module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Analytic.Binomial | {
"line": 216,
"column": 2
} | {
"line": 218,
"column": 73
} | {
"line": 219,
"column": 2
} | [
{
"pp": "a b : ℂ\n⊢ HasFPowerSeriesOnBall (fun x ↦ (b - a) / (1 - x) + a / (1 - x) ^ 2)\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ a * ↑n + b) 0 1",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSp... | [
"case e'_9\na b x✝ : ℂ\n⊢ (b - a) / (1 - x✝) + a / (1 - x✝) ^ 2 = (((b - a) • fun x ↦ 1 / (1 - x)) + a • fun x ↦ 1 / (1 - x) ^ 2) x✝",
"case e'_10\na b : ℂ\n⊢ (FormalMultilinearSeries.ofScalars ℂ fun n ↦ a * ↑n + b) =\n (b - a) • FormalMultilinearSeries.ofScalars ℂ 1 + a • FormalMultilinearSeries.ofScalars ℂ f... | convert
(one_div_one_sub_hasFPowerSeriesOnBall_zero.const_smul (c := b - a)).add
(one_div_one_sub_sq_hasFPowerSeriesOnBall_zero.const_smul (c := a)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Topology.MetricSpace.MetricSeparated | {
"line": 142,
"column": 4
} | {
"line": 143,
"column": 23
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : PseudoEMetricSpace X\ns t s' : Set X\nr : ℝ≥0∞\nr0 : r ≠ 0\nhr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y\nr' : ℝ≥0∞\nr0' : r' ≠ 0\nhr' : ∀ x ∈ s', ∀ y ∈ t, r' ≤ edist x y\n⊢ min r r' ≠ 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"ENNRea... | [] | rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.MetricSeparated | {
"line": 142,
"column": 4
} | {
"line": 143,
"column": 23
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : PseudoEMetricSpace X\ns t s' : Set X\nr : ℝ≥0∞\nr0 : r ≠ 0\nhr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y\nr' : ℝ≥0∞\nr0' : r' ≠ 0\nhr' : ∀ x ∈ s', ∀ y ∈ t, r' ≤ edist x y\n⊢ min r r' ≠ 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"ENNRea... | [] | rw [← pos_iff_ne_zero] at r0 r0' ⊢
exact lt_min r0 r0' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.AperiodicOrder.Delone.Basic | {
"line": 106,
"column": 23
} | {
"line": 106,
"column": 69
} | {
"line": 107,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : MetricSpace Y\nD : DeloneSet X\ncarrier : Set X\npackingRadius coveringRadius : ℝ≥0\nh_carrier : carrier = D.carrier\nh_packing : packingRadius = D.packingRadius\nh_covering : coveringRadius = D.coveringRadius\n⊢ 0 < packingRadius",
"ppTer... | [] | by simpa [h_packing] using D.packingRadius_pos | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 84
} | {
"line": 168,
"column": 4
} | [
{
"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": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"ENNReal.zero_rpow_of_pos",
"Real",
"Preorder.toLT",
"HMul.hMul",
"congrAr... | [
"case inr.inr.inr.inl\ny : ℝ\ny_pos : 0 < y\n⊢ (∞ ^ y).log = ↑y * ∞.log",
"case inr.inr.inr.inr\nx : ℝ≥0∞\ny : ℝ\ny_pos : 0 < y\nx_real : 0 < x.toReal\n⊢ (x ^ y).log = ↑y * x.log"
] | · rw [ENNReal.zero_rpow_of_pos y_pos, log_zero, EReal.mul_bot_of_pos]; norm_cast | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 269,
"column": 57
} | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 91
} | [
{
"pp": "case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ (linearGrowthInf v + linearGrowthSup fun x ↦ b) = linearGrowthInf v",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
... | [
"case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v + 0 = linearGrowthInf v"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 269,
"column": 57
} | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 91
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v ≠ ⊥ ∨ (linearGrowthSup fun x ↦ b) ≠ ⊤",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMo... | [
"u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v ≠ ⊥ ∨ 0 ≠ ⊤"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 269,
"column": 57
} | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 91
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v ≠ ⊤ ∨ (linearGrowthSup fun x ↦ b) ≠ ⊥",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMo... | [
"u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v ≠ ⊤ ∨ 0 ≠ ⊥"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 279,
"column": 56
} | {
"line": 279,
"column": 90
} | {
"line": 279,
"column": 90
} | [
{
"pp": "case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ (linearGrowthSup v + linearGrowthSup fun x ↦ b) = linearGrowthSup v",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
... | [
"case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v + 0 = linearGrowthSup v"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 279,
"column": 56
} | {
"line": 279,
"column": 90
} | {
"line": 279,
"column": 90
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊥ ∨ (linearGrowthSup fun x ↦ b) ≠ ⊤",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMo... | [
"u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊥ ∨ 0 ≠ ⊤"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 279,
"column": 56
} | {
"line": 279,
"column": 90
} | {
"line": 279,
"column": 90
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊤ ∨ (linearGrowthSup fun x ↦ b) ≠ ⊥",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMo... | [
"u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥"
] | linearGrowthSup_const b_bot.ne' hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 56
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u,... | [
"case inr\nα : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩ B c)... | · refine ⟨a, ⟨hat, a_disj⟩, ?_⟩
