module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 67
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ chaar K₀ ⊥ = 0",
"usedConstants": [
"MeasureTheory.Measure.haar.clPrehaar",
"Eq.mpr",
"TopologicalSpace.OpenNhdsOf.toOpens",
"MeasureTheory.Measure.haar.chaa... | let eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
have : Continuous eval := continuous_apply ⊥
change chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
unfold clPrehaar; rw [IsClosed.closure_subset_iff]
· rintro _ ⟨U, _, rfl⟩; apply prehaar_empty
· apply continuous_iff_isCl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 67
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ chaar K₀ ⊥ = 0",
"usedConstants": [
"MeasureTheory.Measure.haar.clPrehaar",
"Eq.mpr",
"TopologicalSpace.OpenNhdsOf.toOpens",
"MeasureTheory.Measure.haar.chaa... | let eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
have : Continuous eval := continuous_apply ⊥
change chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
unfold clPrehaar; rw [IsClosed.closure_subset_iff]
· rintro _ ⟨U, _, rfl⟩; apply prehaar_empty
· apply continuous_iff_isCl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LeftRightLim | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 43
} | [
{
"pp": "case h.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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 185,
"column": 74
} | {
"line": 186,
"column": 56
} | [
{
"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.Topology.Order.LeftRightLim | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 50
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 44
} | [
{
"pp": "case h.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 (le... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 50
} | [
{
"pp": "case h.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 (le... | 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": 208,
"column": 4
} | {
"line": 208,
"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))\nh' : (𝓝[<] a).NeBot\nb : α\nhb : b ∈ Iio a\ns : Set β\ns_mem : s ∈ 𝓝 (left... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 34
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 13
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 29
} | [
{
"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⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LeftRightLim | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 29
} | [
{
"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⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 16
} | [
{
"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",
"usedConstants": []
}
] | intro x s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 346,
"column": 78
} | {
"line": 346,
"column": 89
} | [
{
"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 - ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 73
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\ng : G\nK : Compacts G\n⊢ (haarContent K₀) (Compacts.map (fun x ↦ g * x) ⋯ K) = (haarContent K₀) K",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.haar.chaar",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 493,
"column": 2
} | {
"line": 496,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ 0 < (haarContent K₀).outerMeasure ↑K₀",
"usedConstants": [
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"Eq.ge",
"iInf",
"TopologicalSpace.PositiveCompacts.i... | refine zero_lt_one.trans_le ?_
rw [Content.outerMeasure_eq_iInf]
refine le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans ?_ <| le_iSup₂ K₀.toCompacts hK₀
exact haarContent_self.ge | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 493,
"column": 2
} | {
"line": 496,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ 0 < (haarContent K₀).outerMeasure ↑K₀",
"usedConstants": [
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"Eq.ge",
"iInf",
"TopologicalSpace.PositiveCompacts.i... | refine zero_lt_one.trans_le ?_
rw [Content.outerMeasure_eq_iInf]
refine le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans ?_ <| le_iSup₂ K₀.toCompacts hK₀
exact haarContent_self.ge | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 383,
"column": 43
} | {
"line": 383,
"column": 79
