module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Group.Action | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 35
} | {
"line": 99,
"column": 36
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ s ≤ μ ((fun x ↦ c • x) ⁻¹' s)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ s ≤ μ ((fun x ↦ c • x) ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 39
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ (c • s) = μ s",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns : Set α\n⊢ μ (c • s) = μ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 23
} | {
"line": 111,
"column": 24
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∩ t) = μ (s ∩ c • t)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∩ t) = μ (s ∩ c • t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 23
} | {
"line": 119,
"column": 24
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∪ t) = μ (s ∪ c • t)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s ∪ t) = μ (s ∪ c • t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 23
} | {
"line": 127,
"column": 24
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s \\ t) = μ (s \\ c • t)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ (c⁻¹ • s \\ t) = μ (s \\ c • t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 23
} | {
"line": 135,
"column": 24
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ ((c⁻¹ • s) ∆ t) = μ (s ∆ (c • t))",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝² : Group G\ninst✝¹ : MulAction G α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nc : G\ns t : Set α\n⊢ μ ((c⁻¹ • s) ∆ t) = μ (s ∆ (c • t))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Action | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 40
} | {
"line": 305,
"column": 4
} | [
{
"pp": "G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\nμ : Measure α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : MeasurableConstSMul G α\ns : Set α\nhs : NullMeasurableSet s μ\nc : G\n⊢ NullMeasurableSet (c • s) μ",
"ppTerm": "?m.17",
"assigned": true,
"u... | [
"G : Type u\nα : Type w\nm : MeasurableSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\nμ : Measure α\ninst✝¹ : SMulInvariantMeasure G α μ\ninst✝ : MeasurableConstSMul G α\ns : Set α\nhs : NullMeasurableSet s μ\nc : G\n⊢ NullMeasurableSet ((fun x ↦ c⁻¹ • x) ⁻¹' s) μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 167,
"column": 56
} | {
"line": 167,
"column": 67
} | {
"line": 167,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nh : ∀ (s : Set β), IsCompact s → IsCompact (f ⁻¹' s)\ns : Set β\nhs : s ∈ cocompact β\nt : Set β\nht : IsCompact t\nhts : tᶜ ⊆ s\n⊢ (f ⁻¹' t)ᶜ ⊆ f ⁻¹' s",
"ppTerm": "?m.47",
"assigned": false,
"u... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nh : ∀ (s : Set β), IsCompact s → IsCompact (f ⁻¹' s)\ns : Set β\nhs : s ∈ cocompact β\nt : Set β\nht : IsCompact t\nhts : tᶜ ⊆ s\n⊢ (f ⁻¹' t)ᶜ ⊆ f ⁻¹' s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 66
} | {
"line": 177,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\n⊢ ?m.15 ∈ cocompact ?m.13",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\n⊢ ?m.15 ∈ cocompact ?m.13"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 181,
"column": 65
} | {
"line": 181,
"column": 76
} | {
"line": 181,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\nt : Set α\nht : IsCompact t\nhts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t\n⊢ ⇑f ⁻¹' s ⊆ t",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"use... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nh's : IsClosed s\nt : Set α\nht : IsCompact t\nhts : (⇑f ⁻¹' s)ᶜᶜ ⊆ t\n⊢ ⇑f ⁻¹' s ⊆ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 589,
"column": 4
} | {
"line": 589,
"column": 15
} | {
"line": 589,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : μ.OuterRegular\nH : μ.InnerRegularWRT p IsOpen[inst✝¹]\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nr : ℝ≥0∞\nhr : r < μ s\nthis... | [
"α : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\np : Set α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : μ.OuterRegular\nH : μ.InnerRegularWRT p IsOpen[inst✝¹]\nhd : ∀ ⦃s U : Set α⦄, p s → IsOpen[inst✝¹] U → p (s \\ U)\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nr : ℝ≥0∞\nhr : r < μ s\nthis : 0 < μ uni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 717,
"column": 2
} | {
"line": 717,
"column": 23
} | {
"line": 719,
"column": 2
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | [
"case h₁\nγ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpace γ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 663,
"column": 49
} | {
"line": 663,
"column": 67
} | {
"line": 663,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nH✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]\nhfin : ∀ {s : Set α}, μ s ≠ ∞\ns : ℕ → Set α\nhsd : Pairwise (Function.onFun Disjoint s)\nhsm : ∀ (i : ℕ), Meas... | [