simpa only [← mzero, zero_div] using δnonneg a hat | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 552,
"column": 4
} | {
"line": 552,
"column": 55
} | {
"line": 553,
"column": 2
} | [
{
"pp": "u : ℕ → EReal\nv : ℕ → ℕ\na : EReal\nh : Monotone u\nhv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)\nha : a ≠ 0\nha' : a ≠ ⊤\nhv₁ : 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop\nv_top : Tendsto v atTop atTop\nu_0 : ¬u = ⊥\nh' : ∀ (n : ℕ), u n ≤ 0\nu_0' : linearGrowthInf u = 0\n⊢ (linearGrowthInf fun n ↦ 0) ... | [] | exact linearGrowthInf_const zero_ne_bot zero_ne_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.AEEqOfLIntegral | {
"line": 214,
"column": 13
} | {
"line": 218,
"column": 68
} | {
"line": 218,
"column": 68
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhfi : ∫⁻ (x : α), f x ∂μ ≠ ∞\nhfg : μ.withDensity f = μ.withDensity g\n⊢ f =ᵐ[μ] g",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory... | [] | by
refine AEMeasurable.ae_eq_of_forall_setLIntegral_eq hf hg hfi ?_ fun s hs _ ↦ ?_
· rwa [← setLIntegral_univ, ← withDensity_apply g MeasurableSet.univ, ← hfg,
withDensity_apply f MeasurableSet.univ, setLIntegral_univ]
· rw [← withDensity_apply f hs, ← withDensity_apply g hs, ← hfg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 96
} | {
"line": 148,
"column": 96
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_2\ninst✝¹ : Countable ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν\n⊢ sum μ =\n (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).1 +\n ν.withDensity (sum fun i ↦ (μ i).singularPart ν, ... | [] | simp [withDensity_tsum, measurable_rnDeriv, Measure.sum_add_sum, singularPart_add_rnDeriv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 96
} | {
"line": 148,
"column": 96
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_2\ninst✝¹ : Countable ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν\n⊢ sum μ =\n (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).1 +\n ν.withDensity (sum fun i ↦ (μ i).singularPart ν, ... | [] | simp [withDensity_tsum, measurable_rnDeriv, Measure.sum_add_sum, singularPart_add_rnDeriv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 96
} | {
"line": 148,
"column": 96
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ✝ ν : Measure α\nι : Type u_2\ninst✝¹ : Countable ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).HaveLebesgueDecomposition ν\n⊢ sum μ =\n (sum fun i ↦ (μ i).singularPart ν, ∑' (i : ι), (μ i).rnDeriv ν).1 +\n ν.withDensity (sum fun i ↦ (μ i).singularPart ν, ... | [] | simp [withDensity_tsum, measurable_rnDeriv, Measure.sum_add_sum, singularPart_add_rnDeriv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 473,
"column": 14
} | {
"line": 473,
"column": 21
} | {
"line": 473,
"column": 22
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nC : ℝ≥0\nh : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)\n... | [
"α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nC : ℝ≥0\nh : ∀ (x : α), ∃ᶠ (r : ℝ) in 𝓝[>] 0, μ (closedBall x (3 * r)) ≤ ↑C * μ (closedBall x r)\ns : Set α\nf... | fsubset | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 189,
"column": 18
} | {
"line": 189,
"column": 35
} | {
"line": 190,
"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\nε : ℝ≥0\nεpos : ε > 0\ns : Set α := {x | ¬∀ᶠ (a : Set α) ... | [] | rw [ρo, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 189,
"column": 18
} | {
"line": 189,
"column": 35
} | {
"line": 190,
"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\nε : ℝ≥0\nεpos : ε > 0\ns : Set α := {x | ¬∀ᶠ (a : Set α) ... | [] | rw [ρo, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 189,
"column": 18
} | {
"line": 189,
"column": 35
} | {
"line": 190,
"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\nε : ℝ≥0\nεpos : ε > 0\ns : Set α := {x | ¬∀ᶠ (a : Set α) ... | [] | rw [ρo, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.OneDim | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 75
} | {
"line": 55,
"column": 4
} | [
{
"pp": "case refine_2\nx ε : ℝ\nεpos : ε > 0\n⊢ ∀ᶠ (i : ℝ) in 𝓝[<] x, Icc i x ⊆ Metric.closedBall x ε",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAss... | [
"x ε : ℝ\nεpos : ε > 0\ny : ℝ\nhy : y ∈ Icc (x - ε) x\n⊢ Icc y x ⊆ Metric.closedBall x ε"
] | filter_upwards [Icc_mem_nhdsLT <| show x - ε < x by linarith] with y hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 76,
"column": 2
} | {
"line": 83,
"column": 94
} | {
"line": 85,
"column": 0
} | [
{
"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... | [] | symm
refine additive_of_total (· ≤ · : α → α → Prop) (variationOnFromTo f s) (· ∈ s) ?_ ?_ ha hb hc
· rintro x y _xs _ys
simp only [variationOnFromTo.eq_neg_swap f s y x, add_neg_cancel]
· rintro x y z xy yz xs ys zs
rw [variationOnFromTo.eq_of_le f s xy, variationOnFromTo.eq_of_le f s yz,
variation... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 76,
"column": 2
} | {
"line": 83,
"column": 94
} | {
"line": 85,
"column": 0
} | [
{
"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... | [] | symm
refine additive_of_total (· ≤ · : α → α → Prop) (variationOnFromTo f s) (· ∈ s) ?_ ?_ ha hb hc
· rintro x y _xs _ys
simp only [variationOnFromTo.eq_neg_swap f s y x, add_neg_cancel]
· rintro x y z xy yz xs ys zs
rw [variationOnFromTo.eq_of_le f s xy, variationOnFromTo.eq_of_le f s yz,
variation... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.StarOrdered | {
"line": 84,
"column": 6