} | [
{
"pp": "case neg\nX : 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\nht : IntegrableOn f t μ\nH : ¬IntegrableOn f (s ∪ t) μ\n⊢ ¬Integrable f (μ.restrict s)",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 63
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 13
} | [
{
"pp": "case h\nG : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEapprox :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 393,
"column": 40
} | {
"line": 393,
"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 : b ≤ a\n⊢ ofReal (↑f b - ↑f a) = 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 397,
"column": 35
} | {
"line": 397,
"column": 50
} | [
{
"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 :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 477,
"column": 2
} | {
"line": 484,
"column": 56
} | [
{
"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",
"usedConstants": [
"Measure... | 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": 477,
"column": 2
} | {
"line": 484,
"column": 56
} | [
{
"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",
"usedConstants": [
"Measure... | 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": 508,
"column": 71
} | {
"line": 508,
"column": 89
} | [
{
"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",
"usedConstant... | simp only [not_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 515,
"column": 65
} | {
"line": 516,
"column": 53
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Measu... | by
rw [integral_indicator s_meas, ← setIntegral_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 575,
"column": 2
} | {
"line": 575,
"column": 13
} | [
{
"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\nC : ℝ\nhs : (μ.restrict s) univ < ∞\nhC : ∀ᵐ (x : X) ∂μ.restrict s, ‖f x‖ ≤ C\nthis : IsFiniteMeasure (μ.restrict s)\n⊢ ‖∫ (x : X) in s, f x ∂μ‖ ≤ C * μ.real ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 490,
"column": 8
} | {
"line": 490,
"column": 19
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁴ : OrderTopology R\ninst✝³ : CompactIccSpace R\ninst✝² : MeasurableSpace R\ninst✝¹ : BorelSpace R\ninst✝ : DenselyOrdered R\ns : Set R\nt : ℕ → Set R\nht : s ⊆ ⋃ i, t i\nε : ℝ≥0\nε0 : 0 < ε\nh : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 508,
"column": 48
} | {
"line": 508,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁴ : OrderTopology R\ninst✝³ : CompactIccSpace R\ninst✝² : MeasurableSpace R\ninst✝¹ : BorelSpace R\ninst✝ : DenselyOrdered R\ns : Set R\nt : ℕ → Set R\nht : s ⊆ ⋃ i, t i\nε : ℝ≥0\nε0 : 0 < ε\nh : ∑' (i : ℕ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 26
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\ns : Set X\nμ : Measure X\nf : X → ℝ\nhf : 0 ≤ᶠ[ae (μ.restrict s)] f\nhfi : IntegrableOn f s μ\n⊢ NullMeasurableSet (support f) (μ.restrict s)",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Compl.compl",
... | rw [support_eq_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 326,
"column": 81
} | {
"line": 326,
"column": 93
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\n⊢ normSq x * t * t + re ⟪x, y⟫ * t + t * re ⟪y, x⟫ + re ⟪y, y⟫ =\n normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫",
"usedConstants": [
... | mul_comm t _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 543,
"column": 2
} | {
"line": 551,
"column": 53
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na : R\nha : IsBot a\n⊢ f.measure {... | · have : leftLim f a = f a := by
apply leftLim_eq_of_eq_bot
simp [nhdsLT_eq_bot_iff, ha]
simp only [this, sub_self, ofReal_zero]
apply eq_bot_iff.2
rw [StieltjesFunction.measure]
apply (outer_le_length _ _).trans
rw [← length_diff_botSet]
simp [subsingleton_botSet.eq_singleton_of_mem... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 552,
"column": 36
} | {
"line": 552,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na : R\nha : ¬IsBot a\n⊢ ∃ b, b < a",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 559,
"column": 50
} | {
"line": 559,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na : R\nha : ¬IsBot a\nb : R\nhb : b < a\nu :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 567,
"column": 19
} | {
