module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nn : ℕ\n⊢ SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f",
"usedConstants"... | simpa [mul_comm f] using superpolynomialDecay_param_pow_mul_iff f hk n | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nn : ℕ\n⊢ SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f",
"usedConstants"... | simpa [mul_comm f] using superpolynomialDecay_param_pow_mul_iff f hk n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhk : Tendsto k l atTop\nn : ℕ\n⊢ SuperpolynomialDecay l k (f * k ^ n) ↔ SuperpolynomialDecay l k f",
"usedConstants"... | simpa [mul_comm f] using superpolynomialDecay_param_pow_mul_iff f hk n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 60,
"column": 56
} | {
"line": 60,
"column": 64
} | [
{
"pp": "case funProp.discharger\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 60,
"column": 56
} | {
"line": 60,
"column": 64
} | [
{
"pp": "case funProp.discharger\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 60,
"column": 56
} | {
"line": 60,
"column": 64
} | [
{
"pp": "case funProp.discharger\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ closedBall (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ closedBall (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ closedBall (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 42
} | [
{
"pp": "case neg\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)... | rcases lt_or_ge 0 (r a) with (h'a | h'a) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 23
} | [
{
"pp": "case neg.inl\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (... | exact A a ⟨ha, h'a⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 12
} | [
{
"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\n⊢ IsCoinitialFor {d | μ.restrict s ≤ d + ν.restrict s} ((fun μ ↦ μ.restrict s) '' {d | μ ≤ d + ν})",
"usedConstants": [
"MeasureTheory.Measure"
]
}... | intro ν' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 136,
"column": 6
} | {
"line": 137,
"column": 95
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\nhν : ∀ (s : Set α), ν s ≠ ∞\nto_nnreal_μ : ∀ (... | exact fun n m hnm =>
subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 39
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\nh : μ.HaveLebesgueDecomposition ν\n⊢ Measurable (if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).2 else 0) ∧\n (if h : μ.HaveLebesgueDecomposition ν then (Classical.choose ⋯).1 else 0) ⟂ₘ ν ∧\n μ =\n (if h : μ.HaveLebes... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝ : SecondCountableTopology α\ns : Set α := {x | ¬∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a}\nhs : s = {x | ∃ᶠ (x : Set α) in v.filterAt x, μ x = 0}\n⊢ μ s = 0",
"usedConstants": [
"Measur... | let f : α → Set (Set α) := fun _ => {a | μ a = 0} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 420,
"column": 4
} | {
"line": 426,
"column": 31
} | [] | μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ μ (⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a)) :=
measure_mono M
_ ≤ ∑' a : { a // a ∉ w }, μ (closedBall (c a) (3 * r a)) := measure_iUnion_le _
_ ≤ ∑' a : { a // a ∉ w }, C * μ (B a) := (ENNReal.tsum_le_tsum fun a => μB a (ut (vu a.1.2)))
_ = C * ∑' a : ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 483,
"column": 53
} | {
"line": 483,
"column": 57
} | [
{
"pp": "case refine_1.inl\nα : 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... | h'.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 483,
"column": 53
} | {
"line": 483,
"column": 57
} | [
{
"pp": "case refine_1.inr\nα : 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... | h'.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 789,
"column": 6
} | {
"line": 789,
"column": 22
} | [
{
"pp": "α : 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⊢ c = d",
"usedConstants": [
"Eq.mpr",
"congr... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 612,
"column": 6
} | {
"line": 613,
"column": 26
} | [
{
"pp": "case succ\nα : 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 ≤ eVariatio... | simp only [Function.iterate_succ', Function.comp_apply, v]
