l] u ^ n✝ * u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 304,
"column": 19
} | {
"line": 304,
"column": 30
} | {
"line": 304,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ t ^ Int.ofNat a✝ ~[l] u ^ Int.ofNat a✝",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ t ^ a✝ ~[l] u ^ a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 305,
"column": 21
} | {
"line": 305,
"column": 32
} | {
"line": 305,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ t ^ Int.negSucc a✝ ~[l] u ^ Int.negSucc a✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.... | [
"α : Type u_1\nβ : Type u_3\ninst✝ : NormedField β\nt u : α → β\nl : Filter α\nh : t ~[l] u\nz : ℤ\na✝ : ℕ\n⊢ (t ^ (a✝ + 1))⁻¹ ~[l] (u ^ (a✝ + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 878,
"column": 6
} | {
"line": 881,
"column": 23
} | {
"line": 882,
"column": 2
} | [
{
"pp": "case inr.const.inr\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤... | [] | have : μ s < ∞ := SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs Hs
rcases h0P c hs this εpos with ⟨g, hg, Pg⟩
rw [← eLpNorm_neg, neg_sub] at hg
exact ⟨g, hg, Pg⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 878,
"column": 6
} | {
"line": 881,
"column": 23
} | {
"line": 882,
"column": 2
} | [
{
"pp": "case inr.const.inr\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤... | [] | have : μ s < ∞ := SimpleFunc.measure_lt_top_of_memLp_indicator hp_pos hp_ne_top hc hs Hs
rcases h0P c hs this εpos with ⟨g, hg, Pg⟩
rw [← eLpNorm_neg, neg_sub] at hg
exact ⟨g, hg, Pg⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 374,
"column": 24
} | {
"line": 374,
"column": 79
} | {
"line": 375,
"column": 2
} | [
{
"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\nx : α\n| v x + w x",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.to... | [
"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\nx : α\n| |v x| + |w x|"
] | rw [← abs_eq_self.mpr (hu x), ← abs_eq_self.mpr (hw x)] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 34
} | {
"line": 375,
"column": 35
} | [
{
"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⊢ (fun x ↦ (u - v) x + (t - w) x) =o[l] fun x ↦ |v x| + |w x|",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Real",
"abs",
"Real.instSub",
"HSub.hSub",
... | [
"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⊢ (fun x ↦ u x - v x + (t x - w x)) =o[l] fun x ↦ ‖v x‖ + ‖w x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nv v' : V\np p' : P\n⊢ dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"V : Type u_2\nP : Type u_3\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nv v' : V\np p' : P\n⊢ dist (v +ᵥ p) (v' +ᵥ p') ≤ dist v v' + dist p p'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.AddTorsor | {
"line": 201,
"column": 29
} | {
"line": 201,
"column": 40
} | {
"line": 201,
"column": 41
} | [
{
"pp": "α : Type u_1\nV✝ : Type u_2\nP✝ : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁷ : SeminormedAddCommGroup V✝\ninst✝⁶ : PseudoMetricSpace P✝\ninst✝⁵ : NormedAddTorsor V✝ P✝\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : PseudoMetricSpace Q\ninst✝² : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst✝¹ : ... | [
"α : Type u_1\nV✝ : Type u_2\nP✝ : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝⁷ : SeminormedAddCommGroup V✝\ninst✝⁶ : PseudoMetricSpace P✝\ninst✝⁵ : NormedAddTorsor V✝ P✝\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : PseudoMetricSpace Q\ninst✝² : NormedAddTorsor W Q\nV : Type u_6\nP : Type u_7\ninst✝¹ : NormedAddCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 885,
"column": 6
} | {
"line": 885,
"column": 75
} | {
"line": 885,
"column": 75
} | [
{
"pp": "case inr.add\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P... | [
"case inr.add\nα : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g\nh1P : ∀ ... | memLp_add_of_disjoint hff' f.stronglyMeasurable f'.stronglyMeasurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 81
} | {
"line": 700,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhfi : Integrable (fun x ↦ ↑(f x)) μ\n⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"Preorder.toLT",
"PartialOrder.toPreorder",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhfi : Integrable (fun x ↦ ↑(f x)) μ\n⊢ ∫⁻ (a : α), ↑(f a) ∂μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 716,
"column": 60
} | {
"line": 718,
"column": 32
} | {
"line": 720,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhfi : Integrable (fun x ↦ ↑(f x)) μ\nb : ℝ≥0\n⊢ ∫⁻ (a : α), ↑(f a) ∂μ ≤ ↑b ↔ ∫ (a : α), ↑(f a) ∂μ ≤ ↑b",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal... | [] | by
rw [lintegral_coe_eq_integral f hfi, ENNReal.ofReal, ENNReal.coe_le_coe,
Real.toNNReal_le_iff_le_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 729,
"column": 26
} | {
"line": 729,
"column": 60
} | {
"line": 729,
"column": 61
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f μ\n⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ = 0 ∨ ¬∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ < ∞ ↔ f =ᵐ[μ] 0",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : 0 ≤ᵐ[μ] f\nhfi : Integrable f μ\n⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ = 0 ∨ ¬HasFiniteIntegral f μ ↔ f =ᵐ[μ] 0"
