module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 904,
"column": 12
} | {
"line": 904,
"column": 48
} | {
"line": 904,
"column": 49
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (x : α), f x ... | lintegral_add_left measurable_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 436,
"column": 32
} | {
"line": 436,
"column": 57
} | {
"line": 437,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np : ℝ≥0\ns : Set α\nh : s ⊆ {x | v.limRatioMe... | [] | by rw [inter_union_compl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 456,
"column": 44
} | {
"line": 456,
"column": 69
} | {
"line": 457,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\nq : ℝ≥0\ns : Set α\nh : s ⊆ {x | ↑q < v.limRa... | [] | by rw [inter_union_compl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Monotone | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 35
} | {
"line": 235,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\ns : Set ℝ\nhf : MonotoneOn f s\na b : ℝ\nas : a ∈ s\nbs : b ∈ s\na✝ : a < b\ng : ℝ → ℝ\nhg : Monotone g\ngf : EqOn f g (s ∩ Icc a b)\nx : ℝ\nhx : DifferentiableAt ℝ g x\nh'x : x ∈ s ∩ Ioo a b\nthis : Ioo a b ∈ 𝓝[s] x\ny : ℝ\nhy : y ∈ s\nh'y : y ∈ Ioo a b\n⊢ f y = g y",
"ppTerm": "?m.253... | [] | exact gf ⟨hy, h'y.1.le, h'y.2.le⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.StarOrdered | {
"line": 83,
"column": 4
} | {
"line": 111,
"column": 46
} | {
"line": 113,
"column": 0
} | [
{
"pp": "α : 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✝ : Star... | [] | constructor
· rw [le_def, ← ContinuousMap.coe_coe, ← ContinuousMap.coe_coe g, ← ContinuousMap.le_def,
StarOrderedRing.le_iff]
rintro ⟨p, hp_mem, hp⟩
induction hp_mem using AddSubmonoid.closure_induction_left generalizing f g with
| zero => exact ⟨0, zero_mem _, by ext x; congrm($(hp) x)⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.StarOrdered | {
"line": 83,
"column": 4
} | {
"line": 111,
"column": 46
} | {
"line": 113,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹⁰ : TopologicalSpace α\ninst✝⁹ : Zero α\nR : Type u_2\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : CommSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : IsTopologicalSemiring R\ninst✝¹ : ContinuousStar R\ninst✝ : Star... | [] | constructor
· rw [le_def, ← ContinuousMap.coe_coe, ← ContinuousMap.coe_coe g, ← ContinuousMap.le_def,
StarOrderedRing.le_iff]
rintro ⟨p, hp_mem, hp⟩
induction hp_mem using AddSubmonoid.closure_induction_left generalizing f g with
| zero => exact ⟨0, zero_mem _, by ext x; congrm($(hp) x)⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 888,
"column": 2
} | {
"line": 888,
"column": 94
} | {
"line": 889,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\... | [
"α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf : Locall... | filter_upwards [v.ae_tendsto_average_norm_sub hf, v.ae_eventually_measure_pos] with x hx h'x | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 480,
"column": 8
} | {
"line": 481,
"column": 41
} | {
"line": 482,
"column": 4
} | [
{
"pp": "case e'_3.a.inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : MonotoneOn φ t\nx y : β\nhx : x ∈ t\nhy : y ∈ t\nh : x ≤ y\nu : β\nus : u ∈ t\nvφx : φ x ≤ φ u\nvφy : φ u ≤ φ y\nux : u ≤ x\n⊢ φ... | [] | rw [← le_antisymm vφx (hφ us hx ux)]
exact ⟨x, ⟨hx, ⟨le_rfl, h⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 480,
"column": 8
} | {
"line": 481,
"column": 41
} | {
"line": 482,
"column": 4
} | [
{
"pp": "case e'_3.a.inr\nα : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\nβ : Type u_3\ninst✝ : LinearOrder β\nf : α → E\nt : Set β\nφ : β → α\nhφ : MonotoneOn φ t\nx y : β\nhx : x ∈ t\nhy : y ∈ t\nh : x ≤ y\nu : β\nus : u ∈ t\nvφx : φ x ≤ φ u\nvφy : φ u ≤ φ y\nux : u ≤ x\n⊢ φ... | [] | rw [← le_antisymm vφx (hφ us hx ux)]
exact ⟨x, ⟨hx, ⟨le_rfl, h⟩⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 49
} | {
"line": 51,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : Semifield R\ninst✝⁷ : Semifield S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalSpace S\na : A\nf : C(S, R)\nh : SpectrumRestricts a ⇑f\... | [] | rw [← isCompact_iff_compactSpace] at h_cpct ⊢
exact h.image ▸ h_cpct.image (map_continuous f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 49
} | {
"line": 51,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : Semifield R\ninst✝⁷ : Semifield S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : TopologicalSpace R\ninst✝ : TopologicalSpace S\na : A\nf : C(S, R)\nh : SpectrumRestricts a ⇑f\... | [] | rw [← isCompact_iff_compactSpace] at h_cpct ⊢
exact h.image ▸ h_cpct.image (map_continuous f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 100
} | {
"line": 297,
"column": 4
} | [
{
"pp": "case hom_injective\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRin... | [
"case hom_id\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRing S\ninst✝¹⁴ : Cont... | case hom_injective => exact nonUnitalStarAlgHom_injective (cfcₙHom_injective _) _ halg.injective | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 590,
"column": 47
} | {
"line": 590,
"column": 61
} | {
"line": 590,
"column": 62
} | [
{
"pp": "case monomial\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 : ContinuousFunctio... | [
"case monomial\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 : ContinuousFunctionalCalculus ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 41
} | {
"line": 706,
"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... | [] | simpa using cfc_algebraMap (A := A) 0 f | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 41
} | {
"line": 706,
"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... | [] | simpa using cfc_algebraMap (A := A) 0 f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 704,
"column": 2
} | {
"line": 704,
"column": 41
} | {
"line": 706,
"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... | [] | simpa using cfc_algebraMap (A := A) 0 f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 46
} | {
"line": 720,
"column": 0
} | [
{
"pp": "case neg\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... | [] | · simp [cfc_apply_of_not_continuousOn a h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 775,
"column": 2
} | {
"line": 777,
"column": 34
} | {
"line": 779,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra R A\ninst✝¹ : ContinuousFunctionalCalculus R A p... | [] | cases n with