} | {
"line": 85,
"column": 31
} | {
"line": 86,
"column": 6
} | [
{
"pp": "case mp\nα : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : Zero α\nR : Type u_2\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : CommSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : IsTopologicalSemiring R\ninst✝¹ : ContinuousStar R\nins... | [
"case mp\nα : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : Zero α\nR : Type u_2\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : CommSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : IsTopologicalSemiring R\ninst✝¹ : ContinuousStar R\ninst✝ : StarOrd... | rw [le_def, ← ContinuousMap.coe_coe, ← ContinuousMap.coe_coe g, ← ContinuousMap.le_def,
StarOrderedRing.le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 24
} | {
"line": 805,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA ... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA : μ.FiniteSp... | specialize hx n c hc | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 332,
"column": 8
} | {
"line": 338,
"column": 34
} | {
"line": 339,
"column": 6
} | [
{
"pp": "α : 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 ↦ if i < ... | [] | refine add_le_add (add_le_add le_rfl ?_) le_rfl
have A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
have B : N - 1 + 1 < N + 1 := A.symm ▸ N.lt_succ_self
have C : N - 1 < N + 1 := lt_of_le_of_lt N.pred_le N.lt_succ_self
rw [Finset.sum_eq_sum_Ico_succ_bot C, Finset.sum_eq_sum_Ico_succ_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 332,
"column": 8
} | {
"line": 338,
"column": 34
} | {
"line": 339,
"column": 6
} | [
{
"pp": "α : 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 ↦ if i < ... | [] | refine add_le_add (add_le_add le_rfl ?_) le_rfl
have A : N - 1 + 1 = N := Nat.succ_pred_eq_of_pos Npos
have B : N - 1 + 1 < N + 1 := A.symm ▸ N.lt_succ_self
have C : N - 1 < N + 1 := lt_of_le_of_lt N.pred_le N.lt_succ_self
rw [Finset.sum_eq_sum_Ico_succ_bot C, Finset.sum_eq_sum_Ico_succ_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 46
} | {
"line": 357,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCa... | [] | rw [cfc_apply f a, mkD_of_continuousOn hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 46
} | {
"line": 357,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCa... | [] | rw [cfc_apply f a, mkD_of_continuousOn hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 46
} | {
"line": 357,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCa... | [] | rw [cfc_apply f a, mkD_of_continuousOn hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 541,
"column": 4
} | {
"line": 571,
"column": 21
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\na : α\nl : E\nh : (𝓝[s ∩ Iio a] a).NeBot\nha : a ∈ s\nh'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)\n⊢ eVariationOn f (s ∩ Iic a) ≤ e... | [] | rw [eVariationOn_eq_strictMonoOn]
apply iSup_le
rintro ⟨n, u, u_mono, u_mem⟩
have : u n ≤ a := (u_mem n (by simp)).2
rcases this.eq_or_lt with hn | hn; swap
· exact (sum_le_of_monotoneOn_Iic u_mono.monotoneOn (by grind [StrictMonoOn])).trans le_self_add
cases n with
| zero => simp
| succ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 541,
"column": 4
} | {
"line": 571,
"column": 21
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\na : α\nl : E\nh : (𝓝[s ∩ Iio a] a).NeBot\nha : a ∈ s\nh'f : Tendsto f (𝓝[s ∩ Iio a] a) (𝓝 l)\n⊢ eVariationOn f (s ∩ Iic a) ≤ e... | [] | rw [eVariationOn_eq_strictMonoOn]
apply iSup_le
rintro ⟨n, u, u_mono, u_mem⟩
have : u n ≤ a := (u_mem n (by simp)).2
rcases this.eq_or_lt with hn | hn; swap
· exact (sum_le_of_monotoneOn_Iic u_mono.monotoneOn (by grind [StrictMonoOn])).trans le_self_add
cases n with
| zero => simp
| succ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 21
} | {
"line": 638,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\na b : α\nh : (𝓝[>] a).NeBot\nh' : ContinuousWithinAt f (Ici a) a\nhab : a < b\n⊢ (𝓝[Iic b ∩ Ioi a] a).NeBot",
"ppTerm": "?m.53",
"assigned": true... | [
"α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\na b : α\nh : (𝓝[>] a).NeBot\nh' : ContinuousWithinAt f (Ici a) a\nhab : a < b\n⊢ 𝓝[Iic b ∩ Ioi a] a = 𝓝[>] a"
] | convert h using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 693,
"column": 94
} | {
"line": 701,
"column": 10
} | {
"line": 703,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | [] | by
have h₁ : ContinuousOn f (spectrum R (algebraMap R A r)) :=
continuousOn_singleton _ _ |>.mono <| CFC.spectrum_algebraMap_subset r
rw [cfc_apply f (algebraMap R A r) (cfc_predicate_algebraMap r),
← AlgHomClass.commutes (cfcHom (p := p) (cfc_predicate_algebraMap r)) (f r)]
congr
ext ⟨x, hx⟩
apply CFC.... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 16
} | {
"line": 751,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\ninst✝ : CompleteSpace 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\nh : L = ⊥\n⊢ Nonempty E",
"ppTerm": "?inl",
"assigned": true,... | [] | exact ⟨f x₀⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 864,
"column": 6
} | {
"line": 864,
"column": 23
} | {
"line": 865,
"column": 4
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nl : E\nhf : BoundedVariationOn f s\nx : α\nh'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)\nhx : x ∈ s\nh : (𝓝[s ∩ Iio x] x).NeBot\nH : Tendst... | [] | convert h using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 864,
"column": 6
} | {
"line": 864,
"column": 23
} | {
