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.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 67
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²¹ : Semifield R\ninst✝²⁰ : StarRing R\ninst✝¹⁹ : MetricSpace R\ninst✝¹⁸ : IsTopologicalSemiring R\ninst✝¹⁷ : ContinuousStar R\ninst✝¹⁶ : Semifield S\ninst✝¹⁵ : StarRing S\ninst✝¹⁴ : MetricSpace S\ninst✝¹³ : IsTopologicalSemi... | · simpa [halg.dist_eq] using ContinuousMap.dist_apply_le_dist _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 15
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 15
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nha : p a\n⊢ IsGreatest ((fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nx : 𝕜\nhx : x ∈ σₙ 𝕜 a\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nha : p a\n⊢ IsGreatest ((fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nx : 𝕜\nhx : x ∈ σₙ 𝕜 a\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 387,
"column": 6
} | {
"line": 387,
"column": 32
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 67
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\n... | · simpa [halg.dist_eq] using ContinuousMap.dist_apply_le_dist _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.Units | {
"line": 69,
"column": 2
} | {
"line": 76,
"column": 36
} | [
{
"pp": "X : Type u_1\nR : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nf : C(X, R)\nh : ∀ (x : X), IsUnit (f x)\n⊢ Continuous fun x ↦ ⋯.unit",
"usedConstants": [
"Iff.mpr",
"ContinuousMap.continuous",
"Units.val",
"Eq.mpr",
"Continuous... | refine
continuous_induced_rng.2
(Continuous.prodMk f.continuous
(MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
have := NormedRing.inverse_continuousAt (h x).unit
simp only
simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
exact this.comp (f.continuous... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Units | {
"line": 69,
"column": 2
} | {
"line": 76,
"column": 36
} | [
{
"pp": "X : Type u_1\nR : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : NormedRing R\ninst✝ : CompleteSpace R\nf : C(X, R)\nh : ∀ (x : X), IsUnit (f x)\n⊢ Continuous fun x ↦ ⋯.unit",
"usedConstants": [
"Iff.mpr",
"ContinuousMap.continuous",
"Units.val",
"Eq.mpr",
"Continuous... | refine
continuous_induced_rng.2
(Continuous.prodMk f.continuous
(MulOpposite.continuous_op.comp (continuous_iff_continuousAt.mpr fun x => ?_)))
have := NormedRing.inverse_continuousAt (h x).unit
simp only
simp only [← Ring.inverse_unit, IsUnit.unit_spec] at this ⊢
exact this.comp (f.continuous... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 380,
"column": 4
} | {
"line": 393,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\ninst✝¹⁶ : Conti... | obtain ⟨ha', haf⟩ := h a |>.mp ha
have := QuasispectrumRestricts.cfc f halg.isClosedEmbedding h0 h
rw [cfcₙHom_eq_restrict f ha ha' haf]
refine .of_dist_eq fun g₁ g₂ ↦ ?_
simp only [nonUnitalStarAlgHom_apply, isometry_cfcₙHom a ha' |>.dist_eq]
refine le_antisymm ?_ ?_
all_goals refine Continuous... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 380,
"column": 4
} | {
"line": 393,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²⁵ : Semifield R\ninst✝²⁴ : StarRing R\ninst✝²³ : MetricSpace R\ninst✝²² : IsTopologicalSemiring R\ninst✝²¹ : ContinuousStar R\ninst✝²⁰ : Field S\ninst✝¹⁹ : StarRing S\ninst✝¹⁸ : MetricSpace S\ninst✝¹⁷ : IsTopologicalRing S\ninst✝¹⁶ : Conti... | obtain ⟨ha', haf⟩ := h a |>.mp ha
have := QuasispectrumRestricts.cfc f halg.isClosedEmbedding h0 h
rw [cfcₙHom_eq_restrict f ha ha' haf]
refine .of_dist_eq fun g₁ g₂ ↦ ?_
simp only [nonUnitalStarAlgHom_apply, isometry_cfcₙHom a ha' |>.dist_eq]
refine le_antisymm ?_ ?_
all_goals refine Continuous... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 56
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : PseudoMetricSpace P\ninst✝² : NormedAddTorsor V P\n𝕜 : Type u_5\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 V\np₁ p₂ : P\nc : 𝕜\n⊢ dist ((lineMap p₁ p₂) c) p₁ = ‖c‖ * dist p₁ p₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 53