| zero => simp [cfc_const_one R a]
| succ n => simp [cfc_pow f _ a] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 771,
"column": 4
} | {
"line": 772,
"column": 61
} | {
"line": 773,
"column": 2
} | [
{
"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\ns : Set α\nhf : BoundedVariationOn f s\nx : α\n⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Iio x ∩ Ici y)) (𝓝[s ∩ Iio x] x) (𝓝 0)",
"ppTerm": "?m.50",
... | [] | exact (hf.mono inter_subset_left).tendsto_eVariationOn_Ici_zero_of_filter (𝓝[s ∩ Iio x] x)
(fun y hy ↦ inter_mem_nhdsWithin _ (Ici_mem_nhds hy.2)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 89
} | {
"line": 245,
"column": 6
} | [
{
"pp": "case succ.right\nn k : ℕ\nh : k ≤ n\n⊢ bernsteinPolynomial ℚ n k ∉ span ℚ (Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)",
"ppTerm": "?succ.right",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"RingHomSurjective.ids",
"Module.End.instMonoid",
"Se... | [
"case succ.right\nn k : ℕ\nh : k ≤ n\n⊢ (derivative ^ (n - k)) (bernsteinPolynomial ℚ n k) ∉\n span ℚ (⇑(derivative ^ (n - k)) '' Set.range fun k_1 ↦ bernsteinPolynomial ℚ n ↑k_1)"
] | apply notMem_span_of_apply_notMem_span_image (@Polynomial.derivative ℚ _ ^ (n - k)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 13
} | {
"line": 220,
"column": 4
} | [
{
"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 →... | [
"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 → C(X, ℝ)\nhg... | intro x z | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 36
} | {
"line": 142,
"column": 6
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝⁵ : TopologicalSpace X\nA : Type u_2\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsSemitopologicalRing A\nφ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A\nr : ℝ\nhr : 0 ≤ -r\n⊢ φ ((algebraMap ℝ C(X, ℝ)) (- -r)).toNNReal - φ ((algebraMap ℝ C(X,... | [
"case inr\nX : Type u_1\ninst✝⁵ : TopologicalSpace X\nA : Type u_2\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsSemitopologicalRing A\nφ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A\nr✝ : ℝ\nr : ℝ≥0\n⊢ φ ((algebraMap ℝ C(X, ℝ)) (-↑r)).toNNReal - φ ((algebraMap ℝ C(X, ℝ)) ↑r).toNNRea... | lift -r to ℝ≥0 using hr with r | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 59
} | {
"line": 377,
"column": 4
} | [
{
"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... | [
"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 : CompactSpace ... | have : ContinuousMapZero.UniqueHom ℝ A := inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 107,
"column": 4
} | {
"line": 108,
"column": 29
} | {
"line": 109,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Function.Surjective.forall",
"compl_compl",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (x : Set β), IsOpen[inst✝] x → IsOpen[inst✝¹] (f ⁻¹' Iic x)"
] | rw [compl_surjective.forall]
simp [← isOpen_compl_iff] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 107,
"column": 4
} | {
"line": 108,
"column": 29
} | {
"line": 109,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Function.Surjective.forall",
"compl_compl",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (x : Set β), IsOpen[inst✝] x → IsOpen[inst✝¹] (f ⁻¹' Iic x)"
] | rw [compl_surjective.forall]
simp [← isOpen_compl_iff] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 29
} | {
"line": 138,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic u)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Function.Surjective.forall",
"compl_compl",
"_... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (x : Set β), IsOpen[inst✝] x → IsOpen[inst✝¹] (f ⁻¹' Iic xᶜ)ᶜ"
] | rw [compl_surjective.forall]
simp [← isOpen_compl_iff] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 29
} | {
"line": 138,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic u)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Function.Surjective.forall",
"compl_compl",
"_... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (x : Set β), IsOpen[inst✝] x → IsOpen[inst✝¹] (f ⁻¹' Iic xᶜ)ᶜ"
] | rw [compl_surjective.forall]
simp [← isOpen_compl_iff] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 46
} | {
"line": 212,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nx : α\nh : ∀ (u : Set β), IsOpen[inst✝] u → f x ∈ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ∈ u\nt : Set β\nht : t ∈ 𝓝 (f x)\nu : Set β\nhut : u ⊆ t\nhuo : IsOpen[inst✝] u\nhux : f x ∈ u\n⊢ ∀ᶠ (x : α) in ... | [] | exact (h u huo hux).mono fun _ hx' ↦ hut hx' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 60
} | {
"line": 99,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nn : ℕ\nhn : n ≠ 0\nx : 𝕜\nhx : x ∈ σ a\n⊢ ↑‖x‖₊ ^ n ≤ ⨆ k ∈ σ (a ^ n), ↑‖k‖₊",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"ENN... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nn : ℕ\nhn : n ≠ 0\nx : 𝕜\nhx : x ∈ σ a\n⊢ ↑‖x‖₊ ^ n ≤ ↑‖x ^ n‖₊"
] | apply le_iSup₂_of_le (x ^ n) (spectrum.pow_mem_pow a n hx) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 278,
"column": 77
} | {
"line": 278,
"column": 87
} | {
"line": 278,
"column": 87
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : NonUnitalCStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : NonUnitalAlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\nφ : F\na : A\nψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B\nx s : Unitization ℂ A\nhs : IsSelfAdjoint s\... | [
"F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : NonUnitalCStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : NonUnitalAlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\nφ : F\na : A\nψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B\nx s : Unitization ℂ A\nhs : IsSelfAdjoint s\nthis : ‖ψ s... | coe_le_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 358,
"column": 45
} | {
"line": 358,
"column": 54
} | {
"line": 358,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\nhu : IsUnit (1 - u⁻¹⁻¹ • a)\n⊢ IsUnit (1 - z • a)",
"ppTerm": "?m.75",
"assigned": tru... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\na : A\nz : 𝕜\nh : ↑‖z‖₊ < (spectralRadius 𝕜 a)⁻¹\nhz : ¬z = 0\nu : 𝕜ˣ := Units.mk0 z hz\nhu : IsUnit (1 - u • a)\n⊢ IsUnit (1 - z • a)"
] | inv_inv u | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 313,
"column": 8
} | {
"line": 313,
