module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 167,
"column": 10
} | {
"line": 167,
"column": 41
} | {
"line": 167,
"column": 42
} | [
{
"pp": "case inr.inr.inl\ny : ℝ\ny_pos : 0 < y\n⊢ (0 ^ y).log = ↑y * log 0",
"ppTerm": "?inr.inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.zero_rpow_of_pos",
"Real",
"HMul.hMul",
"congrArg",
"ENNReal.instPowReal",
"EReal",
"ENNReal... | [
"case inr.inr.inl\ny : ℝ\ny_pos : 0 < y\n⊢ log 0 = ↑y * log 0"
] | ENNReal.zero_rpow_of_pos y_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 38
} | {
"line": 99,
"column": 2
} | [
{
"pp": "u : ℕ → EReal\na : EReal\n⊢ linearGrowthInf u ≤ a ↔ ∀ b > a, ∃ᶠ (n : ℕ) in atTop, u n ≤ b * ↑n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMu... | [
"u : ℕ → EReal\na : EReal\n⊢ (∀ y > a, ∃ᶠ (a : ℕ) in atTop, u a / ↑a ≤ y) ↔ ∀ b > a, ∃ᶠ (n : ℕ) in atTop, u n ≤ b * ↑n"
] | rw [linearGrowthInf, liminf_le_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 41
} | {
"line": 210,
"column": 2
} | [
{
"pp": "u : ℕ → EReal\nh : 0 < linearGrowthInf u\nM a : ℝ\na_0 : 0 < a\na_v : ↑a < linearGrowthInf u\n⊢ ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → ↑M < ↑a * ↑b",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Real.instIsOrderedRing",
"NonAssocSemiring.... | [
"u : ℕ → EReal\nh : 0 < linearGrowthInf u\nM a : ℝ\na_0 : 0 < a\na_v : ↑a < linearGrowthInf u\nn : ℕ\nhn : M / a ≤ ↑n\n⊢ ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → ↑M < ↑a * ↑b"
] | obtain ⟨n, hn⟩ := exists_nat_ge (M / a) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 57
} | {
"line": 457,
"column": 2
} | [
{
"pp": "case neg\nu : ℕ → EReal\nv : ℕ → ℕ\nh : Monotone u\nhv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0\nhv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤\nu_0 : ¬u = ⊥\nv_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)\na : EReal\nv_a : a > linearGrowthSup fun n ↦ ↑(v n)\nb : EReal\nu_b : b > linearGrowthInf u\na_0 : 0 < a\n⊢... | [
"case neg\nu : ℕ → EReal\nv : ℕ → ℕ\nh : Monotone u\nhv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0\nhv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤\nu_0 : ¬u = ⊥\nv_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)\na : EReal\nv_a : a > linearGrowthSup fun n ↦ ↑(v n)\nb : EReal\nu_b : b > linearGrowthInf u\na_0 : 0 < a\nb_0 : 0 < b\n... | have b_0 := (h.linearGrowthInf_nonneg u_0).trans_lt u_b | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Average | {
"line": 319,
"column": 62
} | {
"line": 319,
"column": 75
} | {
"line": 319,
"column": 75
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\n⊢ ∫ (x : α), 0 ∂(μ univ)⁻¹ • μ = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"in... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure α\n⊢ 0 = 0"
] | integral_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 18
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\n⊢ sInf {d | μ ≤ d + 0} ≤ μ",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"instReflLe",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"setOf... | [
"case a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\n⊢ μ ≤ sInf {d | μ ≤ d + 0}"
] | · simp [sInf_le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 138,
"column": 8
} | {
"line": 138,
"column": 25
} | {
"line": 138,
"column": 26
} | [
{
"pp": "case a\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh_meas_s : MeasurableSet s\nh_nonempty : {d | μ ≤ d + ν}.Nonempty\nt : Measure α\nh_t_in : t ∈ {d | μ ≤ d + ν}\n⊢ t.restrict s ∈ {d | μ.restrict s ≤ d + ν.restrict s}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants... | [
"case a\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh_meas_s : MeasurableSet s\nh_nonempty : {d | μ ≤ d + ν}.Nonempty\nt : Measure α\nh_t_in : t ∈ {d | μ ≤ d + ν}\n⊢ μ.restrict s ≤ t.restrict s + ν.restrict s"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 46
} | {
"line": 208,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nhl : ¬μ.HaveLebesgueDecomposition ν\n⊢ ν.withDensity (μ.rnDeriv ν) ≤ μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
"M... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nhl : ¬μ.HaveLebesgueDecomposition ν\n⊢ 0 ≤ μ"
] | rw [rnDeriv, dif_neg hl, withDensity_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Average | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 82
} | {
"line": 638,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : AEMeasurable f (μ.restrict s)\nh : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞\nthis : 0 < (μ.restrict s) ({a | (f a).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ} \\ {x | f x = ∞})\nx : α\nhfx : (f x)... | [
"case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : AEMeasurable f (μ.restrict s)\nh : ∫⁻ (a : α) in s, f a ∂μ ≠ ∞\nthis : 0 < (μ.restrict s) ({a | (f a).toReal ≤ ⨍ (a : α) in s, (f a).toReal ∂μ} \\ {x | f x = ∞})\nx : α\nhfx : (f x).toReal ≤ ⨍ ... | rwa [← toReal_laverage hf, toReal_le_toReal hx (setLAverage_lt_top h).ne] at hfx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 709,
"column": 2
} | {
"line": 715,
"column": 65
} | {
"line": 716,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\n⊢ ∃ ε, 0 < ε ∧ ∃ E, MeasurableSet E ∧ 0 < ν E ∧ ∀ (A : Set α), MeasurableSet A → ↑ε * ν (A ∩ E) ≤ μ (A ∩ E)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nh_decomp :\n ∀ (n : ℕ),\n ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ s) ≤ μ (t ∩ s)) ∧\n ∀ (t : Set α), MeasurableSet t → μ ... | have h_decomp (n : ℕ) : ∃ s : Set α, MeasurableSet s