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : PseudoMetricSpace P\ninst✝² : NormedAddTorsor V P\n𝕜 : Type u_5\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 V\np₁ p₂ : P\nc : 𝕜\n⊢ dist ((lineMap p₁ p₂) c) p₂ = ‖1 - c‖ * dist p₁ p₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedAlgebra ℝ A\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : Nontrivial A\na : A\nha : 0 ≤ a\n⊢ IsGreatest (σ ℝ≥0 a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : NormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedAlgebra ℝ A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\na : A\nx : ℝ≥0\nhx : x ∈ σ ℝ≥0 a\nha : 0 ≤ a\n⊢ x ≤ ‖a‖₊",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 13
} | [
{
"pp": "V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\np₁ p₂ p₃ p₄ : V\n⊢ dist (midpoint ℝ p₁ p₂) (midpoint ℝ p₃ p₄) ≤ (dist p₁ p₃ + dist p₂ p₄) / 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 15
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 29
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 619,
"column": 8
} | {
"line": 619,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nha : p a\nthis : CompactSpace ↑(σₙ 𝕜 a)\nf : ContinuousMapZero (↑(σₙ 𝕜 a)) 𝕜\nι : C(↑(σ 𝕜 a), ↑(σₙ 𝕜 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 213,
"column": 2
} | {
"line": 222,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set (WeakDual 𝕜 E)\n⊢ Bornology.IsBounded s ↔ IsVonNBounded 𝕜 s",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
... | constructor
· exact fun h => ((NormedSpace.isVonNBounded_iff 𝕜).mpr h).of_topologicalSpace_le
Dual.dual_norm_topology_le_weak_dual_topology
· intro h_vN
have h_ptwise := (withSeminorms 𝕜 E).isVonNBounded_iff_seminorm_bounded.mp h_vN
obtain ⟨C, hC⟩ := banach_steinhaus (g := fun i : s ↦ WeakDual.toStr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 213,
"column": 2
} | {
"line": 222,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set (WeakDual 𝕜 E)\n⊢ Bornology.IsBounded s ↔ IsVonNBounded 𝕜 s",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
... | constructor
· exact fun h => ((NormedSpace.isVonNBounded_iff 𝕜).mpr h).of_topologicalSpace_le
Dual.dual_norm_topology_le_weak_dual_topology
· intro h_vN
have h_ptwise := (withSeminorms 𝕜 E).isVonNBounded_iff_seminorm_bounded.mp h_vN
obtain ⟨C, hC⟩ := banach_steinhaus (g := fun i : s ↦ WeakDual.toStr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UrysohnsLemma | {
"line": 167,
"column": 8
} | {
"line": 167,
"column": 12
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nn : ℕ\nihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0\nc : CU P\nhx : x ∈ c.C\n⊢ midpoint ℝ (approx n c.left x) (approx n c.right x) = 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommR... | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 167,
"column": 13
} | {
"line": 167,
"column": 17
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nn : ℕ\nihn : ∀ (c : CU P), x ∈ c.C → approx n c x = 0\nc : CU P\nhx : x ∈ c.C\n⊢ midpoint ℝ 0 (approx n c.right x) = 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.... | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 12
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nn : ℕ\nihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1\nc : CU P\nhx : x ∉ c.U\n⊢ midpoint ℝ (approx n c.left x) (approx n c.right x) = 1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommR... | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 177,
"column": 13
} | {
"line": 177,
"column": 17
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nn : ℕ\nihn : ∀ (c : CU P), x ∉ c.U → approx n c x = 1\nc : CU P\nhx : x ∉ c.U\n⊢ midpoint ℝ 1 (approx n c.right x) = 1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.... | ihn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : SeparableSpace E\ninst✝ : ProperSpace 𝕜\ns : Set (WeakDual 𝕜 E)\nhb : Bornology.IsBounded s\nhc : IsClosed s\nb_isCompact' : CompactSpace ↑s\nb_isMetrizable : Metri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 26
} | [
{
"pp": "case h.refine_3\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\ng : C(X, ℝ)\nhgs : EqOn (⇑g) 0 s\nhgt : EqOn (⇑g) 1 t\nhicc : ∀ (x : X), 0 ≤ g x ∧ g x ≤ 1\nx : X\n⊢ (1 - g) x ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 411,