"column": 22
} | {
"line": 313,
"column": 23
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β ... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β β\nr : R\nf ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 313,
"column": 23
} | {
"line": 313,
"column": 37
} | {
"line": 313,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β ... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : IsScalarTower R β β\nr : R\nf ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 321,
"column": 8
} | {
"line": 321,
"column": 22
} | {
"line": 321,
"column": 23
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β ... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β β\nr : R\nf ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 321,
"column": 23
} | {
"line": 321,
"column": 37
} | {
"line": 321,
"column": 38
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β ... | [
"F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\nx : α\nR : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : NonUnitalNonAssocSemiring β\ninst✝³ : IsTopologicalSemiring β\ninst✝² : Module R β\ninst✝¹ : ContinuousConstSMul R β\ninst✝ : SMulCommClass R β β\nr : R\nf ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 89,
"column": 38
} | {
"line": 99,
"column": 60
} | {
"line": 101,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\na b x : A\ninst✝¹ : IsStarNormal a\ninst✝ : IsStarNormal b\nh : SemiconjBy x a b\n⊢ SemiconjBy x (star a) (star b)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NormedAlgebra.restrictScalars",
"_private.Mathlib.Analysis.CStarAl... | [] | by
suffices key : ∀ z : ℂ, x * exp (z • star a) = exp (z • star b) * x by
have (a : A) : HasDerivAt (fun z : ℂ ↦ exp (z • a)) a 0 := by
simpa using hasDerivAt_exp_smul_const a (0 : ℂ)
apply (this (star a)).const_mul x |>.unique
simpa [key] using (this (star b)).mul_const x
intro z
let _ : Normed... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 172,
"column": 13
} | {
"line": 172,
"column": 16
} | {
"line": 172,
"column": 16
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nx✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A\nthis : IsAddTorsionFree A\nha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A\nh_deriv :\n ∀ (a b c : A) (y : ℝ),\n deriv (fun x ↦ NormedSpace.... | [
"case refine_2\nA : Type u_1\ninst✝ : CStarAlgebra A\na : A\nx✝ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A\nthis : IsAddTorsionFree A\nha : ∀ (x : ℝ), NormedSpace.exp (x • a) * NormedSpace.exp (-x • star a) ∈ unitary A\nh_deriv :\n ∀ (a b c : A) (y : ℝ),\n deriv (fun x ↦ NormedSpace.exp (x • a) ... | key | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 49
} | {
"line": 692,
"column": 0
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝¹⁰ : Semifield R\ninst✝⁹ : Field S\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module R A\ninst✝⁵ : Module S A\ninst✝⁴ : IsScalarTower S A A\ninst✝³ : SMulCommClass S A A\ninst✝² : IsScalarTower R S A\ninst✝¹ : TopologicalSpace R\ninst✝ : Top... | [] | rw [← isCompact_iff_compactSpace] at h_cpct ⊢
exact h.image ▸ h_cpct.image (map_continuous f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 689,
"column": 2
} | {
"line": 690,
"column": 49
} | {
"line": 692,
"column": 0
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝¹⁰ : Semifield R\ninst✝⁹ : Field S\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module R A\ninst✝⁵ : Module S A\ninst✝⁴ : IsScalarTower S A A\ninst✝³ : SMulCommClass S A A\ninst✝² : IsScalarTower R S A\ninst✝¹ : TopologicalSpace R\ninst✝ : Top... | [] | rw [← isCompact_iff_compactSpace] at h_cpct ⊢
exact h.image ▸ h_cpct.image (map_continuous f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WeakDual | {
"line": 124,
"column": 53
} | {
"line": 124,
"column": 56
} | {
"line": 124,
"column": 57
} | [
{
"pp": "ι : Type u_4\n𝕜 : Type u_5\nE : Type u_6\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\nt𝕜 : TopologicalSpace 𝕜\ninst✝³ : IsTopologicalRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : T0Space 𝕜\nf : ι → E →ₗ[𝕜] 𝕜\nφ : E →ₗ[𝕜] 𝕜\nx✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜\nφ_con... | [
"ι : Type u_4\n𝕜 : Type u_5\nE : Type u_6\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\nt𝕜 : TopologicalSpace 𝕜\ninst✝³ : IsTopologicalRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : T0Space 𝕜\nf : ι → E →ₗ[𝕜] 𝕜\nφ : E →ₗ[𝕜] 𝕜\nx✝ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜\nφ_cont : Continuo... | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.LocallyConvex.WeakDual | {
"line": 144,
"column": 68
} | {
"line": 147,
"column": 45
} | {
"line": 148,
"column": 2
} | [
{
"pp": "ι : Type u_4\n𝕜 : Type u_5\nE : Type u_6\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : ι → E →ₗ[𝕜] 𝕜\nφ : E →ₗ[𝕜] 𝕜\nt𝕜 : TopologicalSpace 𝕜 := inferInstance\nt₁ : TopologicalSpace E := ⨅ i, induced (⇑(f i)) t𝕜\nt₂ : Finset ι → TopologicalSpace E := fun... | [] | by
simp_rw [this, ← mem_span_iff_continuous_of_finite, Submodule.span_range_eq_iSup,
iSup_subtype]
rw [Submodule.mem_iSup_iff_exists_finset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.HahnBanach | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 49
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nS : Subspace ℝ E\nf : Dual ℝ ↥S\np : Seminorm ℝ E\nhp : ∀ (x : ↥S), f x ≤ p ↑x\nx✝¹ : ℝ\nhc : 0 < x✝¹\nx✝ : E\n⊢ p (x✝¹ • x✝) = x✝¹ * p x✝",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Seminorm.... | [] | simp [map_smul_eq_mul, abs_of_nonneg hc.le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.LocallyConvex.HahnBanach | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 49
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nS : Subspace ℝ E\nf : Dual ℝ ↥S\np : Seminorm ℝ E\nhp : ∀ (x : ↥S), f x ≤ p ↑x\nx✝¹ : ℝ\nhc : 0 < x✝¹\nx✝ : E\n⊢ p (x✝¹ • x✝) = x✝¹ * p x✝",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Seminorm.... | [] | simp [map_smul_eq_mul, abs_of_nonneg hc.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.HahnBanach | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 49
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nS : Subspace ℝ E\nf : Dual ℝ ↥S\np : Seminorm ℝ E\nhp : ∀ (x : ↥S), f x ≤ p ↑x\nx✝¹ : ℝ\nhc : 0 < x✝¹\nx✝ : E\n⊢ p (x✝¹ • x✝) = x✝¹ * p x✝",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Seminorm.... | [] | simp [map_smul_eq_mul, abs_of_nonneg hc.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 13
