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.Unitization | {
"line": 36,
"column": 51
} | {
"line": 36,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na b : E\n⊢ ‖((mul 𝕜 E).flip a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na : E\nthis : ‖(mul 𝕜 E) a⋆‖ ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 37,
"column": 51
} | {
"line": 38,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na : E\nthis : ‖(mul 𝕜 E) a⋆‖ ≤ ... | by
simpa only [ge_iff_le, opNorm_mul_apply, norm_star] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 41,
"column": 8
} | {
"line": 41,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na b : E\n⊢ ‖((mul 𝕜 E) a⋆) b‖ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 43,
"column": 8
} | {
"line": 43,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NonUnitalNormedRing E\ninst✝⁵ : StarRing E\ninst✝⁴ : NormedStarGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : IsScalarTower 𝕜 E E\ninst✝¹ : SMulCommClass 𝕜 E E\ninst✝ : RegularNormedAlgebra 𝕜 E\na b : E\n⊢ ‖((mul 𝕜 E).flip a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 392,
"column": 4
} | {
"line": 392,
"column": 19
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nX : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : TopologicalSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nx₁ x₂ : X\nhx : x₁ ≠ x₂\nf : C(X, 𝕜)\nhfA : f ∈ ↑A.toSubalgebra\nhf : (fun f ↦ ⇑f) f x₁ ≠ (fun f ↦ ⇑f) f x₂\nF : C(X, 𝕜) := f - const X (f x₂)\nhFA : F ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 81
} | [
{
"pp": "case refine_3.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : DenselyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing E\ninst✝⁴ : StarRing E\ninst✝³ : CStarRing E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : IsScalarTower 𝕜 E E\ninst✝ : SMulCommClass 𝕜 E E\na : E\nr : NNReal\nhr : r * ‖a‖₊⁻¹ < 1\nha : 0 < ‖a‖₊\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 32
} | [
{
"pp": "case calc_1.h₁\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : NonUnitalNormedRing E\ninst✝⁶ : StarRing E\ninst✝⁵ : CStarRing E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : IsScalarTower 𝕜 E E\ninst✝² : SMulCommClass 𝕜 E E\ninst✝¹ : StarRing 𝕜\ninst✝ : StarModule 𝕜 E\nx : Unitization... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 600,
"column": 4
} | {
"line": 601,
"column": 11
} | [
{
"pp": "case refine_1\nF : Type u_2\nS : Type u_3\nK : Type u_4\nA : Type u_5\ninst✝¹³ : CommRing K\ninst✝¹² : Ring A\ninst✝¹¹ : Algebra K A\ninst✝¹⁰ : TopologicalSpace K\ninst✝⁹ : T1Space K\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : ContinuousSub A\ninst✝⁶ : ContinuousSMul K A\ninst✝⁵ : FunLike F A K\ninst✝⁴ : Al... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 36
} | [
{
"pp": "case hbc\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : DenselyNormedField 𝕜\ninst✝⁸ : NonUnitalNormedRing E\ninst✝⁷ : StarRing E\ninst✝⁶ : CStarRing E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : IsScalarTower 𝕜 E E\ninst✝³ : SMulCommClass 𝕜 E E\ninst✝² : StarRing 𝕜\ninst✝¹ : StarModule 𝕜 E\ninst✝ : CStarRing 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.WeakDual | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 40
} | [
{
"pp": "case h\nα : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : CommSemiring 𝕜\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : ContinuousAdd 𝕜\ninst✝³ : ContinuousConstSMul 𝕜 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : TopologicalSpace E\n⊢ ∀ (y : E →L[𝕜] 𝕜), Conti... | exact ContinuousLinearMap.continuous | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Module.Spaces.WeakDual | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 31
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_4\ninst✝⁶ : CommSemiring 𝕜\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : ContinuousAdd 𝕜\ninst✝³ : ContinuousConstSMul 𝕜 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : TopologicalSpace E\nV : Set E\nhV : IsOpen (⇑(toWeakSpaceCLM 𝕜 E) '' V)\n⊢ IsOpen[inst✝] V",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 143,