"line": 865,
"column": 4
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nl : E\nhf : BoundedVariationOn f s\nx : α\nh'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)\nhx : x ∈ s\nh : (𝓝[s ∩ Iio x] x).NeBot\nH : Tendst... | [] | convert h using 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 864,
"column": 6
} | {
"line": 864,
"column": 23
} | {
"line": 865,
"column": 4
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nl : E\nhf : BoundedVariationOn f s\nx : α\nh'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)\nhx : x ∈ s\nh : (𝓝[s ∩ Iio x] x).NeBot\nH : Tendst... | [] | convert h using 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 32
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [] | rw [cfcₙ_neg .., cfcₙ_id' R a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 32
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [] | rw [cfcₙ_neg .., cfcₙ_id' R a] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 591,
"column": 2
} | {
"line": 591,
"column": 32
} | {
"line": 593,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [] | rw [cfcₙ_neg .., cfcₙ_id' R a] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Weierstrass | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 68
} | {
"line": 114,
"column": 2
} | [
{
"pp": "a b : ℝ\nf : ℝ → ℝ\nc : ContinuousOn f (Set.Icc a b)\nε : ℝ\npos : 0 < ε\nf' : C(↑(Set.Icc a b), ℝ) := { toFun := fun x ↦ f ↑x, continuous_toFun := ⋯ }\n⊢ ∃ p, ∀ x ∈ Set.Icc a b, |Polynomial.eval x p - f x| < ε",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"a b✝ : ℝ\nf : ℝ → ℝ\nc : ContinuousOn f (Set.Icc a b✝)\nε : ℝ\npos : 0 < ε\nf' : C(↑(Set.Icc a b✝), ℝ) := { toFun := fun x ↦ f ↑x, continuous_toFun := ⋯ }\np : ℝ[X]\nb : ‖p.toContinuousMapOn (Set.Icc a b✝) - f'‖ < ε\n⊢ ∃ p, ∀ x ∈ Set.Icc a b✝, |Polynomial.eval x p - f x| < ε"
] | obtain ⟨p, b⟩ := exists_polynomial_near_continuousMap a b f' ε pos | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 374,
"column": 2
} | {
"line": 392,
"column": 38
} | {
"line": 394,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : TopologicalSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\n⊢ (comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra)).SeparatesPoints",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": ... | [] | intro x₁ x₂ hx
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A := by
refine A.sub_mem hfA (@Eq.subst _ (· ∈ A)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 374,
"column": 2
} | {
"line": 392,
"column": 38
} | {
"line": 394,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : TopologicalSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\n⊢ (comap (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) (restrictScalars ℝ A.toSubalgebra)).SeparatesPoints",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": ... | [] | intro x₁ x₂ hx
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx
let F : C(X, 𝕜) := f - const _ (f x₂)
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A := by
refine A.sub_mem hfA (@Eq.subst _ (· ∈ A)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 65
} | {
"line": 636,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\ns : Set 𝕜\ninst✝¹ : Fact (0 ∈ s)\ninst✝ : CompactSpace ↑s\nh0' : 0 ∈ s\n⊢ _root_.toContinuousMap ⁻¹' closure ↑(adjoin 𝕜 {ContinuousMap.restrict s (ContinuousMap.id 𝕜)}) = Set.univ",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\ns : Set 𝕜\ninst✝¹ : Fact (0 ∈ s)\ninst✝ : CompactSpace ↑s\nh0' : 0 ∈ s\n⊢ _root_.toContinuousMap ⁻¹' ↑(RingHom.ker (ContinuousMap.evalStarAlgHom 𝕜 𝕜 ⟨0, h0'⟩)) = Set.univ"
] | ← ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id s h0' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Convex | {
"line": 84,
"column": 11
} | {
"line": 84,
"column": 21
} | {
"line": 84,
"column": 22
} | [
{
"pp": "z w : ℂ\n⊢ z.Rectangle w = (convexHull ℝ) {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w}",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"HMul.hMul",
"ChainCompletePartialOrder.instOfCo... | [
"z w : ℂ\n⊢ uIcc z.re w.re ×ℂ uIcc z.im w.im = (convexHull ℝ) {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w}"
] | Rectangle, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 54,
"column": 55
} | {
"line": 59,
"column": 97
} | {
"line": 61,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nhk : k ∈ resolventSet 𝕜 a\n⊢ HasDerivAt (resolvent a) (-resolvent a k ^ 2) k",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants":... | [] | by
have H₁ : HasFDerivAt Ring.inverse _ (algebraMap 𝕜 A k - a) :=
hasFDerivAt_ringInverse (𝕜 := 𝕜) hk.unit
have H₂ : HasDerivAt (fun k => algebraMap 𝕜 A k - a) 1 k := by
simpa using! (Algebra.linearMap 𝕜 A).hasDerivAt.sub_const a
simpa [resolvent, sq, hk.unit_spec, ← Ring.inverse_unit hk.unit] using!... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 106,
"column": 11
} | {
"line": 106,
"column": 22
} | {
"line": 106,
"column": 23
} | [
{
"pp": "A : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\n⊢ ↑r ≤ (limsup (fun n ↦ ↑‖a ^ n‖₊ ^ (1 / ↑n)) atTop)⁻¹",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"A : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nr_pos : 0 < r\nr_lt : ↑r < (spectralRadius ℂ a)⁻¹\n⊢ ↑r ≤ liminf (fun i ↦ (↑‖a ^ i‖₊ ^ (1 / ↑i))⁻¹) atTop"
] | inv_limsup, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 44
} | {