} | {
"line": 78,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx' : StrongDual 𝕜 E\nc : 𝕜\nhc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖\nc_zero : ¬c = 0\neq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖\nz : E\nhzs : z ∈ s\n⊢ ‖c⁻¹ • x' z‖ ≤ ‖c⁻¹‖ ... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx' : StrongDual 𝕜 E\nc : 𝕜\nhc : ∀ z ∈ s, ‖x' z‖ ≤ ‖c‖\nc_zero : ¬c = 0\neq : ∀ (z : E), ‖c⁻¹ • x' z‖ = ‖c⁻¹‖ * ‖x' z‖\nz : E\nhzs : z ∈ s\n⊢ ‖c⁻¹‖ * ‖x' z‖ ≤ ‖c⁻¹‖ * ‖c‖"
] | rw [eq z] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.WeakDual | {
"line": 184,
"column": 85
} | {
"line": 189,
"column": 7
} | {
"line": 191,
"column": 0
} | [
{
"pp": "𝕜 : Type u_5\nE : Type u_6\nF : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nf : StrongDual 𝕜 (WeakBilin B)\n⊢ ∃ a, (WeakBilin.eval B) a = f",
"ppTerm": "?m.42",
"assigned"... | [] | by
have : f.toLinearMap ∈
Submodule.span 𝕜 (ContinuousLinearMap.coeLM 𝕜 ∘ₗ WeakBilin.eval B).range := by
simpa [coe_range, mem_span_iff_continuous, continuous_iff_le_induced, ← induced_to_pi] using!
f.continuous.le_induced
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UrysohnsLemma | {
"line": 182,
"column": 12
} | {
"line": 182,
"column": 61
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ 0 ≤ approx 0 c x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Preorder.toLE",
... | [] | exact indicator_nonneg (fun _ _ => zero_le_one) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UrysohnsLemma | {
"line": 182,
"column": 12
} | {
"line": 182,
"column": 61
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ 0 ≤ approx 0 c x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Preorder.toLE",
... | [] | exact indicator_nonneg (fun _ _ => zero_le_one) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UrysohnsLemma | {
"line": 182,
"column": 12
} | {
"line": 182,
"column": 61
} | {
"line": 183,
"column": 2
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ 0 ≤ approx 0 c x",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Preorder.toLE",
... | [] | exact indicator_nonneg (fun _ _ => zero_le_one) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 219,
"column": 4
} | {
"line": 237,
"column": 75
} | {
"line": 244,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nI : Ideal C(X, 𝕜)\nf : C(X, 𝕜)\nhf : f ∈ idealOfSet 𝕜 (setOfIdeal I)\nε : ℝ≥0\nhε : 0 < ε\nt : Set X := {x | ε / 2 ≤ ‖f x‖₊}\nht : IsClosed t\nhtI : Disjoint t (setOfId... | [] | · refine lt_of_le_of_lt ?_ (half_lt_self hε)
have :=
calc
‖((1 - (algebraMapCLM ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
Pi.one_apply, Function... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UrysohnsLemma | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 71
} | {
"line": 475,
"column": 2
} | [
{
"pp": "case refine_4\nX : 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 : IsCo... | [
"case refine_5\nX : 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 ... | · exact tsum_nonneg (fun n ↦ mul_nonneg (u_pos n).le (f_range n x).1) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 46
} | {
"line": 260,
"column": 0
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsStarNormal a\nhb : IsStarNormal b\nhcomm : Commute a b\nhab : a * b = 0\nS : NonUnitalStarSubalgebra ℂ A := (adjoin ℂ {a, b}).topologicalClosure\nhS : IsClosed ↑S\nhcomm₁ : Commute (star a) b\nhcomm₂ : Commute a (star b)\nthi... | [] | all_goals apply le_topologicalClosure; aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 92
} | {
"line": 260,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [] | simpa only [star_mul, star_zero, ← map_star, star_trivial] using congr(star $(mul₁ g f)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 321,
"column": 2
} | {
"line": 322,
"column": 6
} | {
"line": 324,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\n⊢ 1⁺ = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"cfcₙ",
"Eq.mpr",
"NonAssoc... | [] | rw [CFC.posPart_def, cfcₙ_eq_cfc]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 321,
"column": 2
} | {
"line": 322,
"column": 6
} | {
"line": 324,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\n⊢ 1⁺ = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"cfcₙ",
"Eq.mpr",
"NonAssoc... | [] | rw [CFC.posPart_def, cfcₙ_eq_cfc]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 6
} | {
"line": 334,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\nr : ℝ\n⊢ ((algebraMap ℝ A) r)⁺ = (algebraMap ℝ A) r⁺",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [] | rw [CFC.posPart_def, cfcₙ_eq_cfc]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 6
} | {
"line": 334,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\nr : ℝ\n⊢ ((algebraMap ℝ A) r)⁺ = (algebraMap ℝ A) r⁺",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [] | rw [CFC.posPart_def, cfcₙ_eq_cfc]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 31
} | {
"line": 273,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x\nha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x\n⊢ spectrum ℝ a ⊆ {0}",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NormedRing.toRing",
"Set.subset_s... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nha₁ : ∀ x ∈ spectrum ℝ a, 0 ≤ x\nha₂ : ∀ x ∈ spectrum ℝ (-a), 0 ≤ x\n⊢ ∀ y ∈ spectrum ℝ a, y = 0"
] | rw [Set.subset_singleton_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 309,
"column": 2
} | {
"line": 313,
"column": 14
} | {
"line": 317,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nb : A\na : A := star b * b\na_def : a = star b * b\nha : IsSelfAdjoint a\nc : A := b * a⁻\nh_eq_negPart_a : -(star c * c) = a⁻ ^ 3\n⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x",
"ppTerm": "?m.196",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nb : A\na : A := star b * b\na_def : a = star b * b\nha : IsSelfAdjoint a\nc : A := b * a⁻\nh_eq_negPart_a : -(star c * c) = a⁻ ^ 3\nh_c_spec₀ : SpectrumRestricts (-(star c * c)) fun x ↦ ContinuousMap.realToNNReal x\n⊢ ∀ x ∈ spectrum ℝ a, 0 ≤ x"
] | have h_c_spec₀ : SpectrumRestricts (-(star c * c)) (ContinuousMap.realToNNReal ·) := by
simp only [SpectrumRestricts.nnreal_iff, h_eq_negPart_a, CFC.negPart_def]
rw [cfcₙ_eq_cfc (hf0 := by simp), ← cfc_pow (ha := ha) .., cfc_map_spectrum (ha := ha) ..]