∧ (∀ t, MeasurableSet t → ((1 / (n + 1) : ℝ≥0) • ν) (t ∩ s) ≤ μ (t ∩ s))
∧ (∀ t, MeasurableSet t → μ (t ∩ sᶜ) ≤ ((1 / (n + 1) : ℝ≥0) • ν) (t ∩ sᶜ)) := by
obtain ⟨s, hs, hs_le, hs_ge⟩ := hahn_decomposition μ ((1 / (n + 1) : ℝ≥0) • ν)
refine ⟨s,... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 742,
"column": 48
} | {
"line": 742,
"column": 62
} | {
"line": 742,
"column": 62
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)\nhf₃ : ∀ (n : ℕ) (t : Set α), Measur... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)\nhf₃ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ... | _root_.inv_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 795,
"column": 4
} | {
"line": 795,
"column": 63
} | {
"line": 796,
"column": 4
} | [
{
"pp": "case right\nα : Sort u_2\nf : ℕ → α → ℝ≥0∞\nm : ℕ\na : α\nc : ℝ≥0∞ := ⋯\nhc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a\nd : ℝ≥0∞ := ⋯\nhd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)\n⊢ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a",
"ppTerm": "?right",
"assigned":... | [
"case right\nα : Sort u_2\nf : ℕ → α → ℝ≥0∞\nm : ℕ\na : α\nc : ℝ≥0∞ := ⨆ k, ⨆ (_ : k ≤ m + 1), f k a\nhc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a\nd : ℝ≥0∞ := max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)\nhd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)\n⊢ f m.succ a ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a"
] | refine sup_le ?_ (biSup_mono fun n hn ↦ hn.trans m.le_succ) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 336,
"column": 8
} | {
"line": 336,
"column": 89
} | {
"line": 337,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np q : ℝ≥0\nhpq : p < q\ns : Set α := {x | ∃ c... | [] | have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 336,
"column": 8
} | {
"line": 336,
"column": 89
} | {
"line": 337,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np q : ℝ≥0\nhpq : p < q\ns : Set α := {x | ∃ c... | [] | have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1013,
"column": 2
} | {
"line": 1026,
"column": 60
} | {
"line": 1028,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0\nhr : r ≠ 0\n⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"Eq.mpr",
"G... | [] | refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
(?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
rw [← withDensity_eq_iff_of_sigmaFinite]
· simp_rw [ENNReal.smul_def]
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1013,
"column": 2
} | {
"line": 1026,
"column": 60
} | {
"line": 1028,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0\nhr : r ≠ 0\n⊢ ν.rnDeriv (r • μ) =ᵐ[μ] r⁻¹ • ν.rnDeriv μ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Iff.mpr",
"Eq.mpr",
"G... | [] | refine (absolutelyContinuous_smul <| ENNReal.coe_ne_zero.2 hr).ae_le
(?_ : ν.rnDeriv (r • μ) =ᵐ[r • μ] r⁻¹ • ν.rnDeriv μ)
rw [← withDensity_eq_iff_of_sigmaFinite]
· simp_rw [ENNReal.smul_def]
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
suffices ν.singularPart (r • μ) + withDensity (r • μ) (rnDeriv ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 6
} | {
"line": 77,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\na b c : α\nha : a ∈ s\nhb : b ∈ s\nhc : c ∈ s\n⊢ variationOnFromTo f s a b + variationOnFromTo f s b c = variationOnFromTo f s a c",
"ppTerm": "?m.31",
"as... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\na b c : α\nha : a ∈ s\nhb : b ∈ s\nhc : c ∈ s\n⊢ variationOnFromTo f s a c = variationOnFromTo f s a b + variationOnFromTo f s b c"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 72
} | {
"line": 500,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ v.limRatioMeas hρ ⁻¹' {0}\no ... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nx : α\nx✝ : x ∈ v.limRatioMeas hρ ⁻¹' {0}\no : Set α\nxo ... | have μs : μ s ≠ ∞ := measure_ne_top_of_subset inter_subset_right μo.ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 39
} | {
"line": 190,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nthis : ∀ {x y : α}, x ∈ s → y ∈ s → y ≤ x → edist (f x) (f y) ≤ eVariationOn f s\nhxy : ¬y ≤ x\n⊢ edist (f y) (f x) ≤ eVariationOn f s",
"ppTerm": "?inr... | [] | exact this hy hx (le_of_not_ge hxy) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 328,
"column": 8
} | {
"line": 328,
"column": 83
} | {
"line": 329,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nx : α\nhx : x ∈ s\nu : ℕ → α\nhu : Monotone u\nus : ∀ (i : ℕ), u i ∈ s\nn : ℕ\nh : x < u n\nexists_N : ∃ N ≤ n, x < u N\nN : ℕ := Nat.find exists_N\nhN : N ≤ n ∧ x < u N\nw : ℕ → α := fun i... | [] | · exact Finset.sum_Ico_add (fun i => edist (f (w (i + 1))) (f (w i))) N n 1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 890,
"column": 2
} | {
"line": 890,
"column": 74
} | {
"line": 891,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\... | [
"α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf : Locall... | refine squeeze_zero' (Eventually.of_forall fun a => norm_nonneg _) ?_ hx | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 304,
"column": 6
} | {
"line": 311,
"column": 39
} | {
"line": 313,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRing S\ninst✝¹⁴ : Conti... | [] | intro g
rw [h]
refine ⟨cfcₙHom_predicate _ _, ?_⟩
refine { rightInvOn := fun s hs ↦ ?_, left_inv := ((h a).mp ha).2.left_inv }
rw [nonUnitalStarAlgHom_apply,
cfcₙHom_map_quasispectrum] at hs
obtain ⟨r, rfl⟩ := hs
simp [((h a).mp ha).2.left_inv _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 304,
"column": 6
} | {
"line": 311,
"column": 39
} | {