"α : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nH✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]\nhfin : ∀ {s : Set α}, μ s ≠ ∞\ns : ℕ → Set α\nhsd : Pairwise (Function.onFun Disjoint s)\nhsm : ∀ (i : ℕ), MeasurableSet (s... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 740,
"column": 6
} | {
"line": 740,
"column": 38
} | {
"line": 740,
"column": 39
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | [
"γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpace γ := upgra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 762,
"column": 8
} | {
"line": 762,
"column": 35
} | {
"line": 762,
"column": 36
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | [
"γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpace γ := upgra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1032,
"column": 11
} | {
"line": 1032,
"column": 34
} | {
"line": 1032,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.prod ν)\n⊢ ∫⁻ (z : α × β), f z ∂μ.prod ν = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν",
"ppTerm": "?m.34",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.prod ν)\n⊢ ∫⁻ (z : β × α), f z.swap ∂ν.prod μ = ∫⁻ (y : β), ∫⁻ (x : α), f (x, y) ∂μ ∂ν"
] | ← lintegral_prod_swap f | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.Measure | {
"line": 624,
"column": 2
} | {
"line": 624,
"column": 62
} | {
"line": 624,
"column": 63
} | [
{
"pp": "G : Type u_1\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : BorelSpace G\nμ : Measure G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : μ.Regular\nhμ : μ ≠ 0\ns : Set G\nhs : IsOpen[inst✝⁵] s\n⊢ μ s ≠ 0 ↔ s.Nonempty",
"ppTerm": "?m.22",
... | [
"G : Type u_1\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : BorelSpace G\nμ : Measure G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : μ.Regular\nhμ : μ ≠ 0\ns : Set G\nhs : IsOpen[inst✝⁵] s\n⊢ ¬s = ∅ ↔ s.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 928,
"column": 4
} | {
"line": 928,
"column": 64
} | {
"line": 928,
"column": 65
} | [
{
"pp": "case a\nα : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\nK : Set α\nhK : IsCompact K\n⊢ ∀ (c : ℝ≥0∞), μ K < c → ⨅ U, ⨅ (_ : K ⊆ U), ⨅ (_ : IsOpen[inst✝⁴... | [
"case a\nα : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\nK : Set α\nhK : IsCompact K\n⊢ ∀ (c : ℝ≥0∞), μ K < c → ∃ i, K ⊆ i ∧ IsOpen[inst✝⁴] i ∧ μ i < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Prod | {
"line": 149,
"column": 4
} | {
"line": 150,
"column": 24
} | {
"line": 150,
"column": 25
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhμs : μ s = 0\nhf : Measurable fun z ↦ (z.2 * z.1, z.1⁻¹)\nthis : (map (fun z ↦ (z.2 * z.1,... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhμs : μ s = 0\nhf : Measurable fun z ↦ (z.2 * z.1, z.1⁻¹)\nthis : (map (fun z ↦ (z.2 * z.1, z.1⁻¹)) (μ.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 787,
"column": 8
} | {
"line": 787,
"column": 35
} | {
"line": 787,
"column": 36
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝¹ : UpgradedIsCompletelyMetrizableSpa... | [
"γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.LIntegral | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 13
} | {
"line": 40,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ninst✝² : InvolutiveInv G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → ℝ≥0∞\n⊢ ∫⁻ (x : G), f x⁻¹ ∂μ = ∫⁻ (x : G), f x ∂μ",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"G : Type u_1\ninst✝³ : MeasurableSpace G\nμ : Measure G\ninst✝² : InvolutiveInv G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nf : G → ℝ≥0∞\n⊢ ∫⁻ (x : G), f x⁻¹ ∂μ = ∫⁻ (x : G), f x ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LConvolution | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 77,
"column": 0
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ 0 ⋆ₘₗ[μ] f = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LConvolution | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 77,
"column": 0
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ 0 ⋆ₘₗ[μ] f = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LConvolution | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 27
} | {
"line": 83,
"column": 0
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ f ⋆ₘₗ[μ] 0 = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LConvolution | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 27
} | {
"line": 83,
"column": 0
} | [
{
"pp": "G : Type u_1\nmG : MeasurableSpace G\ninst✝¹ : Mul G\ninst✝ : Inv G\nf : G → ℝ≥0∞\nμ : Measure G\n⊢ f ⋆ₘₗ[μ] 0 = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.lintegral_const",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
... | [] | ext; simp [mlconvolution] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 75,
"column": 2
} | {
"line": 80,
"column": 79
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.withDensity_apply_le",
"MeasureTheory.Measure.withDensity",