"column": 66
} | {
"line": 143,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : CommRing 𝕜\ninst✝⁶ : NoZeroDivisors 𝕜\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : ContinuousAdd 𝕜\ninst✝³ : ContinuousConstSMul 𝕜 𝕜\ninst✝² : TopologicalSpace A\ninst✝¹ : Semiring A\ninst✝ : Algebra 𝕜 A\nφ : ↑(characterSpace 𝕜 A)\nh₁ : φ 1 * (1 - φ 1) = 0\nh₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing 𝕜\ninst✝⁹ : NoZeroDivisors 𝕜\ninst✝⁸ : TopologicalSpace 𝕜\ninst✝⁷ : ContinuousAdd 𝕜\ninst✝⁶ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Semiring A\ninst✝³ : Algebra 𝕜 A\ninst✝² : Nontrivial 𝕜\ninst✝¹ : T2Space 𝕜\ninst✝ : Cont... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : CommRing 𝕜\ninst✝⁶ : NoZeroDivisors 𝕜\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : ContinuousAdd 𝕜\ninst✝³ : ContinuousConstSMul 𝕜 𝕜\ninst✝² : TopologicalSpace A\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\nφ ψ : ↑(characterSpace 𝕜 A)\nh : RingHom.ker φ = RingHom.ker ψ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\ns : Set α\nx : α\n⊢ (∀ (i : Set β), IsOpen[inst✝] i ∧ f x ⊆ i → ∀ᶠ (x' : α) in 𝓝[s] x, i ∈ 𝓝ˢ (f x')) ↔\n ∀ (u : Set β), IsOpen[inst✝] u → f x ⊆ u → ∀ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ u",
"usedConst... | case mono => exact fun t₁ t₂ ht h ↦ h.mp <| .of_forall fun x' ↦ by gcongr | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\n⊢ UpperHemicontinuousAt f x ↔ ∀ (u : Set β), IsOpen[inst✝] u → f x ⊆ u → ∀ᶠ (x' : α) in 𝓝 x, f x' ⊆ u",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\n⊢ UpperHemicontinuousAt f x ↔ ∀ u ∈ 𝓝ˢ (f x), f ⁻¹' Iic u ∈ 𝓝 x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 249,
"column": 11
} | {
"line": 249,
"column": 45
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → Set β\ns : Set α\nx : α\nγ : Type u_5\ninst✝ : TopologicalSpace γ\ni : γ → β\nhf : UpperHemicontinuousWithinAt f s x\nhi : IsInducing i\nh_cl : IsClosed[inst✝¹] (range i)\nv : Set β\nhv : IsOpen[inst✝¹] v\nhu ... | ← preimage_inter_range (s := f _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 52
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → Set β\nx : α\nγ : Type u_5\ninst✝ : TopologicalSpace γ\ni : γ → β\nhf : UpperHemicontinuousAt f x\nhi : IsInducing i\nh_cl : IsClosed[inst✝¹] (range i)\n⊢ UpperHemicontinuousAt (fun x ↦ i ⁻¹' f x) x",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 57
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\nhx : f x = ∅\nhf : ∀ (x : α), ∀ t ∈ 𝓝ˢ (f x), ∀ᶠ (x' : α) in 𝓝 x, t ∈ 𝓝ˢ (f x')\n⊢ {a | f a = ∅} ∈ 𝓝 x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 286,
"column": 2
} | {
"line": 289,
"column": 64
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nhf : UpperHemicontinuous f\n⊢ IsClosed[inst✝¹] {x | (f x).Nonempty}",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"UpperHemicontinuousAt",
"_private.Mathlib.Topology... | simp only [← isOpen_compl_iff, compl_setOf, not_nonempty_iff_eq_empty, isOpen_iff_mem_nhds]
intro x (hx : f x = ∅)
simp_rw [upperHemicontinuous_iff, upperHemicontinuousAt_iff] at hf
simpa [hx, empty_mem_iff_bot, nhdsSet_eq_bot_iff] using hf x ∅ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 286,
"column": 2
} | {
"line": 289,
"column": 64
} | [
{
"pp": "α : Type u_5\nβ : Type u_6\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nhf : UpperHemicontinuous f\n⊢ IsClosed[inst✝¹] {x | (f x).Nonempty}",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"UpperHemicontinuousAt",
"_private.Mathlib.Topology... | simp only [← isOpen_compl_iff, compl_setOf, not_nonempty_iff_eq_empty, isOpen_iff_mem_nhds]
intro x (hx : f x = ∅)
simp_rw [upperHemicontinuous_iff, upperHemicontinuousAt_iff] at hf
simpa [hx, empty_mem_iff_bot, nhdsSet_eq_bot_iff] using hf x ∅ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Convex | {