exact (y_mem _ ih).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 612,
"column": 6
} | {
"line": 613,
"column": 26
} | [
{
"pp": "case succ\nα : 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 ≤ eVariatio... | simp only [Function.iterate_succ', Function.comp_apply, v]
exact (y_mem _ ih).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 650,
"column": 4
} | {
"line": 650,
"column": 24
} | [
{
"pp": "α : 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.NeBot\nε : ℝ≥0∞\nεpos : ε > 0\nW : Tendsto (fun y ↦ eVariationOn f (s ∩ I... | exact ⟨y, h'y.1, hy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.StarSubalgebra | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 85
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Semiring A\ninst✝¹² : StarRing A\ninst✝¹¹ : IsSemitopologicalSemiring A\ninst✝¹⁰ : ContinuousStar A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSpace... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.StarSubalgebra | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 85
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Semiring A\ninst✝¹² : StarRing A\ninst✝¹¹ : IsSemitopologicalSemiring A\ninst✝¹⁰ : ContinuousStar A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSpace... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.StarSubalgebra | {
"line": 267,
"column": 58
} | {
"line": 267,
"column": 85
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Semiring A\ninst✝¹² : StarRing A\ninst✝¹¹ : IsSemitopologicalSemiring A\ninst✝¹⁰ : ContinuousStar A\ninst✝⁹ : Algebra R A\ninst✝⁸ : StarModule R A\ninst✝⁷ : TopologicalSpace... | simp only [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 19
} | [
{
"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... | rw [cfc_apply ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 605,
"column": 39
} | {
"line": 617,
"column": 21
} | [
{
"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 ... | by
have := hg.comp hf <| (spectrum R a).mapsTo_image f
have sp_eq : spectrum R (cfcHom (show p a from ha) (ContinuousMap.mk _ hf.restrict)) =
f '' (spectrum R a) := by
rw [cfcHom_map_spectrum (by exact ha) _]
ext
simp
rw [cfc_apply .., cfc_apply f a,
cfc_apply _ _ (cfcHom_predicate (show p a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 10
} | [
{
"pp": "case h.e_6.h.e_toFun.h\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 : Continuo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 194,
"column": 8
} | {
"line": 194,
"column": 62
} | [
{
"pp": "case h.mp.refine_2.h.convert_2\na b : ℝ\nh : a < b\nf : C(↑(Set.Icc a b), ℝ)\np : ℝ[X]\nw : ∀ (x : ↑I), Polynomial.eval (↑x) p = f ((iccHomeoI a b h).symm x)\nq : ℝ[X] := p.comp ((b - a)⁻¹ • X + Polynomial.C (-a * (b - a)⁻¹))\nx : ↑(Set.Icc a b)\n⊢ (↑x - a) * (b - a)⁻¹ ∈ I",
"usedConstants": [
... | have w₁ : 0 < (b - a)⁻¹ := inv_pos.mpr (sub_pos.mpr h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 322,
"column": 31
} | {
"line": 360,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℕ\n⊢ ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • X ^ 2",
"usedConstants": [
"Derivation",
"Derivation.map_natCast",
"cond",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"one_pow",
"Fin... | by
-- We calculate the second `x`-derivative of `(x+y)^n`, evaluated at `y=(1-x)`,
-- either directly or by using the binomial theorem.
-- We'll work in `MvPolynomial Bool R`.
let x : MvPolynomial Bool R := MvPolynomial.X true
let y : MvPolynomial Bool R := MvPolynomial.X false
have pderiv_true_x : pderiv t... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.Weierstrass | {
"line": 55,
"column": 2
} | {
"line": 77,
"column": 58
} | [
{
"pp": "a b : ℝ\n⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤",
"usedConstants": [
"locallyCompact_of_proper",
"Real.partialOrder",
"ConditionallyCompleteLinearOrder.toCompactIccSpace",
"Real",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"NonUnital... | rcases lt_or_ge a b with h | h
-- (Otherwise it's easy; we'll deal with that later.)