... | [] | apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFini... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 75,
"column": 2
} | {
"line": 80,
"column": 79
} | {
"line": 82,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\n⊢ (μ.withDensity f) s = ∫⁻ (a : α) in s, f a ∂μ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.withDensity_apply_le",
"MeasureTheory.Measure.withDensity",
... | [] | apply le_antisymm ?_ (withDensity_apply_le f s)
let t := toMeasurable μ s
calc
μ.withDensity f s ≤ μ.withDensity f t := measure_mono (subset_toMeasurable μ s)
_ = ∫⁻ a in t, f a ∂μ := withDensity_apply f (measurableSet_toMeasurable μ s)
_ = ∫⁻ a in s, f a ∂μ := by congr 1; exact restrict_toMeasurable_of_sFini... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LConvolution | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 25
} | {
"line": 126,
"column": 26
} | [
{
"pp": "case hf\nG : Type u_1\nmG : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : SFinite μ\nf g k : G → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhk : AEMeasurable k μ\nx : G\n⊢ AEMeasurable (Function.u... | [
"case hf\nG : Type u_1\nmG : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsMulLeftInvariant\ninst✝ : SFinite μ\nf g k : G → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhk : AEMeasurable k μ\nx : G\n⊢ AEMeasurable (Function.uncurry fun y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 29
} | {
"line": 110,
"column": 30
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : Measurable g\n⊢ μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : Measurable g\n⊢ μ.withDensity (f + g) = μ.withDensity f + μ.withDensity g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 44
} | {
"line": 127,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (r • f)) s = (r • μ.withDensity f) s",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withD... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ"
] | rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul r hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 50
} | {
"line": 282,
"column": 4
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : μ ({x | f x ≠ 0} ∩ s) = 0\nt✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)\nA : s ⊆ t✝ ∪ {x | f x = 0}\ng : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\nt : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}\n⊢ (μ.withDen... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : μ ({x | f x ≠ 0} ∩ s) = 0\nt✝ : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)\nA : s ⊆ t✝ ∪ {x | f x = 0}\ng : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\nt : {x | f x = 0} =ᵐ[μ.withDensity f] {x | g x = 0}\n⊢ (μ.withDensity g) {x |... | rw [measure_congr t, withDensity_congr_ae hfg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 336,
"column": 37
} | {
"line": 336,
"column": 79
} | {
"line": 336,
"column": 80
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nf' : α → ℝ≥0\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : ↑(f a) ≠ 0 ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nf' : α → ℝ≥0\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : ↑(f a) ≠ 0 → g a = g' a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 366,
"column": 2
} | {
"line": 368,
"column": 66
} | {
"line": 370,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ (dirac a).withDensity f = f a • dirac a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"instHSMul",
"MeasureTheory.Measure",
... | [] | ext s hs
classical
simp [withDensity_apply f hs, setLIntegral_dirac, Set.indicator] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 366,
"column": 2
} | {
"line": 368,
"column": 66
} | {
"line": 370,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\na : α\n⊢ (dirac a).withDensity f = f a • dirac a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"instHSMul",
"MeasureTheory.Measure",
... | [] | ext s hs
classical
simp [withDensity_apply f hs, setLIntegral_dirac, Set.indicator] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 51
} | {
"line": 161,
"column": 52
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nc : ε\nhc : ‖c‖ₑ ≠ ∞\n⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Function.L1Space.HasFinit... | [
"α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nc : ε\nhc : ‖c‖ₑ ≠ ∞\n⊢ (¬‖c‖ₑ = 0 → ¬μ univ = ∞) → ‖c‖ₑ = ∞ → μ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 453,
"column": 65
} | {
"line": 456,
"column": 19
} | {
"line": 458,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nhg : AEMeasurable g (μ.withDensity f)\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, g a ∂μ.withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ",
"ppTerm": "?m.32",
"assigned": true,
"use... | [] | by
rw [restrict_withDensity hs, lintegral_withDensity_eq_lintegral_mul₀' hf.restrict]
rw [← restrict_withDensity hs]
exact hg.restrict | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 255,
"column": 35
} | {
