module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 6
} | {
"line": 62,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ bernsteinPolynomial ℤ 3 2 = 3 * X ^ 2 - 3 * X ^ 3",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"Non... | [] | norm_num [bernsteinPolynomial, choose]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 6
} | {
"line": 62,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ bernsteinPolynomial ℤ 3 2 = 3 * X ^ 2 - 3 * X ^ 3",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"Non... | [] | norm_num [bernsteinPolynomial, choose]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 966,
"column": 4
} | {
"line": 966,
"column": 15
} | {
"line": 966,
"column": 16
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ ... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 993,
"column": 2
} | {
"line": 993,
"column": 36
} | {
"line": 993,
"column": 37
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 998,
"column": 2
} | {
"line": 998,
"column": 13
} | {
"line": 998,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1002,
"column": 2
} | {
"line": 1002,
"column": 13
} | {
"line": 1002,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1006,
"column": 2
} | {
"line": 1006,
"column": 13
} | {
"line": 1006,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹³ : CommSemiring R\ninst✝¹² : PartialOrder R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : IsTopologicalSemiring R\ninst✝⁸ : ContinuousStar R\ninst✝⁷ : ContinuousSqrt R\ninst✝⁶ : StarOrderedRing R\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : Ring A\ninst✝³ : Sta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1071,
"column": 2
} | {
"line": 1071,
"column": 13
} | {
"line": 1071,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ ... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1076,
"column": 2
} | {
"line": 1076,
"column": 13
} | {
"line": 1076,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ ... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1080,
"column": 2
} | {
"line": 1080,
"column": 13
} | {
"line": 1080,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ ... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1084,
"column": 2
} | {
"line": 1084,
"column": 13
} | {
"line": 1084,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ ... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 373,
"column": 24
} | {
"line": 373,
"column": 39
} | {
"line": 373,
"column": 40
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : I... | cfcₙ_apply g a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 381,
"column": 24
} | {
"line": 381,
"column": 39
} | {
"line": 381,
"column": 40
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : I... | cfcₙ_apply g a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 436,
"column": 35
} | {
"line": 436,
"column": 46
} | {
"line": 436,
"column": 47
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 438,
"column": 31
} | {
"line": 438,
"column": 42
} | {
"line": 438,
"column": 43
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 61
} | {
"line": 204,
"column": 4
} | [
{
"pp": "case mpr\na b : ℝ\nh : a < b\np : ℝ[X]\n⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈\n Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Real",
"instHSMul",
... | [
"case mpr\na b : ℝ\nh : a < b\np : ℝ[X]\nq : ℝ[X] := p.comp ((b - a) • X + Polynomial.C a)\n⊢ ↑(toContinuousMapOnAlgHom (Set.Icc a b)) p ∈\n Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I)"
] | let q := p.comp ((b - a) • Polynomial.X + Polynomial.C a) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 27
} | {
"line": 514,
"column": 28
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1085,
"column": 28
} | {
"line": 1085,
"column": 39
} | {
"line": 1085,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\ninst✝ : CompleteSpace E\nhE : Nonempty E\nf : α → E\nhf : BoundedVariationOn f univ\n⊢ ∃ x, Tendsto f atTop (𝓝 x)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\ninst✝ : CompleteSpace E\nhE : Nonempty E\nf : α → E\nhf : BoundedVariationOn f univ\n⊢ ∃ x, Tendsto f atTop (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 1090,
"column": 28
} | {
"line": 1090,
"column": 39
} | {
"line": 1090,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\ninst✝ : CompleteSpace E\nhE : Nonempty E\nf : α → E\nhf : BoundedVariationOn f univ\n⊢ ∃ x, Tendsto f atBot (𝓝 x)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"α : Type u_1\ninst✝² : LinearOrder α\nE : Type u_2\ninst✝¹ : PseudoEMetricSpace E\ninst✝ : CompleteSpace E\nhE : Nonempty E\nf : α → E\nhf : BoundedVariationOn f univ\n⊢ ∃ x, Tendsto f atBot (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 570,
"column": 24
} | {
"line": 570,
"column": 39
} | {
"line": 570,
"column": 40
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalar... | cfcₙ_apply g a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 583,
"column": 34
} | {
"line": 583,
"column": 45
} | {
"line": 583,
"column": 46