· -- We can pullback continuous functions on `[a,b]` to continuous functions on `[0,1]`,
-- by precomposing with an affine map.
let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=
compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Weierstrass | {
"line": 55,
"column": 2
} | {
"line": 77,
"column": 58
} | [
{
"pp": "a b : ℝ\n⊢ (polynomialFunctions (Set.Icc a b)).topologicalClosure = ⊤",
"usedConstants": [
"locallyCompact_of_proper",
"Real.partialOrder",
"ConditionallyCompleteLinearOrder.toCompactIccSpace",
"Real",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"NonUnital... | rcases lt_or_ge a b with h | h
-- (Otherwise it's easy; we'll deal with that later.)
· -- We can pullback continuous functions on `[a,b]` to continuous functions on `[0,1]`,
-- by precomposing with an affine map.
let W : C(Set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=
compRightAlgHom ℝ ℝ (iccHomeoI a b h).symm
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 183,
"column": 4
} | {
"line": 187,
"column": 15
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Modu... | simp only [φ, ψ, NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_mk',
NonUnitalAlgHom.coe_mk]
congr
ext x
simp [hff'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 183,
"column": 4
} | {
"line": 187,
"column": 15
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Modu... | simp only [φ, ψ, NonUnitalStarAlgHom.comp_apply, NonUnitalStarAlgHom.coe_mk',
NonUnitalAlgHom.coe_mk]
congr
ext x
simp [hff'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 514,
"column": 90
} | {
"line": 514,
"column": 98
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 514,
"column": 90
} | {
"line": 514,
"column": 98
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 514,
"column": 90
} | {
"line": 514,
"column": 98
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 10
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 96
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalSemiring R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDi... | simp only [ContinuousMapZero.coe_mk, ContinuousMap.coe_mk, Set.restrict_apply, Subtype.forall] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 223,
"column": 56
} | {
"line": 223,
"column": 68
} | [
{
"pp": "X : 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 → X →... | simp [h, w₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 36
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\nA : Type u_2\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : Algebra ℝ A\ninst✝² : TopologicalSpace A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\ns : Set ℝ≥0\nhs : CompactSpace ↑s\nφ ψ : C(↑s, ℝ≥0) →⋆ₐ[ℝ≥0] A\nhφ : Continuous[_, inst✝²] ⇑φ\nhψ : Cont... | obtain ⟨hφ', hφ_id⟩ := this φ hφ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 250,
"column": 6
} | {
"line": 251,
"column": 86
} | [
{
"pp": "case pos\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... | show ∀ a b ε : ℝ, dist a b < ε ↔ a < b + ε ∧ b - ε < a by
intros; simp only [← Metric.mem_ball, Real.ball_eq_Ioo, Set.mem_Ioo, and_comm] | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 42,
"column": 36
} | {
"line": 43,
"column": 98
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na b : E\n⊢ ‖((mul 𝕜 E).flip a) ... | by
simpa only [flip_apply, mul_apply', norm_star] using le_opNorm ((mul 𝕜 E).flip a) (star b) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 36
} | [
{
"pp": "X : Type u_1\ninst✝⁹ : TopologicalSpace X\ninst✝⁸ : Zero X\nA : Type u_2\ninst✝⁷ : NonUnitalRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : IsSemitopologicalRing A\ninst✝² : IsScalarTower ℝ A A\ninst✝¹ : SMulCommClass ℝ A A\ninst✝ : T2Space A\ns : Set ℝ≥0\nhs : C... | obtain ⟨hφ', hφ_id⟩ := this φ hφ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 11
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : DenselyNormedField 𝕜\ninst✝⁸ : NonUnitalNormedRing E\ninst✝⁷ : StarRing E\ninst✝⁶ : CStarRing E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : IsScalarTower 𝕜 E E\ninst✝³ : SMulCommClass 𝕜 E E\ninst✝² : StarRing 𝕜\ninst✝¹ : StarModule 𝕜 E\ninst✝ : CStarRing 𝕜\nx : Unit... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 96