"line": 32,
"column": 6
} | {
"line": 33,
"column": 13
} | [
{
"pp": "s t : Set ℝ\n⊢ (convexHull ℝ) (⇑equivRealProdLm ⁻¹' s ×ˢ t) = ⇑equivRealProdLm ⁻¹' (convexHull ℝ) (s ×ˢ t)",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"LinearEquiv.symm",
"Real",
"NonUnitalComm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 74,
"column": 36
} | {
"line": 74,
"column": 74
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs₁ : {z | r < z.im} ⊆ s\nhs₂ : s ⊆ {z | r ≤ z.im}\n⊢ s ⊆ closure {z | r < z.im}",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"congrArg",
"Complex.im",
"Complex.instNormedField",
"setOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 79,
"column": 36
} | {
"line": 79,
"column": 74
} | [
{
"pp": "r : ℝ\ns : Set ℂ\nhs₁ : {z | z.im < r} ⊆ s\nhs₂ : s ⊆ {z | z.im ≤ r}\n⊢ s ⊆ closure {z | z.im < r}",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"congrArg",
"Complex.im",
"Complex.instNormedField",
"setOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 44
} | [
{
"pp": "U : Set ℂ\nU_convex : Convex ℝ U\nz w : ℂ\nhz : z ∈ U\nhw : w ∈ U\nhzw : ↑z.re + ↑w.im * I ∈ U\nhwz : ↑w.re + ↑z.im * I ∈ U\n⊢ z.Rectangle w ⊆ U",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nhk : k ∈ resolventSet 𝕜 a\nH₁ :\n HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nhk : k ∈ resolventSet 𝕜 a\nH₁ :\n HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nhk : k ∈ resolventSet 𝕜 a\nH₁ :\n HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nhk : k ∈ resolventSet 𝕜 a\nH₁ :\n HasFDerivAt Ring.inverse (-((ContinuousLinearMap.mulLeftRight 𝕜 A) ↑(IsUnit.unit hk)⁻¹) ↑(IsUnit.unit hk)⁻¹)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nr : ℝ≥0\nhr : ↑r < (spectralRadius 𝕜 a)⁻¹\nz : 𝕜\nz_mem : z ∈ Metric.closedBall 0 ↑r\n⊢ ‖z‖₊ ≤ r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 47
} | [
{
"pp": "A : Type u_2\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra ℂ A\ninst✝¹ : CompleteSpace A\ninst✝ : Nontrivial A\na : A\nh : spectrum ℂ a = ∅\nH₀ : resolventSet ℂ a = Set.univ\nH₁ : Differentiable ℂ fun z ↦ resolvent a z\n⊢ Tendsto (fun z ↦ resolvent a z) (cocompact ℂ) (𝓝 ?m.100)",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 44
} | [
{
"pp": "A : Type u_2\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra ℂ A\ninst✝¹ : CompleteSpace A\ninst✝ : Nontrivial A\na : A\nn : ℕ\n⊢ spectrum ℂ (a ^ n) = (fun x ↦ x ^ n) '' spectrum ℂ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℂ A\ninst✝ : CompleteSpace A\nhA : ∀ {a : A}, IsUnit a ↔ a ≠ 0\nnt : Nontrivial A :=\n {\n exists_pair_ne :=\n Exists.intro 1\n (Exists.intro 0 (hA.mp (Exists.intro { val := 1, inv := 1, val_inv := mul_one 1, inv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 51
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : ↥(unitary E)\na✝ : Nontrivial E\nk : 𝕜\nhk : k ∈ σ 𝕜 ↑u\n⊢ ‖k‖ ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : ↥(unitary E)\na✝ : Nontrivial E\nk : 𝕜\nhk : k⁻¹ ∈ σ 𝕜 ↑(toUnits u)⁻¹\nhnk : k ≠ 0\n⊢ ‖k‖⁻¹ ≤ ‖↑(toUnits u)⁻¹‖",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : ↥(unitary E)\na✝ : Nontrivial E\nk : 𝕜\nhk : k⁻¹ ∈ σ 𝕜 ↑(toUnits u)⁻¹\nhnk : k ≠ 0\nthis : ‖k‖⁻¹ ≤ ‖↑(toUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : E\nhu : u ∈ unitary E\nz : 𝕜\nhz : z ∈ σ 𝕜 u\n⊢ ‖z‖ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 38
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na : A\nx : ℝ≥0\nhx : x ∈ σ ℝ≥0 ↑a\n⊢ x ≤ ‖a‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 52
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ σ ℂ a\nthis : NormedAlgebra ℚ A\nhu : NormedSpace.exp (I • a) ∈ unitary A\nIu : ℂˣ := Units.mk0 I I_ne_zero\n⊢ NormedSpace.exp (I • z) ∈ σ ℂ (NormedSpace.exp (I • a))",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 44
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ σ ℂ a\nthis✝ : NormedAlgebra ℚ A\nhu : NormedSpace.exp (I • a) ∈ unitary A\nIu : ℂˣ := Units.mk0 I I_ne_zero\nthis : NormedSpace.exp (I • z) ∈ σ ℂ (NormedSpace.exp (I • a))\n⊢ z.im = (↑z.re).im",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 75