"line": 313,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRing S\ninst✝¹⁴ : Conti... | [] | intro g
rw [h]
refine ⟨cfcₙHom_predicate _ _, ?_⟩
refine { rightInvOn := fun s hs ↦ ?_, left_inv := ((h a).mp ha).2.left_inv }
rw [nonUnitalStarAlgHom_apply,
cfcₙHom_map_quasispectrum] at hs
obtain ⟨r, rfl⟩ := hs
simp [((h a).mp ha).2.left_inv _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 678,
"column": 2
} | {
"line": 678,
"column": 84
} | {
"line": 680,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R ... | [] | exact Set.Nonempty.image_const (⟨0, spectrum.zero_mem (R := R) not_isUnit_zero⟩) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 731,
"column": 8
} | {
"line": 731,
"column": 31
} | {
"line": 732,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nL : Filter α\nhL : ∀ y ∈ s, s ∩ Ici y ∈ L\nx₀ : α\nhx₀ : x₀ ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ < ε\nH : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ ... | [] | exact (y_mem _ (I n)).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 895,
"column": 64
} | {
"line": 896,
"column": 33
} | {
"line": 898,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : CommRing R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalRing R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p\nf :... | [] | by
rw [cfc_comp' .., cfc_neg_id _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 44
} | {
"line": 242,
"column": 6
} | [
{
"pp": "case succ.right\nn k : ℕ\nh : k ≤ n\n⊢ bernsteinPolynomial ℚ n ↑(Fin.last k) ∉ span ℚ (Set.range (Fin.init fun ν ↦ bernsteinPolynomial ℚ n ↑ν))",
"ppTerm": "?succ.right",
"assigned": true,
"usedConstants": [
"Submodule",
"Semiring.toModule",
"CommSemiring.toSemiring",
... | [
"case succ.right\nn k : ℕ\nh : k ≤ n\n⊢ bernsteinPolynomial ℚ n k ∉ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1.castSucc)"
] | simp only [Fin.val_last, Fin.init_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Bernstein | {
"line": 197,
"column": 10
} | {
"line": 197,
"column": 13
} | {
"line": 197,
"column": 14
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis : IsUniformAddGroup E\nU : Set ... | [
"E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis : IsUniformAddGroup E\nU : Set E\nhU₀ : U ∈... | hU₀ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 6
} | {
"line": 141,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : ↥A\n⊢ ↑A = ↑A.topologicalClosure",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"NormedCommRing.toSeminormedCommRing",
"... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nh : IsClosed ↑A\nf g : ↥A\n⊢ ↑A.topologicalClosure = ↑A"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 198,
"column": 10
} | {
"line": 198,
"column": 27
} | {
"line": 198,
"column": 28
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty... | [
"case refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty X\ng : X → ... | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 75
} | {
"line": 420,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nI : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM\nA₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.r... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nI : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM\nA₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.restrictScala... | have h₂ := (A.toSubmodule.restrictScalars ℝ).map_comap_le I.toLinearMap | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 46
} | {
"line": 406,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : TopologicalSpace α\nf : α → Set β\nh : ∀ (b : β), (f ⁻¹' Iic {b}ᶜ)ᶜ = {x | b ∈ f x}\n⊢ HasOpenLowerSections f ↔ ∀ (b : β), IsOpen[inst✝] (f ⁻¹' Iic {b}ᶜ)ᶜ",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"C... | [] | simp_rw [h, hasOpenLowerSections_iff_isOpen] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 136,
"column": 2
} | {
"line": 139,
"column": 6
} | {
"line": 141,
"column": 0
} | [
{
"pp": "A : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\n⊢ Tendsto (fun n ↦ ENNReal.ofReal (‖a ^ n‖ ^ (1 / ↑n))) atTop (𝓝 (spectralRadius ℂ a))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm... | [] | convert! pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a using 1
ext1
rw [← ofReal_rpow_of_nonneg (norm_nonneg _) _, ← coe_nnnorm, coe_nnreal_eq]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 136,
"column": 2
} | {
"line": 139,
"column": 6
} | {
"line": 141,
"column": 0
} | [
{
"pp": "A : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\n⊢ Tendsto (fun n ↦ ENNReal.ofReal (‖a ^ n‖ ^ (1 / ↑n))) atTop (𝓝 (spectralRadius ℂ a))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm... | [] | convert! pow_nnnorm_pow_one_div_tendsto_nhds_spectralRadius a using 1
ext1
rw [← ofReal_rpow_of_nonneg (norm_nonneg _) _, ← coe_nnnorm, coe_nnreal_eq]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 119,
"column": 20
} | {
"line": 119,
"column": 37
} | {
"line": 119,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nh : ‖a‖ * ‖1‖ < ‖k‖\n⊢ k ∈ {r | IsUnit (↑ₐ r - a)}",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedRing.to... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nh : ‖a‖ * ‖1‖ < ‖k‖\n⊢ IsUnit (↑ₐ k - a)"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 26
} | [
{
"pp": "case lower\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ {z | z.im < 0}\nhz' : z ∈ σ ℂ a\n⊢ False",
"ppTerm": "?lower",
"assigned": true,
"usedConstants": [
"Real",
"NormedRing.toRing",
"spectrum",
"Real.instZero",
"congrAr... | [