] | ← hasFiniteIntegral_iff_ofReal hf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 811,
"column": 6
} | {
"line": 811,
"column": 27
} | {
"line": 811,
"column": 28
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\nthis✝ : Tendsto (fun n ↦ ∫ (x : α), -f n x ∂μ) atTo... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\nthis✝ : Tendsto (fun n ↦ ∫ (x : α), -f n x ∂μ) atTop (𝓝 (∫ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RCLike.Basic | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 34
} | {
"line": 114,
"column": 35
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\n𝓕 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : FunLike 𝓕 E F\ninst✝¹ : AddMonoidHomClass 𝓕 E F\ninst✝ : MulActionHomClass 𝓕 𝕜 E F\nf ... | [
"case inr\n𝕜 : Type u_1\ninst✝⁷ : RCLike 𝕜\n𝓕 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜 F\ninst✝² : FunLike 𝓕 E F\ninst✝¹ : AddMonoidHomClass 𝓕 E F\ninst✝ : MulActionHomClass 𝓕 𝕜 E F\nf : 𝓕\nK : NN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 15
} | {
"line": 98,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\n⊢ ‖c‖ < 1 * R",
"ppTerm... | [
"𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\n⊢ ‖c‖ < R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | {
"line": 144,
"column": 14
} | [
{
"pp": "case refine_2.ha\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nE : Type u_5\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\nx₀ : E\nhx₀ : x₀ ∉ F\nh : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - ... | [
"case refine_2.ha\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nE : Type u_5\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\nx₀ : E\nhx₀ : x₀ ∉ F\nh : ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖\nhx₀' : x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1132,
"column": 4
} | {
"line": 1132,
"column": 37
} | {
"line": 1132,
"column": 38
} | [
{
"pp": "case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᵐ[μ] f\nhf_int : Integrable f μ\nε : ℝ\nhμ : μ {x | ε ≤ f x} = ∞\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instL... | [
"case inl\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᵐ[μ] f\nhf_int : Integrable f μ\nε : ℝ\nhμ : μ {x | ε ≤ f x} = ∞\n⊢ 0 ≤ ∫ (x : α), f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1131,
"column": 2
} | {
"line": 1139,
"column": 72
} | {
"line": 1141,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᵐ[μ] f\nhf_int : Integrable f μ\nε : ℝ\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [] | rcases eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ | hμ
· simpa [measureReal_def, hμ] using integral_nonneg_of_ae hf_nonneg
· have := Fact.mk hμ
calc
ε * μ.real { x | ε ≤ f x } = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
_ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
integral_mono_ae (integrable_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1131,
"column": 2
} | {
"line": 1139,
"column": 72
} | {
"line": 1141,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf_nonneg : 0 ≤ᵐ[μ] f\nhf_int : Integrable f μ\nε : ℝ\n⊢ ε * μ.real {x | ε ≤ f x} ≤ ∫ (x : α), f x ∂μ",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [] | rcases eq_top_or_lt_top (μ {x | ε ≤ f x}) with hμ | hμ
· simpa [measureReal_def, hμ] using integral_nonneg_of_ae hf_nonneg
· have := Fact.mk hμ
calc
ε * μ.real { x | ε ≤ f x } = ∫ _ in {x | ε ≤ f x}, ε ∂μ := by simp [mul_comm]
_ ≤ ∫ x in {x | ε ≤ f x}, f x ∂μ :=
integral_mono_ae (integrable_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 47
} | {
"line": 323,
"column": 48
} | [
{
"pp": "case mk.mk\n𝕜 : Type u_1\nV : Type u_2\nV₁ : Type u_3\nV₁' : Type u_4\nV₂ : Type u_5\nV₃ : Type u_6\nV₄ : Type u_7\nP₁ : Type u_8\nP₁' : Type u_9\nP : Type u_10\nP₂ : Type u_11\nP₃ : Type u_12\nP₄ : Type u_13\ninst✝²⁴ : NormedField 𝕜\ninst✝²³ : SeminormedAddCommGroup V\ninst✝²² : NormedSpace 𝕜 V\nin... | [
"case mk.mk\n𝕜 : Type u_1\nV : Type u_2\nV₁ : Type u_3\nV₁' : Type u_4\nV₂ : Type u_5\nV₃ : Type u_6\nV₄ : Type u_7\nP₁ : Type u_8\nP₁' : Type u_9\nP : Type u_10\nP₂ : Type u_11\nP₃ : Type u_12\nP₄ : Type u_13\ninst✝²⁴ : NormedField 𝕜\ninst✝²³ : SeminormedAddCommGroup V\ninst✝²² : NormedSpace 𝕜 V\ninst✝²¹ : Pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 69
} | {
"line": 195,
"column": 70
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nh : ¬∃ s, Nonempty (Basis (↥s) 𝕜 E)\n⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦\n (if H : ∃ s, Nonempty (Basi... | [
"case neg\n𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nh : ¬∃ s, Nonempty (Basis (↥s) 𝕜 E)\n⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 19
} | {
"line": 92,
"column": 20
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝² : Countable ι\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhc : ∀ (i : ι), c i ≠ ∞\nh : Summable fun i ↦ (c i).toReal * ‖f (x i)‖\n⊢ Summable fun i ↦ ∫ (x : X), ‖f x‖ ∂c i... | [
"ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝² : Countable ι\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhc : ∀ (i : ι), c i ≠ ∞\nh : Summable fun i ↦ (c i).toReal * ‖f (x i)‖\n⊢ Summable fun i ↦ (c i).toReal * ‖f (x i)‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 14