"line": 158,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\n⊢ (spectralRadius ℂ (a⋆ * a)).toReal = ‖a‖ ^ 2",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"IsSelfAdjoint.star_mul_self",
"NonUnitalNormedR... | [] | rw [(IsSelfAdjoint.star_mul_self a).toReal_spectralRadius_complex_eq_norm,
CStarRing.norm_star_mul_self, ← pow_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 44
} | {
"line": 158,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\n⊢ (spectralRadius ℂ (a⋆ * a)).toReal = ‖a‖ ^ 2",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"IsSelfAdjoint.star_mul_self",
"NonUnitalNormedR... | [] | rw [(IsSelfAdjoint.star_mul_self a).toReal_spectralRadius_complex_eq_norm,
CStarRing.norm_star_mul_self, ← pow_two] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 44
} | {
"line": 158,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\n⊢ (spectralRadius ℂ (a⋆ * a)).toReal = ‖a‖ ^ 2",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"IsSelfAdjoint.star_mul_self",
"NonUnitalNormedR... | [] | rw [(IsSelfAdjoint.star_mul_self a).toReal_spectralRadius_complex_eq_norm,
CStarRing.norm_star_mul_self, ← pow_two] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 50
} | {
"line": 418,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : Zero β\n⊢ IsClosed (range toBCF)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"ZeroAtInftyContinuousMap.toBCF",
"PseudoMetricSpace.toUniformSpace",
"Mem... | [
"α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : Zero β\nf : α →ᵇ β\nhf : ClusterPt f (𝓟 (range toBCF))\n⊢ f ∈ range toBCF"
] | refine isClosed_iff_clusterPt.mpr fun f hf => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 473,
"column": 4
} | {
"line": 473,
"column": 38
} | {
"line": 474,
"column": 4
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nh : ∀ x ∈ σ ℝ≥0 a, f x ≤... | [
"case inl\nA : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nh : ∀ x ∈ σ ℝ≥0 a, f x ≤ c\nh✝ : Sub... | rw [Subsingleton.elim (cfc f a) 0] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 38
} | {
"line": 489,
"column": 4
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nhc : 0 < c\nh : ∀ x ∈ σ ... | [
"case inl\nA : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nf : ℝ≥0 → ℝ≥0\na : A\nc : ℝ≥0\nhc : 0 < c\nh : ∀ x ∈ σ ℝ≥0 a, f x <... | rw [Subsingleton.elim (cfc f a) 0] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 91
} | {
"line": 140,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nr : ℝ\nhr : 0 < r\n⊢ ⋂₀ (polar 𝕜 '' {F | F.Finite ∧ F ⊆ closedBall 0 r⁻¹}) = closedBall 0 r⁻¹⁻¹",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedC... | [] | rw [← polar_closedBall (inv_pos_of_pos hr), StrongDual.polar,
(topDualPairing 𝕜 E).flip.sInter_polar_finite_subset_eq_polar (closedBall (0 : E) r⁻¹)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.Barrelled | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 32
} | {
"line": 104,
"column": 32
} | [
{
"pp": "ι : Sort u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : BarrelledSpace 𝕜 E\np : ι → Seminorm 𝕜 E\nhp : ∀ (i : ι), Continuous[inst✝¹, _] ⇑(p i)\nbdd : BddAbove (range p)\n⊢ Continuous[inst✝¹, _] (⨆ i, ⇑(p i... | [
"ι : Sort u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : BarrelledSpace 𝕜 E\np : ι → Seminorm 𝕜 E\nhp : ∀ (i : ι), Continuous[inst✝¹, _] ⇑(p i)\nbdd : BddAbove (range p)\n⊢ Continuous[inst✝¹, _] ⇑(⨆ i, p i)"
] | ← Seminorm.coe_iSup_eq bdd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 70
} | {
"line": 177,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ ContinuousLinearMap.uniformSpace.toTopologicalSpace ≤ instTopologicalSpaceWeakDual 𝕜 E",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case e'_4\n𝕜 : Type u_1\nE : Type u_3\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ instTopologicalSpaceWeakDual 𝕜 E =\n TopologicalSpace.induced (fun x' ↦ StrongDual.toWeakDual x') (instTopologicalSpaceWeakDual 𝕜 E)"
] | convert! (@toWeakDual_continuous _ _ _ _ (by assumption)).le_induced | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.UrysohnsLemma | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 47
} | {
"line": 438,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\n⊢ ∃ f, s = ⇑f ⁻¹' {1} ∧ EqOn (⇑f) 0 t ∧ HasCompactSupport ⇑f ∧ ∀ (x : X), f x ∈ Icc 0 1",
"ppTerm": "?m.45",
... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\n⊢ ∃ f, s = ⇑f ⁻¹' {1} ∧ EqOn (⇑f) 0 t ∧ HasCompactS... | rcases h's.eq_iInter_nat with ⟨U, U_open, hU⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 301,
"column": 2
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\ns : Finset ι\nh : ∀ i ∈ s, IsSelfAdjoint (f i)\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\n⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"F... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert j s hj ih =>
suffices f j * ∑ i ∈ s, f i = 0 by
simp_all [(h j (by simp)).nnnorm_add_eq_max (by cfc_tac) this]
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 301,
"column": 2
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\ns : Finset ι\nh : ∀ i ∈ s, IsSelfAdjoint (f i)\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\n⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"F... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert j s hj ih =>
suffices f j * ∑ i ∈ s, f i = 0 by
simp_all [(h j (by simp)).nnnorm_add_eq_max (by cfc_tac) this]
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 301,
"column": 2
} | {
"line": 307,
"column": 76
} | {