rintro - ⟨x, -, rfl⟩
positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 50
} | {
"line": 488,
"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 :... | [] | simpa using rpow_rpow_inv a x⁻¹ (inv_ne_zero hx) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 50
} | {
"line": 488,
"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 :... | [] | simpa using rpow_rpow_inv a x⁻¹ (inv_ne_zero hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 50
} | {
"line": 488,
"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 :... | [] | simpa using rpow_rpow_inv a x⁻¹ (inv_ne_zero hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 19
} | {
"line": 64,
"column": 19
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhab : ↑a ≤ ↑b\nha : 0 ≤ ↑a\nhb : 0 ≤ ↑b\nc : Unitization ℂ A\nhc : 0 ≤ c\n⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + cfc (fun x ↦ x) c) = 1 - cfc (fun x ↦ x⁻¹) (1 + c)",
"ppTerm": "?m.491",
"assigned"... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhab : ↑a ≤ ↑b\nha : 0 ≤ ↑a\nhb : 0 ≤ ↑b\nc : Unitization ℂ A\nhc : 0 ≤ c\n⊢ 1 - cfc (fun x ↦ x⁻¹) (1 + c) = 1 - cfc (fun x ↦ x⁻¹) (1 + c)",
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : Partia... | cfc_id' ℝ≥0 c | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 72
} | {
"line": 277,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝ m : A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nε : ℝ≥0\nhε : 0 < ε\nx : A\nhx₁ : cfcₙ (fun y ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m) ≤ x\nhx₂ : x ∈ closedBall 0 1\n⊢ x * m ∈ closedBall m ↑ε",
"ppTerm": "?m.183",
"... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝ m : A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nε : ℝ≥0\nhε : 0 < ε\nx : A\nhx₁ : cfcₙ (fun y ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m) ≤ x\nhx₂ : ‖x‖ ≤ 1\n⊢ ‖m - x * m‖ ≤ ↑ε"
] | simp only [mem_closedBall, dist_eq_norm', zero_sub, norm_neg] at hx₂ ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 277,
"column": 20
} | {
"line": 277,
"column": 30
} | {
"line": 277,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝ m : A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nε : ℝ≥0\nhε : 0 < ε\nx : A\nhx₁ : cfcₙ (fun y ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m) ≤ x\nhx₂ : ‖x‖ ≤ 1\n⊢ ↑‖m - x * m‖₊ ≤ ↑ε",
"ppTerm": "?m.192",
"assigned": true,... | [
"A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝ m : A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nε : ℝ≥0\nhε : 0 < ε\nx : A\nhx₁ : cfcₙ (fun y ↦ 1 - (1 + y)⁻¹) (ε⁻¹ ^ 2 • m) ≤ x\nhx₂ : ‖x‖ ≤ 1\n⊢ ‖m - x * m‖₊ ≤ ε"
] | coe_le_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 207,
"column": 91
} | {
"line": 224,
"column": 92
} | {
"line": 225,
"column": 4
} | [
{
"pp": "A : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nx y : E\nh : x ≠ 0\na : A\n⊢ 0 ≤\n ‖x‖ ^ 2 •> (a * star a) - ‖x‖ ^ 2 •> (a * ⟪... | [] | by
calc (0 : A) ≤ ⟪a • x - ‖x‖ ^ 2 • y, a • x - ‖x‖ ^ 2 • y⟫_A := by
exact inner_self_nonneg
_ = a * ⟪x, x⟫ * star a - ‖x‖ ^ 2 • (a * ⟪y, x⟫)
- ‖x‖ ^ 2 • (⟪x, y⟫ * star a) + ‖x‖ ^ 2 • (‖x‖ ^ 2 • ⟪y, y⟫) := by
simp only [inner_sub_right, inn... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Module.Defs | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 98
} | {
"line": 285,
"column": 2
} | [
{
"pp": "case refine_1\nA : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nv : E\ninstNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCor... | [
"case refine_2\nA : Type u_1\nE : Type u_2\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℂ E\ninst✝³ : SMul A E\ninst✝² : Norm E\ninst✝¹ : CStarModule A E\ninst✝ : StarOrderedRing A\nv : E\ninstNACG : NormedAddCommGroup E := NormedAddCommGroup.ofCore ⋯\ninstNS ... | · simpa only [norm_smul, norm_inv, norm_norm] using inv_mul_le_one_of_le₀ le_rfl (by positivity) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 40
} | {
"line": 110,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\ninst✝⁶ : TopologicalSpace Y\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\nt : Set Y\ninst✝⁴ : CompactSpace ↑t\ninst✝³ : Zero ↑t\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : SecondCountableTopologyEither X C... | [] | rw [← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff]
refine aestronglyMeasurable_congr ?_ |>.mp <|
ContinuousMap.aeStronglyMeasurable_mkD_restrict_of_uncurry f g f_cont
filter_upwards [f_zero] with x zero_x
rw [mkD_eq_mkD_of_map_zero _ _ zero_x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.SpecificCodomains.ContinuousMapZero | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 40
} | {
"line": 110,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁷ : MeasurableSpace X\nμ : Measure X\ninst✝⁶ : TopologicalSpace Y\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\nt : Set Y\ninst✝⁴ : CompactSpace ↑t\ninst✝³ : Zero ↑t\ninst✝² : TopologicalSpace X\ninst✝¹ : OpensMeasurableSpace X\ninst✝ : SecondCountableTopologyEither X C... | [] | rw [← ContinuousMapZero.isEmbedding_toContinuousMap.aestronglyMeasurable_comp_iff]
refine aestronglyMeasurable_congr ?_ |>.mp <|
ContinuousMap.aeStronglyMeasurable_mkD_restrict_of_uncurry f g f_cont
filter_upwards [f_zero] with x zero_x
rw [mkD_eq_mkD_of_map_zero _ _ zero_x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 774,
"column": 2
} | {
"line": 780,
"column": 10
} | {
"line": 781,
"column": 2
} | [
{
"pp": "case pos\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 A p... | [
"case neg\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 A p\ns : Set 𝕜... | · rw [continuousOn_iff_continuous_restrict]
convert!