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhf : Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))\n⊢ Summable fun i ↦ (c i).toReal * ‖f (x i)‖",
"ppTerm": "?m.33"... | [
"ι : Type u_1\nX : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\nf : X → E\ninst✝ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\nhf : Integrable f (Measure.sum fun i ↦ c i • Measure.dirac (x i))\n⊢ Summable fun i ↦ (c i).toReal * ‖f (x i)‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1165,
"column": 23
} | {
"line": 1165,
"column": 79
} | {
"line": 1165,
"column": 79
} | [
{
"pp": "α : 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‖ₑ) *... | [
"α : 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‖ₑ) * fun x ↦ ‖g ... | ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) hpq.nonneg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 13
} | {
"line": 164,
"column": 14
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : Countable ι\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\nf : X → E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\ninst✝ : CompleteSpace E\nhc : ∀ (i : ι), c i ≠ ∞\nhf : Summable fun i ↦ (c i).toReal ... | [
"ι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : Countable ι\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\nf : X → E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\ninst✝ : CompleteSpace E\nhc : ∀ (i : ι), c i ≠ ∞\nhf : Summable fun i ↦ (c i).toReal * ‖f (x i)‖\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 36
} | {
"line": 119,
"column": 6
} | [
{
"pp": "R : Type u_2\ninst✝² : Bornology R\ninst✝¹ : Monoid R\ninst✝ : BoundedMul R\ns : Set R\ns_bdd : Bornology.IsBounded s\nn : ℕ\nhn : Bornology.IsBounded ((fun x ↦ x ^ n) '' s)\nx y : R\ny_in_s : y ∈ s\nypow_eq_x : y ^ (n + 1) = x\n⊢ x ∈ (fun x ↦ x ^ n) '' s * s",
"ppTerm": "?m.97",
"assigned": tr... | [
"R : Type u_2\ninst✝² : Bornology R\ninst✝¹ : Monoid R\ninst✝ : BoundedMul R\ns : Set R\ns_bdd : Bornology.IsBounded s\nn : ℕ\nhn : Bornology.IsBounded ((fun x ↦ x ^ n) '' s)\nx y : R\ny_in_s : y ∈ s\nypow_eq_x : y ^ (n + 1) = x\n⊢ y ^ n * y ∈ (fun x ↦ x ^ n) '' s * s"
] | rw [← ypow_eq_x, pow_succ y n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 26
} | {
"line": 147,
"column": 27
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Monoid R\ninst✝ : LipschitzMul R\ns t : Set R\ns_bdd : Bornology.IsBounded s\nt_bdd : Bornology.IsBounded t\nbdd : Bornology.IsBounded (s ×ˢ t)\nC : ℝ≥0\nmul_lip : LipschitzWith C fun p ↦ p.1 * p.2\np a b : R\na_in_s : a ∈ s\nb_in_t : b ∈ t... | [
"case mpr\nR : Type u_1\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Monoid R\ninst✝ : LipschitzMul R\ns t : Set R\ns_bdd : Bornology.IsBounded s\nt_bdd : Bornology.IsBounded t\nbdd : Bornology.IsBounded (s ×ˢ t)\nC : ℝ≥0\nmul_lip : LipschitzWith C fun p ↦ p.1 * p.2\np a b : R\na_in_s : a ∈ s\nb_in_t : b ∈ t\neq_p : a *... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 35
} | {
"line": 198,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝ : SeminormedAddCommGroup R\nx : R\n⊢ Tendsto (fun x_1 ↦ x_1 - x) (cobounded R) (cobounded R)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoMetricSpace.toBornology",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq... | [
"R : Type u_1\ninst✝ : SeminormedAddCommGroup R\nx : R\n⊢ Tendsto (fun x_1 ↦ x_1 + -x) (cobounded R) (cobounded R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 254,
"column": 22
} | {
"line": 254,
"column": 53
} | {
"line": 254,
"column": 54
} | [
{
"pp": "s t : Set ℝ≥0\nhs : Bornology.IsBounded s\nht : Bornology.IsBounded t\nAf : ℝ\nhAf : s ⊆ closedBall 0 Af\nAg : ℝ\nhAg : t ⊆ closedBall 0 Ag\nkey : IsCompact (closedBall 0 Af ×ˢ closedBall 0 Ag)\na✝ x : ℝ≥0\nx_in_s : x ∈ s\ny : ℝ≥0\ny_in_t : y ∈ t\nxy_eq : (fun x1 x2 ↦ x1 * x2) x y = a✝\n⊢ (x, y) ∈ clos... | [
"s t : Set ℝ≥0\nhs : Bornology.IsBounded s\nht : Bornology.IsBounded t\nAf : ℝ\nhAf : s ⊆ closedBall 0 Af\nAg : ℝ\nhAg : t ⊆ closedBall 0 Ag\nkey : IsCompact (closedBall 0 Af ×ˢ closedBall 0 Ag)\na✝ x : ℝ≥0\nx_in_s : x ∈ s\ny : ℝ≥0\ny_in_t : y ∈ t\nxy_eq : (fun x1 x2 ↦ x1 * x2) x y = a✝\n⊢ (x ∈ closedBall 0 Af ∧ y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 780,
"column": 2
} | {
"line": 780,
"column": 13
} | {
"line": 780,
"column": 14
} | [
{
"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\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\n⊢ setToFun μ (-T) ⋯ f = -setToFu... | [
"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\nhT : DominatedFinMeasAdditive μ T C\nf : α → E\n⊢ setToFun μ (-T) ⋯ f = -setToFun μ T hT f"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 224,
"column": 8
} | {
"line": 224,
"column": 23
} | {
"line": 224,
"column": 24
} | [
{
"pp": "case inl\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nh✝ : IsEmpty α\n⊢ dist f g = ⨆ x, dist (f x) (g x)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"iSup",
"Real.ins... | [
"case inl\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nh✝ : IsEmpty α\n⊢ dist f g = sSup ∅"