"line": 255,
"column": 76
} | {
"line": 255,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (-f) μ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroCl... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 263,
"column": 39
} | {
"line": 263,
"column": 80
} | {
"line": 263,
"column": 81
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegra... | [
"α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhfi : HasFiniteIntegral f μ\n⊢ ∫⁻ (x : α), ‖f x‖ₑ ∂μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 267,
"column": 47
} | {
"line": 267,
"column": 88
} | {
"line": 267,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun a ↦ ‖f a‖) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.to... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 551,
"column": 2
} | {
"line": 557,
"column": 6
} | {
"line": 559,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\n⊢ (μ.withDensity f).withDensity f⁻¹ ≤ μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"le_refl",
... | [] | change (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) ≤ μ
rw [← withDensity_mul₀ hf hf.fun_inv]
suffices (f * fun x ↦ (f x)⁻¹) ≤ᵐ[μ] 1 by
refine (withDensity_mono this).trans ?_
rw [withDensity_one]
filter_upwards with x
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 551,
"column": 2
} | {
"line": 557,
"column": 6
} | {
"line": 559,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\n⊢ (μ.withDensity f).withDensity f⁻¹ ≤ μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"le_refl",
... | [] | change (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) ≤ μ
rw [← withDensity_mul₀ hf hf.fun_inv]
suffices (f * fun x ↦ (f x)⁻¹) ≤ᵐ[μ] 1 by
refine (withDensity_mono this).trans ?_
rw [withDensity_one]
filter_upwards with x
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 410,
"column": 4
} | {
"line": 410,
"column": 70
} | {
"line": 411,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nF : ℕ → α → β\nf : α → β\nbound : α → ℝ\nF_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ\nbound_hasFiniteIntegral : HasFiniteIntegral bound μ\nh_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a\nh_... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nF : ℕ → α → β\nf : α → β\nbound : α → ℝ\nF_measurable : ∀ (n : ℕ), AEStronglyMeasurable (F n) μ\nbound_hasFiniteIntegral : HasFiniteIntegral bound μ\nh_bound : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a\nh_lim : ∀ᵐ (a ... | refine h_lim.mono fun a h => (continuous_ofReal.tendsto _).comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Real.Sqrt | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 29
} | {
"line": 251,
"column": 30
} | [
{
"pp": "case mp\nx y : ℝ\nh : 0 ≤ y\n⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nx y : ℝ\nh : 0 ≤ y\n⊢ x ^ 2 ≤ y → -√y ≤ x ∧ x ≤ √y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Sqrt | {
"line": 266,
"column": 56
} | {
"line": 266,
"column": 67
} | {
"line": 266,
"column": 68
} | [
{
"pp": "x : ℝ\nh : 0 ≤ x\n⊢ √x = 0 ↔ x = 0",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : 0 ≤ x\n⊢ √x = 0 ↔ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Sqrt | {
"line": 440,
"column": 63
} | {
"line": 444,
"column": 20
} | {
"line": 446,
"column": 0
} | [
{
"pp": "x : ℝ\nh : -1 ≤ x\n⊢ √(1 + x) ≤ 1 + x / 2",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | by
refine sqrt_le_iff.mpr ⟨by linarith, ?_⟩
calc 1 + x
_ ≤ 1 + x + (x / 2) ^ 2 := le_add_of_nonneg_right <| sq_nonneg _
_ = _ := by ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Sqrt | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 17
} | {
"line": 491,
"column": 18
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Sqrt | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 86
} | {
"line": 493,
"column": 0
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Real.Sqrt | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 86
} | {
"line": 493,
"column": 0
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Sqrt | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 86
} | {
"line": 493,
"column": 0
} | [
{
"pp": "ι : Type u_2\ns : Finset ι\nf g : ι → ℝ≥0\n⊢ ∑ i ∈ s, sqrt (f i) * sqrt (g i) ≤ sqrt (∑ i ∈ s, f i) * sqrt (∑ i ∈ s, g i)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | simpa [*] using sum_mul_le_sqrt_mul_sqrt _ (fun x ↦ sqrt (f x)) (fun x ↦ sqrt (g x)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Sqrt | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 17
} | {
"line": 508,
"column": 18
} | [
{
"pp": "ι : Type u_2\nf g : ι → ℝ\ns : Finset ι\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\n⊢ ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\nf g : ι → ℝ\ns : Finset ι\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\n⊢ ∑ i ∈ s, √(f i) * √(g i) ≤ √(∑ i ∈ s, f i) * √(∑ i ∈ s, g i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 27
} | {
"line": 182,
"column": 28