"line": 309,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\ns : Finset ι\nh : ∀ i ∈ s, IsSelfAdjoint (f i)\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\n⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"F... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert j s hj ih =>
suffices f j * ∑ i ∈ s, f i = 0 by
simp_all [(h j (by simp)).nnnorm_add_eq_max (by cfc_tac) this]
simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h0 (by grind) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 48
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nf g : α →ᵤ[𝔖] β\n⊢ edist f g = ⨆ s ∈ 𝔖, ⨆ x ∈ s, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetric... | [] | simp [edist_def, iSup_and, iSup_comm (ι := α)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 48
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nf g : α →ᵤ[𝔖] β\n⊢ edist f g = ⨆ s ∈ 𝔖, ⨆ x ∈ s, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetric... | [] | simp [edist_def, iSup_and, iSup_comm (ι := α)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 48
} | {
"line": 216,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nf g : α →ᵤ[𝔖] β\n⊢ edist f g = ⨆ s ∈ 𝔖, ⨆ x ∈ s, edist ((toFun 𝔖) f x) ((toFun 𝔖) g x)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetric... | [] | simp [edist_def, iSup_and, iSup_comm (ι := α)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 285,
"column": 58
} | {
"line": 285,
"column": 76
} | {
"line": 286,
"column": 4
} | [
{
"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... | [] | simp [cfcₙ_id ℝ b] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 285,
"column": 58
} | {
"line": 285,
"column": 76
} | {
"line": 286,
"column": 4
} | [
{
"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... | [] | simp [cfcₙ_id ℝ b] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 285,
"column": 58
} | {
"line": 285,
"column": 76
} | {
"line": 286,
"column": 4
} | [
{
"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... | [] | simp [cfcₙ_id ℝ b] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 471,
"column": 33
} | {
"line": 476,
"column": 24
} | {
"line": 478,
"column": 0
} | [
{
"pp": "A : 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✝ : T2Space A\na :... | [] | by
have ha₁' : 0 ∉ spectrum ℝ≥0 a := spectrum.zero_notMem _ ha.isUnit
simp only [rpow_def]
rw [← cfc_comp _ _ a ha.nonneg]
refine cfc_congr fun _ _ => ?_
simp [NNReal.rpow_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 638,
"column": 2
} | {
"line": 639,
"column": 57
} | {
"line": 641,
"column": 0
} | [
{
"pp": "A : 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✝ : T2Space A\na :... | [] | have : a ^ (1 / 2 : ℝ) = a ^ ((1 / 2 : ℝ≥0) : ℝ) := rfl
rw [this, ← nnrpow_eq_rpow (by simp), sqrt_eq_nnrpow a] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 638,
"column": 2
} | {
"line": 639,
"column": 57
} | {
"line": 641,
"column": 0
} | [
{
"pp": "A : 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✝ : T2Space A\na :... | [] | have : a ^ (1 / 2 : ℝ) = a ^ ((1 / 2 : ℝ≥0) : ℝ) := rfl
rw [this, ← nnrpow_eq_rpow (by simp), sqrt_eq_nnrpow a] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 16
} | {
"line": 350,
"column": 2
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra 𝕜 A\ninst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nf : �... | [
"X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra 𝕜 A\ninst✝³ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nf : 𝕜 → 𝕜\na : ... | have hs' := hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 116,
"column": 10
} | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 24
} | [
{
"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 •> x = ↑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 •> x = ↑z •> x\n⊢ sta... | ← star_inner, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 122,
"column": 10
} | {
"line": 122,
"column": 23
} | {
"line": 122,
"column": 24
} | [
{
"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... | ← star_inner, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 16
} | {
"line": 535,
"column": 2
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹¹ : NormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedAlgebra ℝ A\ninst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : T2Space A\ni... | [
"X : Type u_1\nA : Type u_2\ninst✝¹¹ : NormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedAlgebra ℝ A\ninst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : T2Space A\ninst✝² : IsSe... | have hs' := hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 192,
"column": 10
} | {
"line": 192,
"column": 25
} | {
"line": 192,
"column": 25
} | [
{
"pp": "A : Type u_2\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : IsTopologicalRing A\ninst✝ : T2Space A\na b : A\nhb : Com... | [
"A : Type u_2\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : Module ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : IsTopologicalRing A\ninst✝ : T2Space A\na b : A\nhb : Commute a b\nf ... | ← cfcₙ_apply .. | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 894,
"column": 2
} | {
"line": 894,
"column": 16
} | {
"line": 895,
"column": 2
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : ContinuousStar A\ninst✝² : NonUnitalIsometricContinuousFunctionalCalculus �... | [
"X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : ContinuousStar A\ninst✝² : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝... | have hs' := hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 470,