] | iSup_of_empty', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 232,
"column": 25
} | {
"line": 232,
"column": 39
} | {
"line": 232,
"column": 39
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ↑(nndist f g) = ⨆ x, ↑(nndist (f x) (g x))",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"ENNReal.ofNNReal",
"congrArg",
"iS... | [
"α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ↑(⨆ x, nndist (f x) (g x)) = ⨆ x, ↑(nndist (f x) (g x))"
] | nndist_eq_iSup | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 807,
"column": 4
} | {
"line": 808,
"column": 53
} | {
"line": 810,
"column": 0
} | [
{
"pp": "case neg\nα : 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 g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᵐ[μ] g\nhF : ... | [] | have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [setToFun_undef hT hfi, setToFun_undef hT hgi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 807,
"column": 4
} | {
"line": 808,
"column": 53
} | {
"line": 810,
"column": 0
} | [
{
"pp": "case neg\nα : 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 g : α → E\nhT : DominatedFinMeasAdditive μ T C\nh : f =ᵐ[μ] g\nhF : ... | [] | have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
rw [setToFun_undef hT hfi, setToFun_undef hT hgi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1252,
"column": 41
} | {
"line": 1260,
"column": 83
} | {
"line": 1262,
"column": 0
} | [
{
"pp": "F : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nβ : Type u_6\nm m0 : MeasurableSpace β\nμ : Measure β\nhm : m ≤ m0\nf : β →ₛ F\nhf_int : Integrable (⇑f) μ\n⊢ ∫ (x : β), f x ∂μ = ∫ (x : β), f x ∂μ.trim hm",
"ppTerm": "?m.33",
"assigned": true,
... | [] | by
have hf : StronglyMeasurable[m] f := @SimpleFunc.stronglyMeasurable β F m _ f
have hf_int_m := hf_int.trim hm hf
rw [integral_simpleFunc_larger_space (le_refl m) f hf_int_m,
integral_simpleFunc_larger_space hm f hf_int]
congr with x
simp only [measureReal_def]
congr 2
exact (trim_measurableSet_eq h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 634,
"column": 6
} | {
"line": 634,
"column": 40
} | {
"line": 635,
"column": 6
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : AddMonoid β\ninst✝ : LipschitzAdd β\nf g : α →ᵇ β\nx : α\nC : ℝ\nC_nonneg : 0 ≤ ↑(LipschitzAdd.C β)\n⊢ LipschitzWith (LipschitzAdd.C β) fun p ↦ p.1 + p.2",
"ppTerm": "?m.24",
"a... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : PseudoMetricSpace β\ninst✝¹ : AddMonoid β\ninst✝ : LipschitzAdd β\nf g : α →ᵇ β\nx : α\nC : ℝ\nC_nonneg : 0 ≤ ↑(LipschitzAdd.C β)\n⊢ ∀ (x y : (α →ᵇ β) × (α →ᵇ β)), dist (x.1 + x.2) (y.1 + y.2) ≤ ↑(LipschitzAdd.C β) * 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.Analysis.Normed.Module.FiniteDimension | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 15
} | {
"line": 435,
"column": 16
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nh : ¬FiniteDimensional 𝕜 E\ns : Finset E\nF : Submodule 𝕜 E := Submodule.span 𝕜 ↑s\nhF : F.FG\nthis✝ : FiniteDi... | [
"𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace 𝕜\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nh : ¬FiniteDimensional 𝕜 E\ns : Finset E\nF : Submodule 𝕜 E := Submodule.span 𝕜 ↑s\nhF : F.FG\nthis✝ : FiniteDimensional 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 475,
"column": 41
} | {
"line": 475,
"column": 52
} | {
"line": 475,
"column": 53
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : CompleteSpace 𝕜\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Module 𝕜 V\ninst✝ : ContinuousSMul 𝕜 V\nr : ℝ\nrpos : 0 < r\nc : V\nh : IsCompact (closedBall c r)\n⊢ IsCompact (closedBall 0 r)",
"ppTerm": "?m.29",
"assigned": ... | [
"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : CompleteSpace 𝕜\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : Module 𝕜 V\ninst✝ : ContinuousSMul 𝕜 V\nr : ℝ\nrpos : 0 < r\nc : V\nh : IsCompact (closedBall c r)\n⊢ IsCompact (closedBall 0 r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 46
} | {
"line": 484,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\n⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))",
... | [
"𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\n⊢ IsCompact (closedBall 0 (‖c‖ ^ n * r))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 480,
"column": 2
} | {
"line": 487,
"column": 57
} | {
"line": 489,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\n⊢ ProperSpace E",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace"... | [] | rcases exists_isCompact_closedBall (0 : E) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC : ∀ n, IsCompact (closedBall (0 : E) (‖c‖ ^ n * r)) := fun n ↦ by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
simpa [_root_.smul_closedBall' this... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 480,
"column": 2
} | {
"line": 487,
"column": 57
} | {
"line": 489,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : LocallyCompactSpace E\n⊢ ProperSpace E",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace"... | [] | rcases exists_isCompact_closedBall (0 : E) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC : ∀ n, IsCompact (closedBall (0 : E) (‖c‖ ^ n * r)) := fun n ↦ by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