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScalarTower R A ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 587,
"column": 62
} | {
"line": 587,
"column": 82
} | {
"line": 589,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹² : CommRing R\ninst✝¹¹ : Nontrivial R\ninst✝¹⁰ : StarRing R\ninst✝⁹ : MetricSpace R\ninst✝⁸ : IsTopologicalRing R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module R A\ninst✝² : IsScal... | [] | exact (cfcₙ_neg f a) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 623,
"column": 4
} | {
"line": 623,
"column": 15
} | {
"line": 623,
"column": 16
} | [
{
"pp": "case mpr\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalSemiring R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ ... | [
"case mpr\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalSemiring R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDivi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Bernstein | {
"line": 202,
"column": 4
} | {
"line": 203,
"column": 11
} | {
"line": 203,
"column": 12
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis✝ : IsUniformAddGroup E\nU : Se... | [
"E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis✝ : IsUniformAddGroup E\nU : Set E\nhU₀ : U... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 646,
"column": 2
} | {
"line": 646,
"column": 30
} | {
"line": 646,
"column": 31
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalSemiring R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDi... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommSemiring R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalSemiring R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDivisors R\nin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 652,
"column": 4
} | {
"line": 652,
"column": 15
} | {
"line": 652,
"column": 16
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : PartialOrder R\ninst✝¹⁶ : Nontrivial R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : IsTopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ContinuousSqrt R\ninst✝¹⁰ : StarOrderedRing R\ninst✝⁹ :... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : PartialOrder R\ninst✝¹⁶ : Nontrivial R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : IsTopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ContinuousSqrt R\ninst✝¹⁰ : StarOrderedRing R\ninst✝⁹ : NoZeroDivis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 662,
"column": 4
} | {
"line": 662,
"column": 15
} | {
"line": 662,
"column": 16
} | [
{
"pp": "case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : PartialOrder R\ninst✝¹⁶ : Nontrivial R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : IsTopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ContinuousSqrt R\ninst✝¹⁰ : StarOrderedRing R\ninst✝⁹ :... | [
"case pos\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁸ : CommSemiring R\ninst✝¹⁷ : PartialOrder R\ninst✝¹⁶ : Nontrivial R\ninst✝¹⁵ : StarRing R\ninst✝¹⁴ : MetricSpace R\ninst✝¹³ : IsTopologicalSemiring R\ninst✝¹² : ContinuousStar R\ninst✝¹¹ : ContinuousSqrt R\ninst✝¹⁰ : StarOrderedRing R\ninst✝⁹ : NoZeroDivis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 687,
"column": 22
} | {
"line": 687,
"column": 37
} | {
"line": 687,
"column": 38
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalRing R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDivisors R... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : PartialOrder R\ninst✝¹⁷ : Nontrivial R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : MetricSpace R\ninst✝¹⁴ : IsTopologicalRing R\ninst✝¹³ : ContinuousStar R\ninst✝¹² : ContinuousSqrt R\ninst✝¹¹ : StarOrderedRing R\ninst✝¹⁰ : NoZeroDivisors R\ninst✝⁹ : T... | cfcₙ_apply g a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 797,
"column": 15
} | {
"line": 797,
"column": 65
} | {
"line": 797,
"column": 66
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : Semifield 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✝ : ContinuousFunctionalCalculus R A p\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : Semifield 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✝ : ContinuousFunctionalCalculus R A p\na : A\nha : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Bernstein | {
"line": 240,
"column": 8
} | {
"line": 240,
"column": 23
} | {
"line": 240,
"column": 24
} | [
{
"pp": "case hbc\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis : IsUniformAddGroup E... | [
"case hbc\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nf : C(↑I, E)\nthis✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E\nthis : IsUniformAddGroup E\nU : Set E\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 61
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty X\ng... | [
"case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty X\ng : X → X → C... | let U : X → X → Set X := fun x y => {z | f z - ε < g x y z} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 13
} | {