continuous_cfcₙHomSuperset_left hs (hs0 := ⟨hs0⟩) ⟨⟨_, hf.restrict⟩, hf0⟩ (X :=
{a : A | p a ∧ quasispectrum 𝕜 a ⊆ s}) continuous_subtype_val (fun x ↦ x.2.2) with
x
rw [cfcₙHomSuperset_apply, Set.restrict_apply, cfcₙ_apply _ _ (hf.mono x.... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 57
} | {
"line": 456,
"column": 0
} | [
{
"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✝⁷ : Fintype n\ninst✝⁶ : Semiring R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : Module R A\ninst✝³ : Star A\ninst✝² : NonUnitalNonAssocSemiring B\ninst✝¹ : Module R B\ninst✝ : Star B\nf : A →⋆ₙₐ[R] B\nM : CStarM... | [] | by ext; simp [map, star_apply, map_star] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.NullSubmodule | {
"line": 43,
"column": 8
} | {
"line": 43,
"column": 11
} | {
"line": 43,
"column": 12
} | [
{
"pp": "M : Type u_1\ninst✝ : SeminormedCommGroup M\nx y : M\nhx : ‖x‖ = 0\nhy : ‖y‖ = 0\n⊢ ‖x‖ + ‖y‖ = 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"id",
"SeminormedCommGroup.toSemi... | [
"M : Type u_1\ninst✝ : SeminormedCommGroup M\nx y : M\nhx : ‖x‖ = 0\nhy : ‖y‖ = 0\n⊢ 0 + ‖y‖ = 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.NullSubmodule | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 11
} | {
"line": 64,
"column": 12
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : SeminormedCommGroup M\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nc : 𝕜\nx : E\nhx : ‖x‖ = 0\n⊢ ‖c‖ * ‖x‖ = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\ninst✝⁴ : SeminormedCommGroup M\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nc : 𝕜\nx : E\nhx : ‖x‖ = 0\n⊢ ‖c‖ * 0 = 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Extreme | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 23
} | {
"line": 259,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : DenselyOrdered 𝕜\ninst✝ : IsTorsionFree 𝕜 E\nA : Set E\nx : E\n⊢ x ∈ extremePoints 𝕜 A ↔ x ∈ A ∧ ∀ x₁ ∈ A, ∀ x₂ ∈ A, x ∈ segment 𝕜 x₁ x₂ →... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : DenselyOrdered 𝕜\ninst✝ : IsTorsionFree 𝕜 E\nA : Set E\nx : E\n⊢ (x ∈ A ∧ ∀ x₁ ∈ A, ∀ x₂ ∈ A, x ∈ openSegment 𝕜 x₁ x₂ → x₁ = x ∧ x₂ = x) ↔\n x ∈ A ∧... | mem_extremePoints | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 529,
"column": 2
} | {
"line": 531,
"column": 57
} | {
"line": 533,
"column": 0
} | [
{
"pp": "case neg\nB : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne : Trivialization F proj\nα : Type u_5\nl : Filter α\nf : α → Z\nz : Z\nhz : z ∈ e.source\nhl : ¬∀ᶠ (x : α) in l, f x ∈ e.source\n⊢ ((Tendsto (fun n ↦... | [] | · simp only [hl, and_false, false_iff, not_and]
rw [e.source_eq] at hl hz
exact fun h _ ↦ hl <| h <| e.open_baseSet.mem_nhds hz | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 460,
"column": 2
} | {
"line": 461,
"column": 22
} | {
"line": 463,
"column": 0
} | [
{
"pp": "case snd\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\n⊢ (↑𝕜 x).toProd.2 = ((algebraMap 𝕜 𝓜(𝕜, 𝕜)) x).toProd.2",
"ppTerm": "?snd",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Algebra.to_smulCommClass",
... | [] | · refine ContinuousLinearMap.ext fun y => ?_
exact mul_comm y x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.FiberBundle.Basic | {
"line": 718,
"column": 26
} | {
"line": 718,
"column": 36
} | {
"line": 718,
"column": 36
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb✝ : B\na : F\nb : B\nx : Z.Fiber b\n⊢ 𝓝 x = 𝓝 x ⊓ ⊤",
"ppTerm": "?m.194",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\nb✝ : B\na : F\nb : B\nx : Z.Fiber b\n⊢ 𝓝 x = 𝓝 x"
] | inf_top_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 623,
"column": 8
} | {
"line": 623,
"column": 83
} | {
"line": 624,
"column": 6
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(𝕜, A)\nh... | [] | exact key x y (mem_closedBall_zero_iff.1 hx) (mem_closedBall_zero_iff.1 hy) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.FiberBundle.Basic | {
"line": 878,
"column": 2
} | {
"line": 890,
"column": 47
} | {
"line": 892,
"column": 0
} | [
{
"pp": "B : Type u_2\nF : Type u_3\nE : B → Type u_5\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\nX : Type u_6\ninst✝ : TopologicalSpace X\nf : TotalSpace F E → X\ns : Set B\nhs : IsOpen[inst✝³] s\nhf : ∀ b ∈ s, ContinuousOn (f ∘ ... | [] | letI := a.totalSpaceTopology
intro z hz
let e : Trivialization F (π F E) :=
a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj)
refine (e.continuousAt_of_comp_right ?_
((hf z.proj hz).continuousAt (IsOpen.mem_nhds ?_ ?_))).continuousWithinAt
· exact a.mem_base_pretrivializ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Basic | {
"line": 878,
"column": 2
} | {
"line": 890,
"column": 47
} | {
"line": 892,
"column": 0
} | [
{
"pp": "B : Type u_2\nF : Type u_3\nE : B → Type u_5\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\ninst✝¹ : (x : B) → TopologicalSpace (E x)\na : FiberPrebundle F E\nX : Type u_6\ninst✝ : TopologicalSpace X\nf : TotalSpace F E → X\ns : Set B\nhs : IsOpen[inst✝³] s\nhf : ∀ b ∈ s, ContinuousOn (f ∘ ... | [] | letI := a.totalSpaceTopology
intro z hz
let e : Trivialization F (π F E) :=
a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj)
refine (e.continuousAt_of_comp_right ?_
((hf z.proj hz).continuousAt (IsOpen.mem_nhds ?_ ?_))).continuousWithinAt