simpa [_root_.smul_closedBall' this... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 915,
"column": 4
} | {
"line": 916,
"column": 63
} | {
"line": 917,
"column": 4
} | [
{
"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\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nf : α → E\nhf : AEStronglyMea... | [
"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\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nf : α → E\nhf : AEStronglyMeasurable f μ\... | obtain ⟨i, hi, h'i⟩ : ∃ i, ∫⁻ x, ‖fs i x - f x‖ₑ ∂μ < 1 ∧ Integrable (fs i) μ :=
(((tendsto_order.1 hfs).2 _ zero_lt_one).and hfsi).exists | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 611,
"column": 4
} | {
"line": 611,
"column": 15
} | {
"line": 611,
"column": 16
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nf : α → E\nhf : Summable f\nthis : ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x ↦ ‖g x‖\nv : Basis (Fin (finrank ℝ E)) ℝ E\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := v.equivFunL\... | [
"α : Type u_1\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nf : α → E\nhf : Summable f\nthis : ∀ {N : ℕ} {g : α → Fin N → ℝ}, Summable g → Summable fun x ↦ ‖g x‖\nv : Basis (Fin (finrank ℝ E)) ℝ E\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ := v.equivFunL\nH : Summabl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 661,
"column": 6
} | {
"line": 661,
"column": 32
} | {
"line": 662,
"column": 6
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nk : ℕ\nr : F\nhr : ‖r‖ < 1\nu : ℕ → F\nhu : u =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ (fun n ↦ ‖u n‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖↑n ^ k‖ * ‖r‖ ^ n",
"ppTerm": "?m.151",
"assigned":... | [
"F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nk : ℕ\nr : F\nhr : ‖r‖ < 1\nu : ℕ → F\nhu : u =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖r‖ < r'\nh : r' < 1\n⊢ (fun n ↦ ‖u n‖ * ‖r‖ ^ n) =O[atTop] fun n ↦ ‖↑n‖ ^ k * ‖r‖ ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.FiniteDimension | {
"line": 690,
"column": 29
} | {
"line": 690,
"column": 66
} | {
"line": 690,
"column": 67
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ F\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : FiniteDimensional ℝ F\nf : ℕ → E\ng : ℕ → F\nh : f =Θ[atTop] g\n⊢ f =Θ[cofinite] g",
"ppTerm": "?m.37",
"assigned... | [
"E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ F\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : FiniteDimensional ℝ F\nf : ℕ → E\ng : ℕ → F\nh : f =Θ[atTop] g\n⊢ f =Θ[cofinite] g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 190,
"column": 25
} | {
"line": 190,
"column": 36
} | {
"line": 190,
"column": 37
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nn : ℤ\nf : α →ᵇ β\n⊢ ∃ C, ∀ (x y : α), dist ((n • f.toContinuousMap).toFun x) ((n • f.toContinuousMap).toFun y) ≤ C",
"ppTerm": "?m.26",
"assigned": true,
"usedCon... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf✝ g : α →ᵇ β\nx : α\nC : ℝ\nn : ℤ\nf : α →ᵇ β\n⊢ ∃ C, ∀ (x y : α), dist (n • f x) (n • f y) ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 58
} | {
"line": 185,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE✝ : Type u_3\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : CompactSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : SeminormedAddCommGroup E✝\ninst✝² : Nonempty α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : Nontrivial E\n⊢ NontrivialTopology C(α, E)",
"ppTerm": "?m.5",... | [
"α : Type u_1\nβ : Type u_2\nE✝ : Type u_3\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : CompactSpace α\ninst✝⁴ : PseudoMetricSpace β\ninst✝³ : SeminormedAddCommGroup E✝\ninst✝² : Nonempty α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : Nontrivial E\n⊢ ∃ x, ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 313,
"column": 42
} | {
"line": 313,
"column": 85
} | {
"line": 313,
"column": 86
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ∀ (x : α), f x = 0 ∨ g x = 0",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ∀ (x : α), f x = 0 ∨ g x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 30
} | {
"line": 379,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : C(α, R)\nh : f * g = 0\n⊢ ‖f - g‖ = max ‖f‖ ‖g‖",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : C(α, R)\nh : f * g = 0\n⊢ ‖f + -g‖ = max ‖f‖ ‖g‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 395,
"column": 39
} | {
"line": 395,
"column": 61
} | {
"line": 395,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nth... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nthis : f j * ∑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 32
} | {
"line": 396,
"column": 33
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x... | [
"case insert\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\nR : Type u_4\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_5\nf : ι → C(α, R)\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 327,
"column": 2
} | {
"line": 327,
"column": 30
} | {
"line": 327,
"column": 31