"line": 567,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\na : R\nha : ¬IsBot a\nb : R\nhb : b < a\nu :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 787,
"column": 2
} | {
"line": 787,
"column": 24
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\ns : Set X\nf : X → ℝ\nc : ℝ\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nhf : ∀ x ∈ s, c ≤ f x\nhfint : IntegrableOn (fun x ↦ f x) s μ\n⊢ c * μ.real s ≤ ∫ (x : X) in s, f x ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\ncd : Core 𝕜 F\nx : F\n⊢ normSq x = 0 ↔ ⟪x, x⟫ = 0",
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"InnerProductSpace.Core.toC... | simp only [normSq, ext_iff, map_zero, inner_self_im, and_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\ncd : Core 𝕜 F\nx : F\n⊢ normSq x = 0 ↔ ⟪x, x⟫ = 0",
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"InnerProductSpace.Core.toC... | simp only [normSq, ext_iff, map_zero, inner_self_im, and_true] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\ncd : Core 𝕜 F\nx : F\n⊢ normSq x = 0 ↔ ⟪x, x⟫ = 0",
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"InnerProductSpace.Core.toC... | simp only [normSq, ext_iff, map_zero, inner_self_im, and_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 847,
"column": 4
} | {
"line": 847,
"column": 15
} | [
{
"pp": "case hs\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nf_int : Integrable f μ\nf_nonneg : 0 ≤ᶠ[ae μ] f\ns : Set X\nhs : ∀ x ∈ s, 1 ≤ f x\nx : X\nhx : x ∈ s\n⊢ 1 ≤ ENNReal.ofReal (f x)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 865,
"column": 6
} | {
"line": 865,
"column": 21
} | [
{
"pp": "case pos\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH✝ : Integrable f μ\ng : X → ℝ := fun x ↦ max (f x) 0\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\nx : X\nH : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 867,
"column": 6
} | {
"line": 867,
"column": 21
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH✝ : Integrable f μ\ng : X → ℝ := ⋯\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\nx : X\nH : x ∉ s\n⊢ g x ≤ 0",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 869,
"column": 4
} | {
"line": 869,
"column": 19
} | [
{
"pp": "case pos.h'f\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH : Integrable f μ\ng : X → ℝ := fun x ↦ max (f x) 0\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\nx : X\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 911,
"column": 2
} | {
"line": 911,
"column": 42
} | [
{
"pp": "X : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\nι : Type u_5\ninst✝⁵ : Countable ι\nμ : Measure X\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : TopologicalSpace X\ninst✝² : BorelSpace X\ninst✝¹ : T2Space X\ninst✝ : IsLocallyFiniteMeasure μ\nf : C(X, E)\ns : ι → Compacts X\nhf : Summable fun i ↦ ‖Continu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 685,
"column": 4
} | {
"line": 685,
"column": 15
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nx : R\nthis : Nonempty R\nval✝ : O... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 15
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 747,
"column": 2
} | {
"line": 754,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg : f.meas... | ext x
have hf := measure_Iic f hfl x
rw [hfg, measure_Iic g hgl x, ENNReal.ofReal_eq_ofReal_iff, eq_comm] at hf
· simpa using hf
· rw [sub_nonneg]
exact Monotone.le_of_tendsto g.mono hgl x
· rw [sub_nonneg]
exact Monotone.le_of_tendsto f.mono hfl x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 747,
"column": 2
} | {
"line": 754,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg : f.meas... | ext x
have hf := measure_Iic f hfl x
rw [hfg, measure_Iic g hgl x, ENNReal.ofReal_eq_ofReal_iff, eq_comm] at hf
· simpa using hf
· rw [sub_nonneg]
exact Monotone.le_of_tendsto g.mono hgl x
· rw [sub_nonneg]
exact Monotone.le_of_tendsto f.mono hfl x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 17
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 772,
"column": 6
} | {
"line": 772,
"column": 17
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 27
} | [
{
"pp": "case h.h\nF : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\n⊢ ((innerₗ F).flip v) w = ((innerₗ F) v) w",
"usedConstants": [
"real_inner_comm"
]
}
