module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 13
} | {
"line": 592,
"column": 14
} | [
{
"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 ... | [
"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 s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 499,
"column": 48
} | {
"line": 499,
"column": 59
} | {
"line": 499,
"column": 60
} | [
{
"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)... | [
"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), f.length (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 543,
"column": 36
} | {
"line": 543,
"column": 80
} | {
"line": 543,
"column": 81
} | [
{
"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",
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 26
} | {
"line": 640,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\ns : Set X\nμ : Measure X\nf : X → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhfi : IntegrableOn f s μ\n⊢ NullMeasurableSet (support f) (μ.restrict s)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
... | [
"X : Type u_1\nmX : MeasurableSpace X\ns : Set X\nμ : Measure X\nf : X → ℝ\nhf : 0 ≤ᵐ[μ.restrict s] f\nhfi : IntegrableOn f s μ\n⊢ NullMeasurableSet (f ⁻¹' {0}ᶜ) (μ.restrict s)"
] | rw [support_eq_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 550,
"column": 50
} | {
"line": 550,
"column": 61
} | {
"line": 550,
"column": 62
} | [
{
"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 :... | [
"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 : ℕ → R\nu_mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 558,
"column": 19
} | {
"line": 558,
"column": 49
} | {
"line": 558,
"column": 50
} | [
{
"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 :... | [
"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 : ℕ → R\nu_mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 24
} | {
"line": 804,
"column": 25
} | [
{
"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",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"use... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 676,
"column": 4
} | {
"line": 676,
"column": 15
} | {
"line": 676,
"column": 16
} | [
{
"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... | [
"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✝ : OrderBot R\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 864,
"column": 4
} | {
"line": 864,
"column": 15
} | {
"line": 864,
"column": 16
} | [
{
"pp": "case hs\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nf_int : Integrable f μ\nf_nonneg : 0 ≤ᵐ[μ] f\ns : Set X\nhs : ∀ x ∈ s, 1 ≤ f x\nx : X\nhx : x ∈ s\n⊢ 1 ≤ ENNReal.ofReal (f x)",
"ppTerm": "?hs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",... | [
"case hs\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\nf_int : Integrable f μ\nf_nonneg : 0 ≤ᵐ[μ] f\ns : Set X\nhs : ∀ x ∈ s, 1 ≤ f x\nx : X\nhx : x ∈ s\n⊢ 1 ≤ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 882,
"column": 6
} | {
"line": 882,
"column": 21
} | {
"line": 882,
"column": 22
} | [
{
"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 : ... | [
"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 : x ∈ s\n⊢ f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 884,
"column": 6
} | {
"line": 884,
"column": 21
} | {
"line": 884,
"column": 22
} | [
{
"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",
"ppTe... | [
"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 := 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 : x ∉ s\n⊢ f x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 886,
"column": 4
} | {
"line": 886,
"column": 19
} | {
"line": 886,
"column": 20
} | [
{
"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... | [
"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\nhx : x ∈ sᶜ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 11
} | {
"line": 164,
"column": 12
} | [
{
"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... | [
"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 : QuasiMeasurePreserving f ν μ\na ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 741,
"column": 4
} | {
"line": 741,
"column": 15
} | {
"line": 741,
"column": 16
} | [
{
"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... | [
"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.measure = g.meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 264,
"column": 2
} | {
"line": 265,
"column": 87
} | {
"line": 266,
"column": 4
} | [
{
"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 ≪... | [
"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 ≪ μ\n⊢ ν t = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 37
} | {
"line": 276,
"column": 38
} | [
{
"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)",
"ppTe... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 900,
"column": 6
} | {
"line": 900,
"column": 68
} | {
"line": 901,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\ng : X → ℝ\nhf : ∀ x ∈ s, 0 ≤ f x\nh : ∀ x ∈ s, f x ≤ g x\nhg : IntegrableOn g s μ\nh'f : AEStronglyMeasurable f (μ.restrict s)\n⊢ NullMeasurableSet {x | 0 x ≤ f x} (μ.restrict s)",
"ppTerm": "?m.146",
"assigned": true,
... | [] | exact nullMeasurableSet_le aemeasurable_const h'f.aemeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 900,
"column": 6
} | {
"line": 900,
"column": 68
} | {
"line": 901,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\ng : X → ℝ\nhf : ∀ x ∈ s, 0 ≤ f x\nh : ∀ x ∈ s, f x ≤ g x\nhg : IntegrableOn g s μ\nh'f : AEStronglyMeasurable f (μ.restrict s)\n⊢ NullMeasurableSet {x | 0 x ≤ f x} (μ.restrict s)",