"column": 56
} | {
"line": 411,
"column": 67
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nk : Set X\nk_comp : IsCompact k\nk_closed : IsClosed[inst✝²] k\nsk : s ⊆ interior k\nkt : k ⊆ tᶜ\nf : X → ℝ\nhf : Continuous[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 412,
"column": 18
} | {
"line": 412,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nk : Set X\nk_comp : IsCompact k\nk_closed : IsClosed[inst✝²] k\nsk : s ⊆ interior k\nkt : k ⊆ tᶜ\nf : X → ℝ\nhf : Continuous[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 419,
"column": 35
} | {
"line": 419,
"column": 46
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nk : Set X\nk_comp : IsCompact k\nk_closed : IsClosed[inst✝²] k\nsk : s ⊆ interior k\nkt : k ⊆ tᶜ\nf : X → ℝ\nhf : Continuous[inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 453,
"column": 8
} | {
"line": 453,
"column": 19
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ContinuousSqrt | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nx : 𝕜 × 𝕜\nhx : 0 ≤ x.2 - x.1\nhx' : ↑(re (x.2 - x.1)) = x.2 - x.1\n⊢ 0 ≤ re (x.2 - x.1)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real.instLE",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"NormedRing.toR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ContinuousSqrt | {
"line": 47,
"column": 22
} | {
"line": 47,
"column": 33
} | [
{
"pp": "⊢ ∀ (x : ℝ≥0 × ℝ≥0), x.1 ≤ x.2 → x.2 = x.1 + (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) x * (⇑NNReal.sqrt ∘ fun x ↦ x.2 - x.1) x",
"usedConstants": [
"Eq.mpr",
"NNReal.instCommSemiring",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"PartialOrder.toPre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 62
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 465,
"column": 40
} | {
"line": 465,
"column": 56
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 178,
"column": 4
} | {
"line": 180,
"column": 11
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nf : C(X, ℝ≥0)\nc : ℝ≥0\nhc : 0 < c\nx : X\nhx : x ∈ {x | c ≤ f x}\n⊢ ({ toFun := (⇑f ⊔ ⇑(const X c))⁻¹, continuous_toFun := ⋯ } * f) x = 1 x",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Eq.mpr",
"Continuous",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 66
} | [
{
"pp": "X : 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\nx : X\nhx : x ∈ (setOfIdeal I)ᶜ\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 217,
"column": 6
} | {
"line": 218,
"column": 54
} | [
{
"pp": "case pos\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 470,
"column": 4
} | {
"line": 470,
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 237,
"column": 24
} | {
"line": 237,
"column": 50
} | [
{
"pp": "X : 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 (setOfIdeal I)ᶜ\ng... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 257,
"column": 10
} | {
"line": 257,
"column": 37
} | [
{
"pp": "case refine_3.refine_2.inl\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 : D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 258,
"column": 10
} | {
"line": 258,
"column": 37
} | [
{
"pp": "case refine_3.refine_2.inr\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 : D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 314,
"column": 6
} | {
"line": 314,
"column": 47
} | [
{
"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⊢ ∀ ⦃x : X⦄, x ∈ sᶜ → { toFun := fun x ↦ ↑(g x), continuous_toF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 91
} | [
{
"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⊢ { toFun := fun x ↦ ↑(g x), continuous_toFun := ⋯ } x ≠ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 83
} | [
{
"pp": "X : 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)\nhI : IsClosed ↑I\nhI' : I.IsMaximal\nx : X\nhx : setOfIdeal I = {x}ᶜ\n⊢ idealOfSet 𝕜 {x}ᶜ = I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 413,
"column": 4
} | {
"line": 414,
"column": 30
} | [
{