] | exact real_inner_comm v w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 19
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nne : K.Nonempty\nh₁ : IsComplete K\nh₂ : Convex ℝ K\nu : F\nδ : ℝ := ⨅ w, ‖u - ↑w‖\nthis✝ : Nonempty ↑K := Set.Nonempty.to_subtype ne\nzero_le_δ : 0 ≤ δ\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖\... | apply δ_le' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhz : ∀ (i : ι), v i ≠ 0\nho : Pairwise fun i j ↦ ⟪v i, v j⟫ = 0\ns : Finset ι\ng : ι → 𝕜\nhg : ∑ i ∈ s, g i • v i = 0\ni : ι\nhi : i ∈ s\nh' : g i * ⟪v i, v i⟫ = ⟪v ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 13
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : F\nh : re ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖\n⊢ ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Inner.inner",
"Real.instRCLike",
"inner_self_eq_norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 13
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nh : ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * re ⟪x, y⟫_ℝ + ‖y‖ ^ 2\n⊢ ‖x + y‖ ^ 2 = ‖x‖ ^ 2 + 2 * ⟪x, y⟫_ℝ + ‖y‖ ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 166,
"column": 4
} | {
"line": 167,
"column": 11
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝⁵ : Group G\ninst✝⁴ : Group H\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace α\ninst✝¹ : MulAction H β\ninst✝ : MeasurableSpace β\ns : Set α\nμ : Measure α\nν : Measure β\nh : IsFundamentalDomain G s μ\nf : β → α\nhf : QuasiMeasurePreservi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 13
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nh : ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖\n⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 267,
"column": 2
} | {
"line": 268,
"column": 87
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nν : Measure α\nh : IsFundamentalDomain G s μ\nhν : ν ≪ μ\nt : Set α\nH : ν.restrict t ≪... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1122,
"column": 4
} | {
"line": 1122,
"column": 65
} | [
{
"pp": "Y : Type u_2\nE : Type u_3\nF : Type u_4\nX : Type u_5\nG : Type u_6\n𝕜 : Type u_7\ninst✝¹¹ : TopologicalSpace X\ninst✝¹⁰ : TopologicalSpace Y\ninst✝⁹ : MeasurableSpace Y\ninst✝⁸ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 37
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nh : IsFundamentalDomain G s μ\nt : Set α\n⊢ μ t = ∑' (g : G), μ (g • t ∩ s)",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 13
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nh : ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖\n⊢ ‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 67
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nhs : IsFundamentalDomain G s μ\nht : IsFundamentalDomain G t μ\nA : Set α\nhA₀ : Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1126,
"column": 4
} | {
"line": 1126,
"column": 36
} | [
{
"pp": "Y : Type u_2\nE : Type u_3\nF : Type u_4\nX : Type u_5\nG : Type u_6\n𝕜 : Type u_7\ninst✝¹¹ : TopologicalSpace X\ninst✝¹⁰ : TopologicalSpace Y\ninst✝⁹ : MeasurableSpace Y\ninst✝⁸ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 37
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ninst✝² : MeasurableConstSMul G α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : Countable G\nhs : IsFundamentalDomain G s μ\nht : IsFundamentalDomain G t μ\n⊢ μ s = μ t",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 97
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁷ : Group G\ninst✝⁶ : MulAction G α\ninst✝⁵ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ninst✝⁴ : MeasurableConstSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nβ : Type u_6\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nhs : IsF... | simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 97
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁷ : Group G\ninst✝⁶ : MulAction G α\ninst✝⁵ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ninst✝⁴ : MeasurableConstSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nβ : Type u_6\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nhs : IsF... | simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 97