"ppTerm": "?m.146",
"assigned": true,
... | [] | exact nullMeasurableSet_le aemeasurable_const h'f.aemeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 900,
"column": 6
} | {
"line": 900,
"column": 68
} | {
"line": 901,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\ng : X → ℝ\nhf : ∀ x ∈ s, 0 ≤ f x\nh : ∀ x ∈ s, f x ≤ g x\nhg : IntegrableOn g s μ\nh'f : AEStronglyMeasurable f (μ.restrict s)\n⊢ NullMeasurableSet {x | 0 x ≤ f x} (μ.restrict s)",
"ppTerm": "?m.146",
"assigned": true,
... | [] | exact nullMeasurableSet_le aemeasurable_const h'f.aemeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 738,
"column": 2
} | {
"line": 745,
"column": 45
} | {
"line": 747,
"column": 0
} | [
{
"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": 738,
"column": 2
} | {
"line": 745,
"column": 45
} | {
"line": 747,
"column": 0
} | [
{
"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.Group.FundamentalDomain | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 67
} | {
"line": 313,
"column": 68
} | [
{
"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... | [
"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 : MeasurableSet A\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 37
} | {
"line": 320,
"column": 38
} | [
{
"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",
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 97
} | {
"line": 334,
"column": 4
} | [
{
"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": 333,
"column": 6
} | {
"line": 333,
"column": 97
} | {
"line": 334,
"column": 4
} | [
{
"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": 333,
"column": 6
} | {
"line": 333,
"column": 97
} | {
"line": 334,
"column": 4
} | [
{
"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.MeasureTheory.Measure.Stieltjes | {
"line": 755,
"column": 6
} | {
"line": 755,
"column": 17
} | {
"line": 755,
"column": 18
} | [
{
"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\ng : StieltjesFunction R\ny : R\nhf... | [
"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\ng : StieltjesFunction R\ny : R\nhfg : f.measur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 763,
"column": 6
} | {
"line": 763,
"column": 17
} | {
"line": 763,
"column": 18
} | [
{
"pp": "case 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\nhf... | [
"case 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\nhfg : f.measur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 945,
"column": 2
} | {
"line": 945,
"column": 42
} | {
"line": 945,
"column": 43
} | [
{
"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... | [
"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 ↦ ‖ContinuousMap.restr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 55
} | {
"line": 554,
"column": 56
} | [
{
"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",
"ppTerm": "?ref... | [
"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⊢ y ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 58
} | {
"line": 246,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\n⊢ (fun h ↦ g * h) '' ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V ⊆\n... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\ng₁ g₂ : G\nhg₂ : g₂ ∈ s\nhg₁ : g₁ ∈ (fun h ↦ (fun h ↦ g₂ * h) ⁻¹' V... | rintro _ ⟨g₁, ⟨_, ⟨g₂, rfl⟩, ⟨_, ⟨hg₂, rfl⟩, hg₁⟩⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 327,
"column": 81
} | {
"line": 327,
"column": 93
} | {
"line": 327,
"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⟫",
"ppTerm": "?m.532",... | [
"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 + re ⟪y, x⟫ * t + re ⟪y, y⟫ =\n normSq x * t * t + re ⟪x, y⟫ * t + re ⟪y, x⟫ * t + re ⟪y, y⟫"
] | mul_comm t _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 17
} | {
"line": 806,
"column": 18
} | [
{
"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... | [
"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.map (Quotie... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 871,
"column": 4
} | {
"line": 871,
"column": 41
} | {
"line": 871,
"column": 42
} | [
{
"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✝ : ... | [
"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✝ : IsFiniteMeas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 67
} | {
"line": 377,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ chaar K₀ ⊥ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.haar.clPrehaar",
"Eq.mpr",
"TopologicalSpace.OpenNhdsOf.toO... | [] | 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
} | {
"line": 377,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\n⊢ chaar K₀ ⊥ = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.haar.clPrehaar",
"Eq.mpr",
"TopologicalSpace.OpenNhdsOf.toO... | [] | 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.Analysis.InnerProductSpace.Defs | {
"line": 460,
"column": 8
} | {
"line": 460,
"column": 70
} | {
"line": 460,
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoi... | [] | 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": 460,
"column": 8
} | {
"line": 460,
"column": 70
} | {
"line": 460,
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoi... | [] | 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": 460,
"column": 8
} | {
"line": 460,
"column": 70
} | {
"line": 460,
"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",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"InnerProductSpace.Core.toInner'",
"Real",