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ σ ℂ a\n⊢ (ofReal ∘ re) z ∈ σ ℂ a",
"usedConstants": [
"Eq.mpr",
"Real",
"NormedRing.toRing",
"spectrum",
"congrArg",
"Complex.instNormedField",
"CSta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 226,
"column": 60
} | {
"line": 226,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ x✝ ∈ {z | z.im < 0} → x✝ ∈ {z | z.im ≤ 0}",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Complex.im",
"setOf",
"Real.instLT",
"Membership.me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 226,
"column": 60
} | {
"line": 226,
"column": 71
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CStarAlgebra A\ninst✝ : StarModule ℂ A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ x✝ ∈ {z | 0 < z.im} → x✝ ∈ {z | 0 ≤ z.im}",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Complex.im",
"setOf",
"Real.instLT",
"Membership.me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 26
} | [
{
"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\nh : ∀ (ψ : Unitization ℂ A →⋆ₐ[ℂ] Unitization ℂ B) (x : Unitization ℂ A), ‖ψ x‖₊ ≤ ‖x‖₊\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 294,
"column": 52
} | {
"line": 294,
"column": 78
} | [
{
"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\n⊢ ∀ (x : A), ‖φ x‖ ≤ 1 * ‖x‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 15
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : NonUnitalCStarAlgebra A\ninst✝³ : NonUnitalCStarAlgebra B\ninst✝² : EquivLike F A B\ninst✝¹ : NonUnitalAlgEquivClass F ℂ A B\ninst✝ : StarHomClass F A B\nφ : F\na : A\n⊢ ‖a‖₊ ≤ ‖φ a‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 126,
"column": 42
} | {
"line": 126,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nh : ‖a‖ < ‖1‖⁻¹ * ‖k‖\na✝ : Nontrivial A\nhk : k ≠ 0\nku : Aˣ := (Units.map ↑↑ₐ) (Units.mk0 k hk)\n⊢ ‖-a‖ < ‖↑ku⁻¹‖⁻¹",
"usedConstants": [
"Norm.no... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nk : 𝕜\nh : ‖a‖ < ‖1‖⁻¹ * ‖k‖\na✝ : Nontrivial A\nhk : k ≠ 0\nku : Aˣ := (Units.map ↑↑ₐ) (Units.mk0 k hk)\nhku : ‖-a‖ < ‖↑ku⁻¹‖⁻¹\n⊢ IsUnit (k • 1 - a)",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 237,
"column": 54
} | {
"line": 237,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : ProperSpace 𝕜\na : A\nha : (σ a).Nonempty\nr : ℝ≥0\nhr : ∀ k ∈ σ a, ‖k‖₊ < r\n⊢ ∀ x ∈ σ a, ‖x‖ₑ < ↑r",
"usedConstants": [
"Eq.mpr",
"NormedCommRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 249,
"column": 4
} | {
"line": 250,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nn : ℕ\nk : 𝕜\nhk : k ∈ σ a\n⊢ k ^ (n + 1) ∈ σ (a ^ (n + 1))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 253,
"column": 4
} | {
"line": 254,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nn : ℕ\nk : 𝕜\nhk : k ∈ σ a\npow_mem : k ^ (n + 1) ∈ σ (a ^ (n + 1))\n⊢ ↑(‖k‖₊ ^ (n + 1)) ≤ ↑‖a ^ (n + 1)‖₊ * ↑‖1‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\n⊢ (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 300,
"column": 12
} | {
"line": 300,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\n⊢ Tendsto (fun x ↦ x⁻¹ • a) (cobounded 𝕜) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 176,
"column": 15
} | {
"line": 176,
"column": 42
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : MulZeroClass β\ninst✝ : ContinuousMul β\nf g : α →C₀ β\n⊢ Tendsto (↑f * ↑g).toFun (cocompact α) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 193,
"column": 25
} | {
"line": 193,
"column": 52
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddZeroClass β\ninst✝ : ContinuousAdd β\nf g : α →C₀ β\n⊢ Tendsto (↑f + ↑g).toFun (cocompact α) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 207,
"column": 25
} | {
"line": 207,
"column": 48