· exact a.mem_base_pretrivializ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Quotient | {
"line": 193,
"column": 6
} | {
"line": 197,
"column": 75
} | {
"line": 198,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G E\nhf : IsQuotientMap f\ninst✝² : ContinuousConstSMul G E\nhfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ninst✝¹ : Topologi... | [] | have ⟨e', he', hfe⟩ := hWU hW
obtain ⟨g', rfl⟩ := hfG.mp hfe
refine ⟨_, ⟨g⁻¹ * g', rfl⟩, ?_, ?_⟩
· apply Set.mem_of_eq_of_mem (pGE _ e) hW
· apply Set.mem_of_eq_of_mem _ he'; simp_rw [mul_smul, smul_inv_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Quotient | {
"line": 193,
"column": 6
} | {
"line": 197,
"column": 75
} | {
"line": 198,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G E\nhf : IsQuotientMap f\ninst✝² : ContinuousConstSMul G E\nhfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ninst✝¹ : Topologi... | [] | have ⟨e', he', hfe⟩ := hWU hW
obtain ⟨g', rfl⟩ := hfG.mp hfe
refine ⟨_, ⟨g⁻¹ * g', rfl⟩, ?_, ?_⟩
· apply Set.mem_of_eq_of_mem (pGE _ e) hW
· apply Set.mem_of_eq_of_mem _ he'; simp_rw [mul_smul, smul_inv_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Quotient | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 37
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case refine_5\nE : Type u_1\nX : Type u_2\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G E\nhf : IsQuotientMap f\ninst✝² : ContinuousConstSMul G E\nhfG : ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂\ninst✝¹ : Topologi... | [] | · simp_rw [mul_smul, inv_smul_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 508,
"column": 70
} | {
"line": 508,
"column": 86
} | {
"line": 509,
"column": 4
} | [
{
"pp": "n : ℤ\ninst✝ : NeZero n\nhn : IsUnit ↑n\ne : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯\nx✝ : ℝ\n⊢ ∀ ⦃a₂ : ℝ⦄, (fun x ↦ n • x) x✝ = (fun x ↦ n • x) a₂ → x✝ = a₂",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"zsmul_e... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 508,
"column": 70
} | {
"line": 508,
"column": 86
} | {
"line": 509,
"column": 4
} | [
{
"pp": "n : ℤ\ninst✝ : NeZero n\nhn : IsUnit ↑n\ne : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯\nx✝ : ℝ\n⊢ ∀ ⦃a₂ : ℝ⦄, (fun x ↦ n • x) x✝ = (fun x ↦ n • x) a₂ → x✝ = a₂",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"zsmul_e... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 508,
"column": 70
} | {
"line": 508,
"column": 86
} | {
"line": 509,
"column": 4
} | [
{
"pp": "n : ℤ\ninst✝ : NeZero n\nhn : IsUnit ↑n\ne : AddCircle 1 ≃ₜ Circle := homeomorphCircle ⋯\nx✝ : ℝ\n⊢ ∀ ⦃a₂ : ℝ⦄, (fun x ↦ n • x) x✝ = (fun x ↦ n • x) a₂ → x✝ = a₂",
"ppTerm": "?m.224",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"zsmul_e... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 45
} | {
"line": 210,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nt : ℝ\nht : t ∈ Set.Icc 0 1\nhu : ‖↑u - 1‖ < 2\nkey : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖\n⊢ 2 * (1 - Real.cos ‖t • argSelfAdjoint u‖) ≤ ‖↑u - 1‖ ^ 2",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Norm.norm",... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nt : ℝ\nht : t ∈ Set.Icc 0 1\nhu : ‖↑u - 1‖ < 2\nkey : ‖t • argSelfAdjoint u‖ ≤ ‖argSelfAdjoint u‖\n⊢ 2 * (1 - Real.cos ‖t • argSelfAdjoint u‖) ≤ 2 * (1 - Real.cos ‖argSelfAdjoint u‖)",
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nt : ℝ\nht : t ∈... | trans 2 * (1 - Real.cos ‖argSelfAdjoint u‖) | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 198,
"column": 64
} | {
"line": 200,
"column": 6
} | {
"line": 202,
"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\nT : E →L[𝕜] F\n⊢ (↑T).rangeᗮ = (↑(adjoint T)).ker",
"ppT... | [] | by
rw [← T†.ker.orthogonal_orthogonal, T†.orthogonal_ker]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 778,
"column": 52
} | {
"line": 780,
"column": 67
} | {
"line": 782,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\n⊢ IsStarProjection T ↔ T.IsSymmetricProjection",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"InnerProductSpace.... | [] | by
simp [← isStarProjection_toContinuousLinearMap_iff,
ContinuousLinearMap.isStarProjection_iff_isSymmetricProjection] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 216,
"column": 67
} | {
"line": 216,
"column": 93
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case e'_2.e'_5.e'_9\nk : 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\nm : ℕ\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (m + 1)\n⊢ univ = Finset.map e.toEmbedding univ",
"ppTerm": "?e'_2.e'_5.e'_9",
"assigned... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 216,
"column": 67
} | {
"line": 216,
"column": 93
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case e'_3.e'_5\nk : 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\nm : ℕ\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (m + 1)\n⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"ppTerm": "?e'_3.e'_5",
"assign... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 53
} | {
"line": 304,
"column": 53
} | [
{
"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\ni j : Fin (n + 1)\n⊢ ((↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) +ᵥ s.points i) -ᵥ ((↑n)⁻¹ • ∑ x, (s.... | [