"AddMonoidHom.instAddMonoi... | [] | simp only [normSq, ext_iff, map_zero, inner_self_im, and_true] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 27
} | {
"line": 150,
"column": 0
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\n⊢ ((innerₗ F).flip v) w = ((innerₗ F) v) w",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"real_inner_comm"
],
"usedFVars": [
"F",
"inst✝¹",
"inst✝",
"v"... | [] | exact real_inner_comm v w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1158,
"column": 4
} | {
"line": 1158,
"column": 65
} | {
"line": 1158,
"column": 66
} | [
{
"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... | [
"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 : NormedSpace ℝ E\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1162,
"column": 4
} | {
"line": 1162,
"column": 36
} | {
"line": 1162,
"column": 37
} | [
{
"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... | [
"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 : NormedSpace ℝ E\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 22
} | {
"line": 365,
"column": 23
} | [
{
"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 ... | [
"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 i, ∑ j ∈ s, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 13
} | {
"line": 397,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Inner.inner",
... | [
"F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx : F\nh : re ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖\n⊢ ‖x‖ ^ 2 = ‖x‖ * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 13
} | {
"line": 414,
"column": 14
} | [
{
"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",
"ppTerm": "?m.71",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 13
} | {
"line": 428,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 13
} | {
"line": 453,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.60",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 19
} | {
"line": 104,
"column": 8
} | [
{
"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‖\... | [
"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 := ⋯\nthis✝ : Nonempty ↑K := ⋯\nzero_le_δ : 0 ≤ δ\nδ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖\nδ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖\nw : ℕ → ↑K\nhw : ∀ (n : ℕ), ‖u - ↑(w n)‖ < δ + 1... | apply δ_le' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 493,
"column": 2
} | {
"line": 496,
"column": 27
} | {
"line": 498,
"column": 0
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"Eq.ge",
"iInf... | [] | 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
} | {
"line": 498,
"column": 0
} | [
{
"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",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"Eq.ge",
"iInf... | [] | 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.Measure.Haar.Basic | {
"line": 609,
"column": 2
} | {
"line": 609,
"column": 13
} | {
"line": 609,
"column": 14
} | [
{
"pp": "G : 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 : ∃ K ⊆ E... | [
"G : 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 : ∃ K ⊆ E, IsCompact ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Continuous | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | {
"line": 111,
"column": 14
} | [
{
"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",
"ppTerm": "?m.93",
"assigned": fals... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 711,
"column": 4
} | {
"line": 711,
"column": 64
} | {
"line": 712,
"column": 4
} | [
{
"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",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNor... | [
"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\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x"
] | 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.Projection.Minimal | {
"line": 153,
"column": 21
} | {
"line": 153,
"column": 32
} | {
"line": 154,
"column": 12
} | [
{
"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... | [
"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 := ⋯\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‖\nθ : ℝ\nhθ₁ : 0 < θ... | apply δ_le' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 25
} | {
"line": 155,
"column": 12
} | [
{
"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 := ⋯\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‖\nθ : ℝ\... | [
"case a\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‖\nθ : ℝ\nhθ₁... | apply h hw hv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 751,
"column": 4
} | {
"line": 751,
"column": 15
} | {
"line": 751,
"column": 16
} | [
{
"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",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 752,
"column": 4
} | {
"line": 754,
"column": 77
} | {
"line": 756,
"column": 0
} | [
{
"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",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [] | 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": 752,
"column": 4
} | {
"line": 754,
"column": 77
} | {
"line": 756,
"column": 0
} | [
{
"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",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [] | 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.MeasureTheory.Integral.Bochner.Set | {
"line": 1222,
"column": 2
} | {
"line": 1222,
"column": 13
} | {
"line": 1222,
"column": 14
} | [
{
"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... | [
"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 μ\nhk : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 23