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\nx : α\ninst✝³ : Zero β\nR : Type u_2\ninst✝² : Zero R\ninst✝¹ : SMulWithZero R β\ninst✝ : ContinuousConstSMul R β\nr : R\nf : α →C₀ β\n⊢ Tendsto (r • ↑f).toFun (cocompact α) (𝓝 0)",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 235,
"column": 20
} | {
"line": 235,
"column": 47
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g f : α →C₀ β\n⊢ Tendsto (-↑f).toFun (cocompact α) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 245,
"column": 25
} | {
"line": 245,
"column": 52
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nx : α\ninst✝¹ : AddGroup β\ninst✝ : IsTopologicalAddGroup β\nf✝ g✝ f g : α →C₀ β\n⊢ Tendsto (↑f - ↑g).toFun (cocompact α) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 34
} | [
{
"pp": "case h\n𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nh : (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1\nz : 𝕜ˣ\nhz : ↑z ∈ {0}ᶜ\n⊢ resolvent a ↑z = (↑z)⁻¹ • resolvent ((↑z)⁻¹ • a) 1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nh : (fun z ↦ resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ 1\n⊢ (fun z ↦ z⁻¹ • resolvent (z⁻¹ • a) 1) =O[cobounded 𝕜] fun x ↦ ‖x⁻¹‖",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 339,
"column": 10
} | {
"line": 339,
"column": 48
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\na : A\nr : ℝ≥0\nhr : r * ‖a‖₊ < 1\nn : ℕ\nh : ¬‖a‖₊ = 0\n⊢ ‖a‖₊ ^ n.succ * r ^ (n + 1) ≤ 1",
"usedConstants": [
"Eq.mpr",
"NNR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 18
} | [
{
"pp": "case h\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : Zero β\nf g : α →C₀ β\nh : f.toBCF = g.toBCF\nx : α\n⊢ f x = g x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 40
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : Zero β\nι : Type u_2\nF : ι → α →C₀ β\nf : α →C₀ β\nl : Filter ι\n⊢ Tendsto F l (𝓝 f) ↔ TendstoUniformly (fun i ↦ ⇑(F i)) (⇑f) l",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 346,
"column": 12
} | {
"line": 346,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\na : A\ny : 𝕜\nhy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹\nh : ¬‖a‖₊ = 0\n⊢ ‖y‖₊ < ‖a‖₊⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\na : A\ny : 𝕜\nhy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹\nnorm_lt : ‖y • a‖ < 1\n⊢ HasSum (fun n ↦ (ContinuousMultilinearMap.mkPiRing 𝕜 (Fin n) (a ^ n)) fun x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 424,
"column": 4
} | {
"line": 427,
"column": 51
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : Zero β\nf : α →ᵇ β\nhf : ∀ U ∈ 𝓝 f, (U ∩ range toBCF).Nonempty\nε : ℝ\nhε : ε > 0\ng : α →C₀ β\nhg : g.toBCF ∈ ball f (ε / 2)\nx : α\nhx : dist (g x) 0 < ε / 2\n⊢ dist (f x) 0 < ε",
"usedConstants": [
... | calc
dist (f x) 0 ≤ dist (g.toBCF x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
_ < dist g.toBCF f + ε / 2 := add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx
_ ≤ ε := by grw [mem_ball.1 hg, add_halves ε] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 514,
"column": 8
} | {
"line": 514,
"column": 36
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : AddMonoid β\ninst✝¹ : StarAddMonoid β\ninst✝ : ContinuousStar β\nf : α →C₀ β\n⊢ Tendsto (fun x ↦ star (f x)) (cocompact α) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Exponential | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NormedRing A\ninst✝⁴ : NormedAlgebra ℂ A\ninst✝³ : StarRing A\ninst✝² : ContinuousStar A\ninst✝¹ : CompleteSpace A\ninst✝ : StarModule ℂ A\na b : ↥(selfAdjoint A)\nh : Commute ↑a ↑b\nthis : NormedAlgebra ℚ A\n⊢ expUnitary (a + b) = expUnitary a * expUnitary b",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nz : 𝕜\nhz : z ∈ spectrum 𝕜 a\nthis : NormedAlgebra ℚ A\nhexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)\nb : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n\nhb ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na : A\nz : 𝕜\nhz : z ∈ spectrum 𝕜 a\nthis : NormedAlgebra ℚ A\nhexpmul : exp a = exp (a - ↑ₐ z) * ↑ₐ (exp z)\nb : A := ∑' (n : ℕ), (↑(n + 1).factorial)⁻¹ • (a - ↑ₐ z) ^ n\nhb ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 449,