"line": 170,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nA : Type u_2\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra ℝ A\nφ ψ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A\nh : φ.realContinuousMapOfNNReal = ψ.realContinuousMapOfNNReal\nf : C(X, ℝ≥0)\n⊢ φ f = ψ f",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": ... | [
"X : Type u_1\ninst✝³ : TopologicalSpace X\nA : Type u_2\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra ℝ A\nφ ψ : C(X, ℝ≥0) →⋆ₐ[ℝ≥0] A\nh : φ.realContinuousMapOfNNReal = ψ.realContinuousMapOfNNReal\nf : C(X, ℝ≥0)\n⊢ φ f = ψ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 392,
"column": 4
} | {
"line": 392,
"column": 19
} | {
"line": 392,
"column": 20
} | [
{
"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 ∈... | [
"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 ∈ A\n⊢ ¬f x₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Unitization | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 33
} | {
"line": 157,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\nx : Unitization 𝕜 A\n⊢ ‖(addEquiv 𝕜 A) x‖ ≤ 2 * ‖x‖",
"ppTerm": "?m.38",
... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\nx : Unitization 𝕜 A\n⊢ max ‖((addEquiv 𝕜 A) x).1‖ ‖((addEquiv 𝕜 A) x).2‖ ≤\n 2 * max ‖x... | rw [norm_eq_sup, Prod.norm_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.Unitization | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 71
} | {
"line": 167,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\nx : Unitization 𝕜 A\na✝ : Nontrivial A\n⊢ ‖(mul 𝕜 A) x.toProd.2‖ ≤ ‖(algebraMap... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\nx : Unitization 𝕜 A\na✝ : Nontrivial A\n⊢ ‖(mul 𝕜 A) x.toProd.2‖ ≤ ‖(mul 𝕜 A) x.toProd.2 +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 48
} | {
"line": 253,
"column": 49
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : Zero X\nr : ℝ≥0\nf : C(X, ℝ)₀\nx : X\nh : 0 ≤ f x\n⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"NonAssocSemir... | [
"case pos\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : Zero X\nr : ℝ≥0\nf : C(X, ℝ)₀\nx : X\nh : 0 ≤ f x\n⊢ 0 ≤ ↑r * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 11
} | {
"line": 255,
"column": 12
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : Zero X\nr : ℝ≥0\nf : C(X, ℝ)₀\nx : X\nh : f x < 0\n⊢ ↑((r • f).toNNReal x) = ↑((r • f.toNNReal) x)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"NonAssocSemir... | [
"case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : Zero X\nr : ℝ≥0\nf : C(X, ℝ)₀\nx : X\nh : f x < 0\n⊢ ↑r * f x ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Unitization | {
"line": 247,
"column": 17
} | {
"line": 248,
"column": 34
} | {
"line": 248,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\n⊢ ‖1‖ = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NonUnitalNormedRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : RegularNormedAlgebra 𝕜 A\n⊢ ‖1‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 13
} | {
"line": 348,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : Zero X\nA : Type u_2\ninst✝² : NonUnitalRing A\ninst✝¹ : StarRing A\ninst✝ : Module ℝ A\nφ ψ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A\nh : φ.realContinuousMapZeroOfNNReal = ψ.realContinuousMapZeroOfNNReal\nf : C(X, ℝ≥0)₀\n⊢ φ f = ψ f",
"ppTerm": "?m.31",
"ass... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : Zero X\nA : Type u_2\ninst✝² : NonUnitalRing A\ninst✝¹ : StarRing A\ninst✝ : Module ℝ A\nφ ψ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A\nh : φ.realContinuousMapZeroOfNNReal = ψ.realContinuousMapZeroOfNNReal\nf : C(X, ℝ≥0)₀\n⊢ φ f = ψ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 600,
"column": 4
} | {
"line": 601,
"column": 11
} | {
"line": 601,
"column": 12
} | [
{
"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... | [
"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 : AlgHomClass F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 36
} | {
"line": 163,
"column": 37
} | [
{
"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... | [
"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\nx : Unitiz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.WeakDual | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 40
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : 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), Continuous[in... | [] | exact ContinuousLinearMap.continuous | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Module.Spaces.WeakDual | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 31
} | {
"line": 274,
"column": 32
} | [
{
"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",
"... | [
"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.Semicontinuity.Hemicontinuity | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 75
} | {
"line": 33,
"column": 2
} | [
{
"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",
"ppTerm": ... | [
"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"
] | 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": 49,
"column": 2
} | {
"line": 49,
"column": 52
} | {