} | [
{
"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",
"usedConstants": [
"HasFDerivAt",
"NormedCommRing... | 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 H₁.com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 96
} | [
{
"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",
"usedConstants": [
"HasFDerivAt",
"NormedCommRing... | 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 H₁.com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 12
} | [
{
"pp": "case lower\nA : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : 0 < 0\nhz' : z ∈ σ ℂ a\n⊢ False",
"usedConstants": [
"False",
"Real",
"Real.instZero",
"False.elim",
"Real.instLT",
"lt_self_iff_false._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 12
} | [
{
"pp": "case upper\nA : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : 0 < 0\nhz' : z ∈ σ ℂ a\n⊢ False",
"usedConstants": [
"False",
"Real",
"Real.instZero",
"False.elim",
"Real.instLT",
"lt_self_iff_false._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 245,
"column": 49
} | {
"line": 261,
"column": 42
} | [
{
"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\nn : ℕ\n⊢ spectralRadius 𝕜 a ≤ ↑‖a ^ (n + 1)‖₊ ^ (1 / (↑n + 1)) * ↑‖1‖₊ ^ (1 / (↑n + 1))",
"usedConstants": [
"Iff.mpr",
"zero_le",
"NormedComm... | by
refine iSup₂_le fun k hk => ?_
-- apply easy direction of the spectral mapping theorem for polynomials
have pow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1)) := by
simpa only [one_mul, Algebra.algebraMap_eq_smul_one, one_smul, aeval_monomial, one_mul,
eval_monomial] using subset_polynomial_aeval a (@monomial �... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.Barrelled | {
"line": 148,
"column": 9
} | {
"line": 150,
"column": 37
} | [] | p y = p (x + y - x) := by rw [add_sub_cancel_left]
_ ≤ p (x + y) + p x := map_sub_le_add _ _ _
_ ≤ n + n := add_le_add hy hxn' | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 141,
"column": 6
} | {
"line": 145,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\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 : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | rw [isometry_inr (𝕜 := 𝕜) |>.isEmbedding.continuous_iff]
have := continuous_cfcₙAux hp₁ a ha
simp only [coe_comp, NonUnitalStarAlgHom.coe_coe, Function.comp_def,
inrRangeEquiv_symm_apply, coe_codRestrict, ψ]
fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 141,
"column": 6
} | {
"line": 145,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\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 : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | rw [isometry_inr (𝕜 := 𝕜) |>.isEmbedding.continuous_iff]
have := continuous_cfcₙAux hp₁ a ha
simp only [coe_comp, NonUnitalStarAlgHom.coe_coe, Function.comp_def,
inrRangeEquiv_symm_apply, coe_codRestrict, ψ]
fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : PartialOrder A\ninst✝⁷ : StarRing A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Module ℝ A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : SMulCommClass ℝ A A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | have hx := nonneg_iff_isSelfAdjoint_and_quasispectrumRestricts.mpr ⟨hx₁, hx₂⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 257,
"column": 4
} | {
"line": 258,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist (k • x +ᵥ c) (k • y +ᵥ c) = ↑‖k‖₊ * edi... | rw [show edist (k • x +ᵥ c) (k • y +ᵥ c) = _ from (IsometryEquiv.vaddConst c).isometry ..]
exact edist_smul₀ .. | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 257,
"column": 4
} | {
"line": 258,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedDivisionRing 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\nP : Type u_3\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor E P\nc : P\nk : 𝕜\nhk : k ≠ 0\nx y : E\n⊢ edist (k • x +ᵥ c) (k • y +ᵥ c) = ↑‖k‖₊ * edi... | rw [show edist (k • x +ᵥ c) (k • y +ᵥ c) = _ from (IsometryEquiv.vaddConst c).isometry ..]