"case lower\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z.im < 0\nhz' : z ∈ σ ℂ a\n⊢ False"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 26
} | [
{
"pp": "case upper\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ {z | 0 < z.im}\nhz' : z ∈ σ ℂ a\n⊢ False",
"ppTerm": "?upper",
"assigned": true,
"usedConstants": [
"Real",
"NormedRing.toRing",
"spectrum",
"Real.instZero",
"congrAr... | [
"case upper\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : 0 < z.im\nhz' : z ∈ σ ℂ a\n⊢ False"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 359,
"column": 24
} | {
"line": 359,
"column": 57
} | {
"line": 359,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\n⊢ IsUnit (↑u⁻¹ • 1 - a)",
"ppTerm": "?neg✝",
"assigned": true,
"usedCons... | [
"case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\n⊢ IsUnit ((algebraMap 𝕜 A) ↑u⁻¹ - a)"
] | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 68
} | {
"line": 409,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Ring A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : IsSemitopologicalRing A\ninst✝² : T2Space A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na :... | [] | apply (SpectrumRestricts.nnreal_of_nonneg ha).cfcHom_eq_restrict _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 68
} | {
"line": 409,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Ring A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : IsSemitopologicalRing A\ninst✝² : T2Space A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na :... | [] | apply (SpectrumRestricts.nnreal_of_nonneg ha).cfcHom_eq_restrict _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 68
} | {
"line": 409,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Ring A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : StarRing A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : IsSemitopologicalRing A\ninst✝² : T2Space A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na :... | [] | apply (SpectrumRestricts.nnreal_of_nonneg ha).cfcHom_eq_restrict _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 112,
"column": 60
} | {
"line": 112,
"column": 87
} | {
"line": 112,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\na : E\ny : F\nhy : y ∈ {y | ‖(B a) y‖ ≤ 1}\nx✝ : E\nhb : x✝ ∈ {a}\n⊢ ‖(B x✝) y‖ ≤ 1",
"ppTerm": "?m.73",
... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\na : E\ny : F\nhy : y ∈ {y | ‖(B a) y‖ ≤ 1}\nx✝ : E\nhb : x✝ ∈ {a}\n⊢ ‖(B a) y‖ ≤ 1"
] | Set.mem_singleton_iff.mp hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 93,
"column": 25
} | {
"line": 96,
"column": 56
} | {
"line": 98,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\nx : 𝕜\nhx : x ∈ σ 𝕜 a\nhf : ContinuousOn f (σ 𝕜 a)\nha : p a\n⊢ ‖f x‖ ≤ ‖cfc f a‖",
"p... | [] | by
revert hx
nontriviality A
exact (IsGreatest.norm_cfc f a hf ha |>.2 ⟨x, ·, rfl⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 308,
"column": 2
} | {
"line": 310,
"column": 16
} | {
"line": 312,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\n... | [] | · simp only [← cfcₙ_apply f a, (IsGreatest.norm_cfcₙ f a hf hf0 ha |>.lt_iff)]
rintro - ⟨x, hx, rfl⟩
exact h x hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 69
} | {
"line": 330,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nr : ℝ\nhr : ‖r‖ ≤ 1\ny : E\nhy : y ∈ s\nha : (fun x ↦ r • x) y ∈ balancedHull ℝ s\n⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [
"E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nr : ℝ\nhr : ‖r‖ ≤ 1\ny : E\nhy : y ∈ s\nha : (fun x ↦ r • x) y ∈ balancedHull ℝ s\nthis : 0 ≤ 1 + r\n⊢ (fun x ↦ r • x) y ∈ segment ℝ y (-y)"
] | have : 0 ≤ 1 + r := neg_le_iff_add_nonneg'.mp (neg_le_of_abs_le hr) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 340,
"column": 46
} | {
"line": 340,
"column": 52
} | {
"line": 340,
"column": 52
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx✝¹ : E\nx✝ : x✝¹ ∈ -s\n⊢ ∃ r, ‖r‖ ≤ 1 ∧ x✝¹ ∈ r • s",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"SeminormedRing.toNorm",
"Real.ins... | [
"case h\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nx✝¹ : E\nx✝ : x✝¹ ∈ -s\n⊢ ‖-1‖ ≤ 1 ∧ x✝¹ ∈ -1 • s"
] | use -1 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.LocallyConvex.HahnBanach | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 39
} | {
"line": 125,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NormedField 𝕜\ninst✝⁸ : IsRCLikeNormedField 𝕜\nF : Type u_3\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : IsTopologicalAddGroup F\ninst✝⁴ : Module 𝕜 F\ninst✝³ : ContinuousSMul 𝕜 F\ninst✝² : T2Space F\ninst✝¹ : PolynormableSpace 𝕜 F\nS : Submodule 𝕜 F\nins... | [] | exact ⟨g, DFunLike.congr_fun hg.symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.HahnBanach | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 35
} | {
"line": 94,
"column": 0
} | [
{
"pp": "𝕜 : Type v\ninst✝² : RCLike 𝕜\nE : Type u\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\n⊢ ∃ g, ‖g‖ ≤ 1 ∧ g x = ↑‖x‖",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomCl... | [] | by_cases hx : ‖x‖ = 0
· exact ⟨0, by simp, by simp [hx]⟩
· obtain ⟨g, hg⟩ := exists_dual_vector 𝕜 x hx
exact ⟨g, hg.left.le, hg.right⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.HahnBanach | {
"line": 89,
"column": 2
} | {
"line": 92,
"column": 35
} | {
"line": 94,
"column": 0
} | [
{
"pp": "𝕜 : Type v\ninst✝² : RCLike 𝕜\nE : Type u\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\n⊢ ∃ g, ‖g‖ ≤ 1 ∧ g x = ↑‖x‖",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomCl... | [] | by_cases hx : ‖x‖ = 0
· exact ⟨0, by simp, by simp [hx]⟩