"line": 50,
"column": 4
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 42
} | {
"line": 176,
"column": 43
} | [
{
"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... | [
"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✝ : ContinuousMul 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 52
} | {
"line": 80,
"column": 4
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 20
} | {
"line": 128,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nthis : ∀ (u : Set β), (f ⁻¹' Iic uᶜ)ᶜ = {x | (f x ∩ u).Nonempty}\n⊢ LowerHemicontinuous f ↔ ∀ (u : Set β), IsOpen[inst✝] u → IsOpen[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ",
"ppTerm": "?m.37",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nthis : ∀ (u : Set β), (f ⁻¹' Iic uᶜ)ᶜ = {x | (f x ∩ u).Nonempty}\n⊢ LowerHemicontinuous f ↔ ∀ (u : Set β), IsOpen[inst✝] u → IsOpen[inst✝¹] {x | (f x ∩ u).Nonempty}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 308,
"column": 11
} | {
"line": 308,
"column": 45
} | {
"line": 308,
"column": 46
} | [
{
"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 ... | [
"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 : IsOpen[ins... | ← preimage_inter_range (s := f _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 52
} | {
"line": 326,
"column": 4
} | [
{
"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",
"ppT... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 74,
"column": 36
} | {
"line": 74,
"column": 74
} | {
"line": 74,
"column": 75
} | [
{
"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}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"congrArg",
"Complex.im",
... | [
"r : ℝ\ns : Set ℂ\nhs₁ : {z | r < z.im} ⊆ s\nhs₂ : s ⊆ {z | r ≤ z.im}\n⊢ s ⊆ {z | r ≤ z.im}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 79,
"column": 36
} | {
"line": 79,
"column": 74
} | {
"line": 79,
"column": 75
} | [
{
"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}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"congrArg",
"Complex.im",
... | [
"r : ℝ\ns : Set ℂ\nhs₁ : {z | z.im < r} ⊆ s\nhs₂ : s ⊆ {z | z.im ≤ r}\n⊢ s ⊆ {z | z.im ≤ r}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Convex | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 44
} | {
"line": 93,
"column": 45
} | [
{
"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",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
... | [
"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⊢ (convexHull ℝ) {z, ↑z.re + ↑w.im * I, ↑w.re + ↑z.im * I, w} ⊆ U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 55
} | {
"line": 93,
"column": 6
} | [
{
"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",
"ppTerm": "?m.87",
"assigned": false,
"u... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 47
} | {
"line": 157,
"column": 48
} | [
{
"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)",
"ppTerm": "?m.10... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 44
} | {
"line": 184,
"column": 45
} | [
{
"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",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 51
} | {
"line": 81,
"column": 52
} | [
{
"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",
"ppTerm": "?refine_1",
"assigne... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 33
} | {
"line": 86,
"column": 34
} | [
{
"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‖",
"ppTer... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"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... | [
"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 ≤ ‖↑(toUnits u)⁻¹‖\n⊢ 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | {
"line": 95,
"column": 14
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 126,
"column": 42
} | {
"line": 126,
"column": 75
} | {
"line": 126,
"column": 76
} | [
{
"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",
"ppTerm": "?m.156",
"assigned... | [
"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‖ < ‖1‖⁻¹ * ‖k‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 39
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na : A\nx : ℝ≥0\nhx : x ∈ σ ℝ≥0 ↑a\n⊢ x ≤ ‖a‖₊",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\na : A\nx : ℝ≥0\nhx : x ∈ σ ℝ≥0 ↑a\n⊢ x ≤ ‖a‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 66
} | {
"line": 127,
"column": 67
} | [
{
"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)",
"ppTerm... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 237,
"column": 54
} | {
"line": 237,
"column": 65
} | {
"line": 237,
"column": 66
} | [
{
"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",
"ppTerm": "?m.69",
"assigned": true,
"usedConst... | [
"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‖₊ < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 249,
"column": 4
} | {
"line": 250,
"column": 26
} | {
"line": 250,
"column": 27
} | [
{
"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))",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"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))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 253,