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁷ : Group G\ninst✝⁶ : MulAction G α\ninst✝⁵ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ninst✝⁴ : MeasurableConstSMul G α\ninst✝³ : SMulInvariantMeasure G α μ\ninst✝² : Countable G\nβ : Type u_6\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nhs : IsF... | simp only [smul_set_inter, inter_comm, smul_inv_smul, aestronglyMeasurable_sum_measure_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\n⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid... | have : ‖x‖ ^ 2 ≠ 0 := pow_ne_zero _ (norm_ne_zero_iff.2 hx₀) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 15
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ‖⟪x, y⟫ / (↑‖x‖ * ↑‖y‖)‖ = 1\nhx₀ : x ≠ 0\nhy₀ : y ≠ 0\n⊢ ‖⟪x, y⟫‖ / (‖x‖ * ‖y‖) = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 751,
"column": 4
} | {
"line": 753,
"column": 77
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ (x ≠ 0 ∧ ∃ r, r ≠ 0 ∧ y = r • x) → ‖⟪x, y⟫ / (↑‖x‖ * ↑‖y‖)‖ = 1",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Inne... | rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩
simp only [norm_div, norm_mul, norm_ofReal, abs_norm]
exact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 751,
"column": 4
} | {
"line": 753,
"column": 77
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ (x ≠ 0 ∧ ∃ r, r ≠ 0 ∧ y = r • x) → ‖⟪x, y⟫ / (↑‖x‖ * ↑‖y‖)‖ = 1",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Inne... | rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩
simp only [norm_div, norm_mul, norm_ofReal, abs_norm]
exact norm_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_ne_zero_mul hx hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 153,
"column": 21
} | {
"line": 153,
"column": 32
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nh : Convex ℝ K\nu v : F\nhv : v ∈ K\nthis : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩\neq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nw : F\nhw : w ∈ K\nδ : ℝ := ⨅ w, ‖u - ↑w‖\np : ℝ := ⟪u - v, w - v⟫_ℝ\nq : ℝ := ‖w - v‖ ^ 2\nδ_le : ∀ (w... | apply δ_le' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 25
} | [
{
"pp": "case hw\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Set F\nh : Convex ℝ K\nu v : F\nhv : v ∈ K\nthis : Nonempty ↑K := ⋯\neq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nw : F\nhw : w ∈ K\nδ : ℝ := ⋯\np : ℝ := ⋯\nq : ℝ := ⋯\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w... | apply h hw hv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 820,
"column": 2
} | {
"line": 820,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ ⟪x, y⟫ = 1 ↔ x = y",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpa... | convert! inner_eq_norm_mul_iff (𝕜 := 𝕜) (E := E) using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 22
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ ⟪x, y⟫_ℝ ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 836,
"column": 2
} | {
"line": 836,
"column": 22
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ -1 ≤ ⟪x, y⟫_ℝ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nhle : ‖x‖ ≤ ‖y‖\nh : re ⟪x, y⟫ = ‖y‖ ^ 2\nH₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2\nH₂ : re ⟪y, x⟫ = ‖y‖ ^ 2\n⊢ re (⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫)) ≤ 0",
"usedConstants": [
"NormedCommR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 55
} | [
{
"pp": "case refine_2\nG : Type u_1\nα : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G α\ns : Set α\na b : G\nhab : a ≠ b\nx : α\nhx : x ∈ s ∧ ∀ (g : G), g ≠ 1 → x ∉ g • s\ny : α\nhy : y ∈ s ∧ ∀ (g : G), g ≠ 1 → y ∉ g • s\nhxy : (fun x ↦ b • x) y = (fun x ↦ a • x) x\n⊢ x ∈ (a⁻¹ * b) • s",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 585,
"column": 2
} | {
"line": 586,
"column": 25
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝³ : Countable G\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\n⊢ μ (fundamentalFrontier G s) = 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Continuous | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | [
{
"pp": "E : Type u_4\n𝕜 : Type u_7\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx : E\nS : Set E\nhS : Dense S\nh : ∀ v ∈ S, ⟪x, v⟫ = 0\nK : Submodule 𝕜 E := span 𝕜 S\nhK : Dense ↑K\nthis : (fun x_1 ↦ ⟪x, x_1⟫) = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 604,