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ≤ |z.re| + |z.im|",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\n⊢ ‖z‖ ≤ |z.re| + |z.im|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 66
} | {
"line": 446,
"column": 67
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : HasFiniteIntegral f μ\nx : α\n⊢ ‖min (f x) 0‖ ≤ ‖f x‖",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Neg... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : HasFiniteIntegral f μ\nx : α\n⊢ -|f x| ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 13
} | {
"line": 200,
"column": 14
} | [
{
"pp": "z : ℂ\n⊢ |z.im| < ‖z‖ ↔ z.re ≠ 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"Complex.im",
"Real.instLT",
"Complex.instNorm",
"id",
"Real.instAddGroup",
... | [
"z : ℂ\n⊢ |z.im| < ‖z‖ ↔ ¬z.re = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 57
} | {
"line": 484,
"column": 58
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_7\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\nh : HasFiniteIntegral (↑c • f) μ\n⊢ HasFiniteIntegral f μ",
"ppTerm": "?m... | [
"case mp\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\n𝕜 : Type u_7\ninst✝² : NormedRing 𝕜\ninst✝¹ : MulActionWithZero 𝕜 β\ninst✝ : IsBoundedSMul 𝕜 β\nf : α → β\nc : 𝕜ˣ\nh : HasFiniteIntegral (↑c • f) μ\n⊢ HasFiniteIntegral f μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 523,
"column": 2
} | {
"line": 523,
"column": 37
} | {
"line": 523,
"column": 38
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nh : HasFiniteIntegral f μ\ns : Set α\n⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nh : HasFiniteIntegral f μ\ns : Set α\n⊢ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ ≤ ∫⁻ (a : α), ‖f a‖ₑ ∂μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 277,
"column": 23
} | {
"line": 277,
"column": 34
} | {
"line": 277,
"column": 35
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nx✝ : ℝ\nε0 : x✝ > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ (↑f n).re) j - (fun n ↦ (↑f n).re) i| ≤ ‖↑f j - ↑f i‖",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Real.lattice",
... | [
"f : CauSeq ℂ fun x ↦ ‖x‖\nx✝ : ℝ\nε0 : x✝ > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < x✝\nj : ℕ\nij : j ≥ i\n⊢ |(↑f j).re - (↑f i).re| ≤ ‖↑f j - ↑f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 62
} | {
"line": 281,
"column": 63
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nε : ℝ\nε0 : ε > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ (↑f n).im) j - (fun n ↦ (↑f n).im) i| < ε",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Preorder.toLT",
"Real.lattice"... | [
"f : CauSeq ℂ fun x ↦ ‖x‖\nε : ℝ\nε0 : ε > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε\nj : ℕ\nij : j ≥ i\n⊢ |(↑f j).im - (↑f i).im| < ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 377,
"column": 51
} | {
"line": 377,
"column": 62
} | {
"line": 377,
"column": 63
} | [
{
"pp": "z : ℂ\nhz : z ∈ Metric.sphere 0 1\n⊢ ‖z‖ = 1",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\nhz : z ∈ Metric.sphere 0 1\n⊢ ‖z‖ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Norm | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 13
} | {
"line": 387,
"column": 14
} | [
{
"pp": "x : ℝ\nhx : ‖x‖ ≤ 1\n⊢ normSq (-↑x + I * ↑√(1 - x ^ 2)) = 1",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : ‖x‖ ≤ 1\n⊢ normSq (-↑x + I * ↑√(1 - x ^ 2)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 30
} | {
"line": 78,
"column": 45
} | {
"line": 78,
"column": 46
} | [
{
"pp": "z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0)",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"and_true",
"Real.instZero",
"congrArg",
"sub_zero",
"Complex.im",... | [
"z : ℂ\nh : z.im = 0\n⊢ 0 ≤ z.re ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Order | {
"line": 83,
"column": 30
} | {
"line": 83,
"column": 45
} | {
"line": 83,
"column": 46
} | [
{
"pp": "z : ℂ\nh : z.re = 0\n⊢ z.re ^ 2 - z.im ^ 2 ≤ 0 ∧ (z.re = 0 ∨ z.im = 0)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"and_true",
"Real.instZero",
"congrArg",... | [
"z : ℂ\nh : z.re = 0\n⊢ 0 ≤ z.im ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 86,
"column": 37
} | {
"line": 86,
"column": 81
} | {
"line": 86,
"column": 82
} | [
{
"pp": "V : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nK : ℝ≥0\nh : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ... | [
"V : Type u_1\nV₁ : Type u_2\nV₂ : Type u_3\nV₃ : Type u_4\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf✝ g : NormedAddGroupHom V₁ V₂\nf : V₁ →+ V₂\nK : ℝ≥0\nh : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 150,
"column": 49
} | {
"line": 150,
"column": 89
} | {
"line": 150,
"column": 90
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nh : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖\nx y : V₁\n⊢ dist x y ≤ ↑K * dist (f x) (f y)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.nor... | [