exact edist_smul₀ .. | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 620,
"column": 8
} | {
"line": 620,
"column": 24
} | [
{
"pp": "case inl\n𝕜 : 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\na : A\nha : p a\nthis : CompactSpace ↑(σₙ 𝕜 a)\nf : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜\nι : C(↑(σ 𝕜 a)... | change ‖f 0‖ ≤ _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Topology.UrysohnsLemma | {
"line": 449,
"column": 2
} | {
"line": 455,
"column": 15
} | [
{
"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\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\nm : Set X\nm_comp : IsCompact m\nsm : s... | have hgmc : EqOn g 0 mᶜ := by
intro x hx
have B n : f n x = 0 := by
have : mᶜ ⊆ (U n ∩ interior m)ᶜ := by
simpa using inter_subset_right.trans interior_subset
exact fm n (this hx)
simp [g, B] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 312,
"column": 2
} | {
"line": 316,
"column": 20
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\ns : Set X\nx : X\nhx : x ∈ (closure sᶜ)ᶜ\ng : C(X, ℝ)\nhgs : Set.EqOn (⇑g) 0 (closure sᶜ)\nhgx : Set.EqOn (⇑g) 1 {x}\n⊢ ∃ f, (∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0) ∧ f x ≠ 0",
"usedConst... | exact
⟨⟨fun x => g x, continuous_ofReal.comp (map_continuous g)⟩, by
simpa only [coe_mk, ofReal_eq_zero] using fun x hx => hgs (subset_closure hx), by
simpa only [coe_mk, hgx (Set.mem_singleton x), Pi.one_apply, RCLike.ofReal_one] using
one_ne_zero⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 45
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nA : ι → Type u_4\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : StarRing R\ninst✝¹³ : MetricSpace R\ninst✝¹² : IsTopologicalSemiring R\ninst✝¹¹ : ContinuousStar R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\ninst✝⁸ : (i : ι) → Ring (A i)\ninst✝⁷ : (i : ι) → Algebra S (A ... | Pi.spectrum_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 21
} | [
{
"pp": "case pos\nX : Type u_1\nR : 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 : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : Continuou... | rw [cfc_apply ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 12
} | [
{
"pp": "case pos.h.hq\nX : Type u_1\nR : 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 : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : Cont... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 29
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedRing A\ninst✝³ : StarRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ : ContinuousStar A\ns : Set 𝕜\nhs : IsCompact s\nf : 𝕜 → 𝕜\na : X → A\na₀ : A\nl : Filter... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 29
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝⁹ : NormedRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : NormedAlgebra ℝ A\ninst✝⁶ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : ContinuousStar A\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : T2Space A\nins... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 698,
"column": 31
} | {
"line": 698,
"column": 34
} | [
{
"pp": "case inr\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... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 12
} | [
{
"pp": "case pos.h.hq\nX : Type u_1\nR : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : Nontrivial R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 713,
"column": 2
} | {
"line": 715,
"column": 19
} | [
{
"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 :... | rw [nnrpow_eq_rpow (pos_of_ne_zero hy)]
refine isUnit_rpow_iff a y ?_ ha
exact_mod_cast hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 713,
"column": 2
} | {
"line": 715,
"column": 19
} | [
{
"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 :... | rw [nnrpow_eq_rpow (pos_of_ne_zero hy)]
refine isUnit_rpow_iff a y ?_ ha
exact_mod_cast hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 799,
"column": 6
} | {
"line": 799,
"column": 45
} | [
{
"pp": "case neg.inl\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✝ : ... | rw [sqrt_of_not_nonneg H, inverse_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 29
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : ContinuousStar A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 451,
"column": 2
} | {
"line": 454,
"column": 63
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nha : 0 ≤ a\nhab : a ≤ b\n⊢ ‖a‖ ≤ ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real.instLE",