"column": 20
} | {
"line": 449,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : CompleteSpace A\ninst✝ : NormOneClass A\nφ : A →ₐ[𝕜] 𝕜\nx✝¹ : ℝ\nx✝ : x✝¹ ≥ 0\nh : ∀ (x : A), ‖φ.toContinuousLinearMap x‖ ≤ x✝¹ * ‖x‖\n⊢ 1 ≤ x✝¹",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 35
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 521,
"column": 2
} | {
"line": 536,
"column": 30
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSp... | have : CompleteSpace S := hS.completeSpace_coe
intro μ hμ
by_contra h
rw [spectrum.notMem_iff] at h
rw [← frontier_compl, (spectrum.isClosed _).isOpen_compl.frontier_eq, mem_diff] at hμ
obtain ⟨hμ₁, hμ₂⟩ := hμ
rw [mem_closure_iff_clusterPt] at hμ₁
apply hμ₂
rw [mem_compl_iff, spectrum.notMem_iff]
refi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 521,
"column": 2
} | {
"line": 536,
"column": 30
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSp... | have : CompleteSpace S := hS.completeSpace_coe
intro μ hμ
by_contra h
rw [spectrum.notMem_iff] at h
rw [← frontier_compl, (spectrum.isClosed _).isOpen_compl.frontier_eq, mem_diff] at hμ
obtain ⟨hμ₁, hμ₂⟩ := hμ
rw [mem_closure_iff_clusterPt] at hμ₁
apply hμ₂
rw [mem_compl_iff, spectrum.notMem_iff]
refi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 643,
"column": 63
} | {
"line": 643,
"column": 74
} | [
{
"pp": "A : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nt : ℝ≥0\nht : spectralRadius ℝ a ≤ ↑t\nthis : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t\nh : ∀ x ∈ spectrum ℝ a, 0 ≤ x\nx : ℝ\nhx : x ∈ {↑t} - spectrum ℝ a\n⊢ ∃ y ∈ spectrum ℝ a, ↑t - y = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 649,
"column": 6
} | {
"line": 649,
"column": 77
} | [
{
"pp": "A : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nt : ℝ≥0\nht : spectralRadius ℝ a ≤ ↑t\nthis : spectrum ℝ a ⊆ Set.Icc (-↑t) ↑t\nh : spectralRadius ℝ ((algebraMap ℝ A) ↑t - a) ≤ ↑t\n⊢ ∀ x ∈ spectrum ℝ a, ‖↑t - x‖₊ ≤ t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 718,
"column": 4
} | {
"line": 718,
"column": 30
} | [
{
"pp": "case hbc.bc\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NormedField 𝕜\ninst✝³ : ProperSpace 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : CompleteSpace A\na₀ a : A\nha : a ∈ Metric.closedBall a₀ 1\n⊢ ‖a - a₀‖ ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 749,
"column": 2
} | {
"line": 749,
"column": 42
} | [
{
"pp": "A : Type u_2\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace ℝ A\ninst✝² : SMulCommClass ℝ A A\ninst✝¹ : IsScalarTower ℝ A A\ninst✝ : CompleteSpace A\nh₂ : IsClosed (range NNReal.toReal)\nh₁ : IsInducing NNReal.toReal\n⊢ UpperHemicontinuous (quasispectrum ℝ≥0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 28
} | [
{
"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\nz : ℂ\nhf : Differentiable ℂ (expMulMulExp a b x)\nthis : IsBounded (Set.range (expMulMulExp a b x))\n⊢ expMulMulExp a b x z = x",
"usedConstants": [
"_private.Mathlib.Anal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 17
} | [
{
"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\nkey : ∀ (z : ℂ), x * NormedSpace.exp (z • star a✝) = NormedSpace.exp (z • star b) * x\na : A\n⊢ HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 21
} | [
{
"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\nkey : ∀ (z : ℂ), x * NormedSpace.exp (z • star a) = NormedSpace.exp (z • star b) * x\nthis : ∀ (a : A), HasDerivAt (fun z ↦ NormedSpace.exp (z • a)) a 0\n⊢ HasDerivAt (fun y ↦ x * No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 54
} | [
{
"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\nz : ℂ\nx✝¹ : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℂ A\nx✝ : Invertible (NormedSpace.exp (z • star a)) := invertibleExp (z • star a)\n⊢ x * NormedSpace.exp (z • star a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 26
} | [
{