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ‖f - g‖ = max ‖f‖ ‖g‖",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrA... | [
"α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ‖f + -g‖ = max ‖f‖ ‖g‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 343,
"column": 39
} | {
"line": 343,
"column": 61
} | {
"line": 343,
"column": 62
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nthis : f j * ∑ i ∈ s, f i = 0\... | [
"α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nthis : f j * ∑ i ∈ s, f i = 0\n⊢ ‖f j + ∑ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 32
} | {
"line": 344,
"column": 33
} | [
{
"pp": "case insert\nα : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\n⊢ f j * ∑ i ∈ s, ... | [
"case insert\nα : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nι : Type u_2\nf : ι → α →ᵇ R\nh : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\n⊢ ∑ i ∈ s, f j * f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 374,
"column": 25
} | {
"line": 374,
"column": 50
} | {
"line": 374,
"column": 51
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\nR : Type u_1\ninst✝ : SeminormedRing R\nf : α →ᵇ R\nn : ℕ\n⊢ ∃ C, ∀ (x y : α), dist ((f.toContinuousMap ^ n).toFun x) ((f.toContinuousMap ^ n).toFun y) ≤ C",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real.... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\nR : Type u_1\ninst✝ : SeminormedRing R\nf : α →ᵇ R\nn : ℕ\n⊢ ∃ C, ∀ (x y : α), dist (f x ^ n) (f y ^ n) ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 25
} | {
"line": 94,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEDist x E\nkey : 1 - infEDist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEDist x E / ENNReal.ofReal δ ≤ ⊥",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"ENNReal.... | [
"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\nx_out : ENNReal.ofReal δ ≤ infEDist x E\nkey : 1 - infEDist x E / ENNReal.ofReal δ ≤ 1 - 1\n⊢ 1 - infEDist x E / ENNReal.ofReal δ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 589,
"column": 35
} | {
"line": 589,
"column": 46
} | {
"line": 589,
"column": 47
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : IsOrderedAddMonoid β\nf g : α →ᵇ β\nh₁ : f ≤ g\nh : α →ᵇ β\nt : α\n⊢ (fun f ↦ f.toFun) (f + h) t ≤ (fun f ↦ f.toFun) (g + h) t",
"ppTerm": "?m.17",
... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : IsOrderedAddMonoid β\nf g : α →ᵇ β\nh₁ : f ≤ g\nh : α →ᵇ β\nt : α\n⊢ f t ≤ g t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 151,
"column": 40
} | {
"line": 151,
"column": 91
} | {
"line": 151,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nE : Set α\nx : α\nx_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\nε : ℝ\nε_pos : 0 < ε\nε_lt : ENNReal.ofReal ε < infEDist x E\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε\nn : ℕ\nn_large : n ≥ N\n⊢ x ∉ thick... | [
"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nE : Set α\nx : α\nx_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\nε : ℝ\nε_pos : 0 < ε\nε_lt : ENNReal.ofReal ε < infEDist x E\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → |δseq b| < ε\nn : ℕ\nn_large : n ≥ N\n⊢ ENNReal.ofReal ε ≤ in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 13
} | {
"line": 188,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\n⊢ (ENNReal.toNNReal ∘ thickenedIndicatorAux δ E) x ≤ 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Function.comp",
"id",
... | [
"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ : ℝ\nδ_pos : 0 < δ\nE : Set α\nx : α\n⊢ (thickenedIndicatorAux δ E x).toNNReal ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 31
} | {
"line": 170,
"column": 32
} | [
{
"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 : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\nx : (i : δ) → X i\n⊢ (∫⋯∫⁻_s, f ∂μ) x = ∫⁻ (xᵢ : X i), (∫⋯... | [
"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 : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\nx : (i : δ) → X i\n⊢ (∫⋯∫⁻_s, f ∂μ) x = ∫⁻ (xᵢ : X i), (∫⋯∫⁻_s.erase i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 31
} | {
"line": 185,
"column": 32
} | [
{
"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 : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s.erase i, fun x ↦ ∫⁻ (xᵢ : X i), f ... | [
"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 : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ ∫⋯∫⁻_s, f ∂μ = ∫⋯∫⁻_s.erase i, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1098,
"column": 2
} | {
"line": 1100,
"column": 34
} | {
"line": 1102,
"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 C' : ℝ\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : Dominat... | [] | refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf
rw [one_smul]
exact Measure.le_add_left le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1098,
"column": 2
} | {
"line": 1100,
"column": 34
} | {
"line": 1102,
"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 C' : ℝ\nμ' : Measure α\nhT_add : DominatedFinMeasAdditive (μ + μ') T C'\nhT : Dominat... | [] | refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf
rw [one_smul]