"pp": "case h\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : RCLike 𝕜\nx y : X\nhxy : x ≠ y\nf : C(X, ℝ)\nfx : Set.EqOn (⇑f) 0 {x}\nfy : Set.EqOn (⇑f) 1 {y}\n⊢ ((continuousMapEval X 𝕜) x) ({ toFun := fun x ↦ ↑x, continuous_toFun := ⋯ }.comp f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 418,
"column": 4
} | {
"line": 419,
"column": 28
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : RCLike 𝕜\nφ : ↑(characterSpace 𝕜 C(X, 𝕜))\nx : X\nhx : idealOfSet 𝕜 {x}ᶜ = RingHom.ker φ\nf : C(X, 𝕜)\n⊢ f ∈ RingHom.ker ((continuousMapEval X 𝕜) x) ↔ f ∈ RingHom.ker φ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 541,
"column": 2
} | {
"line": 547,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\nx : X\n⊢ ∃ f, HasCompactSupport ⇑f ∧ 0 ≤ ⇑f ∧ f x ≠ 0",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"Filter.instMembership",
"False",
"Real.instLE"... | rcases exists_compact_mem_nhds x with ⟨k, hk, k_mem⟩
rcases exists_continuous_one_zero_of_isCompact hk isClosed_empty (disjoint_empty k)
with ⟨f, fk, -, f_comp, hf⟩
refine ⟨f, f_comp, fun x ↦ (hf x).1, ?_⟩
have := fk (mem_of_mem_nhds k_mem)
simp only [Pi.one_apply] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UrysohnsLemma | {
"line": 541,
"column": 2
} | {
"line": 547,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\nx : X\n⊢ ∃ f, HasCompactSupport ⇑f ∧ 0 ≤ ⇑f ∧ f x ≠ 0",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"Filter.instMembership",
"False",
"Real.instLE"... | rcases exists_compact_mem_nhds x with ⟨k, hk, k_mem⟩
rcases exists_continuous_one_zero_of_isCompact hk isClosed_empty (disjoint_empty k)
with ⟨f, fk, -, f_comp, hf⟩
refine ⟨f, f_comp, fun x ↦ (hf x).1, ?_⟩
have := fk (mem_of_mem_nhds k_mem)
simp only [Pi.one_apply] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝ : CommCStarAlgebra A\na : A\nthis : ‖(gelfandTransform ℂ A) a‖₊ ^ 2 = ‖a‖₊ ^ 2\n⊢ ‖(gelfandTransform ℂ A) a‖ = ‖a‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 210,
"column": 44
} | {
"line": 210,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCommCStarAlgebra A\na b : A\nh : a * b = 0\nf : A → C(↑(characterSpace ℂ (Unitization ℂ A)), ℂ) :=\n ⇑(gelfandStarTransform (Unitization ℂ A)) ∘ ⇑(inrNonUnitalAlgHom ℂ A)\nhf : Isometry f\n⊢ f a * f b = 0",
"usedConstants": [
"NormedCommRing.toSeminormedCommRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCommCStarAlgebra A\na b : A\nh : a * b = 0\n⊢ ‖a - b‖ = max ‖a‖ ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalCommCStarAlgebra.toNonUnitalCStarAlgebra",
"NonUnitalNormedRing.toNorm",
"Real",
"NonUnitalCommRing.toNonUnita... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 32
} | [
{
"pp": "case insert\nA : Type u_1\ninst✝ : NonUnitalCommCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\n⊢ f j * ∑ i ∈ s, f i = 0",
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\nx : ℝ\nx✝ : x ∈ quasispectru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsStarNormal a\nhb : IsStarNormal b\nhcomm : Commute a b\nhab : a * b = 0\n⊢ ‖a - b‖ = max ‖a‖ ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real",
"NonUnitalCStarAlgebra.toNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 266,
"column": 29
} | {
"line": 268,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsStarNormal a\nhb : IsStarNormal b\nhcomm : Commute a b\nhab : a * b = 0\n⊢ ‖a - b‖ = max ‖a‖ ‖b‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalNorm... | by
simpa [sub_eq_add_neg] using
ha.norm_add_eq_max hb.neg hcomm.neg_right (by simpa) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na b : A\nha : IsSelfAdjoint a\nhb : IsSelfAdjoint b\nhab : a * b = 0\n⊢ ‖a - b‖ = max ‖a‖ ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonUnitalNormedRing.toNorm",
"Real",
"NonUnitalCStarAlgebra.toNonUnitalNormedRing",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 32
} | [
{
"pp": "case insert\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : (∀ i ∈ s, IsSelfAdjoint (f i)) → ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nh : ∀ i ∈ insert j s, IsSelfAdjoint (f i)\n⊢ f j * ∑ i ∈ s,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 15