"V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nh : ∀ (x : V₁), ‖x‖ ≤ ↑K * ‖f x‖\nx y : V₁\n⊢ ‖x - y‖ ≤ ↑K * ‖f x - f y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 20
} | {
"line": 165,
"column": 21
} | [
{
"pp": "case pos\nV₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC C' : ℝ\nh : f.SurjectiveOnWith K C\nH : C ≤ C'\ng : V₁\nk_in : f g ∈ K\nhg : ‖g‖ ≤ C * ‖f g‖\nHg : ‖f g‖ = 0\n⊢ ‖g‖ ≤ C' * ‖f g‖",
"ppTer... | [
"case pos\nV₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : AddSubgroup V₂\nC C' : ℝ\nh : f.SurjectiveOnWith K C\nH : C ≤ C'\ng : V₁\nk_in : f g ∈ K\nhg : ‖g‖ ≤ C * ‖f g‖\nHg : ‖f g‖ = 0\n⊢ ‖g‖ ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 254,
"column": 36
} | {
"line": 254,
"column": 80
} | {
"line": 254,
"column": 81
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nhf : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ‖x‖",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": [... | [
"V₁ : Type u_2\nV₂ : Type u_3\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nK : ℝ≥0\nhf : LipschitzWith K ⇑f\nx : V₁\n⊢ ‖f x‖ ≤ ↑K * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 714,
"column": 4
} | {
"line": 714,
"column": 25
} | {
"line": 714,
"column": 26
} | [
{
"pp": "case refine_1\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : f.NormNoninc\nv : V\n⊢ ‖f v‖ ≤ 1 * ‖v‖",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.... | [
"case refine_1\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : f.NormNoninc\nv : V\n⊢ ‖f v‖ ≤ ‖v‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 715,
"column": 4
} | {
"line": 715,
"column": 15
} | {
"line": 715,
"column": 16
} | [
{
"pp": "case refine_2\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : ‖f‖ ≤ 1\nv : V\n⊢ ‖f v‖ ≤ ‖v‖",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nV : Type u_1\nW : Type u_2\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : SeminormedAddCommGroup W\nf : NormedAddGroupHom V W\nh : ‖f‖ ≤ 1\nv : V\n⊢ ‖f v‖ ≤ ‖v‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 726,
"column": 17
} | {
"line": 726,
"column": 28
} | {
"line": 726,
"column": 29
} | [
{
"pp": "V₁ : Type u_3\nV₂ : Type u_4\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nh : (-f).NormNoninc\nx : V₁\n⊢ ‖f x‖ ≤ ‖x‖",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V₁ : Type u_3\nV₂ : Type u_4\ninst✝¹ : SeminormedAddCommGroup V₁\ninst✝ : SeminormedAddCommGroup V₂\nf : NormedAddGroupHom V₁ V₂\nh : (-f).NormNoninc\nx : V₁\n⊢ ‖f x‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 57,
"column": 51
} | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 63
} | [
{
"pp": "E : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : StarAddMonoid E\ninst✝ : NormedStarGroup E\nx : E\n⊢ ‖x‖ ≤ ‖x⋆‖",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : StarAddMonoid E\ninst✝ : NormedStarGroup E\nx : E\n⊢ ‖x‖ ≤ ‖x⋆‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 107,
"column": 10
} | {
"line": 107,
"column": 41
} | {
"line": 107,
"column": 42
} | [
{
"pp": "case inr\nE : Type u_2\ninst✝¹ : NonUnitalNormedRing E\ninst✝ : StarRing E\nh : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [
"case inr\nE : Type u_2\ninst✝¹ : NonUnitalNormedRing E\ninst✝ : StarRing E\nh : ∀ (x : E), ‖x‖ * ‖x‖ ≤ ‖x * x⋆‖\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 17
} | {
"line": 119,
"column": 18
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : 0 < ‖x⋆‖\n⊢ ‖x⋆‖ * ‖x⋆‖ ≤ ‖x‖ * ‖x⋆‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 30
} | {
"line": 138,
"column": 31
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : IsSelfAdjoint x\n⊢ ‖x * x‖ = ‖x‖ ^ 2",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real",
"HMul... | [
"E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\nhx : IsSelfAdjoint x\n⊢ ‖x * x‖ = ‖x‖ * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 44
} | {
"line": 154,
"column": 45
} | [
{
"pp": "E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\n⊢ x * x⋆ = 0 ↔ x = 0",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_2\ninst✝² : NonUnitalNormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nx : E\n⊢ x * x⋆ = 0 ↔ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 48
} | {
"line": 189,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\nι : Type u_4\nR₁ : Type u_5\nR₂ : Type u_6\nR : ι → Type u_7\ninst✝⁹ : NonUnitalNormedRing R₁\ninst✝⁸ : StarRing R₁\ninst✝⁷ : CStarRing R₁\ninst✝⁶ : NonUnitalNormedRing R₂\ninst✝⁵ : StarRing R₂\ninst✝⁴ : CStarRing R₂\ninst✝³ : (i : ι) → NonUnitalNormedRing (R ... | [
"𝕜 : Type u_1\nE : Type u_2\nα : Type u_3\nι : Type u_4\nR₁ : Type u_5\nR₂ : Type u_6\nR : ι → Type u_7\ninst✝⁹ : NonUnitalNormedRing R₁\ninst✝⁸ : StarRing R₁\ninst✝⁷ : CStarRing R₁\ninst✝⁶ : NonUnitalNormedRing R₂\ninst✝⁵ : StarRing R₂\ninst✝⁴ : CStarRing R₂\ninst✝³ : (i : ι) → NonUnitalNormedRing (R i)\ninst✝² :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousLinearMap | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 52