exact Measure.le_add_left le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1181,
"column": 71
} | {
"line": 1181,
"column": 82
} | {
"line": 1181,
"column": 83
} | [
{
"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 μ' μ'' : Measure α\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : Domin... | [
"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 μ' μ'' : Measure α\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 15
} | {
"line": 169,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis✝ : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_closed ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis✝ : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_closed : IsClosed[i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 43
} | {
"line": 172,
"column": 44
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\n... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_closed : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 50
} | {
"line": 174,
"column": 51
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\n... | [
"case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nthis : 𝓝[≤] a = 𝓝[<] a ⊔ pure a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_closed : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\n... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_closed : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 44
} | {
"line": 205,
"column": 45
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (left... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 50
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (left... | [
"case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_... | by_cases! h''c : ¬ ∃ y, Tendsto f (𝓝[>] c) (𝓝 y) | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Topology.Order.LeftRightLim | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 51
} | {
"line": 207,
"column": 52
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (left... | [
"case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : T3Space β\nf : α → β\na : α\nh : Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (leftLim f a)\ns_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 34
} | {
"line": 225,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : T3Space β\ninst✝ : NoTopOrder α\nf g : α → β\nb : β\nh : Tendsto f atTop (𝓝 b)\nh' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f\nhα : Nonempty α\ns : S... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : T3Space β\ninst✝ : NoTopOrder α\nf g : α → β\nb : β\nh : Tendsto f atTop (𝓝 b)\nh' : ∀ᶠ (x : α) in atTop, MapClusterPt (g x) (𝓝 x) f\nhα : Nonempty α\ns : Set β\ns_mem ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 29
} | {
"line": 287,
"column": 30
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x ≤ y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[<] x = ⊥\n⊢... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x ≤ y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[<] x = ⊥\n⊢ f x ≤ f y"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 29
} | {
"line": 332,
"column": 30
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[<] y = ⊥\n⊢... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nhf : Monotone f\nx y : α\nh : x < y\nthis✝ : TopologicalSpace α := Preorder.topology α\nthis : OrderTopology α\nh' : 𝓝[<] y = ⊥\n⊢ rightLim f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 16
} | {
"line": 223,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\nf✝ : StieltjesFunction R\ninst✝ : OrderTopology R\nf : R → ℝ\nhf : Monotone f\n⊢ ∀ (x : R), ContinuousWithinAt (rightLim f) (Ici x) x",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : TopologicalSpace R\nf✝ : StieltjesFunction R\ninst✝ : OrderTopology R\nf : R → ℝ\nhf : Monotone f\nx : R\ns : Set ℝ\nhs : s ∈ 𝓝 (rightLim f x)\n⊢ s ∈ map (rightLim f) (𝓝[≥] x)"
] | intro x s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 338,
"column": 78
} | {
"line": 338,
"column": 89
} | {
"line": 338,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na b : R\nc d : ℕ → R\nss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)\nthis :\n ∀ (s : Finset ℕ) (b : R),\n Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑... | [
"R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na b : R\nc d : ℕ → R\nss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)\nthis :\n ∀ (s : Finset ℕ) (b : R),\n Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 193,
"column": 74
} | {
"line": 194,
"column": 56
} | {
"line": 196,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure X\nmY : MeasurableSpace Y\nν : Measure Y\nf : X → Y → E\ns : Set X\nhs : MeasurableSet s\n⊢ ∫ (x : X), ∫ (y : Y), s.indicator (fun x ↦ f x y) x ∂ν ∂μ = ∫ (x : X) in s, ∫... | [] | by
simp_rw [← integral_indicator hs, integral_indicator₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 63
} | {
"line": 352,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na : R\nc d : ℕ → R\ns✝ s : Finset ℕ\nIH :\n ∀ t ⊂ s,\n ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i)... | [
"R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na : R\nc d : ℕ → R\ns✝ s : Finset ℕ\nIH :\n ∀ t ⊂ s,\n ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 13