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\nx : ℝ\nx✝ : x ∈ quasispectru... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : PseudoEMetricSpace β\nx : α\nf g : α →ᵤ β\n⊢ edist ((fun f ↦ toFun f x) f) ((fun f ↦ toFun f x) g) ≤ ↑1 * edist f g",
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"ENNReal.ofNNReal",
"Equiv.instEquivLike",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 93,
"column": 54
} | {
"line": 93,
"column": 65
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalRing A\ninst✝⁶ : Module ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\na : A\nha : ¬IsSelfAdjoint a\nh : IsSelfAdjoint... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : T2Space A\ninst✝ : StarModule ℝ A\nr : ℝ≥0\na : A\n⊢ (r •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhab :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na c : A\nhac :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 9
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nx : α\nhx : x ∈ ⋃₀ 𝔖\nf g : α →ᵤ[𝔖] β\n⊢ edist ((fun f ↦ (toFun 𝔖) f x) f) ((fun f ↦ (toFun 𝔖) f x) g) ≤ ↑1 * edist f g",
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Nonne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 229,
"column": 2
} | {
"line": 231,
"column": 46
} | [
{
"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... | case of_b_eq =>
rintro rfl
exact negPart_eq_of_eq_PosPart_sub habc hc | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 68
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 271,
"column": 8
} | {
"line": 271,
"column": 83
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\na : A\ninst✝ : IsStarNormal a\nx : A\nφ : ↑(characterSpace ℂ ↥(elemental ℂ x))\n⊢ φ ⟨x, ⋯⟩ ∈ spectrum ℂ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 101,
"column": 4
} | {
"line": 102,
"column": 33
} | [
{
"pp": "case refine_3\nA : Type u_1\ninst✝¹ : CStarAlgebra A\na : A\ninst✝ : IsStarNormal a\nφ ψ : ↑(characterSpace ℂ ↥(elemental ℂ a))\nh : characterSpaceToSpectrum a φ = characterSpaceToSpectrum a ψ\n⊢ φ ⟨a, ⋯⟩ = ψ ⟨a, ⋯⟩",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 22
} | [
{
"pp": "A : Type u_2\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : ∀ x ∈ spectrum ℝ a, 0 ≤ x\nr : ℝ\nhr : 0 ≤ r\na✝ : Nontrivial A\nx : ℝ\nhx : x ∈ spectrum ℝ (r • a)\n⊢ 0 ≤ x",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"LE.le",
"dite",
"Zero.toOfNa... | by_cases hr' : r = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 32
} | [
{
"pp": "case neg\nA : Type u_2\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : ∀ x ∈ spectrum ℝ a, 0 ≤ x\na✝ : Nontrivial A\nx : ℝ\nr : ℝˣ\nhr : 0 ≤ ↑r\nhx : r⁻¹ • x ∈ spectrum ℝ a\nhr' : ¬↑r = 0\n⊢ (↑r)⁻¹ • 0 ≤ (↑r)⁻¹ • x",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Real",
"inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : 0 ≤ a\ny : ℝ\nhy : y ∈ spectrum ℝ a\nx✝ : (algebraMap ℝ ℂ) y = 0 ∨ (algebraMap ℝ ℂ) y ∈ spectrum ℂ a\nhx : (algebraMap ℝ ℂ) y ∈ spectrum ℂ a\n⊢ 0 ≤ (algebraMap ℝ ℂ) y",
"usedConstants": [
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 399,
"column": 14
} | {
"line": 399,
"column": 25
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx y z : A\nhxy : IsSelfAdjoint ↑(y - x) ∧ SpectrumRestricts ↑(y - x) ⇑ContinuousMap.realToNNReal\nhyz : IsSelfAdjoint ↑(z - y) ∧ SpectrumRestricts ↑(z - y) ⇑ContinuousMap.realToNNReal\n⊢ IsSelfAdjoint ↑(z - x)",
"usedConstants": [
"AddGroup.toSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 399,
"column": 46
} | {
"line": 399,
"column": 57
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx y z : A\nhxy : IsSelfAdjoint ↑(y - x) ∧ SpectrumRestricts ↑(y - x) ⇑ContinuousMap.realToNNReal\nhyz : IsSelfAdjoint ↑(z - y) ∧ SpectrumRestricts ↑(z - y) ⇑ContinuousMap.realToNNReal\n⊢ SpectrumRestricts ↑(z - x) ⇑ContinuousMap.realToNNReal",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 369,
"column": 2
} | {
"line": 374,
"column": 63
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℂ A\ninst✝⁷ : SMulCommClass ℂ A A\ninst✝⁶ : IsScalarTower ℂ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarModule ℂ A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrde... | refine eq_top_iff.mpr fun x _ => ?_