"column": 4
} | {
"line": 254,
"column": 28
} | {
"line": 254,
"column": 29
} | [
{
"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‖₊",
"ppTerm": "?m.168",
"assigned": false,
... | [
"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‖₊"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 75
} | {
"line": 201,
"column": 76
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ σ ℂ a\n⊢ (ofReal ∘ re) z ∈ σ ℂ a",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NormedRing.toRing",
"spectrum",
"congrArg",
"Complex.instNorme... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nz : ℂ\nhz : z ∈ σ ℂ a\n⊢ z ∈ σ ℂ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 224,
"column": 60
} | {
"line": 224,
"column": 71
} | {
"line": 224,
"column": 72
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ x✝ ∈ {z | z.im < 0} → x✝ ∈ {z | z.im ≤ 0}",
"ppTerm": "?m.261",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Complex.im",
"setOf",
"Real.instLT",... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ x✝.im < 0 → x✝.im ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 224,
"column": 60
} | {
"line": 224,
"column": 71
} | {
"line": 224,
"column": 72
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ x✝ ∈ {z | 0 < z.im} → x✝ ∈ {z | 0 ≤ z.im}",
"ppTerm": "?m.304",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instZero",
"Complex.im",
"setOf",
"Real.instLT",... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsSelfAdjoint a\nx✝ : ℂ\n⊢ 0 < x✝.im → 0 ≤ x✝.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 46
} | {
"line": 299,
"column": 6
} | [
{
"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",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"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⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (cobounded 𝕜) fun x ↦ ‖Ring.inverse (1 - x⁻¹ • a)‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 300,
"column": 12
} | {
"line": 300,
"column": 23
} | {
"line": 300,
"column": 24
} | [
{
"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)",
"ppTerm": "?m.109",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 34
} | {
"line": 305,
"column": 35
} | [
{
"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\nz : 𝕜ˣ\nhz : ↑z ∈ {0}ᶜ\n⊢ resolvent a ↑z = (↑z)⁻¹ • resolvent ((↑z)⁻¹ • a) 1",
"ppTe... | [
"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\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.CStarAlgebra.Spectrum | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 26
} | {
"line": 281,
"column": 27
} | [
{
"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‖₊\... | [
"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‖₊\n⊢ ‖φ a‖₊ ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 17
} | {
"line": 307,
"column": 18
} | [
{
"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‖",
"ppTerm": "?... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 47
} | {
"line": 387,
"column": 48
} | [
{
"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 ... | [
"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 : Summable f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 55
} | {
"line": 389,
"column": 56
} | [
{
"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 ... | [
"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 : Summable f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 449,
"column": 20
} | {
"line": 449,
"column": 92
} | {
"line": 449,
"column": 93
} | [
{
"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",
"ppTerm": "?m.70",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 515,
"column": 2
} | {
"line": 515,
"column": 35
} | {
"line": 515,
"column": 36
} | [
{
"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... | [
"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.toTopologicalSpace] ↑S\nl :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 21
} | {
"line": 150,
"column": 22
} | [
{
"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... | [
"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 : ClosedEmbedding... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.ZeroAtInfty | {
"line": 424,
"column": 4
} | {
"line": 427,
"column": 51
} | {
"line": 428,
"column": 2
} | [
{
"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",
"ppTerm": "?m.152",
"... | [] | 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.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 198,
"column": 40
} | {
"line": 198,
"column": 51
} | {
"line": 198,
"column": 52
} | [
{
"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... | [
"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 (fun x ↦ st... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 262,
"column": 4
} | {
"line": 264,
"column": 13
} | {
"line": 264,
"column": 14
} | [
{
"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... | [
"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 : QuasispectrumRestricts a ⇑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 30