"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\ni j : Fin (n + 1)\n⊢ (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) - (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points j) + (s.p... | vadd_vsub_vadd_comm _ _ (s.points i) (s.points j) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 364,
"column": 65
} | {
"line": 371,
"column": 18
} | {
"line": 373,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ns : AffineSubspace k P\ninst✝ : FiniteDimensional k ↥s.direction\np : P\n⊢ FiniteDimensional k ↥(vectorSpan k (insert p ↑s))",
"ppTerm": "?m.31",
... | [] | by
rw [← direction_affineSpan, ← affineSpan_insert_affineSpan]
rcases (s : Set P).eq_empty_or_nonempty with (hs | ⟨p₀, hp₀⟩)
· rw [coe_eq_bot_iff] at hs
rw [hs, bot_coe, span_empty, bot_coe, direction_affineSpan]
convert! finiteDimensional_bot k V <;> simp
· rw [affineSpan_coe, direction_affineSpan_inse... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 459,
"column": 6
} | {
"line": 459,
"column": 23
} | {
"line": 459,
"column": 24
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < p ∧ p < ∞\nf : ↥(lp E p)\n⊢ (if hp : p = 0 then Eq.rec (motive := fun x x_1 ↦ ↥(lp E x) → ℝ) (fun f ↦ ↑⋯.toFinset.card) ⋯ f\n else if p = ∞ then ⨆ i, ‖↑f i‖ else (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.to... | [
"α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < p ∧ p < ∞\nf : ↥(lp E p)\n⊢ (if p = ∞ then ⨆ i, ‖↑f i‖ else (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.toReal)) =\n (∑' (i : α), ‖↑f i‖ ^ p.toReal) ^ (1 / p.toReal)"
] | dif_neg hp.1.ne', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 611,
"column": 20
} | {
"line": 613,
"column": 67
} | {
"line": 615,
"column": 0
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\ni : α\nx✝¹ x✝ : ↥(lp E p)\n⊢ dist (↑x✝¹ i) (↑x✝ i) ≤ dist x✝¹ x✝",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"lp.norm_apply_le_norm",
"Norm.norm",
"Eq... | [] | by
simp_rw [dist_eq_norm, ← Pi.sub_apply, ← lp.coeFn_sub]
exact norm_apply_le_norm (zero_lt_one.trans_le Fact.out).ne' .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Calculus | {
"line": 360,
"column": 2
} | {
"line": 367,
"column": 60
} | {
"line": 369,
"column": 0
} | [
{
"pp": "n : ℕ∞\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\ny : E\nhy : y ∈ ball 0 1\n⊢ ContDiffWithinAt ℝ (↑n) (↑univUnitBall.symm) (ball 0 1) y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | [] | apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (√(1 - ‖y‖ ^ 2 : ℝ))⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa [mem_ball_zero_iff, ← _root_.abs_one, ← abs_norm, ← sq_lt_sq, one_pow, ← sub_pos] at hy
refine ContDiffAt.inv ?_ (Real.sqrt_ne_zer... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Calculus | {
"line": 360,
"column": 2
} | {
"line": 367,
"column": 60
} | {
"line": 369,
"column": 0
} | [
{
"pp": "n : ℕ∞\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\ny : E\nhy : y ∈ ball 0 1\n⊢ ContDiffWithinAt ℝ (↑n) (↑univUnitBall.symm) (ball 0 1) y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
... | [] | apply ContDiffAt.contDiffWithinAt
suffices ContDiffAt ℝ n (fun y : E => (√(1 - ‖y‖ ^ 2 : ℝ))⁻¹) y from this.smul contDiffAt_id
have h : (0 : ℝ) < (1 : ℝ) - ‖(y : E)‖ ^ 2 := by
rwa [mem_ball_zero_iff, ← _root_.abs_one, ← abs_norm, ← sq_lt_sq, one_pow, ← sub_pos] at hy
refine ContDiffAt.inv ?_ (Real.sqrt_ne_zer... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 45
} | {
"line": 205,
"column": 0
} | [
{
"pp": "case inr\nx : ℝ\nhx : x < 1\n⊢ x.smoothTransition = 1 ↔ 1 ≤ x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"eq_false",
"congrArg",
"PartialOrder.toPreorder",
"id",
... | [] | · simpa [(lt_one_of_lt_one hx).ne] using hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convolution | {
"line": 368,
"column": 11
} | {
"line": 368,
"column": 31
} | {
"line": 368,
"column": 32
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedAddCommGroup F\nf : G → E\ng : G → E'\nx : G\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 E'\ninst✝⁶ : Normed... | [
"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedAddCommGroup F\nf : G → E\ng : G → E'\nx : G\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedSpace 𝕜 E'\ninst✝⁶ : NormedSpace 𝕜 F\n... | ConvolutionExistsAt, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.MetricSpace.ProperSpace.Lemmas | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 34
} | {
"line": 43,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nx : α\nr : ℝ\ns : Set α\nhr : 0 < r\nhs : IsClosed s\nh : s ⊆ ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : α\nhys : y ∈ s\nhy : s ⊆ closedBall x (dist y x)\n⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'",
"ppTerm": "?inr",
"a... | [
"case inr\nα : Type u_1\ninst✝¹ : PseudoMetricSpace α\ninst✝ : ProperSpace α\nx : α\nr : ℝ\ns : Set α\nhr : 0 < r\nhs : IsClosed s\nh : s ⊆ ball x r\nhne : s.Nonempty\nthis : IsCompact s\ny : α\nhys : y ∈ s\nhy : s ⊆ closedBall x (dist y x)\nhyr : dist y x < r\n⊢ ∃ r' ∈ Ioo 0 r, s ⊆ ball x r'"
] | have hyr : dist y x < r := h hys | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
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