"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\nx✝ : CStarMatrix n n A\ni j : n\n⊢ (fun a x y ↦ a) ((fun M ↦ M default default) x✝) i j = x✝ i j",
... | [] | simp [Subsingleton.elim i default, Subsingleton.elim j default] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 1087,
"column": 2
} | {
"line": 1087,
"column": 16
} | {
"line": 1088,
"column": 2
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹³ : NonUnitalNormedRing A\ninst✝¹² : StarRing A\ninst✝¹¹ : NormedSpace ℝ A\ninst✝¹⁰ : IsScalarTower ℝ A A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : PartialOrder A\ninst✝... | [
"X : Type u_1\nA : Type u_2\ninst✝¹³ : NonUnitalNormedRing A\ninst✝¹² : StarRing A\ninst✝¹¹ : NormedSpace ℝ A\ninst✝¹⁰ : IsScalarTower ℝ A A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrde... | have hs' := hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.Matrix | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 28
} | {
"line": 56,
"column": 6
} | [
{
"pp": "case a\n𝕜 : Type u_1\nn : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nU : Matrix n n 𝕜\nhU : U ∈ Matrix.unitaryGroup n 𝕜\ni j : n\nx : ℝ\nh_x : ∃ a ∈ Finset.univ.val, ‖U i a‖ ^ 2 = x\n⊢ 0 ≤ x",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Norm... | [
"case a\n𝕜 : Type u_1\nn : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nU : Matrix n n 𝕜\nhU : U ∈ Matrix.unitaryGroup n 𝕜\ni j : n\nx : ℝ\na : n\nh_a : a ∈ Finset.univ.val ∧ ‖U i a‖ ^ 2 = x\n⊢ 0 ≤ x"
] | obtain ⟨a, h_a⟩ := h_x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 27
} | {
"line": 300,
"column": 27
} | [
{
"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\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | [
"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\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj ⁻¹' s))\nx ... | Prod.map_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 394,
"column": 4
} | {
"line": 395,
"column": 66
} | {
"line": 396,
"column": 4
} | [
{
"pp": "case pos\nB : Type u_2\nF : Type u_3\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace F\nE : B → Type u_5\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (b : B) → TopologicalSpace (E b)\ninst✝² : ConditionallyCompleteLinearOrder B\ninst✝¹ : OrderTopology B\ninst✝ : FiberBundle F E\na b : ... | [
"case pos\nB : Type u_2\nF : Type u_3\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace F\nE : B → Type u_5\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (b : B) → TopologicalSpace (E b)\ninst✝² : ConditionallyCompleteLinearOrder B\ninst✝¹ : OrderTopology B\ninst✝ : FiberBundle F E\na b : B\nea : Triv... | obtain ⟨ed, hed⟩ : ∃ ed : Trivialization F (π F E), d ∈ ed.baseSet :=
⟨trivializationAt F E d, mem_baseSet_trivializationAt F E d⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.SeparatedMap | {
"line": 132,
"column": 35
} | {
"line": 132,
"column": 64
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case mp\nX : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nh : IsLocallyInjective f\nx : X\nU : Set X\nho : IsOpen[inst✝] U\nhm : x ∈ U\nhi : Set.InjOn f U\n⊢ ∃ U ∈ 𝓝 x, Set.InjOn f U",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [] | exact ⟨U, ho.mem_nhds hm, hi⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Covering.Quotient | {
"line": 327,
"column": 6
} | {
"line": 327,
"column": 23
} | {
"line": 328,
"column": 6
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.o... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh :\n IsCoveringMap f ∧\n Function.Surjective f ∧\n ContinuousConstSMul G E ∧ IsCancelSMul G E ∧ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ne... | have := h.2.2.2.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Covering.Basic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 49
} | {
"line": 487,
"column": 2
} | [
{
"pp": "case refine_2\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_i... | [
"case refine_3\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_iff : ∀ (i : ... | · rwa [Set.inter_comm, ← open_iff _ subset_rfl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 125,
"column": 32
} | {
"line": 125,
"column": 43
} | {
"line": 125,
"column": 43
} | [
{
"pp": "r : ℝ\nhr : r ≤ π\n⊢ BijOn (⇑exp) (Ioo (-r) r) (⇑exp '' {x | -r < x ∧ x < r})",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real",
"Preorder.toLT",
"ContinuousMap",
"PartialOrder.toPreorde... | [
"r : ℝ\nhr : r ≤ π\n⊢ BijOn (⇑exp) (Ioo (-r) r) (⇑exp '' Ioo (-r) r)"
] | Set.Ioo_def | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 291,
"column": 24
} | {
"line": 291,
"column": 93
} | {
"line": 291,
"column": 93
} | [
{
"pp": "x y : Circle\nhne : x ≠ y\n⊢ 2⁻¹ ∈ unitInterval",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.... | [] | simp only [mem_Icc, inv_nonneg, Nat.ofNat_nonneg, true_and]; linarith | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 291,
"column": 24
} | {
"line": 291,
"column": 93
} | {
"line": 291,
"column": 93
} | [
{
"pp": "x y : Circle\nhne : x ≠ y\n⊢ 2⁻¹ ∈ unitInterval",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"NonAssocSemiring.... | [] | simp only [mem_Icc, inv_nonneg, Nat.ofNat_nonneg, true_and]; linarith | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 25
} | {
"line": 228,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : dist u 1 < 2\nε : ℝ\nhuε : dist u 1 ^ 2 < ε\nhε2 : ε < 2 ^ 2\nhε : 0 < ε\nhuε' : dist u 1 < √ε\n⊢ Subtype.val '' closedBall 1 √ε ⊆ {a | IsStarNormal a ∧ spectrum ℂ a ⊆ sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε}}",