"pp": "case a.h\nA : Type u_2\ninst✝ : NonUnitalCStarAlgebra A\na b x : A\nha : IsStarNormal a\nhb : IsStarNormal b\nh : SemiconjBy x a b\n⊢ SemiconjBy ↑x ↑a ↑b",
"usedConstants": [
"NormedRing.toRing",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"SemiconjBy",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Barrelled | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 40
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nκ : Type u_3\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_6\nF : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜₁ →+* 𝕜₂\ninst✝⁸ : RingHomIsometric σ₁₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module 𝕜₁ E\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 146,
"column": 22
} | {
"line": 146,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : IsScalarTower 𝕜 A A\ninst✝⁴ : SMulCommClass 𝕜 A A\ninst✝³ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\ninst✝² : Clo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BanachSteinhaus | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 46
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BanachSteinhaus | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 55
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakDual | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 31
} | [
{
"pp": "case mpr\nι : 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)) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 195,
"column": 40
} | {
"line": 195,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module ℂ A\ninst✝² : IsScalarTower ℂ A A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\na : A\nha : IsSelfAdjoint a\nx : ℂ\nhx : x ∈ σₙ ℂ a\nthis : Set.EqOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 259,
"column": 4
} | {
"line": 261,
"column": 13
} | [
{
"pp": "case right.left\nA : Type u_1\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Module ℝ A\ninst✝² : IsScalarTower ℝ A A\ninst✝¹ : SMulCommClass ℝ A A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha₁ : IsSelfAdjoint a\nha₂ : QuasispectrumR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : PartialOrder A\ninst✝⁷ : StarRing A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Module ℝ A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : SMulCommClass ℝ A A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 86
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedRing A\ninst✝² : StarRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsometricContinuousFunctionalCalculus 𝕜 A p\nf : 𝕜 → 𝕜\na : A\nc : ℝ\nhc : 0 ≤ c\nh : ∀ x ∈ σ 𝕜 a, ‖f x‖ ≤ c\nh✝ : Nontrivial A\nhf : ContinuousO... | simp only [← cfc_apply f a, isLUB_le_iff (IsGreatest.norm_cfc f a hf ha |>.isLUB)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.LocallyConvex.WeakDual | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 96
} | [
{
"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⊢ ↑f ∈ Submodule.span 𝕜 ↑(ContinuousLinearMap.coeLM 𝕜 ∘ₗ WeakBilin.ev... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedRing A\ninst✝³ : StarRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ : Nontrivial A\na : A\nha : p a\n⊢ IsGreatest ((fun x ↦ ‖x‖) '' σ 𝕜 a) ‖a‖",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 31
} | [
{
"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\nx : 𝕜\nhx : x ∈ σ 𝕜 a\nha : p a\n⊢ ‖x‖ ≤ ‖a‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedRing A\ninst✝³ : StarRing A\ninst✝² : NormedAlgebra 𝕜 A\ninst✝¹ : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝ : Nontrivial A\na : A\nha : p a\n⊢ IsGreatest ((fun x ↦ ‖x‖₊) '' σ 𝕜 a) ‖a‖₊",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 31
} | [
{
"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\nx : 𝕜\nhx : x ∈ σ 𝕜 a\nha : p a\n⊢ ‖x‖₊ ≤ ‖a‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 32
} | [
{
"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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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