· obtain ⟨g, hg⟩ := exists_dual_vector 𝕜 x hx
exact ⟨g, hg.left.le, hg.right⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UrysohnsLemma | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 45
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ c.Uᶜ.indicator 1 x ≤ c.left.Uᶜ.indicator 1 x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLE",
"instOfNatNat",
"Urysohns.CU.approx_mem_I... | [] | exact (approx_mem_Icc_right_left c 0 x).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 66
} | {
"line": 119,
"column": 0
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nI : Ideal C(X, R)\nx : X\n⊢ x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 66
} | {
"line": 119,
"column": 0
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nI : Ideal C(X, R)\nx : X\n⊢ x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 66
} | {
"line": 119,
"column": 0
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nI : Ideal C(X, R)\nx : X\n⊢ x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | rw [← Set.mem_compl_iff, setOfIdeal, compl_compl, Set.mem_setOf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 61,
"column": 4
} | {
"line": 64,
"column": 28
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ... | [] | rw [← cfcₙ_mul _ _, ← cfcₙ_zero ℝ a]
refine cfcₙ_congr (fun x _ ↦ ?_)
simp only [_root_.posPart_def, _root_.negPart_def]
simpa using le_total x 0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 61,
"column": 4
} | {
"line": 64,
"column": 28
} | {
"line": 65,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n⊢ cfcₙ (fun x ↦ x⁺) a * cfcₙ... | [] | rw [← cfcₙ_mul _ _, ← cfcₙ_zero ℝ a]
refine cfcₙ_congr (fun x _ ↦ ?_)
simp only [_root_.posPart_def, _root_.negPart_def]
simpa using le_total x 0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UrysohnsLemma | {
"line": 509,
"column": 15
} | {
"line": 509,
"column": 49
} | {
"line": 509,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\ns t : Set X\nhs : IsOpen[inst✝¹] s\nhscp : IsCompact (closure[inst✝¹] sᶜᶜ)\nht : IsClosed[inst✝¹] t\nhst : t ⊆ s\nu v : Set X\nhuIsOpen : IsOpen[inst✝¹] u\nhvIsOpen : IsOpen[inst✝¹] v\nhscompl_subset_u : sᶜ ⊆ u\nht_subset_v : t ⊆ v\nhDisjoin... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\ns t : Set X\nhs : IsOpen[inst✝¹] s\nhscp : IsCompact (closure[inst✝¹] sᶜᶜ)\nht : IsClosed[inst✝¹] t\nhst : t ⊆ s\nu v : Set X\nhuIsOpen : IsOpen[inst✝¹] u\nhvIsOpen : IsOpen[inst✝¹] v\nhscompl_subset_u : sᶜ ⊆ u\nht_subset_v : t ⊆ v\nhDisjointuv : u ⊆ vᶜ... | ← subset_compl_iff_disjoint_right, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 509,
"column": 6
} | {
"line": 509,
"column": 85
} | {
"line": 510,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\ns t : Set X\nhs : IsOpen[inst✝¹] s\nhscp : IsCompact (closure[inst✝¹] sᶜᶜ)\nht : IsClosed[inst✝¹] t\nhst : t ⊆ s\nu v : Set X\nhuIsOpen : IsOpen[inst✝¹] u\nhvIsOpen : IsOpen[inst✝¹] v\nhscompl_subset_u : sᶜ ⊆ u\nht_subset_v : t ⊆ v\nhDisjoin... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\ns t : Set X\nhs : IsOpen[inst✝¹] s\nhscp : IsCompact (closure[inst✝¹] sᶜᶜ)\nht : IsClosed[inst✝¹] t\nhst : t ⊆ s\nu v : Set X\nhuIsOpen : IsOpen[inst✝¹] u\nhvIsOpen : IsOpen[inst✝¹] v\nhscompl_subset_u : sᶜ ⊆ u\nht_subset_v : t ⊆ v\nhDisjointuv : u ⊆ vᶜ... | simp_rw [← subset_compl_iff_disjoint_right, compl_subset_comm (s := u0)] at hu1 | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 32
} | {
"line": 81,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| a",
"ppTerm": "?m.110",
"ass... | [
"A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| cfcₙ id a"
] | rw [← cfcₙ_id ℝ a] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 32
} | {
"line": 81,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| a",
"ppTerm": "?m.110",
"ass... | [
"A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| cfcₙ id a"
] | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 32
} | {
"line": 81,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| a",
"ppTerm": "?m.110",
"ass... | [
"A : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\n| cfcₙ id a"
] | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 177,
"column": 14
} | {
"line": 177,
"column": 32
} | {
"line": 178,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 177,
"column": 14
} | {
"line": 177,
"column": 32
} | {
"line": 178,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 177,
"column": 14
} | {
"line": 177,
"column": 32
} | {
"line": 178,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 194,
"column": 14
} | {
"line": 194,
"column": 32
} | {
"line": 195,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 194,
"column": 14
} | {
"line": 194,
"column": 32
} | {
"line": 195,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 194,
"column": 14
} | {
"line": 194,
"column": 32
} | {
"line": 195,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | [
"A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : NonnegSpectrumCla... | rw [← cfcₙ_id ℝ a] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 6
} | {
"line": 284,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 84,
"column": 27
} | {
"line": 84,
"column": 43
} | {
"line": 84,
"column": 44
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra ℝ A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\nr : ℝ\n⊢ {0}... | [
"A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra ℝ A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\nr : ℝ\n⊢ ∀ i ∈ {a | IsSt... | Filter.mem_iSup, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 405,