"ppTerm": "?m.187",
"assi... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : dist u 1 < 2\nε : ℝ\nhuε : dist u 1 ^ 2 < ε\nhε2 : ε < 2 ^ 2\nhε : 0 < ε\nhuε' : dist u 1 < √ε\nv : ↥(unitary A)\nhv : v ∈ closedBall 1 √ε\n⊢ ↑v ∈ {a | IsStarNormal a ∧ spectrum ℂ a ⊆ sphere 0 1 ∩ {z | 2 * (1 - z.re) ≤ ε}}"
] | rintro - ⟨v, hv, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 337,
"column": 85
} | {
"line": 352,
"column": 67
} | {
"line": 354,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\n⊢ u ∈ pathComponent 1 ↔ ∃ l, (List.map expUnitary l).prod = u",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"Real.partialOrder",
"... | [] | by
constructor
· revert u
simp_rw [← Set.mem_range, ← Set.subset_def, pathComponent_eq_connectedComponent]
refine IsClopen.connectedComponent_subset ?_ ⟨[], by simp⟩
refine .of_thickening_subset_self zero_lt_two ?_
intro u hu
rw [mem_thickening_iff] at hu
obtain ⟨v, ⟨⟨l, (hlv : (l.map expUni... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 283,
"column": 34
} | {
"line": 283,
"column": 87
} | {
"line": 285,
"column": 0
} | [
{
"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\n⊢ adjoint A ∘SL A = 0 ↔ A = 0",
"ppTerm":... | [] | by rw [← norm_eq_zero]; simp [norm_adjoint_comp_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 125,
"column": 2
} | {
"line": 135,
"column": 12
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : CharZero k\ns : Simplex k P n\np : P\n⊢ ∑ i, (s.points i -ᵥ p) = 0 → p = s.centroid",
"ppTerm": "?mpr",
"assigned": true,
"used... | [] | · intro h
rw [← vsub_eq_zero_iff_eq]
have : ∑ i, (s.points i -ᵥ p) = ∑ i, ((s.points i -ᵥ s.centroid) - (p -ᵥ s.centroid)) := by
apply sum_congr rfl
intro x hx
rw [vsub_sub_vsub_cancel_right _ _ s.centroid]
rw [this, sum_sub_distrib, centroid_weighted_vsub_eq_zero] at h
simp only [sum_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 1003,
"column": 2
} | {
"line": 1003,
"column": 22
} | {
"line": 1004,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : InnerProductSpace 𝕜 F\nm : Type u_5\nn : Type u_6\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype n\ninst✝² : DecidableE... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : InnerProductSpace 𝕜 E\ninst✝⁶ : InnerProductSpace 𝕜 F\nm : Type u_5\nn : Type u_6\ninst✝⁵ : Fintype m\ninst✝⁴ : DecidableEq m\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ ... | simp_rw [toLin_self] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 1058,
"column": 2
} | {
"line": 1061,
"column": 39
} | {
"line": 1062,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_5\nE' : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace 𝕜 E'\ninst✝ : FiniteDimensional 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\n⊢ LinearMap.adjoint f.toL... | [] | haveI := FiniteDimensional.complete 𝕜 E
haveI := FiniteDimensional.complete 𝕜 E'
ext x
exact congr($(f.adjoint_comp_self) x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 1058,
"column": 2
} | {
"line": 1061,
"column": 39
} | {
"line": 1062,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_5\nE' : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : InnerProductSpace 𝕜 E'\ninst✝ : FiniteDimensional 𝕜 E'\nf : E →ₗᵢ[𝕜] E'\n⊢ LinearMap.adjoint f.toL... | [] | haveI := FiniteDimensional.complete 𝕜 E
haveI := FiniteDimensional.complete 𝕜 E'
ext x
exact congr($(f.adjoint_comp_self) x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 73
} | {
"line": 187,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : HasContDiffBump E\nn : ℕ∞\nc g : X → E\ns : Set X\nf : (x : X) → ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt ℝ (↑n) c s x\nhr : ContD... | [] | exact prod_mem_nhds (Ioi_mem_nhds (f x).one_lt_rOut_div_rIn) univ_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 73
} | {
"line": 187,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : HasContDiffBump E\nn : ℕ∞\nc g : X → E\ns : Set X\nf : (x : X) → ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt ℝ (↑n) c s x\nhr : ContD... | [] | exact prod_mem_nhds (Ioi_mem_nhds (f x).one_lt_rOut_div_rIn) univ_mem | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 73
} | {
"line": 187,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace ℝ X\ninst✝ : HasContDiffBump E\nn : ℕ∞\nc g : X → E\ns : Set X\nf : (x : X) → ContDiffBump (c x)\nx : X\nhc : ContDiffWithinAt ℝ (↑n) c s x\nhr : ContD... | [] | exact prod_mem_nhds (Ioi_mem_nhds (f x).one_lt_rOut_div_rIn) univ_mem | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 518,
"column": 4
} | {
"line": 518,
"column": 76
} | {
"line": 518,
"column": 76
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\n⊢ AffineIndependent k p ↔ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : Fin 3 → P\n⊢ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ 1 ↔ ¬finrank k ↥(vectorSpan k (Set.range p)) ≤ 1"
] | affineIndependent_iff_not_finrank_vectorSpan_le k p (Fintype.card_fin 3) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 574,
"column": 4
} | {
"line": 575,
"column": 27
} | {
"line": 576,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nhmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)\nhmem2 : s.medial.points 0 ∈ a... | [] | rw [this, Submodule.span_smul_eq_of_isUnit]
simpa using NeZero.ne n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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