} | {
"line": 282,
"column": 31
} | [
{
"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... | [
"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✝ : NonnegSpectrumCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 28
} | {
"line": 85,
"column": 29
} | [
{
"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",
"ppTerm": "?m.72",
"assigned": true,
"u... | [
"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⊢ NormedSpace.exp (z • star b) * x * NormedSpace.exp (-(z • star a)) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 18
} | [
{
"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",
"ppTerm": "?m.66",
"assi... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 520,
"column": 78
} | {
"line": 536,
"column": 30
} | {
"line": 538,
"column": 0
} | [
{
"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... | [] | by
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, Set.mem_sdiff] at hμ
obtain ⟨hμ₁, hμ₂⟩ := hμ
rw [mem_closure_iff_clusterPt] at hμ₁
apply hμ₂
rw [mem_compl_iff, spectrum.notMem_i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 21
} | {
"line": 94,
"column": 22
} | [
{
"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... | [
"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 ↦ NormedSpace.exp (y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 54
} | {
"line": 99,
"column": 4
} | [
{
"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... | [
"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) = NormedSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Fuglede | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 26
} | {
"line": 109,
"column": 27
} | [
{
"pp": "A : 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",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NormedRing.toRing",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAdd... | [
"A : Type u_2\ninst✝ : NonUnitalCStarAlgebra A\na b x : A\nha : IsStarNormal a\nhb : IsStarNormal b\nh : SemiconjBy x a b\n⊢ ↑x * ↑a = ↑b * ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 15
} | {
"line": 124,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ B.polar s\n⊢ ‖(B x) y‖ ≤ 1",
"ppTerm": "?m.91",
"assigned... | [] | exact hy x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 35
} | {
"line": 133,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : F\nhx : ∀ (i : Set E), i.Finite → i ⊆ s → x ∈ B.polar i\na : E\nha : a ∈ s\n⊢ ‖(B a) x‖ ≤ 1",
... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : F\nhx : ∀ (i : Set E), i.Finite → i ⊆ s → x ∈ B.polar i\na : E\nha : a ∈ s\n⊢ ‖(B a) x‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 86
} | {
"line": 109,
"column": 4
} | [
{
"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... | [
"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 : ContinuousOn f (σ 𝕜 a)... | 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.Polar | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 18
} | {
"line": 164,
"column": 19
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.po... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.polar ↑m\nx : ... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 16
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.po... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.polar ↑m\nx : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 154,
"column": 46
} | {
"line": 155,
"column": 67
} | {
"line": 157,
"column": 0
} | [
{
"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‖",
"ppTerm": "?m.27",
"... | [] | by
simpa only [cfc_id 𝕜 a] using! IsGreatest.norm_cfc (id : 𝕜 → 𝕜) a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 32
} | {
"line": 196,
"column": 33
} | [
{
"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... | [
"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 : IsTopologicalSemiring S\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 67
} | {
"line": 197,
"column": 4
} | [
{
"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... | [
"case refine_2\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 : IsTopologicalSemiring S\ninst... | · simpa [halg.dist_eq] using ContinuousMap.dist_apply_le_dist _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 643,
"column": 63
} | {
"line": 643,
"column": 74
} | {
"line": 643,
"column": 75
} | [
{
"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",
"ppTerm": "?m.135",
"assigned": false,
"usedConsta... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 649,
"column": 6
} | {
"line": 649,
"column": 77
} | {
"line": 649,
"column": 78
} | [
{
"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",
"ppTerm": "?m.230",
"assigned": false,
"usedConstants": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 15
} | {
"line": 294,
"column": 16
} | [
{
"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... | [
"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\nc : ℝ\nh : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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