} | {
"line": 96,
"column": 24
} | [
{
"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",
"ppTerm": "?mpr.right",... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 832,
"column": 2
} | {
"line": 832,
"column": 22
} | {
"line": 832,
"column": 23
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 22
} | {
"line": 837,
"column": 23
} | [
{
"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",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?po... | [
"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⊢ ⟪v j, v j⟫ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 21
} | {
"line": 865,
"column": 22
} | [
{
"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",
"ppTerm": "?m.145",
"assigned": t... | [
"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⊢ ‖x‖ ^ 2 ≤ ‖y‖ ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?n... | [
"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⊢ ⟪v i, v j⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?p... | [
"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⊢ ⟪v j, v j⟫ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?... | [
"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⊢ ⟪v i, v j⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?p... | [
"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⊢ ⟪v j, v j⟫ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?... | [
"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⊢ ⟪v i, v j⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "?... | [
"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⊢ ⟪v j, v j⟫ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | {
"line": 232,
"column": 25
} | [
{
"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",
"ppTerm": "... | [
"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⊢ ⟪v i, v j⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 259,
"column": 86
} | {
"line": 259,
"column": 97
} | {
"line": 259,
"column": 98
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\ns : Set (Set E)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x\n⊢ ∀ (i : ↑s), Orthonormal 𝕜 fun x ↦ ↑x",
"ppTerm": "?m.46",
"assigned": true,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\ns : Set (Set E)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x\n⊢ ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 394,
"column": 71
} | {
"line": 394,
"column": 82
} | {
"line": 394,
"column": 83
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | [
"𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 46
} | {
"line": 158,
"column": 47
} | [
{
"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 �... | [
"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 𝕜 G V\nα : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 232,
"column": 2
} | {
"line": 234,
"column": 75
} | {
"line": 236,
"column": 0
} | [
{
"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‖",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpa... | [] | 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
} | {
"line": 236,
"column": 0
} | [
{
"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‖",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpa... | [] | 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.Analysis.InnerProductSpace.Subspace | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 51
} | {
"line": 165,
"column": 52
} | [
{
"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... | [
"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 G V\nα : ι ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Baire.CompleteMetrizable | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | {
"line": 69,
"column": 2
} | [
{
"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... | [
"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), 0 < B 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
} | {
"line": 183,
"column": 43
} | [
{
"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... | [
"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\ninst✝ : Decidab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 99,
"column": 18
} | {
"line": 99,
"column": 29
} | {
"line": 99,
"column": 30
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | [
"ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝ : Module R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 116,
"column": 28
} | {
"line": 116,
"column": 39
} | {
"line": 116,
"column": 40
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | [
"ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝ : Module R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 80
} | {
"line": 232,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\nf : M →L[R] N\ns : ClosedSubm... | [] | simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 80
} | {
"line": 232,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\nf : M →L[R] N\ns : ClosedSubm... | [] | simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 61
} | {
"line": 315,
"column": 62
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | [
"ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝ : Module R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 316,
"column": 56
} | {
"line": 316,
"column": 67
} | {
"line": 316,
"column": 68
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | [
"ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝ : Module R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 73,
"column": 39
} | {