"column": 15
} | {
"line": 604,
"column": 31
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Countable G\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : IsFundamentalDomain G s μ\ninst✝¹ : MeasurableConstSMul G α\ninst✝ : SMulInvariantMeasure G α μ\n⊢ (⋃ g, g⁻¹ • s) \\ ⋃ g, g⁻¹ • fundamentalFrontier G s ... | diff_subset_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 809,
"column": 2
} | {
"line": 809,
"column": 17
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nν : Measure α\ninst✝² : SMulInvariantMeasure G α ν\ninst✝¹ : Countable G\ninst✝ : MeasurableConstSMul G α\nμ : Measure (Quotient α_mod_G)\ns : Set α\nfund_dom_s : IsFundamentalDomain G s ν\nh : μ = Measure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 874,
"column": 4
} | {
"line": 874,
"column": 41
} | [
{
"pp": "case pos\nG : Type u_1\nα : Type u_3\ninst✝⁷ : Group G\ninst✝⁶ : MulAction G α\ninst✝⁵ : MeasurableSpace α\nν : Measure α\ninst✝⁴ : SMulInvariantMeasure G α ν\ninst✝³ : Countable G\ninst✝² : MeasurableConstSMul G α\nμ : Measure (Quotient α_mod_G)\ninst✝¹ : QuotientMeasureEqMeasurePreimage ν μ\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 232,
"column": 2
} | {
"line": 234,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nh : IsComplete ↑K\n⊢ ∀ (u : E), ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
... | letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
let K' : Submodule ℝ E := Submodule.restrictScalars ℝ K
exact exists_norm_eq_iInf_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 232,
"column": 2
} | {
"line": 234,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nh : IsComplete ↑K\n⊢ ∀ (u : E), ∃ v ∈ K, ‖u - v‖ = ⨅ w, ‖u - ↑w‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
... | letI : InnerProductSpace ℝ E := InnerProductSpace.rclikeToReal 𝕜 E
let K' : Submodule ℝ E := Submodule.restrictScalars ℝ K
exact exists_norm_eq_iInf_of_complete_convex ⟨0, K'.zero_mem⟩ h K'.convex | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1187,
"column": 2
} | {
"line": 1187,
"column": 13
} | [
{
"pp": "Y : Type u_2\nE : Type u_3\nX : Type u_5\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace Y\ninst✝³ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : X → Y → E\ns : Set X\nk : Set Y\ninst✝ : IsFiniteMeasureOnCompacts... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 23
} | [
{
"pp": "case mpr.right\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝ : DecidableEq ι\nv : ι → E\nh : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0\ni j : ι\nhij : i ≠ j\n⊢ ⟪v i, v j⟫ = 0",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 46
} | [
{
"pp": "case left\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 51
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Baire.CompleteMetrizable | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyPseudoMetrizableSpace X\nx✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\nhd : ∀ (n : ℕ), Dense (f n)\nB : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n\nBpos : ∀ (n : ℕ... | let c : ℕ → X := fun n => (F n).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 61
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhv : Orthonormal 𝕜 v\nl : ι →₀ 𝕜\nhl : (linearCombination 𝕜 v) l = 0\ni : ι\nkey : ⟪v i, (linearCombination 𝕜 v) l⟫ = ⟪v i, 0⟫\n⊢ l i = 0 i",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = v i\nhj : w j = v j\nh : i = j\nhv : ⟪v i, v j⟫ = 1\n⊢ ⟪w i, w j⟫ = 1",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = v i\nhj : w j = v j\nh : ¬i = j\nhv : ⟪v i, v j⟫ = 0\n⊢ ⟪w i, w j⟫ = 0",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = v i\nhj : w j = -v j\nh : i = j\nhv : ⟪v i, v j⟫ = 1\n⊢ ⟪w i, w j⟫ = 1",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = v i\nhj : w j = -v j\nh : ¬i = j\nhv : ⟪v i, v j⟫ = 0\n⊢ ⟪w i, w j⟫ = 0",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = -v i\nhj : w j = v j\nh : i = j\nhv : ⟪v i, v j⟫ = 1\n⊢ ⟪w i, w j⟫ = 1",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = -v i\nhj : w j = v j\nh : ¬i = j\nhv : ⟪v i, v j⟫ = 0\n⊢ ⟪w i, w j⟫ = 0",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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