} | {
"line": 185,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : Ring 𝕜₂\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜₂ F\nσ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝¹ : RingHomInvPair σ σ₂₁\ninst✝ : RingHomInvPair σ₂₁ ... | [
"𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : Ring 𝕜₂\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : SeminormedAddCommGroup F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜₂ F\nσ : 𝕜 →+* 𝕜₂\nσ₂₁ : 𝕜₂ →+* 𝕜\ninst✝¹ : RingHomInvPair σ σ₂₁\ninst✝ : RingHomInvPair σ₂₁ σ\na : ℝ\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Basic | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 35
} | {
"line": 238,
"column": 36
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nA : E\nU : ↥(unitary E)\n⊢ ‖A * ↑U‖ = ‖A‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"norm_star",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"HMul.hMul"... | [
"E : Type u_2\ninst✝² : NormedRing E\ninst✝¹ : StarRing E\ninst✝ : CStarRing E\nA : E\nU : ↥(unitary E)\n⊢ ‖(↑U)⋆ * A⋆‖ = ‖A‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Unitary | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 29
} | {
"line": 166,
"column": 30
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : StarMul G\na b : G\n⊢ a⁻¹ * b ∈ unitary G ↔ a * star a = b * star b",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : StarMul G\na b : G\n⊢ a⁻¹ * b ∈ unitary G ↔ a * star a = b * star b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Unitary | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 13
} | {
"line": 440,
"column": 14
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : StarMul A\na : A\nU : ↥(unitary A)\n⊢ spectrum R (star ↑U * a * ↑U) = spectrum R a",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\nA : Type u_3\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : StarMul A\na : A\nU : ↥(unitary A)\n⊢ spectrum R (star ↑U * a * ↑U) = spectrum R a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 141,
"column": 14
} | {
"line": 141,
"column": 67
} | {
"line": 141,
"column": 68
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\nin... | [
"ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\ninst✝¹ : RingH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 142,
"column": 15
} | {
"line": 142,
"column": 63
} | {
"line": 142,
"column": 64
} | [
{
"pp": "ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\nin... | [
"ι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\ninst✝¹ : RingH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 44
} | {
"line": 161,
"column": 6
} | [
{
"pp": "case neg\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : ... | [
"case neg\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\nin... | simp only [tsum_bot hL, eq_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.Module | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 17
} | {
"line": 168,
"column": 18
} | [
{
"pp": "case neg.refine_2\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* ... | [
"case neg.refine_2\nι : Type u_5\nR : Type u_7\nR₂ : Type u_8\nM : Type u_9\nM₂ : Type u_10\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : Semiring R₂\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommMonoid M₂\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 43
} | {
"line": 96,
"column": 44
} | [
{
"pp": "⊢ Continuous ⇑normSq",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Continuous ⇑normSq"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 13
} | {
"line": 112,
"column": 14
} | [
{
"pp": "z : ℂ\n⊢ ‖equivRealProd z‖ ≤ 1 * ‖z‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Complex.equivRealProd_apply",
"Equiv.instEquivLike",
"HMul.hMul",
"Real.lattice",
"abs",
... | [
"z : ℂ\n⊢ |z.re| ≤ ‖z‖ ∧ |z.im| ≤ ‖z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 9
} | {
"line": 145,
"column": 10
} | [
{
"pp": "⊢ Tendsto (⇑normSq) (cocompact ℂ) atTop",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Tendsto (⇑normSq) (cocompact ℂ) atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 379,
"column": 26
} | {
"line": 379,
"column": 52
} | {
"line": 379,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ -I • ↑a ∈ selfAdjoint A",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Su... | [] | simp [selfAdjoint.mem_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 379,
"column": 26
} | {
"line": 379,
"column": 52
} | {
"line": 379,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ -I • ↑a ∈ selfAdjoint A",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Su... | [] | simp [selfAdjoint.mem_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 379,
"column": 26
} | {
"line": 379,
"column": 52
} | {
"line": 379,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\na : ↥(skewAdjoint A)\n⊢ -I • ↑a ∈ selfAdjoint A",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Su... | [] | simp [selfAdjoint.mem_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Basic | {