rw [← CStarAlgebra.linear_combination_nonneg x]
apply_rules [sub_mem, Submodule.smul_mem, add_mem]
all_goals
refine subset_span ?_
first | apply CFC.negPart_nonneg | apply CFC.posPart_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 369,
"column": 2
} | {
"line": 374,
"column": 63
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℂ A\ninst✝⁷ : SMulCommClass ℂ A A\ninst✝⁶ : IsScalarTower ℂ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarModule ℂ A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrde... | refine eq_top_iff.mpr fun x _ => ?_
rw [← CStarAlgebra.linear_combination_nonneg x]
apply_rules [sub_mem, Submodule.smul_mem, add_mem]
all_goals
refine subset_span ?_
first | apply CFC.negPart_nonneg | apply CFC.posPart_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 15
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝¹⁰ : NonUnitalNormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : NormedSpace ℝ A\ninst✝⁷ : IsScalarTower ℝ A A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : NonUnitalIsometricContinuousFunctionalCalcu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 85,
"column": 22
} | {
"line": 85,
"column": 33
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra ℝ A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\nr : ℝ\na : A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 31
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace ℝ A\ninst✝³ : SMulCommClass ℝ A A\ninst✝² : IsScalarTower ℝ A A\ninst✝¹ : StarRing A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\n⊢ ‖a⁻‖ ≤ ‖a‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 47,
"column": 27
} | {
"line": 47,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MulOneClass α\nm : α\n⊢ Tendsto (fun x ↦ m * x) (pure 1) (𝓝 m)",
"usedConstants": [
"Pure.pure",
"MulOne.toOne",
"HMul.hMul",
"nhds",
"id",
"MulOne.toMul",
"Filter.instPure",
"MulOneClass.toMulOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 48,
"column": 28
} | {
"line": 48,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MulOneClass α\nm : α\n⊢ Tendsto (fun x ↦ x * m) (pure 1) (𝓝 m)",
"usedConstants": [
"Pure.pure",
"MulOne.toOne",
"HMul.hMul",
"nhds",
"id",
"MulOne.toMul",
"Filter.instPure",
"MulOneClass.toMulOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 59,
"column": 27
} | {
"line": 59,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : MulOneClass α\ninst✝ : SeparatelyContinuousMul α\nm : α\n⊢ Tendsto (fun x ↦ m * x) (𝓝 1) (𝓝 m)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 60,
"column": 28
} | {
"line": 60,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : MulOneClass α\ninst✝ : SeparatelyContinuousMul α\nm : α\n⊢ Tendsto (fun x ↦ x * m) (𝓝 1) (𝓝 m)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 65,
"column": 26
} | {
"line": 65,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : MulOneClass α\ninst✝ : SeparatelyContinuousMul α\nl : Filter α\nhl : l.IsApproximateUnit\n⊢ l ≤ 𝓝 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ApproximateUnit | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MulOneClass α\ninst✝¹ : SeparatelyContinuousMul α\nl : Filter α\ninst✝ : l.NeBot\n⊢ l.IsApproximateUnit ↔ l ≤ 𝓝 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"congrArg",
"Filter.NeBot",
"PartialOrder.toPreorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 49
} | [
{
"pp": "case neg\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 49
} | [
{
"pp": "case neg\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 : MetricSpace A\ninst✝¹ : Algebra R A\ninst✝ : IsometricContinuousFunctiona... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 13
} | [
{
"pp": "R : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : MetricSpace A\ninst✝¹ : Algebra R A\ninst✝ : IsometricContinuousFunctionalCalculus ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 49
} | [
{
"pp": "case const\nX : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : TopologicalSpace X\ns : Set 𝕜\nhs : IsCompact s\na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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