"Real",
"NonUnitalCStarAlgeb... | suffices ∀ a b : A⁺¹, 0 ≤ a → a ≤ b → ‖a‖ ≤ ‖b‖ by
have hb := ha.trans hab
simpa only [ge_iff_le, Unitization.norm_inr] using
this a b (by simpa) (by rwa [Unitization.inr_le_iff a b]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 880,
"column": 2
} | {
"line": 880,
"column": 61
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : ContinuousStar A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus �... | exact ha_cont.cfcₙ' hs f (fun x _ ↦ ha x) (fun x _ ↦ ha' x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 54
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\n⊢ 0 ≤ x → (↑x ≤ 1 ↔ ‖x‖ ≤ 1)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real.instLE",
"Real",
"NonUnitalCStarAlgebra.toNonUn... | rw [← norm_inr (𝕜 := ℂ), ← inr_nonneg_iff, iff_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrderedRing ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₀\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 262,
"column": 12
} | {
"line": 262,
"column": 20
} | [
{
"pp": "case h₁\nA : Type u_1\ninst✝⁸ : NonUnitalCStarAlgebra A\ninst✝⁷ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁶ : Fintype ι\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → Module ℂ (E i)\ninst✝³ : (i : ι) → SMul A (E i)\ninst✝² : (i : ι) → CStarModule A (E i)\ninst✝¹ : StarOrd... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 952,
"column": 2
} | {
"line": 952,
"column": 29
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹¹ : NonUnitalNormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedSpace ℝ A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ ... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 102,
"column": 65
} | {
"line": 102,
"column": 73
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁵ : NonUnitalCStarAlgebra A₁\ninst✝⁴ : NonUnitalCStarAlgebra A₂\ninst✝³ : PartialOrder A₁\ninst✝² : StarOrderedRing A₁\ninst✝¹ : PartialOrder A₂\ninst✝ : StarOrderedRing A₂\nf : A₁ →ₚ[ℂ] A₂\nhcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖\nn : ℕ\nx : A₁ := Classi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 102,
"column": 65
} | {
"line": 102,
"column": 73
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁵ : NonUnitalCStarAlgebra A₁\ninst✝⁴ : NonUnitalCStarAlgebra A₂\ninst✝³ : PartialOrder A₁\ninst✝² : StarOrderedRing A₁\ninst✝¹ : PartialOrder A₂\ninst✝ : StarOrderedRing A₂\nf : A₁ →ₚ[ℂ] A₂\nhcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖\nn : ℕ\nx : A₁ := Classi... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.PositiveLinearMap | {
"line": 102,
"column": 65
} | {
"line": 102,
"column": 73
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁵ : NonUnitalCStarAlgebra A₁\ninst✝⁴ : NonUnitalCStarAlgebra A₂\ninst✝³ : PartialOrder A₁\ninst✝² : StarOrderedRing A₁\ninst✝¹ : PartialOrder A₂\ninst✝ : StarOrderedRing A₂\nf : A₁ →ₚ[ℂ] A₂\nhcontra : ∀ (C : ℝ≥0), ∃ a, 0 ≤ a ∧ ↑C * ‖a‖ < ‖f a‖\nn : ℕ\nx : A₁ := Classi... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 637,
"column": 2
} | {
"line": 637,
"column": 50
} | [
{
"pp": "l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype l\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ ‖A * B‖₊ ≤ ‖A‖₊ * ‖B‖₊",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"i... | simp_rw [frobenius_nnnorm_def, Matrix.mul_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 50,
"column": 2
} | {
"line": 65,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\nhb₂ : Commute... | induction f using ContinuousMap.induction_on_of_compact with
| const r =>
conv =>
enter [1, 2]
equals algebraMap 𝕜 _ r => rfl
rw [AlgHomClass.commutes]
exact Algebra.commute_algebraMap_left r b
| id => rwa [cfcHom_id ha]
| star_id => rwa [map_star, cfcHom_id]
| add f g hf hg => rw [map_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 50,
"column": 2
} | {
"line": 65,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\nhb₂ : Commute... | induction f using ContinuousMap.induction_on_of_compact with
| const r =>
conv =>
enter [1, 2]
equals algebraMap 𝕜 _ r => rfl
rw [AlgHomClass.commutes]
exact Algebra.commute_algebraMap_left r b
| id => rwa [cfcHom_id ha]
| star_id => rwa [map_star, cfcHom_id]
| add f g hf hg => rw [map_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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