"line": 532,
"column": 15
} | {
"line": 532,
"column": 70
} | {
"line": 532,
"column": 71
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nx : ℝ\nh : HasSum (fun x ↦ ↑(f x)) (↑x) L\n⊢ HasSum f x L",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nx : ℝ\nh : HasSum (fun x ↦ ↑(f x)) (↑x) L\n⊢ HasSum f x L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 537,
"column": 15
} | {
"line": 537,
"column": 70
} | {
"line": 537,
"column": 71
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nh : Summable (fun x ↦ ↑(f x)) L\n⊢ Summable f L",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → ℝ\nh : Summable (fun x ↦ ↑(f x)) L\n⊢ Summable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 30
} | {
"line": 563,
"column": 4
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → 𝕜\nc : 𝕜\nh₁ : HasSum (fun x ↦ re (f x)) (re c) L\nh₂ : HasSum (fun x ↦ im (f x)) (im c) L\n⊢ HasSum f c L",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\n𝕜 : Type u_2\ninst✝ : RCLike 𝕜\nL : SummationFilter α\nf : α → 𝕜\nc : 𝕜\nh₁ : HasSum (fun x ↦ re (f x)) (re c) L\nh₂ : HasSum (fun x ↦ im (f x)) (im c) L\n⊢ HasSum f c L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 661,
"column": 2
} | {
"line": 661,
"column": 13
} | {
"line": 661,
"column": 14
} | [
{
"pp": "x : ℝ\n⊢ -↑x ∈ slitPlane ↔ x < 0",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\n⊢ -↑x ∈ slitPlane ↔ x < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 670,
"column": 2
} | {
"line": 670,
"column": 31
} | {
"line": 670,
"column": 32
} | [
{
"pp": "n : ℕ\n⊢ ↑n ∈ slitPlane ↔ n ≠ 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Membership.mem",
"id",
"Ne",
"instOfNatNat",
"Complex.instNatCast",
"Nat.cast",
"Iff",
"Nat",
"Complex",
"OfNat.ofNat",
"Set.instMember... | [
"n : ℕ\n⊢ ↑n ∈ slitPlane ↔ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 689,
"column": 2
} | {
"line": 689,
"column": 13
} | {
"line": 689,
"column": 14
} | [
{
"pp": "z : ℂ\nhz : z ∈ Metric.ball 1 1\n⊢ 0 < z.re",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : ℂ\nhz : z ∈ Metric.ball 1 1\n⊢ 0 < z.re"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 702,
"column": 34
} | {
"line": 702,
"column": 45
} | {
"line": 702,
"column": 46
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nhr : -↑r ∈ s\n⊢ -↑r ≤ 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGr... | [
"r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nhr : -↑r ∈ s\n⊢ 0 ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 704,
"column": 23
} | {
"line": 704,
"column": 34
} | {
"line": 704,
"column": 35
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nz : ℂ\nhzs : z ∈ s\nhz : z ≤ 0\n⊢ ‖z‖ = r",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"r : ℝ\ns : Set ℂ\nhs : s ⊆ sphere 0 r\nz : ℂ\nhzs : z ∈ s\nhz : z ≤ 0\n⊢ ‖z‖ = r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Basic | {
"line": 724,
"column": 36
} | {
"line": 724,
"column": 53
} | {
"line": 726,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : SeminormedAddCommGroup A\ninst✝³ : StarAddMonoid A\ninst✝² : NormedSpace ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : NormedStarGroup A\nx : A\n⊢ ‖x‖ + ‖star x‖ ≤ 2 * ‖x‖",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"norm_star",
"Norm.norm",
"S... | [] | by simp [two_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Basic | {
"line": 728,
"column": 18
} | {
"line": 729,
"column": 86
} | {
"line": 730,
"column": 8
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : SeminormedAddCommGroup A\ninst✝³ : StarAddMonoid A\ninst✝² : NormedSpace ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : NormedStarGroup A\nx : A\n⊢ ‖realPart (Complex.I • -x)‖ ≤ ‖x‖",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",... | [
"A : Type u_1\ninst✝⁴ : SeminormedAddCommGroup A\ninst✝³ : StarAddMonoid A\ninst✝² : NormedSpace ℂ A\ninst✝¹ : StarModule ℂ A\ninst✝ : NormedStarGroup A\nx : A\n⊢ ‖imaginaryPart x‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 540,
"column": 2
} | {
"line": 541,
"column": 9
} | {
"line": 541,
"column": 10
} | [
{
"pp": "z : ↥(selfAdjoint ℂ)\n⊢ ↑(selfAdjointEquiv z) = ↑z",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instTrivialStarReal",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instStarRingReal",
"CommRing.... | [
"z : ↥(selfAdjoint ℂ)\n⊢ ↑(↑z).re = ↑z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 13
} | {
"line": 618,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : Module ℂ A\ninst✝ : StarModule ℂ A\na : A\n⊢ a ≤ 0 ↔ ℜ a ≤ 0 ∧ ℑ a = 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instTrivialStarR... | [
"A : Type u_1\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : Module ℂ A\ninst✝ : StarModule ℂ A\na : A\n⊢ a ≤ 0 ↔ ℜ a ≤ 0 ∧ IsSelfAdjoint a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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