"column": 14
} | {
"line": 429,
"column": 63
} | {
"line": 429,
"column": 64
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx✝ : PartialOrder A := spectralOrder A\n⊢ ∀ (x y : A), x ≤ y ↔ ∃ p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s), y = x + p",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"eq_sub_iff... | [] | by
intro x y
constructor
· intro h
obtain ⟨s, hs₁, _, hs₂⟩ :=
CFC.exists_sqrt_of_isSelfAdjoint_of_quasispectrumRestricts h.1 h.2
refine ⟨s * s, ?_, by rwa [eq_sub_iff_add_eq', eq_comm] at hs₂⟩
exact AddSubmonoid.subset_closure ⟨s, by simp [hs₁.star_eq]⟩
· rintro... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 26,
"column": 6
} | {
"line": 26,
"column": 17
} | {
"line": 26,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nr : ℝ\nhr : 0 < r\n⊢ span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r) = ⊤",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCStarAlgebra... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nr : ℝ\nhr : 0 < r\n⊢ ⊤ ≤ span ℂ ({x | 0 ≤ x} ∩ Metric.closedBall 0 r)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 17
} | {
"line": 43,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nr : ℝ\nhr : 0 < r\n⊢ span ℂ ({x | 0 ≤ x} ∩ Metric.ball 0 r) = ⊤",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCStarAlgebra.toNon... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nr : ℝ\nhr : 0 < r\n⊢ ⊤ ≤ span ℂ ({x | 0 ≤ x} ∩ Metric.ball 0 r)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 181,
"column": 46
} | {
"line": 185,
"column": 49
} | {
"line": 187,
"column": 0
} | [
{
"pp": "R : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : MetricSpace A\ninst✝¹ : Algebra R A\ninst✝ : IsometricContinuousFunctionalCalculus ... | [] | by
have h₁ := lipschitzOnWith_cfc_fun R a
have h₂ := lipschitzWith_one_ofFun_toFun' (𝔖 := {spectrum R a}) (𝔗 := {s}) (β := R) (by simpa)
have h₃ := h₂.lipschitzOnWith (s := {f | ContinuousOn (toFun {s} f) (s)})
simpa using! h₁.comp h₃ (fun f hf ↦ hf.mono hs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 679,
"column": 2
} | {
"line": 680,
"column": 70
} | {
"line": 683,
"column": 0
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Sp... | [] | case neg =>
simp [sqrt_eq_cfc, rpow_def, cfc_apply_of_not_predicate a hnonneg] | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 329,
"column": 2
} | {
"line": 337,
"column": 45
} | {
"line": 339,
"column": 0
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : 0 ≤ a\nb : Aˣ\nhb : IsStrictlyPositive ↑b\nhbab : 0 ≤ ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2))\nh : ↑b ^ (-(1 / 2)) * a * ↑b ^ (-(1 / 2)) ≤ 1\n⊢ a ≤ ↑b",
"ppTerm": "?refine_2",
"as... | [] | · calc
a = (sqrt ↑b * ↑b ^ (-(1 / 2) : ℝ)) * a * (↑b ^ (-(1 / 2) : ℝ) * sqrt ↑b) := by
simp only [CFC.sqrt_eq_rpow .., ← CFC.rpow_add b.isUnit]
norm_num
simp [CFC.rpow_zero (b : A)]
_ = sqrt ↑b * (↑b ^ (-(1 / 2) : ℝ) * a * ↑b ^ (-(1 / 2) : ℝ)) * sqrt ↑b := by
simp only [mul_a... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 778,
"column": 4
} | {
"line": 778,
"column": 73
} | {
"line": 779,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : PartialOrder A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarOrderedRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na : A\na✝ : Nontrivial A\nh : IsStrictlyPositive a⁻¹ʳ\nH ... | [
"A : Type u_1\ninst✝⁷ : PartialOrder A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarOrderedRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na : A\na✝ : Nontrivial A\nh : 0 ≤ 0 ∧ IsUnit 0\nH : ¬IsUnit a\n⊢ Fals... | rw [Ring.inverse_non_unit _ H, IsStrictlyPositive.iff_of_unital] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 544,
"column": 2
} | {
"line": 545,
"column": 42
} | {
"line": 547,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ inr ⁻¹' Icc 0 1 = {x | 0 ≤ x} ∩ closedBall 0 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"_priva... | [] | ext
simp [-mem_Icc, inr_mem_Icc_iff_norm_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 544,
"column": 2
} | {
"line": 545,
"column": 42
} | {
"line": 547,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ inr ⁻¹' Icc 0 1 = {x | 0 ≤ x} ∩ closedBall 0 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"_priva... | [] | ext
simp [-mem_Icc, inr_mem_Icc_iff_norm_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 48
} | {
"line": 123,
"column": 2
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℂ A\ninst✝⁶ : Module ℂ E\ninst✝⁵ : PartialOrder A\ninst✝⁴ : SMul A E\ninst✝³ : Norm A\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarModule ℂ A\nz : ℝ\nx y : E\nh₁ : z •> y = ↑z... | [
"A : Type u_1\nE : Type u_2\ninst✝¹⁰ : NonUnitalRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℂ A\ninst✝⁶ : Module ℂ E\ninst✝⁵ : PartialOrder A\ninst✝⁴ : SMul A E\ninst✝³ : Norm A\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarModule ℂ A\nz : ℝ\nx y : E\nh₁ : z •> y = ↑z •> y\n⊢ sta... | rw [h₁, ← star_inner, inner_smul_left_complex] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 25
} | {
"line": 179,
"column": 4
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝¹⁰ : NonUnitalCStarAlgebra A\ninst✝⁹ : PartialOrder A\nE : Type u_2\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : Module ℂ E\ninst✝⁶ : SMul A E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : Module ℂ F\ninst✝³ : SMul A F\ninst✝² : CStarModule A E\ninst✝¹ : CStarModule A F... | [