"line": 73,
"column": 46
} | {
"line": 73,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ (starRingEnd 𝕜) ⟪T x, y⟫ = ⟪T y, x⟫",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProdu... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ (starRingEnd 𝕜) ⟪x, T y⟫ = ⟪T y, x⟫"
] | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 91,
"column": 33
} | {
"line": 91,
"column": 40
} | {
"line": 91,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪T x, y⟫ + ⟪S x, y⟫ = ⟪x, (T + S) y⟫",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪x, T y⟫ + ⟪S x, y⟫ = ⟪x, (T + S) y⟫"
] | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 36
} | {
"line": 95,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_3\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsSymmetric\nx✝¹ x✝ : E\n⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫",
"ppTerm": "?m.33",
"assigned":... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_3\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsSymmetric\nx✝¹ x✝ : E\n⊢ ∑ i ∈ s, ⟪(T i) x✝¹, x✝⟫ = ∑ i ∈ s, ⟪x✝¹, (T i) x✝⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 101,
"column": 33
} | {
"line": 101,
"column": 40
} | {
"line": 101,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪T x, y⟫ - ⟪S x, y⟫ = ⟪x, (T - S) y⟫",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪x, T y⟫ - ⟪S x, y⟫ = ⟪x, (T - S) y⟫"
] | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Lemmas | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 30
} | {
"line": 48,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nthis : Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})\n⊢ Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
... | [
"K : Type u_1\ninst✝ : RCLike K\nthis : Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})\n⊢ Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | {
"line": 111,
"column": 14
} | [
{
"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\nx✝ : E\n⊢ x✝ ∈ Kᗮ ↔ x✝ ∈ ⨅ v, (↑((innerSL 𝕜) ↑v)).ker",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNor... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nx✝ : E\n⊢ x✝ ∈ Kᗮ ↔ ∀ a ∈ K, ⟪a, x✝⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 5
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nf : E ≃ₗᵢ[𝕜] F\n⊢ map (↑f.toLinearEquiv) Kᗮ = (map (↑f.toLinearEquiv) K)ᗮ",
"ppTerm": "?... | [] | refine (map_orthogonal K f.toLinearIsometry).trans ?_
have : f.toLinearIsometry.range = ⊤ := f.range
rw [this, inf_top_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 5
} | {
"line": 137,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nf : E ≃ₗᵢ[𝕜] F\n⊢ map (↑f.toLinearEquiv) Kᗮ = (map (↑f.toLinearEquiv) K)ᗮ",
"ppTerm": "?... | [] | refine (map_orthogonal K f.toLinearIsometry).trans ?_
have : f.toLinearIsometry.range = ⊤ := f.range
rw [this, inf_top_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 33
} | {
"line": 195,
"column": 33
} | [
{
"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\n⊢ K = ⊥ → Kᗮ = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"AddCommGroup.toAd... | [] | rintro rfl
exact bot_orthogonal_eq_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 33
} | {
"line": 195,
"column": 33
} | [
{
"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\n⊢ K = ⊥ → Kᗮ = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"AddCommGroup.toAd... | [] | rintro rfl
exact bot_orthogonal_eq_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 23
} | {
"line": 370,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E ≃ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : Submodule.map (↑↑↑f) U ⟂ Submodule.map (↑↑↑f) V\nhf : ∀ (p : Submodule... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E ≃ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : Submodule.map (↑↑↑f) U ⟂ Submodule.map (↑↑↑f) V\nhf : ∀ (p : Submodule 𝕜 E), Subm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 447,
"column": 2
} | {
"line": 447,
"column": 13
} | {
"line": 447,
"column": 14
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : ClosedSubmodule 𝕜 E\nx : E\n⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ClosedSubmodule.mem_inf._simp_1",
"Eq.mpr",
"Inn... | [
"𝕜 : Type u_4\nE : Type u_5\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : ClosedSubmodule 𝕜 E\nx : E\n⊢ x ∈ K → (∀ u ∈ K, ⟪u, x⟫ = 0) → x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 55
} | {
"line": 223,
"column": 56
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nh : (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsSy... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nh : (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsSymmetric\nx y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 238,
"column": 39
} | {
"line": 238,
"column": 50
} | {
"line": 238,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsSymmetric\nx y : E\n⊢ ⟪↑T.symm x, y⟫ = ⟪x, ↑T.symm y⟫",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSp... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsSymmetric\nx y : E\n⊢ ⟪T.symm x, y⟫ = ⟪x, T.symm y⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 285,
"column": 44
} | {
"line": 285,
"column": 55
} | {
"line": 285,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\n⊢ ∃ y, T y = S (S v)",
"ppTerm": "?m.124",
"assigned": false,
"usedConstants"... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\n⊢ ∃ y, T y = S (S v)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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