} | {
"line": 257,
"column": 14
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nι : Type u_5\nt : Finset ι\ns : ι → Set X\nhs : ∀ i ∈ t, MeasurableSet (s i)\nhf : ∀ i ∈ t, μ (s i) ≠ ∞\ni : ι\nhi : i ∈ t\n⊢ IntegrableOn (fun x ↦ 1) (s i) μ",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nι : Type u_5\nt : Finset ι\ns : ι → Set X\nhs : ∀ i ∈ t, MeasurableSet (s i)\nhf : ∀ i ∈ t, μ (s i) ≠ ∞\ni : ι\nhi : i ∈ t\n⊢ μ (s i) < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 385,
"column": 40
} | {
"line": 385,
"column": 51
} | {
"line": 385,
"column": 52
} | [
{
"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 : b ≤ a\n⊢ ofReal (↑f b - ↑f a) = 0",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"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 : b ≤ a\n⊢ ↑f b ≤ ↑f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 389,
"column": 35
} | {
"line": 389,
"column": 50
} | {
"line": 389,
"column": 51
} | [
{
"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 :=... | [
"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 := ε / 2\n⊢ ¬ε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 437,
"column": 35
} | {
"line": 437,
"column": 86
} | {
"line": 437,
"column": 86
} | [
{
"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\nhts : s ⊆ t\nh't : ∀ᵐ (x : X) ∂μ, x ∈ t \\ s → f x = 0\nhaux : StronglyMeasurable f\nh'aux : IntegrableOn f t μ\nk : Set X := f ⁻¹' {0}\nhk : MeasurableSet ... | [
"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\nhts : s ⊆ t\nh't : ∀ᵐ (x : X) ∂μ, x ∈ t \\ s → f x = 0\nhaux : StronglyMeasurable f\nh'aux : IntegrableOn f t μ\nk : Set X := f ⁻¹' {0}\nhk : MeasurableSet k\n⊢ ∫ (x : ... | integral_inter_add_sdiff hk (h'aux.mono hts le_rfl) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 496,
"column": 2
} | {
"line": 503,
"column": 56
} | {
"line": 505,
"column": 0
} | [
{
"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\n⊢ ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ",
"ppTerm": "?m.38",
"assigned"... | [] | have h_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0} := by
simp_rw [le_iff_lt_or_eq, setOf_or]
rw [h_union]
have B : NullMeasurableSet {x | f x = 0} μ :=
hf.nullMeasurableSet_eq_fun aestronglyMeasurable_zero
symm
refine integral_union_eq_left_of_ae ?_
filter_upwards [ae_restrict_mem₀ B] with x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 496,
"column": 2
} | {
"line": 503,
"column": 56
} | {
"line": 505,
"column": 0
} | [
{
"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\n⊢ ∫ (x : X) in {x | f x < 0}, f x ∂μ = ∫ (x : X) in {x | f x ≤ 0}, f x ∂μ",
"ppTerm": "?m.38",
"assigned"... | [] | have h_union : {x | f x ≤ 0} = {x | f x < 0} ∪ {x | f x = 0} := by
simp_rw [le_iff_lt_or_eq, setOf_or]
rw [h_union]
have B : NullMeasurableSet {x | f x = 0} μ :=
hf.nullMeasurableSet_eq_fun aestronglyMeasurable_zero
symm
refine integral_union_eq_left_of_ae ?_
filter_upwards [ae_restrict_mem₀ B] with x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 525,
"column": 71
} | {
"line": 525,
"column": 89
} | {
"line": 527,
"column": 0
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nhfi : Integrable f μ\nh_meas : NullMeasurableSet {x | 0 ≤ f x} μ\n⊢ ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {a | ¬0 ≤ f a}, f x ∂μ =\n ∫ (x : X) in {x | 0 ≤ f x}, f x ∂μ - ∫ (x : X) in {x | f x < 0}, f x ∂μ",
"ppTerm": "?m... | [] | simp only [not_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 532,
"column": 65
} | {
"line": 533,
"column": 53
} | {
"line": 535,
"column": 0
} | [
{
"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✝ : CompleteSpace E\ne : E\ns : Set X\ns_meas : MeasurableSet s\n⊢ ∫ (x : X), s.indicator (fun x ↦ e) x ∂μ = μ.real s • e",
"ppTerm": "?m.23",
"assigned": true,
"u... | [] | by
rw [integral_indicator s_meas, ← setIntegral_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Content | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 14
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\n⊢ μ K₁ ≤ μ K₂",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : ↑K₁ ⊆ ↑K₂\n⊢ μ K₁ ≤ μ K₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 56
} | {
"line": 108,
"column": 57
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : Disjoint ↑K₁ ↑K₂\nh₁ : IsClosed ↑K₁\nh₂ : IsClosed ↑K₂\n⊢ μ (K₁ ⊔ K₂) = μ K₁ + μ K₂",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\nh : Disjoint ↑K₁ ↑K₂\nh₁ : IsClosed ↑K₁\nh₂ : IsClosed ↑K₂\n⊢ μ (K₁ ⊔ K₂) = μ K₁ + μ K₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 30
} | {
"line": 111,
"column": 31
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\nK₁ K₂ : Compacts G\n⊢ μ (K₁ ⊔ K₂) ≤ μ K₁ + μ K₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Content | {
"line": 116,
"column": 30
} | {
"line": 116,
"column": 64
} | {
"line": 116,
"column": 65
} | [
{
"pp": "G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ ⊥ = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type w\ninst✝ : TopologicalSpace G\nμ : Content G\n⊢ μ ⊥ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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