"case h₂\nA : Type u_1\ninst✝¹⁰ : NonUnitalCStarAlgebra A\ninst✝⁹ : PartialOrder A\nE : Type u_2\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : Module ℂ E\ninst✝⁶ : SMul A E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : Module ℂ F\ninst✝³ : SMul A F\ninst✝² : CStarModule A E\ninst✝¹ : CStarModule A F\ninst✝ : St... | · exact norm_fst_le x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Matrix.Normed | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 16
} | {
"line": 165,
"column": 2
} | [
{
"pp": "case e_f.refine_1.inr\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype n\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : DecidableEq n\nv : n → α\ni j : n\nhj : j ∈ Finset.univ\nhij : i ≠ j\n⊢ ‖diagonal v i j‖₊ ≤ ‖v i‖₊",
"ppTerm": "?e_f.refine_1.inr",
"assigned": true,
"usedConstants": [
"L... | [] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Matrix.Normed | {
"line": 308,
"column": 73
} | {
"line": 308,
"column": 97
} | {
"line": 310,
"column": 0
} | [
{
"pp": "n : Type u_4\nα : Type u_5\nι : Type u_7\ninst✝² : Fintype n\ninst✝¹ : Unique ι\ninst✝ : SeminormedAddCommGroup α\nv : n → α\n⊢ ↑(∑ i, ‖v i‖₊) = ∑ i, ‖v i‖",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Real",
"Fi... | [] | by simp [NNReal.coe_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.Normed | {
"line": 460,
"column": 66
} | {
"line": 462,
"column": 47
} | {
"line": 464,
"column": 0
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : Fintype n\ninst✝² : NontriviallyNormedField α\ninst✝¹ : NormedAlgebra ℝ α\ninst✝ : DecidableEq n\nf : (n → α) →L[α] m → α\n⊢ ‖LinearMap.toMatrix' ↑f‖₊ = ‖f‖₊",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | by
rw [linfty_opNNNorm_eq_opNNNorm]
simp only [← toLin'_apply', toLin'_toMatrix'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Matrix.Normed | {
"line": 553,
"column": 64
} | {
"line": 553,
"column": 88
} | {
"line": 555,
"column": 0
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nA : Matrix m n α\n⊢ ↑((∑ i, ∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2)) = (∑ i, ∑ j, ‖A i j‖ ^ 2) ^ (1 / 2)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | by simp [NNReal.coe_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.SpecificCodomains.ContinuousMap | {
"line": 71,
"column": 2
} | {
"line": 75,
"column": 66
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝³ : MeasurableSpace X\nμ : Measure X\ninst✝² : TopologicalSpace Y\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompactSpace Y\nf : X → C(Y, E)\nbound : X → ℝ\nbound_int : HasFiniteIntegral bound μ\nbound_ge : ∀ᵐ (x : X) ∂μ, ∀ (y : Y), ‖(f x) y‖ ≤ boun... | [] | · have bound_nonneg : 0 ≤ᵐ[μ] bound := by
filter_upwards [bound_ge] with x bound_x using le_trans (norm_nonneg _) (bound_x h.some)
refine .mono' bound_int ?_
filter_upwards [bound_ge, bound_nonneg] with x bound_ge_x bound_nonneg_x
exact ContinuousMap.norm_le _ bound_nonneg_x |>.mpr bound_ge_x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 79,
"column": 43
} | {
"line": 81,
"column": 33
} | {
"line": 83,
"column": 0
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : MeasurableSpace X\nμ : Measure X\ninst✝⁴ : NormedRing A\ninst✝³ : StarRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : ContinuousFunctionalCalculus 𝕜 A p\ninst✝ : CompleteSpace A\nf : X → 𝕜 → 𝕜\na : A\nhf : Integrable... | [] | by
conv in cfc _ _ => rw [cfc_eq_cfcL_mkD _ a]
exact cfcL_integrable _ _ hf ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 468,
"column": 4
} | {
"line": 470,
"column": 67
} | {
"line": 471,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝⁵ : Unique n\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Mul A\ninst✝¹ : Star A\ninst✝ : Module R A\n⊢ Function.RightInverse (fun M ↦ M default default) fun a x y ↦ a",
"ppTerm": "?m.50",
"assi... | [] | intro
ext i j
simp [Subsingleton.elim i default, Subsingleton.elim j default] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 468,
"column": 4
} | {
"line": 470,
"column": 67
} | {
"line": 471,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝⁵ : Unique n\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Mul A\ninst✝¹ : Star A\ninst✝ : Module R A\n⊢ Function.RightInverse (fun M ↦ M default default) fun a x y ↦ a",
"ppTerm": "?m.50",
"assi... | [] | intro
ext i j
simp [Subsingleton.elim i default, Subsingleton.elim j default] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 583,
"column": 22
} | {
"line": 583,
"column": 78
} | {
"line": 584,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nc : ℂ\nM : CStarMatrix m n A\n⊢ ‖c • M‖ = ‖c‖ * ‖M‖",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"WithCS... | [] | rw [norm_def, norm_def, map_smul, norm_smul _ (toCLM M)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 583,
"column": 22
} | {
"line": 583,
"column": 78
} | {
"line": 584,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nc : ℂ\nM : CStarMatrix m n A\n⊢ ‖c • M‖ = ‖c‖ * ‖M‖",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"WithCS... | [] | rw [norm_def, norm_def, map_smul, norm_smul _ (toCLM M)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 583,
"column": 22
} | {
"line": 583,
"column": 78
} | {
"line": 584,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nc : ℂ\nM : CStarMatrix m n A\n⊢ ‖c • M‖ = ‖c‖ * ‖M‖",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"WithCS... | [] | rw [norm_def, norm_def, map_smul, norm_smul _ (toCLM M)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Matrix | {
"line": 241,
"column": 2
} | {
"line": 245,
"column": 41
} | {
"line": 247,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → 𝕜\nT : EuclideanSpace 𝕜 n →L[𝕜] EuclideanSpace 𝕜 n := toEuclideanCLM (diagonal v)\n⊢ ‖v‖ ≤ ‖toEuclideanCLM (diagonal v)‖",
"ppTerm": "?refine_2",
"assigned": true,
"usedCon... | [] | · refine (pi_norm_le_iff_of_nonneg (norm_nonneg T)).mpr fun i ↦ ?_
calc _ = ‖T (toLp 2 (Pi.single i (1 : 𝕜)))‖ := by
rw [toEuclideanCLM_toLp (diagonal v) (Pi.single i (1 : 𝕜))]
simp
_ ≤ _ := by grw [T.le_opNorm]; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.FiberBundle.Basic | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 28
} | {
"line": 273,
"column": 4
} | [
{
"pp": "B : Type u_2\nF : Type u_3\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace F\nE : B → Type u_5\ninst✝³ : TopologicalSpace (TotalSpace F E)\ninst✝² : (b : B) → TopologicalSpace (E b)\ninst✝¹ : FiberBundle F E\ninst✝ : T1Space B\nx : B\n⊢ IsClosed[inst✝³] (range (TotalSpace.mk x))",
"ppTerm":... | [
"B : Type u_2\nF : Type u_3\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : TopologicalSpace F\nE : B → Type u_5\ninst✝³ : TopologicalSpace (TotalSpace F E)\ninst✝² : (b : B) → TopologicalSpace (E b)\ninst✝¹ : FiberBundle F E\ninst✝ : T1Space B\nx : B\n⊢ IsClosed[inst✝³] (TotalSpace.proj ⁻¹' {x})"
] | rw [TotalSpace.range_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 353,
"column": 42
} | {
"line": 353,
"column": 78
} | {
"line": 354,
"column": 2
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj✝ : Z → B\ne✝ : Pretrivialization F proj✝\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\nhs : IsOpen[inst✝²] s\nproj : Z → ↑s\ne : Pretrivialization F... | [] | ext <;> simp [e.apply_symm_apply hx] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 536,
"column": 6
} | {
"line": 536,
"column": 17
} | {
"line": 536,
"column": 18
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne : Trivialization F proj\nz : Z\nhz : z ∈ e.source\nl : Filter Z\n⊢ l ≤ 𝓝 z ↔ l ≤ comap proj (𝓝 (proj z)) ⊓ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2)",
"ppTerm"... | [
"B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne : Trivialization F proj\nz : Z\nhz : z ∈ e.source\nl : Filter Z\n⊢ l ≤ 𝓝 z ↔ l ≤ comap proj (𝓝 (proj z)) ∧ l ≤ comap (Prod.snd ∘ ↑e) (𝓝 (↑e z).2)"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Covering.Basic | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 72
} | {
"line": 486,
"column": 2
} | [
{
"pp": "case refine_1.a\nE : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝³ : Nonempty (X → E)\nι : Type ?u.20\ninst✝² : Nonempty ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : DiscreteTopology ι\nU : ι → Set E\nV : Set X\nopen_V : IsOpen[inst✝⁴] V\nopen... | [] | · dsimp only; rw [dif_pos (by exact he'.2)]; exact ⟨he'.1, idx_U ..⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 44
} | {
"line": 132,
"column": 44
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ ≤ π\na✝ : Nontrivial A\n⊢ IsLeast (re '' (fun x ↦ NormedSpace.exp (I • x)) '' spectrum ℂ ↑x) (Real.cos ‖x‖)",
"ppTerm": "?m.234",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRin... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ ≤ π\na✝ : Nontrivial A\n⊢ IsLeast (re '' (fun x ↦ NormedSpace.exp (I • x)) '' ⇑(algebraMap ℝ ℂ) '' spectrum ℝ ↑x) (Real.cos ‖x‖)"
] | ← x.2.spectrumRestricts.algebraMap_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 46
} | {
"line": 163,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane\ny : ℂ\nhy : y ∈ spectrum ℂ ↑x\n⊢ ↑(NormedSpace.exp (I • y)).arg = y",
"ppTerm": "?m.562",
"assigned": true,
"usedConstants": [
"NormedCommRing.to... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane\ny : ℂ\nhy : y ∈ ⇑(algebraMap ℝ ℂ) '' spectrum ℝ ↑x\n⊢ ↑(NormedSpace.exp (I • y)).arg = y"
] | ← x.2.spectrumRestricts.algebraMap_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 40,
"column": 2
} | {
"line": 47,
"column": 59
} | {
"line": 48,
"column": 2
} | [
{
"pp": "case inr\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsSelfAdjoint a\nha_norm : ‖a‖ ≤ 1\nh✝ : Nontrivial A\n⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"IsSelfAdjoint.sq... | [
"case inr\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsSelfAdjoint a\nha_norm : ‖a‖ ≤ 1\nh✝ : Nontrivial A\nkey : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x ↦ ↑x.re + I * ↑√(1 - x.re ^ 2)) a\n⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A"
] | have key : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x : ℂ ↦ x.re + I * √(1 - x.re ^ 2)) a := by
rw [CFC.sqrt_eq_real_sqrt (1 - a ^ 2) ?nonneg]
case nonneg =>
rwa [sub_nonneg, ← CStarAlgebra.norm_le_one_iff_of_nonneg (a ^ 2), sq, ha.norm_mul_self,
sq_le_one_iff₀ (by positivity)]
rw [cfc_add .., ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 29
} | {
"line": 97,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nv : E\nw : F\n⊢ ⟪w, (adjointAux (adjointAux A... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nv : E\nw : F\n⊢ ⟪(adjointAux A) w, v⟫_𝕜 = ⟪w, A v⟫_𝕜"
] | adjointAux_inner_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 352,
"column": 22
} | {
"line": 352,
"column": 67
} | {
"line": 354,
"column": 0
} | [
{
"pp": "case mpr.cons\nA : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nxs : List ↥(selfAdjoint A)\nih : (List.map expUnitary xs).prod ∈ pathComponent 1\n⊢ (List.map expUnitary (x :: xs)).prod ∈ pathComponent 1",
"ppTerm": "?mpr.cons",
"assigned": true,
"usedConstants": [
"CStarAlg... | [] | simpa using! (joined_one_expUnitary x).mul ih | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
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